Abstract
Chapter 7 develops the theory of waves in excitable neural fields, where the fundamental network element is a local population of cells rather than a single neuron. It is shown how many of the PDE methods and results from the analysis of waves in reaction–diffusion equations considered in Chap. 2 can be extended to the nonlocal equations of neural field theory. First, the existence and stability of solitary traveling fronts and pulses in one-dimensional excitatory neural fields are considered. In the case of traveling pulses, it is necessary to include some form of local negative feedback mechanism such as synaptic depression or spike frequency adaptation. Two approaches to analyzing wave propagation failure in inhomogeneous neural media are then presented: one based on averaging methods and the other on interfacial dynamics. Finally, wave propagation in stochastic neural fields is analyzed, and oscillatory waves in two-dimensional neural media are briefly discussed.
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In Sect. 5, we showed how both solitary pulses and oscillatory phase waves could occur in a synaptically coupled network of spiking neurons, where the fundamental element of the network was a single neuron. Hence, whether the network acted as an excitable or an oscillatory medium depended primarily on the intrinsic properties of the individual neurons. In this chapter, we focus on waves in excitable neural fields, where the fundamental network element is a local population of cells (see Sect. 6), and show how many of the PDE methods and results for the analysis of waves in reaction–diffusion equations (see part I) can be extended to the nonlocal integrodifferential equations of neural field theory. We begin by analyzing the existence and stability of solitary traveling fronts in a 1D scalar neural field. (Since there is strong vertical coupling between layers of a cortical column, it is possible to treat a thin vertical cortical slice as an effective 1D medium.) In order to relate the models to experiments on disinhibited cortical slices (Sect. 5.1), we assume that the weight distribution is purely excitatory. This is also motivated by the observation that epileptic seizures are often associated with greatly enhanced levels of recurrent excitation [430] (Sect. 9.4). We then extend the analysis to the case of traveling pulses, which requires the inclusion of some form of local negative feedback mechanism such as synaptic depression or spike frequency adaptation. Next we describe two approaches to analyzing wave propagation failure in inhomogeneous neural media: one based on homogenization theory [79, 332] and the other on interfacial dynamics [132]. This is followed by a discussion of wave propagation in stochastic neural fields.
It is useful to emphasize here that there are two main approaches to analyzing the spatiotemporal dynamics of neural field equations. The first method is based on the original work of Amari [8], in which one establishes the existence of nonlinear traveling wave solutions by explicit construction. This is possible if one takes the firing rate function F to be the Heaviside (1.15). It is also possible to study the linear stability of such solutions by constructing an associated Evans function, whose zeros determine the spectrum of the resulting linear operator [134, 552, 696]. The constructive approach of Amari [8] has been particularly useful in providing explicit insights into how spatiotemporal network dynamics depends on the structure of the synaptic weight kernel as well as various physiological parameters. Moreover, in certain cases, it is possible to use singular perturbation methods [504, 505] or fixed-point theorems [172, 336] to extend results for neural fields with Heaviside nonlinearities to those with more realistic sigmoidal nonlinearities; see also [136]. The second method is based on bifurcation theory, following the original work of Ermentrout and Cowan [169], in which one investigates the emergence of spatially periodic stationary and oscillatory patterns through a combination of linear stability analysis, weakly nonlinear analysis, symmetric bifurcation theory, and numerical simulations, as reviewed in [67, 71, 167]. Rigorous functional analytical techniques combined with numerical bifurcation schemes have also been used to study the existence and (absolute) stability of stationary solutions for a general class of neural field models with smooth firing rate functions [185, 642]. As far as we are aware, these methods have not yet been applied to traveling wave solutions of neural field equations.
1 Traveling Fronts in a Scalar Neural Field
1.1 Propagating Fronts in a Bistable Neural Field
We begin by using Amari’s constructive method [8] to analyze the existence of traveling front solutions in a scalar neural field equation. Similar analyses are found in Refs. [76, 134, 504]. We assume a Heaviside rate function (1.15) and an excitatory weight distribution of the form w(x, y) = w(x − y) with w(x) ≥ 0 and w(−x) = w(x). We also assume that w(x) is a monotonically decreasing function of x for x ≥ 0. A common choice is the exponential weight distribution
where σ determines the range of synaptic connections. The latter tends to range from 100 μm to 1 mm. The resulting neural field equation is
with F(u) = H(u −κ). We have fixed the units of time by setting τ = 1. If τ is interpreted as a membrane time constant, then τ ∼ 10 msec. In order to construct a traveling front solution of (7.2), we introduce the traveling wave coordinate ξ = x − ct, where c denotes the wave speed, and set u(x, t) = U(ξ) with \(\lim _{\xi \rightarrow -\infty }U(\xi ) = U_{+} > 0\) and lim ξ→∞ U(ξ) = 0 such that U(ξ) only crosses the threshold κ once. Here \(U_{+} =\int _{ -\infty }^{\infty }w(y)dy\) is a spatially uniform fixed-point solution of (7.2). Since Eq. (7.2) is equivariant with respect to uniform translations, we are free to take the threshold crossing point to be at the origin, U(0) =κ, so that U(ξ) <κ for ξ > 0 and U(ξ) >κ for ξ < 0. Substituting this traveling front solution into Eq. (7.2) then gives
where U′(ξ) = dU∕dξ. Multiplying both sides of the above equation by \({\text{e}}^{-\xi /c}\) and integrating with respect to ξ leads to the solution
Finally, requiring the solution to remain bounded as ξ →∞ (ξ →−∞) for c > 0 (for c < 0) implies that κ must satisfy the condition
Thus, one of the useful aspects of the constructive method is that it allows us to derive an explicit expression for the wave speed as a function of physiological parameters such as firing threshold and range of synaptic connections. In the case of the exponential weight distribution (7.1), the relationship between wave speed c and threshold κ is
This establishes the existence of a unique front solution for fixed κ, which travels to the right (c > 0) when κ < 0. 5 and travels to the left (c < 0) when κ > 0. 5. As we will show below, the traveling front is stable.
Given the existence of a traveling front solution for a Heaviside rate function, it is possible to extend the analysis to a smooth sigmoid nonlinearity using a continuation method [172]. We briefly summarize the main result. Consider the scalar neural field equation (7.2) with F given by the sigmoid function (1.14) and w(x) nonnegative and symmetric with normalization \(\int _{-\infty }^{\infty }w(x)dx = 1\). Suppose that the function \(\tilde{F}(u) = -u + F(u)\) has precisely three zeros at \(u = U_{\pm },U_{0}\) with \(U_{-} < U_{0} < U_{+}\) and \(\tilde{F}^{\prime}(U_{\pm }) < 0\). It can then be shown that (modulo uniform translations) there exists a unique traveling front solution u(x, t) = U(ξ), ξ = x − ct, with
and U(ξ) → U ± as \(\xi \rightarrow \mp \infty \) [172]. Moreover, the speed of the wave satisfies
where F′(U) = dF∕dU and
Since the denominator of Eq. (7.8) is positive definite, the sign of c is determined by the sign of the coefficient Γ. In particular, if the threshold κ = 0. 5 and the gain of the sigmoid η > 4 (see Eq. (1.14), then there exists a pair of stable homogeneous fixed points with U − = −U +, which in turn implies that Γ = 0 and the front solution is stationary. Note that this analysis has been extended to a more general form of nonlocal equations by Chen [108].
1.2 Wave Stability and Evans Functions
Suppose that the scalar neural field equation (7.2) has a traveling wave solution u(x, t) = U(ξ), ξ = x − ct with c > 0. Following Coombes and Owen [134], it is convenient to rewrite the neural field equation in the integral form
with \(\varPhi (t) ={ \text{e}}^{-t}H(t)\). For this representation, the front solution satisfies
In order to determine the stability of the front solutions, we transform to traveling wave coordinates by setting \(u(x,t) = U(\xi,t) = U(\xi ) +\varphi (\xi,t)\), and Taylor expand to first order in \(\varphi\). This leads to the linear integral equation
We now seek solutions of Eq. (7.12) of the form \(\varphi (\xi,t) =\varphi (\xi ){\text{e}}^{\lambda t}\), \(\lambda \in \mathbb{C}\), which leads to the eigenvalue equation \(\varphi = \mathbb{L}(\lambda )\varphi\). That is,
where we have performed the change of variables cs +ξ − y → s. The linear stability of the traveling front can then be determined in terms of the spectrum \(\sigma({\mathbb L}(\lambda))\).
Following appendix section 2.7, we assume that the eigenfunctions \(\varphi \in {L}^{2}(\mathbb{R})\) and introduce the resolvent operator \(\mathcal{R}(\lambda ) = {[\mathbb{L}(\lambda ) - I]}^{-1}\), where I denotes the identity operator. We can then decompose the spectrum \(\sigma (\mathcal{L})\) into the disjoint sum of the discrete spectrum and the essential spectrum. Given the spectrum of the linear operator defined by Eq. (7.13), the traveling wave is said to be linearly stable if [551]
for some K > 0, and λ = 0 is a simple eigenvalue of λ. The existence of at least one zero eigenvalue is a consequence of translation invariance. Indeed, differentiating equation (7.11) with respect to ξ shows that \(\varphi (\xi ) = U^{\prime}(\xi )\) is an eigenfunction solution of Eq. (7.13) with λ = 0. As in the case of PDEs (see Sect. 2.4), the discrete spectrum may be associated with the zeros of an Evans function. A number of authors have applied the Evans function construction to neural field equations [134, 198, 506, 536, 552, 696], as well as more general nonlocal problems [314]. Moreover, for neural fields with Heaviside firing rate functions, the Evans function can be calculated explicitly. This was first carried out by Zhang [696], who applied the method of variation of parameters to the linearized version of the integrodifferential Eq. (7.2), and was subsequently extended using a more direct integral formulation by Coombes and Owen [134].
Construction of Evans function. Setting F(U) = H(U −κ) in Eq. (7.12) and using the identity
$$\displaystyle{ H^{\prime}(U(\xi )-\kappa ) =\delta (U(\xi )-\kappa ) = \frac{\delta (\xi )} {\vert U^{\prime}(0)\vert } }$$(7.15)gives
$$\displaystyle\begin{array}{rcl} \varphi (\xi ) = \frac{\varphi (0)} {c\vert U^{\prime}(0)\vert }\int _{-\infty }^{\infty }w(y)\varPhi ((y-\xi )/c){\text{e}}^{-\lambda (y-\xi )/c}dy.& & {}\end{array}$$(7.16)In order to obtain a self-consistent solution at ξ = 0, we require that
$$\displaystyle\begin{array}{rcl} \varphi (0) = \frac{\varphi (0)} {c\vert U^{\prime}(0)\vert }\int _{0}^{\infty }w(y)\varPhi (y/c){\text{e}}^{-\lambda y/c}dy,& & {}\end{array}$$(7.17)We have used the fact that Φ(y) = 0 for y < 0, which is a consequence of causality. Hence, a nontrivial solution exists provided that \(\mathcal{E}(\lambda ) = 0\), where
$$\displaystyle{ \mathcal{E}(\lambda ) = 1 - \frac{1} {c\vert U^{\prime}(0)\vert }\int _{0}^{\infty }w(y)\varPhi (y/c){\text{e}}^{-\lambda y/c}dy. }$$(7.18)Equation (7.18) can be identified with the Evans function for the traveling front solution of the scalar neural field equation (7.10). It is real valued if λ is real. Furthermore, (i) the complex number λ is an eigenvalue of the operator \(\mathcal{L}\) if and only if \(\mathcal{E}(\lambda ) = 0\), and (ii) the algebraic multiplicity of an eigenvalue is equal to the order of the zero of the Evans function [134, 552, 696]. We briefly indicate the proof of (i) for \(\varPhi (t) ={ \text{e}}^{-t}H(t)\). Equation (7.16) becomes
$$\displaystyle\begin{array}{rcl} \varphi (\xi )& =& \frac{\varphi (0)} {c\vert U^{\prime}(0)\vert }{\text{e}}^{(\lambda +1)\xi /c}\int _{\xi }^{\infty }w(y){\text{e}}^{-(\lambda +1)y/c}dy, {}\\ & =& \varphi (0)\left [1 - \frac{1} {c\vert U^{\prime}(0)\vert }\int _{0}^{\xi }w(y){\text{e}}^{-(\lambda +1)y/c}dy\right ]{\text{e}}^{(\lambda +1)\xi /c}, {}\\ \end{array}$$which in the limit ξ →∞ gives
$$\displaystyle{ \lim _{\xi \rightarrow \infty }\varphi (\xi ) =\varphi (0)\mathcal{E}(\lambda )\lim _{\xi \rightarrow \infty }{\text{e}}^{(\lambda +1)\xi /c}. }$$Assuming that Reλ > −1 (which turns out to be to the right of the essential spectrum), then \(\varphi (\xi )\) will be unbounded as ξ →∞ unless \(\mathcal{E}(\lambda ) = 0\). That is, if \(\mathcal{E}(\lambda ) = 0\), then \(\varphi (\xi )\) is normalizable, the resolvent operator is not invertible and λ is an eigenvalue.
It is also straightforward to show that \(\mathcal{E}(0) = 0\), which we expect from translation invariance. First, setting F(U) = H(U −κ) in Eq. (7.11) and differentiating with respect to ξ show that
$$\displaystyle{ U^{\prime}(\xi ) = -\frac{1} {c}\int _{-\infty }^{\infty }w(y)\varPhi ((y-\xi )/c)dy. }$$(7.19)Thus, defining
$$\displaystyle{ \mathcal{H}(\lambda ) =\int _{ 0}^{\infty }w(y)\varPhi (y/c){\text{e}}^{-\lambda y/c}dy, }$$(7.20)we see that \(c\vert U^{\prime}(0)\vert = \mathcal{H}(0)\) and, hence,
$$\displaystyle{ \mathcal{E}(\lambda ) = 1 - \frac{\mathcal{H}(\lambda )} {\mathcal{H}(0)}. }$$(7.21)It immediately follows that \(\mathcal{E}(0) = 0\).
In order to determine the essential spectrum, consider the inhomogeneous equation
$$\displaystyle\begin{array}{rcl} \varphi (\xi ) - \frac{\varphi (0)} {c\vert U^{\prime}(0)\vert }\int _{-\infty }^{\infty }w(y)\varPhi ((y-\xi )/c){\text{e}}^{-\lambda (y-\xi )/c}dy = h(\xi )& & {}\end{array}$$(7.22)for some normalizable smooth function h on \(\mathbb{R}\). Assuming that λ does not belong to the discrete spectrum, \(\mathcal{E}(\lambda )\neq 0\), we can expresses the constant \(\varphi (0)\) in terms of h(0) by setting ξ = 0 in Eq. (7.22): \(\varphi (0) = h(0)/\mathcal{E}(\lambda )\). Thus,
$$\displaystyle{ \varphi (\xi ) = h(\xi ) + \frac{1} {\mathcal{E}(\lambda )} \frac{h(0)} {c\vert U^{\prime}(0)\vert }\int _{-\infty }^{\infty }w(y)\varPhi ((y-\xi )/c){\text{e}}^{-\lambda (y-\xi )/c}dy. }$$(7.23)Fourier transforming this equation using the convolution theorem gives
$$\displaystyle{ \hat{\varphi }(k) =\hat{ h}(k) + \frac{1} {\mathcal{E}(\lambda )} \frac{h(0)} {c\vert U^{\prime}(0)\vert }\hat{\omega }(k)\hat{\varPhi }(kc + i\lambda ), }$$(7.24)where
$$\displaystyle{ \hat{\varphi }(k) =\int _{ -\infty }^{\infty }\varphi (y){\text{e}}^{iky}dy }$$(7.25)etc. Now suppose that for a given value of k, there exists λ =λ(k) for which \({[\hat{\varPhi }(kc + i\lambda (k))]}^{-1} = 0\). It follows that the right-hand side of Eq. (7.24) blows up if λ =λ(k), that is, the dispersion curve belongs to the essential spectrum.
For the sake of illustration, let us calculate the zeros of the Evans function in the case of the exponential weight function (7.1). Substituting \(\varPhi (t) ={ \text{e}}^{-t}\) and \(w(y) ={ \text{e}}^{-\vert y\vert /\sigma }/2\sigma\) in Eq. (7.20) gives
so that [134]
It follows that λ = 0 is the only zero of the Evans function and it is a simple root (since \(\mathcal{E}^{\prime}(0) > 0\)). Furthermore, in the particular case \(\varPhi (t) ={ \text{e}}^{-t}\), we have \({[\tilde{\varPhi }(kc + i\lambda )]}^{-1} = 1 - ikc+\lambda\) so that the essential spectrum is λ(k) = −1 + ikc, that is, a vertical line in the complex plane at Reλ = −1. It follows that the corresponding traveling front (it it exists) is stable. This example illustrates one of the powerful features of the constructive method based on Heavisides. Not only is it possible to construct exact traveling wave solutions and derive formulae for the speed of the wave, but one can also explicitly construct the Evans function that determines wave stability. The method extends to multi-population neural field models, neural fields with axonal propagation delays, and adaptive neural fields [134]. (Although taking the high-gain limit of a smooth firing rate function is not very realistic from a biological perspective, one finds that many of the basic features of traveling waves persist for finite gain.) In the particular case of axonal delays, it can be shown that delays reduce the speed of a wave but do not affect its stability properties. For example, given a right-moving traveling front solution of the scalar neural field equation (6.118) with τ = 1 and exponential weights, one finds that the speed of the wave is [134, 139]
where v is the propagation speed along an axon, and the Evans function is
1.3 Pulled Fronts
So far we have assumed that the scalar neural field operates in a bistable regime analogous to the FitzHugh–Nagumo equations; see Sect. 2. However, as we explored within the context of CaMKII translocation waves (Sect. 3.2), Fisher-like reaction–diffusion equations support traveling waves propagating into unstable states, resulting in pulled fronts (Sect. 3.3). It turns out that it is also possible to observe pulled fronts in an activity-based version of a scalar neural field equation [70, 132]:
with a(x, t) ≥ 0 for all (x, t). Note that the restriction to positive values of a is a feature shared with population models in ecology or evolutionary biology, for example, where the corresponding dependent variables represent number densities. Indeed, Eq. (7.27) has certain similarities with a nonlocal version of the Fisher–KPP equation, which takes the form [238]
One major difference from a mathematical perspective is that Eq. (7.28) supports traveling fronts even when the range of the interaction kernel K goes to zero, that is, K(x) →δ(x), since we recover the standard local Fisher–KPP equation (3.44) [191, 345]. In particular, as the nonlocal interactions appear nonlinearly in Eq. (7.28), they do not contribute to the linear spreading velocity in the leading edge of the front. On the other hand, nonlocal interactions play a necessary role in the generation of fronts in the neural field equation (7.27).
Plots of firing rate function. Intercepts of y = F(W 0 a) with the straight line y = a determine homogeneous fixed points. (a) Piecewise linear rate function (7.29) showing the existence of an unstable fixed point at a = 0 and a stable fixed point at a =κ. (b) Sigmoidal rate function \(F(a) = 2/(1 +{ \text{e}}^{-2[a-\kappa ]})\) showing the existence of two stable fixed points separated by an unstable fixed point
Suppose that F(a) in Eq. (7.27) is a positive, bounded, monotonically increasing function of a with F(0) = 0, \(\lim _{a\rightarrow {0}^{+}}F^{\prime}(a) = 1\) and \(\lim _{a\rightarrow \infty }F(a) =\kappa\) for some positive constant κ. For concreteness, we take
A homogeneous fixed-point solution a ∗ of Eq. (7.27) satisfies
In the case of the given piecewise linear firing rate function, we find that if W 0 > 1, then there exists an unstable fixed point at a ∗ = 0 (absorbing state) and a stable fixed point at a ∗ =κ; see Fig. 7.1(a). The construction of a front solution linking the stable and unstable fixed points differs considerably from that considered in neural fields with sigmoidal or Heaviside nonlinearities [8, 167], where the front propagates into a metastable state; see Fig 7.1(b). Following the PDE theory of fronts propagating into unstable states [544] (see Sect. 3.3), we expect there to be a continuum of front velocities and associated traveling wave solutions.
Recall that a conceptual framework for studying such solutions is the linear spreading velocity c ∗, which is the asymptotic rate with which an initial localized perturbation spreads into an unstable state based on the linear equations obtained by linearizing the full nonlinear equations about the unstable state. Therefore, linearizing equation (7.27) about a = 0 gives
Note that if a(x, 0) ≥ 0 for all x, then Eq. (7.31) ensures that a(x, t) ≥ 0 for all x and t > 0. One way to see this is to note from Eq. (7.31) that \(a(x,t +\varDelta t) = (1 -\varDelta t)a(x,t) +\varDelta t\int _{-\infty }^{\infty }w(x - x^{\prime})a(x^{\prime},t)dx^{\prime}\). Assuming positivity of the solution at time t and using the fact that the neural field is purely excitatory (w(x) ≥ 0 for all x), it follows that a(x, t +Δ t) is also positive. An arbitrary initial condition a(x, 0) will evolve under Eq. (7.31) as
where G(x, t) is Green’s function
and \(\tilde{w}(k)\) is the Fourier transform of the weight distribution w(x). Hence, the solution can be written in the form of Eq. (3.102):
with \(\tilde{a}_{0}\), the Fourier transform of the initial condition a(x, 0).
Given a sufficiently steep initial condition, for which the Fourier transform \(\tilde{a}(k)\) is analytic, the asymptotic behavior of a(x, t) can be obtained from the large-time asymptotics of G(x, t) based on steepest descents. It immediately follows from the analysis of the Fisher equation in Sect. 3.3 that the linear spreading velocity c ∗ is given by c ∗ = c(λ ∗) where
Using the fact that the Fourier transform of the weight distribution is real valued, we find that
where \(\mathcal{W}(\lambda ) =\hat{ W}(\lambda ) +\hat{ W}(-\lambda )\) and \(\hat{W}(\lambda )\) is the Laplace transform of w(x):
We are assuming that w(y) decays sufficiently fast as |y|→∞ so that the Laplace transform \(\hat{W}(\lambda )\) exists for bounded, negative values of λ. This holds in the case of a Gaussian weight distribution
since
Hence,
If W 0 > 1 (necessary for the zero activity state to be unstable), then c(λ) is a positive unimodal function with c(λ) →∞ as λ → 0 or λ →∞ and a unique minimum at λ =λ 0 with λ 0 the solution to the implicit equation
Example dispersion curves are shown in Fig. 7.2(a) for various values of the Gaussian weight amplitude W 0. Combining Eqs. (7.39) and (7.40) shows that
so that
Assuming that the full nonlinear system supports a pulled front (see Sect. 3.3), then a sufficiently localized initial perturbation (one that decays faster than \({\text{e}}^{-\lambda _{0}x}\)) will asymptotically approach the traveling front solution with the minimum wave speed \(c_{0} = c(\lambda _{0})\). Note that c 0 ∼σ and \(\lambda _{0} {\sim \sigma }^{-1}\). In Fig. 7.2(b), we show an asymptotic front profile obtained by numerically solving the neural field equation (7.27) when W 0 = 1. 2. The corresponding displacement of the front is a linear function of time with a slope consistent with the minimal wave speed c 0 ≈ 0. 7 of the corresponding dispersion curve shown in Fig. 7.2(a). This wave speed is independent of κ.
(a) Velocity dispersion curves c = c(λ) for a pulled front solution of the neural field equation (7.27) with piecewise linear firing rate function (7.29) and a Gaussian weight distribution with amplitude W 0 and width σ. Here σ = 1. 0, κ = 0. 4 and W 0 = 1, 2, 1. 5, 2. 0, 2. 5, 3. 0. Black dots indicate minimum wave speed c 0 for each value of W 0. (b) Snapshots of the front profile evolving from an initial condition consisting of a steep sigmoid function of unit amplitude (gray curve). Here W 0 = 1. 2
The asymptotic analysis of the linear equation (7.31) also shows that, given a sufficiently localized initial condition, \(\vert a(x,t)\vert \sim {\text{e}}^{{-\lambda }^{{\ast}}\xi }\psi (\xi,t)\) as t →∞, where ξ = x − c ∗ t and the leading-edge variable ψ(ξ, t) is given by
with
Positivity of \(\mathcal{D}\) follows from the fact that λ ∗ is a minimum of c(λ). However, as shown by Ebert and van Saarloos [162], although the spreading of the leading edge under linearization gives the right qualitative behavior, it fails to match correctly the traveling front solution of the full nonlinear system. In particular, the asymptotic front profile takes the form \(\mathcal{A}(\xi ) \sim \xi {\text{e}}^{{-\lambda }^{{\ast}}\xi }\) for \(\xi \gg 1\). The factor of ξ reflects the fact that at the saddle point the two branches of the velocity dispersion curve c(λ) meet, indicating a degeneracy. In order to match the \(\xi {\text{e}}^{{-\lambda }^{{\ast}}\xi }\) asymptotics of the front solution with the leading-edge solution, it is necessary to take the leading-edge function ψ(x, t) to be the so-called dipole solution of the diffusion equation [162]:
Putting all of this together, if the neural field equation supports a pulled front, then the leading edge should relax asymptotically as
with ξ = x − c ∗ t. Finally, writing
suggests that to leading order, the velocity relaxes to the pulled velocity c ∗ according to (see also [162])
2 Traveling Pulses in Adaptive Neural Fields
Traveling fronts are not particularly realistic, since populations of cells do not stay in the excited state forever. Hence, rather than a traveling front, propagating activity in cortex is usually better described as a traveling pulse. (One example where fronts rather than pulses occur is wave propagation during binocular rivalry [83, 312, 369, 678]; see Sect. 8.) One way to generate a traveling pulse is to include some form of synaptic inhibition, provided that it is not too strong [8]. However, even in the absence of synaptic inhibition, most neurons possess intrinsic negative feedback mechanisms that slowly bring the cell back to resting voltages after periods of high activity. Possible nonlinear mechanisms include synaptic depression or spike frequency adaptation as discussed in Sect. 6.1. However, most analytical studies of traveling pulses in neural field models have been based on a simpler linear form of adaptation introduced by Pinto and Ermentrout [504]. (For an analysis of waves in neural fields with nonlinear adaptation, see, e.g., [135, 329].) The linear adaptation model is given by
where ε and β determine the rate and amplitude of linear adaptation. We first show how to construct a traveling pulse solution of Eq. (7.49) in the case of a Heaviside rate function F(u) = H(u −κ), following the particular formulation of [198, 696]. We then indicate how singular perturbation methods can be used to construct a traveling pulse for smooth F, as carried out by Pinto and Ermentrout [504]. The introduction of adaptation means that the neural field can support fronts or pulses, depending on whether there exist one or two stable homogeneous fixed points; see Fig. 7.3. We will focus on the latter here. Note, however, that linear (or nonlinear) adaptation can have a nontrivial effect on the propagation of traveling fronts [76, 80]. This is due to the occurrence of a symmetry breaking front bifurcation analogous to that found in reaction–diffusion systems [251, 252, 524, 561]. That is, a stationary front can undergo a supercritical pitchfork bifurcation at a critical rate of adaptation, leading to bidirectional front propagation. As in the case of reaction–diffusion systems, the front bifurcation acts as an organizing center for a variety of nontrivial dynamics including the formation of oscillatory fronts or breathers. The latter can occur, for example, through a Hopf bifurcation from a stationary front in the presence of a weak stationary input inhomogeneity [76].
Plot of nullclines for space-clamped planar system \(\dot{u} = -u + F(u) -\beta u\), \({\epsilon }^{-1}\dot{q} = -q + u\) with \(F(u) = 1/(1 +{ \text{e}}^{-\eta (u-\kappa )})\). Nullcline q = −u + F(u)]∕β for β = 1. 0 (β = 2. 5) intercepts straight nullcline q = u at three fixed points (one fixed point) and the corresponding spatially extended network acts as a bistable (excitable) medium. Other parameters are η = 20, κ = 0. 25
2.1 Exact Traveling Pulse Solution
Without loss of generality, let us consider a right-moving traveling pulse solution of the form (u(x, t), q(x, t)) = (U(x − ct), Q(x − ct)) with U(±∞), Q(±∞) = 0 and U(−Δ) = U(0) =κ; see Fig. 2.1(b). Here c, Δ denote the speed and width of the wave, respectively. We also assume that U(ξ) >κ for ξ ∈ (−Δ, 0) and U(ξ) <κ for ξ < −Δ and ξ > 0. Substituting this solution into Eq. (7.49) with ξ = x − ct then gives
It is useful to rewrite Eq. (7.50) in the matrix form
with \(W(\xi ) =\int _{ \xi }^{\infty }w(x)dx\). We proceed by diagonalizing the left-hand side of Eq. (7.51) using the right eigenvectors v of the matrix
These are given by \(\mathbf{v}_{\pm } = {(\epsilon -\lambda _{\pm },\epsilon )}^{T}\) with corresponding eigenvalues
We will assume that ε is sufficiently small so that β < (1 −ε)2∕4ε and consequently λ ± are real. (For a discussion of the effects of complex eigenvalues λ ± see [580].) Note that \(\mathbf{v}_{\pm }{\text{e}}^{\lambda _{\pm }\xi /c}\) are the corresponding null vectors of the linear operator on the left-hand side of Eq. (7.51). Performing the transformation
then gives the pair of equations
with \(\eta _{\pm } = \mp 1/(\lambda _{+} -\lambda _{-})\). Integrating the equation for \(\tilde{U}\) from −Δ to ∞, we have
Finiteness of \(\tilde{U}\) in the limit ξ →∞ requires the term in square brackets to cancel. Hence, we can eliminate \(\tilde{U}(-\varDelta )\) to obtain the result
Similarly,
Performing the inverse transformation \(U = (\epsilon -\lambda _{+})\tilde{U} + (\epsilon -\lambda _{-})\tilde{Q}\) we have
with \(\chi _{\pm } = (\epsilon -\lambda _{\pm })\eta _{\pm }\). The threshold conditions U(−Δ) =κ and U(0) =κ then yield a pair of equations whose solutions determine existence curves relating the speed c and width Δ of a pulse to the threshold κ [134, 198, 504].
For the sake of illustration, let w be given by the exponential function (7.1). In the domain ξ > 0, there is a common factor of \({\text{e}}^{-\xi /\sigma }\) in the integrand of Eq. (7.57) so that \(U(\xi ) =\kappa { \text{e}}^{-\xi /\sigma }\) for ξ > 0 provided that
(Note that for zero negative feedback (β = 0), Eq. (7.58) reduces to the formula for wave speed of a front in the limit Δ →∞.) On the other hand, when ξ < 0, one has to partition the integral of Eq. (7.57) into the separate domains ξ′ > |ξ|, |ξ|−Δ <ξ′ < |ξ| and ξ′ < |ξ|−Δ. This then determines the second threshold condition as well as the asymptotic behavior of U(ξ) in the limit ξ →−∞:
Existence of right-moving traveling pulses in the case of the excitatory network (7.49) with linear adaptation for an exponential weight distribution (7.1). Here σ = 1, ε = 0. 01 and β = 2. 5. (a) Plot of pulse width Δ against threshold κ. (b) Plot of wave speed c against threshold κ. Stable (unstable) branches indicated by black (gray) curves
(a) Rat cortical slices are bathed in picrotoxin (a GABA A blocker) and a stimulation electrode (SE) is placed in layers 5–6 to initiate epileptiform bursts. An electric field is applied globally or locally across the slice using Ag/AgCl electrodes (FE1,FE2). Layer five neurons have long apical dendrites and are easily polarizable by an electric field, which controls the effective firing threshold of the neuron. (b) The time for an activity pulse to travel between two recording electrodes R1 and R2 depends on the applied electric field, reflecting the dependence of wave speed on the effective firing threshold. [Adapted from Richardson, Schiff and Gluckman [521]]
where the amplitudes A ± and A 0 can be determined from matching conditions at the threshold crossing points [198, 504]. Note that the leading edge of the pulse is positive, whereas the trailing edge is negative due to the effects of adaptation. One finds that for sufficiently slow negative feedback (small ε) and large β there exist two pulse solutions: one narrow and slow and the other wide and fast. This is illustrated in Fig. 7.4. Note that a numerical value of c ∼ 1 in dimensionless units (σ =τ = 1) translates into a physical speed of 60–90 mm/s if the membrane time constant τ = 10 msec and the range of synaptic connections is σ = 600–900 μm. Numerically, the fast solution is found to be stable [504], and this can be confirmed analytically using an Evans function construction [134, 198, 507]; see below. Finally, note that one of the predictions of the neural field model is that the speed of wave propagation should increase as the threshold decreases [504]. Interestingly, this has been confirmed experimentally by applying electric fields to a disinhibited rat cortical slice [521]. The experimental setup is shown in Fig. 7.5. A positive (negative) electric field increases (decreases) the speed of wave propagation by altering the effective excitability of layer V pyramidal neurons. Such neurons have long apical dendrites and are easily polarizable by the electric field.
Construction of Evans function. Rewrite the neural field equation (7.49) in the integral form
$$\displaystyle\begin{array}{rcl} u(x,t) =\int _{ -\infty }^{\infty }\int _{ 0}^{\infty }w(y)\varPhi (s)F(u(x - y,t - s))dsdy -\beta \int _{ 0}^{\infty }\varPsi (s)u(x,t - s)ds,& & {}\end{array}$$(7.60)with \(\varPhi (t) ={ \text{e}}^{-t}H(t)\) and \(\varPsi (t) =\int _{ 0}^{t}\varPhi (s){\text{e}}^{-\epsilon (t-s)}ds\). Linearizing about the pulse solution by setting \(u(x,t) = U(\xi ) +\varphi (\xi ){\text{e}}^{\lambda t}\) gives
$$\displaystyle\begin{array}{rcl} \varphi (\xi )& =& \int _{-\infty }^{\infty }\int _{ \xi -y}^{\infty }w(y)\varPhi ((s + y-\xi )/c){\text{e}}^{-\lambda (s+y-\xi )/c}F^{\prime}(U(s))\varphi (s)\frac{ds} {c} dy \\ & & -\beta \int _{\xi }^{\infty }\varPsi ((s-\xi )/c){\text{e}}^{-\lambda (s-\xi )/c}\varphi (s)\frac{ds} {c}. {}\end{array}$$(7.61)Proceeding along similar lines to the analysis of front stability in Sect. 7.1, we set F(U) = H(U −κ) and use the identity
$$\displaystyle{ H^{\prime}(U(\xi )-\kappa ) =\delta (U(\xi )-\kappa ) = \frac{\delta (\xi )} {\vert U^{\prime}(0)\vert } + \frac{\delta (\xi +\varDelta )} {\vert U^{\prime}(-\varDelta )\vert }. }$$(7.62)This gives
$$\displaystyle\begin{array}{rcl} & & \varphi (\xi ) +\beta \int _{ \xi }^{\infty }\varPsi ((s-\xi )/c){\text{e}}^{-\lambda (s-\xi )/c}\varphi (s)\frac{ds} {c} \\ & & = \frac{\varphi (0)} {c\vert U^{\prime}(0)\vert }\mathcal{H}(\lambda,\xi ) + \frac{\varphi (-\varDelta )} {c\vert U^{\prime}(-\varDelta )\vert }\mathcal{H}(\lambda,\xi +\varDelta ) {}\end{array}$$(7.63)where
$$\displaystyle{ \mathcal{H}(\lambda,\xi ) =\int _{ \xi }^{\infty }w(y)\varPhi ((y-\xi )/c){\text{e}}^{-\lambda (y-\xi )/c}dy. }$$(7.64)Let \(\hat{\mathcal{H}}(\lambda,k)\) denote the Fourier transform of \(\mathcal{H}(\lambda,\xi )\) and \(\hat{\mathcal{G}}(\lambda,k)\) denote the Fourier transform of \(\varPsi (\xi /c){\text{e}}^{-\xi /c}\). Using Fourier transforms and the convolution theorem, Eq. (7.63) can then be rewritten as
$$\displaystyle\begin{array}{rcl} \varphi (\xi ) = \frac{\varphi (0)} {c\vert U^{\prime}(0)\vert }\mathcal{B}(\lambda,\xi ) + \frac{\varphi (-\varDelta )} {c\vert U^{\prime}(-\varDelta )\vert }\mathcal{B}(\lambda,\xi +\varDelta ),& & {}\end{array}$$(7.65)with \(\mathcal{B}(\lambda,\xi )\) the inverse transform of
$$\displaystyle{ \hat{\mathcal{B}}(\lambda,k) = \frac{\hat{\mathcal{H}}(\lambda,k)} {[1 +\beta \hat{ \mathcal{G}}(\lambda,-k)/c]}. }$$(7.66)Finally, the eigenvalues λ are determined by setting ξ = 0, −Δ and solving the resulting matrix equation \(\mathbf{f} = \mathcal{M}(\lambda )\mathbf{f}\) with \(\mathbf{f} = (\varphi (0),\varphi (-\varDelta ))\) and
$$\displaystyle\begin{array}{rcl} \mathcal{M}(\lambda ) = \frac{1} {c}\left (\begin{array}{cc} \frac{\mathcal{B}(\lambda,0)} {\vert U^{\prime}(\xi _{1})\vert }& \frac{\mathcal{B}(\lambda,\varDelta )} {\vert U^{\prime}(-\varDelta )\vert } \\ \frac{\mathcal{B}(\lambda,-\varDelta )} {\vert U^{\prime}(0)\vert }& \frac{\mathcal{B}(\lambda,0)} {\vert U^{\prime}(-\varDelta )\vert } \end{array} \right ).& & {}\end{array}$$(7.67)It follows that the eigenvalues λ are zeros of the Evans function
$$\displaystyle{ \mathcal{E}(\lambda ) = \mbox{ Det}[\mathbf{1} -\mathcal{M}(\lambda )], }$$(7.68)where 1 denotes the identity matrix.
2.2 Singularly Perturbed Pulse Solution
In the case of slow adaptation (ε ≪ 1), Pinto and Ermentrout [504] showed how to construct a traveling pulse solution of Eq. (7.49) for a smooth firing rate function F by exploiting the existence of traveling front solutions of the corresponding scalar equation (7.2). The method is analogous to the construction of traveling pulses in the FitzHugh–Nagumo equation [316]; see Sect.2.3. The basic idea is to analyze separately the fast and slow time behavior of solutions to Eq. (7.49) expressed in traveling wave coordinates:
We will assume the normalization \(\int _{-\infty }^{\infty }w(y)dy = 1\). In the case of fast time, the slow adaptation is taken to be constant by setting ε = 0 so that we have the inner layer equations
Since Q(ξ) = Q 0 is a constant, the term β Q 0 can be absorbed into the threshold of the firing rate function F by making the shift \(U(\xi ) \rightarrow U(\xi ) +\beta Q_{0}\). Hence Eq. (3.55) is equivalent to the scalar equation (7.7), which supports the propagation of traveling fronts. In the case of slow time, we introduce the compressed variable ζ =ε ξ so that
In the limit ε → 0, we have
so that first equation becomes
Inverting this equation yields two branches U = g ±(Q). Hence we obtain a slow time or outer layer equation on each branch (see Fig. 7.6):
The construction of the traveling pulse now proceeds by matching inner and outer solutions [504]. This can be visualized by considering the nullclines of the space-clamped version of Eq. (7.49); see Fig. 7.6. We assume that the gain of F and the strength β of adaptation are such that there is only a single fixed point of the space-clamped system:
-
I
Starting at the unique fixed point, use the fast inner equations and the existence results of [172] to construct a leading front solution at Q = Q 0 with speed c 0 and matching conditions \(\lim _{\xi \pm \infty }U(\xi ) = g_{\pm }(Q_{0})\).
-
II
Use the slow outer equations to determine dynamics of Q along upper branch U = g +(Q)
-
III
The solution leaves upper branch at some point Q 1. Once again use the fast inner equations and [172] to construct a trailing front solution with speed c 1 and matching conditions
$$\displaystyle{\lim _{\xi \pm \infty }U(\xi ) = g_{\mp }(Q_{1})}$$ -
IV
Finally, use the slow outer equations to determine the return to the fixed point along the lower branch.
In order to establish the existence of a traveling pulse solution, it remains to find a value Q 1 for which c 1 = −c 0 so that the leading and trailing edges of the pulse move at the same speed and thus the pulse maintains its shape as it propagates. (Since Q 0 is known, so is c 0.) Adapting the formula for the wave speed obtained in [172], we have
Unfortunately, it is not possible to derive a closed form expression for the wave speed. However, the existence of a matching speed can be established provided that certain additional assumptions are made regarding the shape of the firing rate function; see [504] for more details.
Singular perturbation construction of a traveling pulse in (a) the phase plane and (b) traveling wave coordinates. See text for details
3 Wave Propagation in Heterogeneous Neural Fields
Most studies of neural field theory assume that the synaptic weight distribution only depends upon the distance between interacting populations, that is, w(x, y) = w(|x − y|). This implies translation symmetry of the underlying integrodifferential equations (in an unbounded or periodic domain) and an excitatory network can support the propagation of solitary traveling waves. However, if one looks more closely at the anatomy of cortex, it is clear that its detailed microstructure is far from homogeneous. For example, to a first approximation, primary visual cortex (V1) has a periodic-like microstructure on the millimeter length scale, reflecting the existence of various stimulus feature maps; see Sect. 8.1. This has motivated a number of studies concerned with the effects of a periodically modulated weight distribution on wave propagation in neural fields [64, 132, 332].
We first consider the voltage-based neural field equation (6.115) with periodically modulated weight distribution
where ρ is the amplitude of the periodic modulation and \(\varepsilon\) is the period with K(x) = K(x + 1) for all x. It will also be assumed that if ρ = 0 (no periodic modulation), then the resulting homogeneous network supports a traveling front solution of speed c 0 as analyzed in Sect. 7.1.1. We will describe two alternative methods for analyzing the effects of periodic wave modulation: one based on averaging theory for small \(\varepsilon\) [79], which adapts the method use to study the propagation failure in myelinated axons (Sect. 2.5), and the other based on analyzing interfacial dynamics [132]. Both approaches make use of the observation that for sufficiently small ρ, numerical simulations of the inhomogeneous network show a front-like wave separating high and low activity states. However, the wave does not propagate with constant speed, but oscillates periodically in an appropriately chosen moving frame. This pulsating front solution satisfies the periodicity condition \(u(x,t) = u(x+\varepsilon,t + T)\) so that we can define the mean speed of the wave to be \(c =\varepsilon /T\). We will then consider the effects of periodically modulated weights on the propagation of pulled fronts in an activity-based neural field equation, extending the Hamilton–Jacobi method used in the analysis of CaMKII waves in Sect. 3.2.
3.1 Averaging Theory
Suppose that the period \(\varepsilon\) of weight modulations is much smaller than the range of synaptic interactions \(\varepsilon \ll \sigma\). (We fix the length scales by setting σ = 1.) Following the analysis of saltatory waves along myelinated axons (Sect. 2.5) and [318, 319], we want any inhomogeneous terms to be \(\mathcal{O}(\epsilon )\). Therefore, after substituting Eq. (7.78) into (6.115), we integrate by parts to obtain the equation
Here \(\mathcal{K}^{\prime}(x) =\rho K(x)\) with \(\mathcal{K}\) only having to be defined up to an arbitrary constant. Motivated by the existence of pulsating front solutions, we perform the change of variables ξ = x −ϕ(t) and τ = t. Equation (7.79) becomes
Next perform the perturbation expansions
where U(ξ) is the unique traveling wave solution of the corresponding homogeneous equation (7.7) with unperturbed wave speed c = c 0. The first-order term u 1 satisfies the inhomogeneous linear equation
where
and
The linear operator \(\mathbb{L}\) has a one-dimensional null-space spanned by U′. The existence of U′ as a null vector follows immediately from differentiating both sides of Eq. (7.7) with respect to ξ, whereas its uniqueness can be shown using properties of positive linear operators [172]. Therefore, a bounded solution of Eq. (7.83) with respect to ξ and τ will only exist if the right-hand side of Eq. (7.83) is orthogonal to all elements of the null-space of the adjoint operator \({\mathbb{L}}^{\dag }\). The latter is defined with respect to the inner product
where u(ξ) and v(ξ) are arbitrary integrable functions. Hence,
It can be proven that \({\mathbb{L}}^{\dag }\) also has a one-dimensional null-space [172], that is, it is spanned by some function V (ξ). Equation (7.83) thus has a bounded solution if and only if
where
Note that B 0 is strictly positive since V and U′ can be chosen to have the same sign [172]. Substituting for h 1 using Eqs. (7.85) and (7.82) and performing an integration by parts leads to a differential equation for the phase ϕ:
where
The phase equation (7.90) is identical in form to the one derived in Sect. 2.5 for wave propagation along myelinated axons; see Eq. (2.72). It implies that there are two distinct types of behavior. If the right-hand side of Eq. (7.90) is strictly positive, then there exists a pulsating front of the approximate form U(x −ϕ(t)) and the average speed of propagation is \(c = \varepsilon /T\) with
On the other hand, if the right-hand side of Eq. (7.90) vanishes for some ϕ, then there is wave propagation failure.
In the case of a Heaviside firing rate function F(u) = H(u −κ), it is possible to derive an explicit expression for the wave speed c [64]. The solution for the unperturbed wave front U(ξ) was derived in Sect. 7.1, so it is only necessary to determine the solution V (ξ) of the adjoint equation (7.87), which becomes
This can be integrated to give
Given the solutions for U(ξ) and V (ξ), it can then be shown that (7.91) reduces to the form
where
and
Keeping only the lowest-order contribution to Φ 1, Eq. (7.92) reduces to
with Γ(c 0) = W(0)∕B 0. For the sake of illustration, suppose that the periodic modulation functions K and A are pure sinusoids. Setting A(x) =ρsin(2π x)∕(2π) in Eq. (7.98) we find that
and, hence,
This establishes that a sinusoidally varying heterogeneous neural medium only supports a propagating wave if the velocity c 0 of the (unique) solution of the corresponding homogeneous medium satisfies the inequality
For the particular example of an exponential distribution (7.1) with σ = 1, we have c 0 = (1 − 2κ)∕(2κ) and Γ(c 0) = 1 + c 0 so that
Pulsating pulse solutions in a 1D excitatory neural field with linear adaptation and Heaviside firing rate function; see Eq. (7.49). The threshold is κ = 0. 2, strength of adaptation is β = 2. 0, and adaptation rate constant is ε = 0. 04. The weight distribution is given by \(w(x,y) =\rho w(x - y)\sin (2\pi x/\varepsilon )\) with \(2\pi \varepsilon = 0.3\) and w(x) an exponential weight function. (a) Single-bump solution for ρ = 0. 3. The interior of the pulse consists of non-propagating, transient ripples. (b) Multi-bump solution for ρ = 0. 8. The solitary pulse corresponds to the envelope of a multiple bump solution, in which individual bumps are non-propagating and transient. The disappearance of bumps at one end and the emergence of new bumps at the other end generate the propagation of activity [332]
The above averaging method can also be extended to the case of periodically modulated traveling pulses (pulsating pulses) (see [332]), in which there are two threshold crossing points. One simplifying assumption of the analysis is that, in the presence of periodically modulated weights, additional threshold crossing points do not occur. However, numerical solutions of a neural field equation with linear adaptation have shown that in the case of large amplitude modulations, a pulsating pulse can develop multiple threshold crossing points [332]. That is, the traveling wave represents the envelope of a multi-bump solution, in which individual bumps are non-propagating and transient; see Fig. 7.7. The appearance (disappearance) of bumps at the leading (trailing) edge of the pulse generates the coherent propagation of the pulse. Wave propagation failure occurs when activity is insufficient to maintain bumps at the leading edge.
3.2 Interfacial Dynamics
The averaging method provides a reasonable estimate for the mean wave speed and the critical amplitude ρ for wave propagation failure, provided that the spatial period \(\varepsilon \ll 1\). As shown by Coombes and Laing [132] in the case of a Heaviside firing rate function, a more accurate estimate for the wave speed for larger values of \(\varepsilon\) can be obtained by analyzing the dynamics of the interface between high and low activity states, provided that the amplitude of periodic modulations is not too large [132]. The basic idea is to change to a co-moving frame of the unperturbed system, u = u(ξ, t) with ξ = x − c 0 t such that Eq. (6.115) becomes
with w given by Eq. (7.78) and F(u) = H(u −κ). The moving interface (level set) is then defined according to the threshold condition
Differentiating with respect to t then determines the velocity of the interface in the co-moving frame according to
As in the previous averaging method, suppose that for ρ = 0, there exists a traveling front solution U(ξ) of the homogeneous equation (7.7) with speed c 0. Now make the approximation \(u_{\xi }(\xi _{0}(t),t) = U^{\prime}(0)\), which is based on the assumption that for small amplitudes ρ, the slope of the traveling front varies sufficiently slowly. Setting ξ =ξ 0(t) in Eq. (7.103) and using Eq. (7.3), it is then straightforward to show that [132]
In order to match up with the previous method, let \(K(x) =\sin (2\pi x/\varepsilon )\) and \(w(x) ={ \text{e}}^{-\vert x\vert }/2\). Then c 0 = (1 − 2κ)∕(2κ) and [132]
with
The final step is to look for a T-periodic solution of Eq. (7.107) such that \(\xi _{0}(t) =\xi _{0}(t + T)\). Setting \(x_{0} =\xi _{0} + c_{0}t\) with \(x_{0} \in [0,\varepsilon ]\) and integrating gives
This may be evaluated using a half-angle substitution,
where \(z_{0}(t) =\tan [(2\pi x_{0}(t)/\varepsilon +\phi )/2]\) and x 0(0) = 0. A self-consistent pulsating front solution is then obtained by imposing the condition \(\varepsilon = x_{0}(T)\), which then determines the effective speed \(c =\varepsilon /T\) to be
Note that on Taylor expanding \(\gamma (\varepsilon )\) to first order in \(\varepsilon\), Eq. (7.111) recovers the corresponding result (7.102) obtained using averaging theory. However, the expression derived using interfacial dynamics is more accurate when the period \(\varepsilon\) increases, provided that the amplitude ρ does not become too large.
3.3 Hamilton–Jacobi Dynamics and Slow Spatial Heterogeneities
We now turn to the effects of periodically modulated weights on the propagation of pulled front solutions of the activity-based neural field equation (7.27). In the case of high-frequency modulation, Coombes and Laing [132] adapted previous work by Shigesada et al. on pulsating fronts in reaction–diffusion models of the spatial spread of invading species into heterogeneous environments [574, 575]. (In Sect. 3.2.2 we applied the theory of pulsating fronts to CaMKII translocation waves along spiny dendrites.) We briefly sketch the basic steps in the analysis. First, substitute the periodically modulated weight distribution (7.78) into Eq. (7.27) and linearize about the leading edge of the wave where a(x, t) ∼ 0:
Now assume a solution of the form a(x, t) = A(ξ)P(x), ξ = x − ct with A(ξ) → 0 as ξ →∞ and \(P(x + 2\pi \varepsilon ) = P(x)\). Substitution into Eq. (7.112) then gives
Taking \(A(\xi ) \sim {\text{e}}^{-\lambda \xi }\) and substituting into the above equation yields a nonlocal version of the Hill equation:
In order to determine the minimal wave speed, it is necessary to find a bounded periodic solution P(x) of Eq. (7.114), which yields a corresponding dispersion relation c = c(λ), whose minimum with respect to λ can then be determined (assuming it exists). One way to obtain an approximate solution to Eq. (7.114) is to use Fourier methods to derive an infinite matrix equation for the Fourier coefficients of the periodic function P(x) and then to numerically solve a finite truncated version of the matrix equation. This is the approach followed in [132]. The matrix equation takes the form
where \(K(x/\varepsilon ) =\sum _{n}K_{n}{\text{e}}^{imx/\varepsilon }\), \(P(x) =\sum _{n}P_{n}{\text{e}}^{imx/\varepsilon }\), and \(\mathcal{W}(p) =\hat{ W}(p) +\hat{ W}(-p)\) with \(\hat{W}(p)\), the Laplace transform of w(x). One finds that the mean velocity of a pulsating front increases with the period \(2\pi \varepsilon\) of the synaptic modulations [132]. This is illustrated in Fig. 7.8, which shows space–time plots of a pulsating front for \(\varepsilon = 0.5\) and \(\varepsilon = 0.8\).
Space–time contour plots of a pulsating front solution of the neural field equation (7.112) with piecewise linear firing rate function (7.29), Gaussian weight distribution (7.38), and a \(2\pi \varepsilon\)-periodic modulation of the synaptic weights, \(K(x) =\cos (x/\varepsilon )\). (a) \(\varepsilon = 0.5\) and (b) \(\varepsilon = 0.8\). Other parameters are W 0 = 1. 2, σ = 1. 0, and κ = 0. 4
Now suppose that there is a slowly varying spatial modulation of the synaptic weight distribution (relative to the range of synaptic interactions). (Although we do not have a specific example of long-wavelength modulations in mind, we conjecture that these might be associated with inter-area cortical connections. For example, it has been shown that heterogeneities arise as one approaches the V1/V2 border in visual cortex, which has a number of effects including the generation of reflected waves [688].) In the case of slow modulations, one can extend the Hamilton–Jacobi theory of sharp interfaces developed originally for PDEs (see [178, 203, 204, 212, 421] and Sect. 3.3.2) to the case of neural fields [70]. In order to illustrate this, consider a heterogeneous version of the activity-based neural field equation (7.27) of the form
in which there is a slow (nonperiodic) spatial modulation \(J(\varepsilon x)\) of the synaptic weight distribution with \(\varepsilon \ll 1\).
Recall from Sect. 3.3.2 that the first step in the Hamilton–Jacobi method is to rescale space and time in Eq. (7.116) according to \(t \rightarrow t/\varepsilon\) and \(x \rightarrow x/\varepsilon\) [178, 204, 421]:
Under this hyperbolic rescaling, the front region where the activity a(x, t) rapidly increases as x decreases from infinity becomes a step as \(\varepsilon \rightarrow 0\); see Fig. 7.2(b). This motivates introducing the WKB approximation
with G(x, t) > 0 for all x > x(t) and G(x(t), t) = 0. The point x(t) determines the location of the front and \(c =\dot{ x}\). Substituting (3.85) into Eq. (7.117) gives
We have used the fact that for x > x(t) and \(\varepsilon \ll 1\), the solution is in the leading edge of the front so that F can be linearized. Equation (7.119) can be simplified using the method of steepest descents [70]; see below. This yields the equation
where \(\tilde{w}(k)\) is the Fourier transform of w(x):
Equation (7.120) is formally equivalent to the Hamilton–Jacobi equation
with corresponding Hamiltonian
where p = ∂ x G is interpreted as the conjugate momentum of x, and \(\tilde{w}(ip) = \mathcal{W}(p)\). It follows that the Hamilton–Jacobi equation (7.122) can be solved in terms of the Hamilton equations
Let X(s; x, t), P(s; x, t) denote the solution with x(0) = 0 and x(t) = x. We can then determine G(x, t) according to
Here
which is independent of s due to conservation of “energy,” that is, the Hamiltonian is not an explicit function of time.
Steepest descent calculation of G. The derivation of Eq. (7.120) using steepest descents proceeds as follows. First, substituting the Fourier transfer (7.121) into (7.119) and reversing the order of integration gives
$$\displaystyle\begin{array}{rcl} -\partial _{t}G(x,t) = -1 + \frac{1} {\varepsilon } \int _{-\infty }^{\infty }\int _{ -\infty }^{\infty }\tilde{w}(k)J(x^{\prime}){\text{e}}^{-S(k,x^{\prime};x,t)/\varepsilon }dx^{\prime}\frac{dk} {2\pi },& & {}\end{array}$$(7.128)where
$$\displaystyle{ S(k,x^{\prime}; x,t) = ik(x^{\prime} - x) + G(x^{\prime},t) - G(x,t). }$$(7.129)Exploiting the fact that \(\varepsilon\) is small, we perform steepest descents with respect to the x′ variable with (k, x, t) fixed. Let x′ = z(k, t) denote the stationary point for which ∂ S∕∂ x′ = 0, which is given by the solution to the implicit equation
$$\displaystyle{ ik + \partial _{x}G(x^{\prime},t) = 0. }$$(7.130)Taylor expanding S about this point (assuming it is unique) gives to second order
$$\displaystyle\begin{array}{rcl} S(k,x^{\prime}; x,t)& \approx & S(k,z(k,t); x,t) + \left.\frac{1} {2} \frac{{\partial }^{2}S} {\partial {x^{\prime}}^{2}} \right \vert _{x^{\prime}=z(k,t)}{(x^{\prime} - z(k,t))}^{2} \\ & =& ik[z(k,t) - x] + G(z(k,t),t) - G(x,t) \\ & & \qquad -\frac{1} {2}\partial _{xx}G(z(k,t),t){(x^{\prime} - z(k,t))}^{2}. {}\end{array}$$(7.131)Substituting into Eq. (7.128) and performing the resulting Gaussian integral with respect to x′ yields the result
$$\displaystyle\begin{array}{rcl} -\partial _{t}G(x,t)& =& -1 + \frac{1} {\varepsilon } \int _{-\infty }^{\infty }\sqrt{ \frac{2\pi \varepsilon } {\partial _{xx}G(z(k,t),t)}}\tilde{w}(k)J(z(k,t)) \\ & & \qquad \times {\text{e}}^{-\left (ik[z(k,t) - x] + G(z(k,t),t) - G(x,t)\right )/\varepsilon }\frac{dk} {2\pi }. {}\end{array}$$(7.132)This can be rewritten in the form
$$\displaystyle\begin{array}{rcl} -\partial _{t}G(x,t)& =& -1 + \frac{1} {\sqrt{2\pi \varepsilon }}\int _{-\infty }^{\infty }\tilde{w}(k)J(z(k,t)){\text{e}}^{-\hat{S}(k;x,t)/\varepsilon }dk, {}\end{array}$$(7.133)where
$$\displaystyle{ \hat{S}(k; x,t) = ik[z(k,t) - x] + G(z(k,t),t) - G(x,t) + \frac{\varepsilon } {2}\ln \partial _{xx}G(z(k,t),t). }$$(7.134)The integral over k can also be evaluated using steepest descents. Thus, Taylor expand \(\hat{S}\) to second order about the stationary point k = k(x, t), which is the solution to the equation
$$\displaystyle\begin{array}{rcl} 0 = \frac{\partial \hat{S}} {\partial k} = i[z(k,t) - x] + \frac{\partial z(k,t)} {\partial k} \left [ik + \partial _{x}G(z(k,t),t) + \frac{\varepsilon } {2} \frac{\partial _{xxx}G(z(k,t),t)} {\partial _{xx}G(z(k,t),t)} \right ].& & {}\end{array}$$(7.135)It follows from Eqs. (7.130) and (7.135) that \(z(k(x,t),t) = x + \mathcal{O}(\varepsilon )\) and so
$$\displaystyle{ k(x,t) = i\partial _{x}G(x,t) + \mathcal{O}(\varepsilon ). }$$(7.136)Moreover,
$$\displaystyle{ \hat{S}(k; x,t) \approx \frac{1} {2}\left.\frac{{\partial }^{2}\hat{S}} {\partial {k}^{2}} \right \vert _{k=k(x,t)}{(k - k(x,t))}^{2}. }$$(7.137)Substituting into Eq. (7.133) and performing the Gaussian integral with respect to k gives to leading order in \(\varepsilon\)
$$\displaystyle\begin{array}{rcl} -\partial _{t}G(x,t) = -1 + \frac{1} {\sqrt{i\partial _{xx } G(x,t)\partial _{k } z(k(x,t),t)}}\tilde{w}(k(x,t))J(x).& & {}\end{array}$$(7.138)Finally, setting x′ = z(k, t) in Eq. (7.130) and differentiating with respect to k show that \(\partial _{xx}G(z(k,t),t)\partial _{k}z(k,t) = -i\) and we obtain Eq. (7.120).
Given G(x, t), the location x(t) of the front at time t is determined by the equation G(x(t), t) = 0. Differentiating with respect to t shows that \(\dot{x}\partial _{x}G + \partial _{t}G = 0\). Let us begin by rederiving the wave speed for a homogeneous neural field by setting J(x) ≡ 1. In this case, dp∕ds = 0 so that p =λ 0 independently of s. Hence, x(s) = xs∕t, which implies that
On the other hand,
Combining these two results means that λ 0 is given by the minimum of the function
and c 0 = c(λ 0). This recovers the result of Sect. 7.1.3. Thus, in the case of a Gaussian weight distribution, λ 0 is related to c 0 according to Eq. (7.42). Now suppose that there exists a small-amplitude, slow modulation of the synaptic weights J(x) = 1 +β f(x) with β ≪ 1. We can then obtain an approximate solution of Hamilton’s Eqs. (7.124) and (7.125) and the corresponding wave speed using regular perturbation theory along analogous lines to a previous study of the F–KPP equation [421]. We find (see below) that
Here c 0 is the wave speed of the homogeneous neural field (β = 0), which is given by c 0 = c(λ 0) with λ 0 obtained by minimizing the function c(λ) defined by Eq. (7.141); see Eq. (7.42). Finally, differentiating both sides with respect to t and inverting the hyperbolic scaling yields
(a) Propagating front in a homogeneous network with Gaussian weights (7.38) and piecewise linear rate function (7.29). Parameter values are W 0 = 1. 2, σ = 1, κ = 0. 4. The initial condition is taken to be a steep sigmoid a(x, 0) = 0. 5∕(1 + exp(−η(x − l))) with η = 5 and l = 10. (a) Snapshots of wave profile at time intervals of width Δ t = 5 from t = 10 to t = 40. (b) Space–time contour plot. Wave speed asymptotically approaches the minimum c 0 of the velocity dispersion curve given by Eq. (7.39). (c,d) Propagating front in a network with a linear heterogeneity in the synaptic weights, \(J(x) = 1 +\varepsilon (x - l)\), l = 10, and \({\varepsilon }^{2} = 0.005\). Other parameters as in (a,b). (c) Snapshots of wave profile at time intervals of width Δ t = 5 from t = 10 to t = 40. (d) Space–time contour plot. Wave speed increases approximately linearly with time, so the position x(t) of front evolves according to a downward parabola. Theoretical curve based on the perturbation calculation is shown by the solid curve. The trajectory of the front in the corresponding homogeneous case is indicated by the dashed curve
The analytical results agree reasonably well with numerical simulations, provided that \(\varepsilon\) is sufficiently small [70]. In Fig. 7.9(a) we show snapshots of a pulled front in the case of a homogeneous network with Gaussian weights (7.38) and piecewise linear firing rate function (7.29). Space and time units are fixed by setting the range of synaptic weights σ = 1 and the time constant τ = 1. A corresponding space–time plot is given in Fig. 7.9(b), which illustrates that the speed of the front asymptotically approaches the calculated minimal wave speed c 0. (Note that pulled fronts take an extremely long time to approach the minimal wave speed at high levels of numerical accuracy, since the asymptotics are algebraic rather than exponential in time [162].) In Figs. 7.9(c,d) we plot the corresponding results in the case of an inhomogeneous network. For the sake of illustration, the synaptic heterogeneity is taken to be a linear function of displacement, that is, \(J(x) = 1 +\varepsilon (x - l)\), and \(\beta =\varepsilon\). Equation (7.142) implies that
where we have used Eq. (7.141) and assumed that the initial position of the front is x(0) = l. Hence, the perturbation theory predicts that a linearly increasing modulation in synaptic weights results in the leading edge of the front tracing out a downward parabola in a space–time plot for times \(t \ll \mathcal{O}(1{/\varepsilon }^{2})\). This is consistent with numerical simulations for \({\varepsilon }^{2} = 0.005\), as can be seen in the space–time plot of Fig. 7.9(d).
Perturbation calculation of wave speed. Introduce the perturbation expansions
$$\displaystyle{ x(s) = x_{0}(s) +\beta x_{1}(s) + \mathcal{O}{(\beta }^{2}),\quad p(s) = p_{ 0}(s) +\beta p_{1}(s) + \mathcal{O}{(\beta }^{2}) }$$(7.145)and substitute into Eqs. (7.124) and (7.125). Taylor expanding the nonlinear function f(x) about x 0 and \(\mathcal{W}(p) =\hat{ W}(p) +\hat{ W}(-p)\) about p 0 then leads to a hierarchy of equations, the first two of which are
$$\displaystyle{ \dot{p}_{0}(s) = 0,\quad \dot{x_{0}}(s) = \mathcal{W}^{\prime}(p_{0}), }$$(7.146)and
$$\displaystyle{ \dot{p}_{1}(s) = -f^{\prime}(x_{0})\mathcal{W}(p_{0}),\quad \dot{x_{1}}(s) = \mathcal{W}^{\prime\prime}(p_{0})p_{1}(s) + f(x_{0})\mathcal{W}^{\prime}(p_{0}), }$$(7.147)These are supplemented by the Cauchy conditions x 0(0) = 0, x 0(t) = x and x n (0) = x n (t) = 0 for all integers n ≥ 1. Equations (7.146) have solutions of the form
$$\displaystyle{ p_{0}(s) =\lambda,\quad x_{0}(s) = \mathcal{W}^{\prime}(\lambda )s + B_{0} }$$(7.148)with λ, B 0 independent of s. Imposing the Cauchy data then implies that B 0 = 0 and λ satisfies the equation
$$\displaystyle{ \mathcal{W}^{\prime}(\lambda ) = x/t. }$$(7.149)At the next order,
$$\displaystyle\begin{array}{rcl} p_{1}(s) = -\mathcal{W}(\lambda ) \frac{t} {x}f(xs/t) + A_{1},& & {}\end{array}$$(7.150)$$\displaystyle\begin{array}{rcl} x_{1}(s)& =& -\mathcal{W}^{\prime\prime}(\lambda )\mathcal{W}(\lambda ) \frac{{t}^{2}} {{x}^{2}}\int _{0}^{xs/t}f(y)dy \\ & & \qquad +\int _{ 0}^{xs/t}f(y)dy + \mathcal{W}^{\prime\prime}(\lambda )A_{ 1}s + B_{1}, {}\end{array}$$(7.151)with A 1, B 1 independent of s. Imposing the Cauchy data then implies that B 1 = 0 and
$$\displaystyle{ A_{1} = A_{1}(x,t) = \mathcal{W}(\lambda ) \frac{t} {{x}^{2}}\int _{0}^{x}f(y)dy - \frac{1} {t\mathcal{W}^{\prime\prime}(\lambda )}\int _{0}^{x}f(y)dy. }$$(7.152)Given these solutions, the energy function E(x, t) is
$$\displaystyle\begin{array}{rcl} E(x,t)& =& -1 + [1 +\beta f(x_{0} +\beta x_{1}+\ldots )]\mathcal{W}(\lambda +\beta p_{1}+\ldots ) \\ & =& -1 + \mathcal{W}(\lambda ) +\beta [\mathcal{W}^{\prime}(\lambda )p_{1}(s) + f(x_{0}(s))\mathcal{W}(\lambda )] + \mathcal{O}{(\beta }^{2}). {}\end{array}$$(7.153)Substituting for x 0(s) and p 1(s) and using the condition \(\mathcal{W}^{\prime}(\lambda ) = x/t\), we find that
$$\displaystyle{ E(x,t) = -1 + \mathcal{W}(\lambda ) +\beta \frac{x} {t} A_{1}(x,t) + \mathcal{O}{(\beta }^{2}), }$$(7.154)which is independent of s as expected. Similarly,
$$\displaystyle\begin{array}{rcl} \int _{0}^{t}p(s)\dot{x}(s)ds& =& \lambda x +\beta \mathcal{W}^{\prime}(\lambda )\int _{ 0}^{t}p_{ 1}(s)ds + \mathcal{O}{(\beta }^{2}) \\ & =& \lambda x +\beta \frac{\mathcal{W}^{\prime}(\lambda )} {\mathcal{W}^{\prime\prime}(\lambda )}\int _{0}^{t}\left [\dot{x}_{ 1}(s) -\mathcal{W}^{\prime}(\lambda )f(\mathcal{W}^{\prime}(\lambda )s)\right ]ds + \mathcal{O}{(\beta }^{2}) \\ & =& \lambda x -\beta \frac{\mathcal{W}^{\prime}(\lambda )} {\mathcal{W}^{\prime\prime}(\lambda )}\int _{0}^{x}f(y)dy + \mathcal{O}{(\beta }^{2}). {}\end{array}$$(7.155)Hence, to first order in β,
$$\displaystyle{ G(x,t) = t -\mathcal{W}(\lambda )t +\lambda x -\beta \mathcal{W}(\lambda ) \frac{t} {x}\int _{0}^{x}f(y)dy. }$$(7.156)We can now determine the wave speed c by imposing the condition G(x(t), t) = 0 and performing the perturbation expansions \(x(t) = x_{0}(t) +\beta x_{1}(t) + \mathcal{O}{(\beta }^{2})\) and \(\lambda =\lambda _{0} +\beta \lambda _{1} + \mathcal{O}{(\beta }^{2})\). Substituting into Eq. (7.156) and collecting terms at \(\mathcal{O}(1)\) and \(\mathcal{O}(\beta )\) leads to Eq. (7.142).
4 Wave Propagation in Stochastic Neural Fields
In Sect. 6.4 we constructed stochastic neuronal population models based on a master equation formulation. However, continuum versions of these models are difficult to analyze even under a diffusion approximation, due to the nonlocal nature of the multiplicative noise terms; see Eq. (6.71). Therefore, in this section, we analyze the effects of noise on wave propagation in stochastic neural fields with local multiplicative noise, extending the PDE methods outlined in Sect. 2.6. This form of noise can also be interpreted in terms of parametric fluctuations in the firing threshold [58].
4.1 Spontaneous Front Propagation
Consider the following stochastic neural field equation: U(x, t)
We assume that dW(x, t) represents an independent Wiener process such that
where ⟨⋅ ⟩ denotes averaging with respect to the Wiener process. Here λ is the spatial correlation length of the noise such that C(x∕λ) →δ(x) in the limit λ → 0, and ε determines the strength of the noise, which is assumed to be weak. Moreover, the multiplicative noise term is taken to be of Stratonovich form. The analysis of Eq. (7.157) proceeds along similar lines to the analysis of the stochastic bistable equation in Sect. 2.6; see also [70]. First, using Novikov’s theorem, we rewrite Eq. (7.157) so that the fluctuating term has zero mean:
where
and
The stochastic process R has the variance
The next step in the analysis is to express the solution U of Eq. (7.159) as a combination of a fixed wave profile U 0 that is displaced by an amount Δ(t) from its uniformly translating position \(\xi = x - c_{\varepsilon }t\) and a time-dependent fluctuation Φ in the front shape about the instantaneous position of the front:
The wave profile U 0 and associated wave speed \(c_{\varepsilon }\) are obtained by solving the modified deterministic equation
As in Sect. 2.6, Eq. (7.164) is chosen so that that to leading order, the stochastic variable Δ(t) undergoes unbiased Brownian motion with
where the diffusion coefficient \(D(\varepsilon ) = \mathcal{O}(\varepsilon )\) can be calculated using perturbation analysis (see below).
Perturbation calculation of diffusivity \(D(\varepsilon )\) . Substitute the decomposition (7.163) into Eq. (7.159) and expand to first order in \(\mathcal{O}{(\varepsilon }^{1/2})\):
$$\displaystyle\begin{array}{rcl} & & -c_{\varepsilon }U_{0}^{\prime}(\xi -\varDelta (t))dt - U_{0}^{\prime}(\xi -\varDelta (t))d\varDelta (t) {+\varepsilon }^{1/2}\left [d\varPhi (\xi -\varDelta (t),t) - c_{\varepsilon }\varPhi ^{\prime}(\xi -\varDelta (t),t)dt\right ] {}\\ & & \quad {-\varepsilon }^{1/2}\varPhi ^{\prime}(\xi -\varDelta (t),t)d\varDelta (t) {}\\ & & \qquad = h(U_{0}(\xi -\varDelta (t)))dt + h^{\prime}{(U_{0}(\xi -\varDelta (t)))\varepsilon }^{1/2}\varPhi (\xi -\varDelta (t),t)dt {}\\ & & \quad +\int _{ -\infty }^{\infty }w(\xi -\xi ^{\prime})\left (F(U_{ 0}(\xi ^{\prime} -\varDelta (t))) + F^{\prime}{(U_{0}(\xi ^{\prime} -\varDelta (t)))\varepsilon }^{1/2}\varPhi (\xi ^{\prime} -\varDelta (t),t)\right )d\xi ^{\prime}dt {}\\ & & \quad {+\varepsilon }^{1/2}dR(U_{ 0}(\xi -\varDelta (t)),\xi,t) + \mathcal{O}(\varepsilon ). {}\\ \end{array}$$Imposing Eq. (7.164), after shifting \(\xi \rightarrow \xi -\varDelta (t)\), and dividing through by \({\varepsilon }^{1/2}\) then gives
$$\displaystyle\begin{array}{rcl} d\varPhi (\xi -\varDelta (t),t)& =& \hat{L} \circ \varPhi (\xi -\varDelta (t),t)dt {+\varepsilon }^{-1/2}U_{ 0}^{\prime}(\xi -\varDelta (t))d\varDelta (t) \\ & & +dR(U_{0}(\xi -\varDelta (t),\xi,t) + \mathcal{O}{(\varepsilon }^{1/2}), {}\end{array}$$(7.166)where \(\mathbb{L}\) is the non-self-adjoint linear operator
$$\displaystyle\begin{array}{rcl} \mathbb{L} \circ A(\xi ) = c_{\varepsilon }\frac{dA(\xi )} {d\xi } + h^{\prime}(U_{0}(\xi ))A(\xi ) +\int _{ -\infty }^{\infty }w(\xi -\xi ^{\prime})F^{\prime}(U_{ 0}(\xi ^{\prime}))A(\xi ^{\prime})d\xi ^{\prime}& & {}\end{array}$$(7.167)for any function \(A(\xi ) \in {L}^{2}(\mathbb{R})\). Finally, for all terms in Eq. (7.166) to be of the same order, we require that \(\varDelta (t) = \mathcal{O}{(\varepsilon }^{1/2})\). It then follows that \(U_{0}(\xi -\varDelta (t)) = U_{0}(\xi ) + \mathcal{O}{(\varepsilon }^{1/2})\) and Eq. (7.166) reduces to
$$\displaystyle\begin{array}{rcl} d\varPhi (\xi,t)& =& \mathbb{L} \circ \varPhi (\xi,t)dt {+\varepsilon }^{-1/2}U_{ 0}^{\prime}(\xi )d\varDelta (t) + dR(U_{0}(\xi ),\xi,t) + \mathcal{O}{(\varepsilon }^{1/2}). {}\end{array}$$(7.168)It can be shown that for a sigmoid firing rate function and exponential weight distribution, the operator \(\mathbb{L}\) has a 1D null-space spanned by U 0′(ξ) [172]. (The fact that U 0′(ξ) belongs to the null-space follows immediately from differentiating equation (7.164) with respect to ξ.) We then have the solvability condition for the existence of a nontrivial solution of Eq. (7.168), namely, that the inhomogeneous part is orthogonal to all elements of the null-space of the adjoint operator \({\mathbb{L}}^{\dag }\). The latter is almost identical in form to Eq. (7.87):
$$\displaystyle{ {\mathbb{L}}^{\dag }B(\xi ) = -c_{\varepsilon }\frac{dB(\xi )} {d\xi } + h^{\prime}(U_{0}(\xi ))B(\xi ) + F^{\prime}(U_{0}(\xi ))\int _{-\infty }^{\infty }w(\xi -\xi ^{\prime})B(\xi ^{\prime})d\xi ^{\prime}. }$$(7.169)Hence, \({\mathbb{L}}^{\dag }\) has a one-dimensional null-space that is spanned by some function \(\mathcal{V}(\xi )\). Taking the inner product of both sides of Eq. (7.168) with respect to \(\mathcal{V}(\xi )\) then leads to the solvability condition
$$\displaystyle{ \int _{-\infty }^{\infty }\mathcal{V}(\xi )\left [U_{ 0}^{\prime}(\xi )d\varDelta (t) {+\varepsilon }^{1/2}dR(U_{ 0},\xi,t)\right ]d\xi = 0. }$$(7.170)Thus Δ(t) satisfies the stochastic differential equation (SDE)
$$\displaystyle{ d\varDelta (t) = {-\varepsilon }^{1/2}\frac{\int _{-\infty }^{\infty }\mathcal{V}(\xi )dR(U_{ 0},\xi,t)d\xi } {\int _{-\infty }^{\infty }\mathcal{V}(\xi )U_{ 0}^{\prime}(\xi )d\xi }. }$$(7.171)Using the lowest-order approximation \(dR(U_{0},\xi,t) = g(U_{0}(\xi ))dW(\xi,t)\), we deduce that Δ(t) is a Wiener process with effective diffusion coefficient
$$\displaystyle\begin{array}{rcl} D(\varepsilon )& =& \varepsilon \frac{\int _{-\infty }^{\infty }\int _{ -\infty }^{\infty }\mathcal{V}(\xi )\mathcal{V}(\xi ^{\prime})g(U_{ 0}(\xi ))g(U_{0}(\xi ^{\prime}))\langle dW(\xi,t)dW(\xi ^{\prime},t)\rangle d\xi d\xi ^{\prime}} {{\left [\int _{-\infty }^{\infty }\mathcal{V}(\xi )U_{ 0}^{\prime}(\xi )d\xi \right ]}^{2}} {}\end{array}$$(7.172)
In the case of a Heaviside rate function F(U) = H(U −κ) and multiplicative noise g(U) = g 0 U, the effective speed \(c_{\varepsilon }\) and diffusion coefficient \(D(\varepsilon )\) can be calculated explicitly [70]. (The constant g 0 has units of \(\sqrt{\mathrm{length/time}}\).) The deterministic equation (7.164) for the fixed profile U 0 then reduces to
with
This is identical in structure to Eq. (7.3) for the deterministic neural field modulo the rescaling of the decay term. The analysis of the wave speeds proceeds along similar lines to Sect. 7.1. Thus, multiplying both sides of Eq. (7.173) by \({\text{e}}^{-\xi \gamma (\varepsilon )/c_{\varepsilon }}\) and integrating with respect to ξ gives
Finally, requiring the solution to remain bounded as ξ →∞ (ξ →−∞) for \(c_{\varepsilon } > 0\) (for \(c_{\varepsilon } < 0\)) implies that κ must satisfy the condition
Hence, in the case of the exponential weight distribution (7.1), we have
for \(c_{\varepsilon } > 0\), and
for \(c_{\varepsilon } < 0\). Assuming that \(0 \leq \gamma (\varepsilon ) \leq 1\), we see that multiplicative noise shifts the effective velocity of front propagation in the positive ξ direction.
Numerical simulation showing the propagation of a front solution of the stochastic neural field equation (7.157) for Heaviside weight function F(U) = H(U −κ) with κ = 0. 35, exponential weight function (7.1) with σ = 2, and multiplicative noise g(U) = U. Noise strength ε = 0. 005 and C(0) = 10. The wave profile is shown at successive times (a) t = 0 (b) t = 12 (c) t = 18 and (d) t = 24, with the initial profile at t = 0 given by the solution to Eq. (7.164). In numerical simulations we take the discrete space and time steps Δ x = 0. 1, Δ t = 0. 01. The deterministic part U 0 of the stochastic wave is shown by the solid gray curves and the corresponding solution in the absence of noise \((\varepsilon = 0\)) is shown by the dashed gray curves
In order to calculate the diffusion coefficient, it is first necessary to determine the null vector \(\mathcal{V}(\xi )\) of the adjoint linear operator \({\mathbb{L}}^{\dag }\) defined by Eq. (7.169). Setting F(U) = H(U −κ) and g(U) = g 0 U, we obtain an adjoint equation almost identical to (7.93):
Hence, this has the solution
and Eq. (7.172) reduces to the form
In the case of an exponential weight distribution, U 0(ξ) has the explicit form
and the integrals in (7.181) can be evaluated explicitly to give
Plot of (a) mean \(\overline{X}(t)\) and (b) variance \(\sigma _{X}^{2}(t)\) of front position as a function of time, averaged over N = 4096 trials. Same parameter values as Fig. 7.10
Plot of (a) wave speed \(c_{\varepsilon }\) and (b) diffusion coefficient \(D(\varepsilon )\) as a function of threshold κ. Numerical results (solid dots) are obtained by averaging over N = 4, 096 trials starting from the initial condition given by Eq. (7.182 ). Corresponding theoretical predictions (solid curves) for \(c_{\varepsilon }\) and \(D(\varepsilon )\) are based on Eqs. (7.177) and (7.181), respectively. Other parameters as in Fig. 7.10
In Fig. 7.10 we show the temporal evolution of a single stochastic wave front, which is obtained by numerically solving the stochastic neural field equation (7.157) for F(U) = H(U −κ), g(U) = U and an exponential weight distribution w. In order to numerically calculate the mean location of the front as a function of time, we carry out a large number of level set position measurements. That is, we determine the positions X a (t) such that U(X a (t), t) = a, for various level set values a ∈ (0. 5κ, 1. 3κ) and then define the mean location to be \(\overline{X}(t) = \mathbb{E}[X_{a}(t)]\), where the expectation is first taken with respect to the sampled values a and then averaged over N trials. The corresponding variance is given by \(\sigma _{X}^{2}(t) = \mathbb{E}[{(X_{a}(t) -\bar{ X}(t))}^{2}]\). In Fig. 7.11 we plot \(\overline{X}(t)\) and σ X 2(t) as a function of t. It can be seen that both vary linearly with t, consistent with the assumption that there is a diffusive-like displacement of the front from its uniformly translating position at long time scales. The slopes of these curves then determine the effective wave speed and diffusion coefficient according to \(\overline{X}(t) \sim c_{\varepsilon }t\) and \(\sigma _{X}^{2}(t) \sim 2D(\varepsilon )t\). In Fig. 7.12 we plot the numerically estimated speed and diffusion coefficient for various values of the threshold κ and compare these to the corresponding theoretical curves obtained using the above analysis. It can be seen that there is excellent agreement with our theoretical predictions provided that κ is not too large. As κ → 0. 5, the wave speed decreases towards zero so that the assumption of relatively slow diffusion breaks down.
4.2 Stimulus-Locked Fronts
So far we have assumed that the underlying deterministic neural field equation is homogeneous in space so that there exists a family of traveling front solutions related by a uniform shift. Now suppose that there exists an external front-like input that propagates at a uniform speed v, so that the deterministic equation (7.2) becomes
where the input is taken to be a positive, bounded, monotonically decreasing function of amplitude \(I_{0} = I(-\infty ) - I(\infty )\). The resulting inhomogeneous neural field equation can support a traveling front that locks to the stimulus, provided that the amplitude of the stimulus is sufficiently large [198]. Consider, in particular, the case of a Heaviside firing rate function F(u) = H(u −κ). (See [174] for an extension to the case of a smooth sigmoid function F.) We seek a traveling wave solution \(u(x,t) = \mathcal{U}(\xi )\) where ξ = x − vt and \(\mathcal{U}(\xi _{0}) =\kappa\) at a single threshold crossing point \(\xi _{0} \in \mathbb{R}\). The front is assumed to travel at the same speed as the input (stimulus-locked front). If I 0 = 0, then we recover the homogeneous equation (7.2) and ξ 0 becomes a free parameter, whereas the wave propagates at the natural speed c(κ) given by Eq. (7.6). Substituting the front solution into Eq. (7.184) yields
This can be solved for v > 0 by multiplying both sides by the integrating factor \({v}^{-1}{\text{e}}^{-v\xi }\) and integrating over the interval [ξ, ∞) with U(ξ) → 0 as ξ →∞ to give
with W(ξ) defined according to Eq. (7.3). Similarly, for v < 0, we multiply by the same integrating factor and then integrate over (−∞, ξ] with U(ξ) → W 0 as ξ →−∞ to find
The threshold crossing condition \(\mathcal{U}(\xi _{0}) =\kappa\) then determines the position ξ 0 of the front relative to the input as a function of speed v, input amplitude I 0, and threshold κ.
One of the interesting features of stimulus-locked fronts is that they are much more robust to noise [70]. In order to show this, consider the following stochastic version of Eq. (7.184):
Proceeding along identical lines to the case of freely propagating fronts, Eq. (7.186) is first rewritten so that the fluctuating term has zero mean:
and h and R are given by Eqs. (7.160) and (7.161), respectively. The stochastic field U(x, t) is then decomposed according to Eq. (7.163) with U 0 a front solution of
It is assumed that the fixed profile U 0 is locked to the stimulus (has speed v). However, multiplicative noise still has an effect on U 0 by generating an \(\varepsilon\)-dependent threshold crossing point \(\xi _{\varepsilon }\) such that \(U_{0}(\xi _{\varepsilon }) =\kappa\).
Proceeding to the next order and imposing equation (7.188), we find that \(\varDelta (t) = \mathcal{O}{(\varepsilon }^{1/2})\) and
where \(\mathbb{L}\) is the non-self-adjoint linear operator (7.167) with \(c_{\varepsilon } \rightarrow v\). The last term on the right-hand side of Eq. (7.189) arises from the fact that in Eq. (7.163), U 0 and Φ are expressed as functions of ξ −Δ(t) and \(I(\xi ) = I(\xi -\varDelta (t) +\varDelta (t)) \approx I(\xi -\varDelta (t)) + I^{\prime}(\xi -\varDelta (t))\varDelta (t)\). A nontrivial solution of Eq. (7.189) exists if and only if the inhomogeneous part is orthogonal to the null vector \(\mathcal{V}(\xi )\) of the adjoint operator \({\mathbb{L}}^{\dag }\) defined by Eq. (7.169) with \(c_{\varepsilon } \rightarrow v\). Taking the inner product of both sides of Eq. (7.189) with respect to \(\mathcal{V}(\xi )\) thus leads to the solvability condition
It follows that, to leading order, Δ(t) satisfies the Ornstein–Uhlenbeck equation
where
and
Note that A > 0 for I 0 > 0, since both U 0(ξ) and I(ξ) are monotonically decreasing functions of ξ. Moreover
with D(ε) given by Eq. (7.172). Using standard properties of an Ornstein–Uhlenbeck process [210], we conclude that
In particular, the variance approaches a constant \(D(\varepsilon )/A\) in the large t limit, rather than increasing linearly with time as found for freely propagating fronts.
Plot of existence regions of a stimulus-locked front without noise (γ = 1, dark gray) and in the presence of noise (γ = 0. 9, light gray) with overlapping regions indicated by medium gray. Stimulus taken to be of the form \(I(x,t) = I_{0}H(-\xi ),\xi = x - vt\) with amplitude I 0 and speed v. Other parameter values as in Fig. 7.10. (a) κ = 0. 95: spontaneous fronts exist in the absence of a stimulus (I 0 = 0). (b) κ = 1. 25: there are no spontaneous fronts
Numerical simulation showing the propagation of a stimulus-locked wave-front solution (black curves) of the stochastic neural field equation (7.186) for Heaviside weight function F(U) = H(U −κ) with κ = 0. 35, exponential weight function (7.1) with σ = 2, and multiplicative noise g(U) = U. The external input (gray curves) is taken to be of the form I(x, t) = I 0Erfc[x − vt] with amplitude I 0 = 0. 4 and speed v = 1. 5. Noise strength ε = 0. 005 and C(0) = 10. The wave profile is shown at successive times (a) t = 0 (b) t = 6 (c) t = 12 and (d) t = 24, with the initial profile at t = 0 given by the solution U 0 of Eq. (7.188). In numerical simulations we take the discrete space and time steps Δ x = 0. 1, Δ t = 0. 01
In order to illustrate the above analysis, take g(U) = g 0 U for the multiplicative noise term and set F(U) = H(U −κ). The deterministic Eq. (7.188) for the profile U 0 then reduces to
Existence of front solution proceeds along identical lines to Sect. 7.1, except now the speed v is fixed, whereas the threshold crossing point ξ 0, say, is no longer arbitrary due to the breaking of translation symmetry. The point ξ 0 is determined by the threshold condition U 0(ξ 0) =κ and will depend on the noise strength \(\varepsilon\). In Fig. 7.13 we show existence regions in the (v, I 0)-plane for stimulus-locked fronts when I(ξ) = I 0 H(−ξ), that is, for a step function input of speed v and amplitude I 0. This illustrates the fact that multiplicative noise leads to an \(\varepsilon\)-dependent shift in the existence regions. In Fig. 7.14 we show the temporal evolution of a single stimulus-locked front, which is obtained by numerically solving the Langevin equation (7.186) for F(U) = H(U −κ), g(U) = U and an exponential weight distribution w. Numerically speaking, it is convenient to avoid discontinuities in the input by taking I(x, t) = I 0Erfc[x − vt] rather than a Heaviside. The corresponding mean \(\overline{X}(t)\) and variance σ X 2(t) of the position of the front, which are obtained by averaging over level sets as outlined in Sect. 7.4.1, are shown in Fig. 7.15. It can be seen that, as predicted by the analysis, \(\overline{X}(t)\) varies linearly with t with a slope equal to the stimulus speed v = 1. 5. Moreover, the variance σ X 2(t) approaches a constant value as t →∞, which is comparable to the theoretical value \(D(\varepsilon )/A\) evaluated for the given input. Thus, we find that stimulus-locked fronts are much more robust to noise than freely propagating fronts, since the variance of the mean position saturates as t →∞. Consequently, stimulus locking persists in the presence of noise over most of the parameter range for which stimulus locking is predicted to occur.
Plot of (a) mean \(\overline{X}(t)\) and (b) variance σ X 2(t) of the position of a stimulus-locked front as a function of time, averaged over N = 4096 trials. Smooth gray curve in (b) indicates theoretical prediction of variance. Stimulus taken to be of the form \(I(x,t) = I(x - ct) = I_{0}\mbox{ Erfc}[x - vt]\) with amplitude I −0 = 0. 4 and speed v = 1. 5. Other parameter values as in Fig. 7.10
4.3 Stochastic Pulled Fronts
In the case of the F–KPP equation with multiplicative noise, one finds that the stochastic wandering of a pulled front about its mean position is subdiffusive with varΔ(t) ∼ t 1∕2, in contrast to the diffusive wandering of a front propagating into a metastable state for which varΔ(t) ∼ t [530]. Such scaling is a consequence of the asymptotic relaxation of the leading edge of the deterministic pulled front. Since pulled front solutions of the neural field equation (7.27) exhibit similar asymptotic dynamics (see Eq. (7.46)), it suggests that there will also be subdiffusive wandering of these fronts in the presence of multiplicative noise. In order to illustrate this, consider the stochastic neural field equation
with W(x, t) a Wiener process satisfying Eq. (2.84). Note that the noise term has to vanish when A(x, t) = 0, since the firing rate A is restricted to be positive. Hence, the noise has to be multiplicative. Formally speaking, one can carry over the analysis of the Langevin equation (7.157). First, decompose the solution along similar lines to Eq. (2.96):
with \(\xi = x - c_{\varepsilon }t\), with the fixed front profile A 0 satisfying the deterministic equation
The effective velocity \(c_{\varepsilon }\) of the front is given by the minimum of the dispersion curve
(a) Plot of variance σ X 2(t) of the position of a stochastic pulled front as a function of time. (b) Log-log plot of variance σ X 2(t) as a function of time t. Noise amplitude \(\varepsilon = 0.005\) and κ = 0. 8. Other parameter values as in Fig. 7.2
Fluctuations thus shift the dispersion curve to higher velocities. However, it is no longer possible to derive an expression for the diffusion coefficient \(D(\varepsilon )\) along the lines of Eq. (7.172), since both numerator and denominator would diverge for a pulled front. This reflects the asymptotic behavior of the leading edge of the front. It is also a consequence of the fact that there is no characteristic time scale for the convergence of the front velocity to its asymptotic value, which means that it is not possible to separate the fluctuations into a slow wandering of front position and fast fluctuations of the front shape [162, 486]. Nevertheless, numerical simulations of Eq. (7.197) with F given by the piecewise linear firing rate (7.29) are consistent with subdiffusive wandering of the front. In Fig. 7.16(a), we plot the variance σ X 2(t) of the position of a pulled front solution of Eq. (7.197), which are obtained by averaging over level sets along identical lines to Sect. 7.4.1. It can be seen that the variance appears to exhibit subdiffusive behavior over longer time scales. This is further illustrated by plotting a log-log plot of σ X 2(t) against time t; see Fig. 7.16(b). At intermediate time scales, the slope of the curve is approximately equal to one, consistent with normal diffusion, but at later times the slope decreases, indicating subdiffusive behavior.
5 Traveling Waves in 2D Oscillatory Neural Fields
Troy and Shusterman [580, 634] have shown how a neural field model with strong linear adaptation (see Eq. (7.49)) can act as an oscillatory network that supports 2D target patterns and spiral waves consistent with experimental studies of tangential cortical slices [292]. (For the analysis of spiral waves in the corresponding excitable regime, see [354].) However, since the linear form of adaptation used in these studies is not directly related to physiological models of adaptation, it is difficult to ascertain whether or not the strength of adaptation required is biologically reasonable. This motivated a more recent study of spiral waves in a 2D neural medium involving a nonlinear, physiologically based form of adaptation, namely, synaptic depression [330]. The latter model is given by
The radially symmetric excitatory weight distribution is taken to be an exponential, \(w(r) ={ \text{e}}^{-r}/2\pi\). It can be shown that the space-clamped model
Limit cycle oscillations in the space-clamped system (7.202) for a piecewise linear firing rate function (1.16) with threshold κ = 0. 01 and gain η = 4. (a) Bifurcation diagram showing fixed points u of the system as a function of β for τ q = 80. (b) Corresponding phase-plane plot of q versus u (gray curve) for β = 4, showing that the system supports a stable limit cycle [329]
Target patterns in a 2D neural field with synaptic depression induced by an initial conditional stimulus specified by Eq. (7.203) at t = 0, where χ = 1 and ζ = 25. Initially, an activated state spreads radially outward, across the entire medium as a traveling front. Then, the localized oscillating core of activity emits a target wave with each oscillation cycle. Eventually, these target waves fill the domain. Each target wave can be considered as a phase shift in space of the oscillation throughout the medium; they travel with the same speed as the initial front. Parameters are τ q = 80, β = 4, η = 4, and κ = 0. 01 [330]
supports limit cycle oscillations provided that the firing rate function has finite gain. For example, in the case of the piecewise linear firing rate function (1.16), oscillations arise via a subcritical Hopf bifurcation of a high activity fixed point; see Fig. 7.17. One then finds that the full network model (7.201) supports a spatially localized oscillating core that periodically emits traveling pulses [330]. Such dynamics can be induced by taking an initial condition of the form
Spiral wave generated by shifting the phase of the top and bottom halves of the target pattern shown in Fig. 7.18. The period of the spiral wave oscillation is roughly the same as the period of the oscillation in the space-clamped system. All patches of neurons are oscillating at the same frequency, but phase-shifted as coordinates are rotated about the central phase singularity [330]
where χ and ζ parameterize the amplitude and spatial constant of the initial state. An example of a pulse-emitting core is shown in Fig. 7.18, which oscillates at a frequency of roughly 3 Hz. Pulses are emitted each cycle and travel at a speed of roughly 30 cm/s, which is determined by the period of the oscillations; the latter is set by the time constant of synaptic depression. The initial emission of spreading activity appears as a traveling front which propagates from the region activated by the input current into the surrounding region of zero activity; it travels at the same speed as the subsequent target waves. The front converts each region of the network into an oscillatory state that is phase-shifted relative to the core, resulting in the appearance of a radially symmetric target pattern. Spiral waves can also be induced by breaking the rotational symmetry of pulse emitter solutions [330]. More specifically, if the target pattern produced by the emitter has the top and bottom halves of its domain phase-shifted, then the dynamics evolves into two counterrotating spirals on the left and right halves of the domain. Closer inspection of one of these spirals reveals that it has a fixed center about which activity rotates indefinitely as shown in Fig. 7.19.
A very different mechanism for generating periodic waves in a 1D or 2D neural field model is through the combination of adaptation and a spatially localized input [196, 197]. Recall from Sect. 7.2 that a 1D excitatory neural field with adaptation supports the propagation of solitary traveling pulse, which can be induced by perturbing the system with a transient localized pulse. (In contrast to the previous example, we are assuming that the neural field operates in an excitable regime.) In the case of a 2D network with radially symmetric weights, such a pulse will produce a single expanding circular wave. Now suppose that a 2D localized pulse persists in the form of a radially symmetric Gaussian input \(I(\mathbf{r}) = I_{0}{\text{e}}^{-{r}^{2}{/\sigma }^{2} }\)—this could either represent an external stimulus or a localized region of depolarization. As one might expect, for sufficiently large input amplitude I 0, the neural field supports a radially symmetric stationary pulse or bump centered about the input. Such a bump is not self-sustaining, however, since if the input is removed, then the bump disappears as well. This then raises the question as to what happens to the stability of the bump as the input amplitude is slowly decreased. One finds that the bump first undergoes a Hopf instability as I 0 is decreased, leading to the formation of a spatially localized oscillating pulse or breather [196]. Interestingly, as the input amplitude is further reduced, the breather can undergo a secondary instability such that it now acts as an oscillating core that emits circular target waves. Thus, a spatially localized stationary input provides a mechanism for the formation of a network pacemaker oscillator. A linear stability analysis establishes that the primary instability is due to the growth of radially symmetric eigenmodes. A similar bifurcation scenario also occurs in a neural field with lateral inhibition, except that now the Hopf bifurcation typically involves the growth of nonradially symmetric eigenmodes, resulting in asymmetric breathers and rotating waves [197].
References
Abbott, L.F., Marder, E.: Modelling small networks. In: Koch, C., Segev, I. (eds.) Methods in Neuronal Modelling, pp. 361–410, 2nd edn. MIT Press, Cambridge (1998)
Abbott, L.F., van Vresswijk, C.: Asynchronous states in networks of pulse–coupled oscillators. Phys. Rev. E 48(2), 1483–1490 (1993)
Abbott, L.F., Fahri, E., Gutmann, S.: The path integral for dendritic trees. Biol. Cybern. 66, 49–60 (1991)
Abbott, L.F., Varela, J.A., Sen, K., Nelson, S.B.: Synaptic depression and cortical gain control. Science 275, 220–224 (1997)
Aertsen, A.M.H.J., Gerstein, G.L., Habib, M.K., Palm, G.: Dynamics of neuronal firing correlation: modulation of ‘effective connectivity’. J. Neurophysiol. 61, 900–917 (1989)
Alexander, G., Crutcher, M.: Functional architecture of basal ganglia circuits: neural substrates of parallel processing. Trends. Neurosci. 13, 266–271 (1990)
Allard, J., Moginler, A.: Traveling waves in actin dynamics and cell motility. Curr. Opin. Cell Biol. 25, 107–115 (2013)
Amari, S.: Dynamics of pattern formation in lateral inhibition type neural fields. Biol. Cybern. 27, 77–87 (1977)
Andersen, S.S., Bi, G.Q.: Axon formation: a molecular model for the generation of neuronal polarity. Bioessays 22, 172–179 (2000)
Andersen, P., Morris, R., Amaral, D., Bliss, T., O’Keefe, J.: The Hippocampus Book. Oxford University Press, Oxford (2008)
Angelucci, A., Levitt, J.B., Walton, E.J.S., Hupe, J.M., Bullier, J., Lund, J.S.: Circuits for local and global signal integration in primary visual cortex. J. Neurosci. 22, 8633–8646 (2002)
Antic, S.D., Zhou, W.L., Moore, A.R., Short, S.M., Ikonomu, K.D.: The decade of the dendritic NMDA spike. J. Neurosci. Res. 88, 2991–3001 (2010)
Arimura, N., Kaibuchi, K.: Neuronal polarity: from extracellular signals to intracellular mechanisms. Nat. Rev. Neurosci 8, 194–205 (2007)
Armero, J., Casademunt, J., Ramirez-Piscina, L., Sancho, J.M.: Ballistic and diffusive corrections to front propagation in the presence of multiplicative noise. Phys. Rev. E 58, 5494–5500 (1998)
Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)
Atay, F.M., Hutt, A.: Stability and bifurcations in neural fields with finite propagation speed and general connectivity. SIAM J. Appl. Math. 65, 644–646 (2005)
Austin, T.D.: The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism. Ann. Appl. Prob. 18, 1279–1325 (2006)
Avoli, M., de Curtis, G.B.M.: Do interictal spikes sustain seizures and epileptogenesis. Epilepsy Currents 6, 203–207 (2006)
Baer, S.M., Rinzel, J.: Propagation of dendritic spikes mediated by excitable spines: a continuum theory. J. Neurophysiol. 65, 874–890 (1991)
Bair, W., Koch, C., Newsome, W.T., Britten, K.H.: Power spectrum analysis of bursting cells in area MT in the behaving monkey. J. Neurosci. 14, 2870–2893 (1994)
Baladron, J., Fasoli, D., Faugeras, O., Touboul, J.: Mean field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons. J. Math. Neuro. 2, 10 (2012)
Ballesteros-Yanez, I., Benavides-Piccione, R., Elston, G.N., Yuste, R., DeFelipe, J.: Density and morphology of dendritic spines in mouse neocortex. Neurosci. 138, 403–409 (2006)
Barabasi, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)
Baras, F., Mansour, M., Malek, M., Pearson, J.E.: Microscopic simulation of chemical bistability in homogeneous systems. J. Chem. Phys. 105, 8257–8261 (1996)
Bart, E., Bao, S., Holcman, D.: Modeling the spontaneous activity of the auditory cortex. J. Comput. Neurosci. 19, 357–378 (2005)
Barthelemy, M., Barrat, A., Pastor-Satorras, R., Vespignani2, A.: Velocity and hierarchical spread of epidemic outbreaks in scale-free networks. Phys. Rev. Lett. 92, 178701 (2004)
Bazhenov, M., Timofeev, I., Steriade, M., Sejnowski, T.J.: Model of thalamocortical slow-wave sleep oscillations and transitions to activated states. J. Neurosci. 22, 8691–8704 (2002)
Bedard, C., Destexhe, A.: Macroscopic models of local field potentials the apparent 1/f noise in brain activity. Biophys. J. 96, 2589–2603 (2009)
Beggs, J.M., Plenz, D.: Neuronal avalanches in neocortical circuits. J. Neurosci. 23, 11167–11177 (2003)
Beggs, J.M., Plenz, D.: Neuronal avalanches are diverse and precise activity patterns that are stable for many hours in cortical slice cultures. J. Neurosci. 24, 5216–5229 (2004)
Beggs, J.M., Timme, N.: Being critical of criticality in the brain. Front. Physiol. 3 (163), 1–14 (2012)
Ben-Ari, Y.: Developing networks play a similar melody. Trend. Neurosci. 24, 353–360 (2001)
Ben-Jacob, E., Brand, H., Dee, G., Kramer, L., Langer, J.S.: Pattern propagation in nonlinear dissipative systems. Physica D 14, 348–364 (1985)
Ben-Yishai, R., Bar-Or, R.L., Sompolinsky, H.: Theory of orientation tuning in visual cortex. Proc. Nat. Acad. Sci. 92, 3844–3848 (1995)
Benda, J., Herz, A.V.M.: A universal model for spike-frequency adaptation. Neural Comput. 15, 2523–2564 (2003)
Benguria, R., Depassier, M.: Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation. Comm. Math. Phys. 175, 221–227 (1996)
Bennett, M.R., Farnell, L., Gibson, W.G.: A quantitative model of purinergic junctional transmission of calcium waves in astrocyte networks. Biophys. J. 89, 2235–2250 (2005)
Bennett, M.R., Farnell, L., Gibson, W.G.: A quantitative model of cortical spreading depression due to puringenic and gap-junction transmission in astrocyte networks. Biophys. J. 95, 5648–5660 (2008)
Benucci, A., Frazor, R., Carandini, M.: Standing waves and traveling waves distinguish two circuits in visual cortex. Neuron 55, 103–17 (2007)
Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model: II-biological invasions and pulsating travelling fronts. J. Math. Biol. 51, 75–113 (2005)
Bernard, C., Anderson, A., Becker, A., Poolos, N.P., Beck, H., Johnston, D.: Acquired dendritic channelopathy in temporal lobe epilepsy. Science 305, 532–535 (2004)
Berridge, M.J.: Calcium signalling and cell proliferation. Bioessays 17, 491–500 (1995)
Berridge, M.J.: Neuronal calcium signaling. Neuron 21, 13–26 (1998)
Berridge, M.J., Lipp, P., Bootman, M.D.: The versatility and universality of calcium signalling. Nat. Rev. Mol. Cell Biol. 1, 11–21 (2000)
Bertram, R., Smith, G.D., Sherman, A.: Modeling study of the effects of overlapping calcium microdomains on neurotransmitter release. Biophys. J. 76, 735–750 (1999)
Blake, R.: A primer on binocular rivalry, including current controversies. Brain Mind 2, 5–38 (2001)
Blake, R., Logothetis, N.: Visual competition. Nat. Rev. Neurosci. 3, 1–11 (2002)
Blake, R., Wilson, H.R.: Binocular vision. Vis. Res. 51, 754–770 (2011)
Blakenship, A.G., Feller, M.B.: Mechanisms underlying spontaneous patterned activity in developing neural circuits. Nat. Rev. Neurosci. 11, 18–29 (2010)
Blasdel, G.G.: Orientation selectivity, preference, and continuity in monkey striate cortex. J. Neurosci. 12, 3139–3161 (1992)
Blasdel, G.G., Salama, G.: Voltage-sensitive dyes reveal a modular organization in monkey striate cortex. Nature 321, 579–585 (1986)
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U.: Complex networks: Structure and dynamics. Phys. Rep. 424, 175–308 (2006)
Bonhoeffer, T., Grinvald, A.: Orientation columns in cat are organized in pinwheel like patterns. Nature 364, 166–146 (1991)
Boonstra, T., Daffertshofer, A., Breakspear, M., Beek, P.: Multivariate time-frequency analysis of electromagnetic brain activity during bimanual motor learning. Neuroimage 36, 370–377 (2007)
Borisyuk, R., Kirillov, A.B.: Bifurcation analysis of a neural network model. Biol. Cybern. 66, 319–325 (1992)
Bosking, W.H., Zhang, Y., Schofield, B., Fitzpatrick, D.: Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex. J. Neurosci. 17, 2112–2127 (1997)
Boustani, S.E., Destexhe, A.: A master equation formalism for macroscopic modeling of asynchronous irregular activity states. Neural Comput. 21, 46–100 (2009)
Brackley, C.A., Turner, M.S.: Random fluctuations of the firing rate function in a continuum neural field model. Phys. Rev. E 75, 041913 (2007)
Brascamp, J.W., van Ee, R., Noest, A.J., Jacobs, R.H., van den Berg, A.V.: The time course of binocular rivalry reveals a fundamental role of noise. J. Vision 6, 1244–1256 (2006)
Breakspear, M., Roberts, J.A., Terry, J.R., Rodrigues, S., Mahant, N., Robinson, P.A.: A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis. Cereb. Cortex 16, 1296–1313 (2006)
Bressloff, P.C.: Resonantlike synchronization and bursting in a model of pulse-coupled neurons with active dendrites. J. Comput. Neurosci. 6, 237–249 (1999)
Bressloff, P.C.: Synaptically generated wave propagation in excitable neural media. Phys. Rev. Lett. 82, 2979–2982 (1999)
Bressloff, P.C.: Traveling waves and pulses in a one-dimensional network of integrate-and-fire neurons. J. Math. Biol. 40, 169–183 (2000)
Bressloff, P.C.: Traveling fronts and wave propagation failure in an inhomogeneous neural network. Physica D 155, 83–100 (2001)
Bressloff, P.C.: Bloch waves, periodic feature maps and cortical pattern formation. Phys. Rev. Lett. 89, 088101 (2002)
Bressloff, P.C.: Spatially periodic modulation of cortical patterns by long-range horizontal connections. Physica D 185, 131–157 (2003)
Bressloff, P.C.: Pattern formation in visual cortex. In: Chow, C.C., Gutkin, B., Hansel, D., Meunier, C., Dalibard, J. (eds.) Les Houches 2003: Methods and Models in Neurophysics, pp. 477–574. Elsevier, Amsterdam (2005)
Bressloff, P.C.: Stochastic neural field theory and the system-size expansion. SIAM J. Appl. Math. 70, 1488–1521 (2009)
Bressloff, P.C.: Metastable states and quasicycles in a stochastic Wilson-Cowan model of neuronal population dynamics. Phys. Rev. E 85, 051903 (2010)
Bressloff, P.C.: From invasion to extinction in heterogeneous neural fields. J. Math. Neurosci. 2(6), 1–27 (2012)
Bressloff, P.C.: Spatiotemporal dynamics of continuum neural fields. J. Phys. A 45, 033001 (109 pp.) (2012)
Bressloff, P.C.: Propagation of CaMKII translocation waves in heterogeneous spiny dendrites. J. Math. Biol. 66, 1499–1525 (2013)
Bressloff, P.C., Coombes, S.: Dynamics of strongly coupled spiking neurons. Neural Comput. 12, 91–129 (2000)
Bressloff, P.C., Cowan, J.D.: Amplitude equation approach to contextual effects in visual cortex. Neural Comput. 14, 493–525 (2002)
Bressloff, P.C., Cowan, J.D.: The visual cortex as a crystal. Physica D 173, 226–258 (2002)
Bressloff, P.C., Folias, S.E.: Front bifurcations in an excitatory neural network. SIAM J. Appl. Math. 65, 131–151 (2005)
Bressloff, P.C., Dwyer, V.M., Kearney, M.J.: Sum-over-paths approach to diffusion on trees. J. Phys. A 29, 1881–1896 (1996)
Bressloff, P.C., Cowan, J.D., Golubitsky, M., Thomas, P.J.: Scalar and pseudoscalar bifurcations: pattern formation on the visual cortex. Nonlinearity 14, 739–775 (2001)
Bressloff, P.C., Cowan, J.D., Golubitsky, M., Thomas, P.J., Wiener, M.: Geometric Visual Hallucinations, Euclidean Symmetry and the Functional Architecture of Striate Cortex. Phil. Trans. Roy. Soc. Lond. B 356, 299–330 (2001)
Bressloff, P.C., Folias, S.E., Prat, A., Li, Y.X.: Oscillatory waves in inhomogeneous neural media. Phys. Rev. Lett. 91, 178101 (2003)
Bressloff, P.C., Newby, J.: Quasi-steady state analysis of motor-driven transport on a two-dimensional microtubular network. Phys. Rev. E 83, 061139 (2011)
Bressloff, P.C., Newby, J.M.: Metastability in a stochastic neural network modeled as a velocity jump Markov process. SIAM J. Appl. Dyn. Sys. 12, 1394–1435 (2013)
Bressloff, P.C., Webber, M.: Neural field model of binocular rivalry waves. J. Comput. Neurosci. 32, 233–252 (2012)
Bressloff, P.C., Webber, M.A.: Front propagation in stochastic neural fields. SIAM J. Appl. Dyn. Syst. 11, 708–740 (2012)
Brown, R.J., Norcia, A.M.: A method for investigating binocular rivalry in real-time with the steady state VEP. Vision Res. 37, 2401–2408 (1997)
Brunel, N.: Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J. Comput. Neurosci 8, 183–208 (2000)
Brunel, N., Hakim, V.: Fast global oscillations in networks of integrate–and–fire neurons with low firing rates. Neural Comput. 11, 1621–1671 (1999)
Buckwar, E., Riedler, M.G.: An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution. J. Math. Biol. 63, 1051–1093 (2011)
Buckwar, E., Riedler, M.G.: Laws of large numbers and langevin approximations for stochastic neural field equations. J. Math. Neurosci. 3(1), 1–54 (2012)
Buice, M.A., Chow, C.C.: Effective stochastic behavior in dynamical systems with incomplete information. Phys. Rev. E 84, 051120 (2011)
Buice, M., Cowan, J.D.: Field-theoretic approach to fluctuation effects in neural networks. Phys. Rev. E 75, 051919 (2007)
Buice, M., Cowan, J.D., Chow, C.C.: Systematic fluctuation expansion for neural network activity equations. Neural Comp. 22, 377–426 (2010)
Bures, J., Buresova, O., Krivanek, J.: The Mechanisms and Applications of Leao’s Spreading Depression of Electrical Activity. Academia, Prague (1974)
Butts, D.A., Feller, M.B., Shatz, C.J., Rokshar, D.S.: Retinal waves are governed by collective network properties. J. Neurosci. 19, 3580–3593 (1999)
Butz, E.G., Cowan, J.D.: Transient potentials in dendritic systems of arbitrary geometry. Biophys. J. 14, 661–689 (1974)
Buzsaki, G.: Hippocampal sharp waves: their origin and significance. Brain Res. 398, 242–252 (1986)
Buzsaki, G.: Theta oscillations in the hippocampus. Neuron 33, 325–40 (2002)
Buzsaki, G.: Rhythms of the Brain. Oxford University Press, Oxford (2006)
Byrne, J.H., Roberts, J.L.: From Molecules to Networks: An Introduction to Cellular and Molecular Neuroscience. Elsevier, Amsterdam (2004)
Cai, D., Tao, L., Shelley, M., McLaughlin, D.W.: An effective kinetic representation of fluctuation–driven neuronal networks with application to simple and complex cells in visual cortex. Proc. Natl. Acad. Sci. USA 101, 7757–7562 (2004)
Campbell, K., Gotz., M.: Radial glia: multi-purpose cells for vertebrate brain development. Trend. Neurosci. 25, 235–238 (2002)
Cantrell, R.S., Cosner, C.: Spatial Ecology Via Reaction-Diffusion Equations. Wiley, Chichester (2003)
Carlsson, A.E.: Dendritic actin filament nucleation causes traveling waves and patches. Phys. Rev. Lett. 104, 228102 (2010)
Carpenter, G.: A geometric approach to singular perturbation problems with applications to nerve impulse equations. J. Diff. Eqns. 23, 335–367 (1977)
Casti, A.R.R., Ormutag, A., Sornborger, A., Kaplan, E., Knight, B., Victor, J., Sirovich, L.: A population study of integrate–and–fire–or burst neurons. Neural Comput. 14, 957–986 (2002)
Charles, A.C., Merrill, J.E., Dickson, E.R., Sanderson, M.J.: Intercellular signaling in glial cells: calcium waves and oscillations in response to mechanical stimulation and glutamate. Neuron 6, 983–992 (1991)
Charles, A.C., Naus, C.C., Zhu, D., Kidder, G.M., Dirksen, E.R., Sanderson, M.J.: Intercellular calcium signaling via gap junctions in glioma cells. J. Cell. Biol. 118, 195–201 (1992)
Chen, X.: Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Diff. Eqns. 2, 125 (1997)
Chen, K.C., Nicholson, C.: Spatial buffering of potassium ions in brain extracellular space. Biophys. J. 78, 2776–2797 (2000)
Cheng, H., Lederer, W.J.: Calcium sparks. Physiol. Rev. 88, 1491–1545 (2008)
Cheng, H., Lederer, M.R., Lederer, W.J., Cannell, M.B.: Calcium sparks and waves in cardiac myocytes. Am. J. Physiol. 270, C148–C159 (1996)
Chervin, R.D., Pierce, P.A., Connors, B.W.: Periodicity and directionality in the propagation of epileptiform discharges across neocortex. J. Neurophysiol. 60, 1695–1713 (1988)
Chialvo, D.: Emergent complex neural dynamics. Nat. Phys. 6, 744–750 (2010)
Chong, S.C., Tadin, D., Blake, R.: Endogenous attention prolongs dominance durations in binocular rivalry. J. Vision 5, 1044–1012 (2005)
Chow, C.C., Buice, M.: Path integral methods for stochastic differential equations. arXiv nlin/105966v1 (2011)
Chow, C.C., White, J.A.: Spontaneous action potentials due to channel fluctuations. Biophys. J. 71, 3013–3021 (1996)
Chu, P.H., Milton, J.G., Cowan, J.D.: Connectivity and the dynamics of integrate-and-fire neural networks. Int. J. Bif. Chaos 4, 237–243 (1994)
Chuquet, J., Hollander, L., Nimchinsky., E.A.: High-resolution in vivo imaging of the neurovascular unit during spreading depression. J. Neurosci. 27, 4036–4044 (2007)
Churchland, M.M., et al.: Stimulus onset quenches neural variability: a widespread cortical phenomenon. Nat. Neurosci. 13, 369–378 (2010)
Citri, A., Malenka, R.C.: Synaptic plasticity: multiple forms, functions, and mechanisms. Neuropsychopharmacol. 33, 18–41 (2008)
Cohen, A.H., Ermentrout, G.B., Kiermel, T., Kopell, N., Sigvardt, K.A., Williams, T.L.: Modeling of intersegmental coordination in the lamprey central pattern generator for motion. Trend. Neurosci. 15, 434–438 (1992)
Compte, A., Sanchez-Vives, M.V., McCormick, D.A., Wang, X.J.: Cellular and network mechanisms of slow oscillatory activity ( < 1 Hz) and wave propagations in a cortical network model. J. Neurophysiol. 89, 2707–2725 (2003)
Collingridge, G.L., Peineau, S., Howland, J.G., Wang, Y.T.: Long-term depression in the CNS. Nat. Rev. Neurosci. 11, 459–473 (2010)
Compte, A., Reig, R., Descalzo, V.F., Harvey, M.A., Puccini, G.D., Sanchez-Vives, M.V.: Spontaneous high-frequency (10–80 Hz) oscillations during up states in the cerebral cortex in vitro. J. Neurosci. 28, 13828–13844 (2008)
Connors, B.W., Amitai, Y.: Epilespy: Models, mechanisms and concepts. In: Schwartkroin, P.A. (ed.) Generation of Epileptiform Discharge by Local Circuits of Neocortex, pp. 388–423. Cambridge University Press, Cambridge (1993)
Connors, B.W., Long, M.A.: Electrical synapses in the mammalian brain. Annu. Rev. Neurosci. 27, 393–418 (2004)
Coombes, S.: The effects of ion pumps on the speed of traveling waves in the fire-diffuse-fire model of Ca2+ release. Bull. Math. Biol. 63, 1–20 (2001)
Coombes, S.: Dynamics of synaptically coupled integrate-and-fire-or-burst neuron. Phys. Rev. E 67, 041910 (2003)
Coombes, S., Bressloff, P.C.: Solitary waves in a model of dendritic cable with active spines. SIAM J. Appl. Math. 61, 432–453 (2000)
Coombes, S., Bressloff, P.C.: Saltatory waves in the spike-diffuse-spike model of active dendritic spines. Phys. Rev. Lett. 91, 028102 (2003)
Coombes, S., Bressloff, P.C. (eds.): Bursting: The Genesis of Rhythm in the Nervous System. World Scientific Press, London (2005)
Coombes, S., Laing, C.R.: Pulsating fronts in periodically modulated neural field models. Phys. Rev. E 83, 011912 (2011)
Coombes, S., Lord, G.J.: Intrinsic modulation of pulse-coupled integrate-and-fire neurons. Phys. Rev. E 56(5), 5809–5818 (1997)
Coombes, S., Owen, M.R.: Evans functions for integral neural field equations with Heaviside firing rate function. SIAM J. Appl. Dyn. Syst. 4, 574–600 (2004)
Coombes, S., Owen, M.R.: Bumps, breathers, and waves in a neural network with spike frequency adaptation. Phys. Rev. Lett. 94, 148102 (2005)
Coombes, S., Schmidt, H.: Neural fields with sigmoidal firing rates: approximate solutions. Discret. Contin. Dyn. Syst. Ser. A 28, 1369–1379 (2010)
Coombes, S., Timofeeva, Y.: Sparks and waves in a stochastic fire-diffue-fire model of Ca2+ release. Phy. Rev. E 68, 021915 (2003)
Coombes, S., Owen, M.R., Smith, G.D.: Mode-locking in a periodically forced integrate-and-fire-or-burst neuron model. Phys. Rev. E 64, 041914 (2001)
Coombes, S., Lord, G.J., Owen, M.R.: Waves and bumps in neuronal networks with axo-dendritic synaptic interactions. SIAM J. Appl. Dyn. Sys. 5, 552–574 (2003)
Coombes, S., Hinch, R., Timofeeva, Y.: Receptors, sparks and waves in a fire-diffuse-fire framework for Ca2+ release. Prog. Biophys. Mol. Biol. 85, 197–216 (2004)
Coombes, S., Timofeeva, Y., Svensson, C.M., Lord, G.J., Josic, K., Cox, S.J., Colbert, C.M.: Branching dendrites with resonant membrane: a “sum-over-trips” approach. Biol. Cybern. 97, 137–149 (2007)
Coombes, S., Venkov, N.A., Bojak, I., Liley, D.T.J., Laing, C.R.: Modeling electrocortical activity through improved local approximations of integral neural field equations. Phys. Rev. E 76, 051901 (2007)
Cornell-Bell, A.H., Finkbeiner, S.M., Cooper, M.S., Smith, S.J.: Glutamate induces calcium waves in cultured astrocytes: long-range glial signaling. Science 247, 470–473 (1990)
Dahlem, M.A., Chronicle, E.P.: A computational perspective on migraine aura. Prog. Neurobiol. 74, 351–361 (2004)
Dahlem, M.A., Muller, S.C.: Migraine aura dynamics after reverse retinotopic mapping of weak excitation waves in the primary visual cortex. Biol. Cybern. 88, 419–424 (2003)
Dale, N.: Dynamic ATP signalling and neural development. J. Physiol. 586, 2429–2436 (2008)
Deco, G., Jirsa, V., Robinson, P.A., Breakspear, M., Friston, K.: The dynamic brain: from spiking neurons to neural masses and cortical fields. PLoS Comput. Biol. 4, e1000092 (2008)
Deco, G., Jirsa, V.K., McIntosh., A.R.: Emerging concepts for the dynamical organization of resting-state activity in the brain. Nat. Rev. Neurosci. 12, 43–56 (2011)
Delaney, K.R., Galperin, A., Fee, M.S., Flores, J.A., Gervais, R., Tank, D.W., Kleinfeld, D.: Waves and stimulus-modulated dynamics in an oscillating olfactory network. Proc. Natl. Acad. Sci. USA 91, 669–673 (1994)
Delorme, V., Machacek, M., DerMardirossian, C., Anderson, K.L., Wittmann, T., Hanein, D., Waterman-Storer, C., Danuser, G., Bokoch, G.M.: Cofilin activity downstream of Pak1 regulates cell protrusion efficiency by organizing lamellipodium and lamella actin networks. Dev. Cell. 13, 646–662 (2007)
Derkach, V., Barria, A., Soderling, T.R.: Ca2+/calmodulin-kinase II enhances channel conductance of α-amino-3-hydroxyl-5-methyl-4-isoxazoleproprionate type glutamate receptors. Proc. Nat. Acad. Sci. (USA) 96, 3269–3274 (1999)
Destexhe, A., Contreras., D.: Neuronal computations with stochastic network states. Science 314, 85–90 (2006)
Destexhe, A., Sejnowski, T.J.: Thalamocortical Assemblies: How Ion Channels, Single Neurons and Large-Scale Networks Organize Sleep Oscillations. Oxford University Press, Oxford (2001)
Destexhe, A., Mainen, Z.F., Sejnowski, T.J.: Synthesis of models for excitable membranes synaptic transmission and neuromodulation using a common kinetic formalism. J. Comput. Neurosci. 1, 195–231 (1994)
Difato, F., Tsushima, H., Pesce, M., Benfenati, F., Blau, A., Chieregatti, E.: The formation of actin waves during regeneration after axonal lesion is enhanced by BDNF. Sci. Rep. 1 (183), 1–18 (2011)
Dockery, J.D., Keener, J.P.: Diffusive effects on dispersion in excitable media. SIAM J. Appl. Dyn. 49, 539–566 (1989)
Doubrovinski, K., Kruse, K.: Cytoskeletal waves in the absence of molecular motors. Europhys. Lett. 83, 18003 (2008)
van Drongelen, W., Lee, H.C., Stevens, R.L., Hereld, M.: Propagation of seizure-like activity in a model of neocortex. J. Clin. Neurophysiol. 24, 182–188 (2007)
Dyhrfjeld-Johnsen, J., Santhakumar, V., Morgan, R.J., Huerta, R., Tsimring, L., Soltesz, I.: Topological determinants of epileptogenesis in large-scale structural and functional models of the dentate gyrus derived from experimental data. J. Neurophysiol. 97, 1566–1587 (2007)
Dykman, M.I., Mori, E., Ross, J., Hunt, P.M.: Large fluctuations and optimal paths in chemical kinetics. J. Chem. Phys. A 100, 5735–5750 (1994)
Earnshaw, B.A., Bressloff, P.C.: Diffusion-activation model of CaMKII translocation waves in dendrites. J. Comput. Neurosci. 28, 77–89 (2010)
Ebert, U., van Saarloos, W.: Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts. Physica D 146, 1–99 (2000)
Edwards, J.R., Gibson, W.G.: A model for Ca2+ waves in networks of glial cells incorporating both intracellular and extracellular communication pathways. J. Theor. Biol. 263, 45–58 (2010)
Elgart, V., Kamenev, A.: Rare event statistics in reaction–diffusion systems. Phys. Rev. E 70, 041106 (2004)
Elmer, C.E., Vleck, E.S.V.: Analysis and computation of traveling waves solutions of bistable differential-difference equations. Nonlinearity 12, 771–798 (1999)
Ermentrout, G.B.: The behaviour of rings of coupled oscillators. Journal of Mathematical Biology 23, 55–74 (1985)
Ermentrout, G.B.: The analysis of synaptically generated travelling waves. J. Comput. Neurosci. 5, 191–208 (1998)
Ermentrout, G.B.: Neural networks as spatio-temporal pattern-forming systems. Rep. Prog. Phy. 61, 353–430 (1998)
Ermentrout, G.B., Cowan, J.: A mathematical theory of visual hallucination patterns. Bio. Cybern. 34, 137–150 (1979)
Ermentrout, G.B., Kleinfeld, D.: Traveling electrical waves in cortex: insights from phase dynamics and speculation on a computational role. Neuron 29, 33–44 (2001)
Ermentrout, G.B., Kopell, N.: Frequency plateaus in a chain of weakly coupled oscillators. SIAM J. Appl. Math. 15, 215–237 (1984)
Ermentrout, G.B., McLeod, J.B.: Existence and uniqueness of travelling waves for a neural network. Proc. Roy. Soc. Edin. A 123, 461–478 (1993)
Ermentrout, G.B., Terman, D.: Mathematical Foundations of Neuroscience. Springer, Berlin (2010)
Ermentrout, G.B., Jalics, J.Z., Rubin, J.E.: Stimulus-driven traveling solutions in continuum neuronal models with a general smooth firing rate function. SIAM J. Appl. Math. 70, 3039–3064 (2010)
Escudero, C., Kamanev, A.: Switching rates of multistep reactions. Phys. Rev. E 79, 041149 (2009)
Escudero, C., Rodriguez, J.A.: Persistence of instanton connections in chemical reactions with time–dependent rates. Phys. Rev. E 77, 011130 (2008)
Evans, J.: Nerve axon equations IV: The stable and unstable impulse. Ind. Univ. Math. J. 24, 1169–1190 (1975)
Evans, L.C., Sougandis, P.E.: A PDE approach to geometric optics for certain semilinear parabolic equations. Ind. Univ. Math. J. 38, 141–172 (1989)
Faisal, A.A., Selen, L.P.J., Wolpert, D.M.: Noise in the nervous system. Nat. Rev. Neurosci. 9, 292 (2008)
Falcke, M.: On the role of stochastic channel behavior in intracellular Ca2+ dynamics. Biophys. J. 84, 42–56 (2003)
Falcke, M.: Reading the patterns in living cells - the physics of Ca2+ signaling. Adv. Phys. 53, 255–440 (2004)
Falcke, M., Tsimiring, L., Levine, H.: Stochastic spreading of intracellular Ca2+ release. Phys. Rev. E 62, 2636–2643 (2000)
Fath, G.: Propagation failure of traveling waves in a discrete bistable medium. Physica D 116, 176–190 (1998)
Faugeras, O., Touboul, J., Cessac, B.: A constructive mean–field analysis of multi–population neural networks with random synaptic weights and stochastic inputs. Frontiers in Comp. Neurosci. 3, 1–28 (2009)
Faugeras, O., Veltz, R., Grimbert, F.: Persistent neural states: Stationary localized activity patterns in nonlinear continuous n-population, q-dimensional neural networks. Neural Comput. 21, 147–187 (2009)
Fellin, T., Pascual, O., Gobbo, S., Pozzan, T., Haydon, P.G., Carmignoto, G.: Neuronal synchrony mediated by astrocytic glutamate through activation of extrasynaptic NMDA receptors. Neuron 43, 729–743 (2004)
Fellin, T., Pozzan, T., Carmignoto., G.: Purinergic receptors mediate two distinct glutamate release pathways in hippocampal astrocytes. J. Biol. Chem. 281, 4274–4284 (2006)
Ferster, D., Miller, K.: Neural mechanisms of orientation selectivity in the visual cortex. Annual review of neuroscience 23(1), 441–471 (2000)
Fianco, T.A., McCarthy, K.D.: Intracellular astrocyte calcium waves in situ increase the frequency of spontaneous AMPA receptor currents in CA1 pyramidal cells. J. Neurosci. 24, 722–732 (2004)
Firth, S.I., Wang, C.T., Feller, M.B.: Retinal waves: mechanisms and function in visual system development. Cell Calcium 37, 425–432 (2005)
Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugenics 7, 353–369 (1937)
FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1960)
Fivaz, M., Bandara, S., Inoue, T., Meyer, T.: Robust neuronal symmetry-breaking by Ras-triggered local positive feedback. Curr. Biol. 18, 44–50 (2008)
Flynn, K.C., Pak, C.W., Shaw, A.E., Bradke, F., Bamburg, J.R.: Growth cone-like waves transport actin and promote axonogenesis and neurite branching. Dev. Neurobiol. 69, 761–779 (2009)
Fogelson, A., Zucker, R.S.: Presynaptic calcium diffusion from various arrays of single channels. implications for transmitter release and synaptic facilitation. Biophys. J. 48, 1003–1017 (1985)
Folias, S.E., Bressloff, P.C.: Breathing pulses in an excitatory neural network. SIAM J. Appl. Dyn. Syst. 3, 378–407 (2004)
Folias, S.E., Bressloff, P.C.: Breathers in two–dimensional neural media. Phys. Rev. Lett. 95, 208107 (2005)
Folias, S.E., Bressloff, P.C.: Stimulus–locked traveling pulses and breathers in an excitatory neural network. SIAM J. Appl. Math. 65, 2067–2092 (2005)
Fox, R., Rasche, F.: Binocular rivalry and reciprocal inhibition. Attention Percept. Psychophys. 5(4), 215–217 (1969)
Fox, R.F., Lu, Y.N.: Emergent collective behavior in large numbers of globally coupled independent stochastic ion channels. Phys. Rev. E 49, 3421–3431 (1994)
Franci, A., Drion, G., Sepulchre, R.: An organizing center in a planar model of neuronal excitability. SIAM J. Appl. Dyn. Sys. 11, 1698–1722 (2013)
Franks, K.M., Sejnowski, T.J.: Complexity of calcium signaling in synaptic spines. Bioessays 24, 1130–1144 (2002)
Freidlin, M.I.: Limit theorems for large deviations and reaction-diffusion equations. Ann. Prob. 13, 639–675 (1985)
Freidlin, M.I.: Geometric optics approach to reaction–diffusion equations. SIAM J. Appl. Math. 46, 222–232 (1986)
Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer, New York (1984)
Friedman, A., Craciun, G.: A model of intracellular transport of particles in an axon. J. Math. Biol. 51, 217–246 (2005)
Friedman, A., Hu, B.: Uniform convergence for approximate traveling waves in linear reaction-hyperbolic systems. Ind. Univ. Math. J. 56, 2133–2158 (2007)
Fusi, S., Drew, P.J., Abbott, L.F.: Cascade models of synaptically stored memories. Neuron 45, 599–611 (2005)
Garcia-Ojalvo, J., Sancho, J.M.: External fluctuations in a pattern-forming instability. Phys. Rev. E 53, 5680–5689 (1996)
Gardiner, C.W.: Handbook of Stochastic Methods, 4th edn. Springer, Berlin (2009)
Gardner-Medwin, A.R.: Analysis of potassium dynamics in mammalian brain tissue. J. Physiol. (Lond) 335, 393–462 (1983)
Gartner, J., Freidlin, M.I.: On the propagation of concentration waves in periodic and random media. Soviet Math. Dokl. 20, 1282–1286 (1979)
Gerstner, W., van Hemmen, J.L.: Coherence and incoherence in a globally coupled ensemble of pulse–emitting units. Phys. Rev. Lett. 71(3), 312–315 (1993)
Gerstner, W., Kistler, W.: Spiking Neuron Models. Cambridge University Press, Cambridge (2002)
Gerstner, W., van Hemmen, J.L., Cowan, J.D.: What matters in neuronal locking? Neural Comput. 8, 1689–1712 (1996)
Gilbert, C.D.: Horizontal integration and cortical dynamics. Neuron 9, 1–13 (1992)
Gilbert, C.D., Das, A., Ito, M., Kapadia, M., Westheimer, G.: Spatial integration and cortical dynamics. Proc. Nat. Acad. Sci. USA 93, 615–622 (1996)
Gilbert, C.D., Wiesel, T.N.: Clustered intrinsic connections in cat visual cortex. J. Neurosci. 3, 1116–1133 (1983)
Gilboa, G., Chen, R., Brenner, N.: History–dependent multiple-time-scale dynamics in a single-neuron model. J. Neurosci. 25, 6479–6489 (2005)
Ginzburg, I., Sompolinsky, H.: Theory of correlations in stochastic neural networks. Phys. Rev. E 50, 3171–3191 (1994)
Glass, L., Mackey, M.C.: From Clocks to Chaos. Princeton University Press, Princeton (1988)
Godfrey, K.B., Eglen, S.J.: Theoretical models of spontaneous activity generation and propagation in the developing retina. Mol. Biosyst. 5, 1527–1535 (2009)
Godfrey, K.B., Swindale, N.V.: Retinal wave behavior through activity-dependent refractory periods. PLoS Comp. Biol. 3 (11), 2408–2420 (2007)
Goel, P., Sneyd, J., Friedman, A.: Homogenization of the cell cytoplasm: the calcium bidomain equations. SIAM Nultiscal. Model Simul. 5, 1045–1062 (2006)
Goldberg, J.L.: How does an axon grow? Genes Dev. 17, 941–958 (2003)
Golding, N.L., Staff, N.P., Spruston, N.: Dendritic spikes as a mechanism for cooperative long-term potentiation. Nature 418, 326–331 (2002)
Goldwyn, J.H., Shea-Brown, E.: The what and where of adding channel noise to the Hodgkin-Huxley equations. PLoS Comp. Biol. 7 (11), e1002247 (2011)
Golomb, D., Amitai, Y.: Propagating neuronal discharges in neocortical slices: Computational and experimental study. J. Neurophysiol. 78, 1199–1211 (1997)
Golomb, D., Ermentrout, G.: Bistability in pulse propagation in networks of excitatory and inhibitory populations. Phys. Rev. Lett. 86, 4179–4182 (2001)
Golomb, D., Ermentrout, G.B.: Continuous and lurching traveling waves in neuronal networks with spatially-decaying connectivity and delay. Proc. Natl. Acad Sci. USA 96, 13480–13485 (1999)
Golomb, D., Rinzel, J.: Clustering in globally coupled inhibitory neurons. Physica D 72, 259–282 (1994)
Golomb, D., Wang, X.J., Rinzel, J.: Propagation of spindle waves in a thalamic slice model. J. Neurophysiol. 75, 750–769 (1996)
Gomez, T.M., Spitzer, N.C.: Regulation of growth cone behavior by calcium: New dynamics to earlier perspectives. J. Neurobiol. 44, 174–183 (1999)
Gomez, T.M., Zheng, J.Q.: The molecular basis for calcium-dependent axon pathfinding. Nat. Rev. Neurosci. 7, 115–125 (2006)
Gonzalez-Billault, C., Munoz-Llancao, P., Henriquez, D.R., Wojnacki, J., Conde, C., Caceres, A.: The role of small GTPases in neuronal morphogenesis and polarity. Cytoskeleton 69, 464–485 (2012)
Gordon, G.R., Baimoukhametova, D.V., Hewitt, S.A., Kosala, W.R.A., Rajapaksha, J.S., Fisher, T.E., Bains, J.S.: Noropinephrine triggers release of glial ATP to increase postsynaptic efficacy. Nat. Neurosci. 8, 1078–1086 (2005)
Gorji, A.: Spreading depression: A review of the clinical relevance. Brain Res. Revs. 38, 33–60 (2001)
Gourley, S.A.: Travelling front solutions of a nonlocal fisher equation. J. Math Biol. 41, 272–284 (2000)
Grannan, E.R., Kleinfeld, D., Sompolinsky, H.: Stimulus dependent synchronization of neuronal assemblies. Neural Comput. 5, 550–569 (1993)
Gray, W.G., Lee, P.C.Y.: On the theorems for local volume averaging of multiphase systems. Int. J. Multiphase Flow 3, 333–340 (1977)
Greenstein, J.L., Hinch, R., Winslow, R.L.: Mechanisms of excitation-contraction coupling in an integrative model of the cardiac ventricular myocyte. Biophys. J. 90, 77–91 (2006)
Grindrod, P.: Patterns and Waves: The Theory and Application of Reaction-Diffusion Equations. Clarendon Press, Oxford (1991)
Grinvald, A., Lieke, E.E., Frostig, R.D., Hildesheim, R.: Cortical point-spread function and long-range lateral interactions revealed by real-time optical imaging of macaque monkey primary visual cortex. J. Neurosci. 14, 2545–2568 (1994)
Groff, J.R., DeRemigio, H., Smith, G.D.: Stochastic methods in neuroscience, chap. 2. In: Markov Chain Models of Ion Channels and Calcium Release Sites, pp. 29–64. Oxford University Press, Oxford (2009)
Gu, X., Olson, E.C., Spitzer, N.: Spontaneous neuronal calcium spikes and waves during early differentiation.. J. Neurosci. 14, 6325–6335 (1994)
Gu, X., Spitzer, N.C.: Breaking the code: regulation of neuronal differentiation by spontaneous calcium transients. Dev. Neurosci. 19, 33–41 (1997)
Guan, C.B., Xu, H.T., Jin, M., Yuan, X.B., Poo, M.M.: Long-range Ca2+ signaling from growth cone to soma mediates reversal of neuronal migration induced by Slit-2. Cell 129, 385–395 (2007)
Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983)
Guthrie, P.B., Knappenberger, J., Segal, M., Bennett, M.V.,, Charles, A.C., Kater, S.B.: ATP released from astrocytes mediates glial calcium waves. J. Neurosci. 19, 520–528 (1999)
Haas, B., Schipke, C.G., Peters, O., Sohl, G., Willecke, K., Kettenmann, H.: Activity-dependent ATP-waves in the mouse neocortex are independent from astrocytic calcium waves. Cereb. Cortex 16, 237–246 (2006)
Hagberg, A., Meron, E.: Pattern formation in non–gradient reaction–diffusion systems: the effects of front bifurcations. Nonlinearity 7, 805–835 (1994)
Hagberg, A., Meron, E., Rubinstein, I., Zaltzman., B.: Controlling domain patterns far from equilibrium. Phys. Rev. Lett. 76, 427–430 (1996)
Halassa, M.M., Fellin, T., Haydon, P.G.: The tripartite synapse: roles for gliotransmission in health and disease. Trend. Mol. Med. 13, 54–63 (2007)
Hall, A.: Rho GTPases and the actin cytoskeleton. Science 279, 509–514 (1998)
Han, F., Caporale, N., Dan, Y.: Reverberation of recent visual experience in spontaneous cortical waves. Neuron 60, 321–327 (2008)
Hanggi, P., Grabert, H., Talkner, P., Thomas, H.: Bistable systems: master equation versus Fokker–Planck modeling. Z. Physik B 28, 135 (1984)
Hanggi, P., Talkner, P., Borkovec, M.: Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 62, 251–341 (1990)
Hansel, D., Mato, G., Meunier, C.: Synchrony in excitatory neural networks. Neural Comput. 7, 2307–2337 (1995)
Hansen, M., Boltano, S., Dirksen, E.R., Sanderson, M.J.: Intercellular calcium signaling induced by extracellular adenosine 5-triphosphate and mechanical stimulation in airway epithelial cells. J. Cell. Sci. 106, 995–1004 (1993)
Hanson, M.G., Landmesser, L.T.: Chracaterization of the circuits that generate spontaneous episodes of activity in the early embryonic mouse spinal cord. J. Neurosci. 23, 587–600 (2003)
Hanson, P.I., Meyer, T., Stryer, L., Schulman, H.: Dual role of calmodulin in autophosphorylation of multifunctional cam kinase may underlie decoding of calcium signal. Neuron 12, 943–956 (1994)
Hardy, J.: A hundred years of Alzheimer’s disease research. Neuron 52, 3–13 (2006)
Harper, J.D., Lansbury, P.T. Jr.: Models of amyloid seeding in Alzheimer’s disease and scrapie: mechanistic truths and physiological consequences of the time-dependent solubility of amyloid proteins. Annu. Rev. Biochem. 66, 385–407 (1997)
Harris, T.E.: The theory of branching processes. Dover, New York (1989)
Harris-White, M.E., Zanotti, S.A., Frautschy, S.A., Charles, A.C.: Spiral intercellular calcium waves in hippocampal slice cultures. J. Neurophysiol. 79, 1045–1052 (1998)
Hassinger, T.D., Guthrie, P.B., Atkinson, P.B., Bennett, M.V., Kater, S.B.: An extracellular signaling component in propagation of astrocytic calcium waves. Proc. Natl. Acad. Sci. USA 93, 13268–13273 (1996)
Hastings, S.P.: The existence of progressive wave solutions to the Hodgkin-Huxley equations. Arc. Rat. Mech. Anal. 60, 229–257 (1975)
Haydon, P.G.: Glia: listening and talking to the synapse. Nat. Rev. Neurosci. 2, 185–193 (2000)
Heitmann, S., Gong, P., Breakspear, M.: A computational role for bistability and traveling waves in motor cortex. Front. Comp. Neurosci. 6 (67), 1–15 (2012)
von Helmholtz, H.: Treatise on physiological optics. Dover, New York (1866)
Hennig, M.H., Adams, C., Willshaw, D., Sernagor, E.: Early-stage waves in the retinal network emerge close to a critical state transition between local and global functional connectivity. J. Neurosci. 29, 1077–1086 (2009)
Hill, S., Tononoi, G.: Modeling sleep and wakefulness in the thalamocortical system. J. Neurophysiol. 93, 1671–1698 (2005)
Hill, A.A.V., Masino, M.A., Calabrese, R.L.: Intersegmental coordination of rhythmic motor patterns. J. Neurophysiol. 90, 531–538 (2003)
Hillen, T., Othmer, H.: The diffusion limit of transport equations derived from velocity-jump processes. SIAM J. Appl. Math. 61, 751–775 (2000)
Hinch, E.J.: Perturbation Methods. Cambridge University Press, Cambridge (1991)
Hinch, R.: A mathematical analysis of the generation and termination of calcium sparks. Biophys. J. 86, 1293–1307 (2004)
Hinch, R., Chapman, S.J.: Exponentially slow transitions on a Markov chain: the frequency of calcium sparks. Eur. J. Appl. Math. 16, 427–446 (2005)
Hirsch, J.D., Gilbert, C.D.: Synaptic physiology of horizontal connections in the cat’s visual cortex. J. Physiol. Lond. 160, 106–154 (1991)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)
Hofer, T., Politi, A., Heinrich, R.: Intercellular Ca2+ wave propagation through gap-junctional Ca2+ diffusion: a theoretical study. Biophys. J. 80, 75–87 (2001)
Hofer, T., Venance, L., Giaume, C.: Control and plasticity of intercellular calcium waves in astrocytes: a modeling approach. J. Neurosci. 22, 4850–4859 (2002)
Holcman, D., Tsodyks, M.: The emergence of up and down states in cortical networks. PLoS Comp. Biol. 2(3), 0174–0181 (2006)
Holcman, D., Schuss, Z., Korkotian, E.: Calcium dynamics in dendritic spines and spine motility. Biophys. J. 87, 81–91 (2004)
Holmes, B.B., Diamond, M.I.: Cellular mechanisms of protein aggregate propagation. Curr. Opin. Neuro. 25, 721–726 (2012)
Holmes, E.E., Lewis, M.A., Banks, J.E., Veit, R.R.: Partial differential equations in ecology: spatial interactions and population dynamics. Ecology 75, 17–29 (1994)
Holmes, W.R., Carlsson, A.E., Edelstein-Keshet, L.: Regimes of wave type patterning driven by refractory actin feedback: transition from static polarization to dynamic wave behaviour. Phys. Biol. 9, 046005 (2012)
Hoogland, T.M., Kuhn, B., Gobel, W., Huang, W., Nakai, J., Helmchen, F., Flint, J., Wang, S.S.: Radially expanding transglial calcium waves in the intact cerebellum. Proc. Natl. Acad. Sci. USA 106, 3496–3501 (2009)
Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two–state neurons. Proc. Natl. Acad. Sci. USA 81, 3088–3092 (1984)
Horton, A.C., Ehlers, M.D.: Neuronal polarity and trafficking. Neuron 40, 277–295 (2003)
Houweling, A.R., Bazhenov, M., Timofeev, I., Steriade, M., Sejnowski, T.J.: Homeostatic synaptic plasticity can explain post-traumatic epileptogenesis in chronically isolated neocortex. Cereb. Cortex 15, 834–845 (2005)
Hsu, D., Tang, A., Hsu, M., Beggs, J.M.: Simple spontaneously active Hebbian learning model: homeostatsis of activity and connectivity, and consequences for learining and epileptogenesis. Phys. Rev. E 76, 041909 (2007)
Huang, X., Troy, W.C., Yang, Q., Ma, H., Laing, C.R., Schiff, S.J., Wu, J.: Spiral waves in disinhibited mammalian neocortex. J. Neurosci. 24, 9897–9902 (2004)
Hubel, D.H., Wiesel, T.N.: Sequence regularity and geometry of orientation columns in the monkey striate cortex. J. Comp. Neurol. 158, 267–294 (1974)
Hubel, D.H., Wiesel, T.N.: Uniformity of monkey striate cortex: A parallel relationship between field size, scatter, and magnification factor. J. Comp. Neurol. 158, 295–306 (1974)
Hudson, A., Schulman, H.: Neuronal Ca2+/calmodulin-dependent protein kinase II: the role of structure and autoregulation in cellular function. Annu. Rev. Biochem. 71, 473–510 (2002)
Hutt, A., Bestehorn, M., Wennekers, T.: Pattern formation in intracortical neuronal fields. Network 14, 351–368 (2003)
Hutt, A., Longtin, A., Schimansky-Geier, L.: Additive noise-induces Turing transitions in spatial systems with application to neural fields and the Swift-Hohenberg equation. Physica D 237, 755–773 (2008)
Iacobas, D.A., Suadicani, S.O., Spray, D.C., Scemes, E.: A stochastic two-dimensional model of intercellular Ca2+ wave spread in glia. Biophys. J. 90, 24–41 (2006)
Idan, S., Rinzel, J., Gordon, M.S. (eds.): The Theoretical Foundations of Dendritic Function: Selected Papers of Wilfrid Rall with Commentaries. MIT Press, Cambridge (1995)
Invernizzi, G., Papaleo, E., Sabate, R., Ventura, S.: Protein aggregation: Mechanisms and functional consequences. Int. J. Biochem. Cell Biol. 44, 1541–1554 (2012)
Izhikevich, E.: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press, Cambridge (2006)
Jacobs, K.: Stochastic Processes for Physicists. Cambridge University Press, Cambridge (2010)
Jaffe, D.B., Brown, T.H.: Metabotropic glutamate receptor activation induces calcium waves within hippocampal dendrites. J. Neurophysiol. 72, 471–474 (1994)
Jahr, C., Stevens, C.: A quantitative description of NMDA receptor-channel kinetic behavior. J. Neurosci. 10, 1830–1837 (1990)
Jilkine, A., Maree, A.F.M., Edelstein-Keshet, L.: Mathematical model for spatial segregation of the Rho-family GTPases based on inhibitory crosstalk. Bull. Math. Biol. 68, 1169–1211 (2007)
Jin, D.Z., Dragoi, V., Sur, M., Seung, H.S.: Tilt aftereffect and adaptation-induced changes in orientation tuning and visual cortex. J. Neurosci. 94, 4038–4050 (2005)
Jirsa, V.K., Haken, H.: A derivation of a macroscopic field theory of the brain from the quasi-microscopic neural dynamics. Physica D 99, 503–526 (1997)
Jirsa, V.K., Kelso, J.A.S.: Spatiotemporal pattern formation in neural systems with heterogeneous topologies. Phys. Rev. E 62, 8462–8465 (2000)
van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1992)
Kang, M., Othmer, H.G.: Spatiotemporal characteristics of calcium dynamics in astrocytes. Chaos 19, 037116 (2009)
Kang, N., Xu, J., Xu, Q., Nedergaard, M., Kang, J.S.: Astrocytic glutamate release-induced transient depolarization and epileptiform discharges in hippocampal CA1 pyramidal neuron. J. Neurophysiol. 94, 4121–4130 (2005)
Kang, M., Heeger, D.J., Blake, R.: Periodic perturbations producing phase-locked fluctuations in visual perception. J. Vision 9(2):8, 1–12 (2009)
Kang, M., Lee, S.H., Kim, J., Heeger, D.J., Blake, R.: Modulation of spatiotemporal dynamics of binocular rivalry by collinear facilitation and pattern-dependent adaptation. J. Vision 10(11):3, 1–15 (2010)
Kapitula, T., Kutz, N., Sandstede, B.: The Evans function for nonlocal equations. Ind. Univ. Math. J. 53, 1095–1126 (2004)
Katz, L.C., Shatz, C.J.: Synaptic activity and the construction of cortical circuits. Science 274, 1133–1138 (1996)
Keener, J.P.: Waves in excitable media. SIAM J. Appl. Math. 39, 528–548 (1981)
Keener, J.P.: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math 31, 269–276 (1987)
Keener, J.P.: Homogenization and propagation in the bistable equation. Physica D 136, 1–17 (2000)
Keener, J.P.: Propagation of waves in an excitable medium with discrete release sites. SIAM J. Appl. Math. 61, 317–314 (2000)
Keener, J.P.: Stochastic calcium oscillations. Math. Med. Biol. 23, 1–25 (2006)
Keener, J.P., Newby, J.M.: Perturbation analysis of spontaneous action potential initiation by stochastic ion channels. Phy. Rev. E 84, 011918 (2011)
Keener, J., Sneyd, J.: Mathematical Physiology I: Cellular Physiology, 2nd edn. Springer, New York (2009)
Keener, J.P., Hoppensteadt, F.C., Rinzel, J.: Integrate–and–fire models of nerve membrane response to oscillatory input. SIAM J. Appl. Math. 41(3), 503–517 (1981)
Keizer, J., Smith, G.D.: Spark-to-wave transition: saltatory transmission of calcium waves in cardiac myocytes. Biophys. Chem. 72, 87–100 (1998)
Keizer, J., Smith, G.D., Ponce-Dawson, S., Pearson, J.E.: Saltatory propagation of Ca2+ waves by Ca2+ sparks. Biophys. J. 75, 595–600 (1998)
Khazipov, R., Luhmann, H.J.: Early patterns of electrical activity in the developing cerebral cortex of humans and rodents. Trend. Neurosci. 29, 414–419 (2006)
Khovanov, I.A., Polovinkin, A.V., Luchinsky, D.G., McClintock, P.V.E.: Noise-induced escape in an excitable system. Phys. Rev. E 87, 032116 (2013)
Kilpatrick, Z.P., Bressloff, P.C.: Binocular rivalry in a competitive neural network with synaptic depression. SIAM J. Appl. Dyn. Syst. 9, 1303–1347 (2010)
Kilpatrick, Z.P., Bressloff, P.C.: Effects of synaptic depression and adaptation on spatiotemporal dynamics of an excitatory neuronal network. Physica D 239, 547–560 (2010)
Kilpatrick, Z.P., Bressloff, P.C.: Spatially structured oscillations in a two-dimensional neuronal network with synaptic depression. J. Comp. Neurosci. 28, 193–209 (2010)
Kilpatrick, Z.P., Bressloff, P.C.: Stability of bumps in piecewise smooth neural fields with nonlinear adaptation. Physica D 239, 1048–1060 (2010)
Kilpatrick, Z.P., Folias, S.E., Bressloff, P.C.: Traveling pulses and wave propagation failure in inhomogeneous neural media. SIAM J. Appl. Dyn. Syst. 7, 161–185 (2008)
Kim, H.G., Connors, B.W.: Apical dendrites of the neocortex: correlation between sodium and calcium-dependent spiking and pyramidal cell morphology. J. Neurosci. 13, 5301–5311 (1993)
Kim, U., Bal, T., McCormick, D.A.: Spindle waves are propagating synchronized oscillations in the ferret LGN in vitro. J. Neurophysiol. 74, 1301–1323 (1995)
Kinezaki, N., Kawasaki, K., Takasu, F., Shigesada, N.: Modeling biological invasions into periodically fragmented environments. Theor. Popul. Biol. 64, 291–302 (2003)
Kishimoto, K., Amari, S.: Existence and stability of local excitations in homogeneous neural fields. J. Math. Biol. 7, 303–318 (1979)
Kistler, W.M., Seitz, R., van Hemmen, J.L.: Modeling collective excitations in cortical tissue. Physica D 114, 273–295 (1998)
Kleinfeld, D., Delaney, K.R., Fee, M.S., Flores, J.A., Tank, D.W., Gelperin, A.: Dynamics of propagating waves in the olfactory network of a terrestrial mollusc: an electrical and optical study. J. Neurophysiol. 72, 1402–1419 (1994)
Knessl, C., Mangel, M., Matkowsky, B.J., Schuss, Z., Tier, C.: An asymptotic solution of the Kramers-Moyal equation and first passage time problems for Markov jump processes. Phys. Rev. E 29, 3359–3369 (1984)
Knessl, C., Matkowsky, B.J., Schuss, Z., Tier, C.: An asymptotic theory of large deviations for Markov jump processes. SIAM J. Appl. Math. 46, 1006–1028 (1985)
Koch, C.: Cable theory in neurons with active, linearized membranes. Biol. Cybern. 50, 15–33 (1984)
Koch, C.: Biophysics of Computation. Oxford University Press, New York (1999)
Koch, C., Zador, A.M.: The function of dendritic dendritic spines. devices subserving biochemical rather than electrical compartmentalization. J. Neurosci. 13, 413–422 (1993)
Koch, C., Poggio, T., Torre, V.: Nonlinear interactions in a dendritic tree: localization, timing and role in information processing. Proc. Nat. Acad. Sci. (USA) 80, 2799–2802 (1983)
Kolmogorff, A., Petrovsky, I., Piscounoff, N.: Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Moscow Univ. Bull. Math. 1, 1–25 (1937)
Konur, S., Rabinowitz, D., Fenstermaker, V.L., Yuste, R.: Systematic regulation of spine sizes and densities in pyramidal neurons. J. Neurobiol. 56, 95–112 (2003)
Kopell, N., Ermentrout, G.B.: Symmetry and phase-locking in chains of weakly coupled oscillators. Comm. Pure Appl. Math. 39, 623–660 (1986)
Kramer, M.A., Cash, S.S.: Epilepsy as a disorder of cortical network organization. Neuroscientist 18, 360–372 (2012)
von Krosigk, M., Bal, T., McCormick, D.: Cellular mechanisms of a synchronized oscillation in the thalamus. Science 261, 361–364 (1993)
Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Springer, New-York (1984)
Kurtz, T.G.: Limit theorems for a sequence of jump Markov processes approximating ordinary differential equations. J. Appl. Prob. 8, 344–356 (1971)
Kurtz, T.G.: Limit theorems and diffusion approximations for density dependent Markov chains. Math. Prog. Stud. 5, 67 (1976)
Lago-Fernandez, L.F., Huerta, R., Corbacho, F., Siguenza, J.A.: Fast response and temporal coherent oscillations in small-world networks. Phys. Rev. Lett. 84, 2758–2761 (2000)
Laing, C.R.: Spiral waves in nonlocal equations. SIAM J. Appl. Dyn. 4, 588–606 (2005)
Laing, C.R., Chow, C.C.: A spiking neuron model for binocular rivalry. J. Comput. Neurosci. 12, 39–53 (2002)
Laing, C.R., Troy, W.C.: PDE methods for nonlocal models. SIAM J. Appl. Dyn. Syst. 2, 487–516 (2003)
Lam, Y.W., Cohen, L.B., Wachowiak, M., Zochowski, M.R.: Odors elicit three different oscillations in the turtle olfactory bulb. J. Neurosci. 20, 749–762 (2000)
Lance, J.W.: Current concepts of migraine pathogenesis. Neurology 43, 11–15 (1993)
Largo, C., Cuevas, P., Somjen, G., Rio, R.M., Herreras, O.: The effect of depressing glial function in rat brain in situ on ion homeostasis, synaptic transmission, and neuronal survival. J. Neurosci. 16, 1219–1229 (1996)
Largo, C., Ibarz, J.M., Herreras, O.: Effects of the gliotoxin fluorocitrate on spreading depression and glial membrane potential in rat brain in situ. J. Neurophysiol. 78, 295–307 (1997)
Larkum, M.E., Zhu, J.J., Sakmann, B.: A new cellular mechanism for coupling inputs arriving at different cortical layers. Nature 398, 338–341 (1999)
Larkum, M.E., Nevian, T., Sandler, M., Polsky, A., Schiller, J.: Synaptic integration in tuft dendrites of layer 5 pyramidal neurons: a new unifying principle. Science 325, 756–760 (2009)
Lashley, K.: Patterns of cerebral integration indicated by scotomas of migraine. Arch. Neurol. Pyschiatry 46, 331–339 (1941)
Lauritsen, K.B., Zapperi, S., Stanley, H.E.: Self-organized branching processes: avalanche models with dissipation. Phys. Rev. E 54, 2483–2488 (1996)
Leao, A.A.P.: Spreading depression of activity in the cerebral cortex. J. Neurophysiol. 10, 409–414 (1944)
Leao, A.A.P., Morrisson, R.S.: Propagation of spreading cortical depression. J. Neurophysiol. 8, 33–45 (1945)
Lechleiter, J., Girard, S., Peralta, E., Clapham, D.: Spiral calcium wave propagation and annihilation in xenopus laevis oocytes. Science 252, 123–126 (1991)
Leclerc, C., Neant, I., Moreau, M.: The calcium: an early signal that initiates the formation of the nervous system during early embryogenesis. Front. Mol. Neurosci. 5 (64), 1–12 (2012)
Lee, S.H., Blake, R., Heeger, D.J.: Traveling waves of activity in primary visual cortex during binocular rivalry. Nat. Neurosci. 8, 22–23 (2005)
Lee, S.J., Escobedo-Lozoya, Y., Szatmari, E.M., Yasuda, R.: Activation of CaMKII in single dendritic spines during long-term potentiation. Nature 458, 299–304 (2009)
Lee, S.J., Lim, H.S., Masliah, E., Lee, H.J.: Protein aggregate spreading in neurodegerative diseases: problems and perspectives. Neuro. Res. 70, 339–348 (2011)
Lehky, S.R.: Binocular rivalry is not chaotic. Proc. Roy. Soc. Lond. Biol Sci. 259, 71–76 (1995)
Lehky, S.R.: No binocular rivalry in the LGN of alert macaque monkeys. Vis. Res. 36, 1225–1234 (1996)
Lehner, F.K.: On the validity of Fick’s law for transient diffusion through a porous medium. Chem. Eng. Sci. 34, 821–825 (1977)
Lemieux, L., Daunizeau, J., Walker, M.C.: Concepts of connectivity and human epileptic activity. Front. Syst. Neurosci. 5 (12), 1–13 (2011)
Leopold, D.A., Logothetis, N.K.: Activity changes in early visual cortex reflect monkeys’ percepts during binocular rivalry. Nature 379, 549–553 (1996)
LeVay, S., Nelson, S.B.: Columnar organization of the visual cortex. In: Leventhal, A.G. (ed.) The Neural Basis of Visual Function, pp. 266–315. CRC Press, Boca Raton (1991)
Levelt, W.J.M.: On Binocular Rivalry. Institute for Perception RVO–TNO, Soesterberg, The Netherlands (1965)
Leybaert, L., Paemeleire, K., Strahonja, A., Sanderson, M.J.: Inositol-trisphosphate-dependent intercellular calcium signaling in and between astrocytes and endothelial cells. Glia 24, 398–407 (1998)
Leybaert, L., Sanderson, M.J.: Intercellular Ca2+ waves: mechanisms and function. Physiol. Rev. 92, 1359–1392 (2012)
Li, Y., Rinzel, J.: Equations for InsP3 receptor-mediated calcium oscillations derived from a detailed kinetic model: a Hodgkin-Huxley like formalism. J. Theor. Biol. 166, 461–473 (1994)
Liley, D.J.T., Cadusch, P.J., Dafilis, M.P.: A spatially continuous mean field theory of electrocortical activity. Network 13, 67–113 (2002)
Linder, B., Garcia-Ojalvo, J., Neiman, A., Schimansky-Geier, L.: Effects of noise in excitable systems. Phys. Rep. 392, 321–424 (2004)
Lisman, J.E.: Relating hippocampal circuitry to function: recall of memory sequences by reciprocal dentate-CA3 interactions. Neuron 22, 233–242 (1999)
Lisman, J.: Long-term potentiation: outstanding questions and attempted synthesis. Phil. Trans. R. Soc. Lond. B 358, 829–842 (2003)
Lisman, J.E., Schulman, H., Cline, H.: The molecular basis of CaMKII function in synaptic and behavioral memory. Nat. Rev. Neurosci. 3, 175–190 (2002)
Litwin-Kumar, A., Doiron, B.: Slow dynamics and high variability in balanced cortical networks with clustered connections. Nat. Neurosci. 15, 1498–1505 (2012)
Liu, Q.S., Xu, Q., Kang, G.A.J., Nedergaard, M.: Astrocyte-mediated activation of neuronal kainate receptors. Proc. Natl. Acad Sci. USA 101, 3172–3177 (2004)
Llano, I., González, J., Caputo, C., Lai, F.A., Blayney, L.M., Tan, Y.P., Marty, A.: Presynaptic calcium stores underlie large-amplitude miniature IPSCs and spontaneous calcium transients. Nat. Neurosci. 3, 1256–1265 (2000)
Logothetis, N.K., Leopold, D.A., Sheinberg, D.L.: What is rivalling during binocular rivalry? Nature 380, 621–624 (1996)
London, M., Roth, A., Beeren, L., Hausser, M., Latham, P.E.: Sensitivity to perturbations in vivo implies high noise and suggests rate coding in cortex. Nature 466, 123–127 (2010)
Loxley, P., Robinson, P.: Soliton model of competitive neural dynamics during binocular rivalry. Phys. Rev. Lett. 102(25), 258701 (2009)
Lubenov, E.V., Siapas, A.G.: Hippocampal theta oscillations are travelling waves. Nature 459, 534–539 (2009)
Luo, L.: Actin cytoskeleton regulation in neuronalmorphogenesis and structural plasticity. Ann. Rev. Cell Dev. Biol. 18, 601–635 (2002)
Ly, C., Tranchina, D.: Critical analysis of a dimension reduction by a moment closure method in a population density approach to neural network modeling. Neural Comput. 19, 2032–2092 (2007)
Madison, D.V., Nicoll, R.A.: Control of the repetitive discharge of rat CA1 pyramidal neurones in vitro. J. Physiol. 354, 319–331 (1984)
Magee, J.C., Johnston, D.: Characterization of single voltage-gated Na+ and Ca2+ channels in apical dendrites of rat CA1 pyramidal neurons. J. Physiol. 487, 67–90 (1995)
Maier, R.S., Stein, D.L.: Limiting exit location distribution in the stochastic exit problem. SIAM J. Appl. Math 57, 752–790 (1997)
Maier, N., Nimmrich, V., Draguhn, A.: Cellular and network mechanisms underlying spontaneous sharp wave–ripple complexes in mouse hippocampal slices. J. Physiol. 550, 873–887 (2003)
Major, G., Polsky, A., Denk, W., Schiller, J., Tank, D.W.: Spatiotemporally graded NMDA spike/plateau potentials in basal dendrites of neocortical pyramidal neurons. J. Neurophysiol. 99, 2584–2601 (2008)
Malach, R., Harel, Y.A.M., Grinvald, A.: Relationship between intrinsic connections and functional architecture revealed by optical imaging and in vivo targeted biocytin injections in primate striate cortex. Proc. Natl. Acad. Sci. USA 90, 0469–10473 (1993)
Malenka, R.C., Bear, M.F.: LTP and LTD: an embarrassment of riches. Neuron 44, 5–21 (2004)
Manita, S., Ross, W.N.: Synaptic activation and membrane potential changes modulate the frequency of spontaneous elementary Ca2+ releases events in the dendrites of pyramidal neurons. J. Neurosci. 29, 7833–7845 (2009)
Marder, E., Calabrese, R.L.: Principles of rhythmic motor pattern generation. Physiol. Rev. 76, 687–717 (1996)
Markram, H., Tsodyks, M.: Redistribution of synaptic efficacy between neocortical pyramidal neurons. Nature 382, 807–810 (1996)
Martins-Ferreira, H., Ribeiro, I.J.: Biphasic effects of gap junctional uncoupling agents on the propagation of retinal spreading depression. Braz. J. Med. Biol. Res. 28, 991–994 (1995)
Martins-Ferreira, H., Nedergaard, M., Nicholson, C.: Perspectives on spreading depression. Brain Res. Rev. 32, 215–234 (2000)
Masel, J., Jansen, V.A.A., Nowak, M.A.: Quantifying the kinetic parameters of prion replication. Biophys. Chem. 77, 139–15 (1999)
Masuda, N., Aihara, K.: Global and local synchrony of coupled neurons in small-world networks. Biol. Cybern. 90, 302–309 (2004)
Mataga, N., Mizuguchi, Y., Hensch, T.K.: Experience-dependent pruning of dendritic spines in visual cortex by tissue plasminogen activator. Neuron 44, 1031–1041 (2004)
Matthaus, F.: Diffusion versus network models as descriptions for the spread of prion diseasess in the brain. J. Theor. Biol. 240, 104–113 (2006)
Mattia, M., Guidice, P.D.: Population dynamics of interacting spiking neurons. Phys. Rev. E 66, 051917 (2002)
Matus, A.: Actin plasticity in dendritic spines. Science 290, 754–758 (2000)
Mayer, M.L., Westbrook, G.L., Guthrie, P.B.: Voltage-dependent block by Mg2+ of NMDA responses in spinal cord neurone. Nature 309, 261–263 (1984)
McCormick, D.A., Contreras, D.: On the cellular and network bases of epileptic seizures. Annu. Rev. Physiol. 63, 815–846 (2001)
McLaughlin, T., Torborg, C.L., Feller, M.B., O’Leary, D.D.M.: Retinotopic map refinement requires spontaneous retinal waves during a brief critical period of development. Neuron 40, 1147–1160 (2003)
Meinhardt, H.: Orientation of chemotactic cells and growth cones: models and mechanisms. J. Cell. Sci. 112, 2867–2874 (1999)
Meisel, C., Storch, A., Hallmeyer-Elgner, S., Bullmore, E., Gross, T.: Failure of adaptive self-organized criticality during epileptic seizure attacks. PLoS Comp. Biol. 8, e1002312 (2012)
Melamed, O., Barak, O., Silberberg, G., Markram, H., Tsodyks, M.: Slow oscillations in neural networks with facilitating synapses. J. Comp. Neurosci. 25, 308–316 (2008)
Menchon, S.A., Gartner, A., Roman, P., Dotti, C.G.: Neuronal (bi)polarity as a self-organized process enhanced by growing membrane. PLoS One 6(9), e24190 (2011)
Mendez, V., Fort, J., Rotstein, H.G., Fedotov, S.: Speed of reaction-diffusion fronts in spatially heterogeneous media. Phys. Rev. E 68, 041105 (2003)
Mendez, V., Fedotov, S., Horsthemke, W.: Reaction-Transport Systems. Springer, Berlin (2010)
Meng, X., Huguet, G., Rinzel, J.: Type III excitability, slope sensitivity and coincidence detection. Disc. Cont. Dyn. Systs, 32, 2729–2757 (2012)
Meunier, C., d’Incamps, B.L.: Extending cable theory to heterogeneous dendrites. Neural Comput. 20, 1732–1775 (2008)
Meyer, C., van Vreeswijk, C.: Temporal correlations in stochastic networks of spiking neurons. Neural Comput. 14, 369–404 (2002)
Mikhailov, A.S., Davydov, V.A., Zykov, V.S.: Complex dynamics of spiral waves and motion of curves. Physica D 70, 1–39 (1994)
Miller, J.P., Rall, W., Rinzel, J.: Synaptic amplification by active membrane in dendritic spines. Brain Res. 325, 325–330 (1985)
Miller, L.D., Golarai, J.J.P., Connor, J.: Ca2+ release from intracellular stores induced by afferent stimulation of CA3 pyramidal neurons in hippocampal slices. J. Neurophysiol. 76, 554–562 (1996)
Millhauser, G.L., Salpeter, E.E., Oswald, R.E.: Diffusion model of ion–channel gating and the origin of power–law distributions from single–channel recordings. Proc. Natl. Acad. Sci. USA 85, 1503–1507 (1988)
Milton, J., Jung, P.: Epilepsy as a Dynamic Disease. Springer, Berlin (2003)
Mirollo, R.E., Strogatz, S.H.: Synchronisation of pulse–coupled biological oscillators. SIAM J. Appl. Math. 50(6), 1645–1662 (1990)
Mitchison, T., Kirschner., M.: Cytoskeletal dynamics and nerve growth. Neuron 1, 761–772 (1988)
Miura, R., Huang, H., Wylie, J.: Cortical spreading depression: An enigma. Eur. Phys. J. Special Topics 147, 287–302 (2007)
Mogilner, A., Oster, G.: Cell motility driven by actin polymerization. Biophys. J. 71, 3030–3045 (1996)
Moldakarimov, S., Rollenhagen, J.E., Olson, C.R., Chow, C.C.: Competitive dynamics in cortical responses to visual stimuli. J. Neurophysiol. 94, 3388–3396 (2005)
Moreno-Bote, R., Rinzel, J., Rubin, N.: Noise-induced alternations in an attractor network model of perceptual bistability. J. Neurophysiol. 98, 1125–1139 (2007)
Morgan, R.J., Soltesz, I.: Nonrandom connectivity of the epileptic dentate gyrus predicts a major role for neuronal hubs in seizures. Proc. Natl. Acad. Sci. USA 105, 6179–6184 (2008)
Mori, Y., Jilkine, A., Edelstein-Keshet, L.: Wave-pinning and cell polarity from a bistable reaction-diffusion system. Biophys. J. 94, 3684–3697 (2008)
Mori, Y., Jilkine, A., Edelstein-Keshet, L.: Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization. SIAM J. Appl. Math. 71, 1401–1427 (2011)
Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. J. Biophys. 35, 193–213 (1981)
Mulleners, W.M., Chronicle, E.P., Palmer, J.E., Koehler, P.J., Vredeveld, J.W.: Visual cortex excitability in migraine with and without aura. Headache 41, 565–572 (2001)
Muller, L., Destexhe, A.: Propagating waves in thalamus, cortex and the thalamocortical system: experiments and models. J. Physiol. (Paris) 106, 222–238 (2012)
Mullins, O.J., Hackett, J.T., Buchanan, J.T., Friesen, W.O.: Neuronal control of swimming behavior: Comparison of vertebrate and invertebrate model systems. Prog. Neurobiol. 93, 244–269 (2011)
Murray, J.D.: Mathematical Biology, vol. I, II. Springer, Berlin (2002)
Naeh, T., Klosek, M.M., Matkowsky, B.J., Schuss, Z.: A direct approach to the exit problem. SIAM J. Appl. Math. 50, 595–627 (1990)
Nagumo, J., Armoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. Inst. Radio Eng. 50, 2061–2070 (1964)
Nakamura, T., Barbara, J.G., Nakamura, K., Ross, W.N.: Synergistic release of Ca2+ from IP3 sensitive stores evoked by synaptic activation of mGluRs paired with backpropagating action potentials. Neuron 24, 727–737 (1999)
Nedergaard, M., Cooper, A.J., Goldman, S.A.: Gap junctions are required for the propagation of spreading depression. J. Neurobiol. 28, 433–444 (1995)
Nedergaard, M., Ransom, B., Goldman, S.A.: New roles for astrocytes: redefining the functional architecture of the brain. Trend. Neurosci. 26, 523–530 (2003)
Netoff, T.I., Schiff, S.J.: Decreased neuronal synchronization during experimental seizures. J. Neurosci. 22, 7297–7307 (2002)
Netoff, T.I., Clewely, R., Arno, S., Keck, T., White, J.A.: Epilepsy in small-world networks. J. Neurosci. 24, 8075–8083 (2004)
Neukirchen, D., Bradke, F.: Neuronal polarization and the cytoskeleton. Sem. Cell. Dev. Biol. 22, 825–833 (2011)
Newby, J.M., Bressloff, P.C.: Local synaptic signalling enhances the stochastic transport of motor-driven cargo in neurons. Phys. Biol. 7, 036004 (2010)
Newby, J.M., Bressloff, P.C.: Quasi-steady state reduction of molecular-based models of directed intermittent search. Bull. Math. Biol. 72, 1840–1866 (2010)
Newby, J., Chapman, J.: Metastable behavior in Markov processes with internal states: breakdown of model reduction techniques. J. Math. Biol. published online, DOI: 10.1007/s00285-013-0723-1 (2013)
Newby, J.M., Bressloff, P.C., Keeener, J.P.: The effect of potassium channels on spontaneous action potential initiation by stochastic ion channels. Phys. Rev. Lett. 111, 128–101 (2013)
Newman, E.A.: Propagation of intercellular calcium waves in retinal astrocytes and Muller cells. J. Neurosci. 21, 2215–2223 (2001)
Newman, M.E.J.: Spread of epidemic disease on networks. Phys. Rev. E 66, 016128 (2002)
Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003)
Newman, E.A., Zahs, K.R.: Calcium waves in retinal glial cells. Science 275, 844–847 (1997)
Nicholson, C.: Diffusion and related transport mechanisms in brain tissue. Rep. Prog. Phys. 64, 815–884 (2001)
Nimchinsky, E.A., Sabatini, B.L., Svoboda, K.: Structure and function of dendritic spines. Ann. Rev. Physiol. 64, 313–353 (2002)
Nita, D.A., Cisse’, Y., Timofeev, I., Steriade, M.: Increased propensity to seizures after chronic cortical deafferentation in vivo. J. Neurophysiol. 95, 902–913 (2006)
Noble, J.V.: Geographic and temporal development of plagues. Nature 250, 726–729 (1974)
Novikov, E.A.: Functionals and the random-force method in turbulence theory. Sov. Phys. JETP 20, 1290 (1965)
Nowak, L., Bregestovski, P., Ascher, P., Herbet, A., Prochiantz, A.: Magnesium gates glutamate-activated channels in mouse central neurones. Nature 307, 462–465 (1984)
Nunez, P.I.: Neocortical dynamics and human EEG rhythms. Oxford University Press, New York (1995)
Nykamp, D., Tranchina, D.: A population density method that facilitates large–scale modeling of neural networks: analysis and application to orientation tuning. J. Comp. Neurosci. 8, 19–50 (2000)
Obermayer, K., Blasdel, G.G.: Geometry of orientation and ocular dominance columns in monkey striate cortex. J. Neurosci. 13, 4114–4129 (1993)
O’Donnell, M., Chance, R.K., Bashaw, G.J.: Axon growth and guidance: Receptor regulation and signal transduction. Ann. Rev. Neurosci. 32, 383–412 (2009)
O’Donovan, M.J.: The origin of spontaneous activity in developing networks of the vertebrate nervous system. Curr. Opin. Neuro. 9, 94–104 (1999)
O’Donovan, M.J., Chub, N., Wenner, P.: Mechanisms of spontaneous activity in the developing retina. J. Neurobiol. 37, 131–145 (1998)
Ohira, T., Cowan, J.D.: Stochastic neurodynamics and the system size expansion. In: Ellacott, S., Anderson, I.J. (eds.) Proceedings of the First International Conference on Mathematics of Neural Networks, pp. 290–294. Academic, New York (1997)
O’Keefe, J., Dostrovsky, J.: The hippocampus as a spatial map. preliminary evidence from unit activity in the freely moving rat. Brain Res. 34, 171–174 (1971)
O’Keefe, J., Recce, M.L.: Phase relationship between hippocampal place units and the EEG theta rhythm. Hippocampus 3, 317–330 (1993)
Oksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Springer, New York (2007)
Omurtag, A., Knight, B.W., Sirovich, L.: On the simulation of large populations of neurons. J. Comput. Neurosci. 8, 51–63 (2000)
Onsum, M., Rao, C.V.: A mathematical model for neutrophil gradient sensing and polarization. PLoS Comput. Biol. 3, e36 (2007)
Ooyen, A.V., Graham, B.P., Ramakers, G.J.A.: Competition for tubulin between growing neurites during development. Neurocomputing 38, 73–78 (2001)
Oray, S., Majewska, A., Sur, M.: Dendritic spine dynamics are regulated by monocular deprivation and extracellular matrix degredation. Neuron 44, 1021–1030 (2004)
Othmer, H., Dunbar, S., Alt, W.: Models of dispersal in biological systems. J. Math. Biol. 26, 263–298 (1988)
Otsuji, M., Ishihara, S., Co, C., Kaibuchi, K., Mochizuki, A., Kuroda, S.: A mass conserved reaction-diffusion system captures properties of cell polarity. PLoS Comput. Biol. 3, e108 (2007)
Pak, C.W., Flynn, K.C., Bamburg, J.R.: Actin-binding proteins take the reins in growth cones. Nat. Rev. Neurosci. 9, 136–147 (2008)
Pakdaman, K., Thieullen, M., Wainrib, G.: Fluid limit theorems for stochastic hybrid systems with application to neuron models. J. Appl. Prob. 24, 1 (2010)
Panatier, A., Theodosis, D.T., Mothet, J.P., Touquet, B., Pollegioni, L., Poulain, D.A., Oliet, S.H.: Glia-derived D-serine controls NMDA receptor activity and synaptic memory. Cell 125, 775–784 (2006)
Panja, D.: Effects of fluctuations on propagating fronts. Phys. Rep. 393, 87–174 (2004)
Papanicolaou, G.C.: Asymptotic analysis of transport processes. Bull. Amer. Math. Soc. 81, 330–392 (1975)
Parga, N., Abbott, L.F.: Network model of spontaneous activity exhibiting synchronous transitions between up and down states. Front. Neurosci. 1, 57–66 (2007)
Parker, I., Ivorra, I.: Localized all-or-none calcium liberation by inositol triphosphate. Science 250, 977–979 (1990)
Parker, I., Yao, Y.: Regenerative release of calcium from functionally discrete subcellular stores by inositol triphosphate. Proc. Roy. Soc. Lond. B 246, 269–274 (1991)
Parpura, V., Basarsky, T.A., Liu, F., Jeftinija, K., Jeftinija, S., Haydon, P.G.: Glutamate-mediated astrocyte-neuron signalling. Nature 369, 744–747 (1994)
Parri, H.R., Gould, T.M., Crunelli, V.: Spontaneous astrocytic Ca2+ oscillations in situ drive NMDAR-mediated neuronal excitation. Nat. Neurosci. 4, 803–81 (2001)
Pascual, O., Casper, K.B., Kubera, C., Zhang, J., Revilla-Sanchez, R., Sul, J.Y., Takano, H., Moss, S.J., McCarthy, K., Haydon, P.G.: Astrocytic purinergic signaling coordinates synaptic networks. Science 310, 113–116 (2005)
de Pasquale, F., Gorecki, J., Poielawski., J.: On the stochastic correlations in a randomly perturbed chemical front. J. Phys. A 25, 433 (1992)
Paullet, J.E., Ermentrout, G.B.: Stable rotating waves in two-dimensional discrete active media. SIAM J. Appl. Math. 54, 1720–1744 (1994)
Pavliotis, G.A., Stuart, A.M.: Multiscale Methods: Averaging and Homogenization. Springer, New York (2008)
Pearson, J.E., Ponce-Dawson, S.: Crisis on skid row. Physica A 257, 141–148 (1998)
Pearson, R., Catsicas, M., Becker, D., Mobbs, P.: Purinergic and muscarinic modulation of the cell cycle and calcium signaling in the chick retinal ventricular zone. J. Neurosci. 22, 7569–7579 (2002)
Pearson, R.A., Dale, N., Llaudet, E., Mobbs, P.: ATP released via gap junction hemichannels from the pigment epithelium regulates neural retinal progenitor proliferation. Neuron 46, 731–744 (2005)
Pereda, E., Quiroga, R., Bhattacharya, J.: Nonlinear multivariate analysis of neurophysiological signals. Prog. Neurobiol. 77, 1–37 (2005)
Peters, O., Schipke, C.G., Hashimoto, Y., Kettenmann, H.: Different mechanisms promote astrocyte Ca2+ waves and spreading depression in the mouse neocortex. J. Neurosci. 23, 9888–9896 (2003)
Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2003)
Pinder, G.F., Gray, W.G.: Essentials of Multiphase Flow and Transport in Porous Media. Wiley, New York (2008)
Pinto, D., Ermentrout, G.B.: Spatially structured activity in synaptically coupled neuronal networks I: traveling fronts and pulses. SIAM J. Appl. Math 62, 206–225 (2001)
Pinto, D., Ermentrout, G.B.: Spatially structured activity in synaptically coupled neuronal networks II: lateral inhibition and standing pulses. SIAM J. Appl. Math. 62, 226–243 (2001)
Pinto, D., Jackson, R.K., Wayne, C.E.: Existence and stability of traveling pulses in a continuous neuronal network. SIAM J. Appl. Dyn. Syst. 4, 954–984 (2005)
Pinto, D., Patrick, S.L., Huang, W.C., Connors, B.W.: Initiation, propagation, and termination of epileptiform activity in rodent neocortex in vitro involve distinct mechanisms. J. Neurosci. 25, 8131–8140 (2005)
Plenz, D., Thiagarajan, T.C.: The organizing principles of neuronal avalanches: cell assemblies in the cortex? Trend. Neurosci. 30, 101–110 (2007)
Polleux, F., Snider, W.: Initiating and growing an axon. Cold Spring Harb Perspect Biol 2, a001925 (2010)
Polonsky, A., Blake, R., Braun, J., Heeger, D.J.: Neuronal activity in human primary visual cortex correlates with perception during binocular rivalry. Nat. Neurosci. 3, 1153–1159 (2000)
Ponce, S.J., Keizer, J., Pearson, J.E.: Fire-diffuse-fire model of dynamics of intracellular calcium waves. Proc. Nat. Acad. Sci. USA 96, 6060–6063 (1999)
Prechtl, J.C., Cohen, L.B., Pesaran, B., Mitra, P.P., Kleinfeld, D.: Visual stimuli induce waves of electrical activity in turtle cortex. Proc. Natl. Acad. Sci. USA 94, 7621–7626 (1997)
Prusiner, S.B.: Prions. Proc. Nat. Acad. Sci. (USA) 95, 13363–13383 (1998)
Rakic, P.: Elusive radial glial cells: historical and evolutionary perspective. Glia 43, 19–32 (2003)
Rall, W., Snir, H.A.: Cable theory for dendritic neurons. In: Koch, C., Segev, I. (eds.) Methods in Neuronal Modelling, 2nd edn. MIT Press, Cambridge (1998)
Ramirez, O.A., Couve, A.: The endoplasmic reticulum and protein trafficking in dendrites and axons. Trend. Cell. Biol. 21, 219–227 (2011)
Rangan, A.V., Kovacic, G., Cai, D.: Kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train. Phys. Rev. E 77, 041915 (2008)
Reed, M.C., Venakides, S., Blum, J.J.: Approximate traveling waves in linear reaction-hyperbolic equations. SIAM J. Appl. Math. 50(1), 167–180 (1990)
Renart, A., de la Rocha, J., Bartho, P., Hollender, L., Parga, N., Reyes, A., Harris, K.D.: The asynchronous state in cortical circuits. Science 327, 587–590 (2010)
Rhodes, P.: The properties and implications of NMDA spikes in neocortical pyramidal cells. J. Neurosci. 26, 6704–6715 (2006)
Richardson, K.A., Schiff, S.J., Gluckman, B.J.: Control of traveling waves in the mammalian cortex. Phys. Rev. Lett. 94, 028103 (2005)
Richardson, M.P.: Large scale brain models of epilepsy: dynamics meets connectomics. Epilepsy (2012)
Rinzel, J., Maginu, K.: Kinematic analysis of wave pattern formation in excitable media. In: Pacault, A., Vidal, C. (eds.) Non-Equilibrium Dynamics in Chemical Systems. Springer, Berlin (1984)
Rinzel, J., Terman, D.: Propagation phenomena in a bistable reaction–diffusion system. SIAM J. Appl. Math. 42, 1111–1137 (1982)
Rinzel, J., Terman, D., Wang, X., Ermentrout, B.: Propagating activity patterns in large-scale inhibitory neuronal networks. Science 279, 1351–1355 (1998)
Robinson, P.A.: Patchy propagator, brain dynamics, and the generation of spatially structured gamma oscillations. Phys. Rev. E 73, 041904 (2006)
Robinson, P.A., Rennie, C.J., Wright, J.J.: Propagation and stability of waves of electrical activity in the cerebral cortex. Phys. Rev. E 56, 826–840 (1997)
Robinson, P.A., Rennie, C.J., Wright, J.J., Bahramali, H., Gordon, E., Rowe, D.I.: Prediction of electroencephalographic spectra from neurophysiology. Phys. Rev. E 63, 021903 (2001)
Robinson, P.A., Rennie, C.J., Rowe, D.L.: Dynamics of large-scale brain activity in normal arousal states and epileptic seizures. Phys. Rev. E 65, 041924 (2002)
Rocco, A., Ebert, U., van Saarloos, W.: Subdiffusive fluctuations of “pulled” fronts with multiplicative noise. Phys. Rev. E. 65, R13–R16 (2000)
Rockland, K.S., Lund, J.: Intrinsic laminar lattice connections in primate visual cortex. J. Comp. Neurol. 216, 303–318 (1983)
Rose, J., Jin, S.X., Craig, A.M.: Heterosynaptic molecular dynamics: locally induced propagating synaptic accumulation of CaM Kinase II. Neuron 61, 351–358 (2009)
Ross, C.A., Poirier, M.A.: Protein aggregation and neurodegenerative disease. Nat. Med. 10(Suppl.), S10–S17 (2004)
Ross, W.N.: Understanding calcium waves and sparks in central neurons. Nat. Rev. Neurosci. 13, 157–168 (2012)
Roxin, A., Riecke, H., Solla, S.A.: Self-sustained activity in a small-world network of excitable neurons. Phys. Rev. Lett. 92, 198101 (2004)
Rubin, J.E.: A nonlocal eigenvalue problem for the stability of a traveling wave in a neuronal medium. Discret. Contin. Dyn. Syst. 10, 925–940 (2004)
Rubin, J., Terman, D.: Geometric singular perturbation analysis of neuronal dynamics. In: Fiedler, B. (ed.) Handbook of Dynamical Systems: Towards Applications, vol. 2. Elsevier, Amsterdam (2002)
Rubin, J.E., Terman, D.: Explicit maps to predict activation order in multiphase rhythms of a coupled cell network. J. Math. Neurosci. 2, 4 (2012)
Rubin, J., Wechselberger, M.: Giant squid-hidden canard: the 3d geometry of the Hodgkin–Huxley model. Biol. Cybern. 97, 5–32 (2007)
Rubin, J., Wechselberger, W.: The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales. Chaos 18, 015105 (2008)
Rubinov, M., Sporns, O.: Complex network measures of brain connectivity: Uses and interpretations. Neuroimage 52, 1059–1069 (2010)
Ruthel, G., Banker, G.: Actin-dependent anterograde movement of growth-cone-like structures along growing hippocampal axons: A novel form of axonal transport? Cell Motil. Cytoskel. 40, 160–173 (1998)
Ruthel, G., Banker, G.: Role of moving growth cone-like wave structures in the outgrowth of cultured hippocampal axons and dendrites. J. Neurobiol. 39, 97–106 (1999)
van Saarloos, W.: Front propagation into unstable states. Phys. Rep. 386, 29–222 (2003)
Saez, J.C., Berthoud, V.M., Branes, M.C., Martinez, A.D., Beyer., E.C.: Plasma membrane channels formed by connexins: their regulation and functions. Physiol. Rev. 83, 1359–1400 (2003)
Sagues, F., Sancho, J.M., Garcia-Ojalvo, J.: Spatiotemporal order out of noise. Rev. Mod. Phys. 79, 829–882 (2007)
Saitoh, T., Schwartz, J.H.: Phosphorylation-dependent subcellular translocation of a Ca2+/calmodulin-dependent protein kinase produces an autonomous enzyme in aplysia neurons. J. Cell. Biol. 100, 835–842 (1985)
Samuels, D.C., Hentschel, H.G.E., Fine, A.: The origin of neuronal polarization: a model of axon formation. Phil. Trans. R. Soc. Lond. B 351, 1147–1156 (1996)
Sanchez-Vives, M.V., McCormick, D.A.: Cellular and network mechanisms of rhythmic recurrent activity in neocortex. Nat. Neurosci. 3, 1027–1034 (2000)
Sanderson, M.J., Charles, A.C., Dirksen, E.R.: Mechanical stimulation and intercellular communication increases intracellular Ca2+ in epithelial cells. Cell. Regul. 1, 585–596 (1990)
Sandstede, B.: Stability of travelling waves. In: Handbook of Dynamical Systems, vol. 2, pp. 983–1055. North-Holland, Amsterdam (2002)
Sandstede, B.: Evans functions and nonlinear stability of traveling waves in neuronal network models. Int. J. Bifur. Chaos Appl. Sci. Eng. 17, 2693–2704 (2007)
Sato, T.K., Nauhaus, I., Carandini, M.: Traveling waves in visual cortex. Neuron 75, 218–229 (2012)
Scemes, E., Giaume, C.: Astrocyte calcium waves: what they are and what they do. Glia 54, 716–725 (2006)
Schiller, J., Schiller, Y., Stuart, G., Sakmann, B.: Calcium action potentials restricted to distal apical dendrites of rat neocortical pyramidal neurons. J. Physiol. 505, 605–616 (1997)
Schiller, J., Major, G., Koester, H.J., Schiller, Y.: NMDA spikes in basal dendrites of cortical pyramidal neurons. Nature 404, 285–289 (2000)
Schimansky-Geier, L., Mikhailov, A.S., Ebeling., W.: Effects of fluctuations on plane front propagation in bistable nonequilibrium systems. Ann. Phys. 40, 277 (1983)
Schock, S.C., Munyao, N., Yakubchyk, Y., Sabourin, L.A., Hakim, A.M., Ventureyra, E.C.G., Thompson, C.S.: Cortical spreading depression releases ATP into the extracellular space and purinergic receptor activation contributes to the induction of ischemic tolerance. Brain Res. 1168, 129–138 (2007)
Schuss, Z.: Theory and applications of stochastic processes: an analytical approach. In: Applied Mathematical Sciences, vol. 170. Springer, New York (2010)
Schuster, H.G., Wagner, P.: A model for neuronal oscillations in visual cortex. Biol. Cybern. 64, 77–82 (1990)
Schutz, P., Bode, M., Purwins, H.G.: Bifurcations of front dynamics in a reaction-diffusion system with spatial inhomogeneities. Physica D 82, 382–397 (1995)
Schwartz, E.: Spatial mapping in the primate sensory projection: analytic structure and relevance to projection. Biol. Cybern. 25, 181–194 (1977)
Seely, J., Chow, C.C.: Role of mutual inhibition in binocular rivalry. J. Neurophysiol. 106, 2136–2150 (2011)
Seifert, G.: Astrocyte dysfunction in neurological disorders; a molecular perspective. Nat. Rev. Neurosci. 7, 194–206 (2006)
Selkoe, D.J.: Folding proteins in fatal way. Nature 426, 900–904 (2003)
Selkoe, D.J.: Cell biology of protein misfolding: the examples of Alzheimer’s and Parkinson’s diseases. Nat. Cell Biol. 6, 1054–1061 (2004)
Shadlen, M.N., Newsome, W.T.: Noise, neural codes and cortical organization. Curr. Opin. Neurobiol. 4, 569–579 (1994)
Shapiro, B.E.: Osmotic forces and gap junctions in spreading depression: A computational model. J. Comp. Neurosci. 10, 99–120 (2001)
Shen, K., Meyer, T.: Dynamic control of CaMKII translocation and localization in hippocampal neurons by NMDA receptor simulation. Science 284, 162–166 (1999)
Shen, K., Tereul, M.N., Subramanian, K., Meyer, T.: CaMKIIβ functions as an F-actin targeting module that localizes CaMKIIα/β heterooligomers to dendritic spines. Neuron 21, 593–606 (1998)
Shepherd, G.M.: The dendritic spine: a multifunctional unit. J. Neurophysiol. 75, 2197–2210 (1996)
Shepherd, G.M., Brayton, R.K., Miller, J.P., Segev, I., Rinzel, J., Rall, W.: Signal enhancement in distal cortical dendrites by means of interactions between active dendritic spines. Proc. Natl. Acad. Sci. USA 82, 2192–2195 (1985)
Shew, W.L., Plenz, D.: The functional benefits of criticality in the cortex. Neuroscientist 19, 88–100 (2012)
Shigesada, N., Kawasaki, K., Teramoto, E.: Traveling periodic waves in heterogeneous environments. Theor. Popul. Biol. 30, 143–160 (1986)
Shigesada, N., Kawasaki, K.: Biological invasions: theory and practice. Oxford University Press, Oxford (1997)
Shimada, T., Toriyama, M., Uemura, K., Kamiguchi, H., Sugiura, T., Watanabe, N., Inagaki, N.: Shootin1 interacts with actine retrograde flow and L1-CAM to promote axon outgrowth. J. Cell. Biol. 181, 817–829 (2008)
Shpiro, A., Moreno-Bote, R., Rubin, N., Rinzel, J.: Dynamical characteristics common to neuronal competition models. J. Neurophysiol. 97, 37–54 (2007)
Shpiro, A., Curtu, R., Rinzel, J., Rubin, N.: Balance between noise and adaptation in competition models of perceptual bistability. J Comp. Neurosci. 27, 462–473 (2009)
Shuai, J.W., Jung, P.: Stochastic properties of Ca2+ release of Inositol 1,4,5-Triphosphate receptor clusters. Biophys. J. 83, 87–97 (2002)
Shusterman, V., Troy, W.C.: From baseline to epileptiform activity: a path to synchronized rhythmicity in large-scale neural networks. Phys. Rev. E 77, 061911 (2008)
Silva, J.S.D., Dotti, C.G.: Breaking the neuronal sphere: regulation of the actin cytoskeleton in neuritogenesis. Nat. Rev. Neurosci. 3, 694–704 (2002)
Silva, F.H.L.D., Blanes, W., Kalitzin, S.N., Parra, J., Suffczynski, P., Velis, D.N.: Dynamical diseases of brain systems: different routes to epileptic seizures. IEEE Trans. Biomed. Eng. 50, 540–548 (2003)
Sincich, L.C., Blasdel, G.G.: Oriented axon projections in primary visual cortex of the monkey. J. Neurosci. 21, 4416–4426 (2001)
Sinha, S.R.: Basic mechanisms of sleep and epilespy. J. Clin. Neurophysiol. 28, 103–110 (2011)
Sjostrom, P.J., Rancz, E.A., Roth, A., Hausser, M.: Dendritic excitability and synaptic plasticity. Physiol. Rev. 88, 769–840 (2008)
Skinner, F., Kopell, N., Marder, E.: Mechanisms for oscillation and frequency control in reciprocally inhibitory model neural networks. J. Comput. Neurosci. 1, 69–87 (1994)
Smaily, M.E., Hamel, F., Roques, L.: Homogenization and influence of fragmentation in a biological invasion model. Discret. Contin. Dyn. Syst. Ser. A 25, 321–342 (2009)
Smith, G.D.: Modeling the stochastic gating of ion channels. In: Fall, C., Marland, E.S., wagner, J.M., Tyson, J.J. (eds.) Computational Cell Biology, chap. 11. Springer, New York (2002)
Smith, G.D., Cox, C.L., Sherman, S.M., Rinzel, J.: Fourier analysis of sinusoidally driven thalamocortical relay neurons and a minimal integrate-and-burst model. J. Neurophysiol. 83, 588–610 (2000)
Sneyd, J., Charles, A.C., Sanderson, M.J.: A model for the propagation of intercellular calcium waves. Amer. J. Physiol. cell physiol. 266, C293–C302 (1994)
Sneyd, J., Wetton, B.T., Charles, A.C., Sanderson, M.J.: Intercellular calcium waves mediated by diffusion of inositol trisphosphate: a two-dimensional model. Am. J. Physiol. Cell Physiol. 268, C1537–C1545 (1995)
Sneyd, J., Dale, P.D., Duffy, A.: Traveling waves in buffered systems: applications to calcium waves. SIAM J. Appl. Math. 58, 1178–1192 (1998)
Softky, W.R., Koch, C.: Cortical cell should spike regularly but do not. Neural Comput. 4, 643–646 (1992)
Soltesz, I., Staley, K.: Computational Neuroscience in Epilepsy. Academic, San Diego (2008)
Somers, D.C., Nelson, S., Sur, M.: An emergent model of orientation selectivity in cat visual cortical simple cells. J. Neurosci. 15, 5448–5465 (1995)
Somjen, G.G.: Mechanisms of spreading depression and hypoxic spreading depression-like depolarization. Physiol. Rev. 81, 1065–1096 (2001)
Soula, H., Chow, C.C.: Stochastic dynamics of a finite-size spiking neural network. Neural Comput. 19, 3262–3292 (2007)
Sporns, O.: Networks of the Brain. MIT Press, Cambridge (2010)
Staley, K.J., Dudek, F.E.: Interictal spikes and epileptogenesis. Epilepsy Currents 6, 199–202 (2006)
Stam, C.J., Reijeneveld, J.C.: Graph theoretical analysis of complex networks in the brain. Nonlin. Biomed. Phys. 1 (3), 1–19 (2007)
Steinberg, B., Wang, Y.Q., Huang, H., Miura, R.: Spatial buffering mechanism: Mathematical model and computer simulations. Math. Biosci. Eng. 2, 675–702 (2005)
Steriade, M., McCormick, D.A., Sejnowski, T.J.: Thalamocortical oscillations in the sleeping and aroused brain. Science 262, 679–685 (1993)
Steyn-Ross, M.L., Sleigh, D.A.S.R.J.W., Whiting, D.R.: Theoretical predictions for spatial covariance of the electroencephalographic signal during the anesthetic-induced phase transition: Increased correlation length and emergence of spatial self-organization. Phys. Rev. E 68, 021902 (2003)
Stocker, M., Krause, M., Pedarzani, P.: An apamin-sensitive Ca2+-activated K+ current in hippocampal pyramidal neurons. Proc. Natl. Acad. Sci. USA 96, 4662–4667 (1999)
Strack, S., Choi, S., Lovinger, D.M., Colbran, R.J.: Translocation of autophosphorylated calcium/calmodulin-dependent protein kinase II to the postsynaptic density. J. Biol. Chem. 272, 13467–13470 (1997)
Stuart, G.J., Sakmann, B.: Active propagation of somatic action potentials into neocortical pyramidal cell dendrites. Nature 367, 69–72 (1994)
Stuart, G., Spruston, N., Hausser, M.: Dendrites. Oxford University Press, New York (1999)
Stuart, G., Spruston, N., Hausser, M. (eds.): Dendrites. Oxford University Press, Oxford (2007)
Subramanian, K., Narang, A.: A mechanistic model for eukaryotic gradient sensing: spontaneous and induced phosphoinositide polarization. J. Theor. Biol. 231, 49–67 (2004)
Sun, W., McConnell, E., Pare, J.F., Xu, Q., Chen, M., Peng, W., Lovatt, D., Han, X., Y.Smith, Nedergaard, M.: Glutamate-dependent neuroglial calcium signaling differs between young and adult brain. Science 339, 197–200 (2013)
Sutherland, G.R., McNaughton, B.: Memory trace reactivation in hippocampal and neocortical neuronal ensembles. Curr. Opin. Neurobiol. 10, 180–186 (2000)
Swillens, S., Dupont, G., Combettes, L., Champeil, P.: From calcium blips to calcium puffs: theoretical analysis of the requirement for interchannel communication. Proc. Nat. Acad. Sci. (USA) 96, 13750–13755 (1999)
Swindale, N.V.: The development of topography in the visual–cortex: A review of models. Network 7, 161–274 (1996)
Syed, M.M., Lee, S., He, S., Zhou, Z.J.: Spontaneous waves in the ventricular zone of developing mammalian retina. J. Neurophysiol. 91, 1999–2009 (2004)
Syed, M.M., Lee, S., Zheng, J., Zhou, Z.J.: Stage-dependent dynamics and modulation of spontaneous waves in the developing rabbit retina. J. Physiol. 560, 533–549 (2004)
Sykova, E., Nicholson, C.: Diffusion in brain extracellular space. Physiol. Rev. 88, 1277–1340 (2008)
Tabak, J., Senn, W., O’Donovan, M.J., Rinzel, J.: Modeling of spontaneous activity in developing spinal cord using activity-dependent depression in an excitatory network. J. Neurosci. 20, 3041–3056 (2000)
Tauber, U.C.: Field-theory approaches to nonequilibrium dynamics. Lect. Notes Phys. 716, 295–348 (2007)
Taylor, A., Cottrell, G., Kristan Jr, W.: Analysis of oscillations in a reciprocally inhibitory network with synaptic depression. Neural comput. 14(3), 561–581 (2002)
Terman, D.H., Ermentrout, G.B., Yew, A.C.: Propagating activity patterns in thalamic neuronal networks. SIAM J. Appl. Math 61, 1578–1604 (2003)
Thul, R., Coombes, S., Smith, G.D.: Sensitisation waves in a bidomain fire-diffuse-fire model of intracellular Ca2+ dynamics. Physica D 238, 2142–2152 (2009)
Thul, R., Smith, G.D., Coombes, S.: A bidomain threshold model of propagating calcium waves. J. Math. Biol. 56, 435–463 (2008)
Tian, G.F.,, Hooman, A., Takano, T., Xu, Q., Peng, W., Lin, J., Oberheim, N., Lou, N., Zielke, R., Kang, J., Nedergaard, M.: An astrocytic basis of epilepsy. Nat. Med. 11, 973–981 (2005)
Timofeeva, Y.: Traveling waves in a model of quasi-active dendrites with active spines. Physica D 239, 494–503 (2010)
Timofeeva, Y., Lord, G.J., Coombes, S.: Spatio-temporal filtering properies of a dendritic cable with active spines: a modeling study in the spike-diffuse-spike framework. J. Comput. Neurosci. 21, 293–306 (2006)
Tong, F., Engel, S.A.: Interocular rivalry revealed in the human cortical blind-spot representation. Nature 411, 195–199 (2001)
Tong, F., Meng, M., Blake, R.: Neural bases of binocular rivalry. Trend. Cog. Sci. 10, 502–511 (2006)
Toriyama, M., Shimada, T., Kim, K.B., Mitsuba, M., Nomura, E., Katsuta, K., Sakumura, Y., Roepstorff, P., Inagaki, N.: Shootin1: a protein involved in the organization of an asymmetric signal for neuronal polarization. J. Cell Biol. 175, 147–157 (2006)
Toriyama, M., Sakumura, Y., Shimada, T., Ishii, S., Inagaki, N.: A diffusion-based neurite length-sensing mechanism involved in neuronal symmetry breaking. Mol. Syst. Biol. 6, 1–16 (2010)
Torquato, S.: Random Heterogeneous Materials. Springer, New York (2002)
Touboul, J., Destexhe, A.: Can power-law scaling and neuronal avalanches arise from stochastic dynamics? PLoS One 5, 14 (2010)
Touboul, J., Hermann, G., Faugeras, O.: Noise–induced behaviors in neural mean field dynamics. SIAM J. Appl. Dyn. Sys. 11, 49–81 (2012)
Tritsch, N.X., Yi, E., Gale, J.E., Glowatzki, E., Bergles, D.E.: The origin of spontaneous activity in the developing auditory system. Nature 450, 50–55 (2007)
Troy, W.C., Shusterman, V.: Patterns and features of families of traveling waves in large-scale neuronal networks. SIAM J. Appl. Dyn. Syst. 6, 263–292 (2007)
Tsai, J.C., Sneyd, J.: Are buffers boring? uniqueness and asymptotical sttability of traveling wave fronts in the buffered bistable equation. J. Math. Biol. 54, 513–553 (2007)
Tsai, J.C., Sneyd, J.: Traveling waves in the buffered FitzHugh-Nagumo model. SIAM J. Appl. Math. 71, 1606–1636 (2011)
Tsodyks, M., Pawelzik, K., Markram, H.: Neural networks with dynamic synapses. Neural Comput. 10, 821–835 (1998)
Tuckwell, H.C., Miura, R.M.: A mathematical model for spreading cortical depression. Biophys. J. 23, 257–276 (1978)
Vecchia, D., Pietrobon, D.: Migrain: a disorder of brain excitatory-inhibitory balance? Trend. Neurosci. 35, 507–520 (2012)
Veen, M.V., Pelt, J.V.: Neuritic growth rate described by modelling microtubule dynamics. Bull. Math. Biol. 56, 249–273 (1994)
Vellela, M., Qian, H.: Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited. J. R. Soc. Interface 6, 925–940 (2009)
Veltz, R., Faugeras, O.: Local/global analysis of the stationary solutions of some neural field equations. SIAM J. Appl. Dyn. Syst. 9, 954–998 (2010)
Vicker, M.G.: Eukaryotic cell locomotion depends on the propagation of self-organized reaction-diffusion waves and oscillations of actin filament assembly. J. Expt. Cell Res. 275, 54–66 (2002)
Vladimirski, B.B., Tabak, J., ODonovan, M.J., Rinzel, J.: Episodic activity in a heterogeneous excitatory network, from spiking neurons to mean field. J. Comp. Neurosci. 25, 39–63 (2008)
Vogels, T.P., Abbott, L.F.: Signal propagation and logic gating in networks of integrate-and-fire neurons. J. Neurosci. 25, 10786–10795 (2005)
Volman, V., Sejnowski, T.J., Bazhenov, M.: Topological basis of epileptogenesis in a model of severe cortical trauma. J. Neurophysiol. 106, 1933–1942 (2011)
Volman, V., Bazhenov, M., Sejnowski, T.J.: Computational models of neuron-astrocyte interaction in epilepsy. Front. Comp. Neurosci. 6(58), 1–10 (2012)
Volpert, V., Petrovskii, S.: Reaction-diffusion waves in biology. Phys. Life Rev. 6, 267–310 (2009)
van Vreeswijk, C.: Partial synchronization in populations of pulse-coupled oscillators. Phys. Rev. E. 54(5), 5522–5537 (1996)
van Vreeswijk, C., Sompolinsky, H.: Chaotic balanced state in a model of cortical circuits.. Neural Comput. 10, 1321–1371 (1998)
van Vreeswijk, C., Ermentrout, G.B., Abbott, L.F.: When inhibition not excitation synchronizes neural firing. J. Comput. Neurosci. 1, 313–321 (1994)
Vuyst, E.D., Wang, N., Decrock, E., Bock, M.D., Vinken, M., Moorhem, M.V., Lai, C., Culot, M., Rogiers, V., Cecchelli, R., Naus, C.C., Evans, W.H., Leybaert, L.: Ca2+ regulation of connexin 43 hemichannels in C6 glioma and glial cells. Cell Calcium 46, 176–187 (2009)
Wagner, J., Sneyd, J.: Effects of rapid buffers on Ca2+ diffusion and Ca2+ oscillations. Biophys. J. 67, 447–456 (1994)
Wainrib, G., Thieullen, M., Pakdaman, K.: Reduction of stochastic conductance-based neuron models with time-scales separation. J. Comput. Neurosci. 32, 327–46 (2012)
Walther, G.R., Maree, A.F., Edelstein-Keshet, L., Grieneisen, V.A.: Deterministic versus stochastic cell polarisation through wave-pinning. Bull. Math. Biol. 74, 2570–2599 (2012)
Wang, X.J.: Decision making in recurrent neuronal networks. Neuron 60, 215–234 (2008)
Wang, X.J., Rinzel, J.: Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Comput. 4, 84–97 (1992)
Wang, S.S., Denk, W., Hausser, M.: Coincidence detection in single dendritic spines mediated by calcium release. Nat. Neurosci. 3, 1266–1273 (2000)
Wang, Z., Haydon, P.G., Yeung, E.S.: Direct observation of calcium-independent intercellular ATP signaling in astrocytes.. Anal. Chem. 72, 2001–2007 (2000)
Ward, M.J., Lee, J.: On the asymptotic and numerical-analyses of exponentially ill-conditioned singularly perturbed boundary value problems. Stud. Appl. Math. 94, 271–326 (1995)
Watt, A.J.: Traveling waves in developing cerebellar cortex mediated by asymmetrical purkinje cell connectivity. Nat. Neurosci. 12, 463–473 (2009)
Watts, D.J., Strogatz, S.H.: Collective dynamics of “small-world” networks. Nature 393, 440–442 (1998)
Webber, M., Bressloff, P.C.: The effects of noise on binocular rivalry waves: a stochastic neural field model. J. Stat. Mech 3, P03001 (2013)
Weinberger, H.F.: On spreading speeds and traveling waves for growth and migration in a periodic habitat. J. Math. Biol. 45, 511–548 (2002)
Weiner, O.D., Marganski, W.A., Wu, L.F., Altschuler, S.J., Kirschner, M.W.: An actin-based wave generator organizes cell motility. PLoS Biol. 5, 2053–2063 (2007)
Weissman, T.A., Riquelme, P.A., Ivic, L., Flint, A.C., Kriegstein, A.R.: Calcium waves propagate through radial glial cells and modulate proliferation in the developing neocortex. Neuron 43, 647–661 (2004)
Werry, E.L., Liu, G.J., Bennett., M.R.: Glutamate-stimulated ATP release from spinal cord astrocytes is potentiated by substance P. J. Neurochem. 99, 924–936 (2006)
Wetherington, J., Serranoa, G., Dingledine, R.: Astrocytes in the epileptic brain. Neuron 58, 168–178 (2008)
White, J., Southgate, E., Thomson, J., Brenner, S.: The structure of the nervous system of the nematode Caenorhabditis elegans. Phil. Trans. R. Soc. Lond. B Biol. Sci. 314, 1–340 (1986)
White, J.A., Rubinstein, J.T., Kay, A.R.: Channel noise in neurons. Trends. Neurosci. 23, 131–137 (2000)
Wiley, D.A., Strogatz, S.H., Girvan, M.: The size of the sync basin. Chaos 16, 015103 (2006)
Williams, G.S.B., Huertas, M.A., Sobie, E.A., Jafri, M.S., Smith, G.D.: A probability density approach to modeling local control of calcium-induced calcium release in cardiac myocytes. Biophys. J. 92, 2311–2328 (2007)
Williams, G.S.B., Huertas, M.A., Sobie, E.A., Jafri, M.S., Smith, G.D.: Moment closure for local control models of calcium-induced calcium release in cardiac myocytes. Biophys. J. 95, 1689–1703 (2008)
Wilson, H.R.: Computational evidence for a rivalry hierarchy in vision. Proc. Natl. Acad Sci. USA 100, 14499–144503 (2003)
Wilson, H.R., Cowan, J.D.: Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12, 1–23 (1972)
Wilson, H.R., Cowan, J.D.: A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13, 55–80 (1973)
Wilson, M.A., McNaughton, B.L.: Reactivation of hippocampal ensemble memories during sleep. Science 265, 676–679 (1994)
Wilson, H.R., Blake, R., Lee, S.H.: Dynamics of traveling waves in visual perception. Nature 412, 907–910 (2001)
Winfree, A.: The Geometry of Biological Time. Springer, New York (1980)
Wissner-Gross, Z.D., Scott, M.A., Steinmeyer, J.D., Yanik, M.F.: Synchronous symmetry breaking in neurons with different neurite counts. PLoS One 8 (2), e54905 (2013)
Wolszon, L.R., Rehder, V., Kater, S.B., Macagno, E.R.: Calcium wave fronts that cross gap junctions may signal neuronal death during development. J. Neurosci. 14, 3437–3448 (1994)
Wong, R.O.L.: Retinal waves and visual development. Ann. Rev. Neurosci. 22, 29–47 (1999)
Wu, J.: Propagating waves of activity in the neocortex: what they are, what they do. Neuroscientist 14, 487–502 (2008)
Wu, J.Y., Guan, L., Tsau, Y.: Propagating activation during oscillations and evoked responses in neocortical slices. J. Neurosci. 19, 5005–5015 (1999)
Wu, J., Guan, L., Bai, L., Yang, Q.: Spatiotemporal properties of an evoked population activity in rat sensory cortical slices. J. Neurophysiol. 86, 2461–74 (2001)
Wylie, J.J., Miura, R.M.: Traveling waves in coupled reaction-diffusion models with degenerate sources. Phys. Rev. E 74, 021909 (2006)
Xin, J.: Front propagation in heterogeneous media. SIAM Rev. 42, 161–230 (2000)
Xu, W., Huang, X., Takagaki, K., Wu, J.Y.: Compression and reflection of visually evoked cortical waves. Neuron 55, 119–129 (2007)
Yang, E., Schulman, H.: Structural examination of autoregulation of multifunctional calcium/calmodulin-dependent protein kinase II. J. Biol. Chem. 274, 26199–26208 (1999)
Yao, Y., Choi, J., Parker, I.: Quantal puff of intracellular Ca2+ evoked by inositol triphosphate in Xenopus oocytes. J. Physiol. 482, 533–553 (1995)
Yoshioka, T., Blasdel, G.G., Levitt, J.B., Lund, J.S.: Relation between patterns of intrinsic lateral connectivity, ocular dominance, and cytochrome oxidase–reactive regions in macaque monkey striate cortex. Cereb. Cortex 6, 297–310 (1996)
Young, G.W.D., Keizer, J.: A single pool IP3-receptor model for agonist stimulated Ca2+ oscillations. Proc. Nat. Acad. Sci. (USA) 89, 9895–9899 (1992)
Yuste, R., Bonhoeffer, T.: Morphological changes in dendritic spines associated with LTP. Ann. Rev. Neurosci. 24, 1071–1089 (2001)
Yuste, R., Denk, W.: Dendritic spines as a basic functional units of neuronal integration. Nature 375, 682–684 (1995)
Zahradnikova, A., Zahradnik, I.: A minimal gating model for the cardiac Ca2+ release channel. Biophys. J. 71, 2996–3012 (1996)
Zhang, L.: On the stability of traveling wave solutions in synaptically coupled neuronal networks. Diff. Int. Eq. 16, 513–536 (2003)
Zimmermann, H.: Purinergic signaling in neural development. Sem. Cell Dev. Biol. 22, 194–204 (2011)
Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena, 4th edn. Oxford University Press, Oxford (2002)
Zinner, B.: Existence of traveling wavefront solutions for the discrete Nagumo equation. J. Diff. Eqns. 96, 1–27 (1992)
Zucker, R.S.: Short term synaptic plasticity. Ann. Rev. Neurosci. 12, 13–31 (1989)
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Bressloff, P.C. (2014). Waves in Excitable Neural Fields. In: Waves in Neural Media. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8866-8_7
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