1 Introduction

Oscillatory patterns of electrical activity in the neocortex, usually measured on the scalp by electroencephalography (EEG) and magnetoencephalography (MEG), or measured intracranially by electrocorticography (ECoG), correlate with numerous states of the operation of the brain. These oscillations can emerge as sustained rhythms, or as transient bursts of fluctuations which last only for few cycles. They are presumed to play a fundamental role in cognition and overall functioning of the brain, possibly by facilitating efficient coordination of neuronal activity across the neocortex. Distinct distortions or disruption of these oscillations can hence be used as a marker for specific diseases. Despite their substantial importance, however, the underlying physiological mechanism of the generation of these cortical oscillations is not yet well-understood (Buzsáki and Watson 2012; Jones 2016; van Ede et al. 2018; Jia and Kohn 2011).

Gamma oscillations in the frequency range of 30 to 80 Hz are ubiquitous short-lasting oscillations which usually arise locally in the neocortex and correlate with the operational mode of the underlying cortical network, providing a signature of the engagement of the network. It is hypothesized that gamma oscillations may have a syntactic function in the coordination of information flow across the neocortex. Impaired or irregular gamma oscillations have been observed in psychiatric diseases such as schizophrenia, attention-deficit hyperactivity disorder (ADHD), bipolar disorder, and autism spectrum disorder (ASD), as well as in neurodegenerative diseases such as Alzheimer’s disease (AD) and Parkinson’s disease (PD), and in neurological disorders such as epilepsy (Buzsáki and Watson 2012; Jia and Kohn 2011; Buzsáki and Wang 2012; Lee and Jones 2013; Knoblich et al. 2010; Kopell et al. 2000; Traub et al. 1999; Cardin et al. 2009; Ray and Maunsell 2015). Therefore, understanding the mechanism of the generation of gamma oscillations can provide important information about the healthy operation of the brain.

Dynamic modulation of the responsiveness of neuronal populations is a fundamental property of cortical networks, which is thought to mediate information flow in the neocortex by rapidly altering the cortical excitability and sensitivity. This dynamic control of the engagement of cortical networks can transiently link local populations of neurons together over intervals of time ranging from milliseconds to seconds. It is hence hypothesized that this adaptable control of neuronal activity is a basis of diverse perceptual behaviors in the brain (Haider and McCormick 2009). The responsiveness of neurons can be modulated in several ways, through changes in ionic concentrations as well as variations in postsynaptic parameters such as the membrane conductance, the mean membrane potential, and the variance of membrane potential. These modifications can be made through the ongoing barrages of the synaptic inputs that neurons receive from their surrounding network when the network is involved in a sensory-driven activity (Haider and McCormick 2009; Cardin et al. 2008). In addition, neuromodulators can also alter the responsiveness of neurons (McCormick et al. 1993; McCormick 1992). For example, serotonin and catecholamines such as dopamine and noradrenaline are known to modulate the gain of neurons in the prefrontal cortex (Thurley et al. 2008). Furthermore, acetylcholine can multiplicatively modulate neuronal sensitivity in macaque area V1 neurons by acting on muscarinic and nicotinic receptors (Herrero et al. 2017).

In this paper, the biophysical mean field model of electroencephalographic activity proposed by Liley et al. (2002) is used to demonstrate the emergence and propagation of gamma oscillations in the neocortex as a result of neuronal response modulation. This model has been widely used in the literature to study brain rhythms (Bojak et al. 2004; Bojak and Liley 2007), general anesthesia (Wilson et al. 2006a; Liley and Walsh 2013; Bojak et al. 2013; Foster et al. 2008; Bojak et al. 2015; Steyn-Ross et al. 2004), epileptic seizures (Martinet et al. 2017; Kramer et al. 2012; Kramer et al. 2006; Liley and Bojak 2005; Kramer et al. 2005), and sleep (Lopour et al. 2011; Wilson et al. 2006b; SteynRoss et al. 2005). Tools for numerical implementation of the model and computation of its equilibria and time-periodic solutions are developed by Green and van Veen (2014). Complexity of the dynamic behavior of the model, including the existence of periodic and pseudo-periodic solutions, chaotic behavior, multistability, and dynamic bifurcations are widely studied (Frascoli et al. 2008, 2011; van Veen and Green 2014; van Veen and Lily 2006; Dafilis et al. 2001, 2013, 2015). Rigorous analytical results on well-posedness, regularity, biophysical plausibility, and global dynamics of this model are established by Shirani et al. (2017). Specifically, it is shown that this model can have a noncompact global attracting set and can present a rich variety of spatially localized dynamic behavior.

This paper is organized as follows. In Section 2, the mathematical structure of the model is presented and its variables and parameters are described. In Section 3, it is shown through a codimension-one bifurcation analysis that gamma oscillations emerge robustly in the solutions of the model when neuronal responsiveness is effectively altered. In Section 4, the bifurcation analysis is extended to codimension-two to demonstrate how the emerged gamma oscillations can transition back to resting-state beta oscillations through coordinated modulation of both inhibitory and excitatory neuronal populations. In Section 5, the equations of the model are solved using a finite-element solver and the emergence, spatial propagation, and transition of gamma oscillations are illustrated. Moreover, potential effects of the size and location of electrocortical measurement electrodes on the amplitude and temporal pattern of gamma oscillation recordings are discussed. Finally, the contribution of the results to the understanding of transient gamma oscillations in neocortical activity in the brain are discussed in Section 6.

$$ \begin{array}{@{}rcl@{}} (\tau_{\textsc{e}} \partial_t + 1) v_{\textsc{e}}(x,t) &=& \frac{\mathrm{V}_{\textsc{e} \textsc{e}}-v_{\textsc{e}}(x,t)}{|\mathrm{V}_{\textsc{e} \textsc{e}}|} i_{\textsc{e} \textsc{e}}(x,t) + \frac{\mathrm{V}_{\textsc{i} \textsc{e}}-v_{\textsc{e}}(x,t)}{|\mathrm{V}_{\textsc{i} \textsc{e}}|} i_{\textsc{i} \textsc{e}}(x,t), \\ (\tau_{\textsc{i}} \partial_t + 1) v_{\textsc{i}}(x,t) &=& \frac{\mathrm{V}_{\textsc{e} \textsc{i}}-v_{\textsc{i}}(x,t)}{|\mathrm{V}_{\textsc{e} \textsc{i}}|} i_{\textsc{e} \textsc{i}}(x,t) + \frac{\mathrm{V}_{\textsc{i} \textsc{i}}-v_{\textsc{i}}(x,t)}{|\mathrm{V}_{\textsc{i} \textsc{i}}|} i_{\textsc{i} \textsc{i}}(x,t), \\ (\partial_t +\gamma_{\textsc{e} \textsc{e}} )^2 i_{\textsc{e} \textsc{e}}(x,t) &=& e {\Upsilon}_{\textsc{e} \textsc{e}} \gamma_{\textsc{e} \textsc{e}} \left[ \mathrm{N}_{\textsc{e} \textsc{e}} f_{\textsc{e}}\big(v_{\textsc{e}}(x,t) \big) + w_{\textsc{e} \textsc{e}}(x,t) + g_{\textsc{e} \textsc{e}}(x,t) \right], \\ (\partial_t +\gamma_{\textsc{e} \textsc{i}} )^2 i_{\textsc{e} \textsc{i}}(x,t) &=& e {\Upsilon}_{\textsc{e} \textsc{i}} \gamma_{\textsc{e} \textsc{i}} \left[ \mathrm{N}_{\textsc{e} \textsc{i}} f_{\textsc{e}}\big(v_{\textsc{e}}(x,t) \big) + w_{\textsc{e} \textsc{i}}(x,t) + g_{\textsc{e} \textsc{i}}(x,t) \right], \\ (\partial_t +\gamma_{\textsc{i} \textsc{e}} )^2 i_{\textsc{i} \textsc{e}}(x,t) &=& e {\Upsilon}_{\textsc{i} \textsc{e}} \gamma_{\textsc{i} \textsc{e}} \left[ \mathrm{N}_{\textsc{i} \textsc{e}} f_{\textsc{i}}\big(v_{\textsc{i}}(x,t) \big) + g_{\textsc{i} \textsc{e}}(x,t) \right], \\ (\partial_t + \gamma_{\textsc{i} \textsc{i}} )^2 i_{\textsc{i} \textsc{i}}(x,t) &=& e {\Upsilon}_{\textsc{i} \textsc{i}} \gamma_{\textsc{i} \textsc{i}} \left[ \mathrm{N}_{\textsc{i} \textsc{i}} f_{\textsc{i}}\big(v_{\textsc{i}}(x,t) \big) + g_{\textsc{i} \textsc{i}}(x,t) \right], \\ \left[ (\partial_t + \nu {\Lambda}_{\textsc{e} \textsc{e}} )^2 - \frac{3}{2} \nu^2 {\Delta} \right] w_{\textsc{e} \textsc{e}}(x,t) &=& \nu^2 {\Lambda}_{\textsc{e} \textsc{e}}^2 \mathrm{M}_{\textsc{e} \textsc{e}} f_{\textsc{e}}\big(v_{\textsc{e}}(x,t) \big), \\ \left[ (\partial_t + \nu {\Lambda}_{\textsc{e} \textsc{i}} )^2 - \frac{3}{2} \nu^2 {\Delta} \right] w_{\textsc{e} \textsc{i}}(x,t) &=& \nu^2 {\Lambda}_{\textsc{e} \textsc{i}}^2 \mathrm{M}_{\textsc{e} \textsc{i}} f_{\textsc{e}}\big(v_{\textsc{e}}(x,t) \big), \hspace{10em}(x,t) \in {\Omega} \times (0,T], \end{array} $$
(1)
Table 1 Biophysical parameters of the mean field model (1). All electric potentials are given with respect to the mean resting soma membrane potential vrest = − 70 mV, Bojak and Liley (2005) and Shirani et al. (2017)
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2 Model description

Local networks of cortical neurons are densely interconnected. It is shown, for example, that local networks of inhibitory interneurons within layers II and III in mouse frontal cortex almost form a completely connected graph without any distinct subnetworks (Fino and Yuste 2011; Lee and Huguenard 2011). Such a locally dense structure of the neocortex suggests mean field models as particularly useful models for studying the mesoscopic activity of neocortical neurons being monitored by EEG recordings. The biophysical mean field model developed by Liley et al. (2002) effectively connects the intracortical and corticocortical neuronal activity to EEG measurements from the scalp. For ease of reference, the mathematical structure and description of variables and parameters of this model are given below. Further details of the model are available in the literature (Liley et al. 2002; Bojak and Liley 2005; Shirani et al. 2017; Bojak et al. 2015; Liley and Walsh 2013).

Let \({\Omega } \subset \mathbb {R}^{2}\) be an open rectangle defining the domain of the neocortex, so that each point x = (x1,x2) ∈Ω indicates the location of a local cortical population. Let e and i denote populations of excitatory and inhibitory neurons, respectively. Then, the system of partial differential equations (PDEs) given by (1), with periodic boundary conditions, provides a mean field model of electroencephalographic activity in the neocortex, wherein ∂t denotes the partial derivative with respect to t, and Δ denotes the Laplace operator. For x,y ∈{e,i}, the variable vx(x,t) in (1) denotes the spatially mean soma membrane potential of a population of type x located at x. Moreover, ixy(x,t) denotes the spatially mean postsynaptic activation of the synapses of a population of type x located at x, onto a population of type y located at the same point. In addition, wex(x,t) denotes the mean rate of corticocortical input pulses that a population of type x located at x receives from all other excitatory populations throughout the neocortex. Finally, gxy(x,t) denotes the mean rate of subcortical input pulses of type x received by a population of type y located at x. Note that vx and ixy are measured in mV, whereas wex and gxy are measured in s− 1. Moreover, ixy, wex, and gxy are nonnegative quantities by definition. The constant e in (1) is Napier’s constant, and the nonlinear function fx(⋅) gives the mean firing rate of a population of type x as a sigmoid function of the population’s mean membrane potential, defined as

$$ f_{\textsc{x}}\big(v_{\textsc{x}} \big) := \frac{\mathrm{F}_{\textsc{x}}} {1+\exp\left( -\sqrt{2} \frac{v_{\textsc{x}} - \mu_{\textsc{x}} }{\sigma_{\textsc{x}}} \right)}, \quad \textsc{x} \in \{ \textsc{e}, \textsc{i} \}. $$
(2)

Definition of the biophysical parameters of the model and their range of values are given in Table 1. As in Shirani et al. (2017), in addition to the notational changes to the original equations given by Liley et al. (2002), the reference of electric potentials of each population is set at the resting potential of the population in order to avoid the constant terms that would otherwise appear in (1).

The first two equations in (1) represent the first-order dynamics of the resistive-capacitive membrane of space-averaged neurons; see the details in Liley et al. (2002) and Shirani et al. (2017). The next four critically damped second-order equations generate postsynaptic α-functions in response to impulse inputs. The last two telegraph equations in (1) model the propagation of excitatory axonal pulses along long-range corticocortical fibers. The key variable in this model is the mean membrane potential of excitatory populations ve. It is presumed in the literature that ve is linearly proportional to EEG recordings from the scalp (Liley et al. 2002; Liley and Walsh 2013). Therefore, the spatio-temporal pattern of variations of ve is illustrated throughout this paper as the predictions provided by the model for EEG measurements.

3 Neuronal response modulation and emergence of gamma oscillations

The input-output relationships that characterize neuronal responsiveness in cortical networks usually follow sigmoid curves, that is, changes in intermediate values of the input result in large variations in the neuronal response, whereas changes in the input values below a threshold level or above a saturation level do not evoke substantial variations in the response (Haider and McCormick 2009; Thurley et al. 2008). Such characteristic response curves can be modulated in different ways, resulting in different neuronal behaviors. The sensitivity of a neuron is equally altered at all input levels if its response curve is stretched or shrunk along the output axis by a multiplicative factor (Cardin et al. 2008; Ni et al. 2016; Haider and McCormick 2009). This neuronal gain modulation changes the slope and maximum level of the curve while leaving the threshold and saturation levels unchanged. Shifting the curve right or left along the input axis alters the excitability of a neuron, that is, its responsiveness to different levels of the input. Although the shape of the curve remains unchanged in this case, an input gain modulation is attained by shifts in the threshold and saturation levels (Cardin et al. 2008; Haider and McCormick 2009). Finally, the saliency of a neuron can be altered by stretching or shrinking its response curve along the input axis, which changes the maximum slope of the curve and moves the threshold and saturation levels in opposite directions (Herrero et al. 2017). A combination of these modulatory effects can also result from the action of neuromodulators. For example, it is shown by Thurley et al. (2008) that, mediated by D1 receptors, dopamine can increase the excitability of Layer V pyramidal neurons in rat prefrontal cortex by shifting the firing rate-current response curve of neurons to lower inputs. In addition, it increases the maximum slope (saliency) of the curve at intermediate values of current and decreases the maximum level of the firing rate response.

Table 2 The set of biophysically plausible parameter values used as the nominal values for the computational analysis of Sections 3–5 (Bojak and Liley 2005, Table V, col. 11). The parameters \(\bar {g}_{\textsc {e} \textsc {e}}\), \(\bar {g}_{\textsc {e} \textsc {i}}\), \(\bar {g}_{\textsc {i} \textsc {e}}\), and \(\bar {g}_{\textsc {i} \textsc {i}}\) are the mean values of the physiologically shaped random inputs gee, gei, gie, and gii used by Bojak and Liley (2005), respectively. Electric potentials are given with respect to the mean resting membrane potentials \(\protect v_{\textsc {e}_{\text {rest}}} = -72.293 \text { mV}\) and \(\protect v_{\textsc {i}_{\text {rest}}} = -67.261 \text { mV}\)
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As shown in Fig. 1, the different types of neuronal gain modulation described above can be effectively incorporated into the model (1) using the graph of the sigmoid firing rate functions (2) as the characteristic curves of neuronal responsiveness. In this section, the characteristic curve of inhibitory populations is used to predict the emergence of gamma oscillations in the solutions of (1) when the parameters of the curve are changed effectively. Inhibitory neuronal activity in cortical networks is integral to network oscillations and rapid coordination of functional connectivity across the cortex, (Cardin et al. 2009; Whittington et al. 2000; Haider and McCormick 2009; Buzsáki and Watson 2012; Buzsáki and Wang 2012; Ray and Maunsell 2015; Knoblich et al. 2010; Jia and Kohn 2011). Moreover, neuronal modulations such as modulation through nicotinic and muscarinic receptors occur predominately on inhibitory interneurons (Herrero et al. 2017).

Fig. 1
figure 1

Neuronal gain modulation. Thick curves show the firing rate function (2) for x = i and the nominal parameter values given in Table 2. Other curves in each graph illustrate variations in the shape of nominal curves as Fx, μx and σx, x ∈{e,i}, change over their range of values given in Table 1. Arrows indicate variations corresponding to incremental changes in parameter values

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In this section, predictions on the emergence of gamma oscillations are based on codimension-one bifurcation analysis of a spatially homogeneous version of (1), that is, the model with spatially homogeneous initial values and subcortical inputs. Moreover, to effectively analyze the internal dynamics of the model, the subcortical inputs are assumed to be constant in time. This approximate model is a system of ordinary differential equations (ODEs) obtained from (1) by omitting the Laplacian terms \(-\frac {3}{2} \nu ^{2} {\Delta }\). Bifurcations of equilibria and limit cycles of this system of ODEs can be effectively analyzed using available numerical continuation and bifurcation analysis tools. Moreover, the dynamic behavior of this approximate model can be interpreted as the dynamic behavior of a local cortical network, since omitting the Laplacian terms in the equations is equivalent to considering the corticocortical conduction velocity of zero. Therefore, the possibility of the original model (1) having highly localized dynamic activity, as shown by Shirani et al. (2017), implies that the predictions made by analyzing this approximate model can result in meaningful predictions on the emergence of gamma oscillations in the solutions of the original model. This is demonstrated in Section 5, where the solutions of the full system of PDEs given by (1) are computed using a finite-element solver package.

In this section, as well as in Section 4, the numerical analysis of the ODE system is performed using MatCont (Dhooge et al. 2008) for the nominal parameters values given in Table 2. Note that inhibitory inputs from subcortical regions are rare. Therefore, gie and gii are set equal to zero. The excitatory subcortical inputs gee and gei are set to be constant and, respectively, equal to \(\bar {g}_{\textsc {e} \textsc {e}}\) and \(\bar {g}_{\textsc {e} \textsc {i}}\) given in Table 2. For this set of inputs and parameter values, the ODE system possesses a single stable equilibrium in the biophysically reasonable region of its phase space.

3.1 Neuronal sensitivity modulation

As shown in Fig. 1a, changes in Fi results in multiplicative changes of neuronal sensitivity in inhibitory populations. To observe the effect of this response modulation on the dynamic behavior of the spatially homogeneous system, the equilibrium of the ODE system is continued in both forward and backward directions when Fi is free to change over its range of values given in Table 1. The resulting curve of equilibria is shown in Fig. 2a. The system undergoes two Hopf bifurcations, denoted by H1 and H2, and two fold bifurcations, denoted by F1 and F2. The limit cycles originating from Hopf bifurcation points are also continued and the curves of minimum and maximum values taken by ve on the resulting cycles are shown in Fig. 2a. The frequency of oscillations on these cycles are shown in Fig. 3a. It is observed that, as Fi decreases below the value associated with the Hopf bifurcation point H1, the stable equilibrium becomes unstable. Moreover, the limit cycles originating from H1 are unstable, whereas those originating from H2 are stable. At extremely large values of Fi, the curves of stable and unstable limit cycles collide tangentially at a fold bifurcation of limit cycles. This bifurcation is not made visible in Fig. 2a, since it occurs at biophysically implausible values of Fi.

Fig. 2
figure 2

Codimension-one bifurcation diagrams of the ODE system described in Section 3. Curves of equilibria are shown in blue. Curves of the minimum and maximum values that ve takes on the limit cycles are shown in red. The reference of electric potential for ve is the mean resting soma membrane potential \(\protect v_{\textsc {e}_{\text {rest}}} = -72.293 \text { mV}\). Solid lines indicate stable equilibria and limit cycles. Dashed lines indicate unstable equilibria and limit cycles. Blue dots show the stable equilibrium of the system for the nominal parameter values given in Table 2. Fold and Hopf bifurcation points are indicated by F and H, respectively. Fold bifurcations of limit cycles are indicated by FC. The Hopf bifurcation points and the fold bifurcation of limit cycles that occur at biophysically implausible parameter values are not made visible. a the bifurcation diagram for Fi as a free parameter, b the bifurcation diagram for μi as a free parameter, and c the bifurcation diagram for σi as a free parameter

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Fig. 3
figure 3

Frequency of the limit cycles in the bifurcation diagrams shown in Fig. 2

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The Hopf bifurcation H1 described above is of particular interest. The frequency of unstable cycles originating from this bifurcation point lies between 13 and 15 Hz, suggesting that the damped oscillations around the stable equilibrium at nominal parameter values are in the resting-state low beta (alpha) frequency band. More importantly, as the stable equilibrium switches to an unstable equilibrium at this bifurcation point, orbits of the system depart the vicinity of the equilibrium and converge to a stable limit cycle originated from the Hopf bifurcation H2. As shown in Fig. 3a, stable oscillations on this cycle lie in the gamma frequency band. This predicts that gamma oscillations can emerge in cortical networks when the sensitivity of inhibitory neuronal populations is reduced by modulatory actions.

3.2 Neuronal excitability modulation

Figure 1b shows that changes in μi shifts the characteristic curve of inhibitory neuronal populations along the horizontal axis, thereby changing their excitability in response to variations in the mean membrane potential. Continuation of the equilibrium of the ODE system for the free parameter μi gives the curve of equilibria shown in Fig. 2b. When μi increases from its nominal value, the system undergoes a Hopf bifurcation of equilibrium denoted by H1. The stability of the equilibrium switches at this bifurcation point. Continuation of the limit cycles origination from H1 reveals a fold bifurcation of limit cycles at low values of μi, which is denoted by FC1. This further reveals the existence of a curve of stable limit cycles. Therefore, when μi increases above the bifurcation value, orbits of the system departing the vicinity of the unstable equilibrium converge to a stable limit cycle. As shown in Fig. 3b, the frequency of oscillations on stable cycles lies in the gamma band, and hence this predicts the emergence of gamma oscillations in the electrocortical activity when the neuromodulation decreases the excitability of inhibitory populations by shifting their response curve to higher input values. It should be noted that if μi is free to increase to biophysically implausible values, a second Hopf bifurcation H2 is detected on the curve of equilibria where the unstable equilibrium switches to a stable equilibrium. A second fold bifurcation of limit cycles FC2 is also detected on the curve of limit cycles. These bifurcations are not made visible in Fig. 2b as they appear at extremely large values of μi.

3.3 Neuronal saliency modulation

The maximum slope of response curves can be altered by changing σi, as shown in Fig. 1c. The bifurcation diagram shown in Fig. 2c implies that a Hopf bifurcation at lower values of σi makes the equilibrium unstable. Continuation of the unstable limit cycles originating from this bifurcation point reveals the existence of a stable limit cycle and a fold bifurcation of limit cycles. The orbits of the system converge to a stable limit cycle when σi decreases below the Hopf bifurcation value. Figure 3c shows that the oscillations on this stable cycle are in the gamma frequency band. Therefore, it is predicted by these results that enhancement of the saliency of inhibitory neuronal networks can induce gamma oscillations in the network. Note that at implausibly small values of σi the system undergoes a second Hopf bifurcation and a fold bifurcation of limit cycles, which are not made visible in Fig. 2c.

4 Transition to beta oscillations

Oscillatory patterns of electrocortical activity in the neuronal networks that are involved in cognitive functions, such as gamma oscillations, are usually transient phenomena which emerge briefly in the underlying network and disappear once the cognitive task is processed and the initial state of the network is restored (Buzsáki and Watson 2012; Jones 2016; Traub et al. 1999). However, the bifurcation analysis performed in Section 3 does not explain a biophysically reasonable mechanism for transition from the emerged gamma oscillations back to the resting-state low beta oscillations. This can be seen through the bifurcation diagrams shown in Fig. 2.

Once Fi decreases below the value corresponding to the first Hopf bifurcation in Fig. 2a, orbits of the system converge to a stable limit cycle and oscillate at gamma frequency. However, increasing Fi back to its nominal value cannot force the orbits to depart a stable cycle and converge back to the equilibrium. Additional increments of Fi only shift the orbits to another stable cycle of larger ve-amplitude along the curve of limit cycles. Although this predicts significant robustness for the emergence of gamma oscillations, it cannot explain their transitions back to the initial state. As stated in Section 3.1, further continuation of the limit cycles for larger values of Fi identifies a value at which a fold bifurcation of limit cycles occurs as the curves of stable and unstable limit cycles collide. Therefore, if Fi increases beyond this bifurcation value, the sustained oscillations are terminated and orbits converge to a stable equilibrium. This further allows restoring the initial state through a hysteresis loop. However, the fold bifurcation value of Fi involved in this loop is too large to be biophysically plausible. Alternatively, the bifurcation diagram shown in Fig. 2a also predicts the possibility of the termination of gamma oscillations at very small values of Fi, below the value corresponding to the Hopf bifurcation point H2. At this range of values of Fi, however, the dynamics of the system is extremely sensitive to small variations in Fi so that the orbits of the system can easily converge to equilibrium values which are not biologically realistic.

When gamma oscillations arise through the modulation of μi, the initial resting-state can be restored through a hysteresis loop. This is shown in Fig. 4a. Stable oscillations on limit cycles are terminated when μi decreases below the value corresponding to the fold bifurcation of limit cycles FC1. Orbits of the system converge to a stable equilibrium which moves along the curve of equilibria as μi increases back to its nominal value. Similar to the hysteresis loop described above for modulation of Fi, this mechanism of transition to initial beta oscillations may not be realistic as it involves very small values of μi, corresponding to extreme enhancement of neuronal excitability. Similarly, Fig. 4b implies the possibility of restoring the initial state through a hysteresis loop that involves drastic modulation of σi that substantially diminishes the neuronal saliency.

Fig. 4
figure 4

Restoration of the initial state by modulation of the inhibitory parameters through large hysteresis loops. Step 1: induction of gamma oscillations, Step 2: termination of the gamma oscillations, and Step 3: restoration of the initial parameter value. a modulatory actions on μi and b modulatory actions on σi

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The mechanisms of the emergence of gamma oscillations described in Section 3 and discussed above are based only on the modulation of inhibitory neuronal populations. However, gamma oscillations typically arise from coordinated interaction of inhibitory and excitatory networks (Buzsáki and Wang 2012; Haider and McCormick 2009). In general, cortical networks remarkably maintain a balance of excitation and inhibition during their operation, so that increases in the level of excitation in a cortical region is often accompanied by increases in the level of inhibition. Since long-range corticocortical connections are predominantly excitatory, such an effective balancing of excitation and inhibition must be controlled locally (Haider and McCormick 2009; Dehghani et al. 2016). Due to this locality, the approximate ODE system used in Section 3 can also be helpful in predicting the results of coordinated modulation of both inhibitory and excitatory populations. In what follows, the codimension-one bifurcation analysis of Section 3 is extended to codimension two by additionally considering the parameters of excitatory populations as free parameters. The analysis provides a biologically reasonable explanation for restoration of the initial state after the emergence of gamma oscillations.

Two-parameter continuation of the codimension-one bifurcation points shown in Fig. 2 results in the bifurcation curves shown in Fig. 5. Specifically, continuation of the adjacent fold and Hopf bifurcation points F1 and H2 shown in Fig. 2a reveals a fold-Hopf (or zero-Hopf) codimension-two bifurcation at low values of Fe and Fi, where the resulting curves of fold and Hopf bifurcations intersect each other. This is shown in Fig. 5a. No codimension-two bifurcations are detected along the curve of bifurcations as μe and μi change over their biophysically reasonable range of values. Similarly, continuation of the bifurcation points as σe and σi change over their biophysically plausible range of values does not detect any codimension-two bifurcations.

Fig. 5
figure 5

Two-parameter continuation of the bifurcation points shown in Fig. 2. The starting bifurcation points are indicated by dots with the same colors as they appear in Fig. 2. The curves shown in gray are the result of the continuation of the fold bifurcation of limit cycles that are not made visible in Fig. 2. a continuation for Fi and Fe as free parameters. The detected codimension-two zero-Hopf bifurcation point is indicated by ZH. b continuation for μi and μe as free parameters. The curve shown in purple is the result of the continuation of the Hopf bifurcation point H2 that is not made visible in Fig. 2b. c continuation for σi and σe as free parameters

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The absence of codimension-two bifurcations along nearly the entire extent of the bifurcation curves in Fig. 5 implies that the overall picture of the codimension-one bifurcation diagrams shown in Fig. 2 remains the same when the characteristic parameters of the excitatory populations are allowed to change freely. Moreover, the moderate slope of the bifurcation curves suggest a fair level of robustness for predictions made based on these diagrams. Therefore, to illustrate a global picture of the dynamics of the ODE system when both excitatory and inhibitory characteristic parameters are free to change, a number of points are picked from the curves of Hopf bifurcations in Fig. 5 and the equilibria and limit cycles originated from these points are continued with respect to inhibitory parameters. The resulting sequence of bifurcation diagrams are shown in Fig. 6, and the frequency of oscillations on the limit cycles are shown in Fig. 7. It is observed that the frequency of stable cycles lies predominantly in the gamma band for almost entire range of parameter values of the characteristic curves, which further implies the robustness of emergent gamma oscillations to perturbations in parameters.

Fig. 6
figure 6

Sequences of codimension-one bifurcation diagrams for different values of excitatory parameters. Inhibitory parameters are used as free parameters for generating each diagram. The diagrams shown by thick lines are the same codimension-one diagrams shown in Fig. 2 for the nominal value of excitatory parameters. a the sequence of diagrams for different values of Fe when Fi is used as the free parameter. b the sequence of diagrams for different values of μe when μi is used as the free parameter. c the sequence of diagrams for different values of σe when σi is used as the free parameter

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Fig. 7
figure 7

Frequency of the limit cycles in the sequences of bifurcation diagrams shown in Fig. 6

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Although the bifurcation diagrams within each sequence of diagrams shown in Fig. 6 generally resemble each other, the location of the bifurcation points and, in particular, the fold bifurcations of limit cycles changes significantly when the parameters of excitatory populations change. As shown in Fig. 6a, the fold bifurcation of limit cycles, which occurs at nonphysiologically large values of Fi when Fe takes its nominal value, moves to lower and biophysically plausible values when Fe is decreased. As a result, the curves of limit cycles shrink significantly in size. Similarly, it is implied from the sequence of diagrams shown in Fig. 6b that the second Hopf bifurcation of equilibria H2 and the second fold bifurcation of limit cycles FC2, which occur at extremely large values of μi when μe takes its nominal value, move to biophysically plausible values when μe is decreased. Finally, Fig. 6c shows that the fold bifurcation of limit cycles FC1 which occurs at the nominal value of σe moves to biophysically more reasonable values of σi when σe is increased.

The specific changes within each sequence of bifurcation diagrams shown in Fig. 6 suggest biologically reasonable mechanisms for transition between gamma and beta oscillations through coordinated modulation of inhibitory and excitatory neuronal responsiveness. The stepwise parameter modulations shown in Fig. 8 demonstrate these possible mechanisms. As shown in Fig. 8a, in Step 1, decreasing Fi from its nominal value below the value corresponding to the Hopf bifurcation H1 results in the emergence of gamma oscillations. This is the same result predicted in Section 3. In Step 2, sufficient reduction in Fe to the level at which the curve of limit cycles shrinks substantially results in termination of gamma oscillations and transition to low beta oscillations. The initial resting-state can then be restored by increasing Fi back to its nominal value in Step 3, followed by increasing Fe to its nominal value in Step 4. This coordinated modulation of neuronal activity is consistent with the essential operational property of cortical networks, that is, maintaining a balance of excitation and inhibition. The decrement in Fe in Step 2 is in the direction to compensate for the imbalance created in the network by the decremental modulation of Fi in Step 1. Similarly, in Step 3, the balanced neuronal activity is perturbed when the modulatory effect on Fi vanishes and Fi increases back to its nominal value. This imbalance is compensated by the increment in Fe in Step 4, so that the initial balanced state is restored. Figure 8b demonstrates a similar mechanism of inducing transient gamma oscillations through coordinated modulation of neuronal excitability (μi and μe). As shown in Fig. 8c, transition of gamma oscillations through modulation of neuronal saliency (σi and σe), however, still requires modulation of σi through a hysteresis loop. However, when coordinated with the intermediate excitatory modulation shown in Fig. 8c, the hysteresis loop is reduced in size and becomes biophysically more reasonable.

Fig. 8
figure 8

Mechanisms of transient emergence of gamma oscillations through coordinated modulation of inhibitory and excitatory neuronal responsiveness. Step 1: induction of gamma oscillations by modulation of inhibitory populations. Step 2: termination of the gamma oscillations by modulation of excitatory populations. Steps 3 and 4: restoration of the initial parameter values. a neuronal sensitivity modulation through actions on Fi and Fe. b neuronal excitability modulation through actions on μi and μe. c neuronal saliency modulation through actions on σi and σe. An additional step of modulation of σi to larger values is still necessary to terminate the oscillations in Step 2

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The mechanisms described above are expressed as stepwise procedures for simplicity of exposition. In fact, there can be overlaps between the steps and the neuronal modulation in each step is not necessitated to begin after completion of modulatory actions in previous steps. However, as implied by the bifurcation diagrams of Figs. 5 and 6, in order for these mechanisms to work properly, it is necessary that the modulation of excitatory populations in Step 2 sufficiently lags the initial modulation of inhibitory populations in Step 1, so that the system undergoes the Hopf bifurcation and gamma oscillations emerge. Otherwise, the system simply exhibits damped oscillations of beta frequency about the stable equilibrium which moves as the neuronal parameters change.

The courses of modulatory actions described above are initiated by the modulation of inhibitory populations first, which results in a Hopf bifurcation and gives rise to gamma oscillations. Likewise, the curves of Hopf bifurcations shown in Fig. 5 also predict the emergence of gamma oscillations induced by modulation of excitatory populations. However, the bifurcation diagrams of Fig. 6 imply that, unlike the oscillations induced by inhibitory modulations, transition to beta oscillations in this case cannot analogously result from a subsequent modulation of inhibitory populations that re-establishes the network balance. In fact, modulation of the excitatory parameters in the direction shown in Step 2 in Fig. 8 is necessary for termination of the emerged oscillations, regardless of the type of modulation (inhibitory or excitatory) used initially in the first step to give rise to the gamma oscillations. This can be best understood through the sequence of diagrams shown in Fig. 6a. When Fe increases sufficiently from its nominal value, the system undergoes a Hopf bifurcation and gamma oscillations emerge. However, subsequent modulation of Fi to a biophysically reasonable level, either incremental or decremental, does not terminate the oscillations in a way that allows for restoration of the initial state. In order for this transition to be possible, Fe must be subsequently decreased to the level at which curves of limit cycles shrink substantially in size, so that the oscillations are terminated and orbits converge to a physiological equilibrium. Therefore, when gamma oscillations are induced by modulation of excitatory parameters, restoration of the initial resting-state involves modulatory actions on these parameters through large hysteresis loops.

5 Spatial propagation of gamma oscillations

The numerical analysis of Sections 3 and 4 is based on a spatially homogeneous version of (1), which equivalently represents the dynamics of local cortical networks as a system of ODEs. In this section, the full system of PDEs given by (1) is used to demonstrate the emergence and spatial propagation of gamma oscillations across the neocortex based on the mechanisms predicted in Sections 3 and 4. The computational results are presented only for neuronal sensitivity modulation, that is, the multiplicative modulation of neuronal responsiveness by changing Fi and Fe. This type of response modulation which is often referred to as neuronal gain modulation, as well as the neuronal excitability modulation, is widely observed in experimental studies of cortical networks (Cardin et al. 2008; Ni et al. 2016; Haider and McCormick 2009; Disney et al. 2007; Herrero et al. 2017). Although not presented here, the neuronal excitability and saliency modulations described in Sections 3 and 4 yield similar results.

To perform the computations, the domain of the neocortex is set as Ω = (0,60) × (0,60) mm2. This is much smaller than the actual human neocortex, but it is sufficiently large for the computational analysis of this section due to the spatial locality of the emergent activity. The time horizon of computations is set as T = [0,1100] ms. Neuronal gain modulation is assumed to be attained, for example, by diffusion of neuromodulators into two circular regions of radius R = 15 mm centered at locations a = (15,15) and b = (45,45). These regions are denoted as region A and region B, respectively. In both regions, Fi is decreased from its nominal value to \(\tilde {\mathrm {F}}_{\textsc {i}} = 275 \text { s}^{-1}\) at time \(t_{\textsc {i}_{1}} = 50\) ms, and is increased back to its nominal value at \(t_{\textsc {i}_{2}} = 700\) ms. Note that \(\tilde {\mathrm {F}}_{\textsc {i}}\) is below the bifurcation value shown in Fig. 2a, and hence the results of Section 3 predict the emergence of gamma oscillations. To show the necessity of subsequent decrease in Fe for transition to beta oscillations, Fe is decreased only on region A, from its nominal value to \(\tilde {\mathrm {F}}_{\textsc {e}} = 7 \text { s}^{-1}\) at \(t_{\textsc {e}_{1}} = 500\) ms, and is increased back to its nominal value at \(t_{\textsc {e}_{2}} = 1000\) ms. Therefore, the modulatory actions in region A follow the course of modulations shown in Fig. 8a and the initial resting state is expected to be restored in this region.

To include these modulatory actions in the equations of the model, let

$$ \begin{array}{@{}rcl@{}} \eta(t) &:=& \frac{1}{1 + \exp\left( -\frac{t}{\rho}\right)}, \\ \phi(x) &:=& \left\{\begin{array}{ll} \exp\left( 1 + \frac{1}{ \left \| \frac{x}{R} \right \|_{}^{2} - 1} \right), & \text{if } \left \| x \right \|_{} < R ,\\ 0, & \text{if } \left \| x \right \|_{} \geq R, \end{array}\right. \end{array} $$

where ρ adjusts the sharpness of transition from 0 to 1 in η(t), and \( \left \| \cdot \right \|_{}\) denotes the Euclidean norm in \(\mathbb {R}^{2}\). Setting ρ = 5 ms, the value of η(t) smoothly switches from 0 to 1 at t = 0 over an approximate transition interval of 6ρ = 30 ms. The function ϕ takes its maximum value 1 at x = 0 and radially decreases to 0 as x approaches the boundary of the disk \( \left \| x \right \|_{}\leq R\). The neuronal gain modulations described above are then incorporated into (1) by substituting the following two smooth functions for parameters Fi and Fe,

$$ \begin{array}{@{}rcl@{}} \mathrm{F}_{\textsc{i}}(x,{\kern-.5pt} t{\kern-.5pt}) &=& \mathring{\mathrm{F}}_{\textsc{i}} + (\tilde{\mathrm{F}}_{\textsc{i}} - \mathring{\mathrm{F}}_{\textsc{i}}) \left[\eta(t-t_{\textsc{i}_{1}}) - \eta(t-t_{\textsc{i}_{2}}) \right] \phi(x-a) \\ && + (\tilde{\mathrm{F}}_{\textsc{i}} - \mathring{\mathrm{F}}_{\textsc{i}}) \left[\eta(t-t_{\textsc{i}_{1}}) - \eta(t-t_{\textsc{i}_{2}}) \right] \phi(x-b),\\ \mathrm{F}_{\textsc{e}}{\kern-.5pt}({\kern-.5pt}x{\kern-.5pt},{\kern-.5pt}t{\kern-.5pt}) &=& \mathring{\mathrm{F}}_{\textsc{e}} {\kern-.5pt} +{\kern-.5pt} (\tilde{\mathrm{F}}_{\textsc{e}}{\kern-.5pt} -{\kern-.5pt} \mathring{\mathrm{F}}_{\textsc{e}}) \left[\eta(t{\kern-.5pt} -{\kern-.5pt} t_{\textsc{e}_{1}}) - \eta(t{\kern-.5pt} -{\kern-.5pt} t_{\textsc{e}_{2}}) \right] \phi{\kern-.5pt}(x{\kern-.5pt} -{\kern-.5pt} a{\kern-.5pt}), \end{array} $$

where F̈i and F̈e denote the nominal values of Fi and Fe given in Table 2, respectively.

Fig. 9
figure 9

Emergence and spatial propagation of gamma oscillations in the solutions of the model (1). Each frame illustrates the spatial profile of ve over Ω at the time t shown on the top of the frame. The reference of electric potential for ve is the mean resting soma membrane potential \(\protect v_{\textsc {e}_{\text {rest}}} = -72.293 \text { mV}\). Insets in the bottom of the frames show the modulation state of parameters, with Fi shown on the left and Fe shown on the right. The color bar only indicates the value of ve and does not correspond to the value of parameters shown in the insets. Transient oscillations on the lower left corner of the frames take place in region A. Sustained oscillations on the upper right corner of the frames take place in region B, where the excitatory modulation is not applied

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Depending on the specific purpose of numerical computations, subcortical inputs are usually modeled as physiologically shaped random inputs (Bojak and Liley 2005). However, for clearer illustration of the emergence and propagation pattern of gamma oscillations, and for better comparison of the results with predictions of Sections 3 and 4, in the computations of this section gee, gei, gie, and gii are set to take the same constant values as considered for the approximate ODE system in Sections 3 and 4. Moreover, other than Fi and Fe which are modulated as described above, the rest of the parameters of the model take their nominal values given in Table 2, the same as that considered for the analysis of Sections 3 and 4. For this set of inputs and nominal parameters, the spatially homogeneous stable equilibrium of (1) can be calculated as

$$ \begin{array}{@{}rcl@{}} (v_{\textsc{e}}^{\ast}, v_{\textsc{i}}^{\ast} ) &=& (12.6326, 13.319),\\ (i_{\textsc{e} \textsc{e}}^{\ast}, i_{\textsc{e} \textsc{i}}^{\ast} , i_{\textsc{i} \textsc{e}}^{\ast} ,i_{\textsc{i} \textsc{i}}^{\ast} ) &=& (49.0506, 28.3164, 11.4371, 4.1846), \\ (w_{\textsc{e} \textsc{e}}^{\ast}, w_{\textsc{e} \textsc{i}}^{\ast}) &=& (2245.7, 2057.1). \end{array} $$

The initial values of the variables are assumed to be constant over Ω. The initial value of ve is set as \(v_{\textsc {e}_{0}} = 1.2 v_{\textsc {e}}^{\ast }\), and the rest of variables are initially set at their equilibrium given above.

The solutions of the model are computed using the finite-element-based software COMSOL Multiphysics®; version 5.3a. As stated in Section 2, the boundary condition of the problem is set to be periodic. A triangular mesh is generated with maximum element size of 0.2 mm, which results in 387,436 domain elements and 1200 boundary elements. Implicit backward difference formula (BDF) is chosen as the time stepping method used by the solver, with order of accuracy equal to 2. Time steps taken by solver are set to be manual, and equal to 0.1 ms. Computations take about three days to complete, using a mini workstation with a 3.30 GHz quad-core Intel®; Xeon®; processor, 32 GB of 2133 MHz RAM, and sufficient amount of solid state storage space. The solver uses up to 30 GB of physical memory and up to 46 GB of virtual memory.

5.1 Spatio-temporal pattern of gamma oscillations

The computation results are shown in Fig. 9, in which spatial propagation of the emerged gamma oscillations are shown at every 100 ms. Measurements of ve at the center of regions A and B are also shown in the upper panel of Figs. 10 and 11. As predicted by the results of Section 3, modulation of Fi induces gamma oscillations in both regions, which gradually emerge and become prominent approximately at t = 400 ms. This gradual emergence of gamma oscillations, predicted by the results of Section 3 as the time required by the orbits to converge to the stable limit cycle, is usually observed during experimental studies (Ray and Maunsell 2015). Moreover, as discussed in Section 4, the emerged oscillations in region B do not vanish after Fi returns to its nominal value at \(t_{\textsc {i}_{2}} = 700\) ms. However, as predicted by the mechanism described in Section 4, gamma oscillations in region A gradually vanish after the level of excitation is substantially reduced by decreasing Fe at \(t_{\textsc {e}_{1}} = 500\) ms. As a result, transient gamma oscillations are observed in the measurements of region A for about three cycles and the initial resting-state is approximately restored at the end of the simulation.

Fig. 10
figure 10

Measurements of ve in region A. Dashed lines indicate the times of transitions in Fi, and dotted lines indicate the times of transitions in Fe. The reference of electric potential for ve is the mean resting soma membrane potential \(\protect v_{\textsc {e}_{\text {rest}}} = -72.293 \text { mV}\). Upper panel: point-measurements at the center of the region. Lower panel: average measurements over a disk of diameter 10 mm co-centered with the region

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Fig. 11
figure 11

Measurements of ve in region B. The same description as given in Fig. 10 holds for dashed and dotted lines, as well as measurements of upper and lower panels

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Consistent with the spatial locality property of gamma oscillations stated in Section 1, the gamma oscillations appear locally within the regions where modulatory actions are applied and do not propagate to other regions of the neocortex. This is not an obvious observation, since the cortical activity inside the regions of gain modulation is also transmitted to other regions through the corticocortical communications modeled by the telegraph equations in (1). Although a rigorous mathematical proof of the existence of this localized behavior can be very involved, the intuition coming from the analysis of equilibrium sets of the model (Shirani et al. 2017, sect. 7.2) implies that the coupling between the telegraph equations and the rest of the equations of the model is not very strong. As a result, the activity propagating from the regions of gain modulation to the rest of the neocortex is not sufficiently strong to induce phase transitions in cortical networks of other regions, and hence it cannot effectively engage them in gamma oscillations. This intuition also predicts that the model can present a rich variety of localized oscillatory or non-oscillatory behavior.

Note that, a ring of oscillations in region A is still visible in Fig. 9 up to the end of the simulation. However, it can be seen in the lower panel of Fig. 10 that these oscillations are in beta band. Although the analysis of the ODE system in Section 4 is not sufficient to predict the full spatio-temporal behavior of the solutions of (1), it can provide meaningful explanation for the existence of this ring of beta oscillations. Since ϕ(x) decreases radially and smoothly as x approaches the boundary of the disk, the modulatory action on Fe is not as effectively applied at peripheral regions of the disk as it is applied at the central regions. Therefore, the curves of limit cycles and their frequency shown in Figs. 6a and 7a for low values of Fe imply that the dynamics of the system on the persistent ring of oscillations in region A may still be trapped on stable cycles of beta frequency and lower ve-amplitude. Alternatively, the slowly decreasing amplitude of these beta oscillations can imply that the stable oscillations are indeed terminated in this ring-shaped region as well, but the resulting damped oscillations simply require more time to vanish, possibly due to weak damping in the dynamics of the system for the specific values that Fe takes over this region. Therefore, it is expected that these oscillations will also fade away if an stronger modulation of Fe to a lower value \(\tilde {\mathrm {F}}_{\textsc {e}}\) is applied, ϕ(x) is reshaped to present a sharper transition from 1 to 0, and the simulation time horizon is extended. However, these modifications are not carried out for the results presented here, to avoid excessive computational cost and to additionally illustrate part of the rich spatio-temporal behavior of the model predicted by the analytical results established by Shirani et al. (2017).

5.2 Effect of electrode size and electrode location on measurements of gamma oscillations

As shown in the upper panel of Figs. 10 and 11, the gamma oscillations measured at the center of regions A and B have relatively high amplitude. This may sound reasonable since, as stated in Section 1, gamma oscillations usually present engagement of cortical networks in cognitive activity and hence higher levels of neuronal excitation. However, compared with the high-amplitude oscillations of low frequency that appear over a large region of neocortex during deep sleep, electrocortical measurements of gamma oscillations usually have low amplitude (Ray and Maunsell 2015; Buzsáki and Wang 2012; Jia and Kohn 2011; Ni et al. 2016). A possible explanation for this perhaps unexpected observation can be provided by inspecting the propagation pattern of the oscillations in Fig. 9 and including the effect of electrode size on measurements. Note that the signals shown in the upper panel of Figs. 10 and 11 are point-measurements at the center of regions A and B. However, EEG electrodes are, typically, disks of 5 − 10 mm in diameter. Therefore, they collectively measure the activity of many adjacent cortical networks distributed over the measurement range of the electrode. To approximately incorporate the effect of this collective measurement, the average value of ve over disks of diameter 10 mm co-centered with regions A and B are shown in the lower panel of Figs. 10 and 11. It is observed that, while the frequency of gamma oscillations remains unchanged, their amplitude is drastically reduced in the average measurements. This is due to the propagation pattern of the oscillations within the range of electrodes. The spatial pattern of oscillations in region B in Fig. 9 shows that the gamma oscillations emerging at the center of the region propagate radially to peripheral areas forming spatial waves of very short length. An electrode of diameter 10 mm collectively measures the electrical activity over a number of these waves of oscillations. Consequently, the effect of oscillatory activity of a local network on overall measurement of the electrode can effectively be canceled by the anti-phase oscillations of an adjacent network, so that the amplitude of the overall measurement is substantially reduced.

The spatial locality of gamma oscillations makes the location of electrodes important. To observe this, the 10 mm electrode centered at region B is relocated from the center to the point c = (38,45), so that it overlaps both with the regions where gamma oscillations emerge and with the regions where oscillations remain in beta band. Point-measurements at c and average measurements of the electrode are shown in Fig. 12. Comparing the average measurements before and after relocating the electrode shows that a relatively wide spectrum of low power oscillations, even in high gamma band, are introduced into the measurements as a result of dislocation of the electrode from the center of propagating gamma oscillations. In particular, comparing the two panels of Fig. 12 with each other identifies a time interval over which the waves of gamma oscillations have not reached the center of the electrode but are sufficiently overlapped by the electrode so that they effectively introduce ripples of gamma frequency into the measurements. This interval is marked by a red square in Fig. 12. The measurements over this interval present a ripple of gamma frequency superimposed on a cycle of beta oscillations. This can be misinterpreted as an implication of correlation between beta and gamma oscillations, while it is in fact nothing but the result of collective measurements from different networks, some of which oscillating at beta frequency and others at gamma frequency.

Fig. 12
figure 12

Measurements of ve associated with an electrode which is dislocated from the center of region B. The reference of electric potential for ve is the mean resting soma membrane potential \(\protect v_{\textsc {e}_{\text {rest}}} = -72.293 \text { mV}\). Note the different scales used for the vertical axis of the panels. Upper panel: point-measurements at c = (38,45). Lower panel: average measurements over a disk of diameter 10 mm centered at c = (38,45)

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6 Discussion and conclusion

Using a spatio-temporal model of electrocortical activity, the results presented above explain plausible mechanisms for robust induction of transient gamma oscillations by coordinated modulation of the responsiveness of neuronal populations. The results confirm the locality of gamma oscillations and demonstrate possible patterns of spatial propagation of these oscillations across local regions of the neocortex. Moreover, they predict potential impacts of the size and location of electrocortical measurement electrodes on the amplitude, temporal pattern, and frequency spectrum of the oscillation measurements.

Specifically, the results of Sections 3 and 4 show that sufficient decrease in sensitivity (gain) and excitability (response threshold) of inhibitory populations, as well as sufficient increase in their saliency, can robustly induce gamma oscillations within the regions where the modulatory actions are applied. Similarly, gamma oscillations can also be induced by effectively increasing the sensitivity and excitability of excitatory populations as well as decreasing their saliency. In all these cases, the gamma oscillations emerge as stable limit cycles when the dynamics of the cortical networks presented by the model undergoes a Hopf bifurcation. Regardless of the type of modulations used to induce the gamma oscillation, either excitatory or inhibitory, a subsequent modulation of excitatory population for reduction in the level of excitation is necessary and sufficient for termination of the emerged oscillations and transition to low beta oscillations. This is consistent with the observations available in the literature that identify the excitatory activity of pyramidal cells as the driving force of the oscillations (Buzsáki and Wang 2012; Kopell et al. 2000; Cardin et al. 2009).

The results of this paper further emphasize the importance of developing detailed mechanistic models to unveil the actual sequence of causal relations leading to the emergence and termination of gamma oscillations. Electrophysiological, behavioral, and optogenetic studies of cortical networks have shown that visually and optogenetically induced gamma oscillations modulate neuronal gain (Ni et al. 2016; Cardin et al. 2009). In the seemingly opposite causal direction, however, sensory-driven barrages of synaptic inputs are shown to modulate neuronal gain (Cardin et al. 2008; Haider and McCormick 2009), which consequently can induce gamma oscillations according to the analysis of Section 3. Moreover, termination of the emerged gamma oscillations and their transition to beta oscillations—most likely through substantial reduction in the level of network excitation—is predicted by the results of Section 4 to require specific modulation of excitatory populations. The mechanism of this subsequent modulation and its direct cellular or network causes require further investigation. In particular, when gamma oscillations are initially induced by modulation of inhibitory populations, the analysis of Section 4 shows that this subsequent modulation of excitatory populations is in the direction of restoring the balance of excitation and inhibition in the network. Therefore, it can be postulated that the modulatory actions that result in transition to beta oscillations are the reaction of the cortical network to the imbalance created by the initial, possibly stimulus-driven, modulation. As a result, long-lasting gamma oscillations in the absence of effective neuromodulators or streams of synaptic activity may be assumed as the failure of the network in re-establishing its balance, possibly due to a neuronal disorder.

Propagating waves of gamma oscillations, as demonstrated in Section 5, can be of very short length compared with the diameter of a typical EEG electrode. As a result, the collective measurements of these waves of activity by an electrode shows significantly lower amplitude oscillations compared with point-measurements. Low amplitude of gamma oscillations has led to interpretations that gamma oscillations are nothing but fluctuations at a resonance frequency of cortical networks (Jia and Kohn 2011). The results presented here, however, do not support such interpretations. The gamma oscillations in the solutions of the model emerge as a result of a phase transition in the dynamics of local cortical networks, which is robustly evoked by systematic modulation of neuronal responsiveness.

Due to the locality and propagation pattern of gamma oscillations, the location of measurement electrodes relative to the regions of gamma activity can significantly affect the temporal pattern of measurements so that ripples in broad range of frequency are introduced—even for the simple flat geometry considered for the neocortex in the computational results of Section 5. In particular, as shown in Section 5.2, measurements of an electrode which overlaps with both regions of beta and gamma oscillations can potentially result in misinterpretations about the network dynamics of gamma and beta rhythmic activity, possible correlations between these oscillations, and mechanisms of frequency shifts from gamma to beta in cortical oscillations (Traub et al. 1999; Kopell et al. 2000). Taking further into consideration that the neocortex has a convoluted geometry, these results recommend that the interpretations made solely based on the observation of specific temporal patterns in EEG recordings should be reconsidered, and possibly be validated by using other electrocortical measurement techniques, before being used to establish physiological facts. Moreover, the results discussed in Section 5.2 also demonstrate the limitations of commonly used reduced-network models to fully describe the patterns of rhythmic activity in EEG recordings. In fact, not only cannot such systems of ordinary differential equations explain spatial propagation of oscillatory waves, but also they cannot completely predict the temporal patterns of electrode measurements. Besides measurement noise and other unknown sources of perturbations, the amplitude and pattern of these temporal measurements can be highly affected by the activity of nearby networks and complicated geometry of the neocortex. Such impacts are more significant on measurements of fast oscillations which arise locally and develop waves of very short length.

The numerical analysis of Sections 3 and 4 are based on a single set of biophysically plausible parameter values given in Table 2. An automated search performed by Bojak and Liley (2005) using a stability and spectral analysis of the linearized version of the model has resulted in 73,454 sets of plausible parameter values. However, further bifurcation analysis performed by Frascoli et al. (2011) on a 405 randomly selected subset of these 73,454 parameter sets has identified only two different families of topological structures in the dynamics of the model resulting from these different parameter sets. Moreover, these two families of dynamic structures are shown to be transformable from one family into the other by varying excitatory subcortical inputs. The bifurcation analysis presented here in Sections 3 and 4 involves numerous manual adjustments of continuation parameters in MatCont; see the Appendix. Therefore, an automated search within a large subset of the available 73,454 parameter sets, aimed to identify the parameter values that result in similar qualitative behaviors as shown in Sections 3 and 4, requires specifically developed numerical analysis tools and can be a topic of future research. Instead, the bifurcation analysis of Sections 3 and 4 is repeated here for several other parameter sets, which are generated by randomly perturbing the values given in Table 2 by a magnitude of 10 to 100 percent. The results obtained for a sample of such sets are given in the Appendix. Specifically, it is observed that the analysis is very robust against changes in the membrane time constant of the neurons, whereas it is relatively more sensitive to changes in postsynaptic potential rate constants.

Table 3 A set of parameter values generated by perturbations of magnitude 10 to 100 percent in the values given in Table 2. This set of values is used as nominal parameter values for the computational results presented in the Appendix
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Finally, a note on the numerical computation of the solutions of the model (1) can be of interest for future studies using this model. As the computations of Section 5 proceed in time, the propagating waves in some components of the solutions become exceedingly fine. Although the discretization mesh used for the computation is extremely fine, it still cannot result in completely accurate solutions of these waves. As a result, some irregularities can be observed in the shape of waves in Fig. 9 when the simulation approaches its end. Therefore, more accurate solutions over a longer time horizon requires significantly finer mesh which substantially increases the computational cost of the simulation. These irregularities are observed in the solution components v = (ve,vi) and i = (iee,iei,iie,iii), whereas the solution component w = (wee,wei) does not develop waves of very short length and evolves smoothly in time and space. These observations are confirmed by the analytical results developed by Shirani et al. (2017), which particularly predict that v and i components of solutions can develop drastic asymptotic discontinuities in space, regardless of the smoothness of initial values and subcortical forcing terms.