Calcium Waves and Sparks

Abstract

This chapter gives a comprehensive review of calcium wave modeling, with an emphasis on their role in neuronal calcium signaling. Two models of intracellular waves are considered in some detail: a reaction–diffusion model of calcium dynamics and the fire–diffuse–fire model of calcium release. The latter is formally very similar to the spike–diffuse–spike model of spiny dendrites and is analyzed accordingly. Stochastic models of spontaneous calcium release (calcium puffs and sparks) are then analyzed using the stochastic methods introduced in the first chapter. Finally, several models of intercellular calcium waves in astrocytes are presented. Traditionally, astrocytes were thought to be physiologically passive cells that only play a supporting role in the central nervous system by regulating and optimizing the environment within which neurons operate. However, there is an increasing amount of empirical data indicating that astrocytes play a more active role in modulating synaptic transmission and neuronal signal processing.

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Calcium (Ca²⁺) is one of the most important and well-studied cellular signaling molecules. From a modeling perspective, it attracts a great deal of interest due to the fact that calcium signaling often involves complex spatiotemporal dynamics, including oscillations and waves. There are a number of excellent general reviews on the modeling of calcium dynamics within cells, including Chap. 7 of Keener and Sneyd [322] and Falcke [181]. In this chapter, we focus on the mathematical modeling and analysis of calcium waves and their potential significance in neuronal calcium signaling; see also the reviews by Berridge [43] and Ross [534]. Although there are a number of very detailed whole-cell models of calcium signaling, we will focus on simplified models that are analytically tractable. In particular, we will highlight some of the mathematical methods used to characterize calcium waves.

1 Calcium Signaling in Neurons

In vertebrates, most of the Ca²⁺ is stored in bones, from where it can be released by hormonal stimulation to maintain a high extracellular Ca²⁺ concentration (around 1 mM). On the other hand, active ion pumps and exchangers maintain the cytoplasmic Ca²⁺ concentration at relatively low levels (around 10–100 nM). The resulting steep concentration gradient across the plasma membrane means that cells are able to increase their cytoplasmic Ca²⁺ concentration rapidly by opening either voltage-gated or ligand-gated Ca²⁺ ion channels. Within the context of neuronal processing, the activation of voltage-gated calcium channels (VGCCs) at axon terminals triggers the release of synaptic neurotransmitter (see Sect. 1.3); VGCCs also contribute to the generation of dendritic action potentials (see Sects. 1.4.3 and 3.1). A classical example of ligand gating in neurons is the entry of Ca²⁺ through NMDA receptors on postsynaptic dendritic spines. The resulting transient Ca²⁺ signal is thought to play a major role in the induction of changes in synaptic strength; see below.

Another major mechanism whereby cells, including neurons, regulate their cytoplasmic Ca²⁺ concentration is via the intracellular supply of Ca²⁺ from internal stores such as the endoplasmic reticulum (ER) and mitochondria. Inositol (1,4,5)-trisphosphate (IP₃) receptors and Ryanodine (Ry)

Fig. 4.1

Neural calcium signaling. The entry of Ca²⁺ from outside the cell is mediated by voltage-gated (VGCC) and ligand-gated (LGCC) calcium channels. Stimulation of metabotropic glutamate receptors (mGluRs) produces IP₃ second messengers that bind to IP₃ receptors (IP₃Rs), which subsequently release Ca²⁺ from the endoplasmic reticulum. Both IP₃Rs and Ryanodine receptors (RyRs) are sensitive to Ca²⁺, resulting in Ca²⁺ -induced Ca²⁺ -release (CICR). The latter can sometimes result in the propagation of a Ca²⁺ wave along the dendrites

receptors distributed throughout the ER, for example, mediate the release of Ca²⁺ into the cytoplasm, whereas Ca²⁺ ion pumps maintain the relatively high Ca²⁺ concentration within the ER. The Ry receptor plays a critical role in excitation–contraction coupling in skeletal and cardiac muscle cells, but is also found in nonmuscle cells such as neurons. One important feature of Ry receptors is that they can undergo Ca²⁺-induced Ca²⁺ -release (CICR), in which elevated cytoplasmic Ca²⁺ activates Ry receptors that release further Ca²⁺, which then activates other Ry receptors, resulting in a nonlinear regenerative feedback mechanism. The IP₃ receptor is similar in structure to the Ry receptor, but is found predominantly in nonmuscle cells and is sensitive to the second messenger IP₃. The binding of an extracellular ligand such as a hormone or a neurotransmitter to a metabotropic receptor results in the activation of a G-protein and the subsequent activation of phospholipase C (PLC). This then cleaves phosphatidylinositol bisphosphate (PIP₂) into diacylglycerol (DAG) and IP₃. The water soluble IP₃ is free to diffuse throughout the cell cytoplasm and bind to IP₃ receptors located on the ER membrane, which then open and release Ca²⁺ from the ER. The opening and closing of an IP₃ receptor is also modulated by the concentration of cytoplasmic Ca²⁺, so it too can undergo CICR. Yet another mechanism for controlling cytoplasmic Ca²⁺ is through buffering (binding) to large proteins. Indeed, it is estimated that at least 99 % of the total cytoplasmic Ca²⁺ is bound to buffers. A summary of the basic extracellular and intracellular mechanisms for controlling the concentration of cytoplasmic Ca²⁺ is shown in Fig. 4.1.

Fig. 4.2

The endoplasmic reticulum (ER) is a single and continuous membrane-bound organelle that is distributed throughout the dendrites and axon (not shown) of a neuron [516]. It is responsible for the synthesis and posttranslational modification of most secretory and membrane proteins, as well as the regulation of Ca²⁺ levels. The shape of the ER is heterogeneous, but can be divided into three domains: the nuclear envelope, the ribosome-bound rough ER (RER), and the ribosome-free smooth ER (SER). The RER is constituted primarily by sheets or cisternae, whereas the SER is predominantly composed of 60–100 nm diameter tubules that form irregular polygons with a common luminal space connected via three-way junctions. The ER present in the soma and proximal dendritic compartment is rich in ribosomes, corresponding to the RER, whereas the ER distributed in distal dendrites and axons corresponds mostly to SER and contains only a few sparse ribosomes. The ER comes into close proximity with the plasma membrane at various locations, forming subsurface cisternae within the soma and axon initial segment, and the spine apparatus within dendritic spines

The signaling mechanisms and dynamics of Ca²⁺ release from the ER in neurons have been much less well studied than extracellular sources of Ca²⁺. However, the ER is a continuous network that is distributed throughout a neuron, extending all the way into the dendrites and axon and coming into close contact with the plasma membrane and synapses at many locations; see Fig. 4.2. Hence, as suggested by Berridge [43], it is likely that the integrative and regenerative properties of the ER and plasma membrane provide a binary membrane system that regulates a variety of neuronal process via Ca²⁺ signaling, including excitability, neurotransmitter release, synaptic plasticity, and gene transcription.

• (a) Within the soma and initial axonal segment, the ER forms flattened sheets known as subsurface cisternae that come into close contact with the plasma membrane. These internal structures are thought to play an important role in regulating neuronal excitability. For example, many neurons exhibit significant after-hyperpolarizations (AHPs) following either a single action potential or a burst of action potentials, which suppress subsequent firing of the cell (spike frequency adaptation). Often AHPs have both a fast component and a slow component, which arise from the opening of two separate Ca²⁺-activated K⁺ channels (gK_Ca1 and gK_Ca2). The fast activation of gK_Ca1 is due to the entry of extracellular Ca²⁺ via VCCNs during the course of an action potential, whereas the slow activation of gK_Ca2 is probably due to CICR from Ry receptors located in the subsurface cisternae.

• (b) Within the axon, the ER consists of connecting tubules that run parallel along the axon. The axonal ER network extends into a synapse, coming into close contact with the plasma membrane and associated neurotransmitter secretory vesicles. It is thought that calcium release from Ry and IP₃ receptors contributes to the formation of localized high-concentration bursts of Ca²⁺ necessary for the exocytosis (transport and fusion) of vesicles to the plasma membrane.

• (c) The ER network is also distributed throughout the dendritic tree and into spines, where it terminates at the so-called spine apparatus. The latter consists of closely packed plates of ER that are located within the middle of the spine and is thus well placed to contribute to Ca²⁺ signaling during synaptic plasticity. The latter refers to the rapid and long-lasting changes in synaptic strength that are thought to provide the subcellular substrate for learning and memory. Much of the information regarding synaptic plasticity in mammalian neurons has been obtained from studies of hippocampal and cerebellar neurons [120, 123, 385, 402]. In particular, it has been found that the same synapses can be switched rapidly and reversibly from an enhanced state of long-term potentiation (LTP) to a reduced state of long-term depression (LTD). Interestingly, both LTP and LTD are induced by a transient increase in the local Ca²⁺ concentration within a spine. It was originally thought that the bidirectional switch only depended on the amplitude of the Ca²⁺ signal, with a stronger stimulus inducing LTP and a weaker one LTD. Now, however, it appears likely that the detailed spatial and temporal structure of the calcium signal may be important. Although the influx of extracellular Ca²⁺ through NMDA receptors or voltage-gated channels is still the major contributor to the induction of LTP and LTD, there is increasing evidence that there is also a component arising from intracellular stores [428, 447, 534]. Activation of the ER could be mediated either by Ca²⁺ itself via CICR or by stimulation of metabotropic glutamate (mGlu) receptors in the plasma membrane, resulting in the production of IP₃. The fact that CICR requires both cytoplasmic IP₃ and Ca²⁺ suggests that IP₃ receptors could act as coincidence detectors for mGluR activation and postsynaptic Ca²⁺ signaling arising from back-propagating action potentials or dendritic Ca²⁺ spikes [658].

• (d) One of the most dramatic consequences of CICR is the propagation of intracellular Ca²⁺ waves mediated primarily by the opening of IP₃ receptors. These waves were first observed in nonneuronal cells such as Xenopus laevis oocytes [367, 490], where the resulting changes in Ca²⁺ concentration across the whole cell provided a developmental signal. The discovery of Ca²⁺ waves in neurons is more recent [303, 428, 447, 534, 658]. Since these cells have extensive dendritic and axonal arborizations, the distribution of the components of the Ca²⁺ signaling apparatus within the cell plays a crucial role in determining whether or not local Ca²⁺ release transforms into a propagating wave, and how far it travels. For example, in pyramidal neurons of the hippocampus and cortex, Ca²⁺ waves are usually observed in the primary apical dendrite and perhaps the soma, rarely reaching beyond the point where thick dendrites begin to branch. Since the ER network is distributed throughout the cell, this suggests that there is a heterogeneous distribution of mGlu receptors and IP₃ receptors along the dendrite. Ca²⁺ waves rarely propagate from the dendrite to the soma, unless there is a sufficiently strong and sustained stimulation of a neuron that results in high levels of cytoplasmic IP₃ or higher concentrations of internally stored Ca²⁺. If a Ca²⁺ wave did reach the soma, then it would generate a strong Ca²⁺ signal in the soma and nucleus due to the high concentration of IP₃ receptors in these regions. It has been suggested that a Ca²⁺ wave could allow a strongly stimulated synapse in the dendrites to signal to the nucleus, where the large increase in Ca²⁺ concentration could activate gene transcription, which is a necessary step in more persistent forms of synaptic plasticity such as late long-term potentiation (L-LTP). However, the role of Ca²⁺ waves in initiating gene transcription is still controversial. On the other hand, the likely contribution of IP₃ receptor-mediated Ca²⁺ release during normal LTP and LTD indicates that Ca²⁺ waves could be important in determining the spatiotemporal pattern of synaptic changes.

• (e) Many nonneuronal cell types exhibit spontaneous localized Ca²⁺ release events known as sparks [110]. Calcium sparks, which are thought to be the building blocks of the large regenerative Ca²⁺ signal that controls contraction in cardiac and skeletal muscle cells, arise from the opening of clusters of RyRs by local CICR. The frequency of calcium spark events is sensitive to changes in membrane potential, although they rarely induce calcium waves [111]. Similar events known as calcium puffs have been found in Xenopus laevis oocytes [489, 490, 690]. They are also spatially and temporally localized, are fast, and occur stochastically. However, in contrast to sparks, they are mediated by IP₃Rs and coalesce more easily to form calcium waves. Localized events in neurons with spark-like and puff-like properties have recently been observed in slice preparations, and their frequency can be modulated by synaptic activity and changes in membrane potential [389, 403]. Although it is likely that they also occur in vivo, it is not yet clear whether or not they have a specific function.

2 Reaction–Diffusion Models of Ca²⁺ Dynamics

One of the major challenges in modeling calcium waves is that the ER and cytoplasm comprise two highly interconnected three-dimensional domains (ignoring, for simplicity, other intracellular compartments such as the mitochondria). Assuming that Ca²⁺ undergoes normal diffusion within each compartment, together with buffering, we can write down the following pair of diffusion equations for the Ca²⁺ concentrations c, c _e in the cytoplasm and ER [322]:

$$\displaystyle{ \frac{\partial c} {\partial t} = \nabla \cdot (D_{c}\nabla c) - J_{\mathrm{on}} + J_{\mathrm{off}},\quad \mathbf{r} \in \varOmega _{c} }$$

(4.1)

and

$$\displaystyle{ \frac{\partial c_{e}} {\partial t} = \nabla \cdot (D_{e}\nabla c_{e}) -\hat{ J}_{\mathrm{on}} +\hat{ J}_{\mathrm{off}},\quad \mathbf{r} \in \varOmega _{e}, }$$

(4.2)

where Ω _c and Ω _e denote the cytoplasmic and ER domains, respectively, D _c and D _e are the corresponding diffusivities in the two domains, and the on/off fluxes take into account the binding/unbinding of Ca²⁺ to buffering proteins. These equations are supplemented by various boundary conditions that take into account the exchange of Ca²⁺ between the cytoplasm and the plasma membrane or ER. Let J _in denote the total inward flux of Ca²⁺ from the plasma membrane to the cytoplasm via voltage-gated and ligand-gated ion channels, and let J _p1 be the outward flux due to ATPase ion pumps in the plasma membrane. Then

$$\displaystyle{ D_{c}\nabla c \cdot \mathbf{n} = J_{\mathrm{in}} - J_{\mathrm{p1}},\quad \mathbf{r} \in \partial \varOmega _{c,m}, }$$

(4.3)

where ∂ Ω _c, m is the surface of the plasma membrane with unit normal n. Similarly, let J _IP and J _Ry denote the inward fluxes from the ER to the cytoplasm via IP₃Rs and RyRs, respectively, and let J _p2 denote the outward flux due to ion pumps in the ER. This yields

$$\displaystyle{ D_{c}\nabla c \cdot \hat{\mathbf{n}} = -D_{e}\nabla c_{e} \cdot \hat{\mathbf{n}} = J_{\mathrm{IP}} + J_{\mathrm{Ry}} - J_{\mathrm{p2}},\quad \mathbf{r} \in \partial \varOmega _{e}, }$$

(4.4)

where ∂ Ω _e is the boundary of the ER with unit normal $\hat{\mathbf{n}}$. A schematic diagram of all the fluxes is shown in Fig. 4.3.

Let us first consider a space-clamped version of this model, in which one treats the cytoplasm and ER as two well-mixed homogeneous compartments. In that case, the various surface and volume fluxes are combined as follows:

$$\displaystyle{ \frac{dc} {dt} = -J_{\mathrm{on}} + J_{\mathrm{off}} + \frac{\vert \partial \varOmega _{c,m}\vert } {\vert \varOmega _{c}\vert } [J_{\mathrm{in}} - J_{\mathrm{p1}}] + \frac{\vert \partial \varOmega _{e}\vert } {\vert \varOmega _{c}\vert } [J_{\mathrm{IP}} + J_{\mathrm{Ry}} - J_{\mathrm{p2}}] }$$

(4.5)

and

$$\displaystyle{ \frac{dc_{e}} {dt} = -\hat{J}_{\mathrm{on}} +\hat{ J}_{\mathrm{off}} -\frac{\vert \partial \varOmega _{e}\vert } {\vert \varOmega _{e}\vert } [J_{\mathrm{IP}} + J_{\mathrm{Ry}} - J_{\mathrm{p2}}]. }$$

(4.6)

Each of the fluxes in these equations corresponds to a component of the so-called Ca²⁺-signaling toolkit [44]. Thus, the equations for Ca²⁺ dynamics have to be coupled to additional biochemical equations describing the dynamics of ion pumps, the Ca²⁺-dependent gating of the Ry and IP₃ receptors, and how the latter are regulated by the second messenger IP₃. Often the external flux J _in is simply taken to be a linear increasing function of the IP₃ concentration. There have been many studies of space-clamped Ca²⁺ models with particular focus on calcium oscillations [181, 322].

Fig. 4.3

Schematic diagram showing various fluxes in reaction–diffusion model of Ca²⁺ dynamics

Although the models differ in their degree of complexity with regards the component fluxes, they can often be reduced to simpler models that have similar structure to the FitzHugh–Nagumo (FN) model of excitability. For example, suppose that we ignore fluxes through the plasma membrane so that the total intracellular Ca²⁺ is conserved, that is, $\vert \varOmega _{c}\vert c + \vert \varOmega _{e}\vert c_{e}$ is a constant. Furthermore, suppose that there are only two intracellular fluxes, J _IP and J _p2, and the IP₃ concentration is fixed. Ignoring the effects of buffering, one can write down an analogous equation to (2.3) of the form

$$\displaystyle{ \frac{dc} {dt} = f(c,h),\quad \tau (c)\frac{dh} {dt} = h_{\infty }(c) - h, }$$

(4.7)

where c is the concentration of free cytosolic Ca²⁺, f(c, h) represents the net flux into the cytoplasm from the ER, and h is an inactivation variable that plays a similar role to the recovery variable w of the original FN equations. One interpretation of h is the fraction of IP₃ receptors that are not inactivated by bound Ca²⁺. Li and Rinzel [381] derived such a two-variable model by reducing the classical kinetic model of Ca²⁺-gated IP₃ receptors due to De Young and Keizer [692]; see Sect. 4.4.1.

Unfortunately, the complex geometry and heterogeneity of the ER means that the spatially distributed model given by (4.1)–(4.4) is far too complicated to analyze, even when simplified models of the various fluxes are considered. A common strategy is to ignore the detailed structure of the ER by assuming that the concentrations c and c _e coexist at every point in space. This is motivated by the idea that since diffusion is fast over short distances, local variations due to heterogeneities are smoothed out. Phenomenologically speaking, this leads to the calcium bidomain equations

$$\displaystyle{ \frac{\partial c} {\partial t} = \nabla \cdot (\overline{D}_{c}\nabla c) - J_{\mathrm{on}} + J_{\mathrm{off}} +\chi _{c}[J_{\mathrm{IP}} + J_{\mathrm{Ry}} - J_{\mathrm{p2}}] }$$

(4.8)

and

$$\displaystyle{ \frac{\partial c_{e}} {\partial t} = \nabla \cdot (\overline{D}_{e}\nabla c_{e}) -\hat{ J}_{\mathrm{on}} +\hat{ J}_{\mathrm{off}} -\chi _{e}[J_{\mathrm{IP}} + J_{\mathrm{Ry}} - J_{\mathrm{p2}}]. }$$

(4.9)

Here $\chi _{c} = \vert \partial \varOmega _{e}\vert /\vert \varOmega _{c}\vert,\chi _{e} = \vert \partial \varOmega _{e}\vert /\vert \varOmega _{e}\vert$ are surface to volume ratios, and $\overline{D}_{c},\overline{D}_{e}$ are effective diffusion coefficients. Note, however, that in order to determine the effective diffusion coefficients, which will depend on the geometry of the ER, and justify the precise form of the factors $\chi _{c},\chi _{e}$, it is necessary to derive these equations from first principles using homogenization theory [224, 322]. Elements of the theory are described in appendix section 4.6. Now suppose that only the fluxes J _IP and J _p2 are included along the lines of the space-clamped model. However, in contrast to the latter, it is now necessary to keep track of both the cytoplasmic and ER Ca²⁺ concentrations, since they are spatially varying. A one-dimensional bidomain model with constant diffusion coefficients then takes the form

$$\displaystyle\begin{array}{rcl} \frac{\partial c} {\partial t}& = D_{c} \frac{{\partial }^{2}c} {\partial {x}^{2}} + f(c,c_{e},h)&{}\end{array}$$

(4.10a)

$$\displaystyle\begin{array}{rcl} \frac{\partial c_{e}} {\partial t} & = D_{e} \frac{{\partial }^{2}c} {\partial {x}^{2}} -\chi _{e}f(c,c_{e},h)&{}\end{array}$$

(4.10b)

$$\displaystyle\begin{array}{rcl} \tau (c)\frac{\partial h} {\partial t} & = h_{\infty }(c) - h,&{}\end{array}$$

(4.10c)

where f(c, c _e , h) represents the net Ca²⁺ flux from the ER to the cytoplasm the spatial dimensions have been fixed such that χ _c = 1. Analysis of traveling wave solutions of (4.10) proceeds along very similar lines to that of the Hodgkin–Huxley equations for membrane voltage excitability; see Chap. 2. Depending on parameter values, the model system can support solitary waves when the physiological state of the cytoplasm is excitable, periodic waves when it is oscillatory, and traveling fronts when it is bistable [322].

However, one important feature that needs to be taken into account in the case of calcium excitability is buffering. Insights into the effects of buffering can be obtained by looking at reduced models of excitability such as the bistable equation or the FN equations [592, 635, 636, 653]. Following [592, 653], consider a simple extension of the bistable equation (Sect. 2.2). First, note that the basic chemical reaction for Ca²⁺ buffering takes the form

$$\displaystyle{P +{ \text{Ca}}^{2+}\mathop{\mathop{ \rightleftharpoons }\limits^{ k_{ +}}}\limits_{k_{-}}B,}$$

where P is the buffering protein and B is the protein/Ca²⁺ complex. Let c and b denote, respectively, the concentration of free cytosolic Ca²⁺ and buffered Ca²⁺. The extended bistable equation is then (in one spatial dimension)

$$\displaystyle\begin{array}{rcl} \frac{\partial c} {\partial t}& = D_{c} \frac{{\partial }^{2}c} {\partial {x}^{2}} + f(c) + k_{-}b - k_{+}c(b_{t} - b),&{}\end{array}$$

(4.11a)

$$\displaystyle\begin{array}{rcl} \frac{\partial b} {\partial t}& = D_{b} \frac{{\partial }^{2}b} {\partial {x}^{2}} - k_{-}b + k_{+}c(b_{t} - b).&{}\end{array}$$

(4.11b)

Here b _t is the total buffer concentration, f(c) takes into account all Ca²⁺ fluxes, and the diffusion coefficients of free and buffered Ca²⁺ are assumed to be constants. Suppose that the buffer reaction rates k _± are faster than the time constants of Ca²⁺ dynamics. The buffer concentration b can be taken to be in the quasi steady state $k_{-}b - k_{+}c(b_{t} - b) = 0$, that is,

$$\displaystyle{ b = \frac{b_{t}c} {K + c},\quad K = \frac{k_{-}} {k_{+}}. }$$

(4.12)

Adding (4.11a) and (4.11b) and eliminating b using (4.12) gives [653]

$$\displaystyle\begin{array}{rcl} \frac{\partial c} {\partial t}& =& \frac{1} {1 +\theta (c)}\left (\frac{{\partial }^{2}w(c)} {\partial {x}^{2}} + f(c)\right ) {}\\ & =& \frac{D_{c} + D_{b}\theta (c)} {1 +\theta (c)} \frac{{\partial }^{2}c} {\partial {x}^{2}} - \frac{2D_{b}\theta (c)} {(K + c)(1 +\theta (c))}{\left ( \frac{\partial c} {\partial x}\right )}^{2} + \frac{f(c)} {1 +\theta (c)}, {}\\ \end{array}$$

where

$$\displaystyle{ w(c) = D_{c}c + D_{b}b_{t} \frac{c} {K + c},\quad \theta (c) = \frac{b_{t}K} {{(K + c)}^{2}}. }$$

(4.13)

It is clear that a mobile buffer (D _b > 0) generates a nonlinear advection–diffusion equation for c; the advection term vanishes if the buffer is immobile (D _b = 0).

The form of the effective reaction–diffusion equation for c suggests making then change of variables w = w(c), with w, a monotone increasing function of c since $w^{\prime}(c) = D_{c} + D_{b}\theta (c) > 0$. Denoting the unique inverse of the function w(c) by c =ϕ(w), we can write

$$\displaystyle{ \frac{\partial w} {\partial t} = \frac{D_{c} + D_{b}\varTheta (w)} {1 +\varTheta (w)} \left (\frac{{\partial }^{2}w} {\partial {x}^{2}} + f(\phi (w))\right ), }$$

(4.14)

where Θ(w) = b _t K∕(K +ϕ(w))². Now suppose that f(c) has two stable zeros c _± separated by an unstable zero c ₀ with $c_{-} < c_{0} < c_{+}$. It immediately follows that f(Θ(w)) has three corresponding zeros $w_{\pm },w_{0}$ with $w_{-} < w_{0} < w_{+}$ and w _± stable. A traveling front solution of wave speed v, w(x, t) = W(x − vt), satisfies the ODE

$$\displaystyle{ V _{\xi \xi } + \frac{v} {D_{\text{e}ff}(W)}W_{\xi } + f(\phi (W)). }$$

(4.15)

Proceeding as in Sect. 2.2, multiply both sides of this equation by W _ξ and integrate to show that

$$\displaystyle{v\int _{-\infty }^{\infty } \frac{W_{\xi }^{2}} {D_{\mathrm{eff}}(W(\xi ))}d\xi =\int _{ w_{-}}^{w_{+} }f(\phi (w))dw.}$$

Since the integral on the left-hand side is positive definite, the sign of v is determined by the sign of the integral on right-hand side. In particular, a right-moving wave (v > 0) exists provided that

$$\displaystyle{ \int _{c_{-}}^{c_{+} }\left (D_{c} + D_{b}\theta (c)\right )f(c)dc > 0, }$$

(4.16)

where we have converted back to the original variable c. It can be seen that immobile buffers (D _b = 0) have no effect on the existence of a traveling wave; this result also holds true for wave stability and uniqueness. On the other hand, mobile buffers can eliminate a traveling wave when they are of high affinity (small K ) or have a large diffusion coefficient (large D _b ).

3 Discrete Calcium Sources and the Fire–Diffuse–Fire Model

3.1 Discrete Calcium Sources and Averaging

The above reaction–diffusion models assumed that Ca²⁺ release from the ER is homogeneous in space. This neglects the fact that IP₃ receptors (as well as Ry receptors) tend to be arranged in clusters. For example, in Xenopus oocytes, the density of clusters is around 1 per 30 μm², with each cluster containing about 25 IP₃ receptors [322]. Consequently, the propagation of Ca²⁺ waves tends to be saltatory, jumping from one cluster to the next. (Clustering of IP₃ receptors is also thought to occur along dendrites [534]). We have previously encountered saltatory waves due to discreteness effects, namely, action potential propagation along myelinated axons (Sect. 2.5) and solitary Ca²⁺ spikes propagating along spiny dendrites (Sect. 3.1). The latter waves depend on the action of VGCCs and other voltage-gated ion channels and are thus distinct from the Ca²⁺ waves considered in this chapter, which depend on Ca²⁺ release from internal pools. Nevertheless, the analytical techniques used to study these other examples can be adapted to the present case. First, we will discuss how the averaging method of Sect. 2.5 can be used to study wave propagation failure in the presence of discrete release sites, following [318]. We will then describe an alternative, fire–diffuse–fire (FDF) model of saltatory Ca²⁺ waves, which is identical in structure to the spike–diffuse–spike (SDS) model of Sect. 3.1. (In contrast to other chapters, we will use c to denote cytosolic Ca²⁺ concentration and v to denote wave speed.)

Consider a one-dimensional model of Ca²⁺ diffusion and release of the form [318]

$$\displaystyle{ \frac{\partial c} {\partial t} = -k_{c}c + D_{c} \frac{{\partial }^{2}c} {\partial {x}^{2}} + L\sum _{n}\delta (x - nL)f(c), }$$

(4.17)

where f(c) represents the Ca²⁺-dependent rate of release from each cluster (taken to be identical), and the decay term represents the effects of ion pumps. The clusters are assumed to be evenly spaced with spatial separation L. For simplicity, the ER concentration c _e and IP₃ concentration p are taken to be fixed. It is tempting to apply homogenization theory to this problem (see appendix section 4.6), assuming diffusion is fast on the length scale L, that is, ${L}^{2}k_{c}/D_{c} \ll 1$. Intuitively, one would expect the L-periodic function $g(x) = L\sum _{n}\delta (x - nL)$ to be replaced by its spatial average $\int _{0}^{L}g(x)dx/L = 1$. However, as highlighted by Keener [318, 319] and discussed in Sect. 2.5, the homogenized system cannot account for the fact that discreteness effects can lead to wave propagation failure in regimes that the continuous model would predict traveling waves exist. This reflects the fact that wave solutions are not structurally stable. The occurrence of propagation failure can be established by constructing standing wave-front solutions [318, 322]. The latter are stationary solutions of (4.17). On the intervals nL < x < (n + 1)L, this gives

$$\displaystyle{ 0 = -k_{c}c + D_{c} \frac{{\partial }^{2}c} {\partial {x}^{2}}, }$$

(4.18)

which is supplemented by jump conditions at x = nL,

$$\displaystyle{ \left.D_{c} \frac{\partial c} {\partial x}\right \vert _{nL_{-}}^{nL_{+} } + Lf(c_{n}) = 0. }$$

(4.19)

The latter is obtained by integrating the stationary equation on either side of x = nL. The general solution of (4.18) is

$$\displaystyle{c(x) = c_{n}\cosh (\beta (x - nL)/L) +\hat{ c}_{n}\sinh (\beta (x - nL)/L),\quad nL < x < (n + 1)L,}$$

where ${\beta }^{2} = k_{c}{L}^{2}/D_{c}$, c _n = c(nL) and $\beta \hat{c}_{n}/L = c^{\prime}(nL_{+})$. Continuity of the solution at x = (n + 1)L shows that

$$\displaystyle{c_{n}\cosh (\beta ) +\hat{ c}_{n}\sinh (\beta ) = c_{n+1},}$$

that is,

$$\displaystyle{\hat{c}_{n} = \frac{c_{n+1} - c_{n}\cosh (\beta )} {\sinh (\beta )}.}$$

It follows that

$$\displaystyle{c^{\prime}(nL_{+}) = (c_{n+1} - c_{n}\cosh (\beta )) \frac{\beta } {L\sinh (\beta )}}$$

and

$$\displaystyle{c^{\prime}(nL_{-}) = -(c_{n-1} - c_{n}\cosh (\beta )) \frac{\beta } {L\sinh (\beta )}.}$$

Substituting these results into the jump condition yields the difference equation

$$\displaystyle{ \frac{k_{s}} {\beta \sinh (\beta )} (c_{n+1} - 2c_{n}\cosh (\beta ) + c_{n-1}) + f(c_{n}) = 0. }$$

(4.20)

It is usually not possible to obtain an explicit solution of a nonlinear difference equation. One exception is if f(c) is piecewise linear,

$$\displaystyle{ f(c) = f_{0}H(c - {c}^{{\ast}}), }$$

(4.21)

where c ^∗ is a release threshold. Without loss of generality, suppose that the standing wave solution crosses threshold from below in the interval 0 < x < L. Try a solution of the form $c_{n} = A{_{-}\mu }^{-n}$ for n ≤ 0 and $c_{n} = C - A{_{+}\mu }^{n}$ for n ≥ 0 with 0 <μ < 1. The constant C is determined by taking the limit n →∞ in (4.20) with $c_{n} \rightarrow C > {c}^{{\ast}}$:

$$\displaystyle{\frac{k_{s}} {\beta \sinh (\beta )} (C - 2C\cosh (\beta ) + C) + f_{0} = 0,}$$

so that

$$\displaystyle{C = \frac{\beta \sinh (\beta )} {2\cosh (\beta ) - 2} \frac{f_{0}} {k_{c}} = \frac{\beta \cosh (\beta /2)} {2\sinh (\beta /2)} \frac{f_{0}} {k_{c}}.}$$

The constant μ then satisfies the quadratic equation $\mu -2\lambda {+\mu }^{-1} = 0$ with λ = cosh(β). Choosing the root that is less than one, we have $\mu =\lambda -\sqrt{{\lambda }^{2 } - 1} ={ \text{e}}^{-\beta }$. Finally, the factors A _± can be determined by considering the difference equation for n = 0, 1, which gives [322]

$$\displaystyle{A_{-}(\beta ) = \frac{\mu (\beta )C(\beta )} {1 +\mu (\beta )},\quad A_{+}(\beta ) ={ \text{e}}^{\beta }A_{-}(\beta ).}$$

Given that C, A _± and μ _± are all functions of β (assuming that $f_{0}/k_{c}$ is fixed), the conditions for the existence of a standing wave-front solution are

$$\displaystyle{ A_{-}(\beta ) < {c}^{{\ast}},\quad C(\beta ) - A_{ +}(\beta )\mu (\beta ) ={ \text{e}}^{\beta }A_{ -}(\beta ) \geq {c}^{{\ast}}. }$$

(4.22)

For a given β, a traveling wave exists if the threshold c ^∗ lies beneath the curve A ₋(β), whereas a standing wave exists if c ^∗ lies between A ₋(β) and ${\text{e}}^{\beta }A_{-}(\beta )$; in the latter case propagation failure occurs. Note, in particular, that increasing the threshold c ^∗ or increasing β makes propagation failure more likely. Moreover, increasing β corresponds to increasing the cluster spacing L or decreasing the Ca²⁺ diffusion coefficient D _c .

Having established that wave propagation failure can occur, we now describe how to calculate the wave speed as a function of “discreteness” using the averaging method of Keener [318, 319]. The analysis proceeds along identical lines to the study of inhomogeneous diffusivity in the discrete bistable equation; see Sect. 2.5. First, rewrite (4.17) in the dimensionless form

$$\displaystyle{ \frac{\partial c} {\partial t} = -c + \frac{{\partial }^{2}c} {\partial {x}^{2}} + (1 + g^{\prime}(x/\varepsilon ))f(c), }$$

(4.23)

where $\varepsilon = \sqrt{{L}^{2 } k_{c } /D} \ll 1$, a factor of $k_{c}^{-1}$ has been absorbed into f, and g(x) is the periodic sawtooth function, g(x) = 1∕2 − x for 0 < x < 1 and g(x + n) = g(x). (More precisely, g(x) is assumed to be the sum of Gaussians (2.69) of width σ, with the limit σ → 0 taken at the end of the calculation.) Suppose that the nonlinear function f(c) is one for which there exists a traveling front solution c(x, t) = C(x − vt) of the homogeneous equation (g ≡ 0). The main result of the averaging method is that for $\varepsilon$ sufficiently small, there exists a wavelike solution or pulsating wave of the form [318]

$$\displaystyle{ c(x,t) = C(x -\phi (t)) + \mathcal{O}(\varepsilon ), }$$

(4.24)

where the phase ϕ(t) evolves according to [see also (2.72)]

$$\displaystyle{ \frac{d\phi } {dt} = v -\varPhi (\phi /\varepsilon ) }$$

(4.25)

with

$$\displaystyle{ \varPhi (\phi /\varepsilon ) = \frac{1} {\varLambda } \int _{-\infty }^{\infty }g^{\prime}([\xi +\phi ]/\varepsilon )f(C(\xi ))C^{\prime}(\xi ){\text{e}}^{v\xi }d\xi, }$$

(4.26)

and

$$\displaystyle{ \varLambda =\int _{ -\infty }^{\infty }{\text{e}}^{v\xi }C^{\prime}{(\xi )}^{2}d\xi. }$$

(4.27)

Equation (4.25) implies that the solution is not translationally invariant, rather it moves with a time-dependent velocity ϕ′. If $v -\varPhi (\phi /\varepsilon )$ is strictly positive then ϕ′(t) is a positive, periodic function of t with period

$$\displaystyle{ T =\int _{ 0}^{\varepsilon } \frac{d\phi } {v -\varPhi (\phi /\varepsilon )}. }$$

(4.28)

The mean speed of the wave is $\bar{v} =\varepsilon /T$. On the other hand. if $v -\varPhi (\phi /\varepsilon )$ vanishes for some ϕ, then propagation failure is expected to occur.

Averaging method for discrete Ca ²⁺ release sites. The first step is to rewrite (4.23) as the first-order system

$$\displaystyle{ c_{x} = u,\quad u_{x} = c_{t} - (1 + g^{\prime}(x/\varepsilon ))f(c) + c. }$$

(4.29)

Introducing the exact change of variables $u = w -\varepsilon g(x/\varepsilon )f(u)$ yields the new system

$$\displaystyle{ c_{x} = w -\varepsilon g(x/\varepsilon )f(c),\quad w_{x} = c_{t} - f(c) + c +\varepsilon g(x/\varepsilon )f^{\prime}(c)c_{x}. }$$

(4.30)

It can be seen that if the $\mathcal{O}(1)$ system of equations is independent of $x/\varepsilon$, then the lowest-order averaged system reduces to the homogeneous equation, which we are assuming supports a traveling front solution C(x − vt). Including the inhomogeneous factors $g(x/\varepsilon )$ means that the system is no longer translationally invariant. However, we can look for solutions that are close to a traveling front by going to a traveling coordinate system with $\xi = x -\phi (t)$ to give [318]

$$\displaystyle\begin{array}{rcl} c_{\xi } - w = -\varepsilon g([\xi +\phi ]/\varepsilon )f(c)& & {}\end{array}$$

(4.31a)

$$\displaystyle\begin{array}{rcl} -c_{t} + w_{\xi } +\phi ^{\prime}c_{\xi } + f(c) - c& =\varepsilon g([\xi +\phi ]/\varepsilon )f^{\prime}(c)c_{\xi }.& {}\end{array}$$

(4.31b)

We now seek a perturbative solution of the form

$$\displaystyle{c = C(\xi ) +\varepsilon c_{1}(\xi,t)+\ldots,\quad w(\xi ) = C^{\prime}(\xi ) +\varepsilon w_{1}(\xi,t)+\ldots,\quad \phi ^{\prime}(t) = v +\varepsilon \phi _{1}^{\prime}(t)+\ldots }$$

Substituting into equations (4.31) and collecting terms of equal powers in $\varepsilon$ yields a hierarchy of equations, the first of which is

$$\displaystyle\begin{array}{rcl} \partial _{\xi }c_{1} - w_{1} = -g([\xi +\phi ]/\varepsilon )f(C)& & {}\end{array}$$

(4.32a)

$$\displaystyle\begin{array}{rcl} -\partial _{t}c_{1} + \partial _{\xi }w_{1} + vw_{1} + f^{\prime}(C)c_{1} - c_{1} = vg([\xi +\phi ]/\varepsilon )f(C) -\phi _{1}^{\prime}(t)C^{\prime} + g([\xi +\phi ]/\varepsilon )f^{\prime}(C)C^{\prime}.& & {}\end{array}$$

(4.32b)

Equations (4.32) can be rewritten as

$$\displaystyle{ -\left (\begin{array}{c} 0\\ \partial _{ t}c_{1}\end{array} \right )+\mathbb{L}\left (\begin{array}{c} c_{1} \\ w_{1}\end{array} \right ) = \left (\begin{array}{c} h_{c} \\ -\phi _{1}^{\prime}C^{\prime} + h_{w}\end{array} \right ),\quad \mathbb{L} = \left (\begin{array}{cc} \partial _{\xi } & \quad - 1 \\ f^{\prime}(C) - c&\quad \partial _{\xi } + c\end{array} \right ), }$$

(4.33)

with h _c and h _w determined by inhomogeneous terms on the right-hand side of equations (4.32a) and (4.32b), respectively. Following our analysis of the linear operator (2.62), we know that the matrix operator appearing in equation (4.33) has a null space spanned by (C′, C″). Similarly, the adjoint operator

$$\displaystyle{{\mathbb{L}}^{\dag } = \left (\begin{array}{cc} - \partial _{\xi }&\quad f^{\prime}(C) - C \\ - 1 & \quad - \partial _{\xi } + c\end{array} \right )}$$

has the null vector ${\text{e}}^{c\xi }(-C^{\prime\prime},C^{\prime})$. Hence, applying the Fredholm alternative, it follows that ϕ′(t) must satisfy the phase equation (4.25) with $\phi (t) = v +\varepsilon \phi _{1}(t)$.

For the particular choice f(c) = H(c − c ^∗) used in the analysis of standing waves with $f_{0}/k_{c} = 1$, the homogeneous equation is identical to the one analyzed at the end of Sect. 2.2. Thus, there exists a unique traveling front solution of the form (2.23) with corresponding wave speed (2.24). The mean wave speed can then be calculated explicitly along similar lines to Sect. 2.5, so that

$$\displaystyle{ \bar{v} = \frac{\varepsilon } {2} \frac{1} {\ln \,\left (\frac{v+\varepsilon /\chi } {v-\varepsilon /\chi }\right )}. }$$

(4.34)

with $v = (1 - 2{c}^{{\ast}})/\sqrt{{c}^{{\ast} } -{ {c}^{{\ast} } }^{2}}$ and $\chi = 4[{c}^{{\ast}}-{ {c}^{{\ast}}}^{2}]$. This would predict propagation failure when $\varepsilon \geq \chi v$, that is,

$$\displaystyle{ \epsilon \geq 4(1 - 2{c}^{{\ast}})(\sqrt{{c}^{{\ast} } -{ {c}^{{\ast} } }^{2}}). }$$

(4.35)

Since the analysis is based on the assumption that $\varepsilon$ is small, we expect this condition to be reasonable when c ^∗≈ 0. 5. Taylor expanding about this point the condition for propagation failure is approximately $\varepsilon > 2(1 - 2{c}^{{\ast}})$. Similarly, the condition (4.22) based on the standing wave analysis predicts propagation failure when $\epsilon /(2({\text{e}}^{\epsilon } - 1)) < {c}^{{\ast}}$, which reduces to ε > 2(1 − 2c ^∗) when c ^∗≈ 1∕2. Thus the two approaches agree for small ε and yield qualitatively similar results for large ε. Finally, as in the analysis of myelinated axons, a cubic nonlinearity for the release function f(c) yields an expression for the mean wave speed that involves exponentially small terms so it is necessary to include higher-order terms in the perturbation analysis in order to obtain sufficient accuracy [318].

3.2 The Fire–Diffuse–Fire Model of Ca²⁺ Release

An alternative approach to studying the effects of discrete Ca²⁺ release sites is to consider the so-called FDF model [127, 137, 140, 325, 497, 511]. The basic idea of the model is that once the Ca²⁺ concentration reaches a threshold value c ^∗ at a given release site, that site fires and instantaneously releases a fixed amount σ of Ca²⁺ into the cytoplasm [325, 497, 511]. It is also straightforward to take into account the effects of ion pumps by including a decay term and replacing instantaneous release by a fixed pulse of finite width [127]. The FDF model is then almost identical to the SDS model of Ca²⁺ spikes propagating along spiny dendrites [129, 130]. Since the latter model was analyzed in detail in Sect. 3.1, we can carry over the previous results after minor modifications. Therefore, consider the following version of the FDF model [127]:

$$\displaystyle{ \frac{\partial c} {\partial t} = -\frac{c} {\tau _{c}} + D_{c} \frac{{\partial }^{2}c} {\partial {x}^{2}} +\sum _{n}\delta (x - x_{n})\eta (t - T_{n}), }$$

(4.36)

where T _n is the firing time of the site x _n , which only fires once:

$$\displaystyle{ T_{n} =\inf \{ t\,\vert \,c(x_{n},t) \geq {c}^{{\ast}}\}. }$$

(4.37)

For concreteness, let

$$\displaystyle{\eta (t) = (\sigma /\tau _{d})H(t)H(\tau _{d} - t),}$$

so that each Ca²⁺ puff or spark is of size σ and duration τ _d . Suppose that the Ca²⁺ release sites are regularly spaced, x _m = mL, and consider a saltatory wave solution T _m = m Δ with speed v = L∕Δ. The existence of such a wave can be determined straightforwardly by noting that (4.36) is identical to (3.33) under the mapping

$$\displaystyle{V \rightarrow c,\quad \frac{\bar{n}} {r_{s}}\eta _{0} \rightarrow \frac{\sigma } {\tau _{d}}.}$$

The one major difference between the two models is that thresholding in the SDS model is mediated by an IF process (3.2), which introduces an additional time constant $\hat{\tau }$. Therefore, in order to apply the analysis of saltatory waves along a spiny dendrite to the FDF model (see in Sect. 3.1.3), we need to take the limit $\hat{\tau }\rightarrow 0$. This then gives the following threshold condition for the existence of a saltatory wave:

$$\displaystyle{ {c}^{{\ast}} =\sum _{ n=1}^{\infty }H(nL,n\varDelta ), }$$

(4.38)

with

$$\displaystyle{ H(x,t) = \frac{\sigma } {\tau _{d}}\int _{0}^{\tau _{d} }G_{0}(x,t - s)\mathrm{d}s,\quad G_{0}(x,t) = \frac{1} {\sqrt{4\pi D_{c } t}}{\text{e}}^{-t/\tau _{c}-{x}^{2}/4D_{ c}t}. }$$

(4.39)

In the limit τ _d → 0, H(x, t) →σ G ₀(x, t) and the threshold condition becomes

$$\displaystyle\begin{array}{rcl}{ c}^{{\ast}}& =& \sigma \sum _{ n=1}^{\infty }G_{ 0}(nL,n\varDelta ) \\ & =& \sigma \sum _{n=1}^{\infty } \frac{1} {\sqrt{4\pi D_{c } n\varDelta }}\exp \left (-n\left [ \frac{{L}^{2}t} {4D_{c}\varDelta } + \frac{\varDelta } {\tau _{c}}\right ]\right ) \equiv g(\varDelta ).{}\end{array}$$

(4.40)

In the additional limit τ _c →∞ (i.e., neglecting the flux due to ion pumps), the results of [497] are recovered. In particular, g(Δ) becomes a monotone function of Δ, and the speed of the traveling waves scales linearly as D _c ∕L. The latter follows after introducing the dimensionless threshold $\tilde{{c}}^{{\ast}} = {c}^{{\ast}}L/\sigma$ and noting that Lg∕σ is then only a function of the dimensionless quantity $\varDelta D_{c}/{L}^{2}$. However, this result is inconsistent with the analysis of standing waves, which showed that propagation failure can occur by decreasing the diffusion coefficient. The discrepancy is resolved by taking into account the effects of ion pumps, that is, taking τ _c to be finite. Moreover, g(Δ) is now non-monotone and consequently one finds two solution branches with the faster one corresponding to stable waves [127]. This result also holds in the case of finite pulse width τ _d , for which

$$\displaystyle{ H(x,t) = \frac{\sigma } {\tau _{d}}\left [A(x,t -\min (t,\tau _{d})) - A(x,t)\right ], }$$

(4.41)

with A(x, t) given by (3.40). In summary, the main qualitative results of the SDS model carry over to the FDF model, for example, one finds that propagation failure occurs if the spacing between release sites becomes too large.

One of the limitations of the above FDF model is that it neglects variations in the Ca²⁺ concentration within the ER. However, in many cells, Ca²⁺ release can result in a significant local depletion of ER Ca²⁺. In order to take this into account, the FDF model has been extended in the form of a bidomain threshold-release model [621, 622]:

$$\displaystyle\begin{array}{rcl} \frac{\partial c} {\partial t}& = D_{c} \frac{{\partial }^{2}c} {\partial {x}^{2}} + J_{\text{r}el}(c,c_{e}) - J_{2}(c,c_{e})&{}\end{array}$$

(4.42a)

$$\displaystyle\begin{array}{rcl} \frac{\partial c_{e}} {\partial t} & = D_{e} \frac{{\partial }^{2}c} {\partial {x}^{2}} -\chi _{e}\left [J_{\text{I}P}(c,c_{e}) - J_{2}(c,c_{e})\right ],&{}\end{array}$$

(4.42b)

where the release and uptake fluxes, $J_{\text{r}el}(c,c_{e}) - J_{2}(c,c_{e})$, are functions of the cytosolic and ER Ca²⁺ concentrations. For simplicity, assume that there is a continuous and uniform distribution of Ca²⁺ release sites and that the release flux depends on the difference in the cytosolic and ER Ca²⁺ concentrations:

$$\displaystyle{ J_{\text{r}el}(x,t) \equiv J_{\text{r}el}(c(x,t),c_{e}(x,t)) = (c_{e}(x,t) - c(x,t))\eta (t - T(x)), }$$

(4.43)

with

$$\displaystyle{ T(x) =\inf \{ t\,\vert \,c(x,t) \geq {c}^{{\ast}}\}. }$$

(4.44)

The release sites could be either RyRs or IP₃Rs; in the latter case the IP₃ concentration p is held fixed. Finally, the reuptake flux due to the action of ion pumps is taken to have the linear form

$$\displaystyle{ J_{2}(c,c_{e}) = \frac{c} {\tau _{c}} -\frac{c_{e}} {\tau _{e}}. }$$

(4.45)

The existence and stability of a solitary pulse solution of (4.42) can be investigated along similar lines to Sect. 3.1.1.

4 Stochastic Models of Ca²⁺ Release

The fluorescent imaging of localized Ca²⁺ puffs and sparks has established that Ca²⁺ release is a stochastic process that occurs at spatially discrete sites consisting of clusters of IP₃Rs and RyRs, respectively. In Xenopus oocytes, Ca²⁺ puffs have an amplitude ranging from around 50 to 600 nM, a spatial spread of approximately 6 μm and a typical duration of 1 s [489, 490, 690]. For sufficiently high levels of IP₃ concentration, the amplification of Ca²⁺ puffs by CICR can lead to the formation of Ca²⁺ waves. Ca²⁺ sparks in heart and skeletal muscle tend to be of shorter duration and less spatial spread and are less likely to result in wave propagation [110, 111]. As we have already mentioned Ca²⁺ puffs and sparks have also been observed in neurons [534], suggesting that one needs to take into account stochastic release events when modeling Ca²⁺ waves. Typically, one combines a stochastic model of localized release through RyRs or IP₃Rs with a deterministic reaction–diffusion model of Ca²⁺ waves such as the FDF model [137, 320]. The stochastic modeling of the Ca²⁺-dependent opening and closing of these receptors proceeds along analogous lines to the modeling of voltage-gated ion channels considered in Sect. 1.5.

4.1 Stochastic Model of Ca²⁺ Puffs in a Cluster of IP₃Rs

Stochastic models of Ca²⁺ puffs typically treat a cluster of IP₃Rs as a set of N channels that open and close independently, but are indirectly coupled by the common cytoplasmic Ca²⁺ concentration [180, 182, 579, 612]. Models differ in the level of detail regarding individual receptors. The first deterministic kinetic model of Ca²⁺-gated IP₃Rs was proposed by De Young and Keizer, in their study of agonist-induced Ca²⁺ oscillations. This model assumes that the IP₃ receptor consists of three equivalent receptor subunits, all of which have to be in a conducting state in order to generate a Ca²⁺ flux. Each subunit is taken to have an IP₃-binding site, an activating Ca²⁺-binding site, and an inactivating Ca²⁺-binding site; the conducting state corresponds to the state in which all subunits have the first two binding sites occupied but the third unoccupied. Although the De Young–Keizer model is simple to describe, it involves a relatively large number of variables that have to be coupled to the Ca²⁺ and IP₃ concentrations. A simplified version of the model was subsequently developed by Li and Rinzel [381]. They exploited the fact that the binding of IP₃ and activating Ca²⁺ are fast relative to inactivating Ca²⁺ and used a quasi-steady-state argument to reduce the eight-state subunit model to a model that simply keeps track of whether or not the inactivating Ca²⁺-binding site of a subunit is occupied. More specifically, the Li–Rinzel model is a two-variable model given by

$$\displaystyle\begin{array}{rcl} \frac{dc} {dt} = J_{\mathrm{IP}} + J_{\mathrm{leak}} - J_{\mathrm{p2}}& &{}\end{array}$$

(4.46a)

$$\displaystyle\begin{array}{rcl} \frac{dh} {dt} =\alpha _{h}(1 - h) -\beta _{h}h,& &{}\end{array}$$

(4.46b)

where c is the cytoplasmic Ca²⁺ concentration, h is the fraction of receptors in a cluster not inactivated by Ca²⁺, and p is the IP₃ concentration, which is assumed fixed. The three Ca²⁺ fluxes included in the model are the channel flux J _IP and leakage flux J _leak from the ER to the cytoplasm, and the flux J _p2 pumped back into the ER; see also Fig. 4.2. The expressions for the various fluxes are

$$\displaystyle{ J_{\mathrm{IP}} = f(c,p){h}^{3}[c_{ e} - c],\quad J_{\mathrm{leak}} =\nu _{0}[c_{e} - c],\quad J_{\mathrm{P2}} = \frac{\nu _{1}{c}^{2}} {k_{3}^{2} + {c}^{2}}, }$$

(4.47)

with

$$\displaystyle{ f(c,p) ={ \left ( \frac{p} {p + k_{4}}\right )}^{3} \cdot {\left ( \frac{c} {c + k_{5}}\right )}^{3},\quad \alpha _{ h} =\nu _{2}\frac{p + k_{4}} {p + k_{6}},\quad \beta _{h} =\nu _{3}c. }$$

(4.48)

The various cubic terms reflect the existence of three subunits. Parameter values of the model can be found in [381]. We see that the simplified model resembles the Hodgkin–Huxley model (see Sect. 1.1), after replacing Ca²⁺ concentration c by membrane voltage v and c _e by a reversal potential.

We now describe a stochastic version of the Li–Rinzel model for a cluster of IP₃Rs due to Shuai and Jung [579]. For stochastic versions of the full De Young–Keizer model, see, for example, [180, 182, 244, 612]. The deterministic equations (4.46) describe the mean behavior of a large cluster of Ca²⁺ channels, just as the Hodgkin–Huxley equations for membrane voltage apply to a large number of voltage-gated ion channels. As we discussed in Sect. 1.5, if the number of channels is relatively small, then it is necessary to take into account thermally driven fluctuations in the opening and closing of individual channels. In the case of the Li–Rinzel model, one only needs to consider the opening and closing process for the gating variable h of each subunit. The latter is modeled as the two-state Markov process

$$\displaystyle{ C(\mathrm{closed})\mathop{\mathop{ \rightleftharpoons }\limits^{\alpha _{h}}}\limits_{\beta _{h}(c)}O(\mathrm{open}). }$$

(4.49)

Suppose that there are N independent IP₃Rs, each with three independent subunits labeled i = 1, 2, 3 that are described by the above two-state Markov process. Let N _i (t) (i = 1, 2, 3) denote the number of receptors at time t that have the ith subunit open. Under the adiabatic assumption that the Ca²⁺ concentration c evolves much more slowly than the state transitions of the channels, we can write down a master equation for the probability $P(n_{i},t) = \text{Prob}[N_{i}(t) = n_{i}\vert N_{i}(0) = n_{0}]$ according to

$$\displaystyle\begin{array}{rcl} \frac{dP(n_{i},t)} {dt} & =& (N - n_{i} + 1)\alpha _{h}P(n_{i} - 1,t) + (n_{i} + 1)\beta _{h}P(n_{i} + 1,t){} \\ & & \quad - (n_{i}\beta _{h} + (N - n_{i})\alpha _{h})P(n_{i},t),\quad i = 1,2,3. \\ \end{array}$$

(4.50)

As with voltage-gated ion channels, we have a stochastic hybrid system, since the transition rate β _h depends on the Ca²⁺ concentration c(t), which evolves according to a piecewise deterministic equation of the form (4.46a). The latter, in turn, couples to the discrete stochastic variables N _i (t) through the flux

$$\displaystyle{ J_{\mathrm{IP}} = f(c(t),p)[c_{e} - c(t)]\prod _{i=1}^{3}\frac{N_{i}(t)} {N}. }$$

(4.51)

[Note that one should really write down a differential Chapman–Kolmogorov equation for the joint probability density $p(n_{1},n_{2},n_{3},c,t)$ along the lines of (1.126)]. Finally, for large N, one can obtain a further simplification by carrying out a Kramers–Moyal expansion of the master equation (4.50) along the lines of Sect. 1.5.2. This yields the following SDE for $H_{i}(t) = N_{i}(t)/N$ with H _i treated as a continuous stochastic variable:

$$\displaystyle{ dH_{i} =\alpha _{h}(1 - H_{i}) -\beta _{h}H_{i} + \frac{1} {\sqrt{N}}b(H_{i})dW_{i}, }$$

(4.52)

where

$$\displaystyle{b(H_{i}) = \sqrt{\alpha _{h } (1 - H_{i } ) +\beta _{h } H_{i}},}$$

and W _i (t) is an independent Wiener process with

$$\displaystyle{\langle dW_{i}(t)\rangle = 0,\quad \langle dW_{i}(t)dW_{j}(t^{\prime})\rangle =\delta (t - t^{\prime})dt\,dt^{\prime}\delta _{i,j}.}$$

Shuai and Jung [579] simulated the stochastic Li–Rinzel model in order to investigate the effects of noise on Ca²⁺ oscillations in a space-clamped model. They assumed that the deterministic system (4.46) was monostable at low and high IP₃ concentrations and exhibited limit cycle oscillations (occurring via a Hopf bifurcation) at intermediate concentrations. They showed that noise can enlarge the range of IP₃ concentrations over which oscillations occur—an effect known as coherence resonance. They also found a broad distribution of puff amplitudes, lifetimes, and interpuff intervals. In particular, at low IP₃ concentrations, the amplitude distribution is a monotonically decaying function, whereas at higher concentrations, it is unimodal. This suggests that Ca²⁺ puffs become more significant as IP₃ concentration is increased and hence could impact the spontaneous generation of Ca²⁺ waves. This issue was investigated numerically by Falcke [180] using a stochastic version of the De Young–Keizer model that was incorporated into a reaction–diffusion model of spatially distributed channel clusters. He showed that there is indeed a transition from Ca²⁺ puffs to waves as the IP₃ concentration is increased. At low concentrations, only puffs occur, since there is not enough Ca²⁺ released to stimulate neighboring clusters, which means that the response is purely local. However, as IP₃ concentration increases, global Ca²⁺ waves can emerge from local nucleation sites of high Ca²⁺ concentration. At intermediate levels of IP₃, global events are rare and waves only progress a short distance before dying out. On the other hand, for higher IP₃ concentrations, global waves occur regularly with a well-defined period. Again this oscillatory-like behavior can occur in parameter regimes for which the deterministic model is non-oscillatory.

4.2 Stochastic Model of Ca²⁺ Sparks in a Cluster of RyRs

We now turn to a stochastic model of Ca²⁺ sparks due to Hinch [276]; see also [241]. This model was originally developed for cardiac myocytes (heart muscle cells) and includes details of the geometry of Ca²⁺ release units, in particular, the narrow junctional gap known as the diadic space that separates the sarcoplasmic reticulum (SR) from the plasma membrane; see Fig. 4.4. (In smooth muscle cells the smooth ER is referred to as the sarcoplasmic reticulum.) However, it is possible that a similar mechanism occurs in neurons at the specialized subsurface cisternae of the ER, which also form narrow junctions with the plasma membrane; see Fig. 4.2. In a typical myocyte, there could be up to 10,000 Ca²⁺ release units, each one containing a cluster of around N = 50 RyRs on the surface of the SR. The cluster of RyRs is apposed to L-type Ca²⁺ channels located on so-called t-tubules, which are invaginations of the plasma membrane into the myocyte. (The Ca²⁺ channels are not involved in the spontaneous generation of Ca²⁺ sparks so are ignored in the model.) The diadic space separating the SR from the t-tubules is a region of the mytoplasm (intracellular fluid of myocytes), which is approximately cylindrical in shape with width 10 nm and radius 100 nm. Since the diadic space is a small enclosed volume, it supports an elevation in Ca²⁺ concentration relative to the bulk mytoplasm following the release of Ca²⁺ from an RyR. Such a local elevation plays a crucial role in the Ca²⁺ -induced Ca²⁺ -release (CICR) that results in a Ca²⁺ spark. The SR in a neighborhood of the RyRs is known as the junctional SR (JSR), which may have a different Ca²⁺ concentration from the bulk or network SR (NSR).

We now briefly introduce the model of Hinch in nondimensional form; details of model approximations and estimates of experimentally based model parameters can be found in [276]. First, the diadic space is modeled as a single compartment with Ca²⁺ concentration c satisfying the current conservation equation

$$\displaystyle{ \tau _{D}\frac{dc} {dt} = J_{\text{R}yR} - J_{\text{D}}. }$$

(4.53)

Here τ _D is a time constant, J _RyR is the total Ca²⁺ current through the RyRs, and J _D is the diffusive current from the diadic space to the bulk mytoplasm. The latter is modeled as the Fickian current

$$\displaystyle{ J_{\mathrm{D}} = c - c_{m}, }$$

(4.54)

where c _m is the bulk mytoplasm Ca²⁺ concentration. The total current through the RyRs is taken to be proportional to the number n of open RyRs times the Ca²⁺ concentration c _sr in the JSR:

$$\displaystyle{ J_{\mathrm{RyR}} = c_{\mathrm{sr}}x,\quad x = \frac{n} {N}, }$$

(4.55)

with N the total number of RyRs in the cluster. Each RyR has Ca²⁺-binding sites, which can be activating or deactivating. When an RyR is in an activated state it is promoted to a mode where it continuously opens and closes according to a Markov process (see Sect. 1.5), with a mean open time of 1 ms [695]. The opening of an RyR channel results in an extra Ca²⁺ current flowing into the diadic space, which increases the rate at which Ca²⁺ binds to the other RyRs via CICR, thus creating a positive feedback loop. This feedback loop provides a mechanism for bistability. Note that the RyRs also contain inactivating Ca²⁺-binding sites, but these do not play a role in initiating a Ca²⁺ spark so are not included in the model. For simplicity, the RyRs are modeled using a two-state Markov process involving a single closed state and a single open state (see also [324]):

$$\displaystyle{ C(\mathrm{closed})\mathop{\mathop{ \rightleftharpoons }\limits^{ k_{+}(c)}}\limits_{k_{-}}O(\mathrm{open}), }$$

(4.56)

with transition rates

$$\displaystyle{ k_{+}(c) = \frac{1} {k\tau _{\mathrm{o}}} \frac{{c}^{\alpha }} {{c}^{\alpha } + 1},\quad k_{-} = \frac{1} {\tau _{\mathrm{o}}}. }$$

(4.57)

Fig. 4.4

Schematic diagram of a Ca²⁺ release unit in a cardiac myocyte. It is divided into four compartments: the network SR, the junctional SR, the diadic space, and the bulk mytoplasm. See text for details

Here τ _o is the mean open time of an RyR, α is the number of Ca²⁺ ions that are needed to open an RyR, and k determines the proportion of time the RyRs are open. It is assumed that the RyRs are gated independently of each other. They are, however, indirectly coupled via the Ca²⁺ concentration in the diadic space. The time constant τ _D of diffusive flux from the diadic space is several orders of magnitude smaller than the mean open time τ _o of an RyR, that is, τ _D ∼ 3 μs whereas τ _o ∼ 1 ms. Therefore, the Ca²⁺ concentration in the diadic space can be taken to be in quasi-equilibrium, τ _D → 0, so that

$$\displaystyle\begin{array}{rcl} c = c_{m} + c_{\mathrm{sr}}x.& &{}\end{array}$$

(4.58)

It follows that the transition rate can be reexpressed as a function of the fraction of open channels and the Ca²⁺ concentration in the SR, $k_{+} = k_{+}(c_{m} + c_{\mathrm{sr}}x)$.

Now consider N independent RyRs within a Ca²⁺ release unit, each described by the above two-state Markov process. Let N(t) be the number of open channels at time t and set $p_{n}(t) = \text{Prob}[N(t) = n\vert N(0) = n_{0}]$. Following along similar lines to Sect. 1.5.1, it can be shown that p _n (t) evolves according to a master equation with transitions rates $\omega _{\pm }(n/N): n \rightarrow n \pm 1$:

$$\displaystyle{ \omega _{+}(x) = N(1 - x) \frac{{(c_{m} + c_{\mathrm{sr}}x)}^{\alpha }} {k({(c_{m} + c_{\mathrm{sr}}x)}^{\alpha } + 1)},\quad \omega _{-}(x) = Nx. }$$

(4.59)

For the moment, it is assumed that c _m and c _sr are fixed so that ω ₊ can be treated as a function of x alone. (In Sect. 4.4.4, the dynamics of c _sr following initiation of a Ca²⁺ spark will also be taken into account.) The units of time are fixed by setting τ _o = 1. The master equation for the RyR cluster is then

$$\displaystyle{ \frac{dp_{n}} {dt} =\omega _{+}((n - 1)/N)p_{n-1} +\omega _{-}((n + 1)/N)p_{n+1} - (\omega _{+}(n/N) +\omega _{-}(n/N))p_{n}, }$$

(4.60)

with boundary conditions $p_{N+1} = p_{-1} \equiv 0$. Multiplying both sides of the master equation (4.60) by n∕N and summing over n gives

$$\displaystyle{ \frac{d\langle n/N\rangle } {dt} =\langle \varOmega _{+}(n/N)\rangle -\langle \varOmega _{-}(n/N)\rangle, }$$

(4.61)

where $\omega _{\pm } = N\varOmega _{\pm }$, and the brackets ⟨…⟩ denote a time-dependent ensemble averaging over realizations of the stochastic dynamics, that is, $\langle A(n/N)\rangle =\sum _{n}p_{n}A(n/N)$ for any function of state A(n∕N). In the thermodynamic limit N →∞ (infinite cluster size), statistical correlations can be ignored so that one can take the mean-field limit $\langle \varOmega _{\pm }(n/N)\rangle \rightarrow \varOmega _{\pm }(\langle n/N\rangle )$. This then yields a deterministic equation for the fraction x of open RyRs:

$$\displaystyle{ \frac{dx} {dt} =\varOmega _{+}(x) -\varOmega _{-}(x) = (1 - x) \frac{{(c_{m} + c_{\mathrm{sr}}x)}^{\alpha }} {k({(c_{m} + c_{\mathrm{sr}}x)}^{\alpha } + 1)} - x. }$$

(4.62)

It can be shown that, for physiologically reasonable parameter values, this equation exhibits bistability [276], that is, there exists a pair of stable fixed points x _± separated by an unstable fixed point x ₀. The fixed point x ₋≈ 0 represents a quiescent state, whereas the other fixed point x ₊ represents a Ca²⁺ spark in which a significant fraction of RyRs are in the active mode and can be interpreted as a burst phase. Hence, this model operates in a completely different regime to the model considered by Shuai and Jung [579], which was in a monostable state (a fixed point or limit cycle) in the deterministic limit. Bistability of the Hinch model means that a Ca²⁺ spark is a distinct event rather than a broad distribution of events.

Noise-induced transitions from x ₋ to x ₊ determine the distribution of inter-spark intervals, just as noise-induced transitions from x ₊ to x ₋ determine the distribution of spark lifetimes. Hence, estimating the mean time for the occurrence of a spark event reduces to the problem of calculating the MFPT to reach x ₊, starting from a neighborhood of x ₋, by crossing x ₀. (The dominant contribution to this MFPT is the time to reach x ₀, since the system then quickly relaxes to x ₊.) We encountered an analogous problem in Sect. 1.5, where we considered the mean time for spontaneous action potential initiation due to ion-channel fluctuations. We first estimated the MFPT by carrying out a diffusion approximation of the underlying master equation (1.104), and calculating the MFPT of the corresponding Fokker–Planck equation. However, the diffusion approximation resulted in exponentially large errors. Therefore, we considered an alternative approach in Sect. 1.6, which was based on WKB methods and asymptotic analysis. Here we consider a corresponding WKB approximation of the master equation (4.60). Such an approach has been used increasingly to analyze escape problems in chemical and biological systems [160, 164, 175, 176, 256, 339, 340], including the Ca²⁺ release model of Hinch [277]. Note, however, that in contrast to the analysis of membrane voltage fluctuations in Sect. 1.6, the analysis of calcium sparks is performed with respect to the small parameter 1∕N rather than ε. That is, we carry out a system-size expansion rather than assuming fast kinetics. For the moment, we simply state the results and leave the details to Sect. 4.4.3.

If x ₋ is not on the boundary x = 0 then a standard analysis shows that the mean time τ _i to initiate a Ca²⁺ spark starting from the quiescent state x ₋ is

$$\displaystyle{ \tau _{i} = \frac{2\pi } {\varOmega _{+}(x_{-})\sqrt{\vert \gamma (x_{0 } )\vert \gamma (x_{- } )}}{\text{e}}^{N[\varPhi (x_{0})-\varPhi (x_{-})]}. }$$

(4.63)

with

$$\displaystyle{\gamma (x) = \frac{\varOmega _{-}^{\prime}(x)} {\varOmega _{-}(x)} -\frac{\varOmega _{+}^{\prime}(x)} {\varOmega _{+}(x)},\quad \varPhi (x) {=\int }^{x}\ln \left (\frac{\varOmega _{-}(y)} {\varOmega _{+}(y)}\right )dy.}$$

Similarly, the mean duration τ _f of a spark, which corresponds to the mean time to transition back from x ₊ to x ₋, is given by

$$\displaystyle{ \tau _{f} = \frac{2\pi } {\varOmega _{+}(x_{+})\sqrt{\vert \gamma (x_{0 } )\vert \gamma (x_{+ } )}}{\text{e}}^{N[\varPhi (x_{0})-\varPhi (x_{+})]}. }$$

(4.64)

Note that at a fixed point x _j , $\varOmega _{+}(x_{j}) =\varOmega _{-}(x_{j})$ so that $\text{sign}(\gamma (x_{j})) = \text{sign}(\varOmega _{-}^{\prime}(x_{j}) -\varOmega _{+}^{\prime}(x_{j}))$. Thus, γ(x ₀) < 0 (unstable) and γ(x _±) > 0 (stable). One also finds that $\varPhi (x_{0}) >\varPhi (x_{\pm })$ so that $\tau _{i},\tau _{f}$ are exponentially large. It turns out that in the case of Ca²⁺ release, the quiescent state x ₋ is in an $\mathcal{O}(1/N)$ neighborhood of the boundary x = 0, so that the prefactor of the MFPT has to be modified accordingly; see [276, 277] for details. Nevertheless, the leading order exponential is unchanged. From the perspective of modeling stochastic Ca²⁺ waves using the FDF model of Sect. 4.3.2, the main result that emerges from the analysis of Ca²⁺ sparks is that the deterministic threshold mechanism needs to be replaced by a probabilistic mechanism. In the deterministic case, a cluster of receptors fires whenever the intracellular Ca²⁺ crosses a fixed threshold. On the other hand, in a stochastic FDF model, Ca²⁺ release is a probabilistic process. Given the MFPT τ _i , the probability of a spark event in a time interval t is given by

$$\displaystyle{ P_{i}(t) = 1 -{\text{e}}^{-t/\tau _{i} }. }$$

(4.65)

Recall that the transition rates Ω _±, and thus τ _i , are functions of the Ca²⁺ concentration c _m of the mytoplasm, which has so far been assumed fixed. Indeed, one finds that the probability of release in a given time interval t is a sigmoidal function of c _m [276]. This stochastic release process can now be incorporated into a stochastic version of the FDF model by taking the sequence of release times T _n in (4.36) to be generated according to a sigmoidal probability distribution that is parameterized by the Ca²⁺ concentration in the mytoplasm, which itself evolves according to (4.36); see [137, 320] for details. The resulting model can be used to simulate the spontaneous occurrence of Ca²⁺ waves.

4.3 WKB Method and Quasistationary Approximation

We now present the details of how to calculate the rate of escape from a metastable state. Although, we focus here on the particular problem of Ca²⁺ release, the basic approach applies to the master equation of any bistable system in the large N (weak noise) limit. Throughout the analysis we will switch between n∕N and x, with x treated as a continuous variable; this is a reasonable approximation when N is large. First, note that the master equation (4.60) with the given reflecting boundary conditions at x = 0, 1 has a unique stationary state given by (see also Sect. 1.5.1)

$$\displaystyle{ p_{n}^{{\ast}} = p_{ k}^{{\ast}}\prod _{ m=k+1}^{n}\frac{\varOmega _{+}((m - 1)/N)} {\varOmega _{-}(m/N)} = p_{k}^{{\ast}}\exp \left (\sum _{ m=1}^{n}\ln \left (\frac{\varOmega _{+}((m - 1)/N)} {\varOmega _{-}(m/N)} \right )\right ). }$$

(4.66)

The functions Ω _±(x) are smooth and the second derivatives of lnΩ _±(x) are bounded (except in the limits x → 0 and x → 1). Therefore, away from the boundaries, the sums can be approximated by the trapezium rule [276]

$$\displaystyle{\sum _{m=k}^{n}\ln (f(m/N)) = \frac{\ln (f(k/N) +\ln f(n/N)} {2} + N\int _{k/N}^{n/N}\ln (f(y))dy + \mathcal{O}(1/N),}$$

so that

$$\displaystyle{ p_{n}^{{\ast}} = \frac{\mathcal{N}} {\sqrt{\varOmega _{+ } (x)\varOmega _{- } (x)}}{\text{e}}^{-N\varPhi (x)},\quad x = n/N, }$$

(4.67)

where Φ(x) is the effective potential

$$\displaystyle{ \varPhi (x) {=\int }^{x}\ln \left (\frac{\varOmega _{-}(y)} {\varOmega _{+}(y)}\right )dy, }$$

(4.68)

and $\mathcal{N}$ is a normalization factor such that $\sum _{n=0}^{N}p_{n}^{{\ast}} = 1$.

Now suppose that we place an absorbing boundary at the unstable fixed point $x_{0} = n_{0}/N$. Although there no longer exists a stationary solution, the flux through the absorbing boundary is exponentially small for large N, so that we can use a spectral projection method analogous to Sect. 1.6. First, rewrite the master equation (4.60) for n = 0, …, n ₀ as the linear system

$$\displaystyle{ \frac{d\mathbf{p}} {dt} = \mathbf{Q}\mathbf{p}, }$$

(4.69)

where $\mathbf{p} = {(p_{0}(t),p_{1}(t),\ldots,p_{n_{0}}(t))}^{T}$, Q is the matrix of transition rates, and $p_{n_{0}}(t) = 0$ (absorbing boundary condition). Suppose that the eigenvalues of Q are ordered according to $0 >\lambda _{0} \geq \mbox{ Re}[\lambda _{1}] \geq \mbox{ Re}[\lambda _{2}] \geq \ldots$ with corresponding eigenfunctions $\phi _{n}^{(j)}$, and consider the eigenfunction expansion

$$\displaystyle{ p_{n}(t) =\sum _{ r=0}^{n_{0} }C_{r}{\text{e}}^{-\lambda _{r}t}\phi _{ n}^{(r)}. }$$

(4.70)

We assume that $\vert \lambda _{0}\vert \sim {\text{e}}^{-\eta N}$ for $\eta = \mathcal{O}(1)$, whereas λ _r for r > 0 are only weakly dependent on N. It follows that all other eigenmodes decay to zero much faster than the perturbed stationary density. Thus at large times, we have the quasistationary approximation

$$\displaystyle{ p_{n}(t) \sim C_{0}{\text{e}}^{-\lambda _{0}t}\phi _{ n}^{(0)}. }$$

(4.71)

One can now use a WKB ansatz (see below) to generate a quasistationary solution $\phi _{n}^{\epsilon }$ for which Q ϕ ^ε = 0 and $\phi _{n_{0}}^{\epsilon } \sim \mathcal{O}({\text{e}}^{-\eta N})$. Since the WKB solution does not satisfy the absorbing boundary condition, it is necessary to perform an asymptotic expansion in order to match the quasistationary solution with the solution in a neighborhood of x ₀. In the process this determines λ ₀, whose inverse can be identified as the MFPT to escape from the basin of attraction of the metastable state x ₋.

Dropping exponentially small terms and writing $\phi _{n}^{\epsilon } {=\phi }^{\epsilon }(x)$ with x treated as a continuous variable, we have

$$\displaystyle\begin{array}{rcl} 0& =& \varOmega _{+}{(x - 1/N)\phi }^{\epsilon }(x - 1/N) \\ & & +\varOmega _{-}{(x + 1/N)\phi }^{\epsilon }(x + 1/N) - {(\varOmega _{+}(x) +\varOmega _{-}(x))\phi }^{\epsilon }(x).{}\end{array}$$

(4.72)

We seek a WKB solution of the form

$$\displaystyle{{ \phi }^{\epsilon }(x) \sim K(x;\epsilon ){\text{e}}^{-\varPhi (x)/\epsilon }, }$$

(4.73)

with $K(x;\epsilon ) \sim \sum _{m=0}^{\infty }{\epsilon }^{m}K_{m}(x)$. Substituting (4.73) into (4.72), Taylor expanding with respect to ε, and collecting the $\mathcal{O}(1)$ terms gives

$$\displaystyle{ \varOmega _{+}(x)({\text{e}}^{{\varPhi }^{^{\prime}}(x) } - 1) +\varOmega _{-}(x)({\text{e}}^{{-\varPhi }^{^{\prime}}(x)} - 1) = 0, }$$

(4.74)

where Φ′ = d Φ∕dx. Solving this quadratic equation in e^Φ′ shows that

$$\displaystyle{ \varPhi {=\int }^{x}\ln \frac{\varOmega _{-}(y)} {\varOmega _{+}(y)}dy\quad \text{or}\quad \varPhi = \text{constant}. }$$

(4.75)

Proceeding to the next level, equating terms at $\mathcal{O}(\epsilon )$ gives

$$\displaystyle{\varOmega _{+}{\text{e}}^{\varPhi ^{\prime}}\left (-\frac{K_{0}^{\prime}} {K_{0}} + \frac{\phi ^{\prime\prime}} {2}\right ) +\varOmega _{-}{\text{e}}^{-\varPhi ^{\prime}}\left (\frac{K_{0}^{\prime}} {K_{0}} + \frac{\phi ^{\prime\prime}} {2}\right ) -\varOmega _{+}^{\prime}{\text{e}}^{\varPhi ^{\prime}} +\varOmega _{-}{\text{e}}^{-\varPhi ^{\prime}} = 0.}$$

Substituting for Φ using (4.74) and solving for K ₀ yields the following leading order forms for ϕ ^ε:

$$\displaystyle{{ \phi }^{\epsilon }(x) = \frac{A} {\sqrt{\varOmega _{+ } (x)\varOmega _{- } (x)}}{\text{e}}^{-N\varPhi (x)}. }$$

(4.76)

with Φ given by (4.75), which is sometimes called the activation solution, and

$$\displaystyle{{ \phi }^{\epsilon }(x) = \frac{B} {\varOmega _{+}(x) -\varOmega _{-}(x)}, }$$

(4.77)

which is sometimes called the relaxation solution. The constants A, B are determined by matching solutions around x ₀. Clearly, (4.77) is singular at any fixed point x _j , where $\varOmega _{+}(x_{j}) =\varOmega _{-}(x_{j})$, so is not a valid solution for the required quasistationary density. On the other hand, it does have an interpretation in terms of a corresponding Hamiltonian–Jacobi formulation of the WKB ansatz.

The WKB ansatz was also used in the analysis of metastability in stochastic ion-channel models in Sect. 1.6 and in the analysis of sharp interfaces for slowly modulated pulled fronts; see Sect. 3.2.4. Following along similar lines to the latter example, we can formally interpret (4.74) as a stationary Hamilton–Jacobi equation H(x, Φ ^′(x)) = 0 for Φ, with Hamiltonian

$$\displaystyle\begin{array}{rcl} H(x,p) =\sum _{r=\pm }\varOmega _{r}(x)\left [{\text{e}}^{rp} - 1\right ].& &{}\end{array}$$

(4.78)

This suggests a corresponding classical mechanical interpretation, in which H determines the motion of a particle with position x and conjugate momentum p. A trajectory of the particle is given by the solution of Hamilton’s equations

Fig. 4.5

Phase portrait of Hamiltonian equations of motion for $\varOmega _{\pm } =\omega _{\pm }/N$ given by equation (4.59) with $c_{m} = 0.01,c_{e} = 4,\alpha = 4$ and k = 0. 8. The zero-energy solutions are shown as thicker curves

$$ \displaystyle\begin{array}{rcl} \dot{x}& =& \frac{\partial H} {\partial p} =\sum _{r=\pm 1}r\varOmega _{r}(x){\text{e}}^{rp}{}\end{array}$$

(4.79)

$$\displaystyle\begin{array}{rcl} \dot{p} = -\frac{\partial H} {\partial x} =\sum _{r=\pm 1} \frac{\partial \varOmega _{r}} {\partial x}(x)\left [{\text{e}}^{rp} - 1\right ].& &{}\end{array}$$

(4.80)

Here the time t should be viewed as a parameterization of paths rather than as a real time variable. Introducing the Lagrangian

$$\displaystyle{ L(x,\dot{x}) = p \cdot \dot{ x} - H(x,p), }$$

(4.81)

it follows that Φ(x) with $\varPhi (\bar{x}) = 0$ corresponds to the classical action evaluated along the least-action trajectory from $\bar{x}$ to x:

$$\displaystyle{ \varPhi (x) =\inf _{x(t_{0})=\bar{x},x(T)=x}\int _{0}^{T}L(x,\dot{x})dt. }$$

(4.82)

In terms of the underlying stochastic process X(t), the least-action path can be interpreted as the most probable fluctuational path from $\bar{x}$ to x (in the large-N limit) [205, 398]. Since p = S′ everywhere along this path, we have

$$\displaystyle{ \varPhi (x) =\int _{ \bar{x}}^{x}p(x^{\prime})dx^{\prime}, }$$

(4.83)

with the integral taken along the trajectory. It follows that the leading order term in the WKB approximation is determined by finding zero-energy solutions p = p(x) such that H(x, p(x)) = 0. One solution is p = 0 or Φ = constant, which represents the classical action along a deterministic (or relaxation) trajectory. For example, once the system escapes from the metastable state x ₋, it tends to rapidly converge to the other metastable state x ₊ along such a deterministic path. (The contribution of relaxation trajectory to the mean escape time is usually neglected.) Another solution for Φ is (4.75), which can be interpreted as the action along a non-deterministic path that represents the most probable path of escape from x ₋ to x ₀ [160, 164, 175, 176]. In Fig. 4.5 we illustrate the Hamiltonian phase space for the model showing the constant energy solutions of the Hamiltonian given by (4.78); the zero-energy activation and relaxation trajectories through the fixed points of the deterministic system are highlighted as thicker curves

Given the quasistationary approximation, the rate of escape from the metastable state centered about x = x ₋ can be calculated by matching it with an appropriate inner solution in a neighborhood of the point x = x ₀ [160, 164, 175, 256, 277]. This is necessary since the quasistationary solution (4.76) does not satisfy the absorbing boundary condition at the point x ₀ separating the two metastable states. There are a number of different ways of carrying out the matched asymptotics; see, for example, [277]. Here we will follow an approach based on fixing the probability flux J ₀ through x ₀ and then matching the activation solution for x < x ₀ with the relaxation solution for x > x ₀ using a diffusion approximation of the full master equation (4.60) in the vicinity of x ₀ [164, 175, 256]. The latter yields the Fokker–Planck equation (1.111), which can be rewritten in the form of a conservation equation

$$\displaystyle{ \frac{\partial } {\partial t}P(x,t) = - \frac{\partial } {\partial x}J(x,t) }$$

(4.84)

with

$$\displaystyle\begin{array}{rcl} J(x,t)& =& (\varOmega _{+}(x) -\varOmega _{-}(x))P(x,t) - \frac{1} {2N} \frac{\partial } {\partial x}\left [(\varOmega _{+}(x) +\varOmega _{-}(x))P(x,t)\right ]. {}\\ \end{array}$$

Substituting the quasistationary solution $p(x,t) = C_{0}{e}^{-\lambda _{0}t}\varPi (x)$ into (4.84) and using the fact that λ ₀ is exponentially small give

$$\displaystyle\begin{array}{rcl} J_{0}& =& (\varOmega _{+}(x) -\varOmega _{-}(x))\varPi (x) - \frac{1} {2N} \frac{\partial } {\partial x}\left [(\varOmega _{+}(x) +\varOmega _{-}(x))\varPi (x)\right ], {}\\ \end{array}$$

where J ₀ is the constant flux through x ₀. In a neighborhood of x ₀, this equation can be Taylor expanded to leading order in x − x ₀ and integrated to obtain the solution

$$\displaystyle{ \varPi (x) = \frac{J_{0}N} {\varOmega _{+}(x_{0})}{\text{e}}^{{(x-x_{0})}^{2}{/\sigma }^{2} }\int _{x}^{\infty }{\text{e}}^{-{(y-x_{0})}^{2}{/\sigma }^{2} }dy, }$$

(4.85)

where

$$\displaystyle{ \sigma = \sqrt{ \frac{2\varOmega _{+ } (x_{0 } )} {N[\varOmega _{+}^{^{\prime}}(x_{0}) -\varOmega _{-}^{^{\prime}}(x_{0})]}} }$$

(4.86)

determines the size of the boundary layer around x ₀.

In order to match the activation and relaxation solutions, the following asymptotic behavior of the inner solution (4.85) is used:

$$\displaystyle{ \varPi (x) = \left \{\begin{array}{cc} \frac{NJ{_{0}\sigma }^{2}} {(x - x_{0})\varOmega _{+}(x_{0})}, & x - x_{0} \gg \sigma \\ \\ \frac{NJ_{0}\sigma \sqrt{\pi }} {\varOmega _{+}(x_{0})}{ \text{e}}^{{(x-x_{0})}^{2}{/\sigma }^{2} },&x_{0} - x \gg \sigma. \end{array} \right. }$$

(4.87)

The solution to the right of the saddle matches the relaxation solution (4.77) since $\varOmega _{+}(x) -\varOmega _{-}(x) \approx (x - x_{0})[\varOmega _{+}^{\prime}(x_{0}) -\varOmega _{-}^{\prime}(x_{0})]$ for x ≈ x ₀ such that B = J ₀. In order to match the solution on the left-hand side of x ₀ with the activation solution (4.76), Taylor expand Φ(x) about x ₀ using ${\varPhi }^{^{\prime}}(x_{0}) = 0$ and ${\varPhi }^{^{\prime\prime}}(x_{0}) = 2/{N\sigma }^{2}$. It follows that

$$\displaystyle{ J_{0} = \frac{A\varOmega _{+}(x_{0})} {\sqrt{\varOmega _{+ } (x_{0 } )\varOmega _{- } (x_{0 } )}}\sqrt{\frac{{\vert \varPhi }^{^{\prime\prime}}(x_{0 } )\vert } {2\pi N}}{ \text{e}}^{-N\varPhi (x_{0})}. }$$

(4.88)

The final step in the analysis is to link the flux J ₀ with the escape rate λ ₀. This is achieved by substituting the quasistationary solution into the continuity equation (4.84) and integrating over the interval x ∈ [0, x ₀] with a reflecting boundary condition at x = 0:

$$\displaystyle{ \frac{1} {\lambda _{0}} = \frac{1} {J_{0}}\int _{0}^{x_{0} }{\phi }^{\epsilon }(y)dy. }$$

(4.89)

Since the activation solution is strongly peaked around the fixed point x ₋, a Gaussian approximation of ϕ ^ε(x) around x ₋ yields the final result

$$\displaystyle{ \lambda _{0} = \frac{\varOmega _{+}(x_{-})} {2\pi } \sqrt{{\vert \varPhi }^{^{\prime\prime}}(x_{0 } ){\vert \varPhi }^{^{\prime\prime}}(x_{- } )}{\text{e}}^{-N[\varPhi (x_{0})-\varPhi (x_{-})]}. }$$

(4.90)

Hence, we obtain (4.63) with $\tau _{i} =\lambda _{ 0}^{-1}$ and

$$\displaystyle{\varPhi ^{\prime\prime}(x) = \frac{d} {dx}\ln \left (\frac{\varOmega _{-}(x)} {\varOmega _{+}(x)}\right ) = \frac{\varOmega _{-}^{\prime}(x)} {\varOmega _{-}(x)} -\frac{\varOmega _{+}^{\prime}(x)} {\varOmega _{+}(x)} =\gamma (x).}$$

Similarly, we can obtain (4.64) for the mean time τ _f to terminate a spark.

4.4 Stochastic Phase-Plane Analysis

Fig. 4.6

Sketch of nullclines in the deterministic planar Ca²⁺ spark model with x denoting the fraction of open RyRs and c _SR the Ca²⁺ concentration in the junctional SR. The c _sr nullcline is a monotonically decreasing function x(c _sr ), whereas the x nullcline is cubic-like with three branches $x_{\pm }(c_{\text{s}r})$ and $x_{0}(c_{\text{s}r})$. (Note that the branch $x_{-}(c_{\text{s}r}) \approx 0$; we have moved it away from the vertical axis for the sake of illustration.) In the given diagram there is a single, stable fixed point on the left-hand branch. In the stochastic version of the model a Ca²⁺ spark initiates a jump to the right-hand branch $x_{+}(c_{\text{s}r})$ along the lines outlined in Sect. 4.4.2. This is followed by a stochastic trajectory in which the slow variable c _sr (t) moves down the nullcline until it undergoes a noise-induced transition back to the left-hand branch before the knee at x = x _c . In the deterministic case, the return transition occurs at the knee (dashed curve)

In the above analysis of Ca²⁺ sparks, the concentration c _sr in the JSR was held fixed. This is a reasonable approximation when considering the initiation of a Ca²⁺ spark. However, following Ca²⁺ release from the RyRs, the Ca²⁺ concentration c _sr slowly changes according to

$$\displaystyle{ \tau _{\text{s}r}\frac{dc_{\text{s}r}} {dt} = -c_{\text{s}r}x + k_{\text{s}r}[c_{0} - c_{\text{s}r}]. }$$

(4.91)

where $\tau _{\text{s}r} \gg \tau _{\text{o}} \gg \tau _{D}$. The first term on the right-hand side is the loss of Ca²⁺ through the RyRs, whereas the second terms are the influx J _SR of Ca²⁺ from the NSR with fixed Ca²⁺ concentration c ₀; see Fig. 4.4. The variation of c _sr means that one has to modify the analysis of the time to terminate the Ca²⁺ spark. Following Hinch [276], this can be achieved by combining the theory of stochastic transitions outlined in Sect. 4.4.2 with the classical phase-plane analysis of slow–fast excitable systems such as the FitzHugh–Nagumo equations (see Sect. 2.1). That is, (4.62) and (4.91) form an excitable system with the fraction x of open RyRs acting as the fast variable and c _sr acting as the slow variable. In Fig. 4.6 we sketch the nullclines of the deterministic system in a parameter regime where there is a single, stable fixed point $({x}^{{\ast}},c_{\text{s}r}^{{\ast}})$. In the full stochastic model, the initiation of a Ca²⁺ spark induces a transition to the right-hand x-nullcline according to $x_{-}(c_{\text{s}r}^{{\ast}}) \rightarrow x_{+}(c_{\text{s}r}^{{\ast}})$ as outlined in Sect. 4.4.2. The slow variable then moves down the right-hand nullcline $x_{+}(c_{\text{s}r})$ according to the equation

$$\displaystyle{ \tau _{\text{s}r}\frac{dc_{\text{s}r}} {dt} = -c_{\text{s}r}x_{+}(c_{\text{s}r}) + k_{\text{s}r}[c_{0} - c_{\text{s}r}]. }$$

(4.92)

That is, although x is a stochastic variable, it fluctuates much faster than the dynamics of c _sr so one can substitute a time-averaged value of x in (4.91).

Suppose that c _sr (t) is the solution of (4.92) with $c_{\text{s}r}(0) = c_{\text{s}r}^{{\ast}}$, that is, the Ca²⁺ spark occurs at t = 0. In principle, the spark can terminate at any time t > 0 due to fluctuations in the number of open RyRs. Again using a separation of time scales, we can estimate the rate of transition back to the left-hand branch at time t using

$$\displaystyle{ \lambda (t) = \frac{\varOmega _{+}(x_{+}(t))\sqrt{\vert \gamma (x_{0 } (t))\vert \gamma (x_{+ } (t))}} {2\pi }{ \text{e}}^{N[\varPhi (x_{+}(t))-\varPhi (x_{0}(t))]}, }$$

(4.93)

where $x_{+}(t) = x_{+}(c_{\text{s}r}(t))$, etc. One can now calculate the distribution of spark durations T. Let $P(\tau ) = \mathbb{P}(T >\tau )$ and introduce the spark duration probability density

$$\displaystyle{p(\tau ) = -\frac{dP} {d\tau }.}$$

The probability that a spark terminates in an infinitesimal time interval δ τ is λ(τ)δ τ, so that

$$\displaystyle{P(\tau +\delta \tau ) = P(\tau )(1 -\lambda (\tau )\delta \tau ).}$$

Taking the limit δ τ → 0 and integrating gives $P(\tau ) =\exp \left (-\int _{0}^{\tau }\lambda (t)dt\right )$, and hence

$$\displaystyle{ p(\tau ) =\lambda (\tau )\exp \left (-\int _{0}^{\tau }\lambda (t)dt\right ). }$$

(4.94)

4.5 Whole-Cell Model of Ca²⁺ Sparks

One of the major simplifications of the Hinch model [276] is that the background Ca²⁺ concentrations in the mytoplasm (c _m ) and the NSR (c _nsr) are held fixed. It thus fails to capture the collective behavior of a large population of Ca²⁺ release units (CaRUs) that are coupled via global changes in these background concentrations (assuming diffusion can be neglected on the relevant time scales). This has motivated the development of a whole-cell model of calcium-induced calcium release in cardiac myocytes, based on a system of $\mathcal{N}$ globally coupled CaRUs [672, 673]. We will describe a simplified version of the model in which Ca²⁺ pumps, leakage currents, and voltage-gated Ca²⁺ channels are not modeled explicitly. Let c _j and $\hat{c}_{j}$ denote the Ca²⁺ concentration in the dyadic space and JSR of the jth CaRU, $j = 1,\ldots,\mathcal{N}$. Then

$$\displaystyle\begin{array}{rcl} \tau _{D}\frac{dc_{j}} {dt} = J_{\text{R}yR}^{j} - J_{ \text{D}}^{j},\qquad \tau _{\text{s} r}\frac{d\hat{c}_{j}} {dt} = J_{\text{s}r}^{j} - J_{ \text{R}yR}^{j},& &{}\end{array}$$

(4.95)

where $J_{\mathrm{RyR}}^{j}$ is the total Ca²⁺ current through the RyRs, $J_{\mathrm{D}}^{j}$ is the diffusive current from the diadic space of the jth CaRU to the bulk mytoplasm, and $J_{\mathrm{sr}}^{j}$ is the current from the NSR to the JSR of the jth CaRU. The various fluxes are given by

$$\displaystyle{ J_{\mathrm{D}}^{j} = c_{ j} - c_{m},\quad J_{\text{R}yR}^{j} = x_{ j}\hat{c}_{j},\quad J_{\text{s}r}^{j} = k_{\text{s} r}[c_{\mathrm{nsr}} -\hat{ c}_{j}], }$$

(4.96)

where x _j is the fraction of open RyRs in the jth CaRU. Finally, from conservation of Ca²⁺ ions, the bulk concentrations c _m and c _nsr evolve according to

$$\displaystyle\begin{array}{rcl} \tau _{\mathrm{m}}\frac{dc_{m}} {dt} = -\sum _{j=1}^{\mathcal{N}}J_{ D}^{j} + J_{ m},\qquad \tau _{\text{s}r}\frac{dc_{\mathrm{nsr}}} {dt} = J_{\mathrm{nsr}} -\sum _{j=1}^{\mathcal{N}}J_{\text{s} r}^{j}.& &{}\end{array}$$

(4.97)

Here J _m and J _nsr are the total external currents into the mytoplasm and NSR, respectively, arising from Ca²⁺ pumps and other factors.

When the number $\mathcal{N}$ of CaRUs is large, one can develop a probability density version of the above model, which takes the form of a differential Chapman–Kolmogorov (CK) equation describing the evolution of a single equivalent stochastic CaRU [672]. The basic idea is to introduce a probability density $p(c,\hat{c},n,t)$, with $p(c,\hat{c},n,t)dcd\hat{c}$ interpreted as the fraction of CaRUs at time t that have Ca²⁺ concentrations in the range $c \leq c(t) \leq c + dc,\hat{c} \leq \hat{ c}(t) \leq \hat{ c} + d\hat{c}$ and are in the internal state n, with the latter determined by the number of open RyRs, for example. Introduce the single CaRU equations

$$\displaystyle\begin{array}{rcl} \tau _{D}\frac{dc} {dt}& \equiv & F(c,\hat{c},n) = \frac{n} {N}\hat{c} + c_{m} - c \\ \tau _{\text{s}r}\frac{d\hat{c}} {dt}& \equiv & \hat{F}(\hat{c},n) = k_{\text{s}r}[c_{\mathrm{nsr}} -\hat{ c}] - \frac{n} {N}\hat{c}.{}\end{array}$$

(4.98)

The probability density p then evolves as

$$\displaystyle{ \frac{\partial p} {\partial t} = -\frac{\partial (Fp)} {\partial c} -\frac{\partial (\hat{F}p)} {\partial \hat{c}} +\sum _{n^{\prime}}A(n,n^{\prime};c,\hat{c})p(c,\hat{c},n^{\prime},t), }$$

(4.99)

with $A(n,n^{\prime},c,\hat{c})$ constructed from the transition rates of the master equation describing the opening and closing of RyRs within each CaRU. Note that the CK equation (4.99) is formally similar to the CK equation (1.143) for membrane voltage fluctuations considered; see Sects. 1.5 and 1.6. However, the solution of (1.126) represents the distribution of state trajectories with respect to repeated trails of a single neuron, whereas the solution of (4.99) represents the distribution of state trajectories with respect to a large ensemble of identical CaRUs. Moreover, the latter are globally coupled due to the dependence of $F,\hat{F}$ and A on the Ca²⁺ concentrations $c_{m},c_{\mathrm{nsr}}$. Within the probabilistic framework these evolve according to the equations

$$\displaystyle\begin{array}{rcl} \tau _{\mathrm{m}}\frac{dc_{m}} {dt} & =& -\int _{0}^{\infty }\int _{ 0}^{\infty }\rho (c,\hat{c},t)[c_{ m} - c]dcd\hat{c} + J_{m}{}\end{array}$$

(4.100)

$$\displaystyle\begin{array}{rcl} \tau _{\text{s}r}\frac{dc_{\mathrm{nsr}}} {dt} & =& J_{\mathrm{nsr}} - k_{\text{s}r}\int _{0}^{\infty }\int _{ 0}^{\infty }\rho (c,\hat{c},t)[c_{\mathrm{ nsr}} -\hat{ c}]dcd\hat{c},{}\end{array}$$

(4.101)

with $\rho (c,\hat{c},t) =\sum _{n}p(c,\hat{c},n,t)$. The probability density formulation of CaRUs can increase computational efficiency compared to Monte Carlo simulations of the full ODE model [672, 673].

5 Intercellular Calcium Waves in Astrocytes

So far we have considered intracellular Ca²⁺ waves that propagate within a single cell. However, one also finds intercellular Ca²⁺ waves (ICWs) consisting of increases in cytoplasmic Ca²⁺ concentration that are communicated between cells and appear as waves that spread out from an initiating or trigger cell. The speed and size of ICWs depend on the initiating stimulus and the particular propagation mechanism. An ICW often propagates at a speed of 10–20 μm/s and lasts for periods of up to tens of seconds, indicating that it can involve the recruitment of hundreds of contiguous cells. ICWs were first observed in cultures of astrocytes (discussed below) in response to extracellular glutamate [143], and in airway epithelial cells following mechanical stimulation [550]. They have subsequently been found in a wide range of cell types and under a variety of stimulus conditions, as reviewed in [380, 554].

5.1 Astrocytes and the Tripartite Synapse

Within the context of the central nervous system (CNS), ICWs are emerging as a major long-range signaling mechanism of a particular type of glial (nonneuronal) cell known as an astrocyte; see Fig. 4.7. Other types of glial cells include oligodendrocytes and radial glial cells. The former are responsible for myelination in the CNS just as Schwann cells are responsible for myelination in the peripheral nervous system; see also Sect. 2.5. On the other hand, radial glial cells play a pivotal role during development, with regard to both neuronal migration and neurogenesis [101]; see Sect. 9.1. Radial glial cells usually disappear post-development, but some cells, such as Bergmann glia in the cerebellum and Muller glia in the retina, maintain their radial glia-like properties into adulthood.

Fig. 4.7

Left: Astrocyte in vitro stained with GFAP to show filaments. Right: Astrocytes stained for GFAP, with end-feet ensheathing blood vessels [Public domain figure from Wikipedia Commons]

Traditionally, astrocytes were thought to be physiologically passive cells that only play a supporting role in the CNS by regulating and optimizing the environment within which neurons operate. Their housekeeping duties include maintaining local ion and pH homeostasis, delivering glucose and providing metabolic support via contacts with the vasculature, and clearing neuronal waste such as metabolic products and excess neurotransmitters in the synaptic cleft. However, over the past 20 years there has been an increasing amount of empirical data indicating that astrocytes play an active role in modulating synaptic transmission [253, 268, 449]. That is, astrocytes respond to synaptic neurotransmitters such as glutamate by elevating their intracellular Ca²⁺ levels. This then results in the release of glutamate and other signaling molecules from the astrocytes (gliotransmission), which in turn influences synaptic activity. Thus, astrocytes could potentially contribute to higher brain function and, consequently, behavior. This is consistent with the fact that the number of astrocytes relative to the number of neurons increases dramatically with brain complexity, ranging from 1:25 in the leech, 1:6 in C elegans, 1:3 in lower mammals, to 1.4:1 in the human cortex. Conversely, it is known that following injury to the nervous system or under pathological conditions such as Alzheimer’s disease, epilepsy, and schizophrenia, the structure and protein expression of astrocytes are altered [564]. One interpretation of these structural changes is that they are indicative of the supportive action of astrocytes following injury. On the other hand, given that many CNS disorders are correlated with alterations in synaptic transmission and gliotransmission regulates synaptic processing, it is possible that changes in gliotransmission contribute to these disorders; see Chap. 9.

Fig. 4.8

Astrocytes modulate synaptic transmission and blood flow via Ca²⁺ signaling. Inset: the tripartite synapse. Synaptic transmission of glutamate (A) activates not only postsynaptic receptors such as AMPA but also astrocytic receptors that elevate astrocytic Ca²⁺ levels (B). This then induces the local release of glutamate from the astrocyte (C), which can modulate the synapse. Astrocytes also form a link between a neuron and the local vasculature. Increases in astrocytic Ca²⁺ concentration propagate via astrocyte end feet to small blood vessels, where they control vessel diameter and blood flow

In many higher organisms, astrocytes establish nonoverlapping anatomical and functional domains of size around 100 μm, and their distribution is highly organized. The resulting astrocytic network parcellates the whole neutrophil such that a single astrocyte can make contact with thousands of synapses. The processes of a single astrocyte make contact with a synapse by wrapping around both the presynaptic and postsynaptic domains to form what is known as a tripartite synapse [253, 268, 449]; see Fig. 4.8. Hence, from a geometrical perspective, it is well placed to modulate synaptic transmission. The first component of glial–neuronal interactions at a synapse, namely, the metabotropic glutamate receptor 5 (mGluR5)-dependent elevation in astrocytic Ca²⁺ concentration in response to the release of glutamate neurotransmitters, was discovered in the early 1990s using Ca²⁺ imaging to study cultured glial cells [106, 143]. Both intracellular Ca²⁺ oscillations and ICWs propagating between neighboring astrocytes were observed. Subsequent studies have established that neuroglial signaling also occurs in astrocytes of isolated retina and brain slices [265, 460]. (Note, however, that a recent study of adult rodents has called into question the link between neuroglial Ca²⁺ signaling at the tripartite synapse [610]. It would appear that the astrocytic expression of mGluR5 receptors is developmentally regulated and, in particular, is undetectable in mature rodent brains; almost all previous studies of the tripartite synapse have been in immature animals.) Following the observation that neuronal activity increases Ca²⁺ levels in astrocytes, it was subsequently discovered that the resulting Ca²⁺ elevation induces gliotransmission of glutamate and a variety of other signaling molecules including ATP [249, 491]. Gliotransmission has been demonstrated to modulate synaptic transmission presynaptically and postsynaptically. For example, glutamate can enhance neurotransmitter release presynaptically by binding to metabotropic (mGlu) receptors or kainate receptors [189, 388]. Another example of enhanced synaptic transmission is the binding of ATP to postsynaptic receptors, which induces elevations in Ca²⁺. This, in turn, can drive the insertion of additional AMPA receptors, thus temporarily increasing synaptic efficacy [236]. (The binding of ATP to postsynaptic receptors also provides a mechanism for the propagation of ICWs via paracrine signaling; see below.) On the other hand, hydrolysis of ATP results in the accumulation of adenosine, causing a reduction of synaptic transmission [493].

In addition to directly modulating synaptic transmission, astrocytes appear to regulate NMDA-receptor function. Recall from Sect. 1.3 that NMDA receptors are crucial for the induction of long-term synaptic plasticity, requiring both glutamate and postsynaptic depolarization in order to activate. The opening of an NMDA channel leads to a temporary increase of intracellular Ca²⁺ within the postsynaptic domain, whose amplitude and temporal profile determine whether the synapse is potentiated or depressed. It turns out that there is an additional glycine-binding site that regulates NMDA-receptor activity, whose co-agonist D-serine is released by astrocytes during gliotransmission. Increases in the level of D-serine appear to shift the balance point of long-term plasticity from depression to potentiation [485], a process known as metaplasticity. Finally, not only does astrocytic Ca²⁺ signaling modulate synaptic transmission and plasticity, it also controls blood flow, since intracellular waves propagate to small blood vessels via astrocyte end feet, where they control vessel diameter; see Fig. 4.8.

Fig. 4.9

Two mechanisms for the propagation of ICWs. Local stimulation of a single cell induces an elevation in intracellular IP₃ that generates an intracellular Ca²⁺ wave. Diffusion of IP₃ through gap-junction coupling with an adjacent cell initiates a second intracellular Ca²⁺ wave. In addition, or alternatively, stimulation of the first cell releases ATP via hemichannels in the plasma membrane or vesicular release; this could also be dependent on intracellular Ca²⁺. The extracellular diffusion of ATP to neighboring cells can activate metabotropic receptors that then stimulate the production of IP₃ and the production of downstream intracellular Ca²⁺ waves

Following the original studies of ICWs in astrocyte cultures [106, 143], intracellular Ca²⁺ wave propagation between astrocytes has been observed in vitro in brain slices and whole retina preparations [265, 460, 492]. These empirical findings, combined with the structure of the tripartite synapse, have led to the suggestion that networks of astrocytes could form an extraneuronal pathway for rapid long-distance signaling within the CNS, which interacts reciprocally with neuronal networks. In particular, ICWs could provide a potential mechanism for coordinating and synchronizing the activity of a large group of neuronal and nonneuronal cells. It should be pointed out, however, that the function of ICWs in astrocytes remains controversial, since there have been very few in vivo observations of ICWs in the CNS, excluding the case of spontaneous ICWs during early development where astrocytes play a role in the generation, differentiation, and migration of neurons [666]; see Sect. 9.1. One recent exception is a study demonstrating in vivo spontaneous ICWs (termed glissandi) in mouse hippocampal astrocytes. (ICWs have also been observed in vivo in other glial networks, e.g., Bergmann glia in the cerebellum of rodents [287].) Another concern regarding the functional role of ICWs, either in vitro or in vivo, is whether they require strong inducing stimuli that do not have a normal physiological counterpart. On the other hand, certain pathological conditions such as brain trauma, brain ischemia (oxygen deprivation), and seizures are often associated with strong stimuli. They are all coupled with the release of ATP and glutamate and reductions in the level of extracellular Ca²⁺ that collectively could promote ICW generation. Such ICWs could exacerbate an injury by propagating signals that initiate cell apoptosis (cell death) in surrounding cells [681]. There is also growing evidence that ICWs provide a source of excitation during epileptic episodes [564, 647, 668]; see Sect. 9.4. For example, studies of a brain slice that exhibits epileptiform activity, showed that there was also a corresponding increase in the frequency of Ca²⁺ oscillations in astrocytes [623]. Moreover, the release of glutamate from astrocytes has been observed to cause synchronous neuronal depolarizations [186].

5.2 Mechanisms and Models of Intercellular Calcium Waves

There are thought to be two basic mechanisms for the propagation of ICWs between adjacent cells that occur in astrocytes and other cell types: (a) diffusion of the second messenger IP₃ and, to a lesser extent, Ca²⁺ via gap junctions; (b) paracrine signaling via a diffusing extracellular messenger such as ATP. These mechanisms are illustrated in Fig. 4.9.

5.2 Gap-Junction Signaling

Gap junctions are arrays of transmembrane channels that connect the cytoplasm of two neighboring cells and thus provide a direct diffusion pathway between the cells. Cells sharing a gap-junction channel each provides a hemichannel (also known as a connexon) that connect head-to-head [545]. Each hemichannel is composed of proteins called connexins that exist as various isoforms named Cx23 through Cx62, with Cx43 being the most common. The physiological properties of a gap junction, including its permeability and gating characteristics, are determined by the particular connexins forming the channel. Although gap junctions are readily permeable to Ca²⁺ and IP₃, the diffusion of Ca²⁺ through a channel is greatly restricted due to buffering, so that the propagation of ICWs is primarily mediated by the diffusion of IP₃ when gap junctions are the main mechanism. The possible role of gap junctions in the propagation of ICWs was initially suggested by the observation in epithelial cells that waves only spread in adjacent cells from localized membrane sites in direct contact with the initiating cell. Moreover, the ICWs were not influenced by fluid flow over the apical surface of cells, that would have interfered with extracellular signals [259]. Further evidence for the role of gap junctions came from the study of glioma cells that lacked gap junctions [107]. These cells only supported ICWs when they were transfected with connexins; however, the presence of connexins could also enhance ATP and thus contribute to paracrine signaling (see below). Subsequent experimental and modeling studies have established that the diffusion of IP₃ through gap junctions can support the propagation of ICWs [379, 591]. In particular, ICWs could be initiated by an elevation of IP₃ in a single cell followed by diffusion to other cells without any regeneration of IP3. On the other hand, the range of the waves tends to be limited to only a few cells, suggesting that some other mechanism may be needed in order to generate longer-range waves.

The first model of ICWs mediated by gap junctions was based on studies of mechanical stimulation of cultured epithelial cells [590, 591]. It has subsequently been extended to models of ICWs in networks of astrocytes of varying complexity [163, 281, 310]. The models all have the same basic structure, but differ in terms of whether or not regenerative mechanisms or extracellular signaling mechanisms are included. A simple 1D version of an ICW model is shown in Fig. 4.10. Each cell is treated as a square of size L with local coordinates (x, y), 0 ≤ x, y ≤ L. There are N cells labeled n = 1, …, N arranged on a regular lattice with nearest neighbor coupling. The intracellular dynamics within each cell is described by reaction–diffusion equations of the form given in Sect. 4.2. For example, in the original model of Sneyd et al. [591], this included equations for cytoplasmic Ca²⁺ concentration (c _n ), IP₃ concentration p _n , and a slow variable h _n representing the fraction of IP3 receptors that have not been inactivated by Ca²⁺. An example of an intracellular model for the nth cell is (see, e.g., [280])

$$\displaystyle\begin{array}{rcl} \frac{\partial p_{n}} {\partial t} = D_{p}{\nabla }^{2}p_{ n} - k_{p}p_{n},& &{}\end{array}$$

(4.102a)

$$\displaystyle\begin{array}{rcl} \frac{\partial c_{n}} {\partial t} = D_{c}{\nabla }^{2}p_{ n} - k_{c}c_{n} + f(c_{n},h_{n},p_{n}),& &{}\end{array}$$

(4.102b)

$$\displaystyle\begin{array}{rcl} \tau _{h}\frac{dh_{n}} {dt} = h_{\infty }(c_{n}) - h_{n}.& &{}\end{array}$$

(4.102c)

Fig. 4.10

Schematic diagram of a 1D model of intercellular Ca²⁺ waves mediated by gap junctions

Here the nonlinear function f represents the net flux due to Ca²⁺ release from IP₃ receptors. It is usually assumed that $D_{p} \gg D_{c}$ due to Ca²⁺ buffering. Coupling between adjacent cells in incorporated into the model using the boundary conditions for IP₃. Thus along the common vertical edge between cells n − 1 and n, 1 < n < N, the IP₃ flux J _n is taken to be of the form

$$\displaystyle\begin{array}{rcl} -D_{p}\left.\frac{\partial p_{n-1}(x,y)} {\partial x} \right \vert _{x=L} = D_{p}\left.\frac{\partial p_{n}(x,y)} {\partial x} \right \vert _{x=0} \equiv \mathcal{K}[p_{n-1}(L,y) - p_{n}(0,y)].\qquad & &{}\end{array}$$

(4.103)

These are supplemented by no-flux boundary conditions for IP₃ across all other edges; all cell borders are assumed to be impermeable to Ca²⁺. In a 2D version of the model arranged on a square grid, IP₃ can also flow across common horizontal edges of neighboring cells, while no-flux boundary conditions are imposed along the exterior boundary of the grid. A typical simulation of such a model is to inject IP₃ in the cell at one end of the lattice, which is then allowed to diffuse from cell to cell, resulting in the propagation of an intracellular calcium wave. It is important to note that the ICW is not a traveling wave in the mathematical sense used throughout this book. Rather, the spread of Ca²⁺ is mediated by passive diffusion of intracellular Ca²⁺ and intra-/intercellular IP₃. Thus the rate of spread decreases with distance from the stimulated cell, and the arrival time of the wave increases exponentially with distance. In order to create more persistent ICWs, it is necessary to incorporate some regenerative mechanism. One example of such a mechanism is to assume that the rate of production of IP₃ can be amplified by intracellular Ca²⁺ so that (4.102a) becomes [281]

$$\displaystyle{\frac{\partial p_{n}} {\partial t} = D_{p}{\nabla }^{2}p_{ n} + \frac{\nu c_{n}^{2}} {K + c_{n}^{2}} - k_{p}p_{n}.}$$

Hofer et al. [281] showed that for an intermediate range of values of ν, partial regeneration of IP₃ supports ICWs that propagate further than those relying solely on passive diffusion of IP₃. Finally, an alternative mechanism for increasing the range of ICWs, which does not require some form of regeneration or active process, is to combine gap-junction coupling with an extracellular pathway involving ATP [163, 310]; see below.

5.2 Paracrine Signaling

Early evidence for the involvement of an extracellular component to ICWs arose from the observation that ICWs can propagate between disconnected groups of astrocytes, provided that the degree of separation does not exceed around 100 μm [266]. More recently, advanced imaging techniques have revealed that ICWs are associated with a cloud of extracellular ATP [249, 457]. There are a variety of pathways that could mediate ATP release, including hemichannels in the plasma membrane and vesicular discharge. Modeling studies suggest that a combination of gap-junction and paracrine signaling can significantly increase the range of ICWs [298]. It is also possible that there is a regenerative mechanism that boosts the propagation of ICWs. One candidate for active rather than passive (diffusive) wave propagation is Ca²⁺-dependent ATP release via connexin hemichannels [652]. In the case of astrocytes, the dominant propagation mechanism depends on where they are located within the nervous system. For example, ICWs in the neocortex tend to be mediated by gap junctions, whereas in the hippocampus, both gap-junction and paracrine signaling play a role.

Here we will describe a mathematical model of paracrine signaling in astrocyte networks due to Bennett et al. [37]. Suppose that a(t) denotes the extracellular concentration of ATP in the vicinity of a single cell. ATP binds to metabotropic P2Y receptors in the cell membrane according to the first-order kinetic scheme

$$\displaystyle{ ATP + R\mathop{\mathop{ \rightleftharpoons }\limits^{ k_{+}}}\limits_{k_{-}}R. }$$

In the case of fast binding kinetics, the fraction of P2Y receptors that are bound by ATP is given by the quasi-equilibrium solution

$$\displaystyle{ r = \frac{a} {K_{R} + a},\quad K_{R} = \frac{k_{-}} {k_{+}}. }$$

(4.104)

This then drives activation of G-proteins according to the kinetic scheme

$$\displaystyle{ G_{0}\mathop{\mathop{ \rightleftharpoons }\limits^{ k_{a}(r + r_{0})}}\limits_{k_{d}}G_{1}, }$$

where G _0, 1 represent the inactive and active states of the G-protein. Denoting the amount of activated protein by g and the total G-protein by g _T , we have

$$\displaystyle{ \frac{dg} {dt} = k_{a}(r + r_{0})(g_{T} - g) - k_{d}g. }$$

(4.105)

Again assuming fast kinetics, the quasi-equilibrium solution is

$$\displaystyle{ g = g_{T} \frac{r + r_{0}} {K_{G} + r + r_{0}},\quad K_{G} = \frac{k_{d}} {k_{a}}. }$$

(4.106)

A major simplification of the model is to assume that the release of IP₃ within the cell is mainly driven by activation of G-protein so that Ca²⁺ dependence is ignored. If p is the intracellular IP₃ concentration then

$$\displaystyle{ \frac{dp} {dt} =\kappa g - k_{\mathrm{deg}}p, }$$

(4.107)

for constants κ, k _deg. The next stage of paracrine signaling, namely, the release of ATP by astrocytes is not yet well understood. However, IP₃ as a triggering mechanism, possibly via Ca²⁺ release from intracellular stores, is probably involved. In the model of Bennett et al. [37], IP₃ is assumed to directly cause ATP release from the cell surface into the extracellular space at the rate

$$\displaystyle{ J_{\mathrm{ATP}}(p,t) =\kappa _{\mathrm{ATP}}\chi (t)\left [ \frac{p - p_{\min }} {K_{\mathrm{rel}} + p}\right ]_{0}. }$$

(4.108)

Here κ _ATP and K _rel are constants, [z]₀ = z for z > 0 and [z]₀ = 0 for z < 0, and χ(t) takes into account depletion of ATP stores within the cell. That is χ(0) = 1 and

$$\displaystyle{ \frac{d\chi } {dt} = -\gamma J_{\mathrm{ATP}}(t). }$$

(4.109)

The rate of ATP release is thresholded in order to insure that very small levels of ATP are not amplified, and the depletion factor χ(t) is included in order to terminate ATP release; the latter is motivated by the observation that ATP release is suppressed by repeated stimulation of a cell.

Finally, coupling between cells is modeled in terms of a diffusion equation for the extracellular ATP concentration that takes into account the IP₃-mediated release of ATP from the astrocytes. One way to proceed would be to use a continuum bidomain model, in which astrocytes and extracellular ATP coexist at every point in space. Taking a = a(x, t), p = p(x, t) and denoting the density of astrocytes by ρ(x), we would then have

$$\displaystyle{ \frac{\partial a} {\partial t} = D_{\mathrm{ATP}}{\nabla }^{2}a +\rho (\mathbf{x})J_{\mathrm{ ATP}}(p,t) - \frac{a} {K_{\mathrm{deg}} + a}. }$$

(4.110)

Alternatively, one could explicitly distinguish between extracellular space and the intracellular space of astrocytes, in which case the fluxes J _ATP only exist at the boundaries of the cells. The latter approach is used in the computer simulations of Bennett et al. [37], who take a simplified cubic geometry for the astrocytes and also allow for intracellular diffusion of IP₃. Note that under a quasi-steady-state approximation, (4.107) and (4.110) form a closed system of differential equations, since (4.104) and (4.107) can be used to express g in terms of a. Hence, one can solve for the ATP and IP₃ concentrations independently of Ca²⁺ and then incorporate ICWs by using the IP₃ to drive a standard model of Ca²⁺ release from intracellular stores such as Li–Rinzel. However, if there is regenerative Ca²⁺-dependent production of IP₃ or a Ca²⁺-dependent mechanism for gliotransmission of ATP, then it is necessary to couple the Ca²⁺ dynamics to that of IP₃ and ATP.

Fig. 4.11

Illustration of a 2D periodic heterogeneous medium consisting of a cytosolic domain Ω _c and an ER domain Ω _e separated by a boundary Γ

6 Appendix: Homogenization

As we noted in Sect. 4.2, it is not practical to try modeling the detailed microstructure of the cytoplasmic and ER boundaries, which are highly interleaved. This motivates consideration of a mean-field description of the Ca²⁺ concentrations, in which the concentrations c, c _e are taken to coexist at every point in space with effective diffusion coefficients. As shown by Goel et al. [224], the resulting bidomain model can be derived from first principles using homogenization theory. In this appendix we sketch the basic steps in their derivation. Note that we also encountered a 1D example of homogenization theory in Sect. 3.2.3, where we spatially averaged over a discrete distribution of dendritic spines.

Consider a domain Ω ⊂ ℝ ³ in which the ER is treated as a periodic network that occupies a fraction of Ω, as illustrated in Fig. 4.11. For simplicity, assume that the unit cell is a cube of length ε. The ER Ca²⁺ occupies a connected domain $\varOmega _{e}^{\epsilon }$, and the complementary connected domain $\varOmega _{c}^{\epsilon } =\varOmega /\varOmega _{e}^{\epsilon }$ is occupied by cytosolic Ca²⁺. The concentrations c ^ε and $c_{e}^{\epsilon }$ satisfy the 3D diffusion equation in their respective domains:

$$\displaystyle\begin{array}{rcl} \frac{\partial c} {\partial t}& = \nabla \cdot (A_{\epsilon }(\mathbf{x}/\epsilon )\nabla c(\mathbf{x})),\quad \mathbf{x} \in \varOmega _{c}^{\epsilon }&{}\end{array}$$

(4.111a)

$$\displaystyle\begin{array}{rcl} \frac{\partial c_{e}} {\partial t} & = \nabla \cdot (B_{\epsilon }(\mathbf{x}/\epsilon )\nabla c_{e}(\mathbf{x})),\quad \mathbf{x} \in \varOmega _{e}^{\epsilon },&{}\end{array}$$

(4.111b)

where A _ε and B _ε are the diffusion tensors for Ca²⁺ in the cytosol and ER, respectively. The boundary conditions on the membrane Γ _ε separating the cytosol from the ER take the form

$$\displaystyle\begin{array}{rcl} A_{\epsilon }(\mathbf{x}/\epsilon )\nabla c(\mathbf{x}) \cdot \mathbf{n}_{c}^{\epsilon }& =& \epsilon f({c}^{\epsilon },c_{ e}^{\epsilon }),\quad \text{on}\,\varGamma _{\epsilon }{}\end{array}$$

(4.112a)

$$\displaystyle\begin{array}{rcl} -B_{\epsilon }(\mathbf{x}/\epsilon )\nabla c_{e}(\mathbf{x}) \cdot \mathbf{n}_{c}^{\epsilon } =\epsilon f({c}^{\epsilon },c_{ e}^{\epsilon }),\quad \text{on}\,\varGamma _{\epsilon },& &{}\end{array}$$

(4.112b)

where $n_{c}^{\epsilon },n_{e}^{\epsilon }$ denote the unit exterior normals to the boundaries $\partial \varOmega _{c}^{\epsilon }$ and $\partial \varOmega _{e}^{\epsilon }$, respectively, satisfying $n_{c}^{\epsilon } = -n_{e}^{\epsilon }$ on Γ _ε , and f determines the net flux from the ER to the cytosol.

The basic idea of the homogenization method is to supplement the macroscopic spatial variables $\mathbf{x} = (x_{1},x_{2},x_{3})$ with a set of microscopic variables $\mathbf{y} = (y_{1},y_{2},y_{3}),0 \leq y_{i} \leq 1$, that are the coordinates of a unit cube [496, 630]. Denote by Ω _c (Ω _e ) the set of points y = x∕ε in the unit cube for which $\mathbf{x} \in \varOmega _{c}^{\epsilon }$ ($\mathbf{x} \in \varOmega _{e}^{\epsilon }$). The concentrations ${c}^{\epsilon },c_{e}^{\epsilon }$ are then taken to be functions of x and y with x ∈Ω, y ∈Ω _c for c ^ε, and x ∈Ω, y ∈Ω _e for $c_{e}^{\epsilon }$:

$$\displaystyle{ {c}^{\epsilon } = c(\mathbf{x},\mathbf{y},t),\quad c_{e}^{\epsilon } = c_{ e}(\mathbf{x},\mathbf{y},t) }$$

(4.113)

with y = x∕ε. The next step is to introduce formal asymptotic expansion for c ^ε and $c_{e}^{\epsilon }$ of the form

$$\displaystyle\begin{array}{rcl}{ c}^{\epsilon } = {c}^{0}(\mathbf{x},\mathbf{y},t) +\epsilon {c}^{1}(x,y,t) {+\epsilon }^{2}{c}^{2}(\mathbf{x},\mathbf{y},t) +\ldots & &{}\end{array}$$

(4.114a)

$$\displaystyle\begin{array}{rcl} c_{e}^{\epsilon } = c_{ e}^{0}(\mathbf{x},\mathbf{y},t) +\epsilon c_{ e}^{1}(\mathbf{x},\mathbf{y},t) {+\epsilon }^{2}c_{ e}^{2}(\mathbf{x},\mathbf{y},t)+\ldots,& &{}\end{array}$$

(4.114b)

where c ^k and $c_{e}^{k}$ are 1-periodic functions of $\mathbf{y}$. Setting

$$\displaystyle{\nabla u = \nabla _{\mathbf{x}}u(\mathbf{x},\mathbf{y}) {+\epsilon }^{-1}\nabla _{\mathbf{ y}}u(\mathbf{x},\mathbf{y}),}$$

it follows that (4.111) becomes

$$\displaystyle\begin{array}{rcl} \frac{\partial {c}^{\epsilon }} {\partial t}& =& \left({\epsilon }^{-2}\nabla _{\mathbf{ y}} \cdot (A_{\epsilon }(\mathbf{y})\nabla _{\mathbf{y}}) {+\epsilon }^{-1}[\nabla _{\mathbf{ x}} \cdot (A_{\epsilon }(\mathbf{y})\nabla _{\mathbf{y}}) + \nabla _{\mathbf{y}} \cdot (A_{\epsilon }(\mathbf{y})\nabla _{\mathbf{x}})] + A_{\epsilon }(\mathbf{y})\nabla _{\mathbf{x}}^{2}\right ){c}^{\epsilon } \\ & & \text{for}\,\mathbf{x} \in \varOmega,\mathbf{y} \in \varOmega _{c} {}\end{array}$$

(4.115a)

$$\displaystyle\begin{array}{rcl} \frac{\partial c_{e}^{\epsilon }} {\partial t} & =& \left({\epsilon }^{-2}\nabla _{\mathbf{ y}} \cdot (B_{\epsilon }(\mathbf{y})\nabla _{\mathbf{y}}) {+\epsilon }^{-1}[\nabla _{\mathbf{ x}} \cdot (B_{\epsilon }(\mathbf{y})\nabla _{\mathbf{y}}) + \nabla _{\mathbf{y}} \cdot (B_{\epsilon }(\mathbf{y})\nabla _{\mathbf{x}})] + A_{\epsilon }(\mathbf{y})\nabla _{\mathbf{x}}^{2}\right )c_{ e}^{\epsilon } \\ & & \text{for}\,\mathbf{x} \in \varOmega,\mathbf{y} \in \varOmega _{e}, {}\end{array}$$

(4.115b)

and the boundary conditions (4.112) become

$$\displaystyle\begin{array}{rcl} A_{\epsilon }(\mathbf{y})\left [\nabla _{\mathbf{x}} {+\epsilon }^{-1}\nabla _{\mathbf{ y}}\right ]c \cdot \mathbf{n}_{c}^{\epsilon }& =& \epsilon f({c}^{\epsilon },c_{ e}^{\epsilon }),\quad \text{for}\,\mathbf{x} \in \varOmega,\mathbf{y} \in \varGamma _{\epsilon }{}\end{array}$$

(4.116a)

$$\displaystyle\begin{array}{rcl} -B_{\epsilon }(\mathbf{y})\left [\nabla _{\mathbf{x}} {+\epsilon }^{-1}\nabla _{\mathbf{ y}}\right ]c_{e} \cdot \mathbf{n}_{c}^{\epsilon }& =& \epsilon f({c}^{\epsilon },c_{ e}^{\epsilon }),\quad \text{for}\,\mathbf{x} \in \varOmega,\mathbf{y} \in \varGamma _{\epsilon }.{}\end{array}$$

(4.116b)

Substituting the asymptotic expansions for c ^ε and $c_{e}^{\epsilon }$ and collecting terms of the same order in ε then yields a hierarchy of equations for ${c}^{k},c_{e}^{k}$. At $\mathcal{O}{(\epsilon }^{-2})$, we have

$$\displaystyle\begin{array}{rcl} \nabla _{\mathbf{y}} \cdot (A_{\epsilon }(\mathbf{y})\nabla _{\mathbf{y}}){c}^{0} = 0,\quad \mathbf{y} \in \varOmega _{ c}& &{}\end{array}$$

(4.117a)

$$\displaystyle\begin{array}{rcl} \nabla _{\mathbf{y}} \cdot (B_{\epsilon }(\mathbf{y})\nabla _{\mathbf{y}})c_{e}^{0} = 0,\quad \mathbf{y} \in \varOmega _{ e}.& &{}\end{array}$$

(4.117b)

together with the boundary conditions

$$\displaystyle{ A_{\epsilon }(\mathbf{y})\nabla _{\mathbf{y}}c \cdot \mathbf{n} = 0 = B_{\epsilon }(\mathbf{y})\nabla _{\mathbf{y}}c_{e}^{0} \cdot \mathbf{n},\quad \mathbf{y} \in \varGamma, }$$

(4.118)

where n is a unit normal on Γ. For fixed x, the only periodic solution to these equations is ${c}^{0} = \text{constant},c_{e}^{0} = \text{constant}$, that is,

$$\displaystyle{ {c}^{0} = {c}^{0}(x,t),\quad c_{ e}^{0} = c_{ e}^{0}(x,t). }$$

(4.119)

The $\mathcal{O}{(\epsilon }^{-1})$ equations are

$$\displaystyle\begin{array}{rcl} \nabla _{\mathbf{y}} \cdot \left (A_{\epsilon }(\mathbf{y})\left [\nabla _{\mathbf{x}}{c}^{0} + \nabla _{\mathbf{ y}}{c}^{1}\right ]\right ) = 0,\quad \mathbf{y} \in \varOmega _{ c}& &{}\end{array}$$

(4.120a)

$$\displaystyle\begin{array}{rcl} \nabla _{\mathbf{y}} \cdot \left (B_{\epsilon }(\mathbf{y})\left [\nabla _{\mathbf{x}}c_{e}^{0} + \nabla _{\mathbf{ y}}c_{e}^{1}\right ]\right ) = 0,\quad \mathbf{y} \in \varOmega _{ e},& &{}\end{array}$$

(4.120b)

together with the boundary conditions

$$\displaystyle{ A_{\epsilon }(\mathbf{y})[\nabla _{\mathbf{x}}{c}^{0} + \nabla _{\mathbf{ y}}{c}^{1}] \cdot \mathbf{n} = 0 = B_{\epsilon }(\mathbf{y})[\nabla _{\mathbf{ x}}c_{e}^{0} + \nabla _{\mathbf{ y}}c_{e}^{1}] \cdot \mathbf{n},\quad \mathbf{y} \in \varGamma. }$$

(4.121)

The solutions for c ¹ and $c_{e}^{1}$ can be written in the form

$$\displaystyle\begin{array}{rcl}{ c}^{1}(x,y,t)& =& \sum _{ i=1}^{3}V _{ i}^{c}(y)\frac{\partial {c}^{0}(x,t)} {\partial x_{i}} +\bar{ {c}}^{1}(x,t),{}\end{array}$$

(4.122a)

$$\displaystyle\begin{array}{rcl} c_{e}^{1}(x,y,t)& =& \sum _{ i=1}^{3}V _{ i}^{e}(y)\frac{\partial c_{e}^{0}(x,t)} {\partial x_{i}} +\bar{ c}_{e}^{1}(x,t),{}\end{array}$$

(4.122b)

with the 1-periodic vectors V ^c and V ^e satisfying the “cell equations”

$$\displaystyle\begin{array}{rcl} \nabla _{\mathbf{y}} \cdot \left (A_{\epsilon }(\mathbf{y})\left [\nabla _{\mathbf{y}}V _{k}^{c}(\mathbf{y}) + \mathbf{e}_{ k}\right ]\right ) = 0,\quad \mathbf{y} \in \varOmega _{c}& &{}\end{array}$$

(4.123a)

$$\displaystyle\begin{array}{rcl} \nabla _{\mathbf{y}} \cdot \left (B_{\epsilon }(\mathbf{y})\left [\nabla _{\mathbf{y}}V _{k}^{e}(\mathbf{y}) + \mathbf{e}_{ k}\right ]\right ) = 0,\quad \mathbf{y} \in \varOmega _{e},& &{}\end{array}$$

(4.123b)

and

$$\displaystyle{ A_{\epsilon }(\mathbf{y})[\nabla _{\mathbf{y}}V _{k}^{c}(\mathbf{y}) + \mathbf{e}_{ k}] \cdot \mathbf{n} = 0 = B_{\epsilon }(\mathbf{y})[\nabla _{\mathbf{y}}V _{k}^{e}(\mathbf{y}) + \mathbf{e}_{ k}] \cdot \mathbf{n},\quad \mathbf{y} \in \varGamma, }$$

(4.124)

for k = 1, 2, 3, where e _k is the vector with components $e_{jk} =\delta _{jk}$.

In order to derive mean-field equations for the homogenized concentrations ${c}^{0},c_{e}^{0}$, it is necessary to proceed to $\mathcal{O}(1)$, for which

$$\displaystyle\begin{array}{rcl} \nabla _{\mathbf{x}} \cdot \left [A_{\epsilon }(\mathbf{y})(\nabla _{\mathbf{x}}{c}^{0} + \nabla _{\mathbf{ y}}{c}^{1})\right ] + \nabla _{\mathbf{ y}} \cdot \left [A_{\epsilon }(\mathbf{y})(\nabla _{\mathbf{x}}{c}^{1} + \nabla _{\mathbf{ y}}{c}^{2})\right ] = \frac{\partial {c}^{0}} {\partial t},\quad \mathbf{y} \in \varOmega _{c}& &{}\end{array}$$

(4.125a)

$$\displaystyle\begin{array}{rcl} \nabla _{\mathbf{x}} \cdot \left [B_{\epsilon }(\mathbf{y})(\nabla _{\mathbf{x}}c_{e}^{0} + \nabla _{\mathbf{ y}}c_{e}^{1})\right ] + \nabla _{\mathbf{ y}} \cdot \left [B_{\epsilon }(\mathbf{y})(\nabla _{\mathbf{x}}c_{e}^{1} + \nabla _{\mathbf{ y}}c_{e}^{2})\right ] = \frac{\partial c_{e}^{0}} {\partial t},\quad \mathbf{y} \in \varOmega _{e}& &{}\end{array}$$

(4.125b)

with boundary conditions

$$\displaystyle\begin{array}{rcl} A_{\epsilon }(\mathbf{y})[\nabla _{\mathbf{x}}{c}^{1} + \nabla _{\mathbf{ y}}{c}^{2}] \cdot \mathbf{n}& = f({c}^{0},c_{ e}^{0}),\quad \mathbf{y} \in \varGamma &{}\end{array}$$

(4.126a)

$$\displaystyle\begin{array}{rcl} B_{\epsilon }(\mathbf{y})[\nabla _{\mathbf{x}}{c}^{1} + \nabla _{\mathbf{ y}}{c}^{2}] \cdot \mathbf{n}& = -f({c}^{0},c_{ e}^{0}),\quad \mathbf{y} \in \varGamma.&{}\end{array}$$

(4.126b)

Integrating (4.125a) over Ω _c gives

$$\displaystyle{ \int _{\varOmega _{c}}\left (\nabla _{\mathbf{x}} \cdot \left [A_{\epsilon }(\mathbf{y})(\nabla _{\mathbf{x}}{c}^{0} + \nabla _{\mathbf{ y}}{c}^{1})\right ] + \nabla _{\mathbf{ y}} \cdot \left [A_{\epsilon }(\mathbf{y})(\nabla _{\mathbf{x}}{c}^{1} + \nabla _{\mathbf{ y}}{c}^{2})\right ]\right )d\mathbf{y} = \vert \varOmega _{ c}\vert \frac{\partial {c}^{0}} {\partial t}, }$$

(4.127)

where |Ω _c | is the volume fraction of the unit cell occupied by the cytosol. Using (4.122a), the first term on the left-hand side becomes

$$\displaystyle\begin{array}{rcl} & & \int _{\varOmega _{c}}\nabla _{\mathbf{x}} \cdot \left [A_{\epsilon }(\mathbf{y})(\nabla _{\mathbf{x}}{c}^{0} + \nabla _{\mathbf{ y}}{c}^{1})\right ]d\mathbf{y} {}\\ & & \quad =\int _{\varOmega _{c}}\nabla _{\mathbf{x}} \cdot \left [A_{\epsilon }(\mathbf{y})(\nabla _{\mathbf{x}}{c}^{0} + \nabla _{\mathbf{ y}}\left (\sum _{i=1}^{3}V _{ i}^{c}(y)\frac{\partial {c}^{0}} {\partial x_{i}}\right )\right ]d\mathbf{y} {}\\ & & \quad = \nabla _{\mathbf{x}} \cdot \left (\tilde{A}_{\epsilon }\nabla _{\mathbf{x}}{c}^{0}\right ), {}\\ \end{array}$$

where

$$\displaystyle{ [\tilde{A}_{\epsilon }]_{ik} =\int _{\varOmega _{c}}\sum _{j}[A_{\epsilon }(\mathbf{y})]_{ij}\left [\frac{\partial V _{k}^{c}(\mathbf{y})} {\partial y_{j}} +\delta _{j,k}\right ]d\mathbf{y}. }$$

(4.128)

Finally, applying the divergence theorem to the second term on the left-hand side of (4.127),

$$\displaystyle\begin{array}{rcl} \int _{\varOmega _{c}}\nabla _{\mathbf{y}} \cdot \left [A_{\epsilon }(\mathbf{y})(\nabla _{\mathbf{x}}{c}^{1} + \nabla _{\mathbf{ y}}{c}^{2})\right ]d\mathbf{y}& =& \int _{\varGamma }A_{\epsilon }(\mathbf{y})(\nabla _{\mathbf{ x}}{c}^{1} + \nabla _{\mathbf{ y}}{c}^{2}) \cdot \mathbf{n}d\varGamma (\mathbf{y}) {}\\ & =& \int _{\varGamma }f({c}^{0},c_{ e}^{0})d\varGamma (y) {}\\ & =& \vert \varGamma \vert f({c}^{0},c_{ e}^{0}). {}\\ \end{array}$$

Combining all of the results, we obtain a macroscopic equation for c ⁰ given by

$$\displaystyle{ \vert \varOmega _{c}\vert \frac{\partial {c}^{0}} {\partial t} = \nabla _{\mathbf{x}} \cdot \left (\tilde{A}_{\epsilon }\nabla _{\mathbf{x}}{c}^{0}\right ) + \vert \varGamma \vert f({c}^{0},c_{ e}^{0}). }$$

(4.129)

Proceeding in an identical fashion, we also obtain a macroscopic equation for $c_{e}^{0}$:

$$\displaystyle{ \vert \varOmega _{e}\vert \frac{\partial c_{e}^{0}} {\partial t} = \nabla _{\mathbf{x}} \cdot \left (\tilde{B}_{\epsilon }\nabla _{\mathbf{x}}c_{e}^{0}\right ) -\vert \varGamma \vert f({c}^{0},c_{ e}^{0}), }$$

(4.130)

where $\tilde{B}_{\epsilon }$ is defined according to (4.128) with A _ε replaced by B _ε on the right-hand side. Equations (4.129) and (4.130) constitute a bidomain model with homogenized diffusion tensors $\tilde{A}_{\epsilon }$ and $\tilde{B}_{\epsilon }$. The latter can be determined by solving the cell equations (4.123) and (4.124) for V ^c and V ^e. Goel et al. [224] calculated effective diffusion tensors for several different ER microstructures. They found that diffusion in the cytoplasm could be halved as the ER volume fraction increased from 0 to 0.9.