Population Density and Moment-based Approaches to Modeling Domain Calcium-mediated Inactivation of L-type Calcium Channels.

We present a population density and moment-based description of the stochastic dynamics of domain [Formula: see text]-mediated inactivation of L-type [Formula: see text] channels. Our approach accounts for the effect of heterogeneity of local [Formula: see text] signals on whole cell [Formula: see text] currents; however, in contrast with prior work, e.g., Sherman et al. (Biophys J 58(4):985-995, 1990), we do not assume that [Formula: see text] domain formation and collapse are fast compared to channel gating. We demonstrate the population density and moment-based modeling approaches using a 12-state Markov chain model of an L-type [Formula: see text] channel introduced by Greenstein and Winslow (Biophys J 83(6):2918-2945, 2002). Simulated whole cell voltage clamp responses yield an inactivation function for the whole cell [Formula: see text] current that agrees with the traditional approach when domain dynamics are fast. We analyze the voltage-dependence of [Formula: see text] inactivation that may occur via slow heterogeneous domain [[Formula: see text]]. Next, we find that when channel permeability is held constant, [Formula: see text]-mediated inactivation of L-type channels increases as the domain time constant increases, because a slow domain collapse rate leads to increased mean domain [[Formula: see text]] near open channels; conversely, when the maximum domain [[Formula: see text]] is held constant, inactivation decreases as the domain time constant increases. Comparison of simulation results using population densities and moment equations confirms the computational efficiency of the moment-based approach, and enables the validation of two distinct methods of truncating and closing the open system of moment equations. In general, a slow domain time constant requires higher order moment truncation for agreement between moment-based and population density simulations.

Ca(2+)-activation kinetics modulate successive puff/spark amplitude, duration and inter-event-interval correlations in a Langevin model of stochastic Ca(2+) release.

Through theoretical analysis of the statistics of stochastic calcium (Ca(2+)) release (i.e., the amplitude, duration and inter-event interval of simulated Ca(2+) puffs and sparks), we show that a Langevin description of the collective gating of Ca(2+) channels may be a good approximation to the corresponding Markov chain model when the number of Ca(2+) channels per Ca(2+) release unit (CaRU) is in the physiological range. The Langevin description of stochastic Ca(2+) release facilitates our investigation of correlations between successive puff/spark amplitudes, durations and inter-spark intervals, and how such puff/spark statistics depend on the number of channels per release site and the kinetics of Ca(2+)-mediated inactivation of open channels. When Ca(2+) inactivation/de-inactivation rates are intermediate-i.e., the termination of Ca(2+) puff/sparks is caused by an increase in the number of inactivated channels-the correlation between successive puff/spark amplitudes is negative, while the correlations between puff/spark amplitudes and the duration of the preceding or subsequent inter-spark interval are positive. These correlations are significantly reduced or change signs when inactivation/de-inactivation rates are extreme (slow or fast) and puff/sparks terminate via stochastic attrition.

Calcium homeostasis in a local/global whole cell model of permeabilized ventricular myocytes with a Langevin description of stochastic calcium release.

Population density approaches to modeling local control of Ca(2+)-induced Ca(2+) release in cardiac myocytes can be used to construct minimal whole cell models that accurately represent heterogeneous local Ca(2+) signals. Unfortunately, the computational complexity of such "local/global" whole cell models scales with the number of Ca(2+) release unit (CaRU) states, which is a rapidly increasing function of the number of ryanodine receptors (RyRs) per CaRU. Here we present an alternative approach based on a Langevin description of the collective gating of RyRs coupled by local Ca(2+) concentration ([Ca(2+)]). The computational efficiency of this approach no longer depends on the number of RyRs per CaRU. When the RyR model is minimal, Langevin equations may be replaced by a single Fokker-Planck equation, yielding an extremely compact and efficient local/global whole cell model that reproduces and helps interpret recent experiments that investigate Ca(2+) homeostasis in permeabilized ventricular myocytes. Our calculations show that elevated myoplasmic [Ca(2+)] promotes elevated network sarcoplasmic reticulum (SR) [Ca(2+)] via SR Ca(2+)-ATPase-mediated Ca(2+) uptake. However, elevated myoplasmic [Ca(2+)] may also activate RyRs and promote stochastic SR Ca(2+) release, which can in turn decrease SR [Ca(2+)]. Increasing myoplasmic [Ca(2+)] results in an exponential increase in spark-mediated release and a linear increase in nonspark-mediated release, consistent with recent experiments. The model exhibits two steady-state release fluxes for the same network SR [Ca(2+)] depending on whether myoplasmic [Ca(2+)] is low or high. In the later case, spontaneous release decreases SR [Ca(2+)] in a manner that maintains robust Ca(2+) sparks.

Discrete-state stochastic models of calcium-regulated calcium influx and subspace dynamics are not well-approximated by ODEs that neglect concentration fluctuations.

Cardiac myocyte calcium signaling is often modeled using deterministic ordinary differential equations (ODEs) and mass-action kinetics. However, spatially restricted "domains" associated with calcium influx are small enough (e.g., 10(-17) liters) that local signaling may involve 1-100 calcium ions. Is it appropriate to model the dynamics of subspace calcium using deterministic ODEs or, alternatively, do we require stochastic descriptions that account for the fundamentally discrete nature of these local calcium signals? To address this question, we constructed a minimal Markov model of a calcium-regulated calcium channel and associated subspace. We compared the expected value of fluctuating subspace calcium concentration (a result that accounts for the small subspace volume) with the corresponding deterministic model (an approximation that assumes large system size). When subspace calcium did not regulate calcium influx, the deterministic and stochastic descriptions agreed. However, when calcium binding altered channel activity in the model, the continuous deterministic description often deviated significantly from the discrete stochastic model, unless the subspace volume is unrealistically large and/or the kinetics of the calcium binding are sufficiently fast. This principle was also demonstrated using a physiologically realistic model of calmodulin regulation of L-type calcium channels introduced by Yue and coworkers.

Reduction of calcium release site models via moment fitting of phase-type distributions.

Models of calcium (Ca(2 +)) release sites derived from continuous-time Markov chain (CTMC) models of intracellular Ca(2 +) channels exhibit collective gating reminiscent of the experimentally observed phenomenon of Ca(2 +) puffs and sparks. In order to overcome the state-space explosion that occurs in compositionally defined Ca(2 +) release site models, we have implemented an automated procedure for model reduction that replaces aggregated states of the full release site model with much simpler CTMCs that have similar within-group phase-type sojourn times and inter-group transitions. Error analysis based on comparison of full and reduced models validates the method when applied to release site models composed of 20 three-state channels that are both activated and inactivated by Ca(2 +). Although inspired by existing techniques for fitting moments of phase-type distributions, the automated reduction method for compositional Ca(2 +) release site models is unique in several respects and novel in this biophysical context.

Spontaneous Ca2+ sparks and Ca2+ homeostasis in a minimal model of permeabilized ventricular myocytes.

Many issues remain unresolved concerning how local, subcellular Ca(2+) signals interact with bulk cellular concentrations to maintain homeostasis in health and disease. To aid in the interpretation of data obtained in quiescent ventricular myocytes, we present here a minimal whole cell model that accounts for both localized (subcellular) and global (cellular) aspects of Ca(2+) signaling. Using a minimal formulation of the distribution of local [Ca(2+)] associated with a large number of Ca(2+)-release sites, the model simulates both random spontaneous Ca(2+) sparks and the changes in myoplasmic and sarcoplasmic reticulum (SR) [Ca(2+)] that result from the balance between stochastic release and reuptake into the SR. Ca(2+)-release sites are composed of clusters of two-state ryanodine receptors (RyRs) that exhibit activation by local cytosolic [Ca(2+)] but no inactivation or regulation by luminal Ca(2+). Decreasing RyR open probability in the model causes a decrease in aggregate release flux and an increase in SR [Ca(2+)], regardless of whether RyR inhibition is mediated by a decrease in RyR open dwell time or an increase in RyR closed dwell time. The same balance of stochastic release and reuptake can be achieved, however, by either high-frequency/short-duration or low-frequency/long-duration Ca(2+) sparks. The results are well correlated with recent experimental observations using pharmacological RyR inhibitors and clarify those aspects of the release-reuptake balance that are inherent to the coupling between local and global Ca(2+) signals and those aspects that depend on molecular-level details. The model of Ca(2+) sparks and homeostasis presented here can be a useful tool for understanding changes in cardiac Ca(2+ )release resulting from drugs, mutations, or acquired diseases.

Models of cardiac excitation-contraction coupling in ventricular myocytes.

Mathematical and computational modeling of cardiac excitation-contraction coupling has produced considerable insights into how the heart muscle contracts. With the increase in biophysical and physiological data available, the modeling has become more sophisticated with investigations spanning in scale from molecular components to whole cells. These modeling efforts have provided insight into cardiac excitation-contraction coupling that advanced and complemented experimental studies. One goal is to extend these detailed cellular models to model the whole heart. While this has been done with mechanical and electrophysiological models, the complexity and fast time course of calcium dynamics have made inclusion of detailed calcium dynamics in whole heart models impractical. Novel methods such as the probability density approach and moment closure technique which increase computational efficiency might make this tractable.

Markov chain models of coupled calcium channels: Kronecker representations and iterative solution methods.

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of stochastic calcium excitability (i.e., calcium puffs and sparks). Calcium release site models are stochastic automata networks that involve many functional transitions, that is, the transition probabilities of each channel depend on the local calcium concentration and thus the state of the other channels. We present a Kronecker-structured representation for calcium release site models and perform benchmark stationary distribution calculations using both exact and approximate iterative numerical solution techniques that leverage this structure. When it is possible to obtain an exact solution, response measures such as the number of channels in a particular state converge more quickly using the iterative numerical methods than occupation measures calculated via Monte Carlo simulation. In particular, multi-level methods provide excellent convergence with modest additional memory requirements for the Kronecker representation of calcium release site models. When an exact solution is not feasible, iterative approximate methods based on the power method may be used, with performance similar to Monte Carlo estimates. This suggests approximate methods with multi-level iterative engines as a promising avenue of future research for large-scale calcium release site models.

Calcium-dependent inactivation and the dynamics of calcium puffs and sparks.

Localized intracellular Ca(2+) elevations known as puffs and sparks arise from the cooperative activity of inositol 1,4,5-trisphosphate receptor Ca(2+) channels (IP(3)Rs) and ryanodine receptor Ca(2+) channels (RyRs) clustered at Ca(2+) release sites on the surface of the endoplasmic reticulum or sarcoplasmic reticulum. When Markov chain models of these intracellular Ca(2+)-regulated Ca(2+) channels are coupled via a mathematical representation of a Ca(2+) microdomain, simulated Ca(2+) release sites may exhibit the phenomenon of "stochastic Ca(2+) excitability" reminiscent of Ca(2+) puffs and sparks where channels open and close in a concerted fashion. To clarify the role of Ca(2+) inactivation of IP(3)Rs and RyRs in the dynamics of puffs and sparks, we formulate and analyze Markov chain models of Ca(2+) release sites composed of 10-40 three-state intracellular Ca(2+) channels that are inactivated as well as activated by Ca(2+). We study how the statistics of simulated puffs and sparks depend on the kinetics and dissociation constant of Ca(2+) inactivation and find that puffs and sparks are often less sensitive to variations in the number of channels at release sites and strength of coupling via local [Ca(2+)] when the average fraction of inactivated channels is significant. Interestingly, we observe that the single channel kinetics of Ca(2+) inactivation influences the thermodynamic entropy production rate of Markov chain models of puffs and sparks. While excessively fast Ca(2+) inactivation can preclude puffs and sparks, moderately fast Ca(2+) inactivation often leads to time-irreversible puffs and sparks whose termination is facilitated by the recruitment of inactivated channels throughout the duration of the puff/spark event. On the other hand, Ca(2+) inactivation may be an important negative feedback mechanism even when its time constant is much greater than the duration of puffs and sparks. In fact, slow Ca(2+) inactivation can lead to release sites with a substantial fraction of inactivated channels that exhibit puffs and sparks that are nearly time-reversible and terminate without additional recruitment of inactivated channels.

Modeling local and global intracellular calcium responses mediated by diffusely distributed inositol 1,4,5-trisphosphate receptors.

Considerable insight into intracellular Ca2+ responses has been obtained through the development of whole cell models that are based on molecular mechanisms, e.g., single channel kinetics of the inositol 1,4,5-trisphosphate (IP3) receptor Ca2+ channel. However, a limitation of most whole cell models to date is the assumption that IP3 receptor Ca2+ channels (IP3Rs) are globally coupled by a "continuously stirred" bulk cytosolic [Ca2+], when in fact open IP3Rs experience elevated "domain" Ca2+ concentrations. Here we present a 2N+2-compartment whole cell model of local and global Ca2+ responses mediated by N=100,000 diffusely distributed IP3Rs, each represented by a four-state Markov chain. Two of these compartments correspond to bulk cytosolic and luminal Ca2+ concentrations, and the remaining 2N compartments represent time-dependent cytosolic and luminal Ca2+ domains associated with each IP3R. Using this Monte Carlo model as a starting point, we present an alternative formulation that solves a system of advection-reaction equations for the probability density of cytosolic and luminal domain [Ca2+] jointly distributed with IP3R state. When these equations are coupled to ordinary differential equations for the bulk cytosolic and luminal [Ca2+], a realistic but minimal model of whole cell Ca2+ dynamics is produced that accounts for the influence of local Ca2+ signaling on channel gating and global Ca2+ responses. The probability density approach is benchmarked and validated by comparison to Monte Carlo simulations, and the two methods are shown to agree when the number of Ca2+ channels is large (i.e., physiologically realistic). Using the probability density approach, we show that the time scale of Ca2+ domain formation and collapse (both cytosolic and luminal) may influence global Ca2+ oscillations, and we derive two reduced models of global Ca2+ dynamics that account for the influence of local Ca2+ signaling on global Ca2+ dynamics when there is a separation of time scales between the stochastic gating of IP3Rs and the dynamics of domain Ca2+.

Calcium release site ultrastructure and the dynamics of puffs and sparks.

When Markov chain models of intracellular Ca(2+)-regulated Ca(2+) channels are coupled via a mathematical representation of a Ca(2+) microdomain, simulated Ca(2+) release sites may exhibit the phenomenon of 'stochastic Ca(2+) excitability' reminiscent of Ca(2+) puffs and sparks. Interestingly, some single-channel models that include Ca(2+) inactivation are not particularly sensitive to channel density, so long as the requirement for inter-channel communication is satisfied, while other single-channel models that do not include Ca(2+) inactivation open and close synchronously only when the channel density is in a prescribed range. This observation led us to hypothesize that single-channel models with Ca(2+) inactivation would be less sensitive to the details of release site ultrastructure than models that lack a slow Ca(2+) inactivation process. To determine if this was the case, we simulated Ca(2+) release sites composed of instantaneously coupled Ca(2+)-regulated Ca(2+) channels whose random spatial locations were chosen from a uniform distribution on a disc of specified radius and compared the resulting release site dynamics to simulations with channels arranged on hexagonal lattices. Analysis of puff/spark statistics confirmed our hypothesis that puffs and sparks are less sensitive to the spatial organization of release sites when the single-channel model includes a slow inactivation process. We also investigated the validity of several different mean-field reductions that do not explicitly account for the details of release site ultrastructure. The most successful approximation maintains a distinction between each channel's substantial influence on its own stochastic gating and the collective contribution of elevated [Ca(2+)] from neighbouring channels.

Markov chain models of coupled intracellular calcium channels: Kronecker structured representations and benchmark stationary distribution calculations.

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of stochastic calcium excitability (i.e., calcium puffs and sparks). We present a Kronecker structured representation for calcium release site models and perform benchmark stationary distribution calculations using numerical iterative solution techniques that leverage this structure. In this context we find multi-level methods and certain preconditioned projection methods superior to simple Gauss-Seidel type iterations. Response measures such as the number of channels in a particular state converge more quickly using these numerical iterative methods than occupation measures calculated via Monte Carlo simulation.

The dynamics of luminal depletion and the stochastic gating of Ca2+-activated Ca2+ channels and release sites.

Single channel models of intracellular calcium (Ca(2+)) channels such as the 1,4,5-trisphosphate receptor and ryanodine receptor often assume that Ca(2+)-dependent transitions are mediated by constant background cytosolic [Ca(2+)]. This assumption neglects the fact that Ca(2+) released by open channels may influence subsequent gating through the processes of Ca(2+)-activation or inactivation. Similarly, the influence of the dynamics of luminal depletion on the stochastic gating of intracellular Ca(2+) channels is often neglected, in spite of the fact that the sarco/endoplasmic reticulum [Ca(2+)] near the luminal face of intracellular Ca(2+) channels influences the driving force for Ca(2+), the rate of Ca(2+) release, and the magnitude and time course of the consequent increase in cytosolic domain [Ca(2+)]. Here we analyze how the steady-state open probability of several minimal Ca(2+)-regulated Ca(2+) channel models depends on the conductance of the channel and the time constants for the relaxation of elevated cytosolic [Ca(2+)] and depleted luminal [Ca(2+)] to the bulk [Ca(2+)] of both compartments. Our approach includes Monte Carlo simulation as well as numerical solution of a system of advection-reaction equations for the multivariate probability density of elevated cytosolic [Ca(2+)] and depleted luminal [Ca(2+)] conditioned on each state of the stochastically gating channel. Both methods are subsequently used to study the role of luminal depletion in the dynamics of Ca(2+) puff/spark termination in release sites composed of Ca(2+) channels that are activated, but not inactivated, by cytosolic Ca(2+). The probability density approach shows that such minimal Ca(2+) release site models may exhibit puff/spark-like dynamics in either of two distinct parameter regimes. In one case, puffs/spark termination is due to the process of stochastic attrition and facilitated by rapid Ca(2+) domain collapse [cf. DeRemigio, H., Smith, G., 2005. The dynamics of stochastic attrition viewed as an absorption time on a terminating Markov chain. Cell Calcium 38, 73-86]. In the second case, puff/spark termination is promoted by the local depletion of luminal Ca(2+).

A multivariate population density model of the dLGN/PGN relay.

Using a population density approach we study the dynamics of two interacting collections of integrate-and-fire-or-burst (IFB) neurons representing thalamocortical (TC) cells from the dorsal lateral geniculate nucleus (dLGN) and thalamic reticular (RE) cells from the perigeniculate nucleus (PGN). Each population of neurons is described by a multivariate probability density function that satisfies a conservation equation with appropriately defined probability fluxes and boundary conditions. The state variables of each neuron are the membrane potential and the inactivation gating variable of the low-threshold Ca2+ current I(T). The synaptic coupling of the populations and external excitatory drive are modeled by instantaneous jumps in the membrane potential of postsynaptic neurons. The population density model is validated by comparing its response to time-varying retinal input to Monte Carlo simulations of the corresponding IFB network composed of 100 to 1,000 cells per population. In the absence of retinal input, the population density model exhibits rhythmic bursting similar to the 7 to 14 Hz oscillations associated with slow wave sleep that require feedback inhibition from RE to TC cells. When the TC and RE cell potassium leakage conductances are adjusted to represent cholingergic neuromodulation and arousal of the network, rhythmic bursting of the probability density model may either persists or be eliminated depending on the number of excitatory (TC to RE) or inhibitory (RE to TC) connections made by each presynaptic cell. When the probability density model is stimulated with constant retinal input (10-100 spikes/sec), a wide range of responses are observed depending on cellular parameters and network connectivity. These include asynchronous burst and tonic spikes, sleep spindle-like rhythmic bursting, and oscillations in population firing rate that are distinguishable from sleep spindles due to their amplitude, frequency, or the presence of tonic spikes. In this context of dLGN/PGN network modeling, we find the population density approach using 2,500 mesh points and resolving membrane voltage to 0.7 mV is over 30 times more efficient than 1,000-cell Monte Carlo simulations.

The dynamics of stochastic attrition viewed as an absorption time on a terminating Markov chain.

Localized Ca(2+) elevations known as Ca(2+) puffs and sparks are cellular signals that arise from the cooperative activity of clusters of inositol 1,4,5-trisphosphate receptors and ryanodine receptors clustered at Ca(2+) release sites on the surface of the endoplasmic reticulum or sarcoplasmic reticulum. When Markov chain models of these intracellular Ca(2+)-regulated Ca(2+) channels are coupled via a mathematical representation of Ca(2+) microdomain, simulated Ca(2+) release sites may exhibit the phenomenon of "stochastic Ca(2+) excitability" where the inositol 1,4,5-trisphosphate receptors (IP(3)Rs) or ryanodine receptors (RyRs) open and close in a concerted fashion. Interestingly, under some conditions simulated puffs and sparks can be observed even when the single-channel model used does not include slow Ca(2+) inactivation or, indeed, any long-lived closed/refractory state [V. Nguyen, R. Mathias, G. Smith, Stochastic automata network descriptor for Markov chain models of instantaneously-coupled intracellular Ca(2+) channels, Bull. Math. Biol. 67 (2005) 393-432]. In this case, termination of the localized Ca(2+) elevation occurs when all of the intracellular channels at a release site simultaneously close through a process referred to as stochastic attrition [M. Stern, Theory of excitation-contraction coupling in cardiac muscle, Biophys. J. 63 (1992) 497-517]. In this paper, we investigate the statistical properties of stochastic attrition viewed as an absorption time on a terminating Markov chain that represents a Ca(2+) release site composed of N two-state channels that are activated by Ca(2+). Assuming that the local [Ca(2+)] experienced by a channel depends only on the number of open channels at the Ca(2+) release site (i.e., instantaneous mean-field coupling [ibid.], we derive the probability distribution function for the time until stochastic attrition occurs and present an analytical formula for the expectation of this random variable. We explore how the contribution of stochastic attrition to the termination of Ca(2+) puffs and sparks depends on the number of channels at a release site, the source amplitude of the channels (i.e., the strength of the coupling), the background [Ca(2+)], channel kinetics, and the cooperactivity of Ca(2+) binding. Because we explicitly model the Ca(2+) regulation of the intracellular channels, our results differ markedly from (and in fact generalize) preliminary analyses that assume the intracellular Ca(2+) channels are uncoupled and consequently independent.

The effect of residual Ca2+ on the stochastic gating of Ca2+-regulated Ca2+ channel models.

Single-channel models of intracellular Ca(2+) channels such as the inositol 1,4,5-trisphosphate receptor and ryanodine receptor often assume that Ca(2+)-dependent transitions are mediated by a constant background [Ca(2+)] as opposed to a dynamic [Ca(2+)] representing the formation and collapse of a localized Ca(2+) domain. This assumption neglects the fact that Ca(2+) released by open intracellular Ca(2+) channels may influence subsequent gating through the processes of Ca(2+)-activation or -inactivation. We study the effect of such "residual Ca(2+)" from previous channel opening on the stochastic gating of minimal and realistic single-channel models coupled to a restricted cytoplasmic compartment. Using Monte Carlo simulation as well as analytical and numerical solution of a system of advection-reaction equations for the probability density of the domain [Ca(2+)] conditioned on the state of the channel, we determine how the steady-state open probability (p(open)) of single-channel models of Ca(2+)-regulated Ca(2+) channels depends on the time constant for Ca(2+) domain formation and collapse. As expected, p(open) for a minimal model including Ca(2+) activation increases as the domain time constant becomes large compared to the open and closed dwell times of the channel, that is, on average the channel is activated by residual Ca(2+) from previous openings. Interestingly, p(open) for a channel model that is inactivated by Ca(2+) also increases as a function of the domain time constant when the maximum domain [Ca(2+)] is fixed, because slow formation of the Ca(2+) domain attenuates Ca(2+)-mediated inactivation. Conversely, when the source amplitude of the channel is fixed, increasing the domain time constant leads to elevated domain [Ca(2+)] and decreased open probability. Consistent with these observations, a realistic De Young-Keizer-like IP(3)R model responds to residual Ca(2+) with a steady-state open probability that is a monotonic function of the domain time constant, though minimal models that include both Ca(2+)-activation and -inactivation show more complex behavior. We show how the probability density approach described here can be generalized for arbitrarily complex channel models and for any value of the domain time constant. In addition, we present a comparatively simple numerical procedure for estimating p(open) for models of Ca(2+)-regulated Ca(2+) channels in the limit of a very fast or very slow Ca(2+) domain. When the ordinary differential equation for the [Ca(2+)] in a restricted cytoplasmic compartment is replaced by a partial differential equation for the buffered diffusion of intracellular Ca(2+) in a homogeneous isotropic cytosol, we find the dependence of p(open) on the buffer time constant is qualitatively similar to the above-mentioned results.

A stochastic automata network descriptor for Markov chain models of instantaneously coupled intracellular Ca2+ channels.

Although there is consensus that localized Ca(2+) elevations known as Ca(2+) puffs and sparks arise from the cooperative activity of intracellular Ca(2+) channels, the precise relationship between single-channel kinetics and the collective phenomena of stochastic Ca(2+) excitability is not well understood. Here we present a formalism by which mathematical models for Ca(2+)-regulated Ca(2+) release sites are derived from stochastic models of single-channel gating that include Ca(2+) activation, Ca(2+) inactivation, or both. Such models are stochastic automata networks (SANs) that involve a large number of functional transitions, that is, the transition probabilities of the infinitesimal generator matrix of one of the automata (i.e., an individual channel) may depend on the local [Ca(2+)] and thus the state of the other channels. Simulation and analysis of the SAN descriptors representing homogeneous clusters of intracellular Ca(2+) channels show that (1) release site density can modify both the steady-state open probability and stochastic excitability of Ca(2+) release sites, (2) Ca(2+) inactivation is not a requirement for Ca(2+) puffs or sparks, and (3) a single-channel model with a bell-shaped open probability curve does not lead to release site activity that is a biphasic function of release site density. These findings are obtained using iterative, memory-efficient methods (novel in this biophysical context and distinct from Monte Carlo simulation) that leverage the highly structured SAN descriptor to unambiguously calculate the steady-state probability of each release site configuration and puff statistics such as puff duration and inter-puff interval. The validity of a mean field approximation that neglects the spatial organization of Ca(2+) release sites is also discussed.