Perturbation analysis of spontaneous action potential initiation by stochastic ion channels.

A stochastic interpretation of spontaneous action potential initiation is developed for the Morris-Lecar equations. Initiation of a spontaneous action potential can be interpreted as the escape from one of the wells of a double well potential, and we develop an asymptotic approximation of the mean exit time using a recently developed quasistationary perturbation method. Using the fact that the activating ionic channel's random openings and closings are fast relative to other processes, we derive an accurate estimate for the mean time to fire an action potential (MFT), which is valid for a below-threshold applied current. Previous studies have found that for above-threshold applied current, where there is only a single stable fixed point, a diffusion approximation can be used. We also explore why different diffusion approximation techniques fail to estimate the MFT.

Exact reductions of Markovian dynamics for ion channels with a single permissive state.

We show that the cooperative model for the kinetics of a tetrameric potassium ion channel derived in Nekouzadeh et al. (Biophys J 95(7):3510-3520, 2008) is an invariant manifold reduction of the full master equation for the channel kinetics. We further establish the validity of this reduction for ion channel models consisting of multiple independent subunits with cooperative transitions from a single permissive state to a conducting state. Finally, we conclude that solutions of the reduced model are globally asymptotically stable solutions of the full master equation system.

Invariant manifold reductions for Markovian ion channel dynamics.

We show that many Markov models of ion channel kinetics have globally attracting stable invariant manifolds, even when the Markov process is time dependent. The primary implication of this is that, since the dimension of the invariant manifold is often substantially smaller than the full master equation system, simulations of ion channel kinetics can be substantially simplified, with no approximation. We show that this applies to certain models of potassium channels, sodium channels, ryanodine receptors and IP(3) receptors. We also use this to show that the original Hodgkin-Huxley formulations of potassium channel conductance and sodium channel conductance are the exact solutions of full Markov models for these channels.

Stochastic calcium oscillations.

While the oscillatory release of calcium from intracellular stores is comprised of fundamentally stochastic events, most models of calcium oscillations are deterministic. As a result, the transition to calcium oscillations as parameters, such as IP(3) concentration, are changed is not described correctly. The fundamental difficulty is that whole-cell models of calcium dynamics are based on the assumptions that the calcium concentration is spatially homogeneous, and that there are a sufficiently large number of release sites per unit volume so that the law of large numbers is applicable. For situations where these underlying assumptions are not applicable, a new modelling approach is needed. In this paper, we present a model and its analysis of calcium dynamics that incorporates the fundamental stochasticity of release events. The model is based on the assumptions that release events are rapid, while reactivation is slow. The model presented here is comprised of two parts. In the first, a stochastic version of the fire-diffuse-fire model is studied in order to understand the spark-to-wave transition and the probability of sparks resulting in abortive waves versus whole-cell calcium release. In the second, this information about the spark-to-wave transition is incorporated into a stochastic model (a Chapman-Kolmogorov equation) that tracks the number of activated and inactivated calcium release sites as a function of time. By solving this model numerically, information about the timing of whole-cell calcium release is obtained. The results of this analysis show a transition to oscillations that agrees well with data and with Monte Carlo simulations.