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  • Spontaneous excitability in the Morris–Lecar model with ion channel noise


    Noise induced excitability is studied in type I and II Morris–Lecar neurons subject to constant sub threshold input, where fluctuations arise from sodium and potassium ion channels. Ion channels open and close randomly, creating current fluctuations that can induce spontaneous firing of action potentials. Both noise sources are assumed to be weak so that spontaneous action potentials occur on a longer timescale than ion channel fluctuations. Asymptotic approximations of the stationary density function and most probable paths are developed to understand the role of channel noise in spontaneous excitability. Even though the deterministic dynamical behavior of type I and II action potentials differ, results show that a single mechanism explains how ion channel noise generates spontaneous action potentials.


    The Morris–Lecar (ML) equations were originally developed as a model of calcium dynamics in muscle fibers of the barnacle Balanus nubilus (Morris 1981). The ML equations can also be interpreted as simplified version of the Hodgkin–Huxley equations, a well known model of single neuron transmembrane voltage dynamics. The most widely used simplified version of the Hodgkin–Huxley equations is the so-called FitzHugh–Nagumo equations. Unlike the simpler FitzHugh–Nagumo equations, ML displays a richer set of dynamics, in particular, several different types of excitability.

    The ML equations are given by, \[\label{eq:28} C_{\rm m}\frac{dv}{dt} = x_{\infty}(v)f_{{\rm Na}}(v) + wf_{{\rm K}}(v) + f_{{\mathrm{leak}}}(v) + I_{{\mathrm{app}}}\] \[\frac{dw}{dt} = \frac{w_{\infty}(v) - w}{\tau_{w}(v)},\] where \(v\) is the transmembrane voltage and \(w\) represents the fraction of open \(\text{K}^{{+}}\) channels. The functions \(f_{i}(v) = g_{i}(v_{i}-v)\) determine the ionic currents. The fraction of open \(\text{Na}^{{+}}\) channels is assumed to be an instantaneous function of \(v\) with \[\label{eq:31} x_{\infty}(v) = (1 + \tanh(2(\gamma_{{\rm Na}}v + \kappa_{{\rm Na}})))/2.\] The steady state fraction and time scale for \(w\) are given by \[\label{eq:32} w_{\infty}(v)=(1 + \tanh(2(\gamma_{{\rm K}}v + \kappa_{{\rm K}})))/2,\quad \tau_{w}(v) = 2\beta_{{\rm K}}\cosh(\gamma_{{\rm K}}v+\kappa_{{\rm K}}),\] respectively. (See Appendix \ref{sec:appendix_params} for parameter values.)

    The deterministic ML model should be viewed as a mean field limit of a stochastic model that includes the random opening and closings of ion channels. Single channel opening and closing statistics can be measured experimentally. The channel variables that modify the ionic conductances represent the fraction of open channels. The fraction of open channels is a continuous, deterministic quantity if (i) the number of ion channels is taken to be infinite while the conductance of an single channel vanishes or (ii) channels open and close infinitely fast so that the fraction of open channels is an instantaneous function of the voltage. For the ML model, the potassium channel variable \(w\) is obtained by the former while the sodium channel variable \(x_{\infty}(v)\) is determined by the latter. Note that \(w\) is a dynamic variable with its own governing equation while \(x_{\infty}\) can be viewed as the quasi-steady-state fraction of open sodium channels.

    The deterministic ML model can display several different types of excitable behavior. In every case, there is a single stable fixed point \((v_{A}, w_{A})\) representing the resting state and we assume that the applied current \(I_{{\mathrm{app}}}\) is below threshold so that the deterministic system does not exhibit repetitive firing. Below threshold, only current fluctuations from stochastic ion channels can induce an excitable event. If current fluctuations push the system over a voltage threshold to the excited state, the voltage undergoes a transient spike called an action potential before returning to the resting voltage. We consider two situations. A type I neuron has three fixed points: a stable fixed point corresponding to the resting state, an unstable saddle, and an unstable fixed point corresponding to the excited state. A type II neuron has one fixed point corresponding to the stable resting state. Deterministic repetitive firing can occur in type I and II neurons when the input current is increased above threshold (Izhikevich 2000).

    In most physically relevant cases, the system is close to the deterministic limit so that the ion channel fluctuations are weak compared to the deterministic forces. In other words, a deterministic trajectory and a stochastic trajectory that share the same initial conditions are likely to remain close over sufficiently small time scales. This situation is commonly referred to as weak noise. Under weak noise conditions, a rare sequence of fluctuations can cause metastable dynamical behavior that the deterministic model cannot describe. Metastable behavior occurs on long timescales.

    A spontaneous excitable event can naturally be separated into two phases: the initiation phase and the excitation phase. The initiation phase is driven by ion channel fluctuations and is therefore a metastable transition. The excitation phase begins once fluctuations increase the voltage to a threshold. Then, the system undergoes a transient increase in voltage before returning to the stable fixed point. Unlike the initiation phase, the excitation phase is not metastable, instead being driven primarily by deterministic forces. If we can derive a description of the metastable initiation phase then it can be combined with the deterministic description of the excitation phase to obtain a complete picture of the spontaneous excitable event.

    In the weak noise limit, the probability that the process takes a particular path from point A to point B is sharply peaked along a most probable path (MPP). A MPP is a statistic, similar to the mode, of a probability distribution functional over the function space of continuous paths. Although they describe a stochastic process, MPPs themselves are not stochastic. One can show using large deviations theory (Freidlin 2012, Feng 2006) that the likelihood of deviating from the MPP is an exponentially decreasing function of the magnitude of the deviation. In other words, stochastic trajectories are highly likely to closely follow the MPP. In general, there are two kinds of MPPs. Any deterministic trajectory connecting point A to point B is a MPP. If there is no deterministic path that connects the two points and the transition is noise induced metastable transition, MPPs build action and become more improbable as the path gets longer. The action is a measure of how improbable the MPP is. By themselves, MPPs provide a good qualitative description of how different metastable transitions occur; they can be thought of as describing noise induced dynamical behavior.

    MPPs can also be used to approximate other important properties of the stochastic process. One can show that there is a nontrivial connection between MPPs that start at the stable fixed point and the stationary probability density function, which describes the relative fraction of time the system spends in different dynamical regimes and determines how rare excitable events are. Since stochastic trajectories leading from the stable resting voltage to the threshold of an excitable event are described by MPPs, it is no surprise that MPPs determine the asymptotics of the average metastable transition time, also called the mean first passage time (MFPT) or mean exit time.

    Several groups have studied stochastic conductance based single neuron models using large deviation theory (Berglund 2006, Khovanov 2013, Chow 1996). Until recently, it has only been possible to examine conductance models perturbed by a continuous Markov process. Channel noise can be approximated by a continuous Markov process, however it is well known that this can generate significant errors for metastable dynamics (Newby 2011, Newby 2013a). Recently, the authors have studied type II excitability in the stochastic ML model with channel noise, deriving MPPs using the WKB method (Keener 2011, Newby 2013). While the results showed excellent agrement with Monte Carlo simulations, a systematic connection between large deviation theory and the WKB method was not established.

    The WKB method is a practical tool used extensively to study metastability in continuous Markov processes and birth-death processes (Ludwig 1975, Dykman 1996, Maier 1997, Schuss 2010, Tél 1989). The connection between WKB and large deviation theory is well studied for continuous Markov process (Ludwig 1975) and for birth-death processes (Hanggi 1984, Peliti 1985). Establishing such a connection for the stochastic ML model is complicated by the presence of fast and slow variables in a stochastic process that has both continuous and discrete elements. However, due to recent advances in this area (Bressloff 2014, Kifer 2009), a systematic analysis is now possible. From a practical perspective, establishing a link between large deviation theory and the WKB method facilitates the development of numerical algorithms. MPPs are computed using the geometric minimum action method (GMAM) (Heymann 2008). We also develop an ordered upwind method (OUM) to compute the asymptotic approximation of the stationary density function, based on a similar algorithm for continuous Markov processes (Cameron 2012).

    The goal of this paper is to develop theory to describe the metastable behavior leading to different types of spontaneous excitation in the stochastic ML model. In particular, we develop asymptotic approximations of the MPP for a metastable excitable event and the stationary probability density function.

    The paper is organized as follows. In Section \ref{sec:stoch-model} we introduce the stochastic ML model. Then, in Section \ref{sec:mpp} we formulate an approximation of most probable paths, and show how they are connected to the stationary density function. We show how all relevant quantities can be calculated using the WKB method in Section \ref{sec:wkb}. Results for type I excitability are presented in Section \ref{sec:sp1}, including spontaneous action potentials and spontaneous bursting. In Section \ref{sec:sp2}, we present results for the ML model showing type II excitability.