International Journal of Stochastic Analysis

Volume 2015, Article ID 658342, 13 pages

http://dx.doi.org/10.1155/2015/658342

## Stochastic Nonlinear Equations Describing the Mesoscopic Voltage-Gated Ion Channels

^{1}Centro de Análisis Estocástico, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, 7820436 Santiago, Chile^{2}Facultad de Ciencias Naturales y Exactas, Universidad de Playa Ancha, Leopoldo Carvallo 270, 2360696 Valparaíso, Chile

Received 7 August 2014; Revised 22 January 2015; Accepted 5 March 2015

Academic Editor: Nikolai N. Leonenko

Copyright © 2015 Mauricio Tejo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose a stochastic nonlinear system to model the gating activity coupled with the membrane potential for a typical neuron. It distinguishes two different levels: a macroscopic one, for the membrane potential, and a mesoscopic one, for the gating process through the movement of its voltage sensors. Such a nonlinear system can be handled to form a Hodgkin-Huxley-like model, which links those two levels unlike the original deterministic Hodgkin-Huxley model which is positioned at a macroscopic scale only. Also, we show that an interacting particle system can be used to approximate our model, which is an approximation technique similar to the jump Markov processes, used to approximate the original Hodgkin-Huxley model.

#### 1. Introduction

In 1952, Hodgkin and Huxley proposed their famous model for the membrane potential dynamic of a typical neuron through observations made in the squid giant axon. Basically, the model is based on the mean behavior of the potassium () and sodium () ion channels, whose states (open/closed) are determined by a* gating process* acting inside those channels. The proposed system is 4-dimensional nonlinear equations, fully coupled through the membrane potential (or voltage) and the probability that a representative gate/ion channel of each species is open (see [1]). As it describes deterministically the general membrane potential through such a mean field approach, it might be indicated as a macroscopic process with respect to all the processes involved in the internal neuronal process.

In order to explain the internal fluctuations or its channel noise in a neuron (see [2]), stochastic versions of the original Hodgkin-Huxley model have been suggested. A bunch of them are based on taking into account the underlying stochasticity in the gating activity, by describing the open/closed processes as jump voltage-coupled Markov processes and by using empirical measures instead of probability measures of the corresponding open times (see [3–6]). In a certain way, such methods are consistent because when the number of channels or gates goes towards infinity the original deterministic Hodgkin-Huxley equations are recovered (see [5, 6]).

In this work, beyond considering the inherent stochasticity of such phenomena, we want to include a continuous dynamical model for the gating activity, because the discrete two-state (open/closed) point of view of its modeling is just an approximation of the corresponding metastable states of the continuous movement of the proteins involved.

The voltage sensors are proteins whose conformational positions are responsible for the states of the gating activity (see [7]). This internal process is known as the* voltage-gated ion channel process*. A continuous state space stochastic process seems to be suitable to describe the position of such proteins, and it is what we are going to propose. Due to the nature and scale on which we will tackle this, we say that the dynamics representing such voltage sensors are located at a mesoscopic scale or level. Thus, our proposed system will consider two different levels, the macroscopic and the mesoscopic one, which are fully coupled through a link voltage-dependent function. We will call our model* Hodgkin-Huxley-like model*, which will conserve the main structural characteristics of the original voltage equation. Thus, placing the gating phenomenon from a continuous point of view is basically the main motivation for conducting this work.

Our model is a stochastic nonlinear system, where its nonlinearity is due to the intervention of the probability law of the stochastic process in its dynamic (see [8–12]). In our case, such a probability will represent the probability that a representative voltage sensor of any species “pulls up” its corresponding gate, which is equivalent to the probability that the corresponding channel is open in the original or classical model.

The mathematical consistence of our proposal is evaluated by means of the approximation via a particle system, that is, a system with several interacting voltage sensors. It means that our model will meet with features similar to those of the original macroscopic case, with respect to the convergence of empirical measures to the corresponding probability law. In our case, this convergence property by using empirical measures from a stochastic particle system approximating our* ideal* system is known as* propagation of chaos*.

Specifically, our propagation of chaos result is as follows: consider the voltage equation depending on empirical measures instead of the probability that the (potassium, sodium) ion channels are in an open state, as in the original Hodgkin-Huxley equations. Suppose that those empirical measures depend on the behavior of certain continuous state space stochastic processes representing the interacting voltage sensors of typical (potassium, sodium) ion channels. Then, when the number of sensors (or channels) goes towards infinity, we will recover a Hodgkin-Huxley-like system which has two remarkable fully coupled components: the voltage equation depending on the probability laws of the corresponding voltage sensor positions (macroscopic part) and the stochastic equations for the position of those voltage sensors (mesoscopic part).

This result extends earlier ideas for the description of the ion channel dynamic: from a discrete space state of open/closed gates to a continuous state space of the position of the voltage sensors. Our extension makes the introduction of the propagation of chaos property necessary, which comes to replace the approximation via jump voltage-coupled Markov processes.

Another nice property of our model is the* recurrence*, which has a biological interpretation. Also, this work generalizes the ideas in [13].

In Section 2, an introduction of those stochastic nonlinear processes, our model, and its main features are set.

In Section 3, the connection of our model with the original Hodgkin-Huxley model is given, as well as its mathematical justification.

Conclusions are given in Section 4.

#### 2. Our Model

##### 2.1. Nonlinear Processes and Propagation of Chaos

We are going to introduce some basic facts about a certain class of nonlinear stochastic differential equations which are useful to describe situations where individuals (particles, components of a system, etc.) are interacting with each other through a mean field force. Let be a continuous-time -valued stochastic process defined on some complete probability space satisfyingwhere is a -Brownian motion and . This kind of systems has been widely studied under different forms of and under extensions of the diffusion part (see, e.g., [8–12, 14]). As example, (1) is a special case of the nonlinear system considered in [11] and the existence and uniqueness of a solution of (1) are proved when (i.e., , where is the set of probability measures on satisfying , for ) and Lipschitz condition on (see Theorem 2.2 therein, which states the existence and uniqueness of a solution in law sense, although a stronger result is proved). That analysis is based on the contraction of the Wasserstein metric on , which is defined aswhere is the set of probability measures on such that the marginal law with respect to is and the marginal law with respect to is . The nice feature of this metric is that the metric space is complete and separable (see [15]). Here we say that (1) has a unique solution pathwise and in law, which for a usual Itô diffusion is analogous to saying that it has a unique strong solution.

The probability law of (1) follows a special parametrization of the McKean-Vlasov equation. This is a nonlinear equation whose general form is given bywhere is a measure on and is a test function with compact support and with derivatives of any order. In this case, it describes the Fokker-Planck equation for the temporal evolution of the law of (1) (also called Fokker-Planck McKean-Vlasov equation), where we have thatOther interesting examples arising in possibly singular cases of can be found in [10].

The law is considered to describe a mean field force. That is, two individuals from (1), and driven by independent and identically distributed (iid, for short) Brownian motions and , are “interacting” through its common law . But it is an* ideal* situation because, probabilistically, they are not interacting ( and are iid). It is a well-known fact that, under some conditions, we can approximate the law of (1) on bounded intervals of time by the interacting particle systemwhere the Brownian motions are iid. Here, the main result is as follows: for some fixed, if solves (1) driven by and , then uniformly on any finite time horizon (see, e.g., Theorem 2.3 in [11]). This property is known as propagation of chaos, term attributed to Kac ([16]).

In this work and under the reference of the Hodgkin-Huxley model ([1]), we are going to suggest that when the gating process in a typical potassium or sodium ion channel is seen as a continuous-space stochastic process describing the movement of a representative voltage sensor, the law of the opening times can be approximated via interacting particle systems as in (5). This comes to replace the classical view of the two-state jump voltage-coupled Markov process for the gating activity (see [3–6]). Due to the dimensional change, we say that the interacting particle system is located at a lower level than the general membrane potential. If the membrane potential is considered as a macroscopic process, which is natural since the voltage equation is describing the deterministic and general membrane potential through a mean field approach, the voltage sensor dynamic will be seen as a mesoscopic process (where for us, the ion dynamics are at microscopic scale).

##### 2.2. The Membrane Potential and the Voltage-Gated Process Seen as a Nonlinear System

To describe the coupled evolution of the membrane potential and the voltage-gated process through the position of the voltage sensors, we distinguish two different scales according to its nature: the first one follows a deterministic general dynamic (macroscopic), and the second one follows a stochastic local dynamic in a continuous state space (mesoscopic). The classical Hodgkin-Huxley equations explicitly describe the mean performance of the and channels depending on the voltage evolution. The paradigm is that each channel contains 4 gates, which can be in one of the two states: open or closed. A channel is in an open state (conductance) if all its gates are open. Otherwise, the channel is in a closed state (nonconductance).

Such gates and its corresponding voltage sensors are proteins and together they form the so-called voltage-gated process. A channel has its 4 gates of the same type and a channel has 3 of one type and 1 of another. Although all of this comes from the classical formulation of Hodgkin-Huxley model, as technology advances (see [17]), more precise descriptions about how those proteins operate have been given (see, e.g., [7]).

For our mathematical treatment, it is enough to consider only a simple system with 1 gate (voltage sensor) type (extension arises naturally as we will see later). A simple scheme for the states of an ion channel is depicted in Figure 1.