Strong convergence with error estimates for a stochastic compartmental model of electrophysiology

TL;DR

This work develops a spatially extended stochastic electrophysiology model based on a PDMP, discretizing space into compartments each containing at most one ion channel. It proves a strong, almost-sure convergence to a PDE-ODE limit as the compartment size shrinks, with an explicit error bound optimized at $p=1/3$ (up to logarithmic factors). The authors introduce a multi-scale homogenization approach, including a corrector term and local spatial averaging, to bridge the stochastic lattice system and its continuum limit, and validate the theory with dedicated numerical simulations using Poisson-thinning based algorithms. The results illuminate how random ion-channel configurations in spatially structured neurons can be faithfully captured by a PDE framework, with potential extensions to more general channel distributions and defects. Overall, the paper advances the rigorous understanding of stochastic effects in spatial neural models and provides practical numerical tools for simulating PDMP-driven electrophysiology.

Abstract

This paper presents a rigorous mathematical analysis, alongside simulation studies, of a spatially extended stochastic electrophysiology model, the Hodgkin-Huxley model of the squid giant axon being a classical example. Although most studies in electrophysiology do not account for stochasticity, it is well known that ion channels regulating membrane voltage open and close randomly due to thermal fluctuations. We introduce a spatially extended compartmental model in which this stochastic behavior is captured through a piecewise-deterministic Markov process (PDMP). Space is discretized into n compartments each of which has at most one ion channel. We also devise a numerical method to simulate this stochastic model and illustrate the numerical method by simulation studies. We show that a classical system of partial differential equations (PDEs) approximates the stochastic system as $n \to \infty$. Unlike existing results, which focus on weak convergence or convergence in probability, we establish an almost sure convergence result with a precise error bound of order $n^{1/3}$. Our findings broaden the current understanding of stochastic effects in spatially structured neuronal models and have potential applications in studying random ion channel configurations in neurobiology. Additionally, our proof leverages ideas from homogenization theory in PDEs and can potentially be applied to other PDMPs or accommodate other ion channel distributions with random spacing or defects.

Figures (8)

• Figure 1: A discretized circle $\mathbb{S}_n:=\{kh\}_{k=0}^{n-1}$ with $n=12$, which consists of $n$ points that divide the circle $\mathbb{S}$ into $n$ pieces of length $h=h_n=L/n$. At each location $kh$, the voltage $V^{(k)}$ and the states of the channels $Z^{(k)}=\{Z^{(k)}_{i,j}\}_{1\leq i\leq I,\,1\leq j\leq J}$ are described by equations \ref{['eq:general_model V']} and \ref{['eq:general_model Z']} respectively
• Figure : Transition Network of Hodgkin-Huxley Ion Channel
• Figure : Quantifying the error using PET
• Figure : Comparison of stochastic and deterministic for $h=1/16$
• Figure : Comparison of stochastic and deterministic for $h=1/4$

Theorems & Definitions (27)

• Remark 1: Related models
• Lemma 1: Existence and uniqueness for PDE
• Remark 2
• Lemma 2: Existence and uniqueness for stochastic model
• Example 1: Wave propagation
• Example 2: Multiple Gates
• Example 3: Hodgkin-Huxley model
• Example 4: Mutually exclusive channel
• Example 5: Macroscopic channel density
• Theorem 1