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Nonlinear multidomain model for nerve bundles with random structure

Irina Pettersson, Antonina Rybalko, Volodymyr Rybalko

TL;DR

This work derives a stochastic homogenization-based macroscopic multidomain model for electrical activity in nerve fascicles with randomly distributed axons and nonlinear conductivities. It combines stochastic two-scale convergence in the mean with the monotone operator method and Palm measure techniques to obtain a non-deterministic effective problem in which the intracellular potential is stationary and constant on axon cross-sections, while the extracellular part is governed by a homogenized conductivity σ^e_{hom}. The main result shows convergence of the microscopic variables (v_ε,g_ε,u_ε) to a limit system for v_0 and g_0, coupled to a homogenized extracellular equation and a constrained intracellular component via the projection P_{K(𝒜)} I_{ion}. This framework enables analyzing how distributions of axon radii and random microstructure influence wave propagation, and provides a well-posed macroscopic description for electrophysiology in randomly structured nerve bundles.

Abstract

We present a derivation of a multidomain model for the electric potential in bundles of randomly distributed axons with different radii. The FitzHugh-Nagumo dynamics is assumed on the axons' membrane, and the conductivity depends nonlinearly on the electric field. Under ergodicity conditions, we study the asymptotic behavior of the potential in the bundle when the number of the axons in the bundle is sufficiently large and derive a macroscopic multidomain model describing the electrical activity of the bundle. Due to the randomness of geometry, the effective intracellular potential is not deterministic but is shown to be a stationary function with realizations that are constant on axons' cross sections. The technique combines the stochastic two-scale convergence and the method of monotone operators.

Nonlinear multidomain model for nerve bundles with random structure

TL;DR

This work derives a stochastic homogenization-based macroscopic multidomain model for electrical activity in nerve fascicles with randomly distributed axons and nonlinear conductivities. It combines stochastic two-scale convergence in the mean with the monotone operator method and Palm measure techniques to obtain a non-deterministic effective problem in which the intracellular potential is stationary and constant on axon cross-sections, while the extracellular part is governed by a homogenized conductivity σ^e_{hom}. The main result shows convergence of the microscopic variables (v_ε,g_ε,u_ε) to a limit system for v_0 and g_0, coupled to a homogenized extracellular equation and a constrained intracellular component via the projection P_{K(𝒜)} I_{ion}. This framework enables analyzing how distributions of axon radii and random microstructure influence wave propagation, and provides a well-posed macroscopic description for electrophysiology in randomly structured nerve bundles.

Abstract

We present a derivation of a multidomain model for the electric potential in bundles of randomly distributed axons with different radii. The FitzHugh-Nagumo dynamics is assumed on the axons' membrane, and the conductivity depends nonlinearly on the electric field. Under ergodicity conditions, we study the asymptotic behavior of the potential in the bundle when the number of the axons in the bundle is sufficiently large and derive a macroscopic multidomain model describing the electrical activity of the bundle. Due to the randomness of geometry, the effective intracellular potential is not deterministic but is shown to be a stationary function with realizations that are constant on axons' cross sections. The technique combines the stochastic two-scale convergence and the method of monotone operators.

Paper Structure

This paper contains 14 sections, 16 theorems, 169 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geometry of nerve fascicles
  3. Stochastic two-scale convergence in the mean
  4. Problem setup
  5. Effective problem and main result
  6. Proof of Theorem \ref{['th:main-short']}
  7. Microscopic problem
  8. Compactness and passage to the limit
  9. Well-posedness of the limit problem
  10. Examples of random geometry
  11. Randomly distributed disks of two different radii
  12. Disks with a finite number of different radii and random shifts
  13. Monotonicity of the nonlinear terms
  14. Nondimensionalization of the problem

Key Result

Theorem 2.1

Let $f\in L^p(\Omega)$, $p\ge 1$. Then for almost all $\omega \in \Omega$, the realization $f(T_x\omega)$ possesses a mean value, that is Moreover, the mean value $\langle f(T_x \omega)\rangle$ considered as a function of $\omega \in \Omega$, is invariant, and If the system $T_x$ is ergodic, then

Figures (2)

  • Figure 1: Fascicle $G$ with the lateral boundary $\Sigma$.
  • Figure 1: Random distribution of discs in Section \ref{['sec:example-without-shifts']}: $p_0=1/3$ (no disk), $p_1=1/2$ (orange disk with radius $r_1=0.2$), $p_2=1/6$ (blue disk with radius $r_2 = 0.3$).

Theorems & Definitions (36)

  • Theorem 2.1: Birkhoff Ergodic Theorem
  • Definition 3.1
  • Lemma 3.2: Palm measure $\mu$ and Campbell's formula
  • Remark 3.3
  • Definition 3.4: Stochastic two-scale convergence on random surfaces
  • Definition 3.5
  • Lemma 3.6
  • Proof 1
  • Definition 3.7
  • Remark 4.1
  • ...and 26 more