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Weak formulation and spectral approximation of a Fokker-Planck equation for neural ensembles

Ling Yan, Pei Zhang, Yanli Wang, Zhennan Zhou

TL;DR

The paper develops a weak formulation and a Laguerre‑Legendre spectral‑Galerkin method (LLSGM) for the Fokker–Planck equation associated with the nonlinear noisy leaky integrate‑and‑fire (NNLIF) model of neural ensembles in a semi‑unbounded domain. By decomposing the trial space into a compact boundary‑fitting part and a complementary space, the method achieves zeroth‑order boundary conditions while implicitly incorporating derivative BCs, yielding a scheme with proven consistency and spectral accuracy in space. The framework is extended to a two‑population model with synaptic delays and refractory states, including a coupled set of PDEs and ODEs for refractory dynamics, with a fully discrete scheme and numerical tests demonstrating accuracy, efficiency, and the ability to capture blow‑up and periodic oscillations. Overall, the LL SGM provides a flexible, structure‑preserving, and computationally efficient tool for simulating large neural networks modeled by Fokker–Planck equations and their extensions.

Abstract

In this paper, we focus on efficiently and flexibly simulating the Fokker-Planck equation associated with the Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) model, which reflects the dynamic behavior of neuron networks. We apply the Galerkin spectral method to discretize the spatial domain by constructing a variational formulation that satisfies complex boundary conditions. Moreover, the boundary conditions in the variational formulation include only zeroth-order terms, with first-order conditions being naturally incorporated. This allows the numerical scheme to be further extended to an excitatory-inhibitory population model with synaptic delays and refractory states. Additionally, we establish the consistency of the numerical scheme. Experimental results, including accuracy tests, blow-up events, and periodic oscillations, validate the properties of our proposed method.

Weak formulation and spectral approximation of a Fokker-Planck equation for neural ensembles

TL;DR

The paper develops a weak formulation and a Laguerre‑Legendre spectral‑Galerkin method (LLSGM) for the Fokker–Planck equation associated with the nonlinear noisy leaky integrate‑and‑fire (NNLIF) model of neural ensembles in a semi‑unbounded domain. By decomposing the trial space into a compact boundary‑fitting part and a complementary space, the method achieves zeroth‑order boundary conditions while implicitly incorporating derivative BCs, yielding a scheme with proven consistency and spectral accuracy in space. The framework is extended to a two‑population model with synaptic delays and refractory states, including a coupled set of PDEs and ODEs for refractory dynamics, with a fully discrete scheme and numerical tests demonstrating accuracy, efficiency, and the ability to capture blow‑up and periodic oscillations. Overall, the LL SGM provides a flexible, structure‑preserving, and computationally efficient tool for simulating large neural networks modeled by Fokker–Planck equations and their extensions.

Abstract

In this paper, we focus on efficiently and flexibly simulating the Fokker-Planck equation associated with the Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) model, which reflects the dynamic behavior of neuron networks. We apply the Galerkin spectral method to discretize the spatial domain by constructing a variational formulation that satisfies complex boundary conditions. Moreover, the boundary conditions in the variational formulation include only zeroth-order terms, with first-order conditions being naturally incorporated. This allows the numerical scheme to be further extended to an excitatory-inhibitory population model with synaptic delays and refractory states. Additionally, we establish the consistency of the numerical scheme. Experimental results, including accuracy tests, blow-up events, and periodic oscillations, validate the properties of our proposed method.

Paper Structure

This paper contains 20 sections, 2 theorems, 119 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Weak solution
  3. Numerical scheme
  4. Weak formulation and basis function
  5. Fully discrete scheme
  6. Consistency of the numerical scheme
  7. Model extensions
  8. Model introduction
  9. Numerical scheme
  10. Numerical test
  11. Study of the order of accuracy and the stability test
  12. Order of accuracy for one-population model
  13. Order of accuracy for two-population model
  14. Stability test for one-population model
  15. Efficiency comparison
  16. ...and 5 more sections

Key Result

Theorem 2.1

Given $p(v,t)\in C^1([0,T];U(\Omega))$, if $(p(v,t), N(t))$ is a classical solution of eq:1-eq:1.1, then $(p(v,t), N(t))$ is a weak solution. Conversely, if $(p(v,t), N(t))$ is a weak solution of eq:1-eq:1.1 and $p(v,t)\in C^1((0,T];C^2((-\infty,V_R)\cup (V_R,V_F]))$, then $(p(v,t), N(t))$ is a clas

Figures (7)

  • Figure 1: (Order of accuracy in Sec. \ref{['sec:accuracy_one_pop']}) The spectral convergence of the explicit-implicit scheme \ref{['eq:44']} in the spatial discretization for the one-population model. (a) $M$ is odd. (b) $M$ is even.
  • Figure 2: (Order of accuracy in Sec. \ref{['sec:accuracy_two_pop']}) The spectral convergence of the scheme \ref{['eq:two_fully_dis']} in the spatial discretization for the two-population model. Here, the blue line is for the excitatory population (E-P) and the red line is for the inhibitory population (I-P). (a) $M$ is odd. (b) $M$ is even.
  • Figure 3: (Blow-up events for the one-population model in Sec. \ref{['sec:blow_up_one']}) The blow-up phenomenon for the one-population model. (a) The evolution of the mean firing rate $N(t)$. (b) The density function $p(v,t)$ at $t = 2.95, 3.15$, and $3.35$.
  • Figure 4: (Blow-up events for the two-population model in Sec. \ref{['sec:blow_up_two']}) The blow-up phenomenon for the two-population model. The first row is the behavior of the mean firing rate $N_E(t)$, and the density function $p_E(v,t)$ at $t = 3.85, 4.05$, and $4.25$ for the excitatory population, and the second row is the behavior of the mean firing rate $N_I(t)$, and the density function $p_I(v,t)$ at $t = 2.25, 3.25$, and $4.25$ for the inhibitory population.
  • Figure 5: (Two-population model with delays and refractory states in Sec. \ref{['sec:dely_refactory']}) The evolution of the mean firing rates $N_{\alpha}(t), \alpha = E, I$ for $b_E^E = 3.5$, where the periodic oscillations are observed.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Definition 2.1: classical solution
  • Definition 2.2: weak solution
  • Remark 2.1
  • Theorem 2.1: Equivalence to the classical solution
  • proof
  • Theorem 3.1: Consistency
  • proof