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Action potential dynamics on heterogenous neural networks: from kinetic to macroscopic equations

Marzia Bisi, Martina Conte, Maria Groppi

TL;DR

In the context of multi-agent systems of binary interacting particles, a kinetic model for action potential dynamics on a neural network is proposed, accounting for heterogeneity in the neuron-to-neuron connections, as well as in the brain structure, to study the influence of the network heterogeneities on the membrane potential propagation and synchronization.

Abstract

In the context of multi-agent systems of binary interacting particles, a kinetic model for action potential dynamics on a neural network is proposed, accounting for heterogeneity in the neuron-to-neuron connections, as well as in the brain structure. Two levels of description are coupled: in a single area, pairwise neuron interactions for the exchange of membrane potential are statistically described; among different areas, a graph description of the brain network topology is included. Equilibria of the kinetic and macroscopic settings are determined and numerical simulations of the system dynamics are performed with the aim of studying the influence of the network heterogeneities on the membrane potential propagation and synchronization.

Action potential dynamics on heterogenous neural networks: from kinetic to macroscopic equations

TL;DR

In the context of multi-agent systems of binary interacting particles, a kinetic model for action potential dynamics on a neural network is proposed, accounting for heterogeneity in the neuron-to-neuron connections, as well as in the brain structure, to study the influence of the network heterogeneities on the membrane potential propagation and synchronization.

Abstract

In the context of multi-agent systems of binary interacting particles, a kinetic model for action potential dynamics on a neural network is proposed, accounting for heterogeneity in the neuron-to-neuron connections, as well as in the brain structure. Two levels of description are coupled: in a single area, pairwise neuron interactions for the exchange of membrane potential are statistically described; among different areas, a graph description of the brain network topology is included. Equilibria of the kinetic and macroscopic settings are determined and numerical simulations of the system dynamics are performed with the aim of studying the influence of the network heterogeneities on the membrane potential propagation and synchronization.
Paper Structure (10 sections, 3 theorems, 34 equations, 5 figures, 2 tables)

This paper contains 10 sections, 3 theorems, 34 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model description
  4. Microscopic rule for action potential propagation
  5. Equilibria of the kinetic and macroscopic systems
  6. Numerical simulations
  7. Test 1: directed and weighted graph
  8. Test 2: heterogenous parameter network
  9. Test 3: heterogenous distribution of the connections
  10. Discussion

Key Result

proposition thmcounterproposition

Let us consider system macro_sis_inhom with $\textbf{i}^i_{ext}\ne \textbf{i}^j_{ext}$ and $\gamma^i\ne \gamma^j$ for $i,j\in\mathbb{I}$, $i\ne j$. It admits a unique equilibrium configuration $({\bf V}^*,{\bf W}^*)$ of components

Figures (5)

  • Figure 1: Test 1. Schematic representation of the graphs described by the adjacent matrices \ref{['B_exA']} (Example A - left plot) and \ref{['B_exB']} (Example B - right plot), respectively.
  • Figure 2: Test 1. Dynamics of the potentials $V_i(t)$, for $i=1,\dots,5$, emerging from system \ref{['macro_sis_inhom']} on the networks schematized in Figure \ref{['Graph_exAB']}. Left column (panels a-c) refers to Example A, while right column (panels d-f) to Example B. The rows show the cases of directed-non weighted (top), undirected-weighted (middle), and directed-weighted (bottom) graphs, respectively.
  • Figure 3: Test 2. Schematic representation of an undirected and non-weighted ring graph (left plot). Dynamics of the potentials $V_i(t)$, for $i=1,\dots,5$ emerging from system \ref{['macro_sis_inhom']} (right plot).
  • Figure 4: Test 2. Dynamics of the potentials $V_i(t)$, for $i=1,\dots,5$, emerging from system \ref{['macro_sis_inhom']} on the network schematized in the left plot of Figure \ref{['GammaIcurr_equal']} when different values of $\gamma^i$ (left plot) and $\textbf{i}^i_{ext}$ (right plot) are assigned to the different nodes.
  • Figure 5: Test 3. Dynamics of the potentials $V_i(t)$, for $i=1,\dots,5$, emerging from system \ref{['macro_sis_inhom_2']} on the networks schematized in the left plot of Figure \ref{['GammaIcurr_equal']} for the same value (left plot) or different values (right plot) of $m_c^i$ across the nodes.

Theorems & Definitions (6)

  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition
  • proof