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

1. Problem Statement

The paper addresses the challenge of modeling action potential dynamics in heterogeneous neural networks. While collective behaviors like neuron synchronization are well-studied, existing models often struggle to simultaneously capture:

• Microscopic stochasticity: The individual interactions between neurons (particle level).
• Macroscopic topology: The structural connectivity between different brain regions (graph level).
• Heterogeneity: Variations in connection degrees, spatial locations, and physiological parameters across different brain areas.

The authors aim to bridge the gap between kinetic theory (statistical description of particle ensembles) and macroscopic network dynamics to understand how network heterogeneities influence membrane potential propagation and synchronization.

2. Methodology

The authors propose a multi-scale modeling framework that couples two levels of description:

A. Microscopic Level (Kinetic Theory)

• State Space: A single neuron is characterized by a quartet $(X, V, W, C)$, where:
  • $X$: Spatial location (belonging to a specific brain macro-area).
  • $V$: Dimensionless membrane potential.
  • $W$: Recovery variable (negative feedback).
  • $C$: Degree of connections (number of synapses).
• Interaction Rule: Neurons interact via binary collisions. The probability of interaction is governed by a collision kernel $B(X, X^*)G(C, C^*)$:
  • $B(X, X^*)$: Describes the probability of connection based on the topological distance between brain areas (graph adjacency).
  • $G(C, C^*)$: Describes the probability based on connection degrees. The authors analyze two cases:
    1. $G=1$: All neurons in an area are equally likely to interact (mean-field within area).
    2. $G=C^*$: The probability of interaction depends on the postsynaptic neuron's degree (relevant for disease propagation where highly connected neurons are more susceptible).
• Microscopic Dynamics: The evolution of $V$ and $W$ follows a modified Morris–Lecar/FitzHugh–Nagumo model, incorporating external currents and coupling terms.

B. Macroscopic Level (Derivation)

• Kinetic Equation: Starting from the microscopic interaction rules, the authors derive a Boltzmann-type kinetic equation for the distribution function $f(x, v, w, c, t)$.
• Moment Closure: By integrating the kinetic equation, they derive systems of coupled ordinary differential equations (ODEs) for macroscopic moments:
  • Case 1 ($G=1$): Yields a closed system for the average membrane potential ($V_i$) and recovery variable ($W_i$) in each macro-area.
  • Case 2 ($G=C^*$): Requires the introduction of mixed first-order moments ($K^v_i, K^w_i$) involving the connection degree $C$, resulting in a larger system of ODEs to maintain closure.

C. Numerical Analysis

The authors employ a fourth-order Runge–Kutta method to simulate the derived macroscopic systems on various graph topologies (directed, undirected, weighted) with heterogeneous parameters.

3. Key Contributions

1. Unified Framework: The paper successfully integrates kinetic theory with graph theory to model neural networks, allowing for a rigorous transition from microscopic stochastic interactions to macroscopic deterministic equations.
2. Handling Heterogeneity: The model explicitly accounts for heterogeneity in:
   • Topology: Directed vs. undirected, weighted vs. unweighted graphs.
   • Parameters: Spatially varying external currents ($i_{ext}$) and relaxation rates ($\gamma$).
   • Connectivity: Distributions of connection degrees ($C$) within and between nodes.
3. Analytical Equilibrium Proofs:
   • The authors prove the existence and uniqueness of equilibrium configurations for both macroscopic systems (Cases 1 and 2) under general conditions (non-identical parameters across nodes).
   • They demonstrate that the equilibrium is determined by the network topology and parameter distribution, independent of initial conditions.
   • They show that the kinetic distribution function converges to a Dirac delta function centered at the macroscopic equilibrium values.
4. Novel Interaction Kernel: The introduction of the degree-dependent kernel $G(C, C^*) = C^*$ provides a mechanism to study how connection density influences signal propagation, which is crucial for modeling pathologies like Alzheimer's.

4. Results

The numerical simulations (three test cases) reveal distinct dynamical behaviors:

• Test 1 (Topology Effects):

  • Directed/Weighted Graphs: The timing of action potential spikes is strongly influenced by the net outflow ($O_i$) and the sum of outgoing weights ($T_i$). Nodes with higher outgoing connectivity spike earlier.
  • Synchronization: Nodes with identical topological properties (same $O_i$ or $T_i$) tend to synchronize, but differences in their specific connection patterns can lead to desynchronization.
  • Conclusion: Directed connections and spatial proximity significantly accelerate action potential propagation.

• Test 2 (Parameter Heterogeneity):

  • Varying the relaxation parameter ($\gamma_i$) across nodes alters the resting potential and the amplitude of the undershoot but does not change the spike timing.
  • Varying the external current ($i_{ext}$) shifts the entire action potential profile (resting state) linearly without changing the spike shape or timing.
  • Result: Heterogeneous parameters lead to asynchronous equilibrium manifolds where neurons stabilize at different potential levels.

• Test 3 (Connection Distribution Heterogeneity):

  • Using the $G(C, C^*) = C^*$ model, the authors show that the mean connection degree ($m_c$) of a node and its neighbors dictates the spike timing.
  • Nodes with higher effective connectivity ($m_i = \sum B_{ij} m_j^c$) spike faster.
  • Result: Even on a symmetric ring graph, heterogeneity in connection distributions creates asynchronous dynamics and slows down the overall spiking rate compared to the $G=1$ case.

5. Significance and Future Outlook

• Theoretical Impact: This work provides a rigorous mathematical foundation for studying complex brain dynamics using kinetic theory, offering a flexible tool to analyze how microscopic rules scale up to macroscopic phenomena.
• Biological Relevance: The model is particularly relevant for understanding disease propagation (e.g., Alzheimer's) where the connectivity of neurons plays a critical role in the spread of pathology. The ability to model heterogeneous connection degrees allows for more realistic simulations of disease progression.
• Future Directions: The authors plan to extend the model to larger, more realistic neural networks (beyond the illustrative 5-node example) and apply the framework to specific disease progression models to predict therapeutic control strategies.

In summary, the paper demonstrates that network topology and connection heterogeneity are fundamental drivers of neural synchronization and signal propagation speed, offering a powerful new lens for analyzing brain dynamics through the intersection of kinetic theory and network science.