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

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.

💡 Research Summary

The paper introduces a novel multiscale framework for modeling action‑potential propagation in heterogeneous neural networks, combining a kinetic description of pairwise neuron interactions with a graph‑theoretic representation of inter‑area connectivity. At the microscopic level each neuron is characterized by a quadruple (X, V, W, C) where X denotes the macro‑area index, V the dimensionless membrane potential, W a recovery variable, and C the degree (number of synaptic connections). Interactions between two neurons occur with probability proportional to Δt · B(X,X*) · G(C,C*), where B captures the influence of the macro‑area locations (the adjacency matrix of the brain‑area graph) and G encodes the dependence on connection degree. Two choices for G are examined: G = 1 (all neurons equally likely to interact) and G = C* (the probability scales with the postsynaptic neuron’s degree, a situation relevant for disease spread models).

The microscopic interaction rule L is a hybrid of the Morris‑Lecar and FitzHugh‑Nagumo models:
L = (i_i^ext + γ_i( \bar v − v ) + (v* − v) − w, v − a w).
Here i_i^ext is an external current, γ_i a damping coefficient, \bar v a relaxation voltage, and a a recovery constant. This rule yields a Boltzmann‑type kinetic equation for the distribution f_i(v,w,c,t) of neurons in each macro‑area i. By testing with φ(v,w,c)=v and φ=v,w respectively, the authors derive evolution equations for the first moments V_i(t)=⟨v⟩_i and W_i(t)=⟨w⟩_i.

When G = 1 the moment closure leads to a 2N‑dimensional ordinary differential system:

dV_i/dt = (i_i^ext + γ_i( \bar v − V_i ) − W_i) ∑j B{ij} + ∑j B{ij}(V_j − V_i)
dW_i/dt = (V_i − a W_i) ∑j B{ij}.

If G = C* the closure requires the degree‑weighted moments K_{v,i}=⟨c v⟩i and K{w,i}=⟨c w⟩i, producing a 4N‑dimensional system in which the mean degree m_i^c of each area appears as a scaling factor. The adjacency matrix B{ij} is defined as 1 for self‑connections, a positive weight b_{ij} when an edge (i,j) exists, and 0 otherwise.

The authors prove existence and uniqueness of equilibria for both systems. The equilibrium conditions reduce to linear algebraic systems M \tilde V = b (and analogous systems for K_{v},K_{w}) where M is strictly diagonally dominant because each diagonal entry contains the sum of all outgoing weights plus the local damping term, guaranteeing nonsingularity. Consequently, even when external currents i_i^ext and damping coefficients γ_i differ across areas, a unique steady state (V*, W*, K_v*, K_w*) exists.

Numerical experiments explore three network topologies: Erdős‑Rényi random graphs, Barabási‑Albert scale‑free graphs, and a data‑driven human brain connectome. For each topology the authors simulate both G = 1 and G = C* cases, measuring propagation speed, global synchrony (via average pairwise correlation), and emergence of asynchronous clusters. Key findings include:

• Greater heterogeneity in degree distribution (as in scale‑free networks) slows down action‑potential spread and hampers global synchrony. Hubs dominate transmission but the overall network remains fragmented.
• Increasing the inter‑area coupling weights B_{ij} promotes rapid propagation and drives the system toward full synchrony, highlighting B_{ij} as a primary control parameter.
• In the G = C* scenario, high‑degree neurons act as transmission hubs, leading to localized synchrony clusters around them while the rest of the network stays asynchronous. This mirrors hypotheses about disease propagation in neurodegenerative disorders where highly connected neurons may serve as vectors.

The paper’s contribution lies in bridging kinetic theory—traditionally used for gases or swarming agents—with neuroscience, providing a mathematically rigorous framework that captures both microscopic electrophysiology and macroscopic network topology. The diagonal‑dominance proof ensures well‑posedness, while the numerical analysis demonstrates how structural heterogeneities shape emergent dynamics. The authors suggest that the framework can be extended to model pathological processes (e.g., Alzheimer’s disease spread), to design artificial neural architectures with desired synchronization properties, and to explore control strategies by tuning external currents or edge weights.