Synchronization, Kinematic Waves and Spike-Phase-Separation in Feedback Ising Neural Networks on Heterogeneous Graphs

Abstract

Structural heterogeneity constrains collective dynamics in complex systems. However, its analytical tractability out of equilibrium remains limited. In this work, we study a class of kinetic Ising neural networks driven out of equilibrium by a homeostatic feedback loop between the neuronal excitability and the population firing rate. Using a Curie-Weiss heterogeneous mean-field approximation validated by Monte Carlo simulations, we provide an analytical characterization of how a macroscopic synchronized limit cycle emerges via an Andronov-Hopf bifurcation on heterogeneous networks. We derive closed-form phase boundaries and show that the onset of oscillations is explicitly controlled by network heterogeneity through the degree moment ratio. Degree heterogeneity decouples the spiking rate per neuron m from the spiking rate per synapse u, generating physical phenomena absent in homogeneous systems. These include (i) kinematic waves of sequential, degree-ordered activations propagating from the network periphery to the hubs, and (ii) a low-temperature phase-separated state emerging via a pitchfork bifurcation. We prove that for highly heterogeneous topologies, this phase-separated fixed point stabilizes and dynamically destroys the synchronized limit cycle. These results provide a mathematical framework for understanding how heterogeneity regulates macroscopic oscillations and out-of-equilibrium transitions in neural networks