Synchronization, Kinematic Waves and Spike-Phase-Separation in Feedback Ising Neural Networks on Heterogeneous Graphs
arXiv:2607.28275
The paper analyzes how structural heterogeneity in neural networks influences collective dynamics, showing that degree variability can trigger synchronized oscillations via a Hopf bifurcation and lead to phase‑separated states through a pitchfork bifurcation.
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
15 pages, 5 figures