Controlling Neural Synchrony Through Variance-Driven Coupling in Complex Network Topologies

Adaptive control of synchrony in neuronal networks is central to understanding both normal brain function and pathological states such as epilepsy and tremor. We study a modified FitzHugh-Nagumo (FHN) network in which the local excitability is extended by a fifth-order nonlinearity and the global coupling strength adapts homeostatically to the spatial variance of neural activity. Using a combination of numerical bifurcation analysis and direct time-domain simulation, we map regimes of quiescence, stable fixed point, and self-sustained oscillation in the two-parameter space of input current and nonlinearity. The analytically predicted Hopf boundary agrees closely with the simulated transition to oscillations. Extending to networks with ring, Watts-Strogatz, and Barabasi-Albert topologies, we show that variance-driven adaptation strongly increases coupling and synchrony in rings, is partly suppressed in small-world graphs, and is almost ineffective in scale-free networks where hubs dominate connectivity. A simple feedback controller regulating the Kuramoto order parameter enables targeted desynchronization or resynchronization. Finally, stochastic forcing produces a non-monotonic impact on coherence, suggesting noise-induced resonance effects. This simulation-based framework links single-neuron excitability, network topology, and adaptive coupling control, and may inform strategies for brain-computer interfaces and neuromodulation therapies.