Statistical Mechanics Of Phase Space Partitioning In Large-Scale Spiking Neuron Circuits

Synaptic interactions structure the phase space of the dynamics of neural circuits and constrain neural computation. Understanding how requires methods that handle those discrete interactions, yet few exist. Recently, it was discovered that even random networks exhibit dynamics that partitions the phase space into numerous attractor basins. Here we utilize this phenomenon to develop theory for the geometry of phase space partitioning in spiking neural circuits. We find basin boundaries structuring the phase space are pre-images of spike-time collision events. Formulating a statistical theory of spike-time collision events, we derive expressions for the rate of divergence of neighboring basins and for their size distribution. This theory reveals that the typical basin diameter grows with inhibitory coupling strength and shrinks with the rate of spike events. Our study provides an analytical and generalizable approach for dissecting how connectivity, coupling strength, single neuron dynamics and population activity shape the phase space geometry of spiking circuits.