Statistical mechanics of spike events underlying phase space partitioning and sequence codes in large-scale models of neural circuits

Cortical circuits operate in an inhibition-dominated regime of spiking activity. Recently, it was found that spiking circuit models in this regime can, despite disordered connectivity and asynchronous irregular activity, exhibit a locally stable dynamics that may be used for neural computation. The lack of existing mathematical tools has precluded analytical insight into this phase. Here we present analytical methods tailored to the granularity of spike-based interactions for analyzing attractor geometry in high-dimensional spiking dynamics. We apply them to reveal the properties of the complex geometry of trajectories of population spiking activity in a canonical model of locally stable spiking dynamics. We find that attractor basin boundaries are the preimages of spike-time collision events involving connected neurons. These spike-based instabilities control the divergence rate of neighboring basins and have no equivalent in rate-based models. They are located according to the disordered connectivity at a random subset of edges in a hypercube representation of the phase space. Iterating backward these edges using the stable dynamics induces a partition refinement on this space that converges to the attractor basins. We formulate a statistical theory of the locations of such events relative to attracting trajectories via a tractable representation of local trajectory ensembles. Averaging over the disorder, we derive the basin diameter distribution, whose characteristic scale emerges from the relative strengths of the stabilizing inhibitory coupling and destabilizing spike interactions. Our study provides an approach to analytically dissect how connectivity, coupling strength, and single-neuron dynamics shape the phase space geometry in the locally stable regime of spiking neural circuit dynamics.