Assemblies and the k-Cap Process: The Effects of Locality on Neural Firing Dynamics

The $k$-cap (or $k$-winners-take-all) process is a simple model of firing activity and inhibition in the brain. The process models the formation of neural assemblies, a longstanding proposal for the representation of abstract concepts in the brain. It is applied to a directed graph as follows: in each iteration, a subset of $k$ vertices of the graph are identified as winners; the next round winners are the vertices that have the highest total degree from the current winners, with ties broken randomly.

Previous theoretical analyses of neural assemblies model the connectome as a directed Erd\"{o}s-R\'{e}nyi random graph, where each synaptic connection is equally likely. While a reasonable starting point, the model does not take into account {\em locality} of synapses. Here we study directed geometric random graphs in any constant dimension, which allow the synapse probability to be a function of spatial locations of the endpoints or other neuronal features. In this setting, we study the the dynamics of firing neurons under inhibition, as modeled by the $k$-cap process. Our work represents the first rigorous analysis of assembly formation in such a more accurate representation of the connectome. The structure which emerges, both theoretically and in simulation, reflects properties of assemblies which have been noted experimentally but have not yet justified theoretically: our analysis reveals that within a small number of time steps ($\polylog(k)$) (1) firing neurons lie in a small ball, and (2) within this ball, the subset of $k$ firing neurons are essentially random.