Diverse mean-field dynamics of clustered, inhibition-stabilized Hawkes networks via combinatorial threshold-linear networks

TL;DR

This paper demonstrates that combinatorial threshold-linear networks (CTLNs) serve as an effective mean-field model for clustered, inhibition-stabilized Hawkes networks, enabling prediction of diverse nonlinear dynamics from network structure.

Contribution

It establishes a link between CTLN models and clustered Hawkes networks, allowing prediction of complex dynamics from connectivity structure, including metastable and chaotic states.

Findings

CTLN accurately predicts dynamics in large, fast-inhibition regimes.

Networks exhibit diverse behaviors like metastable orbits and chaos.

Bifurcations occur with slower inhibition, leading to oscillations.

Abstract

Networks of interconnected neurons display diverse patterns of collective activity. Relating this collective activity to the network's connectivity structure is a key goal of computational neuroscience. We approach this question for clustered networks, which can form via biologically realistic learning rules and allow for the re-activation of learned patterns. Previous studies of clustered networks have focused on metastabilty between fixed points, leaving open the question of whether clustered spiking networks can display more rich dynamics--and if so, whether these can be predicted from their connectivity. Here, we show that in the limits of large population size and fast inhibition, the combinatorial threshold linear network (CTLN) model is a mean-field theory for inhibition-stabilized nonlinear Hawkes networks with clustered connectivity. The CTLN has a large body of ``graph rules'' relating network structure to dynamics. By applying these, we can predict the dynamic attractors of our clustered spiking networks from the structure of between-cluster connectivity. This allows us to construct networks displaying a diverse array of nonlinear cluster dynamics, including metastable periodic orbits and chaotic attractors. Relaxing the assumption that inhibition is fast, we see that the CTLN model is still able to predict the activity of clustered spiking networks with reasonable inhibitory timescales. For slow enough inhibition, we observe bifurcations between CTLN-like dynamics and global excitatory/inhibitory oscillations.