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  • Open Access

Diverse Mean-Field Dynamics of Clustered, Inhibition-Stabilized Hawkes Networks via Combinatorial Threshold-Linear Networks

Caitlin Lienkaemper*

Gabriel Koch Ocker†

  • Department of Mathematics and Statistics and Center for Systems Neuroscience, Boston University, Boston, Massachusetts, 02215, USA

  • *Contact author: clienk@mit.edu
  • †Contact author: gkocker@bu.edu
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PRX Life 3, 043004 – Published 14 October, 2025

DOI: https://doi.org/10.1103/m8hf-q5pl

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 reactivation 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 richer 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.

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PRX Life 3, 043004– Published 14 October, 2025
  • Received 25 June 2025
  • Accepted 3 September 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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