Excitatory-Inhibitory Homeostasis and Bifurcation Control in the Wilson-Cowan Model of Cortical Dynamics
Authors: Páscoa dos Santos Francisco , Verschure Paul FMJ 
Published: June 2024

Although the primary function of excitatory-inhibitory (E-I) homeostasis is the maintenance of mean firing rates, the conjugation of multiple homeostatic mechanisms is thought to be pivotal to ensuring edge-of-bifurcation dynamics in cortical circuits. However, computational studies on E-I homeostasis have focused solely on the plasticity of inhibition, neglecting the impact of different modes of E-I homeostasis on cortical dynamics. Therefore, we investigate how oscillations and edge-of-bifurcation dynamics are shaped by the diverse mechanisms of E-I homeostasis employed by cortical networks. Using the Wilson-Cowan model, we explore how distinct modes of E-I homeostasis maintain stable firing rates in models with varying levels of input and how it affects circuit dynamics. Our results confirm that E-I homeostasis can be leveraged to control edge-of-bifurcation dynamics and that some modes of homeostasis maintain mean firing rates under higher levels of input by modulating the distance to the bifurcation.

Additionally, relying on multiple modes of homeostasis ensures stable activity while keeping oscillation frequencies within a physiological range. Our findings tie relevant features of cortical networks, such as E-I balance, the generation of gamma oscillations, and edge-of-bifurcation dynamics, under the framework of firing-rate homeostasis, providing a mechanistic explanation for the heterogeneity in the distance to the bifurcation found across cortical areas. In addition, we reveal the functional benefits of relying upon different homeostatic mechanisms, providing a robust method to regulate network dynamics with minimal perturbation to the generation of gamma rhythms and explaining the correlation between inhibition and gamma frequencies found in cortical networks.

Author summary

We study how excitatory-inhibitory (E-I) homeostasis controls edge-of-bifurcation dynamics in cortical networks and how it impacts the generation of gamma oscillations. Importantly, while previous studies have limited E-I homeostasis to the plasticity of inhibition, we explore the wide range of mechanisms employed by cortical networks and, more importantly, how they interact. Here, we derive the mathematical solution for the Wilson-Cowan model under distinct modes of homeostasis and study how they shape model dynamics and the generation of gamma oscillations. That said, we demonstrate that E-I homeostasis, particularly of excitation and intrinsic excitability, modulates model dynamics relative to the bifurcation between damped and sustained oscillations in a manner previously unaccounted for, providing a mechanism for the implementation of heterogeneous distances to the bifurcation across cortical areas. Furthermore, our results stress the functional benefits of relying on multiple modes of homeostasis, allowing for the control of firing rates and circuit dynamics while ensuring that gamma oscillations remain within a physiological range and explaining the relationship between inhibition and gamma frequencies found in empirical data. With these results, we unify E-I balance, edge-of-bifurcation dynamics, and gamma oscillations under the lens of firing-rate homeostasis.