Evaluating parameter space stiffness of maximum entropy models and departure from criticality

Introduction

The emergence of collective behaviour in biological systems remains to be fully understood. Models inspired by statistical physics, where neurons are treated as Ising-like variables, provide an appealing approach to fill this conceptual gap. They offer, notably, a principled framework for inferring the effective interactions and constraints that shape the collective activity, as well a method for detecting criticality without relying on avalanche calculations [1].

Methods

We focus here on Maximum Entropy Models (MEM), where a probability distribution over states is inferred by maximizing its entropy while enforcing a match between the expectation values of a given set of observables and their empirical averages. This procedure results in a Boltzmann-like distribution with temperature and a Hamiltonian constrained by the chosen observables, parameterized by a set of Lagrange multipliers. While this approach is typically applied to experimental data, here we apply it to simulation results in order to systematically study how variations of the structural and dynamical model parameters map to changes in the effective parameters of the inferred MEM.

Results

We implement an integrate-and-fire (IF) model known to be poised at criticality [2] and train a MEM consistent with the chosen observables, namely the mean activity and pairwise correlations. The inferred Boltzmann-like distribution is parameterized by so-called local fields and effective couplings. It can be used to calculate the covariance matrix between the observables, which, in turn, is equivalent to the Fisher Information Matrix (FIM). We characterize the stiffness of the model based on the eigenvalue spectrum, and a preliminary analysis consists of imposing incremental parameter changes in the direction of the leading eigenvalue. This leads to a new model whose departure from criticality can then be evaluated.

Discussion

Future work includes the study of the covariance between structural parameters and MEM parameters, with the goal to identify how structural features of the IF model are associated with criticality. Further investigation will also evaluate whether a two-compartiment model endowed with an intrinsic bursting mechanism can be tackled with MEMs.

Acknowledgments

This work was funded by the Natural Sciences and Engineering Research Council of Canada and the New-Brunswick Innovation Foundation.

References

[1] Meshulam, L. & Bialek, W. (2025). Statistical mechanics for networks of real neurons. Reviews of Modern Physics, 97(4). doi: 10.1103/jcrn-3nrc
[2] Simões, T., Filho, C., Herrmann, H., Andrade, J., & de Arcangelis, L. (2024). Thermodynamic analog of integrate-and-fire neuronal networks by maximum entropy modelling. Scientific Reports, 14(1). doi: 10.1038/s41598-024-60117-3