Thermodynamic analog of integrate-and-fire neuronal networks by maximum entropy modelling

Relying on maximum entropy arguments, certain aspects of time-averaged experimental neuronal data have been recently described using Ising-like models, allowing the study of neuronal networks under an analogous thermodynamical framework. Here, we apply for the first time the Maximum Entropy method to an Integrate-and-fire (IF) model that can be tuned at criticality, offering a controlled setting for a systematic study of criticality and finite-size effects in spontaneous neuronal activity, as opposed to experiments. We show that generalized Ising models that accurately predict the average local activities and correlation functions between neurons of the IF model networks in the critical state exhibit a spin glass phase for low temperatures, having mostly negative intrinsic fields and a bimodal distribution of interaction constants that tends to become unimodal for larger networks. Results appear to be affected by sample-to-sample connectivity variations and subsampling. Furthermore, we also found that networks with higher percentage of inhibitory neurons lead to Ising-like systems with reduced thermal fluctuations. Finally, considering only neuronal pairs associated with the largest correlation functions allows the study of larger system sizes. Author summaryBrain activity, either stimulated or spontaneous, in vivo or in vitro, exhibits complex spatiotemporal behavior. Trying to make sense of it, several research groups have analyzed time-averaged experimental neuronal data using maximum entropy arguments, mapping the neuronal dynamics into a generalized Ising-like model and allowing to study neuronal data using tools typical of critical phenomena. However, the intricacy of real biological networks in experimental settings pose challenges in the precision and reliability of the neuronal measurements. Here, we apply for the first time the Maximum Entropy Method to an Integrate-and-fire model with synaptic plasticity, providing a foundation for a more systematic and comprehensive study of spontaneous brain activity. We show that generalized Ising models are able to reproduce the numerical time-averaged data of local activities and correlation functions of integrate-and-fire neurons and predict qualitatively higher-order quantities such as the three-point correlation functions across triplets of neurons. We show that subsampling affects the efficiency of the mapping and that the analogous thermodynamics functions of the Ising-like models depend on sample-to-sample network variations and on the presence of inhibition in the neural network.