An Introduction to the Non-Equilibrium Steady States of Maximum Entropy Spike Trains

Herzog

Corcoran

et al. 2019

Despite their differences, biological systems at different spatial scales tend to exhibit common organizational patterns. Unfortunately, these commonalities are often hard to grasp due to the highly specialized nature of modern science and the parcelled terminology employed by various scientific sub-disciplines. To explore these common organizational features, this paper provides a comparative study of diverse applications of the maximum entropy principle, which has found many uses at different biological spatial scales ranging from amino acids up to societies. By presenting these studies under a common approach and language, this paper aims to establish a unified view over these seemingly highly heterogeneous scenarios.

A Comparison of the Maximum Entropy Principle Across Biological Spatial Scales

Herzog

Corcoran

et al. 2019

Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics

Maldonado

Cessac

2020

The Thermodynamic Formalism provides a rigorous mathematical framework for studying quantitative and qualitative aspects of dynamical systems. At its core, there is a variational principle that corresponds, in its simplest form, to the Maximum Entropy principle. It is used as a statistical inference procedure to represent, by specific probability measures (Gibbs measures), the collective behaviour of complex systems. This framework has found applications in different domains of science. In particular, it has been fruitful and influential in neurosciences. In this article, we review how the Thermodynamic Formalism can be exploited in the field of theoretical neuroscience, as a conceptual and operational tool, in order to link the dynamics of interacting neurons and the statistics of action potentials from either experimental data or mathematical models. We comment on perspectives and open problems in theoretical neuroscience that could be addressed within this formalism.

Linear Response of General Observables in Spiking Neuronal Network Models

Cessac

Ampuero

2021

We establish a general linear response relation for spiking neuronal networks, based on chains with unbounded memory. This relation allow us to predict the influence of a weak amplitude time dependent external stimuli on spatio-temporal spike correlations, from the spontaneous statistics (without stimulus) in a general context where the memory in spike dynamics can extend arbitrarily far in the past. Using this approach, we show how the linear response is explicitly related to the collective effect of the stimuli, intrinsic neuronal dynamics, and network connectivity on spike train statistics. We illustrate our results with numerical simulations performed over a discrete time integrate and fire model.