# Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics
## 1 Introduction
Most neurons produce short electrical pulses, called _action potentials_ or _spikes_, and it is widely believed that the collective spike trains emitted by neuronal networks encode information about the underlying dynamics and response to stimuli [35; 36; 37]. Thus, researchers have devoted a lot of effort to understanding the correlations structure in the statistics of spike trains [38; 39; 40; 41; 42; 43; 44; 45]. Because spikes can be represented as binary variables, it is natural to adapt methods and concepts from statistical physics, and more specifically the statistical physics of spin systems, to analyse spike trains statistics. There have been many successful attempts in this direction. All of the approaches we know about are based on variational principles. The most direct connection from statistical physics to spike train statistics is done via the maximum entropy principle which has attracted a lot of attention during the past years [38; 39; 43; 46; 47] (see [47] for a physicists-oriented review). Unfortunately, most of these articles are limited to the original form of an Ising spin-glass potential (pairwise interactions with random couplings) or variants of it with higher order interactions [40; 41; 42], where successive times are independent, thereby neglecting the time correlations and causality one may expect from a network of neurons with interactions (exceptions can be found in [48; 49; 50]). We focus on this approach and its extension to causal networks in the present review as it is natural to link with the Thermodynamic Formalism. Another approach, which actually appeared earlier in mathematical neuroscience, is the dynamic mean-field theory that takes into account dynamics and time correlations. In this approach, originated in quantum field theory and Martin-Siggia-Rose formalism (Statistical dynamics of classical systems [51]), the variational principle is expressed via the minimisation of an effective action containing the equations for the dynamics. It was introduced in the field of theoretical neuroscience by [PERSON] who initially applied it to the study of spin-glasses dynamics [52; 53; 54], before analysing neuronal networks dynamics [29]. Here, the effective action can be summarised, in the limit when the number of neurons tends to infinity, by dynamic mean-fieldequations depending on neuron's firing rates and pairwise time-correlations. Thus, here, the information about spike statistics is contained in the two first moments (Gaussian limit). This approach has inspired a lot of important works, where temporal correlations are taken into account, see, for example [55; 56; 57], or the recent work by [PERSON] and collaborators (i.e., the review on dynamic mean-field theory in [58] and the link to from dynamic mean-field theory to large deviations from [PERSON] and Guionnet [59] and [60]). Another trend attempts to relate neuronal dynamics to spike statistics, expressed via a maximum-likelihood approach [61] (we apologise if we have forgotten other approaches that we ignore). To our taste, the most achieved work in this direction is [PERSON]'s information geometry [62] making a beautiful link between probability measures (e.g., exponential, like Gibbs measures) and differential geometry. In this review, we show how Thermodynamic Formalism can be used as an alternative way to study the link between the neuronal dynamics and spike statistics, not only from experimental data, but also from mathematical models of neuronal networks, properly handling causal and memory dependent interactions between neurons and their spikes. It is also based on a variational principle (and, actually, the four methods that are discussed in this introduction certainly have a common \"hat\", the large deviations theory [63]), although we depart from this principle at some point where we discuss extensions of this approach to non-stationary situations through a linear response formula.