# Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics
## 1 Introduction  
Initiated by [PERSON] [1, 2], the goal of statistical physics was to establish a link between the microscopic mechanical description of interacting particles in a gas or a fluid, and the macroscopic description that is provided by thermodynamics [3, 4]. Although this program is, even nowadays, far from being completed [1, 5], the work of [PERSON] and his successors opened new avenues of research, not only in physics but also in mathematics. Especially the term \"ergodic\", which was coined by [PERSON] [1], inaugurated an important branch of mathematics that provides a rigorous link between the description of dynamical systems in terms of their trajectories and the description in terms of statistics of orbits and, more generally, between dynamical systems theory and probability theory. At the core of the ergodic theory, there is a set of \"natural\" dynamically invariant probability measures in the phase space, somewhat generalising the Liouville distribution for conservative systems with strong analogies with Gibbs distributions in statistical physics [6, 7]. This strong connection, in particular, gave birth to the so-called Thermodynamic Formalism. The introduction of Thermodynamic Formalism that occurred in the 1970s was primarily due to [PERSON], [PERSON], and [PERSON] [8, 9, 10]. The development of Thermodynamic Formalism initially served to derive rigorous criteria characterising the existence and uniqueness of the Gibbsstates in the infinite volume limit. Although Gibbs states and equilibrium states (see Section 2.1.1) are naturally defined in finite volume systems, the extension to infinite volume (\"thermodynamic limit\") is far from straightforward. Indeed, it does not follow from the Caratheodory or Kolmogorov extension theorems [11; 12], that the equilibrium states of the infinite volume define a measure, as there is no way to express the marginals associated to an infinite-volume Gibbs measure without making explicit reference to the measure itself [12]. When considering conditional probabilities, rather than marginals, [PERSON], Lanford, and Ruelle led to a different consistency condition that affords for the building of infinite volume Gibbs measures [7]. In the context of dynamical systems, [PERSON], [PERSON], and [PERSON] were able to connect the theory of hyperbolic (Anosov) dynamical systems to results in statistical mechanics. Indeed, [PERSON] found an unexpected link between the equilibrium statistical mechanics of spin systems and the ergodic theory of Anosov systems by a codification using Markov partitions (see Section 2.1.1 for details). This idea was later extended for a much more general class of hyperbolic systems [10; 13; 14]. While the Thermodynamic Formalism started as a branch of rigorous statistical mechanics, nowadays it is viewed from different communities as a branch of dynamical systems or stochastic processes. There have been a few attempts to use Thermodynamic Formalism in different ways other than as a natural mathematical foundation of statistical mechanics, for example, studying population dynamics [15; 16], self-organised criticality [17], the relative abundance of amino acids across diverse proteomes [18], analyse the difference between introns and exons in genetic sequences [19; 20], coding sequence density estimation in genomes [21], and statistics of spike trains in neural systems [22; 23; 24; 25; 26; 27; 28], which is the main topic of this review. Neuronal networks are biological systems whose components, such as neurons, synapses, ionic channels,..., are ruled by the laws of physics and are written in terms of differential equations; hence, are dynamical systems. On the other hand, because of their large dimensionality, it is natural to attempt to characterise neuronal networks using methods inspired by statistical physics, for example, mean-field methods [29; 30; 31; 32], density methods [33] or Fokker-Planck equations [34].