# Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics
## 5 Discussion and Perspectives
### What Else Do Thermodynamic Formalism and Gibbs Distributions May Tell Us about Neuroscience? The relationship between mathematics and physics has been historically symbiotic and Thermodynamic Formalism is an interesting example of how ideas from physics may help to solve problems and introduce ideas into the field of mathematics. The history of Thermodynamic Formalism also shows how purely mathematical results can be obtained as corollaries of physical laws, inverting the frequently assumed relationship between physics and mathematics [149]. However, in the case considered in this review--the link between Thermodynamic Formalism and neuroscience--the mathematical problem is motivated by biology, not by physics. While [PERSON] argues in favour of the \"The unreasonable effectiveness of mathematics in the natural sciences\" [150], [PERSON], after spending several years working in mathematical problems related to biology, replied with \"The unreasonable ineffectiveness of mathematics in biology.\" [151]. While there are reasons to argue that this is still the case, it is less clear if one can blame the field of mathematics or just the fact that we have not yet used the right tools or frameworks. In the quest for these \"right tools\", there is a long tradition of using ideas from statistical physics to study neural networks, and in particular, to represent the emergence of collective behaviour from microscopic interactions, with the hope that statistical aspects of the collective behaviour will be independent of the details in these systems. This gave rise to major branches of theoretical neuroscience, like dynamic mean-field methods [29; 152; 153; 154] or the Maximum Entropy approach [38; 49; 148; 155], mainly coming from physicists. Although there are considerably less articles using mathematical methods to rigorously analyse the collective behaviour of neuronal networks some promising approach have been recently proposed based on large deviations [156; 157], Kalikow-typedecomposition [91; 158], stochastic processes [159; 160; 161; 162; 163], dynamical systems [137; 164], etc. As we have developed in this review, Thermodynamic Formalism could also be one of these tools, providing interesting connections between mathematics and physics, dynamics and statistics, applied to neuroscience. Especially, we have described how Thermodynamic Formalism: (1) provides a conceptual and operational (i.e., allowing to develop algorithms and software [99]) framework to analyse experimental spike train data; (2) allows to derive explicit expressions linking spike statistics to neuronal networks dynamics; (3) extends to non stationarity via linear response theory; and, (4) proposes a realm to address questions related to criticality. Here, we would like to propose some other extensions, not yet explored so staying at the level of ideas, all based on the power of Thermodynamic Formalism to make explicit and operational links between dynamics, statistics and symbolic coding. 1. **Geometry of the state space. ** A prominent aspect of Thermodynamic Formalism, that we haven't discussed yet in this review, is its link to the characterisation of the geometry of attractors and, especially, fractal sets [165; 166]. For example, the composition of contracting mappings along symbolic orbits defines the so-called Iterated Function Systems (IFS) [167] generating fractal sets with tunable geometry and structure. Now, it is interesting to remark that Integrate and Fire models are actually piecewise contracting dynamical systems having a structure similar to IFS where the contracting pieces are symbolically encoded by spike blocks [114]. It would be interesting to investigate, along these lines, the structure of attractors in Integrate and Fire models, and how orbits, encoded by spike blocks, are related to the geometry of attractors (the \(\Omega\)-limit set). 2. **Transitions between attractors. ** The concept of attractor is actually central in describing brain dynamics [168; 169].