# Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics
## 5 Discussion and Perspectives
### Phase Transitions
** The occurrence of a second-order phase transition mathematically means that the pressure is \(C^{1}\) but not \(C^{2}\) when some limit is taken. Here, we have two possible limits: the range of the potential \(R\) tends to infinity or the number of neurons \(N\) tends to infinity. These two limits could also be addressed simultaneously and they do not necessarily commute. For potentials associated to finite \(R\) and \(N\) the Perron-Frobenius theorem guarantees the existence and uniqueness of the Gibbs measure and the analyticity of the pressure can also be proved, preventing phase transitions. When \(R\) or \(N\) are infinite, the properties of the RPF operator Section 2.3.2 characterises the presence or absence of phase transitions. Indeed, there are conditions ensuring a spectral gap for this operator, ensuring the exponential decay of correlations. Now, Equation (7) characterise the second derivative of the pressure as a time series of correlations, which converge when the correlations decay exponentially. On the opposite side, the non-summability of time correlation function implies the non-existence of the second derivative, and thus, of a second-order phase transition. Therefore, a possibility to have a second-order phase transition is when the spectral gap property for the RPF operator when \(R\to+\infty\) or \(N\to+\infty\) is absent. In statistical mechanics, second-order phase transitions can be characterised by how the zeros of the partition function, written as a polynomial, pinch the real axis (Lee-Yang phenomenon) [142; 143; 144]. In our case, when \(R>1\), the object of interest is not the partition function, but rather the largest eigenvalue of \(\mathcal{L}_{\phi}\), which has to stay analytic in the limit \(R\), or \(N\), \(\to+\infty\). The absence of the spectral gap property presents an analogy with the Lee-Yang phenomenon, although we do not know about results establishing a deeper link. * **Can we relate known examples of dynamical systems exhibiting phase transitions to models in neuroscience? ** Another possible example to be interpreted in neuroscience is the Dyson model [145], in which there exists a phase transition in the sense of spontaneous magnetisation when the temperature goes to zero, due to an infinite range potential whose correlation does not decay exponentially fast. In our case, the range of the potential should be taken in time, keeping (possibly) the number of neurons finite. Other examples exist of rigorous characterisations of phase transitions in the thermodynamic description of Pomeau-Manneville intermittent maps, passing from an integrable density function associated with the measure to heavy-tailed densities [146]. An interesting result may hint at the connection between the topological Markov map of the interval and stochastic chains of infinite order or chains with complete connections. Ref [147] presents how to build a topological Markov map of the interval whose invariant probability measure is the stationary law of a given stochastic chain of infinite order. This is interesting in this context because as we presented in (27), there are mathematical models of spiking neurons whose spike statistics are represented by chains of infinite order. This result or its inverse i.e, how to build a stochastic chain of infinite order from a topological Markov map may hint at conditions in the parameters or conditions of the mathematical models of spiking neurons to exhibit second order phase transitions. * **What could the dynamical or mechanistic origins of a second-order phase transition be in a spiking neuronal network model? ** Handling experimental data is of course important, but for long experiments with living neuronal tissue, one cannot control the size of the sample, the stationarity of data, and so on. Accordingly, assume that we have been able to find an example of a Gibbs distribution exhibiting a second-order phase transition when \(R\to+\infty\) or \(N\to+\infty\).