# Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics
## 5 Discussion and Perspectives
### Thermodynamic Formalism for More Biologically Plausible Neuron Models  
In this review, we have considered rather academic models of neuronal networks, where, especially, time is considered to be discrete. There are good reasons for that. As we remarked at the beginning of Section 3, we were considering models, like the Integrate-and-Fire, where spikes arise instantaneously, in continuous time, thereby providing a _possibly uncountable_ set of potential spike trains. The question is whether we are dealing here with a realistic property of biological neuronal networks or with an artifact created by the instantaneous reset. Real spikes have a duration (a few ms) and a refractory period (also a few ms), so, for a fixed initial condition, spike trains produced by a continuous-time neuronal network are countable. Now, it might be that the set of spike trains depend continuously on the initial condition, so that we are still left with an uncountable set of spikes. It is out of the scope of this review to discuss from a biological perspective, whether or not neuronal networks have the cardinality of the continuum (see [101; 117; 131] for a discussion on this topic). Instead, we have considered a strategy consisting of discretizing time, avoiding the problem of potentially uncountable spikes. Here, we briefly mention another strategy, allowing for associating a countable set of spike trains to continuous time networks with a countable set of spike trains. We first make the remark that the instantaneous reset of voltage is physical and biological nonsense, inducing pathologies in the dynamics [132]. On this basis, we use a convenient mathematical trick that is explained in the next paragraph, which can certainly be criticized on phenomenological grounds [131], especially when the dynamical system representing the neuronal activity is deterministic. After spiking, a biological neuron stays at rest a certain time (refractory period). Accordingly, the trick is the following. Fix \(\delta>0\) and define a spiking variable \(x_{n}^{k}\in\{\,0,1\,\}\), where \(n\) is an integer, where \(x_{n}^{k}=1\) if neuron \(k\) emits a spike in the time interval \([n\delta,(n+1)\delta[\) and \(x_{n}^{k}=0\) otherwise. Recall that \(t_{(r)}^{k}\) denotes the time at which neuron \(k\) emits its \(r\)-th spike. This reads:  
\[x_{n}^{k}=\left\{ \begin{array}{ll}1,&\quad\text{if}\quad\exists r,t_{(r)}^{k }\in[n\delta,(n+1)\delta[;\\ 0,&\quad\text{otherwise.}\end{array} \right.\]  
Spiking variables are therefore time-discrete events with a time resolution \(\delta\). When \(V^{k}\) reaches the threshold at time \(t_{(r)}^{k}\) it is reset to \(0\), and stays there until time \((n+1)\delta\). After this, follows the sub-threshold evolution (25) until the next time where \(V^{k}\) reaches the threshold. Note that, in this modelling, \(\delta\) can be quite small when compared to the time scales of the dynamics. In this way, the set of spike trains \(x\) becomes at most countable. This trick can be used to generalise the Integrate-and-Fire model into a conductance-based Integrate-and-Fire model, which was introduced by [PERSON] and [PERSON] in [133] and mathematically studied in [23; 24; 117], where the synaptic conductance depends on the spike history. One can still show \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multicolumn{5}{c}{**Thermodynamic Formalism and Gibbs Measures**} \\ \hline \multirow{2}{*}{Number of neurons} & \multicolumn{2}{c}{Memory of the potential} \\ \cline{2-5}  & Memoryless & Finite & Infinite \\ \hline \multirow{2}{*}{Finite} & Boltzmann-Gibbs & Gibbs in the sense of & Chains with complete \\  & & Bowen & connections \\ \hline \multirow{2}{*}{Infinite} & Countable & state & Countable & state & Chains with variable \\  & Bernoulli & Markov & length \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Types of Gibbs measures potentially found in experimental data analysis or in the analysis of mathematical models of networks of interacting spiking neurons.