# Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics
## 4 Spiking Neuronal Network Models: The Leaky Integrate-and-Fire and Beyond
### Markov Partition and Symbolic Coding  
In this section, we consider the deterministic discrete-time neuronal network model obtained considering (25) with \(\sigma_{B}=0\), studied in detail in [114]. The threshold \(\theta\) of the voltage in a network of \(N\) neurons induces a natural partition of \(\mathbb{R}^{N}\), \(\mathcal{P}=\prod_{k=1}^{N}\mathcal{P}_{x^{k}}\), where \(x^{k}\in\left\{\,0,1\,\right\}\), \(\mathcal{P}_{0}=\,]-B,\theta[\), \(\mathcal{P}_{1}=[\theta,B]\) where the bound \(B\) depends on synaptic weights [114; 117]. If \(V_{n}^{k}\in\mathcal{P}_{0}\), it evolves according to the sub-threshold dynamics (25) and it does not emit a spike at time \(n\). In contrast, if \(V_{n}^{k}\in\mathcal{P}_{1}\), it emits a spike at time \(n\) and its trajectory is set back to \(\mathcal{P}_{0}\) at time \(n+1\). Thus, \(\mathcal{P}\) is a natural partition in the sense thatit informs about the spikes of each neuron. Therefore, to each trajectory \(V\equiv\left\{\,V_{n}^{k},\,k=1\ldots N,n\in\mathbb{Z}\,\right\}\), there is associated an infinite spike sequence \(x\) such that \(x_{n}^{k}=0\Longleftrightarrow V_{n}^{k}\in\mathcal{P}_{0}\) and \(x_{n}^{k}=1\Longleftrightarrow V_{n}^{k}\in\mathcal{P}_{0}\). This partition can be used to generate a Markov partition [114], but, in general, the Markov partition is not \(\mathcal{P}\) but a finite refinement \(\mathcal{Q}\) of \(\mathcal{P}\). What ensures that a Markov partition exists is that (27) is contracting. More precisely, it contracts, in one step, at speed \(\gamma\) for directions (neurons) such that \(V^{k}<\theta\), and it contracts, with an infinite speed, for directions such that \(V^{k}\geq\theta\) (reset). However, this generates discontinuities in the mapping (27), and a singularity set \(\mathcal{S}=\left\{\,V\in\mathbb{R}^{N}\,|\,\exists k\in\left\{\,1\ldots N\, \right\},V^{k}=\theta\,\right\}\), where the map associated with (27), hereafter denoted by \(\mathcal{G}\), is discontinuous. Thus, \(\mathcal{G}\) is piecewise continuous and piecewise contracting. Now, recall that \(\mathcal{Q}\) is a Markov partition for the dynamics with contracting map \(\mathcal{G}\) if its elements satisfy \(\mathcal{G}(\mathcal{Q}_{n})\cap\mathcal{Q}_{n^{\prime}}\
eq\mathcal{O} \Rightarrow\mathcal{G}(\mathcal{Q}_{n})\subset\mathcal{Q}_{n^{\prime}}\). In other words, the image of \(\mathcal{Q}_{n}\) is included in \(\mathcal{Q}_{n^{\prime}}\) whenever the transition \(n\to n^{\prime}\) is legal. Here, in general, the elements of \(\mathcal{P}\) do not satisfy this condition. This is because the image of a domain of \(\mathcal{P}\) usually intersects in several domains (in this case, the image intersects the singularity set). From the neural network's perspective this means that, in general, it is not possible to know the spiking pattern at time \(n+1\) knowing the spiking pattern at time \(n\). There are several possibilities, depending on the membrane potential values and not only on the firing state of the neurons. Of course, if, say \(\mathcal{P}_{n}\) is such that \(\mathcal{G}\left(\,\mathcal{P}_{n}\,\right)\) intersects several domains \(\mathcal{P}_{n_{1}},\ldots,\mathcal{P}_{n_{l}}\) one can take the preimages of these domains \(\mathcal{G}^{-1}\left(\,\mathcal{P}_{n_{r}}\,\right)\) to construct a refinement of \(\mathcal{P}\), such that the Markov partition requirement is satisfied in one iteration of the map. However, nothing guarantees that, at the second iteration, some elements of this new partition will not intersect the singularity set under \(\mathcal{G}^{2}\). Can we find a finite refinement of \(\mathcal{P}\), such that the trajectory of the partition elements never intersects several partition elements? It is shown in [114] that this property is satisfied for generic values of the synaptic weights \(W_{ij}\). Essentially, it is based on the fact that the distance between the \(\Omega\)-limit set of (27) and the singularity set \(\mathcal{S}\), is generically positive.