# Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics
## 4 Spiking Neuronal Network Models: The Leaky Integrate-and-Fire and Beyond
### Dynamics and Spikes  
A fundamental equation in neuronal membrane potential dynamics is the conservation of electric charge, written in its most canonical form, as follows:  
\[C\frac{dV}{dt}=-\sum_{X}g_{X}\left(V-V_{X}\right)+I(t),\]  
where \(C\) is the membrane capacitance of the neuron and \(V\) is its membrane potential. The sum \(\sum_{X}\) holds on ionic currents of the form \(i_{X}=-g_{X}\left(V-V_{X}\right)\) involving specific ionic channels permeable to specific ions (e.g., \(Na^{+}\), \(K^{+}\), \(Cl^{-}\), \(Ca^{2+}\)). Here, we include the neuron's intrinsic currents' (e.g., sodium and potassium currents triggering a spike [111]) and synaptic currents [109]. The conductance, \(g_{X}\), of channels of type \(X\) depends, in general, non-linearly on activation variables, themselves dependent on the voltage. \(V_{X}\) is the Nernst reversal potential, i.e., the value of the membrane potential at which the current \(i_{X}\) reverses its direction. Finally, \(I(t)\) is an external current that can mimic, e.g., an injection by an electrode or an external stimulus. In its simplest form, for a single isolated neuron, this equation takes the form:  
\[C\frac{dV}{dt}=-\frac{1}{R}V+I(t), \tag{24}\]  
where \(R\) is the membrane resistance and the term \(g=\frac{1}{R}\) corresponds to a unique passive conductance. In this case, we consider only a leak current where the leak reversal potential is set to 0. This is the equation of an RC circuit, which is quite simple, but quite far from a real neuron, as this equation does not even produce spikes. To circumvent this problem, one introduces a threshold, \(\theta\), such that Equation (24) holds whenever \(V(t)<\theta\) (sub-threshold dynamics), corresponding to the \"integrate\" phase. In contrast, for all times \(t_{(r)}\) such that \(V(t_{(r)})=\theta\), two effects take place: (i) The membrane potential of the neuron is reset instantaneously to a rest value, here 0, without a loss of generality; (ii) a spike is recorded at times \(t_{(r)}\) called \"spike times\". This is the \"fire\" phase (see Figure 2B, bottom). While this is a simple artificial way to generate spikes, there is a huge price to pay on mathematical grounds because the threshold introduces a singularity set in the phase space where the dynamic is not differentiable. We develop this aspect below. The generalisation of (24) to a network of \(N\) neurons is straightforward. Adding the contribution of synaptic currents, \(I^{k}_{(syn)}(t)\), building the network interactions, we obtain:  
\[C^{k}\,\frac{dV^{k}}{dt}+\ \frac{1}{R}V^{k}=I^{k}_{(syn)}(t)+S^{k}(t)+\sigma_{ B}\xi^{k}(t),\quad\text{if}\ V^{k}(t)<\theta, \tag{25}\]  
where \(C^{k}\) is the membrane capacitance of neuron \(k\), \(V^{k}\), its membrane potential. The resistance \(R\) is assumed to be the same for all neurons. In the synaptic current,  
\[I^{k}_{(syn)}(t)=\sum_{j,r}W_{kj}\,\alpha\left(t-t^{j}_{(r)}\right), \tag{26}\]  
the parameter \(W_{kj}\) represents the synaptic strength (\"weight\") from the pre-synaptic neuron \(j\) to the post-synaptic neuron \(k\) (see Figure 2B). Synaptic weights can be negative (inhibition) or positive (excitation). By convention, \(W_{kj}=0\) if there is no connection from \(j\) to \(k\). This way, the sum in (26) holds for \(j=1\ldots N\). The function \(\alpha\) represents the time profile of the postsynaptic current induced by a pre-synaptic spike [112]. It has been experimentally observed that the tail of this function is exponential. On mathematical grounds this is essential. The sum in Equation (26) considers all the spike times \(t^{j}_{(r)}\) emitted by all the pre-synaptic neurons \(j\) before time \(t\). When considering the asymptotic regime \(t\to-\infty\) (to get rid of initial conditions) this sum may contain an infinite number of terms. Thus, in order to ensure the sumability of this series one needs \(\alpha\) to decay sufficiently fast (here exponentially fast). Equation (25) holds in the sub-threshold regime.