# Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics
## 5 Discussion and Perspectives
### Phase Transitions  
Several studies have shown that the population of vertebrate retinal ganglion cells responding to naturalistic stimulus is poised near a \"critical state\" [73; 74]. From the maximum entropy joint distribution (13), a family of Gibbs distributions can be built introducing a parameter \(1/\beta\) (analogous to the inverse temperature), which scales all of the Lagrange multipliers of the inferred Hamiltonian. When \(\beta\to 0\) (infinite temperature), the uniform distribution is obtained, and when \(\beta\to+\infty\) (zero temperature), the Dirac delta supported at the spike configuration(s) of minimal energy is obtained. \(1/\beta=1\) corresponds to the inferred maximum entropy distribution from data. These studies have only analysed memoryless Gibbs distributions (13). From this representation, it is possible to compute the fluctuations (variance) of the energy \(U_{\lambda}\) as a function of the \"temperature\" parameter \(T\). This quantity can be obtained as the second derivative of the pressure, Equation (6), which is, in thermodynamics, related to the heat capacity \(C_{T}\). On numerical grounds, this quantity can be computed while using MonteCarlo simulations and plotted as a function of the \"temperature\" \(T=\frac{1}{\beta}\), for different network sizes, (see Figure 4). The form of \(C_{T}\) versus \(T\) plot, for maximum entropy models of Ising type obtained from the recording of retinal ganglion cells responding to naturalistic stimuli are shown in Figure 4 (redrawn from [74]). It can be observed that there is a clear, increasing peak at \(T=1\), which starts to manifest itself when larger and larger groups of neurons are considered. This presumable divergence of the heat capacity (or variance of \(U\)) when \(N\to\infty\) (thermodynamic limit) is interpreted as a second order phase transition (a so-called \"critical regime\" [140]). The behavior of the specific heat that is observed in Figure 4 suggests that the heat capacity of a maximum entropy distribution, fitted over an increasingly large group of neurons in the retina, diverges. This phenomenon has been considered to be a \"signature of criticality\" (details of this study and a discussion about whether criticality is functional for retinal ganglion cells can be found in Ref. [74]). Some criticism regarding this approach to diagnostic criticality has appeared arguing that the maximum entropy principle is likely to yield models that are close to singular values of parameters, akin to critical points in physics where phase transitions occur. Statistically distinguishable models tend to accumulate close to critical points, where the susceptibility (inverse Fisher Information) diverges in infinite systems [141]. These ideas have also been applied to numerical simulations of a canonical feed-forward population model showing that the specific heat diverges whenever the average correlation strength is independent of the population size [75], as in the random subsampling of correlations used in [74]. Additionally, note that, for spike trains obtained from discrete Markov processes, binning generates a stochastic process with unbounded memory akin to inducing spurious phase transitions [102]. This interesting approach leads, nevertheless, to several questions in the context of Thermodynamic Formalism. Figure 4: Signatures of criticality Generic plot of heat capacity \(C_{T}\) versus temperature \(T\) for maximum entropy models built constraining firing rates and pairwise correlations of retinal ganglion cells responding to naturalistic stimuli [74]. A clear peak appears at \(T=1\) when groups of an increasingly large number of neurons are considered (thermodynamic limit). * **Does this signature of criticality extend to Gibbs distributions with potentials of range \(R>1\), i.e., with memory? ** How does it depend on \(R\)? We are not aware of any experimental results addressing this issue. This question is related to the following:
* **What is this signature of criticality from the point of view of Thermodynamic Formalism?