# Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics
## 2 Mathematical Setting of Thermodynamic Formalism
### Correlations
#### 2.3.2 Ruelle-Perron-Frobenius Operator  
Let \(C(X)\) be the set of continuous functions on \(X\). Consider the potential \(\phi\) and a continuous function \(f\in C(X)\), so one can define a bounded linear operator associated to \(\phi\) (transfer operator), called the Ruelle-Perron-Frobenius (RPF), as follows (There is a close analogy between this operator and the propagator in quantum field theory or the Koopman operator in classical dynamics [5]. All of these operators characterise how measures or observables evolve ruled by the dynamics.):  
\[\mathcal{L}_{\phi}f(x)=\sum_{y\in\sigma^{-1}x}e^{\phi(y)}f(y). \tag{8}\]  
The spectral properties of (8) yields information to characterise the pressure and study the ergodic properties of the system, in particular, the rate of decay of their correlation functions [80]. For instance, if \(1\) is a simple eigenvalue and the modulus of each of the other eigenvalues is smaller than one, this is equivalent to be mixing [80]. When the potential considered is of finite range, then the transfer operator corresponds to a matrix and the whole formalism is equivalent to Markov chains defined on finite alphabets. A potential \(\phi\) is called normalised if \(\mathcal{L}_{\phi}(1)=1\). The log of a normalised potential of range \(R+1\), corresponds to the transition probabilities of a Markov chain with memory depth \(R\). Moreover, in this case \(F(\phi)=0\). For Lipschitz observables in the finite dimensional case, the Perron-Frobenius theorem assures a unique eigenvector associated to the maximal eigenvalue, from which the unique invariant measure (Markov) is obtained. This measure has mixing properties, exponential decay of correlations, central limit theorem, and a large deviations principle (see Section 2.3.4). When the operator acts on an infinite dimensional space (such as the space of continuous functions), then the spectrum of a bounded linear operator \(\mathcal{L}\) is given by the set \(\text{spec}(\mathcal{L})=\{\lambda\in\mathbb{C}\ :\ \text{such that}\ (\lambda I-\mathcal{L})\ \text{has no bounded inverse}\}\), this set may contain points that are not necessarily eigenvalues (see, for instance, [86]). In this case, the strategy is to find a proper subspace where the spectrum of \(\mathcal{L}\) has a finite number of such complex numbers whose norm is the spectral radius, say \(\rho\), and the rest of the spectrum has norm strictly less than \(\rho\) (spectral gap). In this scenario, it is known that there is exponential decay of correlations for a sufficiently regular class of observables (such as Lipschitz), and the central limit theorem holds. In the absence of the spectral gap, then one has sub-exponential decay of correlations, which breaks down the central limit theorem, and the phase transition phenomenon appears (for further details and precise definitions, see [80] and the references therein).  
Note that, given a potential \(\psi\), one can explicitly find a normalised potential \(\phi\) cohomologous to \(\psi\), as follows,  
\[\phi:=\psi+\log R-\log R\circ\sigma-\log\rho, \tag{9}\]  
where \(R\) is the right eigenvector (real and positive) associated to the unique maximum eigenvalue \(\rho\) that is associated to \(\mathcal{L}_{\psi}\).  
**Remark 2**.: _Note that the normalisation of the potential \(\psi\) does not require a partition function. In fact, as discussed below, the classical normalisation by a partition function is a particular case of (9), holding for memory-less potentials that does not generalise to range \(R>1\) potentials._