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Now, denoting the neuronal momentum as , the critical transition between a progressive and an evanescent (reflected) wave corresponds to the propagation vector , respectively.

Following the analysis due to Peretto [38], the set of internal states of a large neural network is designated by Si{i ∈ 1, 2, ..., M}, which represents the internal state marker for the elementary unit i. Pertinent to the set {Si}, an extensive quantity Q(Si) can be specified which is proportional to M, the size the of the system. The internal state has two dichotomous limits, namely, +SU and -SL associated with MU and ML cellular elements (respectively) and (MU + ML) = M. Hence, in the bounded region Ω, (MU/M) and (ML/M) are fractions of neurons at the two dichotomous states, namely, SU and SL, respectively. It may be noted that in the deterministic model due to Wilson and Cowan [44], these fractions represent the proportions of excitatory cells becoming active (per unit time) and the corresponding inactive counterparts.

In terms of the phase space variable x and the associated momentum p, the probability distribution ρ({x, p}, t) refers to the probability of the systems being in the {x,p} phase space, at time t. It is a localized condition and is decided explicitly by the stationary solution of the Boltzmann equation (dp/dt = 0) leading to the Gibbs distribution given by:

where Z refers to the partition function given by the normalization term Σ{x, p}exp[-HN(x, p)/kBT]); here, kB is the pseudo-Boltzmann constant, T is the pseudo-temperature, and HN(x,p) refers to the Hamiltonian which is the single global function describing the dynamic system.

Pertinent to the kinetic picture of the neuronal transmission, the wave function Ψ and its conjugate Ψ* are two independent variables associated with the collective movement of the neuronal process in a generalized coordinate system with p and x being the canonical momentum and positional coordinates, respectively. Hence, the following transformed equations can be written:

where p and x satisfy the classical commutation rule, namely, [x, p] = 1 and, [x, x] = [p, p] = 0. Further, the corresponding Hamiltonian in the transformed coordinate system is:

The canonical momentum p and the canonical coordinate x are related as follows:

which are Hamilton’s first and second equations, respectively; and the Hamiltonian U which refers to the energy density can be stated in terms of an amplitude function Φ as:

The corresponding energy-momentum tensor for the neuronal transmission can be written as:

where

G = |Φ|2 k Momentum density
Energy flux density
Momentum flux density of the
neuronal flow or flux

The above tensor is not symmetric; however, if the momentum density function is defined in terms of the weighting factor W as G = |Φ|2 k/W2 (with the corresponding , then the tensor is rendered symmetric.

The dynamics of the neuronal cellular system at the microscopic level can be described by the Hamiltonian HN, (x1, x2, ..., xM; p1, p2, ..., pM; S1, S2, ..., SM) with the state variables Si’s depicting the kinematic parameters imposed by the synaptic action. The link between the microscopic state of the cellular system and its macroscopic (extensive) behavior can be described by the partition function Z written in terms of Helmholtz free energy associated with the wave function. Hence:

where kBT represents the (pseudo) Boltzmann energy of the neural system as stated earlier. On a discrete basis, the partition function simply represents the sum of the states, namely:

where Ei depicts the free energy of the neuronal domain Φi. Essentially, Z refers to a controlling function which determines the average energy of the macroscopic neuronal system. There are two possible ways of relating the partition function versus the free energy adopted in practice in statistical mechanics. They are:

Helmholtz free energy:

Gibbs free energy:

The corresponding partition functions can be explicitly written as:

and

where f is the force vector.

The relevant Hamiltonians referred to above are related to each other by the relation,

and the following Legendre transformations provide the functional relation between versus the force vector and Si versus Hf.

Physically, for a given set of microscopic variables (x, p), the function describes a system of neuronal “particles” with coordinates xi’s and momenta pi’s in interaction with the environment Ωi having the states (Si) stipulated by a set of kinematic parameters Si’s, i ∈ 1, 2, ..., M; and the Legendre transform Hf (x, p; ) describes the same system in interaction with Ωi with a dynamical force parameter . Thus, and Hf are alternative Hamiltonians to describe the neural dynamics associated with Ωi.


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