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In terms of this transitional probability, the rate of change of the probability distribution function ρ (at a given time, t) of the neuronal state (corresponding to the subregion Φi) can be written via the well-known Boltzmann equation, namely:

The above Equation (7.37) specifies the net change in Φ at the same instant (markovian attribution) as the excitatory process permits the progressive neural transmission and the inhibitory process sets an evanescent condition of inhibiting the neural transmission.

Under the equilibrium condition, setting dρ/dt = 0 yields:

where the superscript o refers to equilibrium status. Assuming that the equilibrium values ρo (S1) and ρo (S2) are decided by Boltzmann distribution, , it follows that w(S2, S1) exp [(ES1 - ES2)/kBT] = w(S1, S2); and this relation (known as the principle of detailed balance) must be satisfied regardless of the microscopic origin of neuronal interactions, as has been observed by Peretto [38]. Further, if the energy of the equilibrium state (S0) is much larger than the other two dichotomous states S1 and S2, w(S2, S0) << w(S0, S2), the solution of master equation (Equation 7.37) leads to: . That is, the probability of neurons being in the state |S0> will decay exponentially with a time-constant τS0; or from the neural network point of view it refers implicitly to the integration process (low-pass action) associated with neuronal input-output relation.

In terms of the macroscopic potential function Φ(S0) in the sample domain Ωi, specified by , the time dependency of the neuronal transition process can also be written as:

where <τ> is the average (energy) relaxation time or the time of integration involved in neural transmission.

In a neuronal aggregate of M cells with the dichotomous state of activity, there are 2M possible different states which could be identified by S = 1, ..., 2M associated with an M-dimensional hypercube. A superficial analogy here with the quantum statistical mechanics situation corresponds to a set of M subsystems, each having two possible quantum states as, for example, a set of M atoms each having a spin 1/2 [13,32].

Each of the 2M states has a definite successor in time so that the progress of the state-transition process (or the neuronal wave motion) can be considered as a sequence i2 = z(i1) → i3 = z(i2) ≚ ..., and so on. Regarding this sequence, Griffith [13] observes that in the terminal cycle of the state-transitional process, there are three possible situations, namely, a state of equilibrium with a probability distribution ρ(S0); and the other two are dichotomous states, identified as S1 ⇒ +SU and S2 ⇒ -SL with the statistics ρ(S1) and ρ(S2), respectively.

In computing the number of states which end up close to the equilibrium at the terminal cycle, Griffith [13] observes the following fundamental difference between the neural and the quantum situation. From the quantum mechanics point of view, the transition probabilities between two states ρ1,2 ⇒ Ψ1,2 with reference to the equilibrium state, namely, ρ0 ⇒ Ψ0 (due to an underlying potential perturbation φ), are equal in both directions because they are proportional, respectively, to the two sides of the equation given by [96]:

which is true inasmuch as φ is Hermitian. That is, in the case of neuronal dynamics, only with the possibilities of i2 = z(i1) and i1 = z(i2) the microscopic reversibility is assured; and then there would be a natural tendency for the microscopic parameter ρ1,2 to move near to ρ0.

7.7 Lattice Gas System Analogy of Neural Assembly

Consistent with the wave-functional characterization, the neuronal transmission can be considered as the interaction of a wave function with the (neuronal) assembly constituted by the repeated translation of the basic unit (neuron) cell. Such an assembly resembles or is analogous to a lattice structure in space comprising a set of locales (or cells) normally referred to as the sites and a set of interconnections between them termed as bonds. Relevant to this model, the neuronal assembly is regarded as translationally invariant. As a consequence, each cell (or site) is like any other in its characteristics, state, and environment. This refers to the equilibrium condition pertaining to the density of neuronal states. The implication of this translational invariance is the existence of extended states (or delocalized states). Further, the neuronal assembly can be regarded as a disordered system wherein the spatial characteristics are random so that the potentials (at each cell) are localized.

In the case of delocalized states inasmuch as all site energies are identical, the neuronal transmission refers to a simple case of the wave proliferating over the entire neural assembly; whereas, the localized situation would make some regions of the neuronal lattice more preferable (energy-wise) than the others so that the neuronal spatial spread is rather confined to these regions.

Considering the neuronal wave function localized on individual sites, the present problem refers to calculating the probability that the neural transmission occurs between two sites i and j with the transition resulting in the output being at one of the two dichotomous limits, namely, +SU or -SL. This would lead to the calculation of the number of such transitions per unit time.


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