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In the localized regime, the neuronal transmission from site i to site j (with energies Ei and Ej) corresponds to a transition probability wij. If the energy gaps between +SU and -SL states of ith and jth units are ΔEi and ΔEj respectively, then regardless of the previous state set, Sk=+SU with probability wk=1/[1 + exp(-ΔEk)/kBT], where k = i or j. This local decision rule in the neuronal system ensures that in thermal equilibrium, the relative probability of two global states is determined solely by their energy difference, dictated by the (pseudo) Boltzmann factor, namely, exp [-(Ej - Ei)/kBT). This also refers to the extent of transmission link (weighting factor) between the sites i and j.

The corresponding intrinsic transition rate (from site i to site j) can be written as: γij = exp[-2αixij - (Ej - Ei)/kBT], if Ej > Ei; or, γij = exp [-2αixij] if Ej < Ei where αi is specified by a simple ansatz for the wave function Ψi(xi) localized at xi, taken as Ψi(xi) = exp(-αi |x - xi|) similar to the tunneling probability for the overlap of the two states; and xij = |xj - xi|. Further, the average transition rate from i to j is: Γij = <(Mi/M)(1 - Mj/M) γij> where the Mi’s are the neural cell population participating in the transition process out of the total population M. Assuming the process is stationary, Γij can also be be specified in terms of the probability distribution function ρi and ρj as a self-consistent approximation. Hence, Γij = ρi(1 - ρjij. Under equilibrium conditions, there is a detailed balance between the transitions i to j and j to i as discussed earlier. Therefore, (Γij)o = (Γji)o, with the superscript o again referring to the equilibrium condition. Hence, ρio(1 - ρjo)(γij)o = ρjo (1 - ρio)(γji)o. However, (γij)o = (γji) exp[-Ej - Ei/kBT] which yields the solution that:

And

where φpB is the cellular (local) barrier potential energy (or the site pseudo-Fermi level.)

The Hamiltonian corresponding to the neuronal activity with the dichotomous limits (+SU and -SL) corresponding to the possible interactions at ith and jth sites is given by Equation (7.28). Suppose the bias θi is set equal to zero. Then the Ising Hamiltonian (Equation 7.28) has a symmetry. That is, it remains unchanged if i and j are interchanged; or for each configuration in which a given Si has the value +SU, there is another dichotomous value, -SL, such that +SU and -SL have the same statistical weight regardless of the pseudo-temperature. This implies that the neuronal transition ought to be zero in this finite system. Hence, within the framework of the Ising model, the only way to obtain a nonzero spontaneous transition (with the absence of external bias) is to consider an infinite system (which takes into account implicitly the thermodynamic limit). Such a limiting case corresponds to the classical continuum concept of wave mechanics attributed to neuronal transmission depicting the McCulloch-Pitts logical limits wherein the neuron purports to be an idealization of a real neuron and has features of being able to be excited by its inputs and of giving a step output (0 or 1) when a threshold is exceeded.

7.8 The Average Rate of Neuronal Transmission Flow

The weighting factor or the connectivity between the cells, namely, Wij of Equation (7.28) is a random variable as detailed in Chapter 6. The probabilistic attribute(s) of Wij can be quantified here in terms of the average transition rate of the neuronal transmission across the ith and jth sites as follows.

The site energies Ei and Ej pertaining to the ith and jth cells are assumed as close to the cellular barrier potential (or pseudo-Fermi level) φpB; and, the following relation(s) are also assumed: |Ei|, |Ej|, |Ej - Ei| >> kBT. Hence, ρio ≈ 1 for Ei < 0 and ρio ≈ exp[-(EipB)/kBT] for Ei > 0; and under equilibrium condition:

The corresponding neuronal transmission across ith and jth cells can be specified by a flow rate, , equal to = (Γijij). Now, suppose the perturbations at the equilibrium are Eij = (Eji)o &43; ΔEij, ρij = (ρij)o + Δρij, and γij = (γij)o + Δij, and assuming that the detailed balance relation, namely, (Γij)o = (Γji)o is satisfied, the following relation can be stipulated:

For a differential change in the local potential barrier energy (at the pseudo-Fermi level), Again, Δ φpB >> kBT, ρi simplifies as equal to approximately with . This relation concurs implicitly with the observation of Thompson and Gibson [37,79] who observed that neuronal firing statistics “depends continuously on the difference between the membrane potential and the threshold”. Further, it is evident that the neuronal transmission rate is decided by: (1) Rate of state-transition between the interconnected cells; and (2) the difference in the local barrier potentials at the contiguous cells concerned.


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