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Assuming the initial condition for Ψ(x, t = 0) as independent of the site coordinate x, then at an, ith site with a random potential Ei(x), the solution for Ψi(x, t) can be written explicitly in terms of a complete set of eigenfunctions and eigenvalues, namely:

where and the eigenvalues are all positive. Further, the asymptotic expansion of Ψi(x, t) is decided predominantly by the lowest eigenvalues. As defined earlier, the total transmission probability for a given site is Ψi(t) = ∫i(x, t)dx. The corresponding ensemble-average behavior of the neuronal transmission over the entire neuronal cell aggregates can be specified by the first cumulant [K2(t) - K12(t)], where K1(t) and K2(t) are the first and second moments of Ψi(x, t) respectively; and this first cumulant refers to the lower bound of the neuronal transmission statistics. It is much larger than the mean value, K1(t). That is, the fluctuations of neuronal transmission dominate the mean level of transmission. In other words, the random neuronal transmission across the interconnected cells over the long-term (memory) space Ω or the collective movement of state-transitions is a zero-mean stochastical process. Correspondingly, the wave function set forth by the random potentials at the cellular sites has a non-self-averaging behavior.

The incident (or incoming) collective movement of neurons in the domain Ω has its energy relation specified again in terms of a wave function Ψ similar to the conventional Schrödinger wave equation:

where , as indicated earlier, refers to the velocity of the “neuronal wave”; and the “connectivity” of neuronal transmission between the cells, therefore, corresponds to a scalar weighting factor (similar to the refractive index of a medium) given by W = [1 - (ΦpB/E)2]1/2.

Considering the analogy with the Ising spin model, the interconnected set of a large number of M neurons (with the state transition at a localized site, i) has a Hamiltonian or the extensive quantity given by [38]:

where Wij is the weighting factor, and Σθiσi is the Hamiltonian introduced by an external field (bias) θi at site i. In terms of the extensive parameter (or the Hamiltonian) HN, the eigenvalue equation can be written as: HNΨi = EiΨi, where i indexing refers to the set of eigenstates.

Considering the neuronal wave equation (Equation 7.29), the progressive (traveling) wave solution given by Equation (7.30) can be explicitly written as:

where the propagation vector (constant) k is decided by the boundary conditions relevant to the wave function Ψ. Further, the quantity θa refers to the phase constant associated with the amplitude of the wave, and Φ refers to the amplitude function of the wave.

The selective wave functions Φi (or mode functions), which are the eigen-solutions of the wave equation and propagate across the domain Ω, correspond to the excitatory neuronal transmission. Those which are cut off can be considered as having faced an inhibitory process and represent the evanescent waves. Over the subregions of with Ωi ∩ Ωj = 0 for i ≠ j, and M as indicated earlier, refers to the total number of subregions or cells (Figure 7.2). For the ith subregion, field Ψi is defined as Ψi = Ψ (Ωi) or zero for outside Ωi; and Ψ = ΣiMΨi with the summation representing the geometrical conjunction or the synaptic junction wherein the superposition of neuronal inputs occurs.

Each Φi being orthonormal to all other mode functions permits the incident (incoming) wave function Ψin to be expanded more explicitly in reference to the two domains Ωi (i ∈ MU) and Ωi’ (i’ ∈ ML) via spatial harmonics as:

where (n = 0, 1, ..., ∞), (v = 0, 1, ..., ∞) and An and Bv are the amplitudes of the relevant modes.

The {(Ψi)} components in the ith subregion correspond to the excitatory process leading to the final outcome as (+SU); and (Ψi’)’s are the reflected waves which are evanescent. They do not contribute to the output, and the relevant inhibitory process renders the dichotomous state as (-SL). The existence of (Ψi)’s and (Ψi’)’s is dictated by the relevant continuity conditions. In terms of energy density functions, the conservation relation can be written in terms of the modal functions as:

If the amplitude of the incident wave function is set equal to a(Ωi/Ω) where (Ωi/Ω) is the fraction depicting the spatial extent (or size) of the ith neuron in the state-space assembly of Ψ, then ΣΦi and ΣΦi’ can be proportionately set equal to CTn a(Ωi/Ω) and CRn a(Ωi/Ω), respectively. Here, CTn and CRn as mentioned earlier are coefficients of transmission and reflection, respectively, so that CTn = (1 - CRn).

7.6 Concepts of Wave Mechanics versus Neural Dynamics

The concepts of wave mechanics described above can be translated into considerations relevant to the neuronal assembly and/or a large (artificial) neural network. Considering the neuronal dynamics, the kinetics of the neuronal state denoted by a variable Si and undergoing a transition from S1 to S2 can be specified by a transition probability (or the probability per unit time that a transition would take place from state S1 to state S2), namely, wi(S1, S2); and, in terms of wave mechanics perspective, wi is given by the Fermi golden rule, namely:

where Φi is the eigen-potential function specified by Equation (7.35), and are energy level transitions, and the delta function guarantees the energy conservation.


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