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Littles model introduces a markovian dynamics to neuronal transmission. That is, the neurons are presumed to have no memory of states older than a specific time (normally taken as the interval between the recurring neuronal action potentials). Corresponding evolution dynamics has been addressed by Peretto [38], and it is shown that only a subclass of markovian processes which obeys the detailed balance principle can be described by Hamiltonians representing an extensive parameter for a fully interconnected system such as a neuronal assembly. This conclusion in the framework of the present model is implicit due to the fact the intrinsic transition rate of a wave functional attribute prevails under equilibrium conditions with the existence of a detailed balance between the interconnection i to j or j to i sites. 7.10 Wave Functional Representation of Hopfields NetworkConsider a unit (say, mth neuron) in a large neuronal assembly, which sets up a potential barrier φpB over a spatial extent apB. Assuming the excitatory situation due to the inputs at the synaptic node of the cell, it corresponds to the neuronal wave transmission across this mth cell, with CTn ≈ 1. The corresponding output or the emergent wave is given by the solution of the wave equation, namely, Equation (7.29) with appropriate boundary conditions. It is given by: where k(m) is the propagation vector, E(m) is the incident wave energy, Φ(m) is the mth mode amplitude function, e(m) = [πE(m)apB/λ(m)φpB] and λ(m) = 2π/k(m). Hence, the net output due to the combined effect of all the interconnected network of M neuron units at the mth synaptic node can be written as a superposition of the wave functions. That is: where Suppose intraneuronal disturbances are present. Then a noise term should be added to Equation (7.47). In terms of wave function notations, this noise term η(m) can be written as: Φη(m)exp[jξη(m))], where the amplitude Φη and the phase term ξη are random variates, (usually taken as zero-mean gaussian). Hence, the noise perturbed neural output can be explicitly specified by: where The nonlinear operation in the neuron culminating in crossing the threshold of the potential barrier corresponds to a detection process decided by the input (random) sequence r(m) so that the summed input exceeds the barrier energy across the neuron. Such a detection process refers to minimizing the mean square functional relationship given by: Written explicitly and rearranging the terms, the above relation (Equation 7.49) simplifies to: with W(m, m) = 0 and m, n ∈ 1, 2, ..., M; further: and The ε of Equation (7.50) depicts a neural network with the weights of interconnection being W and an external (bias) input of θ. Thus, the energy function of the Hopfield network can be constructed synonymously with wave functional parameters. 7.11 Concluding RemarksThe application of the concept of wave mechanics and the use of quantum theory mathematics in neurobiology were advocated implicitly by Gabor as early as in 1946. As stated by Licklider [75], the analogy [to] the position-momentum and energy-time problems that led Heisenberg in 1927 to state his uncertainty principle ... has led Gabor to suggest that we may find the solution [to the problems of sensory processing] in quantum mechanics. Supplemented by the fact that statistical mechanics too can be applied to study the neuronal activity, the foregoing analyses considered can be summarized as follows: The neuronal activity can be represented by the concepts of wave mechanics. Essentially, considering the fact that the interconnected neurons assume randomly one of the dichotomous potentials (0 or φpB), the input sequence at any given neuron would set a progression of state transitions in the interconnected cells. Such a spatial progress or the collective movement of state-transition flux across the neuronal assembly can be regarded as the neuronal transmission represented as a wave motion. Hence, the dynamic state of neurons can be described by a set of extensive quantities vis-a-vis the wave functional attributions to the neuronal transmission with relevant alternative Hamiltonian perspectives presented. Accordingly, the neuronal transmission through a large interconnected set of cells which assume randomly a dichotomous short-term state of biochemical activity can be depicted by a wave equation. In representing the neuronal transmission as a collective movement of neuronal states, the weighting factor across the neural interconnections refers implicitly to a long-term memory activity. This corresponds to a weight space Ω with a connectivity parameter (similar to the refractive index of optical transmission through a medium) decided by the input and local energy functions. The wave mechanical perspectives indicate that the collective movement of state transitions in the neuronal assembly is a zero-mean stochastical process in which the random potentials at the cellular sites force the wave function depicting the neuronal transmission into a nonself averaging behavior. Considering the wave functional aspects of neuronal transmission, the corresponding eigen-energies (whose components are expressed in terms of conventional wave parameters such as the propagation constant) can be specified. The wave mechanical considerations explicitly stipulate the principle of detailed balance as the requisite for microscopic reversibility in the neuronal activity. Specified in terms of the strength of synapses, it refers to Wij = Wji. This symmetry condition restricts the one-to-one analogy of applying the spin-glass model only to a limited subclass of collective processes.
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