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The neuronal assembly can also be regarded as analogous to a lattice gas system. Such a representation enables the elucidation of the probability of state transitions at the neuronal cells. That is, by considering the neuronal assembly as a disordered system with the wave function being localized, there is a probability that the neural transmission occurs between two sites i and j with the transition of the state +SU to -SL (or vice versa) leading to an output; and hence the number of such transitions per unit time can be specified.

In terms of the wave mechanics concept, the McCulloch-Pitts regime refers to the limit of the wavelength being so very short (with φpB << Ei) that any physically realizable potential would change only by a very negligible amount assuring a complete transmission. Within the framework of the Ising spin model, such a nonzero spontaneous transition would, however, warrant an infinite system (in the thermodynamic limit and assuming the external bias being zero) as observed by Peretto.

In the existing studies based on statistical mechanics modeling of neuronal activity, Peretto identifies an extensive parameter (expressed as a Hamiltonian) for a fully interconnected Hopfield network, and relevant state-transitional probability has hence been deduced. The basis for Peretto’s modeling stems from the existence of a long-range persistent order/state in biological neurons (analogous to or mimicking the Ising spin system), as observed by Little.

Peretto, by considering the output of action potentials occurring at regular intervals due to the synchronized excitatory (or inhibitory) synaptic action, has elucidated the markovian aspects of neuronal activity. This is substantiated by the evolution equation of the system described in Chapter 6. Peretto deduced a digital master equation to characterize the markovian structure of neuronal transmission. He has indicated the existence of a Hamiltonian at least for a narrow subclass of markovian processes which obey the detailed balance principle governing the state-transition rate. Similar observations are plausible by considering the Fermi golden rule as applied to the state-transition probability and its dynamics governed by Boltzmann’s equation. It is inferred thereof that regardless of the microscopic origin of neuron interactions, the principle of detailed balance must be satisfied in the neuronal dynamic process.

Consideration of Boltzmann’s equation indicates that the probability of neurons being in the equilibrium state will decay exponentially. From the neural network point of view, it implicitly refers to the well-known integration process (low-pass action) associated with neuronal input/output relation.

The average rate of neuronal transmission flow depends on the rate of state-transition between the interconnected cells and the difference in the local barrier potentials at the contiguous cells concerned. This is in concurrence with a similar observation by Thompson and Gibson. The average rate of transmission has also been shown as an implicit measure of the weighting factor. Further, by considering the spatial “spread” or the “size” of the wave-packet emulating the neuronal transmission, the spread when minimum represents the passage of a wave-packet across the cell. In other words, it represents the neuronal transmission with a state-transition having taken place.

In terms of wave mechanics considerations, the effect of external stimuli refers to altering the neuronal transmission rate. This is in concurrence with Little’s heuristic justifications that the active state proliferation is decided largely by the interneuronal weighting function. Any external bias would perturb this value via an implicit change in the local potentials.

The eigenstates of the “neuronal wave” represent the neuronal information (or memory storage at the sites). The existence of eigenstates presumably warrants an extended state of the cellular sites which is guaranteed by the translational invariancy of the neuronal assembly.

Considering the presence of intraneural disturbances in a Hopfield network the corresponding system can be modeled in terms of wave functional parameters as could be evinced from Equation (7.50).

The spread in the neuronal wave function across a cell is an implicit indicator of whether state transition has occurred or not. The minimum spread assures the transmission of the neuronal wave confirming that transition has occurred; and its zero value (ideally) refers to the McCulloch-Pitts logical transition or the spontaneous transition.

This spread in the wave function can be equated to the “smearing” condition proposed by Shaw and Vasudevan. That is, from the thermodynamics point of view the neuronal transition (when modeled as analogous to the Ising principle of interacting spins), corresponds to the McCulloch-Pitts model with the kBT term tending to zero. This would require, however, a total absence of fluctuations in the postsynaptic potentials; but, in the real neurons, there is an inevitable fluctuation in the summed up postsynaptic potentials, which leads to a finite spread in the wave-functional transmission.

The amplitude of the transmitted wave function, namely, CTna(Ωi/Ω) is an implicit function of the domain fraction (Ωi/Ω). The subset(s) of Ωi across which a preferential wave transmission occurs decide the so-called persistent order (based on a learning mechanism) attributed to neurons by Little.

The quasiparticle and wave-like propagation of neural transmission describe a kinetic view that stems from a phase space kinetic equation derived from the wave equation. The resulting field theory picture of neurons is a wave train corresponding to a system of quasiparticles whose diffusive kinetics permits the elucidation of the amplitude and phase properties of the propagating wave train in a continuum and thereby provide a particle portrait of neuronal dynamics. Griffith once observed the “concept of nervous energy is not theoretically well-based, at least not yet ... However, although there are obstacles it may be argued they are not necessarily insurmountable.”


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