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The McCulloch-Pitts depiction of a neuron (best known as the formal or mathematical neuron) is a logical and idealized representation of neuronal activity. It purports the idealization of a real neuron with the features of being able to be excited by its inputs and of giving an output when a threshold is exceeded. This all-or-none response, although it provides a digital machine logic mathematically with ease of tracking the state-transitions in the neuronal transmission, it is rather unrealistic in relation to its time dependence. That is, the McCulloch-Pitts model presents results on the state-transitional relaxation times which are astronomically large in value, which is not obvious in the real neuron situation as also observed by Griffith [11,12]. In the present modeling strategies, the nonrealistic aspects of the McCulloch-Pitts model vis-a-vis a real neuron is seen due to the fact that only in the limiting case of the (pseudo) de Broglie wavelength approaching zero, the nonlinear gain (Λ) of the state-transition process would approach infinity (corresponding to the McCulloch-Pitts regime). However, this is only possible when the cellular potential barrier value φpB approaches infinity which is rather physically not plausible. As long as φpB has a finite value less than Ei, Λ is finite as per the above analysis confirming a realistic model for the neuronal activity rather than being the McCulloch-Pitts version. In terms of magnetic spins, the spontaneous state transition (McCulloch-Pitts model) corresponds to the thermodynamic limit of magnetization for an infinite system (at temperature below the critical value). This infinite system concept translated into the equivalent neural network considerations, refers to the nonlinear gain (Λ) of the network appproaching infinity.
The foregoing deliberations lead to the following inferences:
- The dynamic state of neurons can be described by a set of extensive quantities vis-a-vis momentum flow analogously equitable to a to quasiparticle dynamics model of the neuronal transmission, with appropriate Hamiltonian perspectives.
- Accordingly, the neural 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 function representing the particle motion.
- Hence, considering the wave functional aspects of neuronal transmission, corresponding eigen-energies can be stipulated.
- Further, in terms of the Hamiltonian representation, of neural dynamics, there are corresponding free-energy (Helmholtz and Gibbs versions) and partition functions.
- Relevant to quasiparticle dynamics representation, the neural transmission when modeled as a random particulate (Brownian) motion subjected to a potential barrier at the neuronal cell, a Langevin force equation can be specified for the state-transition variable in terms of a neuronal mass parameter; and, the transmission (excitatory response) across the cell or nontransmission (inhibitory response) is stipulated by the neuronal particle (dually regarded also as a wave with an eigen-energy) traversing through the cellular potential or reflected by it. The quasiparticulate and wave-like representation of neuronal transmission leads to the explicit determination of transmission and reflection coefficients.
- On the basis of particulate and/or wave-like representation of neuronal transmission and using the modified Langevin function description of the nonlinear state-transition process, a corresponding gain function can be deduced in terms of the neuronal mass and the barrier energy. Relevant formalism specifies that the gain function approaching infinity (McCulloch-Pitts regime) corresponds to the potential barrier level (φpB) at the cell becoming infinitely large in comparison with the eigen-energy (Ei) of the input (which is not however attainable physically).
- In terms of (pseudo) de Broglies concept extended to the dual nature of neurons, the spontaneous transition of McCulloch-Pitts regime refers to the wavelength becoming so very short (with φpB << Ei) that any physically realizable potential would change only by a negligible amount over a wavelength. This corresponds to the classical limit of transmission involving no reflection.
- The size of the wave-packet associated with neuronal quasi-particle transmission can be related to the range of the interaction and the momentum of the incident particle. When the spread is minimum, the passage of a wave packet can be considered as the neuronal transmission with a state-transition having taken place. This can be heuristically proved as follows.
The extent of the cluster of cells participating in the neuronal transmission refers to the average range of interaction (equal to <xij>); and the corresponding size of the wave-packet emulating the neuronal transmission can be decided as follows: Suppose (Δxij)in is the spatial spread of the incident wave-packet at the site, i. After the passage through the cell (local interaction site), the corresponding outgoing wave-packet has the spread given by [3]: where τij is the ith to jth site-trans time; and the velocity term is dictated by the uncertainity principle, namely, . Also ≈ij ≈ <xij>/vij = <xij> mN/p. Hence (τxij)out has a minimum when . The corresponding momentum spread is, therefore, .
The aforesaid parameter spreading implicitly refers to the smearing parameter defined by Shaw and Vasudevan [76] in respect to Littles analogy [33] of neurons and and Ising glass spins. The ad hoc parameter β = 1/kBT in Littles model represents the fluctuation governing the total (summed up) potential gathered by the neuron in a time step taken in the progression of neuronal activity. Shaw and Vasudevan relate the factor 1/kBT to the gaussian statistics of the action potential and the poissonian process governing the occurrence of the rate of chemical quanta reaching the postsynaptic memory and electrically inducing the postsynaptic potential. The relevant statistics elucidated in [76] referred to the variations in size and the probability of release of these quanta manifesting as fluctuations in the postsynaptic potentials (as observed in experimental studies).
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