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7.3 Neural “Particle” Dynamics

The kinetic quasiparticle description (with a microscopic momentum-position attribution) of a neuronal phase space is apropos in depicting the corresponding localizable wave-packet, Ψ(x, t). Considering the neuronal transmission across the ith cell similar to random particulate (Brownian) motion subjected to a quadratic potential, the Langevin force equation depicting the fluctuation of the state variable specifies that [87,94]:

where So is a normalization constant, α is a constant dependent on the width of the potential barrier, mN is the pseudo-mass of the neuronal particle and is the critical velocity at which this “particle” in the absence of the effect of (thermal) random force will just reach the top of the barrier and come to rest. In the event of the neuronal energy E exceeding the barrier potential φpB, or , the corresponding transmission function is given by [87,94]:

The transmission function indicated above specifies implicitly the nonlinear transition process between the input-to-output across the neuronal cell. The motion of a “neuronal particle” can also be described by a wave function . The eigenfunction ⒘i(x) is a solution to:

and

assuming that at x = 0 crossing of the potential barrier occurs depicting a neuronal state transition.

The traveling wave solution of Equation (7.20a) in general form is given by:

where is the momentum equal to ; and CRn is the reflection coefficient.

Similarly, for x > 0:

where ; and (1 - CRn is the transmission coefficient

Classically, the neural state transmission corresponding to the “neuronal particle” entering the (output) region x > 0 has a probability of one. Because of the presumed wave-like properties of the particle, there is a certain probability that the particle may be reflected at the point x = 0 where there is a discontinuous change in the regime of (pseudo) de Broglie wavelength. That is, the probabililty flux incident upon the potential discontinuity can be split into a transmitted flux and a reflected flux. When E ≈ φpB, the probability of reflection would approach unity; and, in the case of E >> φpB, it can be shown that [95]:

or even in the limit of large energies (or E >> φpB), the (pseudo) de Broglie wavelength is so very short that any physically realizable potential φ changes by a negligible amount over a wavelength. Hence, there is total potential transmission with no reflected wave corresponding to the classical limit. Further, the transmission factor (for x > 0) can be decided by a function Func(.) whose argument can be set equal to:

where aPB refers to the width of the potential barrier.

Therefore, the transmission factor of Equation (7.19) can be rewritten in terms of the energy, mass and wave-like representation of neuronal transmission as:

In terms of neuronal network considerations, CTn can be regarded as the time-average history (or state-transitional process) of the activation induced updates on the state-vectors (Si) leading to an output set, σi. This average is decided explicitly by the modified Langevin function as indicated in Chapter 5. That is, by analogy with particle dynamics wherein the collective response is attributed to nonlinear dependence of forces on positions of the particles, the corresponding statistics due to Maxwell-Boltzmann could be extended to neuronal response to describe the stochastic aspects of the neuronal state-vector. The ensemble average which depicts the time-average history thereof (regarding the activation-induced updates on state-vectors) is the modified Langevin function, given by [16]:

where βG is a scaling factor and (βG/kBT) is a nonlinear (dimensionless) gain factor Λ. This modified Langevin function depicts the stochastically justifiable squashing process involving the nonlinear (sigmoidal) gain of the neuron unit as indicated in Chapter 5. Further, the modified Langevin function has the slope of (1/3 + 1/3q) at the origin which can be considered as the order parameter of the system. Therefore, at a specified gain-factor, any other sigmoidal function adopted to depict the nonlinear neuronal response should have its argument multiplied by a factor (1/3 + 1/3q) of the corresponding argument of the Langevin function.

Heuristically, the modified Langevin function denotes the transmission factor across the neuronal cell. Hence, writing in the same format as Equation (7.19), the transmission factor can be written in terms of the modified Langevin function as follows:

Comparing the arguments of Equations (7.24) and (7.26a), it can be noted that:

omitting the order parameter which is a coefficient in the argument of Equation (7.24), due to the reasons indicated above. Hence, it follows that:

Thus, in the nonlinear state-transition process associated with the neuron, the limiting case of the gain Λ → ∞ (McCulloch-Pitts regime) corresponds to the potential barrier level φpB → ∞. This refers to the classical limit of the (pseudo) de Broglie wavelength approaching zero; and each neuronal state Si is equitable to an energy level of Ei. That is, for the set of neuronal state-vectors {Si} ⇔ {Ei}.


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