By Regis Sbudd J. Bates

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Considering f(v, t)dv to denote the probability at time t that the measure v of the deviation from the resting potential lies between v and (v + dv), the following one-dimensional diffusion equation can be specified: where C and D are constant coefficients. Gernstein and Mandelbrot used the above diffusion model equation to elucidate the interspike interval distribution (of random walk principle). The corresponding result is: Upon reaching the threshold and allowing the postsynaptic potential to decay with a time constant specified by exp(- •t), an approximate diffusion equation for f(v, t) can be written as follows: Solution of the above equation portrays an unrestricted random passage of x to xo over the time and an unrestricted path of decay of the potential in the postsynaptic regime.

Pertinent to this arrangement, the pattern can be retrieved in one shot by a feed-forward algorithm; or the correct pattern is deduced via many iterations through the same network by means of a feedback algorithm. The feed-forward network has a linear or nonlinear associative memory wherein the synaptic weights are precomputed and prestored. The feedback associative memory networks are popularly known as Hopfield nets. 4 Net Function and Neuron Function The connection network neurons are mathematically represented by a basis function U (W, x) where W is the weight matrix and x is the input matrix.

A more involved description of the neuronal activity continuum refers to nonlinear integro-differential equations as elucidated by Orguztoreli [45] and Kawahara et al. [46]. Another modeling technique due to Ventriglia [47] incorporating intraneuron excitation, proportion of neurons in refractory state, velocity of impulses, neuronal density, synaptic density, axonic branching, and fraction of excitatory neurons in the continuum description of spatiotemporal neuronal activity has led to the study of informational waves, dynamic activities, and memory effect.

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Advanced Intelligent Networks by Regis Sbudd J. Bates
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