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The poissonian process pertinent to the neuronal spike train specifies that different interspike intervals are independent of each other. Should the neuronal spike activity be non-poissonian, deviations from spike-to-spike independence can be expected. Such deviations can be formalized in terms of the serial correlation coefficients for the observed sequence of interspike intervals. Existence of finite (nonzero) correlation coefficients may not, however, be ruled out altogether. The reasons are:
The variations in the interspike interval, if existing, render the neuronal process nonstationary which could affect the underlying probability regime not being the same at all times. 3.2.2 Random sequence of neural potential spikesPertinent to the neural activity across the interconnected set of cells, the probabilistic attributes of neuronal spikes can be described by the considerations of random walk theory as proposed by Gerstein and Mandelbrot [2]. The major objective of this model is to elucidate the probability distribution for the interspike interval with the assumption of independence for intervals and associated process being poissonian. After the cell fires, the intracellular potential returns to its resting value (resting potential); and, due to the arrival of a random sequence of spikes, there is a probability p at every discrete time interval that the intracellular potential rises towards the threshold value; or there is a probability q = (1 - p) of receding from the threshold potential. The discrete steps of time Δt versus the discrete potential change (rise or fall) Δv constitute a (discrete) random walk stochastical process. If the threshold value is limited, the random walk faces an absorbing barrier and is terminated. The walk could, however, be unrestricted as well, in the sense that in order to reach a threshold v = vo exactly at time t (or after t steps), the corresponding probability, p would decide the probability density of the interspike interval pI given by [14]: This refers to the probability that the interspike interval lies between t and (t + Δt) is approximately given by pI Δt. 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(- Εt), 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. The classical temporal random neurocellular activity as described above can also be extended to consider the spatiotemporal spread of such activities. Relevant algorithms are based on partial differential equations akin to those of fluid mechanics. On these theories one considers the overall mean level of activity at a given point in space rather than the firing rate in any specific neuron, as discussed below.
Copyright © CRC Press LLC
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