Stochastic Processes in the Neurosciences by Henry C. Tuckwell

By Henry C. Tuckwell

This monograph is situated on quantitative research of nerve-cell habit. The paintings is foundational, with many better order difficulties nonetheless last, particularly in reference to neural networks. completely addressed subject matters contain stochastic difficulties in neurobiology, and the remedy of the idea of comparable Markov techniques.

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Calvin and Stevens (1965), (1968) made a careful study of the randomness of the ISIs of cat spinal motoneurons using intracellular recording. The voltage fluctuations in the absence of stimulation were analyzed and found to have peak to peak amplitudes between 2 and 8mV. Their amplitude distribution was found to be approximately Gaussian, and their autocovariance approximately an exponentially decreasing function of time difference with time constant of about 4 msec. It was inferred that the voltage fluctuations were due to randomly arriving postsynaptic potentials (EPSPs and IPSPs) and that they could be represented by a stationary OUP, X(t), as defined above.

Solving the equation for HQ with boundary conditions gives ONE-DIMENSIONAL DIFFUSION PROCESSES 51 As fl—»°°, ju0(jt)-*l. Hence, denoting the firing time by Te, for all // and all o. , Abramowitz and Stegun (1965)). 8) with boundary conditions jU 1 (-a) = ju1(0) = 0 and by letting fl^>°°. ) is the confluent hypergeometric function of the first kind. , Thomas (1975)). Higher-order moments of Te have also been found. The second moment was found in series form by Ricciardi and Sacerdote (1979). Expressions from which the third moment could also be calculated were given by Sato (1978).

This is not without application for the following reason. When the drug tetrodotoxin (TTX) is applied to nerve cells, the sodium channels that are opened during an action potential are blocked. A neuron cannot then emit an action potential. However, the potassium channels are not blocked, and neither are the channels involved in synaptic transmission. Thus with random input to a cell treated with TTX, the voltage will be unrestricted except (locally) by reversal potentials for synaptic action, which do not enter the present model.

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