Some models for interaction of renewal processes related to neuron firing

Abstract

One area of science in which probabilistic models are beginning to play a role is in neuron firing. The firing process may be thought of in terms of electrical potentials: there is a potential difference across the membrane surrounding a neuron; this potential difference is determined by the concentrations of sodium, potassium, and chlorine ions on the two sides of the membrane. When electrical impulses arrive along the many input fibers, the potential difference changes; when it crosses a certain threshold, the neuron “fires”, that is, an impulse is discharged along the axon. Arriving impulses are of two types, those which raise and those which lower the potential inside the neuron; these are called stimuli and inhibitors, respectively. The mechanism by which stimuli and inhibitors interact in changing the neuron potential is not well understood; however, it appears that some of the stimuli cause the neuron to respond, or fire, and the inhibitors play a role in preventing the other stimuli from causing a response. The ability of stimuli to cause responses is further weakened by the decay, in the absence of arriving impulses, of the transmembrane potential toward a “resting level”, the level to which the potential is reset after a firing. While several models for the firing process have been proposed ([3], [4], [5], [6], [7], [11]), the present paper is devoted to introducing some modifications into a model proposed by Ten Hoopen and Reuver ([9], [10]). In particular, we place a time limit on the ability of inhibitors to prevent a stimulus from producing a response. Hopefully, these modifications bring their model closer to physiological reality.