The Stochastic, Markovian, Hodgkin-Huxley Type of Mathematical Model of the Neuron

Abstract

The aim of this paper is to show how the Hodgkin-Huxley model of the neuron's membrane potential can be extended to a stochastic one. This extension can be done either by adding fluctuations to the equations of the model or by using Markov kinetic schemes' formalism. We are presenting a new extension of the model. This modification simplifies computational complexity of the neuron model especially when considering a hardware implementation. The hardware implementation of the extended model as a system on a chip using a field-programmable gate array (FPGA) is demonstrated in this paper. The results confirm the reliability of the extended model presented here.

Training the Stochastic Kinetic Model of Neuron for Calculation of an Object’s Position in Space

“…Among other indicators we have made a comparison of the number of gates that change their state in a specific time step for both versions of the model. In paper [24] we present our results which show that stochastic kinetic model due to the simplification of the differential equations to the mathematical rules can significantly reduce the training time without loss of the neuron's properties.…”

“…The algorithm presented in this paper simplifies and speeds up the process of training of the stochastic kinetic model of neuron. The simulation of the model is faster; moreover, the reduction in the number of weights results in faster training process [10,24].…”

Regularization theory in the study of generalization ability of a biological neural network model

“…Huxley in 1952 [8]. The biological foundations of the model have been thoroughly studied and presented in a number of publications, e.g., in [3,5,8,9,14]; therefore, in this paper, we only provide a concise mathematical description of the model.…”

“…Their values are obtained from the normal distribution. A detailed description of the procedure for obtaining these values is included, among others, in [5,[12][13][14]. The forms of transfer functions α i (V ) and β i (V ), which appear in the kinetic schemes, are given in Table 1 [8].…”

“…In the next step, we equate all derivatives to zero, thus obtaining a system of equations, whose solution is the minimum of function E. For clarity, the derivaties of (12) are given below as three separate systems of Eqs. (13)(14)(15)). Only the system of equations (14) changes depending on the form of regualizer used.…”