Abstract:
This study is a continuation of M. M. Preobrazhenskaya's work "Relay System of Differential Equations with Delay as a Perceptron Model", which aimed to combine approaches related to artificial neural networks and modeling biological neurons using differential equations with delay. The model of a single neuron was proposed, which allows for the existence of special modes called "aging" and "dying" behavior of the neuron. The study found a certain range of parameters where the "dying" mode of the neuron exists and numerically demonstrated the existence of the "aging" mode. Purpose. We will unify the concepts of "aging" and "dying" neurons into the term "freezing" neuron. For this neuron, we will analytically construct a solution and find the range of parameters for its existence and stability, which will extend the results of the reference article. Methods. To study this model, an auxiliary equation obtained by exponential substitution in the original equation is considered. Then, the method of step integration of a differential equation with delay and the introduction of additional functions are used. Results. A solution of the "freezing" neuron type for the original model is constructed, and the range of parameters for the existence and stability of this solution is described. Conclusion. The study obtained an extension of results for solutions of a special type in the model proposed by M. M. Preobrazhenskaya.
Keywords:phenomenological model of a neuron, differential equation with delay, relay equation, step method, parameter range, stability
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