2024/07/05 by Preobrazhenskaia, Margarita M., Preobrazhenskii, Igor E.
#37L15 #37N25 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Biological sciences #FOS: Mathematics #Neurons and Cognition (q-bio.NC)
paper · doi:10.48550/arxiv.2408.16000
We consider the relay version of the generalized Hutchinson's equation as a phenomenological model of an isolated neuron. After an exponential substitution, the equation takes the form of a differential-difference equation with a piecewise-constant right-hand side. In the work of A. Yu. Kolesov and others, this equation was studied with a negative continuous initial function, and the existence and orbital stability of a periodic solution with a period longer than the delay were proven. In the present work, all possible solutions with continuous on the interval of the delay length initial functions, containing no more than two zeros, are constructed. It is proven that the equation has a periodically unstable solution, the period of which is shorter than the delay.