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A method of modeling the phenomenon of bursting behavior in neural systems based on delay equations is proposed. A singularly perturbed scalar nonlinear differentialdifference equation of Volterra type is a mathematical model of a neuron and a separate pulse containing one function without delay and two functions with different lags. It is established that this equation, for a suitable choice of parameters, has a stable periodic motion with any preassigned number of bursts in the time interval of the period length. To prove this assertion we first go to a relay-type equation and then determine the asymptotic solutions of a singularly perturbed equation. On the basis of this asymptotics the Poincare operator is constructed. The resulting operator carries a closed bounded convex set of initial conditions into itself, which suggests that it has at least one fixed point. The Frechet derivative evaluation of the succession operator, made in the paper, allows us to prove the uniqueness and stability of the resulting relax of the periodic solution.

A pair of diffusion connected FitzHugh–Nagumo oscillators with an asymmetric interaction are considered. The problem is investigated in the close to critical case, where the matrix of the linear part of the system has a pair of purely imaginary eigenvalues. The normal form is constructed and its coefficients are determined depending on the initial parameters. The source system may be in two different situations: stable singlefrequency oscillations with two different frequencies coexist or a single-frequency mode branches from the equilibrium. The obtained asymptotic results are supplemented by the numerical analysis.