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We consider chains of identical diffusive weakly coupled oscillation systems with different conditions on coupling on a chain bound. We suppose that every interacting oscillator undergoes Andronov–Hopf bifurcation, and the coefficient of coupling is proportional to the supercriticality value. For this case on a stable integral manifold of the system a normal form is constructed for which, in case of three oscillators interact, we can study the simplest stationary states and their phase transformations. As the coupling parameter changes, for the uniform stationary state corresponding to the uniform cycle of the problem there can be two cases, in the former case it loses stability with the emergence of two stable nonuniform states, and in the latter one it merges with two unstable nonuniform states and transfers to them its stability. For the stationary state corresponding to oscillations in antiphase two cases can be also distinguished. In the first one, this stationary state becomes stable due to the contraction of a stable limit cycle of the system into it (Andronov–Hopf bifurcation), and in the second case it becomes stable after branching from it an unstable limit cycle. If there are four oscillators in the chain, the system of phase difference for a small coupling coefficient was studied.

The statistical analysis of random quantities derived from the dynamics of interaction of a pair of neuron-type oscillators are presented. It is shown that computation of some statistical characteristics of the process allows, with enough precision, to distinguish two types of orbits, while their phase portraits, Lyapunov dimension and time plots are slightly different.

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.

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 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.

The statistical analysis of random quantities derived from the dynamics of interaction of a pair of neuron-type oscillators are presented. It is shown that computation of some statistical characteristics of the process allows, with enough precision, to distinguish two types of orbits, while their phase portraits, Lyapunov dimension and time plots are slightly different.