Dmitry Sinelshchikov

We study a model of three Hindmarsh – Rose neurons with directional electrical connections. We consider two fully-connected neurons that form a slave group which receives the signal from the master neuron via a directional coupling. We control the excitability of the neurons by setting the constant external currents. We study the possibility of excitation of the slave system in the stable resting state by the signal coming from the master neuron, to make it fire spikes/bursts tonically. We vary the coupling strength between the master and the slave systems as another control parameter. We calculate the borderlines of excitation by different types of signal in the control parameter space. We establish which of the resulting dynamical regimes are chaotic. We also demonstrate the possibility of excitation by a single burst or a spike in areas of control parameters, where the slave system is bistable. We calculate the borderlines of excitation by a single period of the excitatory signal.

A quasi-one-dimensional model of flow of a liquid in a viscoelastic tube is considered. A closed system of the nonlinear equations for the description of perturbations of pressure and radius is propose at flow of a liquid in a is viscoelastic tube. For the analysis of system technique of the multiscale method and the perturbation theory is used. The mathematical model was investigated in case of the large Reynolds numbers. In the equation of movement of a wall of a tube the cubic correction to Hooke’s law is considered. Families of the nonlinear evolutionary equations for the description of perturbations of the basic characteristics of flow are obtained. Exact solutions of some nonlinear evolution equations are found.