Ivan Garashchuk

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.

We study a minimal network of two coupled neurons described by the Hindmarsh – Rose model with a linear coupling. We suppose that individual neurons are identical and study whether the dynamical regimes of a single neuron would be stable synchronous regimes in the model of two coupled neurons. We find that among synchronous regimes only regular periodic spiking and quiescence are stable in a certain range of parameters, while no bursting synchronous regimes are stable. Moreover, we show that there are no stable synchronous chaotic regimes in the parameter range considered. On the other hand, we find a wide range of parameters in which a stable asynchronous chaotic regime exists. Furthermore, we identify narrow regions of bistability, when synchronous and asynchronous regimes coexist. However, the asynchronous attractor is monostable in a wide range of parameters. We demonstrate that the onset of the asynchronous chaotic attractor occurs according to the Afraimovich – Shilnikov scenario. We have observed various asynchronous firing patterns: irregular quasi-periodic and chaotic spiking, both regular and chaotic bursting regimes, in which the number of spikes per burst varied greatly depending on the control parameter.