The map with no predetermined firing order for the network of oscillators with time-delayed pulsatile coupling

The network of the pulse-coupled oscillators is studied in the presence of coupling delays. Because of the delays the past activity of the network is capable to influence the future network dynamics. In general case this leads to the infinite dimension of the corresponding dynamical system. We prove the Theorem that states that under certain conditions (weak coupling and appropriate initial conditions) the network can be fully characterized by a finite dimensional state vector. We construct the return map describing the evolution of this state vector over time. This map does not need any presupposed activity pattern in the network and works for any initial conditions.     

Stimulus-evoked activity in clustered networks of stochastic rate-based neurons

Understanding the effect of network connectivity patterns on the relation between the spontaneous and the stimulus-evoked network activity has become one of the outstanding issues in neuroscience. We address this problem by considering a clustered network of stochastic rate-based neurons influenced by external and intrinsic noise. The bifurcation analysis of an effective model of network dynamics, comprised of coupled mean-field models representing each of the clusters, is used to gain insight into the structure of metastable states characterizing the spontaneous and the induced dynamics. We show that the induced dynamics strongly depends on whether the excitation is aimed at a certain cluster or the same fraction of randomly selected units, whereby the targeted stimulation reduces macroscopic variability by biasing the network toward a particular collective state. The immediate effect of clustering on the induced dynamics is established by comparing the excitation rates of a clustered and a homogeneous random network.     

Synchronization of time-delay coupled pulse oscillators

We present a detailed study of the dynamics of pulse oscillators with time-delayed coupling. We get the return maps, obtain strict solutions and analyze their stability. For the case of two oscillators, a periodical structure of synchronization regions is found in parameter space, and the regions corresponding to in-phase and antiphase regimes alternate with growth of time delay. Two types of switching between in-phase and antiphase regimes are studied. We also show that for different parameters coupling delay may have synchronizing or desynchronizing effect. Another novel result is that phase locked regimes exist for arbitrary large values. The specificity of system dynamics with large delay is studied.     

Desynchronization by phase slip patterns in networks of pulse-coupled oscillators with delays

Coupling delays may cause drastic changes in the dynamics of oscillatory networks. In the present paper we investigate how coupling delays alter synchronization processes in networks of all-to-all coupled pulse oscillators. We derive an analytic criterion for the stability of synchrony and study the synchronization areas in the space of the delay and coupling strength. Specific attention is paid to the scenario of destabilization on the borders of the synchronization area. We show that in bifurcation points the system possesses homoclinic loops, which give rise to complex long- or quasi-periodic solutions. These newly born solutions are characterized by a synchronous group, from which an oscillator periodically escapes, laps one period, and rejoins. We call such a dynamical regime “phase slip patterns”.     

Phase reset of complex oscillations by a pulsed action

The effect of phase reset in a complex oscillatory system under the action of an external pulse is investigated. After the pulse action, the phase of oscillations in the system acquires a new value, which is virtually independent of the initial phase and is only determined by the pulse parameters (the amplitude and the duration). The phase reset is observed for pulses of different durations, including those considerably shorter than the characteristic scale of oscillations in the system. The effect occurs without any fundamental difference for the cases of regular and chaotic oscillations.