Power law behavior related to mutual synchronization of chemically coupled map neurons

The widely represented network motif, constituting an inhibitory pair of bursting neurons, is modeled by chaotic Rulkov maps, coupled chemically via symmetrical synapses. By means of phase plane analysis, that involves analytically obtaining the curves guiding the motion of the phase point, we show how the neuron dynamics can be explained in terms of switches between the noninteracting and interacting map. The developed approach provides an insight into the observed time series, highlighting the mechanisms behind the regimes of collective dynamics, including those concerning the emergent phenomena of partial and common oscillation death, hyperpolarization of membrane potential and the prolonged quiescence. The interdependence between the chaotic neuron series takes the form of intermittent synchronization, where the entrainment of membrane potential variables occurs within the sequences of finite duration. The contribution from the overlap of certain block sequences embedding emergent phenomena gives rise to the sudden increase of the parameter characterizing synchronization. We find its onset to follow a power law, that holds with respect to the coupling strength and the stimulation current. It is established how different types of synaptic threshold behavior, controlled by the gain parameter, influence the values of the scaling exponents

Slow rate fluctuations in a network of noisy neurons with coupling delay

We analyze the emergence of slow rate fluctuations and rate oscillations in a model of a random neuronal network, underpinning the individual roles and interplay of external and internal noise, as well as the coupling delay. We use the second-order finite-size mean-field model to gain insight into the relevant parameter domains and the mechanisms behind the phenomena. In the delay-free case, we find an intriguing paradigm for slow stochastic fluctuations between the two stationary states, which is shown to be associated to noise-induced transitions in a double-well potential. While the basic effect of coupling delay consists in inducing oscillations of mean rate, the coaction with external noise is demonstrated to lead to stochastic fluctuations between the different oscillatory regimes

Phase plane approach to cooperative rhythms in neuron motifs with delayed inhibitory synapses

The phenomenon of burst synchronization is analyzed in binary and ternary motifs consisting of Rulkov map neurons coupled via delayed inhibitory synapses. We determine the particular roles and the interplay between the intrinsic neuron and synaptic parameters, as well as the network topology. The developed method, resting on exactly obtaining the curves that guide the neuron orbits in the phase plane, enabled us to identify the motif-specific mechanisms of how the synchronized rhythms emerge, even in the presence of strong delay. It is explained why the location of the parameter space domain optimal for burst synchronization gets shifted with different motif architectures. Further, it is suggested how for each motif a distinct cooperative rhythm may be singled out, that is absent on any of the other considered motifs