Phase models and clustering in networks of oscillators with delayed coupling

We consider a general model for a network of oscillators with time delayed, circulant coupling. We use the theory of weakly coupled oscillators to reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. We use the phase model to study the existence and stability of cluster solutions. Cluster solutions are phase locked solutions where the oscillators separate into groups. Oscillators within a group are synchronized while those in different groups are phase-locked. We give model independent existence and stability results for symmetric cluster solutions. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions. We apply our analytical results to a network of Morris Lecar neurons and compare these results with numerical continuation and simulation studies

Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback

Interaction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator reaches a specific state, it emits a pulse, which returns after propagating through a delay line. The impact of an incoming pulse is described by the oscillator's phase reset curve (PRC). In such a system we discover an unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous "jittering" regimes with non-equal interspike intervals (ISIs). Each of these regimes corresponds to a periodic solution of the system with a period roughly proportional to the delay. The number of different "jittering" solutions emerging at the bifurcation point increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. In particular, we show how a periodic solution exhibiting several distinct ISIs can imply the existence of multiple other solutions obtained by rearranging of these ISIs. We show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback

Clustered Chimera States in Systems of Type-I Excitability

Chimera is a fascinating phenomenon of coexisting synchronized and desynchronized behaviour that was discovered in networks of nonlocally coupled identical phase oscillators over ten years ago. Since then, chimeras were found in numerous theoretical and experimental studies and more recently in models of neuronal dynamics as well. In this work, we consider a generic model for a saddle-node bifurcation on a limit cycle representative for neural excitability type I. We obtain chimera states with multiple coherent regions (clustered chimeras/multi-chimeras) depending on the distance from the excitability threshold, the range of nonlocal coupling as well as the coupling strength. A detailed stability diagram for these chimera states as well as other interesting coexisting patterns like traveling waves are presented

Delay-induced patterns in a two-dimensional lattice of coupled oscillators

We show how a variety of stable spatio-temporal periodic patterns can be created in 2D-lattices of coupled oscillators with non-homogeneous coupling delays. A "hybrid dispersion relation" is introduced, which allows studying the stability of time-periodic patterns analytically in the limit of large delay. The results are illustrated using the FitzHugh-Nagumo coupled neurons as well as coupled limit cycle (Stuart-Landau) oscillators

Comparing Epileptiform Behavior of Mesoscale Detailed Models and Population Models of Neocortex

Two models of the neocortex are developed to study normal and pathologic neuronal activity. One model contains a detailed description of a neocortical microcolumn represented by 656 neurons, including superficial and deep pyramidal cells, four types of inhibitory neurons, and realistic synaptic contacts. Simulations show that neurons of a given type exhibit similar, synchronized behavior in this detailed model. This observation is captured by a population model that describes the activity of large neuronal populations with two differential equations with two delays. Both models appear to have similar sensitivity to variations of total network excitation. Analysis of the population model reveals the presence of multistability, which was also observed in various simulations of the detailed model

Exponential multistability of memristive Cohen-Grossberg neural networks with stochastic parameter perturbations

© 2020 Elsevier Ltd. All rights reserved. This manuscript is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Licence http://creativecommons.org/licenses/by-nc-nd/4.0/.Due to instability being induced easily by parameter disturbances of network systems, this paper investigates the multistability of memristive Cohen-Grossberg neural networks (MCGNNs) under stochastic parameter perturbations. It is demonstrated that stable equilibrium points of MCGNNs can be flexibly located in the odd-sequence or even-sequence regions. Some sufficient conditions are derived to ensure the exponential multistability of MCGNNs under parameter perturbations. It is found that there exist at least (w+2) l (or (w+1) l) exponentially stable equilibrium points in the odd-sequence (or the even-sequence) regions. In the paper, two numerical examples are given to verify the correctness and effectiveness of the obtained results.Peer reviewe

Firing multistability in a locally active memristive neuron model

Funding Information: This work is supported by The Major Research Project of the National Natural Science Foundation of China (91964108), The National Natural Science Foundation of China (61971185), The Open Fund Project of Key Laboratory in Hunan Universities (18K010). Publisher Copyright: © 2020, Springer Nature B.V. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.The theoretical, numerical and experimental demonstrations of firing dynamics in isolated neuron are of great significance for the understanding of neural function in human brain. In this paper, a new type of locally active and non-volatile memristor with three stable pinched hysteresis loops is presented. Then, a novel locally active memristive neuron model is established by using the locally active memristor as a connecting autapse, and both firing patterns and multistability in this neuronal system are investigated. We have confirmed that, on the one hand, the constructed neuron can generate multiple firing patterns like periodic bursting, periodic spiking, chaotic bursting, chaotic spiking, stochastic bursting, transient chaotic bursting and transient stochastic bursting. On the other hand, the phenomenon of firing multistability with coexisting four kinds of firing patterns can be observed via changing its initial states. It is worth noting that the proposed neuron exhibits such firing multistability previously unobserved in single neuron model. Finally, an electric neuron is designed and implemented, which is extremely useful for the practical scientific and engineering applications. The results captured from neuron hardware experiments match well with the theoretical and numerical simulation results.Peer reviewedFinal Accepted Versio