Embedding the dynamics of a single delay system into a feed-forward ring

Delayed-feedback oscillators replicate the dynamics of multiplex networks: Wavefront propagation and stochastic resonance

The widespread development and use of neural networks have significantly enriched a wide range of computer algorithms and promise higher speed at lower cost. However, the imitation of neural networks by means of modern computing substrates is highly inefficient, whereas physical realization of large scale networks remains challenging. Fortunately, delayed-feedback oscillators, being much easier to realize experimentally, represent promising candidates for the empirical implementation of neural networks and next generation computing architectures. In the current research, we demonstrate that coupled bistable delayed-feedback oscillators emulate a multilayer network, where one single-layer network is connected to another single-layer network through coupling between replica nodes, i.e. the multiplex network. We show that all the aspects of the multiplexing impact on wavefront propagation and stochastic resonance identified in multilayer networks of bistable oscillators are entirely reproduced in the dynamics of time-delay oscillators. In particular, varying the coupling strength allows suppressing and enhancing the effect of stochastic resonance, as well as controlling the speed and direction of both deterministic and stochastic wavefront propagation. All the considered effects are studied in numerical simulations and confirmed in physical experiments, showing an excellent correspondence and disclosing thereby the robustness of the observed phenomena.

Adaptive myelination causes slow oscillations in recurrent neural loops

The brain is known to be plastic, i.e., capable of changing and reorganizing as it develops and accumulates experience. Recently, a novel form of brain plasticity was described which is activity-dependent myelination of nerve fibers. Since the speed of propagation of action potentials along axons depends significantly on their degree of myelination, this process leads to adaptive change of axonal delays depending on the neural activity. To understand the possible influence of the adaptive delays on the behavior of neural networks, we consider a simple setup, a neuronal oscillator with delayed feedback. We show that introducing the delay plasticity into this circuit can lead to the occurrence of slow oscillations which are impossible with a constant delay.

Chimera states for directed networks

We demonstrate that chimera behavior can be observed in ensembles of phase oscillators with unidirectional coupling. For a small network consisting of only three identical oscillators (cyclic triple), tiny chimera islands arise in the parameter space. They are surrounded by developed chaotic switching behavior caused by a collision of rotating waves propagating in opposite directions. For larger networks, as we show for a hundred oscillators (cyclic century), the islands merge into a single chimera continent, which incorporates the world of chimeras of different configurations. The phenomenon inherits from networks with intermediate ranges of the unidirectional coupling and it diminishes as the coupling range decreases.

Mode Hopping in Oscillating Systems with Stochastic Delays

We study a noisy oscillator with pulse delayed feedback, theoretically and in an electronic experimental implementation. Without noise, this system has multiple stable periodic regimes. We consider two types of noise: (i) phase noise acting on the oscillator state variable and (ii) stochastic fluctuations of the coupling delay. For both types of stochastic perturbations the system hops between the deterministic regimes, but it shows dramatically different scaling properties for different types of noise. The robustness to conventional phase noise increases with coupling strength. However for stochastic variations in the coupling delay, the lifetimes decrease exponentially with the coupling strength. We provide an analytic explanation for these scaling properties in a linearized model. Our findings thus indicate that the robustness of a system to stochastic perturbations strongly depends on the nature of these perturbations.

Delay-induced switched states in a slow-fast system

We consider the two-component delay system εx'(t) = - x(t) - y(t) + f(x(t - 1)), y'(t) = ηx(t) with small para- meters ε, η and positive feedback function f. Previously, such systems have been reported to model switching in optoelectronic experiments, where each switching induces another one after approximately one delay time, related to one round trip of the signal. In this paper, we study these delay-induced switched states. We provide conditions for their existence and show how the formal limits ε → 0 and/or η → 0 facilitate our understanding of this phenomenon. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.