Complex oscillations in the delayed Fitzhugh-Nagumo equation

Motivated by the dynamics of neuronal responses, we analyze the dynamics of the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system provides a canonical example of a canard explosion for sufficiently small delays. Beyond this regime, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov-Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canard-induced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation

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

Complex partial synchronization patterns in networks of delay-coupled neurons

We study the spatio-temporal dynamics of a multiplex network of delay-coupled FitzHugh–Nagumo oscillators with non-local and fractal connectivities. Apart from chimera states, a new regime of coexistence of slow and fast oscillations is found. An analytical explanation for the emergence of such coexisting partial synchronization patterns is given. Furthermore, we propose a control scheme for the number of fast and slow neurons in each layer.DFG, 163436311, SFB 910: Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und Anwendungskonzept

Periodic Oscillation Analysis for a Coupled FHN Network Model with Delays

The existence of periodic oscillation for a coupled FHN neural system with delays is investigated. Some criteria to determine the oscillations are given. Simple and practical criteria for selecting the parameters in this network are provided. Some examples are also presented to illustrate the result

A Dynamical Systems Analysis of Movement Coordination Models

In this thesis, we present a dynamical systems analysis of models of movement coordination, namely the Haken-Kelso-Bunz (HKB) model and the Jirsa-Kelso excitator (JKE). The dynamical properties of the models that can describe various phenomena in discrete and rhythmic movements have been explored in the models' parameter space. The dynamics of amplitude-phase approximation of the single HKB oscillator has been investigated. Furthermore, an approximated version of the scaled JKE system has been proposed and analysed. The canard phenomena in the JKE system has been analysed. A combination of slow-fast analysis, projection onto the Poincare sphere and blow-up method has been suggested to explain the dynamical mechanisms organising the canard cycles in JKE system, which have been shown to have different properties comparing to the classical canards known for the equivalent FitzHugh-Nagumo (FHN) model. Different approaches to de fining the maximal canard periodic solution have been presented and compared. The model of two HKB oscillators coupled by a neurologically motivated function, involving the effect of time-delay and weighted self- and mutual-feedback, has been analysed. The periodic regimes of the model have been shown to capture well the frequency-induced drop of oscillation amplitude and loss of anti-phase stability that have been experimentally observed in many rhythmic movements and by which the development of the HKB model has been inspired. The model has also been demonstrated to support a dynamic regime of stationary bistability with the absence of periodic regimes that can be used to describe discrete movement behaviours.This work was supported by The Higher Committee For Education Development in Iraq (HCED) and the University of Mosul

Regenerative memory in time-delayed neuromorphic photonic resonators

We investigate a photonic regenerative memory based upon a neuromorphic oscillator with a delayed self-feedback (autaptic) connection. We disclose the existence of a unique temporal response characteristic of localized structures enabling an ideal support for bits in an optical buffer memory for storage and reshaping of data information. We link our experimental implementation, based upon a nanoscale nonlinear resonant tunneling diode driving a laser, to the paradigm of neuronal activity, the FitzHugh-Nagumo model with delayed feedback. This proof-of-concept photonic regenerative memory might constitute a building block for a new class of neuron-inspired photonic memories that can handle high bit-rate optical signals

Novel Modes of Synchronization and Extreme Events in Coupled Chemical Oscillators

We experimentally and computationally investigate dynamical behaviors in coupled chemical oscillators. These networks of chemical oscillators are created using catalytic Ru(bpy)32+ loaded cation exchange beads submerged in catalyst-free Belousov-Zhabotinsky (BZ) solutions. Various network structures are created by utilizing the photosensitive nature of the Ru(bpy) 32+ catalyst. The response of the oscillators due to light stimuli can be characterized by constructing a phase response curve (PRC). The PRC quantifies the excitatory and inhibitory responses of BZ oscillators due to applied light perturbations as a function of the oscillators\u27 phase. Different initial concentrations of reactants in the BZ reaction solutions can vary the degree in the excitatory and inhibitory regions of the PRC. We explore synchronization in star networks in both excitatory and inhibitory systems. We describe experiments, simulations, and analytical theory that provides a detailed characterization of novel modes of synchronization in chemical oscillator networks. Synchronization of peripheral oscillators coupled through a hub oscillator is exhibited at coupling strengths leading to novel synchronization of the hub with the peripheral oscillators. The heterogenous peripheral oscillators have different phase velocities that give rise to divergence; however, the perturbation from the hub acts to realign the phases by delaying the faster oscillators more than the slower oscillators. A theoretical analysis provides insights into the mechanism of the synchronization. Computational studies into extreme events are investigated using a modified four-variable Oregonator model, which describes the BZ system. Extreme events are ubiquitous throughout biological, natural, social, and financial systems. Examples of such events are epileptic seizures, earthquakes, riots, and stock market crashes. These events are considered rare excursions from the normal dynamics of a system, which are considered aperiodic in occurrence. The consequences that these events have on the system makes the development of models and experimental methods to study these events important. We will describe the appearance of extreme events in the Oregonator system using instantaneous and time-delayed coupling. We will also discuss a proposed mechanism for the sudden appearance of extreme events in both instantaneous and time-delayed coupling