Route to Chaos and Chimera States in a Network of Memristive Hindmarsh-Rose Neurons Model with External Excitation

Abstract

In this paper we have introduced and investigated the collective behavior of a network of
 memristive Hindmarsh-Rose (HR) neurons. The proposed model was built considering the memristive autapse
 of the traditional 2D HR neuron. Using the one-parameter bifurcation diagram and its corresponding maximal
 Lyapunov exponent graph, we showed that the proposed model was able to exhibit a reverse period doubling
 route to chaos, phenomenon of interior and exterior crises. Three different configurations of the ring-star
 network of the memristive HR neuron model, including ring-star, ring, and star, have been considered. The
 study of those network configurations revealed incoherent, coherent , chimera and cluster state behaviors.
 Coherent behavior is characterized by synchronization of the neurons of the network, while incoherent
 behaviors are characterized by the absence of synchronization. Chimera states refer to a differet state where
 there is a coexistence of synchroniaed and asynchronized nodes of the network. One of the interesting result
 of the paper is the prevalence of double-well chimera states in both ring and ring-star network and has been
 first mentioned in the case of memrisitve HR neuron model.

Complex spatiotemporal dynamics in a network of locally and magnetically coupled VDPCL oscillators

“…Another type of chimera state namely chimera at the phase flip transition which consists of the coexisting out-ofphase synchronized coherent domains and the incoherent domain in between the coherent domains has also been discovered [57]. The emergence of the so-called double well chimera has been also reported in recent studies [58,59]. Three different categories of chimera states including the basic chimera structure consisting of an incoherent subpopulation in a chaotic state while the coherent subpopulation could be periodic or remain close to a steady state, have also been identified in networks of limit cycle or chaotic oscillators under nonlocal coupling [60].…”

Dynamical investigation and FPGA implementation of a new Heartbeat model based on the Barrio-Varea-Aragon-Maini oscillator

“…It's derived from the reaction-diffusion Turing equations [23]; In this model u and v are the state variables, u represents the action potential or membrane voltage of the cardiac pacemaker cell and v represents its recovery variable, they are dimensionless in the context of this work. u  and v are respectively the time derivative of u and v. 2  is the Laplacian operator. Du and Dv represent the diffusion coefficients of u and v respectively.…”

“…Various physical, chemical, biological or ecological phenomena, etc can be modelled thanks to mathematics, and then be analysed in order to predict their future behaviour and anticipate, for example, the action to be taken depending on whether the occurrence of the said behaviour will prove beneficial or harmful. In the theory of dynamic systems, these behaviours include among others: The route to chaos phenomenon highlighted in a neuron model [1][2][3] the chaotic dynamics characterised by sensitivity to initial conditions and highlighted in a system of interacting nephrons [4]; the coexistence of attractors [5,6], the antimonotonicity [7,8] and the hysteresis [9][10][11], just to name a few. To predict these different phenomena in a given system, several analysis tools for dynamic systems are needed, including : Time series, which allow the trajectory of the system to be observed over time; phase portraits, which characterise the presence of an attractor; bifurcation diagrams, which indicate the values taken asymptotically by a system as a function of its control parameter; the Lyapunov exponent, which provides information on the degree of sensitivity of the system to its initial conditions; and basins of attraction, which provide information on the set of initial conditions for which the trajectories of the system converge towards one of its attractors.…”

Dynamical behavior of memristor-coupled heterogeneous discrete neural networks with synaptic crosstalk

“…[7][8][9] As is widely known, a memristor is a device that describes the relationship between charge and flux. [10] Since its development by HP Laboratory in 2008, memristors have found extensive applications in neural networks, [11][12][13][14][15] image encryption, [16,17] chaotic systems, [18][19][20][21][22][23][24] and more. Due to their unique biomimetic properties, such as nanoscale size, low power consumption, and non-volatility, continuous memristors are considered the optimal choice for simulating synapses in analog form.…”