Chaotic Spiking Research Articles

Investigating different synaptic connections of the Chay neuron model

The study of synaptic connections offers valuable insights into neuron interactions. It can demonstrate the neurons’ collective behaviors, like synchronization. In this paper, the dynamics of two coupled neurons in three different synaptic connections are studied: electrical, chemical, and electrochemical. The Chay neuron model is explored, and its dynamics are analyzed. The Chay model shows various dynamics by changing the bifurcation parameters. The coupling dynamics for various synaptic connections are investigated with the help of bifurcation diagrams, time series, and state space diagrams. The results of the variation of the maximal conductance gI of the mixed Na+−Ca2+ channels and the maximal conductance gK,V of K+ channels reveal various firing dynamics, including periodic and asynchronous bursting, chaotic bursting, and chaotic spiking. The simulations show complete synchronization for the case of electrical and electrochemical couplings. In other words, the two coupled Chay neurons can show different dynamics, which are synchronous or asynchronous, by varying parameters and coupling coefficients.

Superextreme spiking oscillations and multistability in a memristor-based Hindmarsh–Rose neuron model

In this paper, we investigate the occurrence of superextreme spiking (SES) oscillations and multistability behavior in a memristor-based Hindmarsh–Rose neuron model. The presence of SES oscillations has been identified as arising due to the occurrence of an interior crisis. As the membrane current I(t), considered as the control parameter is varied, the system transits from bounded chaotic spiking (BCS) oscillations to SES oscillations. These transitions are captured numerically using geometrical representations like time series plots, phase portraits and inter-spikes interval return maps. The characterization of SES from the BCS oscillations is made using statistical tools such as phase shift analysis and probability density distribution function. The multistability nature has been observed using bifurcation analysis and confirmed by the Lyapunov exponents for two different sets of initial conditions. The numerical simulations are substantiated through real-time hardware experiments realized through a nonlinear circuit constructed using an analog model of the memristor.

Noise-activated barrier crossing in multiattractor dissipative neural networks

Noise-activated transitions between coexisting attractors are investigated in a chaotic spiking network. At low noise level, attractor hopping consists of discrete bifurcation events that conserve the memory of initial conditions. When the escape probability becomes comparable to the intrabasin hopping probability, the lifetime of attractors is given by a detailed balance where the less coherent attractors act as a sink for the more coherent ones. In this regime, the escape probability follows an activation law allowing us to assign pseudoactivation energies to limit cycle attractors. These pseudoenergies introduce a useful metric for evaluating the resilience of biological rhythms to perturbations.

Resonant neuronal groups

We create a Spiking Neural Network (SNN) architecture based on transforming the dynamics – Unstable Periodic Orbits (UPOs) – of a chaotic spiking neuron model to Neuronal Groups composed from Resonant Neurons. An input fed to the SNN will activate one of its neuronal groups. An activated neuronal group represents ‘memory’ or a neural state of the SNN. By exploiting a fundamental principle in chaos theory, which is Chaotic Sensitivity upon Initial Conditions, in conjunction with chaos control, we show that similar inputs, when fed separately to the SNN, will always activate the same neuronal group and different inputs will activate different neuronal groups. In addition, we show that differences between the system responses (i.e. neuronal groups) are proportional to differences between inputs. These features make the system suitable for input discrimination; we give an example of discerning human physical actions. More importantly, we study the capacity of the SNN. We show that the number of neuronal groups that can be reached is extremely large; it grows exponentially with the increase of the network size (i.e. number of neurons). This is due to neurons mixing, which allows the same resonant neuron to belong to other neuronal groups and due to the theoretically infinite number of UPOs available in a chaotic system that can be stabilized through chaos control. Also, our work competes with Izhikevich's polychronous groups, so we compare our results to his. We discuss the relevance of the work in the nonlinear sciences and its relation to chaotic neuro-dynamics, cognitive science, neural computation, machine learning and memory modeling including future considerations and open problems.

On a Statistical Approach to Phase Synchronization in Some Map-Based Neural Chaotic Spiking–Bursting Models

On a Statistical Approach to Phase Synchronization in Some Map-Based Neural Chaotic Spiking–Bursting Models

Hamiltonian energy and coexistence of hidden firing patterns from bidirectional coupling between two different neurons

In this paper, bidirectional-coupled neurons through an asymmetric electrical synapse are investigated. These coupled neurons involve 2D Hindmarsh–Rose (HR) and 2D FitzHugh–Nagumo (FN) neurons. The equilibria of the coupled neurons model are investigated, and their stabilities have revealed that, for some values of the electrical synaptic weight, the model under consideration can display either self-excited or hidden firing patterns. In addition, the hidden coexistence of chaotic bursting with periodic spiking, chaotic spiking with period spiking, chaotic bursting with a resting pattern, and the coexistence of chaotic spiking with a resting pattern are also found for some sets of electrical synaptic coupling. For all the investigated phenomena, the Hamiltonian energy of the model is computed. It enables the estimation of the amount of energy released during the transition between the various electrical activities. Pspice simulations are carried out based on the analog circuit of the coupled neurons to support our numerical results. Finally, an STM32F407ZE microcontroller development board is exploited for the digital implementation of the proposed coupled neurons model.

Analysis and electronic circuit implementation of an integer-and fractional-order Shimizu-Morioka system

The dynamics of an integer-order and fractional-order Lorenz like system called Shimizu-Morioka system is investigated in this paper. It is shown thatinteger-order Shimizu-Morioka system displays bistable chaotic attractors, monostable chaotic attractors and coexistence between bistable and monostable chaotic attractors. For suitable choose of parameters, the fractional-order Shimizu-Morioka system exhibits bistable chaotic attractors, monostable chaotic attractors, metastable chaos (i.e. transient chaos) and spiking oscillations. The bifurcation structures reveal that the fractional-order derivative affects considerably the dynamics of Shimizu-Morioka system. The chain fractance circuit is used to designand implement the integer- and fractional-order Shimizu-Morioka system in Pspice. A close agreement is observed between PSpice based circuit simulations and numerical simulations analysis. The results obtained in this work were not reported previously in the interger as well as in fractional-order Shimizu-Morioka system and thus represent an important contribution which may help us in better understanding of the dynamical behavior of this class of systems.

Firing activities induced by memristive autapse in Fitzhugh–Nagumo neuron with time delay

Due to the natural non-volatility and distinctive plasticity, memristors are considered as ideal devices to mimic biological synapses. In this study, an autapse which is implemented with a locally active memristor, is introduced into Fitzhugh–Nagumo neuron and thus a new neuron model is established. The local stability of the neuron model with and without time delay is analyzed, respectively. Four coexisting firing patterns, including chaotic spiking, periodic spiking, periodic bursting and chaotic bursting, dependent on the memristor initial values, are explored. We find that the neuron has four regular attraction basins and its firing pattern can be regulated by choosing appropriate initial values. The time delay has an important effect on firing activities and the neuron model transits from periodic spiking, to chaotic bursting, and then to chaotic spiking with the increase of the time delay. Furthermore, the influence of the autaptic intensity on firing activities of the neuron is also revealed. In order to verify the complex firing activities, a neuron circuit is constructed and circuit simulations are performed.

Slow Invariant Manifold of Laser with Feedback

Previous studies have demonstrated, experimentally and theoretically, the existence of slow–fast evolutions, i.e., slow chaotic spiking sequences in the dynamics of a semiconductor laser with AC-coupled optoelectronic feedback. In this work, the so-called Flow Curvature Method was used, which provides the slow invariant manifold analytical equation of such a laser model and also highlights its symmetries if any exist. This equation and its graphical representation in the phase space enable, on the one hand, discriminating the slow evolution of the trajectory curves from the fast one and, on the other hand, improving our understanding of this slow–fast regime.

Nonlinear dynamics of a self-mixing thin-slice solid-state laser subjected to Doppler-shifted optical feedback.

Chaotic oscillations of a linearly polarized single longitudinal-mode thin-slice Nd:GdVO_{4} laser placed in a self-mixing laser Doppler velocity scheme were dynamically characterized in terms of the intensity probability distribution, joint time-frequency analysis, and short-term Fourier transformation of temporal evolutions, and the degree of disorder in the amplitude and phase of the long-term temporal evolutions. The transition from chaotic relaxation oscillations (ROs) to chaotic spiking oscillations (SOs) was explored via the chaotic itinerancy (CI) regime by increasing the feedback ratio toward the laser from a rotating scattering object. The intensity probability distribution was found to change from an exponential decay in the RO regime to an inverse power law in the SO regime, which manifests itself in self-organized critical behavior, while stochastic subharmonic frequency locking among the two periodicities of RO and SO takes place in the CI regime featuring quantum-noise (spontaneous-emission)-induced order in the amplitude and phase of the spiking oscillations. All of the experimental results were reproduced by numerical simulations of a model equation of a single-mode self-mixing solid-state laser subjected to Doppler-shifted optical feedback from a rotating scattering object.

Dynamics of synaptically coupled FitzHugh–Nagumo neurons

An extended FitzHugh–Nagumo model is developed to model the synaptic coupling of neurons and the associated dynamics. The neurons are coupled via chemical synapses, which can excite or inhibit the postsynaptic neuron. The neurotransmitter–receptor dynamics is governed with a gating dynamic, which allows for the distinction between fast and slow synapses. Detailed nonlinear dynamics analysis are performed for simple neural motifs of a self-synapsed neuron and two synaptically coupled neurons for the instabilities that lead to neural oscillations, and the associated phase diagrams are obtained. Interesting dynamics such as synchronization of fast and slow firing neurons, frequency enhancement, quiescent neurons coupled to oscillators, chaotic spiking, and periodic bursting are found.

A STDP Rule that Favours Chaotic Spiking over Regular Spiking of Neurons

We compare the number of states of a Spiking Neural Network (SNN) composed from chaotic spiking neurons versus the number of states of a SNN composed from regular spiking neurons while both SNNs implementing a Spike Timing Dependent Plasticity (STDP) rule that we created. We find out that this STDP rule favors chaotic spiking since the number of states is larger in the chaotic SNN than the regular SNN. This chaotic favorability is not general; it is exclusive to this STDP rule only. This research falls under our long-term investigation of STDP and chaos theory.

Parametrically Excited Chaotic Spike Sequences and Information Aspects in an Ensemble of FitzHugh–Nagumo Neurons

Parametrically Excited Chaotic Spike Sequences and Information Aspects in an Ensemble of FitzHugh–Nagumo Neurons

Asynchronous Chaos and Bifurcations in a Model of Two Coupled Identical Hindmarsh – Rose Neurons

We study a minimal network of two coupled neurons described by the Hindmarsh – Rose model with a linear coupling. We suppose that individual neurons are identical and study whether the dynamical regimes of a single neuron would be stable synchronous regimes in the model of two coupled neurons. We find that among synchronous regimes only regular periodic spiking and quiescence are stable in a certain range of parameters, while no bursting synchronous regimes are stable. Moreover, we show that there are no stable synchronous chaotic regimes in the parameter range considered. On the other hand, we find a wide range of parameters in which a stable asynchronous chaotic regime exists. Furthermore, we identify narrow regions of bistability, when synchronous and asynchronous regimes coexist. However, the asynchronous attractor is monostable in a wide range of parameters. We demonstrate that the onset of the asynchronous chaotic attractor occurs according to the Afraimovich – Shilnikov scenario. We have observed various asynchronous firing patterns: irregular quasi-periodic and chaotic spiking, both regular and chaotic bursting regimes, in which the number of spikes per burst varied greatly depending on the control parameter.

Bifurcations to bursting and spiking in the Chay neuron and their validation in a digital circuit

Abstract Due to the great importance of the outward current carried by K+ions and the inward one by Na+ and Ca2+ions in 3-D Chay neuron model, the dynamical behaviors associated with the maximal conductances of K+ions channel and mixed Na+-Ca2+ions channel are explored for better demonstrating dynamical effects on the electrical activities in this paper. By utilizing some common dynamical exploring methods, the representative electrical activities of bursting and spiking behaviors endowed with the fast-slow structure of the 3-D Chay neuron model are revealed and classified on the dependence of the two maximal conductances. Moreover, the bifurcation mechanisms for periodic fold/fold bursting, periodic and chaotic fold/homoclinic bursting, and chaotic spiking behaviors are expounded qualitatively by fast-slow dynamical analysis with some specified model parameters. Finally, based on a field programmable gate array (FPGA), a digitally circuit-implemented electronic neuron is made, from which the chaotic and periodic bursting/spiking behaviors are experimentally captured to confirm the numerical simulations.

Learning Long Temporal Sequences in Spiking Networks by Multiplexing Neural Oscillations.

Many cognitive and behavioral tasks—such as interval timing, spatial navigation, motor control, and speech—require the execution of precisely-timed sequences of neural activation that cannot be fully explained by a succession of external stimuli. We show how repeatable and reliable patterns of spatiotemporal activity can be generated in chaotic and noisy spiking recurrent neural networks. We propose a general solution for networks to autonomously produce rich patterns of activity by providing a multi-periodic oscillatory signal as input. We show that the model accurately learns a variety of tasks, including speech generation, motor control, and spatial navigation. Further, the model performs temporal rescaling of natural spoken words and exhibits sequential neural activity commonly found in experimental data involving temporal processing. In the context of spatial navigation, the model learns and replays compressed sequences of place cells and captures features of neural activity such as the emergence of ripples and theta phase precession. Together, our findings suggest that combining oscillatory neuronal inputs with different frequencies provides a key mechanism to generate precisely timed sequences of activity in recurrent circuits of the brain.

Elimination of spiral waves in a one-layer and two-layer network of pancreatic beta cells using a periodic stimuli

Abstract Regulating the glucose level in human body is achieved through the Pancreaticβcells and the glucose release is mainly governed by the bursting activity of the βcells. The Pernarowski model is one of the well known Pancreatic model and is analogous to some of the well known neuron models. The Pernarowski model exhibits chaotic spiking and periodic bursting. In this paper we investigate the dynamical properties of the model through various tools like stability of equilibrium, Hopf's bifurcation, Lyapunov exponents and bifurcation diagrams. As was discussed in literatures the neuron models exhibit spiral waves and so we are interested in understanding such spiral wave generation in the Pernarowski model. For this we construct One-layer and Two-layer network of βcells and study the impact of the plane waves on the spiral wave existence. We show that by a selection of the amplitude and frequency of the stimuli, we can eliminate the spiral waves in the both One-layer and Two-layer network of βcells.

Firing multistability in a locally active memristive neuron model

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.

Effective low-dimensional dynamics of a mean-field coupled network of slow-fast spiking lasers.

Low-dimensional dynamics of large networks is the focus of many theoretical works, but controlled laboratory experiments are comparatively very few. Here, we discuss experimental observations on a mean-field coupled network of hundreds of semiconductor lasers, which collectively display effectively low-dimensional mixed mode oscillations and chaotic spiking typical of slow-fast systems. We demonstrate that such a reduced dimensionality originates from the slow-fast nature of the system and of the existence of a critical manifold of the network where most of the dynamics takes place. Experimental measurement of the bifurcation parameter for different network sizes corroborates the theory.

Firing patterns of an improved Izhikevich neuron model under the effect of electromagnetic induction and noise

Abstract The neuron's behavior is related to the changes in its complex electrophysiological environment such as variations in ion concentration. In this paper, a three-variable memristive Izhikevich model is proposed to describe the behavior of neurons under electromagnetic induction and noise. This model can represent the effect of internal and external magnetic fields on neurons. The improved model without the external magnetic field can exhibit firing patterns such as regular spiking, resonator, chattering, fast-spiking, chaotic spiking, and chaotic bursts. The presence of the external magnetic field causes the firing pattern of the neuron to change, for example shifting from chaotic firings to the periodic ones or vice versa. Furthermore, the firing rate of the neuron is increased. It is also observed that the external field effect can stimulate the neuron to fire, while in the absence of the external field, it is at rest. Finally, in order to detect the realistic neuron's activities, we examine the effect of Gaussian white noise on the improved model. The results show that the model has more robustness against noise with considering the external magnetic field.