A New Memristive Neuron Map Model and Its Network’s Dynamics under Electrochemical Coupling

A memristor is a vital circuit element that can mimic biological synapses. This paper proposes the memristive version of a recently proposed map neuron model based on the phase space. The dynamic of the memristive map model is investigated by using bifurcation and Lyapunov exponents’ diagrams. The results prove that the memristive map can present different behaviors such as spiking, periodic bursting, and chaotic bursting. Then, a ring network is constructed by hybrid electrical and chemical synapses, and the memristive neuron models are used to describe the nodes. The collective behavior of the network is studied. It is observed that chemical coupling plays a crucial role in synchronization. Different kinds of synchronization, such as imperfect synchronization, complete synchronization, solitary state, two-cluster synchronization, chimera, and nonstationary chimera, are identified by varying the coupling strengths.

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.

Multi-Bifurcation Cascaded Bursting Oscillations and Mechanism in a Novel 3D Non-autonomous Circuit System with Parametric and External Excitation

In this paper, multi-timescale dynamics and the formation mechanism of a 3D non-autonomous system with two slowly varying periodic excitations are systematically investigated. Interestingly, the system shows novel multibifurcation cascaded bursting oscillations (MBCBOs) when the frequency of the two excitations is much lower than the mean frequency of the original system (MFOS). For instance, periodic, quasi-periodic and chaotic bursting oscillations induced by a variety of cascaded bifurcations are first observed, and the phenomenon of spiking transfer is also revealed. Besides, stability and local bifurcations of the system are comprehensively investigated to analyze the mechanism of the observed MBCBOs, in which bifurcation diagram, Lyapunov exponents, time series, phase portraits, and transformed phase diagrams are used. Finally, through a circuit simulation and hardware experiment, these complex dynamics phenomena are verified physically.

Smooth Nonlinear Fitting Scheme for Analog Multiplierless Implementation of Hindmarsh-Rose Neuron Model

The Hindmarsh-Rose (HR) neuron model is built to describe the neuron electrical activities. Due to the polynomial nonlinearities, multipliers are required to implement the HR neuron model in analog. In order to avoid the multipliers, this brief presents a novel smooth nonlinear fitting scheme. We first construct two nonlinear fitting functions using the composite hyperbolic tangent functions and then implement an analog multiplierless circuit for the two-dimensional (2D) or three- dimensional (3D) HR neuron model. To exhibit the nonlinear fitting effects, numerical simulations and hardware experiments for the fitted HR neuron model are provided successively. The results show that the fitted HR neuron model with analog multiplierless circuit can display different operation patterns of resting, periodic spiking, and periodic/chaotic bursting, entirely behaving like the original HR neuron model. The analog multiplierless circuit has the advantage of low implementation cost and thereby it might be suitable for the hardware implementation of large-scale neural networks.

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.

Memristor Synapse-Based Morris–Lecar Model: Bifurcation Analyses and FPGA-Based Validations for Periodic and Chaotic Bursting/Spiking Firings

Electromagnetic induction current sensed by the membrane potential in biological neurons can be characterized with a memristor synapse, which can be employed to demonstrate the real oscillating voltage patterns of Barnacle muscle fibers. This paper presents a 3D autonomous memristor synapse-based Morris–Lecar (abbreviated as m-ML) model, which is implemented through introducing a memristor synapse-based induction current to substitute the externally applied current in an existing 2D nonautonomous Morris–Lecar model. Making use of one- and two-parameter bifurcation plots and time-domain representations, diverse period-adding bifurcations as well as abundant periodic and chaotic burst firings are demonstrated. Through constructing the fold and Hopf bifurcation sets of fast spiking subsystem, bifurcation analyses of these chaotic and periodic burst firings are carried out. Moreover, the periodic and chaotic spiking firings and coexisting firing behaviors are illustrated by using one- and two-parameter bifurcation plots and local attraction basins. Finally, based on a field programmable gate array (FPGA) board, a compact digital electronic neuron is fabricated for the 3D m-ML model, from which periodic and chaotic bursting/spiking firings are experimentally measured to verify the results of the numerical simulations.

Chaotic Bursting Dynamics and Coexisting Multistable Firing Patterns in 3D Autonomous Morris–Lecar Model and Microcontroller-Based Validations

A three-dimensional (3D) autonomous Morris–Lecar (simplified as M–L) neuron model with fast and slow structures was proposed to generate periodic bursting behaviors. However, chaotic bursting dynamics and coexisting multistable firing patterns have been rarely discussed in such a 3D M–L neuron model. For some specified model parameters, MATLAB numerical plots are executed by bifurcation plots, time sequences, phase plane plots, and 0–1 tests, from which diverse forms of chaotic bursting, chaotic tonic-spiking, and periodic bursting behaviors are uncovered in the 3D M–L neuron model. Furthermore, based on the theoretically constructing fold/Hopf bifurcation sets of the fast subsystem, the bifurcation mechanism for the chaotic bursting behaviors is thereby expounded qualitatively. Particularly, through numerically plotting the attraction basins related to the initial states under two sets of specific parameters, coexisting multistable firing patterns are demonstrated in the 3D M–L neuron model also. Finally, a digitally circuit-implemented electronic neuron is generated based on a low-power microcontroller and its experimentally captured results faultlessly validate the numerical plots.