Stochastic oscillations in a neuron model with two-dimensional map

2019 ◽  
Author(s):  
V. M. Nasyrova ◽  
L. B. Ryashko ◽  
I. N. Tsvetkov
Keyword(s):  
Neuron Model ◽  
Two Dimensional ◽  
Stochastic Oscillations

COMPLEX BIFURCATION STRUCTURES IN THE HINDMARSH–ROSE NEURON MODEL

2007 ◽  
Vol 17 (09) ◽  
pp. 3071-3083 ◽  
Author(s):  
J. M. GONZÀLEZ-MIRANDA
Keyword(s):  
Phase Diagram ◽  
Bifurcation Diagram ◽  
Parameter Space ◽  
Neuron Model ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Rapid Responses ◽  
Bifurcation Structures

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

Chaos in the Rulkov Neuron Model Based on Marotto’s Theorem

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Penghe Ge ◽  
Hongjun Cao
Keyword(s):  
Neuron Model ◽  
Initial Conditions ◽  
Stability Conditions ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Fast Subsystem ◽  
One Dimensional ◽  
Sensitive Dependence ◽  
The Stability ◽  
Slow Subsystem

The existence of chaos in the Rulkov neuron model is proved based on Marotto’s theorem. Firstly, the stability conditions of the model are briefly renewed through analyzing the eigenvalues of the model, which are very important preconditions for the existence of a snap-back repeller. Secondly, the Rulkov neuron model is decomposed to a one-dimensional fast subsystem and a one-dimensional slow subsystem by the fast–slow dynamics technique, in which the fast subsystem has sensitive dependence on the initial conditions and its snap-back repeller and chaos can be verified by numerical methods, such as waveforms, Lyapunov exponents, and bifurcation diagrams. Thirdly, for the two-dimensional Rulkov neuron model, it is proved that there exists a snap-back repeller under two iterations by illustrating the existence of an intersection of three surfaces, which pave a new way to identify the existence of a snap-back repeller.

Bipolar Pulse-Induced Coexisting Firing Patterns in Two-Dimensional Hindmarsh–Rose Neuron Model

2019 ◽  
Vol 29 (01) ◽  
pp. 1950006 ◽  
Author(s):  
Han Bao ◽  
Aihuang Hu ◽  
Wenbo Liu
Keyword(s):  
Membrane Potential ◽  
Neuron Model ◽  
Periodic Variation ◽  
Constant Current ◽  
Two Dimensional ◽  
Bipolar Pulse ◽  
Firing Patterns ◽  
Current Stimulus ◽  
Unstable Node ◽  
Stimulus Effect

In this paper, a bipolar pulse (BP) current is taken to mimic a periodic stimulus effect on the membrane potential in the axon of a neuron. By introducing the BP current to substitute the externally applied constant current, a BP-forced two-dimensional Hindmarsh–Rose (HR) neuron model is proposed. Based on the proposed neuron model, the BP-switched equilibrium point and its stability evolution with the periodic variation in time are explored. Furthermore, coexisting asymmetric attractors (or coexisting firing patterns) with bistability are revealed by phase plane orbits, time sequences, and attraction basins, as well as the BP-induced coexisting asymmetric attractors’ behaviors are then elaborated through bifurcation analysis. The research results exhibit that, with the increase of the time, the stabilities of the neuron model are continually switched between an unstable node-focus and a stable point, resulting in the coexisting behaviors of numerous asymmetric attractors under the specified initials. Consequently, the newly introduced BP current stimulus, instead of the original constant current stimulus, allows the two-dimensional HR neuron model to possess complex dynamical behaviors for the membrane potential. Additionally, a hardware breadboard is fabricated and circuit experiments are carried out to validate the numerical simulations.

scholarly journals Bifurcation Analysis of a Two-Dimensional Neuron Model under Electrical Stimulation

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Chunhua Yuan ◽  
Xiangyu Li
Keyword(s):  
Electrical Stimulation ◽  
Hopf Bifurcation ◽  
Dynamic Characteristics ◽  
Neuron Model ◽  
Equilibrium Points ◽  
Model Parameters ◽  
Two Dimensional ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Firing Characteristics

The two-dimensional neuron model can not only reproduce abundant firing patterns, but also satisfy the research of dynamical behavior because of its nonlinear characteristics. It is the most simplified model that includes the fast and slow variables required for neuron firing. In this paper, the dynamic characteristics of two-dimensional neuron model are described by both analytical and numerical methods, and the influence of model parameters and external stimuli on dynamic characteristics is described. The firing characteristics of the Prescott model under external electrical stimulation are studied, and the influence of electrophysiological parameters on the firing characteristics is analyzed. The saddle-node bifurcation and Hopf bifurcation characteristics are studied through the distribution of equilibrium points. It is found that there are critical saddle-node bifurcation and critical Hopf bifurcation in the Prescott model. And the value range of the key parameters that cause the critical bifurcation of the model is obtained by analytical methods.

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

2021 ◽  
Author(s):  
Jianming Cai ◽  
Han Bao ◽  
Quan Xu ◽  
Zhongyun Hua ◽  
Bocheng Bao
Keyword(s):  
Large Scale ◽  
Neuron Model ◽  
Hardware Implementation ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Implementation Cost ◽  
Hyperbolic Tangent ◽  
Nonlinear Fitting ◽  
Chaotic Bursting ◽  
Electrical Activities

Abstract 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.

Critical Analysis of Two Dimensional and Four-Dimensional Spiking Neuron Models

2019 ◽  
Vol 16 (9) ◽  
pp. 3897-3905
Author(s):  
Pankaj Kumar Kandpal ◽  
Ashish Mehta
Keyword(s):  
Error Correction ◽  
Neuron Model ◽  
Learning Algorithm ◽  
Two Dimensional ◽  
Spiking Neuron ◽  
Classification Rate ◽  
Integrate And Fire ◽  
Study Results ◽  
Spiking Neuron Model ◽  
Integrate And Fire Neuron

In the present article, two-dimensional “Spiking Neuron Model” is being compared with the fourdimensional “Integrate-and-fire Neuron Model” (IFN) using error correction back propagation learning algorithm (error correction learning). A comparative study has been done on the basis of several parameters like iteration, execution time, miss-classification rate, number of iterations etc. The authors choose the five-bit parity problem and Iris classification problem for the present study. Results of simulation express that both the models are capable to perform classification task. But single spiking neuron model having two-dimensional phenomena is less complex than Integrate-fire-neuron, produces better results. On the contrary, the classification performance of single ingrate-and-fire neuron model is not very poor but due to complex four-dimensional architecture, miss-classification rate is higher than single spiking neuron model, it means Integrate-and-fire neuron model is less capable than spiking neuron model to solve classification problems.

Two-Dimensional Piecewise-Linear Neuron Model

2020 ◽  
pp. 1-1
Author(s):  
Han Bao ◽  
Chengjie Chen ◽  
Yihua Hu ◽  
Mo Chen ◽  
Bocheng Bao
Keyword(s):  
Neuron Model ◽  
Piecewise Linear ◽  
Two Dimensional
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