The Computation of Two-Dimensional Manifold for Hindmarsh-Rose Neuron Model

2010 ◽  
Author(s):  
Meng Jia ◽  
Yangyu Fan ◽  
Weijian Tian ◽  
Jiong Zhao
Keyword(s):  
Neuron Model ◽  
Dimensional Manifold ◽  
Two Dimensional

scholarly journals Random Assignment Problems on 2d Manifolds

2021 ◽  
Vol 183 (2) ◽  
Author(s):  
D. Benedetto ◽  
E. Caglioti ◽  
S. Caracciolo ◽  
M. D’Achille ◽  
G. Sicuro ◽  
...  
Keyword(s):  
Assignment Problem ◽  
Unit Area ◽  
Constant Term ◽  
Beltrami Operator ◽  
Explicit Calculation ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Random Assignment ◽  
Assignment Problems ◽  
Theoretical Formulation

AbstractWe consider the assignment problem between two sets of N random points on a smooth, two-dimensional manifold $$\Omega $$ Ω of unit area. It is known that the average cost scales as $$E_{\Omega }(N)\sim {1}/{2\pi }\ln N$$ E Ω ( N ) ∼ 1 / 2 π ln N with a correction that is at most of order $$\sqrt{\ln N\ln \ln N}$$ ln N ln ln N . In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first $$\Omega $$ Ω -dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace–Beltrami operator on $$\Omega $$ Ω . We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.

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 Two-dimensional Manifold with Point-like Defects

2015 ◽  
Vol 74 ◽  
pp. 32-35 ◽  
Author(s):  
V.A. Gani ◽  
A.E. Dmitriev ◽  
S.G. Rubin
Keyword(s):  
Dimensional Manifold ◽  
Two Dimensional

Energy transfer in rotating turbulence

1997 ◽  
Vol 337 ◽  
pp. 303-332 ◽  
Author(s):  
CLAUDE CAMBON ◽  
N. N. MANSOUR ◽  
F. S. GODEFERD
Keyword(s):  
Energy Transfer ◽  
Rotation Axis ◽  
Physical Space ◽  
Nonlinear Effects ◽  
Spectral Energy ◽  
Rossby Number ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
Plane Normal

The influence of rotation on the spectral energy transfer of homogeneous turbulence is investigated in this paper. Given the fact that linear dynamics, e.g. the inertial waves regime found in an RDT (rapid distortion theory) analysis, cannot affect a homogeneous isotropic turbulent flow, the study of nonlinear dynamics is of prime importance in the case of rotating flows. Previous theoretical (including both weakly nonlinear and EDQNM theories), experimental and DNS (direct numerical simulation) results are collected here and compared in order to give a self-consistent picture of the nonlinear effects of rotation on turbulence.The inhibition of the energy cascade, which is linked to a reduction of the dissipation rate, is shown to be related to a damping of the energy transfer due to rotation. A model for this effect is quantified by a model equation for the derivative-skewness factor, which only involves a micro-Rossby number Roω=ω′/(2Ω) – ratio of r.m.s. vorticity and background vorticity – as the relevant rotation parameter, in accordance with DNS and EDQNM results.In addition, anisotropy is shown also to develop through nonlinear interactions modified by rotation, in an intermediate range of Rossby numbers (RoL<1 and Roω>1), which is characterized by a macro-Rossby number RoL based on an integral lengthscale L and the micro-Rossby number previously defined. This anisotropy is mainly an angular drain of spectral energy which tends to concentrate energy in the wave-plane normal to the rotation axis, which is exactly both the slow and the two-dimensional manifold. In addition, a polarization of the energy distribution in this slow two-dimensional manifold enhances horizontal (normal to the rotation axis) velocity components, and underlies the anisotropic structure of the integral length-scales. Finally a generalized EDQNM (eddy damped quasi-normal Markovian) model is used to predict the underlying spectral transfer structure and all the subsequent developments of classic anisotropy indicators in physical space. The results from the model are compared to recent LES results and are shown to agree well. While the EDQNM2 model was developed to simulate ‘strong’ turbulence, it is shown that it has a strong formal analogy with recent weakly nonlinear approaches to wave turbulence.

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 Growing two-dimensional manifold of nonlinear maps based on generalized Foliation condition

2012 ◽  
Vol 61 (2) ◽  
pp. 029501
Author(s):  
Li Hui-Min ◽  
Fan Yang-Yu ◽  
Sun Heng-Yi ◽  
Zhang Jing ◽  
Jia Meng
Keyword(s):  
Dimensional Manifold ◽  
Two Dimensional ◽  
Nonlinear Maps

scholarly journals Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere

2019 ◽  
Vol 11 (4) ◽  
pp. 72-79
Author(s):  
Anna Kravchenko ◽  
Sergiy Maksymenko
Keyword(s):  
Homotopy Type ◽  
Morse Function ◽  
Previous Article ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Reeb Graph ◽  
Natural Right ◽  
Morse Functions ◽  
Path Component ◽  
Unique Vertex

Let $M$ be a compact two-dimensional manifold and, $f \in C^{\infty}(M, R)$ be a Morse function, and $\Gamma$ be its Kronrod-Reeb graph.Denote by $O(f)={f o h | h \in D(M)}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $D(M)$ onC^{\infty}$, and by $S(f)={h\in D(M) | f o h = f }$ the coresponding stabilizer of this function.It is easy to show that each $h\in S(f)$ induces an automorphism of the graph $\Gamma$.Let $D_{id}(M)$ be the identity path component of $D(M)$, $S'(f) = S(f) \cap D_{id}(M)$ be the subgroup of $D_{id}(M)$ consisting of diffeomorphisms preserving $f$ and isotopic to identity map, and $G$ be the group of automorphisms of the Kronrod-Reeb graph induced by diffeomorphisms belonging to $S'(f)$. This group is one of key ingredients for calculating the homotopy type of the orbit $O(f)$. In the previous article the authors described the structure of groups $G$ for Morse functions on all orientable surfacesdistinct from $2$-torus and $2$-sphere.  The present paper is devoted to the case $M = S^2$. In this situation $\Gamma$ is always a tree, and therefore all elements of the group $G$ have a common fixed subtree $Fix(G)$, which may even consist of a unique vertex. Our main result calculates the groups $G$ for all Morse functions $f: S^2 \to R$ whose fixed subtree $Fix(G)$ consists of more than one point.

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