NEURONS ARE POISED NEAR THE EDGE OF CHAOS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250098 ◽  
Author(s):  
LEON CHUA ◽  
VALERY SBITNEV ◽  
HYONGSUK KIM
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Point ◽  
Nonlinear Differential Equations ◽  
Complex Conjugate ◽  
Excitation Current ◽  
Hopf Bifurcation Point ◽  
Synaptic Excitation ◽  
Edge Of Chaos ◽  
Subcritical Hopf Bifurcation ◽  
Dc Current

This paper shows the action potential (spikes) generated from the Hodgkin–Huxley equations emerges near the edge of chaos consisting of a tiny subset of the locally active regime of the HH equations. The main result proves that the eigenvalues of the 4 × 4 Jacobian matrix associated with the mathematically intractable system of four nonlinear differential equations are identical to the zeros of a scalar complexity function from complexity theory. Moreover, we show the loci of a pair of complex-conjugate zeros migrate continuously as a function of an externally applied DC current excitation emulating the net synaptic excitation current input to the neuron. In particular, the pair of complex-conjugate zeros move from a subcritical Hopf bifurcation point at low excitation current to a super-critical Hopf bifurcation point at high excitation current. The spikes are generated as the excitation current approaches the vicinity of the edge of chaos, which leads to the onset of the subcritical Hopf bifurcation regime. It follows from this in-depth qualitative analysis that local activity is the origin of spikes.

Spatiotemporal Dynamics in Reaction–Diffusion Neural Networks Near a Turing–Hopf Bifurcation Point

2019 ◽  
Vol 29 (11) ◽  
pp. 1950154 ◽  
Author(s):  
Jiazhe Lin ◽  
Rui Xu ◽  
Xiaohong Tian
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Bifurcation Point ◽  
Reaction Diffusion ◽  
Spatiotemporal Dynamics ◽  
Normal Form Theory ◽  
Hopf Bifurcation Point ◽  
Codimension Two ◽  
Form Theory

Since the electromagnetic field of neural networks is heterogeneous, the diffusion phenomenon of electrons exists inevitably. In this paper, we investigate the existence of Turing–Hopf bifurcation in a reaction–diffusion neural network. By the normal form theory for partial differential equations, we calculate the normal form on the center manifold associated with codimension-two Turing–Hopf bifurcation, which helps us understand and classify the spatiotemporal dynamics close to the Turing–Hopf bifurcation point. Numerical simulations show that the spatiotemporal dynamics in the neighborhood of the bifurcation point can be divided into six cases and spatially inhomogeneous periodic solution appears in one of them.

Dynamics near a symmetric Hopf bifurcation

1988 ◽  
Vol 108 (3-4) ◽  
pp. 249-267 ◽  
Author(s):  
Wayne Nagata
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Symmetry Breaking ◽  
Bifurcation Point ◽  
Unperturbed System ◽  
System Of Differential Equations ◽  
Hopf Bifurcation Point ◽  
Secondary Bifurcations

SynopsisWe consider the effects of a small symmetry breaking perturbation on a system of differential equations near a Hopf bifurcation point, where the unperturbed system has O(2) symmetry. We show that there exist secondary bifurcations of invariant two-tori of solutions and that the flow on the tori can be quasiperiodic or weakly resonant (phase locked), depending on the size of the perturbation.

COHERENCE RESONANCE–INDUCED STOCHASTIC NEURAL FIRING AT A SADDLE-NODE BIFURCATION

2011 ◽  
Vol 25 (29) ◽  
pp. 3977-3986 ◽  
Author(s):  
HUAGUANG GU ◽  
HUIMIN ZHANG ◽  
CHUNLING WEI ◽  
MINGHAO YANG ◽  
ZHIQIANG LIU ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Point ◽  
Firing Pattern ◽  
Neuronal System ◽  
Neural Firing ◽  
Coherence Resonance ◽  
Saddle Node Bifurcation ◽  
Hopf Bifurcation Point ◽  
Firing Patterns ◽  
Saddle Node

Coherence resonance at a saddle-node bifurcation point and the corresponding stochastic firing patterns are simulated in a theoretical neuronal model. The characteristics of noise-induced neural firing pattern, such as exponential decay in histogram of interspike interval (ISI) series, independence and stochasticity within ISI series are identified. Firing pattern similar to the simulated results was discovered in biological experiment on a neural pacemaker. The difference between this firing and integer multiple firing generated at a Hopf bifurcation point is also given. The results not only revealed the stochastic dynamics near a saddle-node bifurcation, but also gave practical approaches to identify the saddle-node bifurcation and to distinguish it from the Hopf bifurcation in neuronal system. In addition, many previously observed firing patterns can be attribute to stochastic firing pattern near such a saddle-node bifurcation.

ESTIMATION OF THE HOPF BIFURCATION POINT FOR AEROELASTIC SYSTEMS

2001 ◽  
Vol 248 (1) ◽  
pp. 31-42 ◽  
Author(s):  
A. SEDAGHAT ◽  
J.E. COOPER ◽  
A.Y.T. LEUNG ◽  
J.R. WRIGHT
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Point ◽  
Hopf Bifurcation Point

Bifurcation Analysis of a Railway Wheelset With Nonlinear Wheel-rail Contact

2021 ◽  
Author(s):  
Jinying Guo ◽  
Huailong Shi ◽  
Ren Luo ◽  
Jing Zeng
Keyword(s):  
Contact Angle ◽  
Hopf Bifurcation ◽  
Bifurcation Point ◽  
Critical Speed ◽  
Nonlinear Term ◽  
Linear Term ◽  
Railway Vehicle ◽  
Hopf Bifurcation Point ◽  
Quintic Polynomial ◽  
The Stability

Abstract Stability is a key factor for the operation safety of railway vehicles, while current work employs linearized and simplified wheel/rail contact to study the bifurcation mechanism and assess the stability. To study the stability and bifurcation characters under real nonlinear wheel/rail contact, a fully parameterized nonlinear railway vehicle wheelset model is built. In modeling, the geometry nonlinearities of wheel and rail profiles come from field measurements, including the rolling radius, contact angle, and curvatures, etc. Firstly, four flange force models and their effects on the stability bifurcations are compared. It shows that an exponent fitting is more proper than a quintic polynomial one to simulate the flange, and works well without changing the Hopf bifurcation type. Then the effects of each term of the nonlinear geometry of wheel/rail contact on the Hopf bifurcation and Limit Circle bifurcation are discussed. Both the linear term and nonlinear term of rolling radius have a significant influence on Hopf bifurcation and Limit Point of Circle (LPC) bifurcation. The linear critical speed (Hopf bifurcation point) and the nonlinear critical speed (LPC bifurcation point) changes times while within the calculated range of the linear term of the rolling radius. Its nonlinear term changes the bifurcation type and the nonlinear critical speed almost by half. The linear term of contact angle, the radius of curvature of wheel, and rail profile should be taken into consideration since they can change both the bifurcation point and type, while the cubic term can be ignored. Furtherly, the field measured wheel profiles for several running mileages are employed to examine the real geometry nonlinearities and the according Hopf bifurcation behavior. The result shows that a larger suspension stiffness would increase the running stability under wheel wear.

Hopf Bifurcation of a Magnetic Bearing System with Time Delay

2004 ◽  
Vol 127 (4) ◽  
pp. 362-369 ◽  
Author(s):  
J. C. Ji ◽  
Colin H. Hansen
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Magnetic Bearing ◽  
Complex Conjugate ◽  
Infinite Dimensional ◽  
System With Time Delay ◽  
Two Degree Of Freedom

This paper is concerned with a study of the influence of a time delay occurring in a PD feedback control on the dynamic stability of a rotor suspended by magnetic bearings. In the presence of geometric coordinate coupling and time delay, the equations of motion governing the response of the rotor are a set of two-degree-of-freedom nonlinear differential equations with time delay coupling in nonlinear terms. It is found that as the time delay increases beyond a critical value, the equilibrium position of the rotor motion becomes unstable and may bifurcate into two qualitatively different kinds of periodic motion. The resultant Hopf bifurcation is associated with two coincident pairs of complex conjugate eigenvalues crossing the imaginary axis. Based on the reduction of the infinite dimensional problem to the flow on a four-dimensional center manifold, the bifurcating periodic solutions are investigated using a perturbation method.

FOOD WEB CHAOS WITHOUT SUBCHAIN OSCILLATORS

2005 ◽  
Vol 15 (11) ◽  
pp. 3481-3492 ◽  
Author(s):  
BRIAN BOCKELMAN ◽  
BO DENG
Keyword(s):  
Hopf Bifurcation ◽  
Singular Perturbation ◽  
Food Web ◽  
Bifurcation Point ◽  
Period Doubling ◽  
Reproductive Rate ◽  
Hopf Bifurcation Point ◽  
Parameter Values ◽  
The Web

A basic food web of four species is considered, of which there is a bottom prey X, two competing predators Y, Z on X, and a super predator W only on Y. The main finding is that population chaos does not require the existence of oscillators in any subsystem of the web. This minimum population chaos is demonstrated by increasing the relative reproductive rate of Z alone without alternating any other parameter nor any nullcline of the system. It occurs as the result of a period-doubling cascade from a Hopf bifurcation point. The method of singular perturbation is used to determine the Hopf bifurcation involved as well as the parameter values.

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