scholarly journals Multiple and Complex Codimension-2 Bifurcations underlying Post-inhibitory Rebound Spike and Excitability Transition

2021 ◽  
Author(s):  
Xianjun Wang ◽  
Huaguang Gu ◽  
Yuye Li ◽  
Bo Lu
Keyword(s):  
Hopf Bifurcation ◽  
Homoclinic Orbit ◽  
Neuronal Model ◽  
Type I ◽  
Type Ii ◽  
Neural Firing ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Snic Bifurcation ◽  
Degenerate Bifurcation

Abstract Neuron exhibits nonlinear dynamics such as excitability transition and post-inhibitory rebound (PIR) spike related to bifurcations, which are associated with information processing, locomotor modulation, or brain disease. PIR spike is evoked by inhibitory stimulation instead of excitatory stimulation, which presents a challenge to the threshold concept. In the present paper, 7 codimension-2 or degenerate bifurcations related to 10 codimension-1 bifurcations are acquired in a neuronal model, which presents the bifurcations underlying the excitability transition and PIR spike. Type I excitability corresponds to saddle-node bifurcation on an invariant cycle (SNIC) bifurcation, and type II excitability to saddle-node (SN) bifurcation or sub-critical Hopf (SubH) bifurcation or sup-critical Hopf (SupH) bifurcation. The excitability transition from type I to II corresponds to the codimension-2 bifurcation, Saddle-Node Homoclinic orbit (SNHO) bifurcation, via which SNIC bifurcation terminates and meanwhile big homoclinic orbit (BHom) bifurcation and SN bifurcation emerge. A degenerate bifurcation via which BHom bifurcation terminates and fold limit cycle (LPC) bifurcation emerges is responsible for spiking transition from type I to II, and the roles of other codimension-2 bifurcations (Cusp, Bogdanov-Takens, and Bautin) are discussed. In addition, different from the widely accepted viewpoint that PIR spike is mainly evoked near Hopf bifurcation rather than SNIC bifurcation, PIR spike is identified to be induced near SNIC or BHom or LPC bifurcations, and threshold curves resemble that of Hopf bifurcation. The complex bifurcations present comprehensive and deep understandings of excitability transition and PIR spike, which are helpful for the modulation to neural firing activities and physiological functions.

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.

Time-Averaged Quantity of a Low-Temperature Semiconductor Experiment Reflects Scaling Behavior of Saddle-Node Bifurcation to Chaos

1994 ◽  
Vol 49 (9) ◽  
pp. 838-842 ◽  
Author(s):  
R. Richter ◽  
A. Kittel ◽  
J. Paris
Keyword(s):  
Magnetic Field ◽  
Low Temperature ◽  
Impact Ionization ◽  
Scaling Behavior ◽  
Type I ◽  
Saddle Node Bifurcation ◽  
Time Resolved ◽  
Saddle Node ◽  
Temperature Impact ◽  
P Type

Abstract Low-temperature impact ionization breakdown in p-type germanium crystals gives rise to spontaneous oscillations of the current flow. We demonstrate experimental evidence of a particularly high-conducting dynamical state that is limited to a finite parameter regime of the current versus magnetic field characteristic. After bifurcation from a coexisting nonoscillatory state to periodicity, one observes a type-I intermittent transition to chaos and, eventually, a jump back to the nonoscil­latory branch upon increasing the magnetic field control parameter. The scaling behavior of the underlying saddle-node bifurcation, already found in time-resolved measurements, also becomes visible in a square-root dependence of the time-averaged current developing both prior to and after the critical point. Our result might be of interest for time-averaged information is accessible.

scholarly journals Complex Behaviors of Epidemic Model with Nonlinear Rewiring Rate

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Ding Fang ◽  
Yongxin Zhang ◽  
Wendi Wang
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Epidemic Model ◽  
Multiple Equilibria ◽  
Homoclinic Bifurcation ◽  
Propagation Model ◽  
Bifurcation Curve ◽  
Saddle Node Bifurcation ◽  
Saddle Node

An SIS propagation model with the nonlinear rewiring rate on an adaptive network is considered. It is found by bifurcation analysis that the model has the complex behaviors which include the transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation. Especially, a bifurcation curve with “S” shape emerges due to the nonlinear rewiring rate, which leads to multiple equilibria and twice saddle-node bifurcations. Numerical simulations show that the model admits a homoclinic bifurcation and a saddle-node bifurcation of the limit cycle.

Bifurcation analysis method of nonlinear traffic phenomena

2015 ◽  
Vol 26 (10) ◽  
pp. 1550111 ◽  
Author(s):  
Wenhuan Ai ◽  
Zhongke Shi ◽  
Dawei Liu
Keyword(s):  
Hopf Bifurcation ◽  
Traffic Flow ◽  
Bifurcation Analysis ◽  
Phase Plane ◽  
Flow Model ◽  
Analysis Method ◽  
Equilibrium Solutions ◽  
Saddle Node Bifurcation ◽  
Traffic Flow Model ◽  
Saddle Node

A new bifurcation analysis method for analyzing and predicting the complex nonlinear traffic phenomena based on the macroscopic traffic flow model is presented in this paper. This method makes use of variable substitution to transform a traditional traffic flow model into a new model which is suitable for the stability analysis. Although the substitution seems to be simple, it can extend the range of the variable to infinity and build a relationship between the traffic congestion and the unstable system in the phase plane. So the problem of traffic flow could be converted into that of system stability. The analysis identifies the types and stabilities of the equilibrium solutions of the new model and gives the overall distribution structure of the nearby equilibrium solutions in the phase plane. Then we deduce the existence conditions of the models Hopf bifurcation and saddle-node bifurcation and find some bifurcations such as Hopf bifurcation, saddle-node bifurcation, Limit Point bifurcation of cycles and Bogdanov–Takens bifurcation. Furthermore, the Hopf bifurcation and saddle-node bifurcation are selected as the starting point of density temporal evolution and it will be helpful for improving our understanding of stop-and-go wave and local cluster effects observed in the free-way traffic.

Bifurcation Analysis of the Wound Rotor Induction Motor

2015 ◽  
Vol 25 (12) ◽  
pp. 1550163 ◽  
Author(s):  
R. Salas-Cabrera ◽  
A. Hernandez-Colin ◽  
J. Roman-Flores ◽  
N. Salas-Cabrera
Keyword(s):  
Hopf Bifurcation ◽  
Induction Motor ◽  
Bifurcation Analysis ◽  
Open Loop ◽  
Experimental Results ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Bifurcation Phenomena ◽  
Three Phase ◽  
The Individual

This work deals with the bifurcation phenomena that occur during the open-loop operation of a single-fed three-phase wound rotor induction motor. This paper demonstrates the occurrence of saddle-node bifurcation, hysteresis, supercritical saddle-node bifurcation, cusp and Hopf bifurcation during the individual operation of this electromechanical system. Some experimental results associated with the bifurcation phenomena are presented.

APPLICATION OF A TWO-DIMENSIONAL HINDMARSH–ROSE TYPE MODEL FOR BIFURCATION ANALYSIS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350055 ◽  
Author(s):  
SHYAN-SHIOU CHEN ◽  
CHANG-YUAN CHENG ◽  
YI-RU LIN
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Sufficient Conditions ◽  
Type Model ◽  
Two Dimensional ◽  
Saddle Node Bifurcation ◽  
Trapping Region ◽  
Saddle Node ◽  
The Stability ◽  
Two Equilibria

In this study, we examine the bifurcation scenarios of a two-dimensional Hindmarsh–Rose type model [Tsuji et al., 2007] with four parameters and simulate some resemblances of neurophysiological features for this model using spike-and-reset conditions. We present possible classifications based on the results of the following assessments: (1) the number and stability of the equilibria are analyzed in detail using a table to demonstrate the matter in which the stability of the equilibrium changes and to determine which two equilibria collapse through the saddle-node bifurcation; (2) the sufficient conditions for an Andronov–Hopf bifurcation and a saddle-node bifurcation are mathematically confirmed; and (3) we elaborately evaluate the sufficient conditions for the Bogdanov–Takens (BT) and Bautin bifurcations. Several numerical simulations for these conditions are also presented. In particular, two types of bistable behaviors are numerically demonstrated: the BT and Bautin bifurcations. Notably, all of the bifurcation curves in the domain of the remaining parameters are similar when the time scale is large. Additionally, to show the potential for a limit cycle, the existence of a trapping region is demonstrated. These results present a variety of diverse behaviors for this model. The results of this study will be helpful in assessing suitable parameters for fitting the resemblances of experimental observations.

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 Bifurcations of a Ratio-Dependent Holling-Tanner System with Refuge and Constant Harvesting

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Xia Liu ◽  
Yepeng Xing
Keyword(s):  
Hopf Bifurcation ◽  
Backward Bifurcation ◽  
Heteroclinic Bifurcation ◽  
Saddle Node Bifurcation ◽  
Predator Prey ◽  
Prey System ◽  
Saddle Node ◽  
Bifurcation Phenomena ◽  
Ratio Dependent

The bifurcation properties of a predator prey system with refuge and constant harvesting are investigated. The number of the equilibria and the properties of the system will change due to refuge and harvesting, which leads to the occurrence of several kinds bifurcation phenomena, for example, the saddle-node bifurcation, Bogdanov-Takens bifurcation, Hopf bifurcation, backward bifurcation, separatrix connecting a saddle-node and a saddle bifurcation and heteroclinic bifurcation, and so forth. Our main results reveal much richer dynamics of the system compared to the system with no refuge and harvesting.

scholarly journals The Twisting Bifurcations of Double Homoclinic Loops with Resonant Eigenvalues

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Xiaodong Li ◽  
Weipeng Zhang ◽  
Fengjie Geng ◽  
Jicai Huang
Keyword(s):  
Periodic Orbit ◽  
Homoclinic Orbit ◽  
Dimensional System ◽  
Bifurcation Diagrams ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Higher Dimensional

The twisting bifurcations of double homoclinic loops with resonant eigenvalues are investigated in four-dimensional systems. The coexistence or noncoexistence of large 1-homoclinic orbit and large 1-periodic orbit near double homoclinic loops is given. The existence or nonexistence of saddle-node bifurcation surfaces is obtained. Finally, the complete bifurcation diagrams and bifurcation curves are also given under different cases. Moreover, the methods adopted in this paper can be extended to a higher dimensional system.

RESONANCE STRUCTURE IN A WEAKLY DETUNED LASER WITH INJECTED SIGNAL

2001 ◽  
Vol 11 (10) ◽  
pp. 2587-2605 ◽  
Author(s):  
CATALINA MAYOL ◽  
MARIO A. NATIELLO ◽  
MARTÍN G. ZIMMERMANN
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbit ◽  
Global Bifurcation ◽  
Homoclinic Orbits ◽  
Resonance Structure ◽  
Saddle Node Bifurcation ◽  
Arnold Tongue ◽  
Saddle Node ◽  
Organizing Center ◽  
Qualitative Dynamics

We describe the qualitative dynamics and bifurcation set for a laser with injected signal for small cavity detunings. The main organizing center is the Hopf-saddle-node bifurcation from where a secondary Hopf bifurcation of a periodic orbit originates. We show that the laser's stable cw solution existing for low injections, also suffers a secondary Hopf bifurcation. The resonance structure of both tori interact, and homoclinic orbits to the "off" state are found inside each Arnold tongue. The accumulation of all the above resonances towards the Hopf-saddle-node singularity points to the occurrence of a highly degenerate global bifurcation at the codimension-2 point.

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