scholarly journals Fast-slow Variable Dissection with Two Slow Variables Related to Calcium Concentrations: A Case Study to Bursting in a Neural Pacemaker Model

2021 ◽  
Author(s):  
Yuye Li ◽  
Huaguang Gu ◽  
Yanbing Jia ◽  
Kaihua Ma
Keyword(s):  
Bifurcation Point ◽  
Slow Variable ◽  
Quiescent State ◽  
Bifurcation Curve ◽  
Fast Subsystem ◽  
Saddle Node Bifurcation ◽  
Initiation Point ◽  
Saddle Node ◽  
Pathological Pain ◽  
Slow Variables

Abstract Neuronal bursting is an electrophysiological behavior participating in physiological or pathological functions and a complex nonlinear alternating between burst and quiescent state modulated by slow variables. Identification of dynamics of bursting modulated by two slow variables is still an open problem. In the present paper, a novel fast-slow variable dissection method with two slow variables is proposed to analyze the complex bursting in a 4-dimensional neuronal model to describe bursting associated with pathological pain. The lumenal (Clum) and intracellular (Cin) calcium concentrations are the slowest variables respectively in the quiescent state and burst duration. Questions encountered when the traditional method with one low variable is used. When Clum is taken as slow variable, the burst is successfully identified to terminate near the saddle-homoclinic bifurcation point of the fast subsystem and begin not from the saddle-node bifurcation. With Cin chosen as slow variable, Clum value of initiation point is far from the saddle-node bifurcation point, due to Clum not contained in the equation of membrane potential. To overcome this problem, both Cin and Clum are regarded as slow variables, the two-dimensional fast subsystem exhibits a saddle-node bifurcation point, which is extended to a saddle-node bifurcation curve by introducing Clum dimension. Then, the initial point of burst is successfully identified to be near the saddle-node bifurcation curve. The results present a feasible method for fast-slow variable dissection and deep understanding to the complex bursting behavior with two slow variables, which is helpful for the modulation to pathological pain.

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.

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.

FOLLOWING A SADDLE-NODE OF PERIODIC ORBITS' BIFURCATION CURVE IN CHUA'S CIRCUIT

2009 ◽  
Vol 19 (02) ◽  
pp. 487-495 ◽  
Author(s):  
E. FREIRE ◽  
E. PONCE ◽  
J. ROS
Keyword(s):  
Periodic Orbits ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Numerical Continuation ◽  
Bifurcation Curve ◽  
Saddle Node Bifurcation ◽  
Leading Role ◽  
Chua’S Oscillator ◽  
Saddle Node ◽  
Parametric Region

Starting from previous analytical results assuring the existence of a saddle-node bifurcation curve of periodic orbits for continuous piecewise linear systems, numerical continuation is done to get some primary bifurcation curves for the piecewise linear Chua's oscillator in certain dimensionless parameter plane. The primary period doubling, homoclinic and saddle-node of periodic orbits' bifurcation curves are computed. A Belyakov point is detected in organizing the connection of these curves. In the parametric region between period-doubling, focus-center-limit cycle and homoclinic bifurcation curves, chaotic attractors coexist with stable nontrivial equilibria. The primary saddle-node bifurcation curve plays a leading role in this coexistence phenomenon.

Fast–Slow Variable Dissection with Two Slow Variables: A Case Study on Bifurcations Underlying Bursting for Seizure and Spreading Depression

2021 ◽  
Vol 31 (06) ◽  
pp. 2150096
Author(s):  
Kaihua Ma ◽  
Huaguang Gu ◽  
Zhiguo Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Spreading Depression ◽  
Slow Variable ◽  
Multiple Time Scales ◽  
Extracellular Potassium ◽  
Fast Subsystem ◽  
Depolarization Block ◽  
Bifurcation Points ◽  
Slow Variables ◽  
Dissection Method

The identification of nonlinear dynamics of bursting patterns related to multiple time scales and pathology of brain tissues is still an open problem. In the present paper, representative cases of bursting related to seizure (SZ) and spreading depression (SD) simulated in a theoretical model are analyzed. When the fast–slow variable dissection method with only one slow variable (extracellular potassium concentration, [Formula: see text]) taken as the bifurcation parameter of the fast subsystem is used, the mismatch between bifurcation points of the fast subsystem and the beginning and ending phases of burst appears. To overcome this problem, both slow variables [Formula: see text] and [Formula: see text] (intracellular sodium concentration) are regarded as bifurcation parameters of the fast subsystem, which exhibits three codimension-2 bifurcation points and multiple codimension-1 bifurcation curves containing the saddle-node bifurcation on an invariant cycle (SNIC), the supercritical Hopf bifurcation (the border between spiking and the depolarization block), and the saddle homoclinic (HC) bifurcation. The bursting patterns for SD are related to the Hopf bifurcation and the depolarization block while for SZ to SNIC. Furthermore, at the intersection points between the bursting trajectory and the bifurcation curves in plane ([Formula: see text], [Formula: see text]), the initial or termination phases of burst match the SNIC or HC point well or the Hopf point to a certain extent due to the slow passage effect, showing that the fast–slow variable dissection method with suitable process is still effective to analyze bursting activities. The results present the complex bifurcations underlying the bursting patterns and a proper performing process for the fast–slow variable dissection with two slow variables, which are helpful for modulation to bursting patterns related to brain disfunction.

Perturbation Solution for Secondary Bifurcation in the Quadratically-Damped Mathieu Equation

2001 ◽  
Author(s):  
Deepak V. Ramani ◽  
Richard H. Rand ◽  
William L. Keith
Keyword(s):  
Perturbation Method ◽  
Limit Cycles ◽  
Bifurcation Point ◽  
Mathieu Equation ◽  
Bifurcation Curve ◽  
Mathieu Functions ◽  
Bifurcation Of Limit Cycles ◽  
Saddle Node Bifurcation ◽  
Secondary Bifurcation ◽  
Transition Curves

Abstract This paper concerns the quadratically-damped Mathieu equation:x..+(δ+ϵcos⁡t)x+x.|x.|=0. Numerical integration shows the existence of a secondary-bifurcation in which a pair of limit cycles come together and disappear (a saddle-node bifurcation of limit cycles). In δ–ϵ parameter space, this secondary bifurcation appears as a curve which emanates from one of the transition curves of the linear Mathieu equation for ϵ ≈ 1.5. The bifurcation point along with an approximation for the bifurcation curve is obtained by a perturbation method which uses Mathieu functions rather than the usual sines and cosines.

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