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.

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 Slow-fast effect and generation mechanism of brusselator based on coordinate transformation

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 261-268 ◽  
Author(s):  
Xianghong Li ◽  
Jingyu Hou ◽  
Yongjun Shen
Keyword(s):  
Time Scales ◽  
Coordinate Transformation ◽  
Generation Mechanism ◽  
Attraction Domain ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Fast Subsystem ◽  
Similar System ◽  
Fast Analysis ◽  
Fold Bifurcation

AbstractThe Brusselator with different time scales, which behaves in the classical slow-fast effect, is investigated, and is characterized by the coupling of the quiescent and spiking states. In order to reveal the generation mechanism by using the slow-fast analysis method, the coordinate transformation is introduced into the classical Brusselator, so that the transformed system can be divided into the fast and slow subsystems. Furthermore, the stability condition and bifurcation phenomenon of the fast subsystem are analyzed, and the attraction domains of different equilibria are presented by theoretical analysis and numerical simulation respectively. Based on the transformed system, it could be found that the generation mechanism between the quiescent and spiking states is Fold bifurcation and change of the attraction domain of the fast subsystem. The results may also be helpful to the similar system with multiple time scales.

DYNAMICS AND DOUBLE HOPF BIFURCATIONS OF THE ROSE–HINDMARSH MODEL WITH TIME DELAY

2009 ◽  
Vol 19 (11) ◽  
pp. 3733-3751 ◽  
Author(s):  
SUQI MA ◽  
ZHAOSHENG FENG ◽  
QISHAI LU
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Value Of Time ◽  
Asymptotically Stable ◽  
Universal Unfolding ◽  
Double Hopf Bifurcation ◽  
Bifurcation Points ◽  
The Rose

In this paper, we are concerned with the Rose–Hindmarsh model with time delay. By applying the generalized Sturm criterion, a number of imaginary roots of the characteristic equation are classified. The absolutely stable regions for any value of time delay are detected. By the continuous software DDE-Biftool, both the Hopf bifurcation curves and double Hopf bifurcation points are illustrated in parametric spaces. The normal form and universal unfolding at double Hopf bifurcation points are considered by the center manifold method. Some examples also indicate that the corresponding unique attractor near each double Hopf point is asymptotically stable.

Optimal control of nonlinear aeroelastic system with non‐semi‐simple eigenvalues at Hopf bifurcation points

2020 ◽  
Vol 41 (5) ◽  
pp. 1524-1542
Author(s):  
Licai Wang ◽  
Yudong Chen ◽  
Chunyan Pei ◽  
Lina Liu ◽  
Suhuan Chen
Keyword(s):  
Optimal Control ◽  
Hopf Bifurcation ◽  
Aeroelastic System ◽  
Bifurcation Points

scholarly journals Spreading Depression Can Be Elicited in Brain Stem of Immature But Not Adult Rats

2003 ◽  
Vol 90 (4) ◽  
pp. 2163-2170 ◽  
Author(s):  
Frank Richter ◽  
Sven Rupprecht ◽  
Alfred Lehmenkühler ◽  
Hans-Georg Schaible
Keyword(s):  
Brain Stem ◽  
Time Course ◽  
Spreading Depression ◽  
Chloride Ions ◽  
Extracellular Potassium ◽  
Peak Time ◽  
Adult Rats ◽  
Extracellular Potassium Concentration ◽  
Nmda Receptor Blocker ◽  
The Brain

Spreading depression (SD), a neuronal mechanism involved in brain pathophysiology, occurs in brain areas with high neuronal density such as the cerebral cortex. By contrast, the brain stem is thought to be resistant to SD. Here we show that DC shifts resembling cortical SD can be elicited in rat brain stem by topical application of KCl but not by pricking the brain stem. However, this was only possible until postnatal day 13, and, in addition, susceptibility for SD had to be enhanced. The latter was achieved by superfusion of the brain stem for 45 min with a solution containing acetate instead of chloride ions. Transient asphyxia or hypoxia by 2 min breathing 6% O2 in N2 had a similar effect. Negative brain stem DC deflections were paralleled by an increase of extracellular potassium concentration ≤40 mM and were spreading, but unlike cortical SD they were not inducible by glutamate and N-methyl-d-aspartate (NMDA). Time course and slope of brain stem SD either resembled cortical SD or were long-lasting and sustained. The latter stopped normal breathing. Different from cortical SD, negative brain stem DC deflections were changed in their slope (mostly converted into sustained shape, peak time was significantly prolonged, decline-time and duration were prolonged), but not abolished by the NMDA receptor blocker MK-801. Thus we demonstrate that the immature brain stem has the capacity to generate negative DC shifts, which could be relevant as a risk factor in newborn brain stem function.

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