Bursting Types and Bifurcation Analysis in the Pre-Bötzinger Complex Respiratory Rhythm Neuron

2017 ◽  
Vol 27 (01) ◽  
pp. 1750010 ◽  
Author(s):  
Jing Wang ◽  
Bo Lu ◽  
Shenquan Liu ◽  
Xiaofang Jiang
Keyword(s):  
Hopf Bifurcation ◽  
Respiratory Rhythm ◽  
Vital Role ◽  
Dynamical Analysis ◽  
Bifurcation Curve ◽  
Excitable Cells ◽  
Theoretical Derivation ◽  
Fold Bifurcation ◽  
Bötzinger Complex ◽  
Lyapunov Coefficient

Many types of neurons and excitable cells could intrinsically generate bursting activity, even in an isolated case, which plays a vital role in neuronal signaling and synaptic plasticity. In this paper, we have mainly investigated bursting types and corresponding bifurcations in the pre-Bötzinger complex respiratory rhythm neuron by using fast–slow dynamical analysis. The numerical simulation results have showed that for some appropriate parameters, the neuron model could exhibit four distinct types of fast–slow bursters. We also explored the bifurcation mechanisms related to these four types of bursters through the analysis of phase plane. Moreover, the first Lyapunov coefficient of the Hopf bifurcation, which can decide whether it is supercritical or subcritical, was calculated with the aid of MAPLE software. In addition, we analyzed the codimension-two bifurcation for equilibria of the whole system and gave a detailed theoretical derivation of the Bogdanov–Takens bifurcation. Finally, we obtained expressions for a fold bifurcation curve, a nondegenerate Hopf bifurcation curve, and a saddle homoclinic bifurcation curve near the Bogdanov–Takens bifurcation point.

scholarly journals Fast-Slow Coupling Dynamics Behavior of the van der Pol-Rayleigh System

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3004
Author(s):  
Danjin Zhang ◽  
Youhua Qian
Keyword(s):  
Hopf Bifurcation ◽  
Phase Portrait ◽  
External Excitation ◽  
Fast Subsystem ◽  
Van Der Pol ◽  
Fold Bifurcation ◽  
Modes Of Vibration ◽  
Lyapunov Coefficient ◽  
The Stability ◽  
Rayleigh System

In this paper, the dynamic behavior of the van der Pol-Rayleigh system is studied by using the fast–slow analysis method and the transformation phase portrait method. Firstly, the stability and bifurcation behavior of the equilibrium point of the system are analyzed. We find that the system has no fold bifurcation, but has Hopf bifurcation. By calculating the first Lyapunov coefficient, the bifurcation direction and stability of the Hopf bifurcation are obtained. Moreover, the bifurcation diagram of the system with respect to the external excitation is drawn. Then, the fast subsystem is simulated numerically and analyzed with or without external excitation. Finally, the vibration behavior and its generation mechanism of the system in different modes are analyzed. The vibration mode of the system is affected by both the fast and slow varying processes. The mechanisms of different modes of vibration of the system are revealed by the transformation phase portrait method, because the system trajectory will encounter different types of attractors in the fast subsystem.

scholarly journals Dynamical analysis of a giving up smoking model with time delay

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zizhen Zhang ◽  
Ruibin Wei ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

Analysis of the non-periodic oscillations of a self-excited friction-damped system with closely-spaced modes

2021 ◽  
Author(s):  
Lukas Woiwode ◽  
Alexander F. Vakakis ◽  
Malte Krack
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Dry Friction ◽  
The Self ◽  
Perturbation Approach ◽  
Friction Damping ◽  
Periodic Oscillations ◽  
Analytical Results ◽  
Fold Bifurcation ◽  
Phase Motion

Abstract It is widely known that dry friction damping can bound the self-excited vibrations induced by negative damping. The vibrations typically take the form of (periodic) limit cycle oscillations. However, when the intensity of the self-excitation reaches a condition of maximum friction damping, the limit cycle loses stability via a fold bifurcation. The behavior may become even more complicated in the presence of any internal resonance conditions. In this work, we consider a two-degree-of-freedom system with an elastic dry friction element (Jenkins element) having closely spaced natural frequencies. The symmetric in-phase motion is subjected to self-excitation by negative (viscous) damping, while the symmetric out-of-phase motion is positively damped. In a previous work, we showed that the limit cycle loses stability via a secondary Hopf bifurcation, giving rise to quasi-periodic oscillations. A further increase of the self-excitation intensity may lead to chaos and finally divergence, long before reaching the fold bifurcation point of the limit cycle. In this work, we use the method of Complexification-Averaging to obtain the slow flow in the neighborhood of the limit cycle. This way, we show that chaos is reached via a cascade of period doubling bifurcations on invariant tori. Using perturbation calculus, we establish analytical conditions for the emergence of the secondary Hopf bifurcation and approximate analytically its location. In particular, we show that non-periodic oscillations are the typical case for prominent nonlinearity, mild coupling (controlling the proximity of the modes) and sufficiently light damping. The range of validity of the analytical results presented herein is thoroughly assessed numerically. To the authors' knowledge, this is the first work that shows how the challenging Jenkins element can be treated formally within a consistent perturbation approach in order to derive closed-form analytical results for limit cycles and their bifurcations.

Pre-Bötzinger Complex Functions as a Central Hypoxia Chemosensor for Respiration In Vivo

2000 ◽  
Vol 83 (5) ◽  
pp. 2854-2868 ◽  
Author(s):  
Irene C. Solomon ◽  
Norman H. Edelman ◽  
Judith A. Neubauer
Keyword(s):  
Short Duration ◽  
Phrenic Nerve ◽  
Excitatory Amino Acid ◽  
Respiratory Rhythm ◽  
High Amplitude ◽  
Unique Region ◽  
Homocysteic Acid ◽  
Bötzinger Complex ◽  
Nerve Discharge

Recently, we identified a region located in the pre-Bötzinger complex (pre-BötC; the proposed locus of respiratory rhythm generation) in which activation of ionotropic excitatory amino acid receptors usingdl-homocysteic acid (DLH) elicits a variety of excitatory responses in the phrenic neurogram, ranging from tonic firing to a rapid series of high-amplitude, rapid rate of rise, short-duration inspiratory bursts that are indistinguishable from gasps produced by severe systemic hypoxia. Therefore we hypothesized that this unique region is chemosensitive to hypoxia. To test this hypothesis, we examined the response to unilateral microinjection of sodium cyanide (NaCN) into the pre-BötC in chloralose- or chloralose/urethan-anesthetized vagotomized, paralyzed, mechanically ventilated cats. In all experiments, sites in the pre-BötC were functionally identified using DLH (10 mM, 21 nl) as we have previously described. All sites were histologically confirmed to be in the pre-BötC after completion of the experiment. Unilateral microinjection of NaCN (1 mM, 21 nl) into the pre-BötC produced excitation of phrenic nerve discharge in 49 of the 81 sites examined. This augmentation of inspiratory output exhibited one of the following changes in cycle timing and/or pattern: 1) a series of high-amplitude, short-duration bursts in the phrenic neurogram (a discharge similar to a gasp), 2) a tonic excitation of phrenic neurogram output, 3) augmented bursts in the phrenic neurogram (i.e., eupneic breath ending with a gasplike burst), or 4) an increase in frequency of phrenic bursts accompanied by small increases or decreases in the amplitude of integrated phrenic nerve discharge. Our findings identify a locus in the brain stem in which focal hypoxia augments respiratory output. We propose that the respiratory rhythm generator in the pre-BötC has intrinsic hypoxic chemosensitivity that may play a role in hypoxia-induced gasping.

Models of Respiratory Rhythm Generation in the Pre-Bötzinger Complex. I. Bursting Pacemaker Neurons

1999 ◽  
Vol 82 (1) ◽  
pp. 382-397 ◽  
Author(s):  
Robert J. Butera ◽  
John Rinzel ◽  
Jeffrey C. Smith
Keyword(s):  
Membrane Potential ◽  
Respiratory Rhythm ◽  
Rhythm Generation ◽  
Minimal Models ◽  
Dynamic Features ◽  
Bursting Neurons ◽  
Wide Range ◽  
Bötzinger Complex ◽  
Voltage Dependent ◽  
Respiratory Rhythm Generation

A network of oscillatory bursting neurons with excitatory coupling is hypothesized to define the primary kernel for respiratory rhythm generation in the pre-Bötzinger complex (pre-BötC) in mammals. Two minimal models of these neurons are proposed. In model 1, bursting arises via fast activation and slow inactivation of a persistent Na+ current I NaP-h. In model 2, bursting arises via a fast-activating persistent Na+ current INaP and slow activation of a K+ current IKS. In both models, action potentials are generated via fast Na+ and K+currents. The two models have few differences in parameters to facilitate a rigorous comparison of the two different burst-generating mechanisms. Both models are consistent with many of the dynamic features of electrophysiological recordings from pre-BötC oscillatory bursting neurons in vitro, including voltage-dependent activity modes (silence, bursting, and beating), a voltage-dependent burst frequency that can vary from 0.05 to >1 Hz, and a decaying spike frequency during bursting. These results are robust and persist across a wide range of parameter values for both models. However, the dynamics of model 1 are more consistent with experimental data in that the burst duration decreases as the baseline membrane potential is depolarized and the model has a relatively flat membrane potential trajectory during the interburst interval. We propose several experimental tests to demonstrate the validity of either model and to differentiate between the two mechanisms.

scholarly journals Dynamical Analysis of a Viral Infection Model with Delays in Computer Networks

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Propagation Model ◽  
Dynamical Analysis ◽  
Infection Model ◽  
Virus Propagation ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
The Stability

This paper is devoted to the study of an SIRS computer virus propagation model with two delays and multistate antivirus measures. We demonstrate that the system loses its stability and a Hopf bifurcation occurs when the delay passes through the corresponding critical value by choosing the possible combination of the two delays as the bifurcation parameter. Moreover, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by means of the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate the obtained results.

scholarly journals Dynamical analysis of a predator-prey interaction model with time delay and prey refuge

2018 ◽  
Vol 5 (1) ◽  
pp. 138-151 ◽  
Author(s):  
Jai Prakash Tripathi ◽  
Swati Tyagi ◽  
Syed Abbas
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Interaction Model ◽  
Dynamical Analysis ◽  
Prey Refuge ◽  
Normal Form Theory ◽  
Predator Species ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model

AbstractIn this paper, we study a predator-prey model with prey refuge and delay. We investigate the combined role of prey refuge and delay on the dynamical behaviour of the delayed system by incorporating discrete type gestation delay of predator. It is found that Hopf bifurcation occurs when the delay parameter τ crosses some critical value. In particular, it is shown that the conditions obtained for the Hopf bifurcation behaviour are sufficient but not necessary and the prey reserve is unable to stabilize the unstable interior equilibrium due to Hopf bifurcation. In particular, the direction and stability of bifurcating periodic solutions are determined by applying normal form theory and center manifold theorem for functional differential equations. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. At the end, we perform some numerical simulations to substantiate our analytical findings.

scholarly journals Influence of Electric Current and Magnetic Flow on Firing Patterns of Pre-Bötzinger Complex Model

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Wenchao Ji ◽  
Moutian Liu ◽  
Lixia Duan
Keyword(s):  
Electric Current ◽  
Respiratory Rhythm ◽  
Complex Model ◽  
Neuronal Firing ◽  
Essential Factor ◽  
Firing Activity ◽  
Magnetic Flow ◽  
Bötzinger Complex ◽  
Bistable Structure ◽  
Slow Decomposition

The dynamics of neuronal firing activity is vital for understanding the pathological respiratory rhythm. Studies on electrophysiology show that the magnetic flow is an essential factor that modulates the firing activities of neurons. By adding the magnetic flow to Butera’s neuron model, we investigate how the electric current and magnetic flow influence neuronal activities under certain parametric restrictions. Using fast-slow decomposition and bifurcation analysis, we show that the variation of external electric current and magnetic flow leads to the change of the bistable structure of the system and hence results in the switch of neuronal firing pattern from one type to another.

scholarly journals Lethal avian influenza A (H5N1) virus induces ataxic breathing in mice with apoptosis of pre-Botzinger complex neurons expressing neurokinin-1 receptor

2017 ◽  
Vol 313 (5) ◽  
pp. L772-L780 ◽  
Author(s):  
Jianguo Zhuang ◽  
Na Zang ◽  
Chunyan Ye ◽  
Fadi Xu
Keyword(s):  
Respiratory Failure ◽  
Influenza A ◽  
Early Stage ◽  
Respiratory Rhythm ◽  
Body Weight Loss ◽  
Neurokinin 1 Receptor ◽  
Arterial Blood ◽  
Bötzinger Complex ◽  
Neurokinin 1 ◽  
Active Caspase

Lethal influenza A (H5N1) induces respiratory failure in humans. Although it also causes death at 7 days postinfection (dpi) in mice, the development of the respiratory failure and the viral impact on pre-Botzinger complex (PBC) neurons expressing neurokinin 1 receptor (NK1R), which is the respiratory rhythm generator, have not been explored. Body temperature, weight, ventilation, and arterial blood pH and gases were measured at 0, 2, 4, and 6 dpi in control, lethal HK483, and nonlethal HK486 viral-infected mice. Immunoreactivities (IR) of PBC NK1R, H5N1 viral nucleoprotein (NP), and active caspase-3 (CASP3; a marker for apoptosis) were detected at 6 dpi. HK483, but not HK486, mice showed the following abnormalities: 1) gradual body weight loss and hypothermia; 2) tachypnea at 2–4 dpi and ataxic breathing with long-lasting apneas and hypercapnic hypoxemia at 6 dpi; and 3) viral replication in PBC NK1R neurons with NK1R-IR reduced by 75% and CASP3-IR colabeled at 6 dpi. Lethal H5N1 viral infection causes tachypnea at the early stage and ataxic breathing and apneas (hypercapnic hypoxemia) leading to death at the late stage. Its replication in the PBC induces apoptosis of local NK1R neurons, contributing to ataxic breathing and respiratory failure.

RT-PCR reveals muscarinic acetylcholine receptor mRNA in the pre-Bötzinger complex

2001 ◽  
Vol 281 (6) ◽  
pp. L1420-L1424 ◽  
Author(s):  
Jiunu Lai ◽  
Xuesi M. Shao ◽  
Richard W. Pan ◽  
Edward Dy ◽  
Cindy H. Huang ◽  
...  
Keyword(s):  
Muscarinic Receptor ◽  
Muscarinic Receptors ◽  
Respiratory Rhythm ◽  
Pcr Amplification ◽  
Cholinergic Receptor ◽  
Receptor Subtype ◽  
Receptor Subtypes ◽  
Muscarinic Receptor Subtypes ◽  
Rt Pcr ◽  
Bötzinger Complex

Muscarinic receptors mediate the postsynaptic excitatory effects of acetylcholine (ACh) on inspiratory neurons in the pre-Bötzinger complex (pre-BötC), the hypothesized site for respiratory rhythm generation. Because pharmacological tools for identifying the subtypes of the muscarinic receptors that underlie these effects are limited, we probed for mRNA for these receptors in the pre-BötC. We used RT-PCR to determine the expression of muscarinic receptor subtypes in tissue punches of the pre-BötC taken from rat medullary slices. Cholinergic receptor subtype M2 and M3 mRNAs were observed in the first round of PCR amplification. All five subtypes, M1–M5, were observed in the second round of amplification. Our results suggest that the majority of muscarinic receptor subtypes in the pre-BötC are M2 and M3, with minor expression of M1, M4, and M5.

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