Bogdanov–Takens singularity in the Hindmarsh–Rose neuron with time delay

doi:10.1016/j.amc.2019.02.046

Applied Mathematics and Computation

Volume 354, 1 August 2019, Pages 180-188

https://doi.org/10.1016/j.amc.2019.02.046 Get rights and content

Abstract

In this paper, we study the Bogdanov–Takens singularity in the Hindmarsh–Rose neuron model with time delay. We use the center manifold reduction and the normal form method, by means of which the dynamics near this nonhyperbolic equilibrium can be reduced to the study of the dynamics of the corresponding normal form restricted to the associated two-dimensional center manifold. We show that changes in the time delay length can lead to the saddle-node bifurcation, to the Hopf bifurcation, and to the homoclinic bifurcation.

Introduction

As one of the most potential research directions in the 21st century, dynamics of phenomenological neuron models have been studied extensively [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. The Hindmarsh–Rose model for the action potential, an improved FitzHugh’s model, was given as a mathematical description of bursting dynamics of neurons [12]. The Hindmarsh–Rose Neuron models [12], [13] may be the derivation of the FitzHugh’s model [14] or the realistic Hodgkin–Huxley model[15]. The study of Hindmarsh–Rose Neuron models is still an important and yet difficult role in nonlinear dynamic system theory [16], [17], [18], [19], [20], [21], [22].

Time delay is a common and important factor in bursting processes, which causes the neural models more intricate, realistic and meaningful. The analysis of time delay neural models has been widely concerned whether single neuron or multiple neurons [2], [23], [24], [25]. In neural processing information, Ma and Feng considered a time-delayed signal as feedback mechanism in classic Hindmarsh–Rose model and obtained the following delayed model [26] $\begin{matrix} {\begin{matrix} \dot{x} = y + a x {(t - τ)}^{2} - b x^{3} - c z + I_{a p p}, \\ \dot{y} = c - y - d x^{2}, \\ \dot{z} = r s (x - \bar{x}) - r z, \end{matrix} \end{matrix}$ where x, y, z, I_app, τ means membrane potential, a recovery variable, varying hyperpolarizing current, applied current and the synaptic transmission delay, respectively. r affects the variable speed of the variable z. All parameters are real and positive constants. In 2017, Lakshmanan et al. [27] introduced time delays into the slow oscillation state of z, and obtained the result of Hopf bifurcation for the following model $\begin{matrix} {\begin{matrix} x^{'} = y + a x^{2} - b x^{3} - z (t - τ) + I_{e x t}, \\ y^{'} = c - y - d x^{2}, \\ z^{'} = r s (x - \bar{x}) - r z . \end{matrix} \end{matrix}$ Time delay τ affects the firing process so that the dynamics of the simple delayed neuron will be more complicated. Lakshmanan et al. mentioned in [27] that it is necessary for us to further study the codimension-two bifurcations for obtaining some deep information if two parameters are simultaneously varied in system (1.2).

As we know, the global bifurcation in Hindmarsh–Rose model will be important in more biophysical research. For example, studying and selecting different parameters in modeling can reflect different types of electrophysiological behavior. In this paper, we aim to derive topological normal form for the relevant bifurcations of (1.2) that two separate parameters are required for singularity analysis. In particular, as a very important codimension two bifurcation, Bogdanov–Takens bifurcation is studied from the tangent points of Hopf and saddle-node bifurcation curves and has been studied in many literatures [2], [23], [28], [29]. By the normal form method [30], we obtain the families of saddle-node, Hopf and homoclinic bifurcation curve.

This article is organized as follows. The distribution of eigenvalues and existence of Bogdanov–Takens bifurcation are discussed in Section 2. In Section 3 the transverse expansion of Bogdanov–Takens bifurcation is obtained. These paradigms are used to predict Bogdanov–Takens bifurcation graphs in Section 4. Finally, Section 5 gives the conclusion.

Section snippets

Eigenvalue analysis of equilibrium

Now we first consider the distribution of eigenvalues of (1.2) at equilibrium and then make a thorough inquiry about existence of Bogdanov–Takens bifurcation, which requires that the corresponding characteristic equation has a zero root of multiplicity two, and no other roots with zero real parts.

Let the equilibrium of the (1.2) is $E_{0} = (x_{0}, y_{0}, z_{0}),$ which satisfies the following equation $\begin{matrix} {\begin{matrix} y - b x^{3} + a x^{2} - z (t - τ) + I_{e x t} = 0, \\ c - d x^{2} - y = 0, \\ r (s (x - \bar{x}) - z) = 0 . \end{matrix} \end{matrix}$ It means that $\begin{matrix} b x_{0}^{3} + (d - a) x_{0}^{2} + s x_{0} - (s \bar{x} + c + I_{e x t}) = 0, \\ y_{0} = c - d x_{0}^{2}, \\ z_{0} = s (x_{0} \end{matrix}$

Normal form and unfolding for Bogdanov–Takens bifurcation

In this section, Bogdanov–Takens bifurcation will be investigated by considering s and τ as bifurcation parameter. If $s = s_{0},$ $τ = τ_{0},$ the characteristic Eq. (2.3) at the equilibrium E₀ has a double zero root from Theorem 2.1.

Taking $t = \frac{t}{τ}$ into system (2.2), we have $\begin{matrix} {\begin{matrix} x^{'} = τ ((2 a x_{0} - 3 b x_{0}^{2}) x + y - z (t - 1)) + τ ((a - 3 b x_{0}) x^{2} - b x^{3}), \\ y^{'} = τ (- 2 d x_{0} x - y) - τ d x^{2}, \\ z^{'} = τ (r s x - r z) . \end{matrix} \end{matrix}$ Let $s = s_{0} + μ_{1}, τ = τ_{0} + μ_{2},$ we have $\begin{matrix} {\begin{matrix} x^{'} = (τ_{0} + μ_{2}) ((2 a x_{0} - 3 b x_{0}^{2}) x + y - z (t - 1)) + τ_{0} ((a - 3 b x_{0}) x^{2} - b x^{3}), \\ y^{'} = (τ_{0} + μ_{2}) (- 2 d x_{0} x - y) - τ_{0} d x^{2}, \\ z^{'} = (τ_{0} + μ_{2}) (r s_{0} x - r z) + τ_{0} r μ_{1} x . \end{matrix} \end{matrix}$ We choose the Banach

Bifurcation diagrams of system (1.2) on the center manifold

In this part, we will further analyze the dynamics for the following second order truncated normal form of system (3) ${\begin{matrix} {\dot{Z}}_{1} = Z_{2}, \\ {\dot{Z}}_{2} = λ_{1} Z_{1} + λ_{2} Z_{2} + η_{1} Z_{1}^{2} + η_{2} Z_{1} Z_{2} . \end{matrix}$ Case A: The first time rescaling and coordinate transformation

Let $\begin{matrix} Z_{1} = \frac{η_{1}}{η_{2}^{2}} ξ_{1} - \frac{λ_{2}}{η_{2}}, Z_{2} = \frac{η_{1} | η_{1} |}{η_{2}^{3}} ξ_{2}, t = \frac{η_{2}}{| η_{1} |} τ, \end{matrix}$ system (4.1) becomes (still using Z₁, Z₂ for simplicity) ${\begin{matrix} {\dot{Z}}_{1} = Z_{2}, \\ {\dot{Z}}_{2} = u_{1} + u_{2} Z_{1} + Z_{1}^{2} + s Z_{1} Z_{2}, \end{matrix}$ where $\begin{matrix} s = \pm 1, \\ u_{1} = \frac{η_{2}^{2}}{η_{1}^{2}} (k_{2} μ_{1} + k_{3} μ_{2}) [(k_{2} - \frac{η_{2}}{η_{1}} k_{1}) μ_{1} + k_{3} μ_{2}], \\ u_{2} = \frac{η_{2}}{η_{1}} [(\frac{η_{2}}{η_{1}} k_{1} - 2 k_{2}) μ_{1} - 2 k_{3} μ_{2}], \\ k_{1} = - r τ_{0}, k_{2} = 2 r τ_{0}, k_{3} = r s_{0} . \end{matrix}$ For $s = - 1,$ complete bifurcation diagrams of (4.2)

Conclusion

In the paper, the dynamic behavior near the Bogdanov–Takens bifurcation point of the Hindmarsh–Rose neuron with time delay is discussed. When the two parameters of the Hindmarsh–Rose neuron with time delay change simultaneously, the codimension-two bifurcation analysis has been studied. Applying classic normal form method and center manifold, we studied the existence conditions of some typical bifurcations. The forthcoming study will provide a deeper discussion and good results.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (grant nos. 11772306, 11832002, 11705122), Guangxi Natural Science Foundation (grant no. 2018GXNSFDA281028), Shandong Provincial Natural Science Foundation, China (grant no. ZR2018MA025), the Guangxi University High Level Innovation Team and Distinguished Scholars Program of China (document no. 2018 35), and by the Slovenian Research Agency (grants J4-9302, J1-9112 and P1-0403).

References (35)

R. FitzHugh
Impulses and physiological states in theoretical models of nerve membrane
Biophys. J.
(1961)
A. Bandyopadhyay et al.
Impact of network structure on synchronization of Hindmarsh–Rose neurons coupled in structured network
Appl. Math. Comput.
(2018)
X. He et al.
Codimension two bifurcation in a simple delayed neuron model
Neural Comput. Appl.
(2013)
X. He et al.
Codimension two bifurcation in a simple delayed neuron model
Neural Comput. Appl.
(2013)
Y. Kuznetsov
Kuznetsov Elements of Applied Bifurcation Theory
(1995)
L. Wang et al.
A gradual noisy chaotic neural network for solving the broadcast scheduling problem in packet radio networks
IEEE Trans. Neural Netw. Learn. Syst.
(2006)
X. He et al.
Bogdanov–Takens singularity in tri-neuron network with time delay
IEEE Trans. Neural Netw. Learn. Syst.
(2013)
H. Wang et al.
Equilibrium analysis and phase synchronization of two coupled HR neurons with gap junction
Cognitive Neurodyn.
(2013)
Z. Song et al.
Codimension-two bursting analysis in the delayed neural system with external stimulations
Nonlinear Dyn.
(2012)
X. Liu et al.
Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model
Nonlinear Dyn.
(2012)

P. Orio et al.

Chaos versus noise as drivers of multistability in neural networks

Chaos

(2018)

M. Perc

Thoughts out of noise

European Journal of Physics

(2006)

Q.Y. Wang et al.

Delay-induced multiple stochastic resonances on scale-free neuronal networks

Chaos

(2009)

D.Q. Guo et al.

Functional importance of noise in neuronal information processing

Europhys. Lett.

(2018)

Z.C. Wei et al.

Nonstationary chimeras in a neuronal network

Europhys. Lett.

(2018)

X.J. Sun et al.

Fast regular firings induced by intra- and inter-time delays in two clustered neuronal networks

Chaos

(2018)

J.L. Hindmarsh et al.

A model of the nerve impulse using two first-order differential equations

Nature

(1982)

Cited by (30)

Complexity analysis and discrete fractional difference implementation of the Hindmarsh–Rose neuron system
2023, Results in Physics
Nowadays, multistability assessment in discrete nonlinear fractional difference systems is an emotive topic. This paper introduces a novel discrete, nonequilibrium, memristor-based Hindmarsh–Rose neuron (HRN) with the Caputo fractional difference scheme. Furthermore, in a setting involving commensurate and incommensurate scenarios, the complex structure of the proposed discrete fractional model, which includes its multistability, concealed chaos and hyperchaotic attractor, is examined using a wide range of computational approaches, such as Lyapunov exponents, phase portraits, bifurcation illustrations, and the 0–1 evaluation emergence of synchronization. These evolving properties imply that the fractional discrete HRN has an undetected multistability. Ultimately, an elaborate investigation is performed to confirm the existence of unpredictability employing approximation entropy (ApEn) and the $ℂ_{0}$ measurements. It is demonstrated that whenever the immediate synchronization indicates that it happens, the neurons’ interactions modify, recurring in decreasing fractional orders. Furthermore, minimizing the derivative order increases the incidence of explosions in the synchronization manifold, which is opposite to the behaviour of one nerve cell.
Dynamics of a network of quadratic integrate-and-fire neurons with bimodal heterogeneity
2021, Physics Letters, Section A: General, Atomic and Solid State Physics
Citation Excerpt :
Dynamic processes in large networks of interacting neurons are the focus of intense research. Examples of collective behavior found in such systems are synchronization and collective oscillations [1–4], multistability [3–7], collective irregular dynamics and chaos [7–13], spatiotemporal patterns [14–17] and others. At the microscopic level, neural networks are described by a huge number of differential equations, and their solution requires significant computer resources.
An exact low-dimensional system of mean-field equations for an infinite-size network of pulse coupled integrate-and-fire neurons with a bimodal distribution of an excitability parameter is derived. Bifurcation analysis of these equations shows a rich variety of dynamic modes that do not exist with a unimodal distribution of the excitability parameter. New modes include multistable equilibrium states with different levels of the spiking rate, collective oscillations and chaos. All oscillatory modes coexist with stable equilibrium states. The mean field equations are a good approximation to the solutions of a microscopic model consisting of several thousand neurons.
Studying the performance of critical slowing down indicators in a biological system with a period-doubling route to chaos
2020, Physica A: Statistical Mechanics and its Applications
Citation Excerpt :
Dynamical systems can show variance dynamics [1–3].
This paper aims to investigate critical slowing down indicators in different situations where the system’s parameters change. Variation of the bifurcation parameter is important since it allows finding bifurcation points. A system’s parameters can vary through different functions. In this paper, five cases of bifurcation parameter variation are considered in a biological model with a period-doubling route to chaos. The first case is a slow and small stepwise variation of the bifurcation parameter. The second case is a cyclic, state-dependent variation of the bifurcation parameter. In the third case, a small cyclic variation is combined with a sizeable stochastic resonance. The fourth case involves variations by a large noise, and finally, in the fifth case, significant stepwise changes in the parameter are studied. To identify the conditions under which critical slowing down occurs, an improved version of four well-known critical slowing down indicators (autocorrelation at lag-1, variance, kurtosis, and skewness) are used. The results show that when bifurcations are caused by a sudden change in a parameter or state, critical slowing down cannot be observed before the bifurcation points. However, in cases with slowly varying parameters, critical slowing down can be detected before the bifurcation points. Thus critical slowing down indicators can predict these bifurcation points. In other words, in three cases, the system approaches bifurcation points slowly. In other cases, the bifurcations occur suddenly because of a significant shift in the parameter or state. Thus critical slowing down indicators cannot predict those bifurcation points. However, critical slowing down indicators can predict the bifurcation points in other cases.
Phase synchronization and mode transition induced by multiple time delays and noises in coupled FitzHugh–Nagumo model
2019, Physica A: Statistical Mechanics and its Applications
Citation Excerpt :
Matias et al. [34] studied the dynamic synchronization by reproducing coherence spectra and experimental time delay. The Bogdanov–Takens singularity was studied in the Hindmarsh–Rose neuron model with time delay [35], and it was shown that changes in the time delay length can lead to the saddle–node bifurcation, to the Hopf bifurcation, and to the homoclinic bifurcation. The phase transitions can occur in various coupled bursting neuron [36], and time delay can induce synchronization in two neurons with antiphase bursting activities.
Noise and time delay are ubiquitous in various physical and biological systems. In this paper, we studied the phase synchronization and mode transition of oscillation mode induced by time delays and noises in a coupled FitzHugh–Nagumo (FHN) neural system. In the case of coupled neurons with single time delay, it is found that the mode oscillation of neurons induced by the noise and time delay undergoes a successive transitions from the synchronization, the out-of phase, the alternating phase-drift and anti-phase state, to the anti-phase state. In the case of coupled neurons with two time delays, when the two delays are comparable, the large amplitude oscillation of potential is in alternating synchronous and asynchronous state in the absence of noises, and the small amplitude oscillation of potential is always synchronized. The oscillation behaviors of coupled neurons with two time delays are similar to the cases with a single time delay in the presence of noises. When one of two time delays is largely dominant over the other, the neurons show completely different dynamic behaviors.
Synchronization in scale-free neural networks under electromagnetic radiation
2024, Chaos
Dynamical behavior of memristor-coupled heterogeneous discrete neural networks with synaptic crosstalk
2024, Chinese Physics B

View all citing articles on Scopus

View full text

Bogdanov–Takens singularity in the Hindmarsh–Rose neuron with time delay

Abstract

Introduction

Section snippets

Eigenvalue analysis of equilibrium

Normal form and unfolding for Bogdanov–Takens bifurcation

Bifurcation diagrams of system (1.2) on the center manifold

Conclusion

Acknowledgements

Biophys. J.

Appl. Math. Comput.

Neural Comput. Appl.

Neural Comput. Appl.

A gradual noisy chaotic neural network for solving the broadcast scheduling problem in packet radio networks

IEEE Trans. Neural Netw. Learn. Syst.

Bogdanov–Takens singularity in tri-neuron network with time delay

IEEE Trans. Neural Netw. Learn. Syst.

Equilibrium analysis and phase synchronization of two coupled HR neurons with gap junction

Cognitive Neurodyn.

Codimension-two bursting analysis in the delayed neural system with external stimulations

Nonlinear Dyn.

Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model

Nonlinear Dyn.

Chaos versus noise as drivers of multistability in neural networks

Chaos

Thoughts out of noise

European Journal of Physics

Delay-induced multiple stochastic resonances on scale-free neuronal networks

Chaos

Functional importance of noise in neuronal information processing

Europhys. Lett.

Nonstationary chimeras in a neuronal network

Europhys. Lett.

Fast regular firings induced by intra- and inter-time delays in two clustered neuronal networks

Chaos

A model of the nerve impulse using two first-order differential equations

Nature