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Cognitive Neurodynamics logoLink to Cognitive Neurodynamics
. 2013 Jan 3;7(4):341–349. doi: 10.1007/s11571-012-9237-6

Hopf bifurcation and bursting synchronization in an excitable systems with chemical delayed coupling

Lixia Duan 1,2,, Denggui Fan 1, Qishao Lu 3
PMCID: PMC3713206  PMID: 24427210

Abstract

In this paper we consider the Hopf bifurcation and synchronization in the two coupled Hindmarsh–Rose excitable systems with chemical coupling and time-delay. We surveyed the conditions for Hopf bifurcations by means of dynamical bifurcation analysis and numerical simulation. The results show that the coupled excitable systems with no delay have supercritical Hopf bifurcation, while the delayed system undergoes Hopf bifurcations at critical time delays when coupling strength lies in a particular region. We also investigated the effect of the delay on the transition of bursting synchronization in the coupled system. The results are helpful for us to better understand the dynamical properties of excitable systems and the biological mechanism of information encoding and cognitive activity.

Keywords: Delay, Chemical coupling, Excitability, Hopf bifurcation, Synchronization

Introduction

Excitability is a character common to many biological systems. For example, we have found some isolated or coupled neurons typically exhibit excitable behavior. From a dynamical system point of view we might paraphrase this as follows: A slight perturbation of the single stable stationary state would lead to a large and long lasting excursion away from stationary point before the system asymptotically returning back to equilibrium. Two general types of synaptic connection between neurons are electrical and chemical, and the chemical synapses are much more common. The time-delay is well known in the information transition between neurons because of the gap junctions. Researchers show that time-delay can change qualitatively the dynamical features of the system (Burić and Ranković 2007; Faria 2000).

The studies related to neural systems are mainly focused on the following two aspects. On the one hand, a lot of researches have been made on the synchronous transition and its application in the coupled neural system and network with no delay. For example, Shi and Lu (2004) studied the complete synchronization of coupled Hindmarsh–Rose (HR) neural network with the ring structure. Wang et al. (2007) gave the types of bursting and synchronous transition in the coupled modified Morris–Lecar (ML) neurons systems. And Yuan et al. (2005) analyzed synchronization and asynchronization in two coupled excitable systems, and so on. On the other hand, time-delay can change qualitatively the dynamical features of the system and consequently will influence the neuron’s biological behavior. Recently, the coupled systems with delay have attracted much attention of researchers in different research fields. Burić and Todorović (2003), Ranković (2011) investigated Hopf bifurcation of coupled excitable FitzHugh–Nagumo (FHN) neurons with delayed coupling and concluded that the time delay can induce the Hopf bifurcation. Wang et al. (2009) elaborated the bifurcation and synchronization of synaptically coupled FHN models with time delay and they confirmed that rich bifurcation behavior can be exhibited with variation of the coupling strength and time delay, as well as the synchronization of the coupled neurons can be achieved in some parameter ranges. Burić et al. (2005) observed different synchronization states in a coupled FHN system with delay with the variation of the coupling strength and delay. Dhamala et al. (2004) studied in detail the effect of the delay on the complete synchronization of the coupled HR systems. Zhen and Xu (2010), Neefs et al. (2010) touched respectively on the stability of the stationary point in a non-chemical coupled FHN and HR neural system with delay. All works but not limited to the mentioned above have promoted better understanding of the dynamics of coupled systems with delay.

Synchronous firing behavior could be widely found during the neural activity in the brain. The observation of synchronous neural activity in the central nervous system suggest that neural activity is a cooperative process of neurons and synchronization plays a vital role in information processing in the brain. On the other hand, bifurcations led by the coupling strength and time-delay can induce different dynamical synchronous behaviors and hence form different synchronization regions. As is known, different synchronization areas imply different neurocognitive function and pathological state. So the investigation of the bifurcation and synchronization is still of vital importance for us to understand the mechanism of how the information is encoded and the cognitive activity happens.

Particularly, excitable systems with delay are very important in theoretical study and application. Hence, bifurcation and synchronization of coupled excitable systems with delay is interesting and necessary to be investigated. To explore the general rule of the effect of delay for the excitable neural system and offer a more sufficient theoretical basis for diagnosing or curing these diseases connected with neural system, we must deeply and extensively analyze the dynamic nature of different systems, especially the excitable neural systems with delay. Most of the papers focus on Hopf bifurcation and synchronization of the electrical coupled general HR neural systems with delay. However, the effects of the coupled strength and delay on the bifurcations and synchronization in chemical coupled excitable HR systems are still important in understanding information processing in the brain. In this paper, motivated by Ranković (2011), Burić et al. (2008), Vasović et al. (2012), we focus on the excitable HR system with chemical couple and time delay. Our studies are mainly about the stability of the stationary point, the bifurcation with one or two parameters and the effect of delay on the synchronization transition of the excitable system.

This paper is organized as follows. In “Main results” section, we discuss the stability of the stationary point, give the conditions for Hopf bifurcations happening of the coupled excitable HR system with no or with time-delay and graphically reveal the effect of the delay on the synchronization transition in the excitable HR system. We conclude the paper in last section.

Main results

The model of two chemical coupled HR excitable neurons with delay is given as follows:

graphic file with name M1.gif 1

where Inline graphic, x is the synaptic membrane potential, y is a recovery variable and z is a slow adaptation current. The variable parameter τ represents the time delay in signal transmission and the variable parameter c is the coupling strength between the first neuron at time t and its neighbor at some previous time t − τ, so they are real and positive. The coupling term that we shall use, is the form of the FTM (Belykh et al. 2005; Somers and Kopell 1993) coupling:

graphic file with name M3.gif

In this paper, the values of the parameters θ, V, and k will be fixed as θ = 0.25, V = 2, k = 10.

The dynamics of the coupled system depends on the properties of each of the units and their interactions. Both the neurons in the two coupled HR excitable system display the excitable behaviors. The single HR excitable neuron model is described as the following equations:

graphic file with name M4.gif 2

where Inline graphic are positive parameters, Inline graphic and usually Inline graphic is satisfied.

It is easy to verify that point Inline graphic is the only one stable stationary solution of system (2) when Inline graphic. The eigenvalues of the linearized system of (2) are Inline graphic, of which one is real and the others are complex. Therefore, system (2) has a stable focus·node attractor when Inline graphic. The phase trajectory and firing pattern of system (2) for parameters Inline graphic are intuitively shown in Fig. 1. The steady state is stable focus·node.

Fig. 1.

Fig. 1

The phase trajectory and firing pattern of single HR excitable system for parameters Inline graphic. a Stable focus·node in 3-D space, b the time series of membrane potential

Local stability and bifurcations of the stationary solution

In this section we study the stability and bifurcations of the stationary solution Inline graphic of the system (1) for varying parameters c > 0 and Inline graphic. Parameters a, b, d, s meet the condition Inline graphic such that each of the units displays the excitable behavior.

Instantaneous coupling τ = 0

Consider the system (1) in the case of instantaneous coupling

graphic file with name M17.gif 3

where Inline graphic

Theorem 2.1.1

Stationary solutionInline graphicof the system (3):

  • (i)

    is stable attractor for everyInline graphic;

  • (ii)

    is stablefocus·nodefocus·node forInline graphicand stablefocus·nodenodeforInline graphic;

  • (iii)

    has supercritical Hopf bifurcation when parameterInline graphic.

Proof

Local stability of the stationary solution is determined by analyzing the linearized system at Inline graphic:

graphic file with name M25.gif 4

where Inline graphic Linear part of (4)

graphic file with name M27.gif

implies the following characteristic equation:

graphic file with name M28.gif

where

graphic file with name M29.gif

Then

graphic file with name M30.gif

hence

graphic file with name M31.gif

where

graphic file with name M32.gif

with solutions

graphic file with name M33.gif
  • (i)

    The sign of the real parts of the six eigenvalues determines the stability type of the trivial stationary point. Obviously, the real part of Inline graphic and Inline graphic is negative for any value of the parameter c, but Inline graphic and Inline graphic becomes pure imaginary for Inline graphic. Furthermore, Inline graphic, so the real parts of Inline graphic and Inline graphic are also negative for Inline graphic. As a result, the real parts of six eigenvalues are all negative for Inline graphic, namely the stationary solution is a stable attractor.

  • (ii)

    We can know from (i) that Inline graphic and Inline graphic is a pair of complex conjugated eigenvalues for Inline graphic, i.e., Inline graphic; while Inline graphic and Inline graphic is a pair of complex-conjugate eigenvalues with negative real part for Inline graphic, and they become real when Inline graphic. Accordingly, the stationary solution is stable focus·nodefocus·node and stable focus·nodenode, respectively.

  • (iii)
    When Inline graphic, Inline graphic and Inline graphic are pure imaginary Inline graphic, where Inline graphic. The real parts of the other four eigenvalues are negative. The type of bifurcation at Inline graphic can be obtained by reducing the system (3) on the corresponding center manifold. Applying the center manifold theorem (Lu 2010) to system (3) and by logical deduction, step after step, we come to the dynamical system on the center manifold with parameter Inline graphic:
    graphic file with name M59.gif 5
    After using the normal form method (Lu 2010) and the conversion of coordinates Inline graphic, we can get the normal form with the small Inline graphic of Hopf bifurcation
    graphic file with name M62.gif 6
    where
    graphic file with name M63.gif

Since Inline graphic, stationary point for Inline graphic has supercritical Hopf bifurcation and that completes the proof of the theorem.

Now we consider an example to illustrate our conclusion. Let Inline graphicInline graphic in system (3), then we have Inline graphic. Then, the Theorem 2.1.1 has supercritical Hopf bifurcation at Inline graphic. The stability of the stationary point for the system (3) in Theorem 2.1.1 with parameters Inline graphic is shown in Fig. 2, in which we can clearly observe the change of the six eigenvalues’ real parts with the variation of the coupling strength in the complex plane.

Fig. 2.

Fig. 2

Computing and plotting stability of the stationary point of the system (3) for parameter a c = 0.04, b c = 0.0418, c c = 0.4185. The five-pointed stars denote the positions of the six eigenvalues of (3) in the complex plane, of which the green/red stars are those eigenvalues with negative/positive real part

Delayed coupling τ > 0

Consider the system (1) in the case of delayed coupling. As in the previous case, local stability of stationary solution is determined by analyzing the system linearized at Inline graphic.

Theorem 2.1.2

The system (1) undergoes a codim-1 Hopf bifurcation at equilibriumInline graphicfor critical time delays given by:

graphic file with name M73.gif

where

graphic file with name M74.gif
Proof

Linearization of the system (1) at Inline graphic

graphic file with name M76.gif 7

where

graphic file with name M77.gif

Because of the existence of delay, system (9) can be written as the form

graphic file with name M78.gif

where

graphic file with name M79.gif

The equation Inline graphic implies

graphic file with name M81.gif

where

graphic file with name M82.gif

Bifurcations due to a nonzero time lag occur when some of the roots of equation above across the imaginary axes. Let us first discuss the nonzero pure imaginary roots. Substitution Inline graphic into Inline graphic gives

graphic file with name M85.gif

hence

graphic file with name M86.gif

or into Inline graphic gives

graphic file with name M88.gif

From the identities Inline graphic and Inline graphic, we can obtain

graphic file with name M91.gif 8

and

graphic file with name M92.gif 9

where

graphic file with name M93.gif

Since Inline graphic, the term Inline graphic can be factored out from (9) to obtain

graphic file with name M96.gif 10

Hence

graphic file with name M97.gif

Substitution of Inline graphic into (10) gives

graphic file with name M99.gif

Differentiation of the characteristic equation

graphic file with name M100.gif

gives

graphic file with name M101.gif

and

graphic file with name M102.gif

Substitution of Inline graphic finally gives

graphic file with name M104.gif

The expressions for Inline graphic in Theorem 2.1.2 are valid if coupling constant c satisfies Inline graphic, i.e. Inline graphic. For this reason, Theorem 2.1.2 is valid for Inline graphic and for Inline graphic, the stationary point is stable for any τ.

Below we use the DDE-BIFTOOL (Engelborghs et al. 2007), a Matlab package for bifurcation analysis of delay differential equations, to simulate the results of Theorem 2.1.2. Parameters in system (1) are the same as in Fig. 2 and then we have Inline graphic, i.e., the Theorem 2.1.2 is valid for Inline graphic. The first few branches of the Hopf bifurcation curves Inline graphic given by Theorem 2.1.2 is shown in Fig. 3, from which we can see that the system (1) has a good robustness for any delay when Inline graphic; while for Inline graphic, it is very sensitive for delay, namely for any fixed value of Inline graphic, the system will undergo a series of supercritical or subcritical Hopf bifurcations as the delay increasing.

Fig. 3.

Fig. 3

First few branches of the Hopf bifurcation curves Inline graphic given by Theorem 2.1.2. Parameters as in Fig. 2. Cyan/blue curves correspond to supercritical/subcritical Hopf bifurcations. (Color figure online)

Synchronization transition of the firing patterns

In this section, corresponding to the different regions of Fig. 3 we mainly numerically investigate when the excitability of coupled system switches on or off and how the synchronous states transit with the changing of coupling strength and time-delays. So in what follows, we still focus on synchronization transitions of two excitable HR neurons coupled by delayed chemical coupling (1), and study the influence of coupling strength and delay on synchronization and neural firing.

As is shown in Fig. 4a, b, when the coupling strength is less than the critical value for Hopf bifurcation (Inline graphic), the system (1) will go to stable state for any delay. When coupling strength c = 0.03, the system (1) slows down to a stable state after going through an attenuated-oscillation for Inline graphic and Inline graphic. The coupled neurons can’t fire in this case.

Fig. 4.

Fig. 4

The firing patterns of the chemically coupled HR neurons for parameters c = 0.03 and aInline graphic, bInline graphic where the red and blue lines represent neuron 1 and neuron 2, respectively. (Color figure online)

When the coupling strength is more than the critical value for Hopf bifurcation (Inline graphic), the coupled neurons begin to fire. To study the synchronous firing of the two neurons with delayed coupling, we introduce a statistic—similarity function: Inline graphic, where Inline graphic represents the mean with respect to time. Similarity function measures the time relatedness of the two signals Inline graphic and Inline graphic. The smaller the Inline graphic is, the lager the relatedness between Inline graphic and Inline graphic is, i.e., in-phase synchronization of the two coupled system becoming stronger from the synchronization point of view. Especially, when the coupled systems are complete synchronization, Inline graphic at Inline graphic.

We give the change of Inline graphic in the plane Inline graphic, as shown in Fig. 5a. Obviously, there are some areas in the plane Inline graphic where Inline graphic. So by numerical calculation, the coupled systems arrive at complete synchronization in some delay areas when Inline graphic (see Fig. 5b). Meanwhile, in other delay areas the coupled systems reach to the inverse synchronization, see Figs. 6 and 7.

Fig. 5.

Fig. 5

The change graph of the similarity functions for the chemically coupled HR neurons. a The variation of Inline graphic in the plane Inline graphic, b the variation of Inline graphic with Inline graphic increasing when Inline graphic

Fig. 6.

Fig. 6

The phase graph in the plane Inline graphic of the chemically coupled HR neurons with parameters C = 0.2 and τ = 5, 10, 30, 45, 90 from a to e

Fig. 7.

Fig. 7

The transition of the synchronous firing patterns of system (1) with different delay τ from top to bottom being aInline graphic, complete bursting synchronization, bInline graphic, inverse spiking synchronization, cInline graphic, inverse bursting synchronization, dInline graphic, complete bursting synchronization; eInline graphic, inverse bursting synchronization

Figure 6a, d show that the membrane potentials of the two coupled neurons are completely linear correlated, indicating complete synchronization happens. Whereas, Fig. 6b, c, e show the two coupled neurons reach to inverse synchronization.

Figure 7 shows the synchronous firing patterns for coupling strength Inline graphic. The system (1) becomes oscillating when the coupling strength above the critical value for Hopf bifurcation (Inline graphic). Taking some values of delay, the transition of synchronization can be shown in Fig. 5. It is observed that when the delay Inline graphic, the coupled neurons reach to complete bursting synchronization, as shown in Fig. 7a. When the delay increases to Inline graphic, the complete bursting synchronization transits to inverse spiking synchronization, as shown in Fig. 7b. As the delay increased to Inline graphic, the inverse spiking synchronization transits to inverse bursting synchronization as shown in Fig. 7c. When the delay further increased to Inline graphic, the coupled neurons reach to complete bursting synchronization again, as shown in Fig. 7d. And in the Fig. 7e, i.e., when the delay is as large as Inline graphic, the coupled system returns to inverse bursting synchronization once more. Therefore, the increasing of delay alternatively and regularly switches the synchronous patterns of system (1) between the complete bursting synchronization and the inverse synchronization.

As above, firstly, we can know when the coupling strength is lower than the critical value the coupled systems always retain to be stable no matter how large the delay is, but the opposite hold false when the coupling strength is larger than the critical value, i.e., at this time, the coupled systems are always excitable. Secondly, when the neurons are coupling, the synaptic strength between actual neurons will increase to tend a stable value, which will keep its stability under certain conditions without drastic changes of the environment. Otherwise, great numbers of synchronous phenomena such as the spread of the virus in epidemiological network and congestions in the transmission of the signals are all influenced by time-delay. Therefore we mainly discuss the effect of time-delay in the synchronization.

Conclusion

In this paper, based on the excitable HR systems with chemical delayed coupling,we analytically investigated the conditions for the Hopf bifurcations by means of bifurcation theory. We numerically gave the supercritical and subcritical Hopf bifurcations which agree with the analytical ones. We also numerically investigated the synchronization transition of the firing patterns and found the complete synchronization and the inverse synchronization changed alternatively with the delay increasing. The reason for the synchronization transition of the firing pattern maybe relate to the supercritical and subcritical Hopf bifurcations which also occur alternatively as the delay increasing. We will give detailed research in the future work.

A certain delay can destroy the complete synchronization of coupled excitable neurons, and as the delay increasing, it can prohibit the inverse synchronization from arising in some particular time region. Moreover, we can control the c varying in a small region to inhibit the system’s excitability. As we know that phase synchronization and synchronization transitions have become more and more important because of their physiological and pathological significance, so the results may be helpful for us to forecast the dynamical behavior of coupled excitable neurons and provide a theoretical direction for developing medical treatment of restraining the synchronization of neurons, and further understand the essence of encoding and decoding of information.

Acknowledgments

The Project was supported by the NSFC (Grant No.10902001 and 11072013).

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