Ma Jun et al- Breakup of Spiral Waves in Coupled Hindmarsh-Rose Neurons

8/3/2019 Ma Jun et al- Breakup of Spiral Waves in Coupled Hindmarsh-Rose Neurons

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CHIN.PHYS.LETT. Vol. 25, No. 12(2008) 4325

Breakup of Spiral Waves in Coupled Hindmarsh-Rose Neurons

MA Jun( ) , JIA Ya( ) , TANG Jun( ), YANG Li-Jian( )1 Department of Physics, Central China Normal University, Wuhan 430079

2

School of Sciences, Lanzhou University of Technology, Lanzhou 730050

(Received 3 September 2008)Breakup of spiral wave in the Hindmarsh-Rose neurons with nearest-neighbour couplings is reported. Appropriate initial values and parameter regions are selected to develop a stable spiral wave and then the Gaussian colourednoise with different intensities and correlation times is imposed on all neurons to study the breakup of spiral wave, respectively. Based on the mean eld theory, The statistical factor of synchronization is dened to analyse the evolution of spiral wave. It is found that the stable rotating spiral wave encounters breakup with increasing intensity of gaussian coloured noise or decreasing correlation time to certain threshold.

PACS: 47.54. r, 05. 45. a

Extensive theoretical and experimental studies onring pattern in neurons have been carried out by es-tablishing some reasonable and biophysical models [1,2]

in recent years. Most of the previous works focusedon the topics of stochastic resonance [3,4] pattern ring,the functions of neurons, communicating signal amongneurons, and synchronization and bifurcation analysisof neurons etc. [5 11] It is such a complex process forreceiving and transmitting signal among neurons thatmany problems are still open to be investigated. Asis known, the neuronal system of our body consists of enormous neurons and it is interesting to study thecoupling action and pattern transition in the presenceof noise so that it can pave a way to know the in-formation encoding in the neurons. More frequently,the interspike intervals (ISIs)of few neurons are easyto be observed and measured but it could be diffi-cult to detect the ISIs of enormous neurons synchron-ically. Therefore, it is more important to investigatethe evolution of spatial ring pattern and nd easyway to identify the transition of ring pattern in thespatiotemporal system, which can be described by thecoupled oscillators or reaction-diffusion system.

In a theoretical way, spiral wave could be simulatedand observed in the reaction-diffusion equation and

the coupled oscillators numerically. As we know, spi-ral wave in two-dimensional spatial space is observedin the physical, biological and chemical systems. [12 15]

Especially, the electrical activity in cell [16] can occurspiral wave in the cardiac tissue, which may indicatea kind of heart disease and the breakup of spiral wave just causes a rapid death of heart. There are muchsimilarities between the spiral wave in the excitablemedia and the cardiac tissue. It is interesting to studythe synchronization of neurons and the development

and transition of ring pattern in the coupled neuronsin two-dimensional space. Up to date, the mechanismof instability [17 22] of the spiral wave has been inves-tigated extensively, and many schemes have also beenproposed to eliminate or remove the spiral wave in thereaction-diffusion system. [23 29] Furthermore, noise-induced transitions of patterns in the excitable me-dia have attracted much attention, for example, noisecan induce instability of spiral wave. [30] For simplic-ity, the Gaussian white noise is often used to studythe transition of pattern in the excitable media andspatial-extended system. In fact, it is more practicalto describe the effect of noise with the coloured noiseby introducing the factor of correlation time insteadof the Gaussian white noise.

There are many famous neuron models, and theHindmarshRose (HR) neuron [1] is investigated in thisstudy. In our works, nearest-neighbour coupling inthe HR neurons with appropriate coefficient and pa-rameters are used to develop a stable rotating spiralwave and the initial values will be offered availably.Then the factor of synchronization is dened and theGaussian coloured noised will introduced into all theneurons to study the breakup of spiral wave. The dis-tribution for factor of synchronization and snapshots

of membrane voltage will be plotted to discuss thebreakup of spiral wave.

It is interesting and important to study the ac-tion and role of noise in the biological and neuronalsystem because the neurons often are sensitive to theexternal noise. It is more practical to simulate theuncertain uctuation in the spatiotemporal system byintroducing coloured noise into the system rather thanthe Gaussian white noise. The widely used colourednoise[31] Q(t) is often produced from the Gaussian

Supported by the National Natural Science Foundation of China under Grant Nos 10747005, 30670529 and 10875049, and theKey Project of MOE under No 108096.

Email: [email protected] Email: [email protected] 2008 Chinese Physical Society and IOP Publishing Ltd
http://www.iop.org/journals/cplhttp://www.cps-net.org.cn/http://cpl.iphy.ac.cn/


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white nose ( t), and the statistical correlation is oftendescribed by

Q(t) = 0 , Q(t)Q(t ) =D 0

e

| t t | , (1)

where D0 is the intensity and is the correlationtime. It describes the case for Gaussian white noise at 0. Clearly, the Gaussian coloured noise can be de-cided by the intensity and the correlation time. Theactivator is more sensitive than the inhibitor to theexternal noise, therefore, the coloured noise is addedto the right side of the formula which describes theevolution of the activator. In this study, we intro-duce a Gaussian coloured noise into the coupled HRneurons and study the instability of spiral patternwith different intensities and correlation times, respec-tively. There is complex mutual coupling and interac-tion among the neurons, and it is important to denesome statistical variables to measure the correlationof all the neurons or sites. It is convenient to calcu-late the amplitude error of few neurons. It could beacceptable to investigate the correlation of enormousneurons based on the mean eld theory. Therefore, anew scale factor is proposed to study the instabilityof spiral wave even though the snapshots showing isavailable. A mean eld F and factor synchronization[32] are dened to discuss the evolution of the voltageof all the neurons. The mean eld F and the factor of synchronization R can be described by

F =1

N 2

N

j =1

N

i =1

V ij = V s , (2)

R =F 2 F 2

1N 2

N

j =1

N

i =1

( V 2ij V ij 2), (3)

where the variable V ij describes the activator of thesystem and V s is the mean value of all neurons. HereV ij represents the variable x ij in the coupled HR neu-rons, which is given in the following. Instant N 2 is thetotal number of all neurons. In the case of completesynchronization of all the neurons, all the neurons willbe the same amplitude and the error of each pair of neurons will equal to zero. As a result, the factorof synchronization is close to zero as well. In otherwords, the factor of synchronization will equal to zerowhen the whole media becomes homogenous, and allthe neurons become complete synchronization. Fur-thermore, it just indicates that these neurons reachgeneral synchronization when the factor of synchro-nization is above and beyond zero.

The three-variable HR neuron model of action po-tential has been proposed as a mathematical represen-tation of the ring behaviour of neurons. [1] Our aim is

to study the instability of spiral wave in the coupledHR neurons with nearest-neighbour couplings [33 35]

by introducing Gaussian coloured noise into all theneurons. When the Gaussian coloured noise is im-posed on the whole neuronal system, the coupled HRneurons with nearest-neighbour coulings are often de-scribed by

dx i,j /dt = yi,j ax 3i,j + bx2i,j zi,j + I

+ D(x i +1 j + x i 1j + x ij +1+ x ij 1 4x ij ) + Q(t),

dyij /dt = c dx2i,j yi,j ,dzi,j /dt = r (s(x i,j + ) zi,j ), (4)

where the variable x used to dene the membrane ac-

tion potential, variable y is associated with the fastcurrent ( Na + or k+ ) and considered as a recoveryvariable. The variable z is a slow adaptation cur-rent maybe linked to Ca 2+ . Parameters a, b,c,d, ands are often invariable, and parameter r is a propor-tional constant, while parameter can be regardedas the reversal potential of Ca 2+ , I is the exter-nal current and D is the coupling coefficient. Q(t)is the Gaussian coloured noise and parameter k isswitched with 0 or 1 to close or open the noise tothe neurons. These parameters are often selected witha = 1 .0, b = 3 .0, c = 1 .0, d = 5 .0, s = 4 .0, = 1 .6 and

different intensities of external current I can inducesingle neuron to become chaotic or periodical and theparameters r and are often treated as bifurcationparameters to analyse the transition of ring patternfor few neurons.

t/t/

Fig. 1. Formation of spiral wave in Hindmarsh-Rose neu-ron. The snapshots of membrane voltage are shown ingray. Here x (91 : 93 , 1 : 100) = 2 .0, y (91 : 93 , 1 : 100) =2.0, z (91 : 93 , 1 : 100) = 1.0; x (94 : 96 , 1 : 100) =0.0, y (94 : 96 , 1 : 100) = 0 .0, z (94 : 96 , 1 : 100) = 0 .0;x (97 : 99 , 1 : 100) = 1.0, y (97 : 99 , 1 : 100) = 1.0, z (97 :99, 1 : 100) = 2 .0; and other neurons are selected withx ( i, j ) = 1.31742 , y ( i, j ) = 7.67799 , z ( i, j ) = 1 .13032.

In our numerical simulation tests, no-ux bound-ary condition is used to nd appropriate solutions forEq. (1), the total number of neurons is N 2 = 200 200and time step h = 0 .02 is used. A stable rotating spiralwave could be developed within 400 time units using


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coupling coefficient D = 0 .5 and I = 1 .315 (or D = 1,D0 = 0 .001 and = 2 .0). A stable rotating spiralwave is plotted in Fig. 1.

The gaussian coloured noise with different intensi-ties and correlation times is imposed on all the cou-pled HR neurons and the factor of synchronizationand membrane voltages in different sites are calcu-lated numerically. In Fig. 2 we show the evolution of synchronization of factor R vs intensity of noise, andin Fig. 3, we illustrate the evolution of synchronizationfactor R vs correlation time.

0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8

0 . 0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8

0 . 0 0

0 . 0 5

0 . 1 0

0 . 1 5

0 . 2 0

( b )

= 1

= 2

= 3

= 4

= 5

R

( a )

= 7

= 1 0

= 1 5

= 2 0

= 2 5

R

I n t e n s i t y o f n o i s e D

0

Fig. 2. Evolution of synchronization factor vs intensity of coloured noise.

Fig. 3. Evolution of synchronization factor vs correlationtime of coloured noise.

The curve in Fig. 2 shows that the factors of syn-chronization begin to increase with increasing inten-sity of coloured noise and some peaks could be founddue to the stochastic resonance when the colourednoise works on all the coupled neurons.

The curve in Fig. 3 illustrates that the factors of synchronization begin to decrease with increasing cor-relation time and certain peaks and sudden turningpoints are also found when the coloured noise is im-posed on all the neurons. Furthermore, the distri-bution for factor of synchronization vs intensity andcorrelation time of noise is plotted in Fig. 4.

D

Fig. 4. Distribution for synchronization factor vs correla-tion time of coloured noise.

D =0.03, =3, t =100 D =0.03, =3, t =300

D =0.04, =15, t =100 D =0.04, =15, t =300

Fig. 5. Evolution of spiral wave when gaussian colourednoised works on all the neurons, the upper snapshots areshown with D 0 = 0 .03 for correlation time = 3 and thebottom snapshots are plotted with D 0 = 0 .04 for correla-tion time = 15.

Within the vicinity of dense region about inten-sity D0 = 0 .03, the factor of synchronization beginsto change suddenly which indicates sudden transitionof spiral wave, that is breakup of spiral wave. In com-parison of the results in Figs.24, it is conrmed thatsmaller correlation time and/or stronger intensity of gaussian coloured noise easier cause breakup of spiralwave, and the sudden changing point in the curve for


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factor of synchronization just indicates the transition.Finally, the snapshots of membrane voltage in differ-ent sites with intensity D0 = 0 .03 for correlation time = 3 and D 0 = 0 .04 for correlation time = 15 areplotted in Fig. 5, which shows the breakup of spiralwave.

The snapshots in Fig. 5 conrm that breakup of spiral could be induced when the gaussian colourednoise is imposed on all the neurons. Smaller correla-tion or stronger intensity usually cause the breakup of spiral wave.

The numerical results conrm that the breakup of spiral wave can occur when the intensity of colourednoise surpasses a upper threshold or the correlationtime is decreased to a bottom threshold. Finally, itis important to discuss the mechanism of the breakupof the spiral wave in the coupled HR neurons. In thiswork, the Gaussian coloured noise is imposed on theright side of the formula which decides the evolutionof activator x. From the point of nonlinear dynamics,the inputting of coloure noise just changes the exter-nal current as I = 1 .315+ Q(t). As mention above, thesingle HR neuron could become chaotic with the in-creasing intensity of external current and it could beconrmed by calculating the spectrum of Lyapunovexponent. The coloured noise is imposed on each neu-ron, all the neurons become chaotic due to nearest-neighbour coupling when the gaussian coloured noisewith appropriate intensity and/or correlation time isimposed on the coupled neurons. The excitable neu-rons response to the external stimuli from noise sen-sitively and it could encounter global instability syn-chronically, as a result, the breakup of spiral wave isobserved.

In summary, we have reported the formation andbreakup of spiral wave in the HR neurons with com-plete nearest-neighbour couplings. The factor of syn-chronization is dened to discuss the transition of spi-ral wave when the gaussian coloured noise in intro-duced into all the neurons. It is found that the spiralwave encounters breakup when the correlation time isdecreased or the intensity of coloured noise is increasedto a certain threshold. The factor of synchronizationis also found to be increased with increasing intensityof coloured noise and/or decreasing correlation timeand peaks are observed in the curve for factor of syn-chronization due to stochastic resonance. It is alsoconrmed that the rst sudden changing point in thecurve R D0 (factor of synchronization vs intensityof noise) indicates the threshold for breakup of spiralwave and it could be conrmed by observing the snap-shots. It is also interesting and challenging to study

the problem when the neurons are connected withother types, for example, small-world connection, [36]

and these interesting problems could be carried out inour forthcoming works in detail.

We would give great thanks to G. Yu for usefuldiscussion.

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Ma Jun et al- Breakup of Spiral Waves in Coupled Hindmarsh-Rose Neurons