Emergence of spiral wave induced by defects block

Commun Nonlinear Sci Numer Simulat 18 (2013) 1665–1675

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Commun Nonlinear Sci Numer Simulat

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Emergence of spiral wave induced by defects block

1007-5704/$ - see front matter � 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cnsns.2012.11.016

⇑ Corresponding author.E-mail address: [email protected] (J. Ma).

Jun Ma a,⇑, Qirui Liu b, Heping Ying c, Ying Wu d

a Department of Physics, Lanzhou University of Technology, Lanzhou 730050, Chinab School of Transportation, Wuhan University of Technology, Wuhan 430063, Chinac Department of Physics, Zhejiang University, Hangzhou 310027, Chinad School of Aerospace, Xian Jiaotong University, Xian 710049, China

a r t i c l e i n f o

Article history:Received 17 May 2012Received in revised form 21 November 2012Accepted 21 November 2012Available online 12 December 2012

Keywords:Spiral waveDefectsFactor of synchronizationNetwork of neurons

a b s t r a c t

The development of spiral wave in the regular network of Hindmarsh–Rose neurons withnearest-neighbor connection is investigated under no-flux and/or periodical boundary con-dition, respectively. At first, specific initial values are selected to detect the formation ofspiral wave, it is found that the developed state is controlled by the bifurcation s, whichcontrols the electric activity of neuron from spiking to burst behavior, and different devel-oped states are observed. Furthermore, the formation of ordered wave induced by thedefect block in the network of neurons is also investigated. In the numerical studies, thefirst step is to generate target-like wave by imposing a discrepant forcing current (not peri-odical signal) on a local region, the second step is to produce an artificial defect by settingthe variables in a local area to zero. The supposed defect is used to block the propagation oftarget wave in the network, and the condition for spiral wave emergence is investigated in anumerical way. It indicates that the existence of defect in the media accounts for the emer-gence of spiral wave though most of the previous works used to simulate the developmentof spiral wave by using specific initial values. A statistical factor of synchronization in thetwo-dimensional space is defined to detect the appearance and robustness of spiral wavein the network of neurons. It is confirmed that the contour profile of the developed patternis dependent on the intensity of coupling and defects block.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Spiral wave is a class of peculiar pattern far from the equilibrium state in spatiotemporal system [1,2] and the arm num-ber often depends on the media property and control condition. Spiral waves can be detected in realistic media, for example,the chemical Belousov–Zhabotinsky reaction system [3], CO oxidation on Pt (110) surface [4,5], propagation of electric wavein cardiac tissue [6–8], intracellular calcium waves in Xenopus oocytes [9–11] and intercellular calcium waves in brain slices(Hippocampal Slice Cultures) [12,13] etc. It was reported that the appearance and breakup of spiral wave in the cardiac tissuecan account for arrhythmia and fibrillation [14], and then many interesting works have been presented to discuss thedynamics of spiral wave and many schemes are proposed to suppress the spiral waves in the media [15–21]. For example,Hildebrand et al. [15] investigated the statistical properties of topological defects in reaction–diffusion system. Zhou et al.[21] discussed the noise-induced coherent patterns in a subexcitable system in theoretical and experimental way. Yangand Zhang [22] gave interesting measurement about the critical size for supporting spiral wave in oscillatory media. Zhanet al. reported the phase synchronization of spiral wave and formation of spiral wave in bilayer system by analyzing thedynamics and high-frequency dominance [23,24]. Gao et al. [25] and Zhang et al. [26] studied the dynamics of inwardly

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1666 J. Ma et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1665–1675

rotating spiral wave in nonuniform excitable media and drift dynamics of two-armed spirals by imposing periodic advectivefield and periodical forcing on the media. More interesting, the attractive results in Ref. [27] are helpful for curing cardiacdefibrillation. Tang et al. [28] presented a useful active scheme to control spiral turbulence in the media. Chen et al.[29,30] investigated the control of spiral wave and turbulence in inhomogeneous excitable media. Some other interestingworks about spiral wave in reaction–diffusion system could be referred to Refs. [31–37] and the references therein.

The formation and control of spiral wave in the two-dimensional array of coupled array and network of neurons was alsodiscussed [38–42]. For example, a perfect spiral wave in a coupled array of Chua circuit [38] was induced to measure thedynamics of spiral wave in coupled oscillators, and then an intermittent scheme [39] was used to control the spiral waveand spatiotemporal chaos in the two-dimensional coupled Chua circuit with nearest neighbor connections. He et al. [40] dis-cussed the development of spiral wave in small-world network with inhomogeneous property, and it was confirmed thatlong-range connection can cut down the effect of inhomogeneity of media on the formation of ordered wave. Woo et al.[41] presented detailed discussion about the formation of spiral wave in the network of sine-circle maps in a numericalway. Perc [42] studied the effect of noise on the formation of spiral wave in the network of neurons in the small-world net-work, and then the stochastic resonance and coherence resonance [43–45] in network were also confirmed. Gu et al. [46]investigated multiple induction of spiral waves with a stochastic signal in a square lattice network model composed of typeI Morris–Lecar (ML) neurons with nearest connection type. Erichsen et al. [47] discussed the multistability in networks ofHindmarsh–Rose neurons with nearest-neighbor connections. Wang et al. [48] presented some interesting results aboutthe delay on the coherence of spiral wave in network of Hodgkin–Huxley neuron with noise being considered. The authorof this paper ever investigated the phase transition and robustness of spiral wave in network of neurons in the presenceof additive noise and channel block [49–51].

The application of voltage-sensitive dye imaging is helpful to observe the formation of spiral wave in rat neocortical slicesin an experimental way, and it was thought that spiral waves might serve as emergent population pacemakers to generateperiodic activity in a nonoscillatory network without individual cellular pacemakers [52,53]. In fact, spiral wave also playsactive role in coordinating oscillation phases over a population of neurons and binding sensory information or dynamicaltemporal storage in working memory. Some interesting works about the development and selection of spiral wave in net-work of neurons have been reported [53–59]. For an example, Schiff [53] observed that oscillatory episodes from isotropicpreparations in the middle layers of a mammalian cortex, which can display irregular and chaotic spatiotemporal waveactivity. For example, Wilkinson and Metta [55] studied the possible functional roles for spiral wave activity in visual cortex.Ma et al. [59] detected the effect of ion channel block on the transition of spiral wave in the small-world network of Hodg-kin–Huxley neurons. However, the effect of defects on the development of spiral wave in the networks could be an interest-ing topic to be investigated. There are many reliable common neuron models as reviewed in Ref. [60], and these commonneuron models can reproduce main properties of electric activities and dynamics of neurons. In this paper, we will discussthe formation and transition of spiral wave due to the block of defects in the two-dimensional array, which the dynamics ofeach node is described by a three-variable Hindmarsh–Rose neuron model [61,62] and each node is connected with nearest-neighbor connection type (regular network), and the periodical boundary, no-flux boundary condition will be used,respectively.

2. Ordered wave in two-dimensional network of neurons

The three-variable Hindmarsh–Rose model [61,62] for a single neuron is often defined by

_x ¼ y� ax3 þ bx2 � zþ Ie; _y ¼ c � dx2 � y; _z ¼ r½sðxþ vÞ � z�; ð1Þ

where a; b; c; d; r; s, and v are parameters and the overdot means the variable to time derivative. The variable xðtÞ representsthe difference of electrical potentials across the neuron’s membrane, and variable y (t) is associated with the fast current(Naþ or Kþ) and considered as a recovery variable, the variable z (t) is a slow adaptation current probably linked to Ca2þ.The parameter b controls the switching between bursting and spiking behaviors and allows to control the spiking frequency,parameter r is the rate of change of the slow variable z, in the case of spiking behaviors, the spiking frequency is dependenton the value r, while the number of spikes per burst is determined by the value r in the case of bursting behaviors. Chaoticstate could be observed in this model for a ¼ 1; b ¼ 3; c ¼ 1; d ¼ 5; s ¼ 4; r ¼ 0:006 , v ¼ 1:6; I e ¼ 3. A two-dimensionalregular network with nearest-neighbor connection is described by

dxij=dt ¼ yij � ax3ij þ bx2

ij � zij þ Ie þ Dðxiþ1j þ xi�1j þ xijþ1 þ xij�1 � 4xijÞ ð2Þdyij=dt ¼ c � dx2

ij � yij

dzij=dt ¼ r½sðxij þ vÞ � zij� ð3Þ

where the constant D is coupling intensity between neurons in the nodes of network, xij defines the membrane potential inthe node (i; j). For simplicity, neuron in each node will be imposed the same external forcing current I e without special state-ment and all the neurons are identical thus can hold the same parameters. To measure the collective behavior and phasetransition of waves in the network of neurons, a statistical factor of synchronization in two-dimensional space is definedin Refs. [51,59] according to the mean field theory.

Fig. 1. Distribution for factors of synchronization R vs. bifurcation parameter s under no-flux boundary condition boundary (a), the snapshots forthe developed pattern under different bifurcation parameters s (b), (c) the tip motion of stable rotating spiral wave in Fig. 1(b) (s = 3.5), at fixedforcing current I ¼ 1:315, the coupling intensity D ¼ 2, transilient period t ¼ 1000 time units. The initial values are selected as follows:xð91 : 93;1 : 100Þ ¼ yð91 : 93;1 : 100Þ ¼ 2; zð91 : 93; 1 : 100Þ ¼ �1:0; xð94 : 96;1 : 100Þ ¼ yð94 : 96; 1 : 100Þ ¼ zð94 : 96; 1 : 100Þ ¼ 0:0; xð97 : 99; 1 : 100Þ ¼yð97 : 99;1 : 100Þ ¼ �1; zð97 : 99;1 : 100Þ ¼ 2:0; xðijÞ ¼ �1:31742; yðijÞ ¼ �7:67799; zðijÞ ¼ 1:12032 for the rest sites.

J. Ma et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1665–1675 1667

F ¼ 1N2

XN

j¼1

XN

i¼1

xij ð4Þ

R ¼ hF2i � hFi2

1N2

PNj¼1

PNi¼1 hx2

iji � hxiji2� � ð5Þ

where the N2 represents the total number of neurons in the network and the factor of synchronization is defined with R.xij denotes the sampled measurable variable in the network. In this paper, the output series for membrane potential ofneurons xij will be calculated to approach the corresponding factor of synchronization. As reported in Refs.[51,59], asmaller value for factor of synchronization is necessary to support the spiral wave, that is to say, factor of synchroniza-tion often holds smaller value when spiral wave is well developed to occupy the network of neuron. More generally, in anumerical way, certain emboliform initial values of variables are often used to generate perfect spiral wave in the reac-tion–diffusion system, coupled oscillators in array and/or network. For example, Figs. 1 and 2 show the distribution offactor of synchronization and snapshots for the developed spiral wave under no-flux and periodical boundary condition,respectively. In the following section, we will investigate the effect of defects with different sizes on blocking the targetwave in the network, and find the critical condition for supporting spiral wave, and the tip position of spiral wave willbe tracked if possible.

3. Numerical results and discussions

In this section, the coupled nonlinear equations for the network will be carried out by using explicit Euler algo-rithms, the time step Dt ¼ 0:02, forcing current Ie ¼ 1:315, coupling intensity along horizontal axis and vertical axisD ¼ Dþ ¼ D� ¼ 2; a ¼ 1; b ¼ 3; c ¼ 1; d ¼ 5; r ¼ 0:006; v ¼ 1:6 without special statements. At first, we will investigatethe development of spiral wave at fixed appropriate initial values (for generating spiral seed), no-flux and periodicalboundary conditions will be considered, respectively. In Fig. 1, the case for no-flux boundary is measured andplotted.

Fig. 2. Distribution for factors of synchronization R vs bifurcation parameter s under periodical boundary condition boundary (a), the snapshots forthe developed pattern under different bifurcation parameters s (b), (c) the tip motion of stable rotating spiral wave in Fig. 1(b) (s = 3.5), at fixedforcing current I ¼ 1:315, the coupling intensity D ¼ 2, transilient period t ¼ 1000 time units. The initial values are selected as follows:xð91 : 93;1 : 100Þ ¼ yð91 : 93;1 : 100Þ ¼ 2; zð91 : 93;1 : 100Þ ¼ �1:0; xð94 : 96;1 : 100Þ ¼ yð94 : 96;1 : 100Þ ¼ zð94 : 96;1 : 100Þ ¼ 0:0; xð97 : 99;1 : 100Þ ¼yð97 : 99;1 : 100Þ ¼ �1; zð97 : 99;1 : 100Þ ¼ 2:0; xðijÞ ¼ �1:31742; yðijÞ ¼ �7:67799; zðijÞ ¼ 1:12032 for the rest sites.


The curve in Fig. 1 shows that the factor of synchronization could be changed by the bifurcation parameter s greatly, thespatiotemporal pattern in the network is adjusted with increasing the value of parameter s. Turbulent state often emerges atsmaller parameter s, spiral wave could be induced under moderate parameter s and smaller factor of synchronization is oftenapproached, then homogeneous state is reached with higher bifurcation parameter s being used. The potential mechanismcould be that the adaptation [64] of single neuron is controlled with increasing the value of bifurcation parameter s. It is con-firmed that moderate parameter s is helpful to induce and develop stable rotating spiral wave in the network of neuronsunder the no-flux boundary condition. And the tip position as shown in Fig. 1(c) confirmed that the induced spiral wave ro-tates stably in the network. Furthermore, the time series of average membrane potentials of neurons in the network are cal-culated and dominant periodicity is observed when spiral wave is developed to occupy the network. The time series willdecrease to stable value when spiral waves (homogenous state) are removed, while time series will appear irregular behav-ior for the turbulent state.

In fact, no-flux boundary condition could be suitable for networks with finite size and the external effect near the bound-ary is often omitted for a larger number of nodes in the network. Therefore, it is important to investigate this case underperiodical boundary case. In Fig. 2, it gives the results for developing spatiotemporal pattern under periodical boundary con-ditions and the same initial values will be used as well.

The results in Fig. 2 show that the factor of synchronization is dependent on the bifurcation parameter s, smaller factor ofsynchronization is often approached when the spatiotemporal pattern is in ordered state. In the region of our numerical re-sults, no perfect spiral wave is developed to occupy the network completely, spiral waves just occupy some area in the net-work and the tip of spiral waves are not in regular orbit. Compared the results in Fig. 2 and Fig. 1, the potential mechanismcould be that the boundary effect accounts for the difference for the final state because the spiral seed is affected by the peri-odical boundary greatly though it develops well in the case of no-flux boundary.

Above all, we discussed the case that the variables of neurons in local area are give with specific initial values to producespiral seed and develop spiral wave under no-flux and periodical boundary conditions. However, it just gives a convenientway for numerical test. As it is known, the origin of spiral wave depends on the defects in the media. For example, electric

Table 1The factor of synchronization vs. defect size with a transient period about 1000 time units, no-flux boundary condition is used.

ny ¼ 100; nx ¼ 1 2 3 4 5 6 7

Coupling intensity D = 2.0; R= 0.000808 0.000756 0.000715 0.000674 0.000672 0.000583 0.000667Coupling intensity D = 2.5; R= 0.002002 0.001938 0.001771 0.001696 0.001540 0.001588 0.001414Coupling intensity D = 3.0; R= 0.002131 0.002220 0.001729 0.002249 0.001488 0.002179 0.001604


signal from the sinus node in heart generates target-like wave and spiral wave could be induced when these target waves areblocked by the defects in the cardiac tissue. As a result, it is important to simulate the process of spiral wave development inthe network of neurons due to the effect of defects block.

In the following numerical studies, the bifurcation parameter is selected as s ¼ 4:0. The problem about blocking targetwave by using defects is described as follows:

(1) Target wave condition is required. In the region 80 6 i 6 85;80 6 85, the forcing current on each node is selected asIe ¼ 2:63 and Ie ¼ 1:315 for the rest nodes in this network. Due to the effect of coupling among neurons, target-likewave is formed in the network. The position or region for generating target wave can select other regions in thenetwork and two different constant forcing currents are powerful to induce target wave in a local area. Surely, itcould also be effective to generating target wave with pinning scheme by imputing periodical signals into local areaof the network. However, it is difficult to select the amplitude and angular frequency of the external periodical forc-ing because the developed traveling wave induced by the defects block will compete with the target wavecompletely.

(2) Defects measurement. For simplicity, in the region i0 6 i 6 100;1 6 j 6 100, the variables in the nonlinear coupledequations as shown in Eq. (1) are set to zero, then a stripe-like area with 100�ð100� i0Þ nodes is set as a defects area.The size of defects block is controlled by selecting different integers i0, in a simple way, it marks nx ¼ 100� i0;ny ¼ 100and the node number in the defect area is nxny.

3.1. Development of spiral wave under no-flux boundary condition

At first, we investigate the case by selecting different coupling intensities under bo-flux boundary condition. The size ofthe defects block area is selected with different values, for example, the coupling intensity Dþ ¼ D� ¼ 2:0;2:5;3:0, the factorsof synchronization are listed in Table 1 as follows:

Table 1 show that smaller factors of synchronization could be approached when the coupling intensity is selected withsmaller value. The factors of synchronization are within certain region 0.000583 � 0.000808, it just indicates that the devel-oped state is similar to each other and spiral wave or ordered state could be approached. The snapshots are plotted in Fig. 3to give a visual understanding. Furthermore, we also check this problem by selecting other coupling intensities, and the re-sults are shown in Figs. 4 and 5, respectively.

It is found that the developed spiral wave is cut into two segments when the coupling intensity is small, and the areaoccupied by the target wave is small when the area of defects block is small, then the intensity of coupling is increased,and the results are shown in Figs. 4 and 5.

Clearly, the developed spiral wave becomes sparse when stronger intensity of coupling is used, and the tip motion orbit issimilar to each other when spiral wave is well developed to occupy the network completely.

The results in Figs. 3–5 confirmed that the size of defects and coupling intensity are critical to develop spiral wave in thenetwork of neuron when no-flux boundary condition is used. Perfect spiral wave could be induced to occupy the networkcompletely at fixed bigger intensity of coupling, otherwise, the spiral wave will be cut into multi-arms such as two-armor three-arm spiral wave etc. Spiral waves are also induced even if the area of defects is very small though the developedspiral wave is imperfect. The tip motion orbit is similar to each other when the stable rotating spiral wave is induced to oc-cupy the network completely, and it is also found that the factor of synchronization is close to each other when spiral wave isobserved in the network of neurons. To detect the boundary effect of network, we also studied this problem under periodicalboundary condition.

3.2. Development of spiral wave under periodical boundary condition

It is claimed that periodical boundary condition could be better than the no-flux boundary condition in the measurementof collective behaviors of neurons because the neurons in each domain of living systems could be affected by the other neu-rons close to the border of the network or domain investigated. As a result, we also calculated the distribution for factors ofsynchronization and the results are listed in Table 2 as follows.

The results in Table 2 show that all the factors of synchronization are close to each other with an order about 10�3, and nodistinct jump for these values, it just indicates that spiral wave could be developed due to the effect of defect block, and thenthe snapshots are plotted as shown in Figs. 6–8 to give a clear illustration and understanding.

Fig. 3. The developed state for nx ¼ 1 (a), nx ¼ 2 (b), nx ¼ 3 (c), nx ¼ 4 (d) at t ¼ 1000 time units and ny ¼ 100, coupling intensity D ¼ 2, the upright striparea marks the defect area and the snapshots are shown in color, the motion of tip for spiral seed or spiral wave is marked with purple. (For interpretation ofthe references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. The developed state for nx ¼ 1 (a), nx ¼ 2 (b), nx ¼ 3 (c), nx ¼ 4 (d) at t ¼ 1000 time units and ny ¼ 100, coupling intensity D ¼ 2:5, the upright striparea marks the defect area and the snapshots are shown in color, the motion of tip for spiral seed or spiral wave is marked with purple. (For interpretation ofthe references to colour in this figure legend, the reader is referred to the web version of this article.)



Table 2The factor of synchronization vs. defect size with a transient period about 1000 time units, periodical boundary condition is used.

ny ¼ 100; nx ¼ 1 2 3 4 5 6 7

Coupling intensity D = 2.0; R= 0.002414 0.002307 0.002151 0.002168 0.002223 0.002028 0.002190Coupling intensity D = 2.5; R= 0.006326 0.007138 0.006570 0.005446 0.007784 0.007196 0.007127Coupling intensity D = 3.0; R= 0.003514 0.003681 0.003243 0.003716 0.003191 0.003885 0.003370


The results in Fig. 6 confirmed that spiral wave could be induced to occupy the network and the target wave coexists withthe spiral wave as well.

The results in Fig. 7 showed that perfect spiral wave is also developed to occupy the network but the spiral wave becomessparse than the ones as shown in Fig. 6.

The results in Figs. 6–8 confirmed that spiral wave could be well developed to occupy the network of neurons, and thespiral wave becomes sparse with increasing the intensity of coupling. It becomes easier to develop perfect spiral wave tooccupy the network by using a bigger intensity of coupling because the generated travelling wave can propagate with a high-er speed to diffract from the defect area. The tip motion of spiral wave is similar to each other when the defect size is chan-ged under the same coupling intensity. There are some differences between the case for no-flux and periodical boundarycondition, for example, in the case of no-flux boundary condition, the developed spiral wave keeps perfect in the coreand border of the network. On the other hand, the developed spiral wave is invaded by the border and the spiral wave isnot smooth in the contour profile under periodical boundary.

In a summary, spiral wave in the network of neurons could be developed by the defects block, and the target wavecould coexist with the developed spiral wave though the spiral wave suppress the target wave in a small local area.Under the periodical boundary condition, spiral wave is easy to be induced by the defects. The developed spiral wavecan diffract the block area and shows certain robustness, and the contour profile of spiral wave can be controlled bychanging the defect size and coupling intensity as well. It could give a potential way to understand the formation ofspiral wave in the excitable media. Furthermore, we also checked the scheme on the network of Hodgkin–Huxley neu-ron model [63], and similar results are approached. That is to say, the scheme is independent of the neuron modelselection.

Fig. 6. The developed state for nx ¼ 1 (a), nx ¼ 2 (b), nx ¼ 3 (c), nx ¼ 4 (d) at t ¼ 1000 time units and ny ¼ 100, coupling intensity D ¼ 2, the upright striparea marks the defect area and the snapshots are shown in color, the motion of tip for spiral seed or spiral wave is marked with purple. (For interpretation ofthe references to colour in this figure legend, the reader is referred to the web version of this article.)





4. Conclusions

In this paper, the formation and development of ordered wave in the network of neurons were investigated in theoreticaland numerical way. In a general way, specific initial values are often selected in the cleat type to make a spiral seed and thusspiral wave could be induced under appropriate parameter conditions. In fact, spiral wave could be induced by blocking thetravelling wave in the media or network. Some appealing results are approached as follows.

(1) A statistical variable defined as factor of synchronization in the two-dimensional network is used to detect the appear-ance of spiral wave and it is confirmed that smaller factor of synchronization is associated to the survival of spiral wave.

(2) The boundary effect is discussed. In the case of developing spiral wave by selecting appropriate initial values, perfectspiral wave could be induced by the bifurcation parameter s under no-flux boundary condition. The developed turbu-lent state, spiral wave and homogeneous state just depend on the selection of bifurcation parameter s, which controlthe adaptation of neuron and the electric activities. However, in the case of periodical boundary condition, no perfectspiral wave could be formed to occupy the network completely but just occupy some region of the network, moreinteresting, target-like state could be observed as well.

(3) Stripe-like defect area is constructed to study the block effect on the target wave, it is found that stable rotating spiralwaves could be developed to occupy the network, the spiral wave can pass through the block area and the contourprofile could be changed by the coupling intensity, a perfect single-arm spiral wave is induced as shown inFig. 5(d) and Fig. 8(d), multi-arm spiral waves is developed when smaller coupled intensity is used.

(4) In the case of periodical boundary condition, spiral wave is easy to emerge due to the defects block. The formation ofspiral wave induced by defects block is independent of the selection of boundary condition. These results could behelpful to understand the formation of spiral wave in the biological systems.

Acknowledgments

This work is partially supported by the National Nature Science of Foundation of China under Grant No. 11265008 (J. Ma).

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Emergence of spiral wave induced by defects block