Development of spiral wave in a regular network of excitatory neurons due to stochastic poisoning of ion channels

Commun Nonlinear Sci Numer Simulat 18 (2013) 3350–3364

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Development of spiral wave in a regular network of excitatory neurons due to stochastic poisoning of ion channels Xinyi Wu a, Jun Ma a,⇑, Fan Li a, Ya Jia b a b

Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China Department of Physics, Huazhong Normal University, Wuhan 430079, China

a r t i c l e

i n f o

Article history: Received 18 December 2012 Received in revised form 8 May 2013 Accepted 12 May 2013 Available online 24 May 2013 Keywords: Spiral wave Target wave Channel noise Defects Morris–Lecar neuron

a b s t r a c t Some experimental evidences show that spiral wave could be observed in the cortex of brain, and the propagation of this spiral wave plays an important role in signal communication as a pacemaker. The profile of spiral wave generated in a numerical way is often perfect while the observed profile in experiments is not perfect and smooth. In this paper, formation and development of spiral wave in a regular network of Morris–Lecar neurons, which neurons are placed on nodes uniformly in a two-dimensional array and each node is coupled with nearest-neighbor type, are investigated by considering the effect of stochastic ion channels poisoning and channel noise. The formation and selection of spiral wave could be detected as follows. (1) External forcing currents with diversity are imposed on neurons in the network of excitatory neurons with nearest-neighbor connection, a target-like wave emerges and its potential mechanism is discussed; (2) artificial defects and local poisoned area are selected in the network to induce new wave to interact with the target wave; (3) spiral wave can be induced to occupy the network when the target wave is blocked by the artificial defects or poisoned area with regular border lines; (4) the stochastic poisoning effect is introduced by randomly modifying the border lines (areas) of specific regions in the network. It is found that spiral wave can be also developed to occupy the network under appropriate poisoning ratio. The process of growth for the poisoned area of ion channels poisoning is measured, the effect of channels noise is also investigated. It is confirmed that perfect spiral wave emerges in the network under gradient poisoning even if the channel noise is considered. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Neurons can generate different kinds of electric activities and signals are communicated among neurons under complex coupling. Normal physical response and activities of body depend on the collective behaviors of neurons in different domains in the neuronal system. Many theoretical models [1–6] have been proposed to study the dynamical properties of neuron and synchronization among neurons. Particularly, stochastic and/coherence resonance [7–28] in neurons can throw light on understanding the information encoding and response to external forcing in neurons. The electric activities of neurons or collective behaviors of neuronal activities become regular due to stochastic or coherence resonance. Noise often plays an important role in inducing stochastic or coherence resonance [29,30], while the spatiotemporal dynamics of neurons can often be changed by time delay, operational modes and other factors [31,32]. Interestingly, spatial coherence resonance often

⇑ Corresponding author. Tel.: +86 15009310193. E-mail addresses: [email protected], [email protected] (J. Ma). 1007-5704/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2013.05.011

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occurs in the network of neurons, and beautiful patterns emerge. For example, spiral wave are induced in the network of neurons due to the multiple spatial coherence [33–35]. The brain cortex contains a large number of neurons, a fraction of neurons keep active in the form of spiking, bursting, while most of the neurons keep silent as asleep. Complex spatiotemporal pattern [36–42] is formed by marking the electric activities of neurons in space. For example, spiral wave could be observed in the cortex of brain [43,44], and the dynamics of spiral wave in neocortex was discussed [45]. Some regular population oscillations like target waves can also emerge in the network. For example, Lewis et al. [46] showed that a self-organizing process can generate regular population oscillations in a network with random spontaneous activity and random gap junction-like coupling, and the network activity underlying the oscillations was topologically similar to target-pattern activity. The author of this paper even confirmed that local periodic forcing can generating target wave in the network of neuron [42], and these powerful target waves are effective to suppress the spiral wave in the network. Spiral wave often emerges in the excitable and oscillatory media, and the formation of spiral wave was simulated by using some theoretical models [47–52]. For instance, reaction–diffusion models have been very helpful in reproducing spiral wave. Winfree turbulence of scroll waves is a special kind of spatiotemporal chaos that emerges exclusively in a three-dimensional excitable media [53], and is considered as one of the principal mechanisms of cardiac fibrillation. As a result, some effective schemes were presented to suppress these turbulences, for example, Alonso et al. [54] confirmed that the turbulence can readily be controlled by weak nonresonant modulation of the medium excitability. Furthermore, it was also confirmed that the scroll wave can be controlled by using a spatially distributed random forcing superimposed on a control parameter of the system [55]. In the two-dimensional case, it was claimed that the appearance of spiral wave in cardiac tissue could be associated with a kind of heart disease called as arrhythmia, and the breakup of spiral wave could induce repaid death of the heart [56]. Interestingly, Weise1 et al. [57] presented a new mechanism of spiral wave initiation by investigating the effect of deformation on the vulnerability of excitable media in a discrete reaction–diffusion-mechanics model, which combined the FitzHugh–Nagumo type equations for cardiac excitation with a discrete mechanical description of a finite-elastic isotropic material to model cardiac excitation–contraction coupling and stretch activated depolarizing current. Alonso et al. [58] even derived a simple effective medium theory for spatially heterogeneous nonlinear reaction–diffusion media, and its validity is detected through comparisons with simulations of front and pulse propagation in systems with spatially varying diffusion coefficients and reaction rates. Many studies have been carried out in this field, and some results are expected to give practical guidance to cure or prevent the occurrence of ventricular fibrillation [59–65]. For example, Gong et al. [63] developed antispiral wave in reaction– diffusion system. Bursac et al. [64] detected the emergence of spiral wave with multiarms in a two-dimensional cardiac substrate. Zhang et al. [65] investigated the development of spiral wave in subexcitable media by imposing appropriate electric field, and it is confirmed that an isolated broken plane wave retracting originally in subexcitable media can propagate continuously and eventually evolve into a rotating spiral wave. Alonso et al. [66] discussed the relation of negative filament tension, tissue excitability, and the effects of discreteness in the tissue on the filament tension, and the differences in the onset of arrhythmias in thin and thick tissue was explained. In most of the previous works, the formation of spiral wave in the media often depends on the special initial values as spiral seed, or noise is used to induce the spiral wave in subexcitable media. In fact, spiral wave can be generated by blocking a target wave, which can be induced by local periodical forcing or external forcing with diversity [67]. In cardiac tissue, spiral wave can be developed when the emitted electric wave from the sino-atrial node is blocked by abnormal tissue such as ischemia and heteromorphosis. The abnormal state in a local area could play as defects, the dynamics of traveling waves could be changed by these defects, and its main properties could be detected by imposing artificial defects in the media. Indeedly, it is more interesting to explore the potential cause for the emergence of these defects in a neuronal system. In this paper, a simplified neuron model will be used to construct a regular network, which each node embedded in the space is connected with nearest-neighbor type, and the development, transition of spiral wave due to artificial defects and ion channels blocking in a random way will be investigated, respectively. The Morris–Lecar Model [68–73] is an abstraction of the Hodgkin–Huxley [1] Model that has two state variables: the voltage within the neuron, and a potassium gating variable. Two types of neuronal excitability can be observed in Morris–Lecar Model, and the excitability of the neurons starts to oscillate at an arbitrarily low frequency were characterized as type I, while those starting at a specific non-zero frequency as type II [68,69]. The bifurcation dynamics corresponding to type I (and type II) excitabilities was elucidated to be a saddlenode bifurcation (and a Hopf bifurcation) [68]. In addition, the effect of channel noise will also be considered. The contents of this paper are arranged as follows. In Section 2, the neuronal model is described, the scheme for developing target wave by imposing forcing currents with diversity is explained, and the concept and realization of artificial defects are presented, the degree of ion channels blocking or poisoning is defined, the description about poisoned area with irregular border lines. In Section 3, numerical results are presented, and gradient poisoning criterions are defined. In Section 4, the conclusions and our results are summarized.

2. Model and schemes Neurons could be connected in small-world type and regular type. In the case of regular connection, each neuron is coupled with the nearest-neighbor neurons, while in the case of small-world type, an additional long-range connection with certain probability is also considered based on a local regular connection. In this section, the case for regular connection

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[74–76] in network is considered. The dynamical description of a Morris–Lecar (ML) network of neurons embedded in a twodimensional space [33] is given by

C

dV ij ¼ g L ðV ij  V L Þ  g Ca M 1 ðV ij ÞðV ij  V Ca Þ  g K N ij ðV ij  V K Þ þ Iij þ DðV i1j þ V iþ1j þ V ij1 þ V ijþ1  4V ij Þ dt

ð1aÞ

dNij ¼ kN ðV ij Þ½N1 ðV ij Þ  Nij  dt

ð1bÞ

  V ij  V 1 M1 ðV ij Þ ¼ 0:5 1 þ tanh V2

ð2aÞ

  V ij  V 3 N1 ðV ij Þ ¼ 0:5 1 þ tanh V4

ð2bÞ

kN ðV ij Þ ¼ kN cosh

  V ij  V 3 2V 4

ð2cÞ

where the variable Vij represents the membrane potential of neuron in node (i, j) with the physical unit mV, Nij denotes the gate variable for potassium in node (i, j), gCa represents the conductance associated to Calcium current, gK measures the conductance associated to Sodium current, while gL is a constant leak conductance, and the corresponding maximal conductance max are marked with g max ; g max , respectively. VCa, VK, and VL are the corresponding reversal potentials, parameter C (lF/ L Ca ; g K 2 cm ) defines the capacitance for membrane of neuron, Iij (lA/cm2) measures the external forcing current on the neuron in node (i, j), D is the coupling intensity between any nearest-neighbor neurons in the regular network embedded in a twodimensional array. The conductances of the potassium and calcium channels vary in a sigmoidal way with the membrane voltage Vij, and the dependence is defined in Eq. (2a), Eq. (2b).kN is a time constant associated to the ratio between the faster and slower variable, and it is often independent of the voltage. V1, V2, V3, V4, are considered as adjustable parameters to reproduce the main properties of electric activities of biological neurons. For more detailed meanings about these parameters, readers can refer to the Refs. [68,69]. The development and transition of spiral wave in the network under blocking in ion channels in a diffusive and random way are investigated as follows. 2.1. Ordered wave is developed At first, a target-like wave is induced by imposing external forcing currents with diversity on the network. Target-like wave could be generated by introducing constant current I1 in a local area in the network, while the other nodes are driven with another constant current I2. 2.2. Properties of ordered wave are detected The wave length of the developed wave (target wave or spiral wave) is estimated by detecting the time series of nodes in a row or line in the network. For simplicity, in the regular network with 400  400 nodes, the time series of membrane potentials of neurons in the nodes (i = 300, j = 1:400) are observed and recorded, the velocity and length of wave could be estimated by detecting the interval time for two successive wave fronts reaching the nodes (i = 300, j = 1:400) from the wave source or center (Fig. 1). 2.3. Ion channels poisoning or blocking The conductance is often changed when the ion channels are blocked, as a result, the modified conductance is given by [77]

g k ¼ g max xk ; k

g ca ¼ g max ca xca

ð3Þ

where xk(and xCa) represents the poisoning ratio of working ion channels of Potassium (and Calcium) to the total ion channels of Potassium (and Calcium). A smaller value for xk(and xCa) denotes that only a fraction of ion channels are working, while a bigger xk(and xCa) means that many ion channels are active and open. Our studies focus on the problem about poisoning in ion channels of potassium, and the ion channels of sodium are open and working. 2.4. Artificial defects The membrane potentials of neurons in a local area are selected with constants, for simplicity, the membrane potentials are set as zero in the local area, thus artifical defects area could be generated.

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Fig. 1. Ordered wave propagates outwardly (a) and inwardly (b). Time series for membrane potentials of neurons in nodes (i = 300, j = 1:400). The velocity of wave could be estimated according to (300–200)/t1, and wave length is estimated according to (300–200)(t2–t1)/t1.

2.5. The process of ion channels poisoning is described in a random (stochastic) way In this process, the size (or number) of poisoned area for neurons is increased and the neurons closed to the poisoned area is poisoned in a stochastic way. In realistic models, ion channels of more and more neurons are poisoned but reach to an upthreshold. In our numerical studies, 160,000 identical neurons are placed on the nodes in two-dimensional array with regular connections uniformly; the maximal upthreshold for poisoned neuron number is selected as 60,000 (Fig. 2). The dynamical equation for the gate variables in the presence of channel noise is often described as follows

dNij ¼ kN ðV ij Þ½N1 ðV ij Þ  Nij  þ nij ðtÞ dt

ð4Þ

While the membrane potentials of neurons are still measured by Eq. (1a), n(t) is the independent Gaussian white noise and the statistical properties [78] are given by

Fig. 2. Schematic diagram of the poisoned area for neurons in ion channels poisoning state. The adjacent neurons will be included into the poisoned area with certain probability (about p = 0.333 in our numerical studies), ion channels in the neuron in the center have been poisoned.

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hnij ðtÞnm;l ðtÞi ¼ cðV ij ; N ij Þdðt  t 0 Þdi;m dj;l ¼ 2Dnoise dðt  t 0 Þdi;m dj;l ;

hnij ðtÞi ¼ 0;

cðV ij ; Nij Þ ¼

1 kN ðV ij Þ½ð1  2N1 ÞNij þ N1  N0

ð5aÞ ð5bÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Dnoise is the intensity of noise, the random variable n(t) is defined aspnffiffiffiffiffi¼ffi cðV; NÞDB. N0 is the number of working and open channels, DB is regarded as an increment for Wiener process (DB ¼ Dt Z, Z  N(0,1)). d(⁄) represents the Dirac-d function. As a result, a single stochastic ML model is redefined as follows

V nþ1 ¼ V n þ

1 ðg Ca m1 ðV n ÞðV n  V Ca Þ  g K ðV n  V K Þ  g L ðV n  V L Þ þ IÞDt C

Nnþ1 ¼ Nn þ ½kN ðV n ÞðN1 ðVÞ  Nn ÞDt þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cðV n ; Nn ÞDtZ

ð6aÞ ð6bÞ

For any node (i, j) in the network of ML neurons, the Eqs. (6a), (6b) could be rewritten by imposing subscript (i, j) on the variable Nn, Nn + 1, Vn, Vn + 1, I in Eqs. (6a), (6b). 3. Numerical results and discussions In this section, the conductance and parameters in the network of neurons are selected as gL = 2.0, gca = 4.0, gk = 8.0, VL = 60.0, VCa = 120.0, VK = 80.0,V1 = 1.2, V2 = 18.0, V3 = 12.0, V4 = 17.4, C = 5.0, kN = 1/15. Type II excitability for ML neurons [68,69] is approached in this parameter region. The time step is 0.01, the regular network is constructed by 400  400 nodes, where neurons are placed on nodes uniformly with nearest-neighbor connection, and no-flux boundary condition is used without special statements. At first, different external forcing currents with diversity are used to induce a perfect target-like wave in the network, the sampled time series of membrane potentials of neurons in a line are recorded, and the profile of target-like wave under different coupling intensity, external forcing currents with diversity are detected and discussed. The membrane potentials of neurons in different nodes could be different at fixed time, thus the color scale in space is different because membrane potential in each node could be in much difference. The snapshots (Fig. 3–16) will illustrate the distribution of membrane potentials of neurons in the network, the difference in color bar depicts the difference of membrane potentials. Fig. 3 shows that a perfect target wave could be induced to occupy the network under appropariate forcing currents with divesity, and target wave becomes sparser when a stronger coupling intensity D is used in the network. The time series in Fig. 3(a1–d1) also confirm that the developed ordered wave will propagate with a higher velocity under a stronger coupling intensity D. Furthermore, we also check this case at fixed coupling intensity D = 4 by changing the values of external forcing currents with diversity, and the results are plotted in Fig. 4. The results in Fig. 4 show that the profile of the target wave is also dependent on the selection of external forcing currents (I1, I2)with diversity (I1I2). It is found that the target wave becomes dense when higher diversity (I1I2) is used, otherwise, a sparser target wave will be induced to occupy the network when the diversity (I1I2) is selected with smaller value. The results in Fig. 4(a1–d1) confirm that the transient period for two sucessive wave fronts reaching the sampled nodes (i = 300,

Fig. 3. Developed target-like wave at t = 1200 time units in the left panel, for D = 1(a), D = 2(b), D = 3(c), D = 4(d); the right panel depicts the time series of membrane potentials for neurons in nodes(i = 300, j = 1:400) for D = 1(a1), D = 2(b1), D = 3(c1), D = 4(d1); external forcing current I1 = 50 is used for nodes at 195 6 i 6 200, 195 6 j 6 200, the other nodes are imposed with I2 = 40. The color bar that shows the membrane potential.

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Fig. 4. Developed target-like wave at t = 1200 time units in the left panel, external forcing current I1 is used for nodes at 195 6 i 6 200, 195 6 j 6 200 for I1 = 50(a), I1 = 60(b), I1 = 70(c), I1 = 80(d); the right panel depicts the time series of membrane potentials for neurons in nodes(i = 300, j = 1:400); the other nodes are imposed with I2 = 40, coupling intensity D = 4.

j = 1:400) is shorter when the target wave is dense, and the velocity is much dependent on the coupling intensity D. Clearly, the developed states could depend on the selection of external forcing current with diversity. Therefore, it is important to investigate the effect of diversity from external forcing currents (I1, I2). For simplicity, we will study this problem by changing the size which the forcing current I1, I2 is imposed, respectively. The results are plotted in Fig. 5.

Fig. 5. The developed states are plotted in the left panel, and the right panel depicts the time series of membrane potentials of neurons under different forcing currents with diversity. Couping intensity D = 2, I2 = 40, I1 = 50, the forced area for current I1 is marked as 198 6 i 6 200, 198 6 j 6 200(a,a1); 197 6 i 6 200, 197 6 j 6 200 (b,b1); 196 6 i 6 200, 196 6 j 6 200 (c,c1); 195 6 i 6 200, 195 6 j 6 200(d,d1) .

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Fig. 6. Developed traveling wave induced by the artificial defects at t = 1200 time units. The defects area is marked as 1  200(a), 2  200(b), 3  200(c), 4  200(d), coupling intensity D = 2.

Fig. 7. The developed sprial wave due to the interaction between the target wave and the defects-induced wave at t = 2000 time units, coupling intensity D = 2, the intensity of forcing currents with diversity is set as I1 = 60 for the nodes (193 6 i 6 200, 193 6 j 6 200), I2 = 40 for the other nodes in the network; the defects area is marked as 1  200(a), 2  200(b), 3  200(c), 4  200(d).

The results in Fig. 5 show that stable target wave is induced when the size of stimulated area with forcing current I1 is beyond certain value, no distinct difference can be observed among the developed target waves when the same coupling

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Fig. 8. The developed wave at t = 1200 time units under ion channels poisoning and external forcing currents with diversity. The poisoning ratio xk = 0.2(a), xk = 0.3(b), xk = 0.4(c), xk = 0.5(d), coupling intensity D = 2, external forcing current I1 = 80 is imposed on the nodes for 193 6 i 6 200, 193 6 j 6 200, and the other nodes are imposed with current I2 = 40.

intensity and forcing currents with diversity are used. The time series in Fig. 5 also confirm that the interval period of the two successive wave fronts reaching the sampled nodes will be close to each other when the size of forced area with current I1 is increased to certain value, and the profile of the target wave is similar with each other. Then we investigate the case for traveling wave induced by artificial defects, and the results are shown in Fig. 6. The results in Fig. 6 show that traveling wave can be induced by defects and the developed wave begins to propagate in the network even no external forcing currents with diversity being imposed. Extensive numerical results confirm that the developed wave induced by the defects can also occupy the network completely within certain long transient period about 2000 time units. To explore the formation mechanism of the spiral wave, external forcing currents with diversity and artificial defects are considered in the network of neuron. And it is found that spiral wave can be generated when the target wave, which is induced by external forcing currents with diversity, competes with the traveling wave induced by the artificial defects, and the results are shown in Fig. 7. The results in Fig. 7 confirm that the developed spiral wave can occupy the network completely. The potential mechanism could be that the developed target wave, which induced by the external forcing currents with diversity, is broken by the defects-induced traveling wave, thus spiral wave is indcued within certain transient period. It is also found that the profile of the spiral wave is indpendpent of width of defects area, but the center of the spiral wave depends on the length of the defects area. Some experiments [79,80] in neuronal cultures have confirmed that the coupling strength between neurons can affect the propagation velocity of an activity front, for instance by applying nerve-growth factors that increase neuronal connectivity. Extensive numerical results confirm that spiral wave becomes sparse when the intensity of coupling is

Fig. 9. The developed state when the target wave (induced by the external forcing currents with diversity) and the plane wave (induced by the poisoned area) at t = 1200 time units. I1 = 80 for the nodes 191 6 i 6 200, 191 6 j 6 200 and I2 = 40 for the other nodes in the network; coupling intensity D = 4 and the poisoning ratio xk = 0.3(a), xk = 0.4(b), xk = 0.5(c), the maximal number of poisoned neuron is 60,000.

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increased, particularly, no perfect spiral wave can be observed in the network when the coupling intensity is beyond certain threshold. Indeed, the membrane potential keeps stable value and no distinct fluctuation can be observed when the ion channels of neurons are poisoned badly. As a result, it is interesting to measure the collective behavior of eletric activities of neurons when ion channels of a large number of neurons are poisoned and blocked. In a realistic neuronal system, ion channels in membrane patch can be blocked by some drugs, the ions (Ca2+, K+) can not pass through the membrane and then the intracellular and extracellular voltage media is stabilized because the intracellular and extracellular ion concentration keeps stable. As a result, it is reasonable to investigate the wave propagation when ion channels of a fraction of neurons are poisoned in the network. Some results are plotted in Fig. 8. The results in Fig. 8 confirm that stable rotating spiral wave can be induced to occupy the network completely under appropriate poisoning ratio xk = 0.4, otherwise, the target wave induced by the external forcing currents with diversity and the plane wave induced by the poisoned area will coexist with each other. In fact, the border lines for poisoned area could be irregular in realistic neuronal system because of certain stochastic factors. According to the criterion presented above, we will investigate this problem when the size of poisoned area is enlarged in a random way. In our numerical studies, the transient period for observing the development of spiral wave is about 1200 time units without special statement. For simplicity, the interval time for next adjacent neurons to be poisoned is about 0.3 ms, which enables the poisoned area with certain size could be formed. Some results in this case are plotted in Fig. 9. The results in Fig. 9 show that the target wave is broken and the network becomes irregular in spatiotemporal pattern, the potential mechanism is that no ordered wave (plane wave) could be induced by the poisoned area due to the irregularity of the border lines for the poisoned area. Spiral wave is also observed in the network as shown in Fig. 9c under appropriate

Fig. 10. The distribution of the poisoned area of ion channels in neurons at t = 1200 time units. Poisoning ratio xk = 0.3(a), xk = 0.4(b), xk = 0.5(c), the maximal number of poisoned neurons is 60,000, the dark blue area marks the poisoned area for neurons. The area marked with a white circle depicts the area supporting the spiral wave as shown in Fig 9c. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. The development of the poisoned area for neurons at fixed poisoning ratio xk = 0.5. For t = 300(a), t = 450(b), t = 600(c), t = 750(d), t = 900(e), t = 1050 time units(f), the maximal number of poisoned neuron is 60,000.

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poisoning ratio xk. To understand the distribution of poisoned area, the poisoned area in the network under different poisoning ratios are plotted in Fig. 10. The results in Fig. 10 show that the border lines of the poisoned area are not smooth and irregular poisoned area is increased vs. time in a random way. Furthermore, the growth process of the poisoned area in a random way is plotted in Fig. 11. The results in Fig. 11 show that a poisoned area with irregular border lines could be induced in the network with certain transient period. And it is found that the poisoned area is formed completely at t = 750 time units when 60,000 neurons are

Fig. 12. The competition between the target wave and the traveling wave induced by the poisoned area. I1 = 80 for the nodes 191 6 i 6 200, 191 6 j 6 200, and I2 = 40 for the other nodes in the network; coupling intensity D = 4, and the poisoning ratio xk = 0.5, for t = 750 time units(a), t = 900 time units(b), t = 1050 time units(c), t = 1200 time units(d), the maximal number of poisoned neuron is 60,000.

Fig. 13. The development of spiral wave under gradient poisoning in ion channels of neurons. The maximal poisoning ratio xk = 0.45, gradient poisoning h = 0.001, I1 = 80 for the nodes 191 6 i 6 200, 191 6 j 6 200 and I2 = 40 for the other nodes in the network; coupling intensity D = 4, for t = 390 time units(a), t = 630 time units(b), t = 810 time units(c), t = 930 time units(d), t = 1050 time units(e), t = 1200 time units(f), the maximal number of poisoned neuron is 30,000.

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included in the poisoned area. It is found that no distinct spiral wave could be formed in the network before t = 750 time units because the poisoned area keeps increasing at t < 750 time units. Now it is interesting to detect the growth and development of spiral wave when the poisoned area-induced wave competes with the target wave induced by the external forcing currents with diversity at t > 750 time units, and the results are plotted in Fig. 12. The results in Fig. 12 show that spiral wave could be induced in the network only when the size of the poisoned area is fixed, otherwise, spiral segments are just generated and driven randomly by the traveling wave induced by the poisoned area or the target wave. Therefore, spiral wave could be induced only when the size of the poisoned area is increased to certain value. In a realistic neuronal system, the poisoning degree could be decreased in certain gradient value when the drug is diffused in living cells. This problem is simplified by considering three or more adjacent neurons in the network as follows.

Fig. 14. The development of spiral wave due to the interaction between target wave and the traveling wave induced by the poisoned area with irregular border lines. The poisoning ratio xk = 0.4, I1 = 80 for the nodes 191 6 i 6 200, 191 6 j 6 200 and I2 = 40 for the other nodes in the network; coupling intensity D = 4, for t = 250 time units(a), t = 450 time units(b), t = 750 time units(c), t = 1050 time units(d), t = 1350 time units(e), t = 1650 time units(f), the maximal number of poisoned neuron is 60,000 and the number of open(working) ion channels is N0 = 1000.

Fig. 15. The development of spiral wave due to the interaction between target wave and the traveling wave induced by the poisoned area with irregular border lines. The poisoning ratio xk = 0.5, I1 = 80 for the nodes 191 6 i 6 200, 191 6 j 6 200 and I2 = 40 for the other nodes in the network; coupling intensity D = 4, for t = 250 time units(a), t = 550 time units(b), t = 950 time units(c), t = 1350 time units(d), t = 1650 time units(e), t = 2000 time units(f), the maximal number of poisoned neuron is 60,000 and the number of open(working) ion channels is N0 = 1000.

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(1) Neuron (i1, j1) is selected with poisoning ratio xk1; (2) Neuron (i2, j2) is selected with poisoning ratio xk2; (3) Neuron (i3, j3) is selected with poisoning ratio xk3 = 1(no ion channel poisoning);

xk1 < xk2 ; xk1 < xk3 ¼ 1 (4) The poisoning ratio xk3 becomes xk1 + h when neuron(i3, j3) is infected by the neuron(i1, j1); (5) The poisoning ratio xk3 becomes xk2 + h when neuron(i3, j3) is infected by the neuron(i2, j2); (6) When neuron (i3, j3) is infected by the neuron(i1, j1) and neuron(i2, j2) the poisoning ratio xk3 = xk1 + h if xk1 + h is no more than xk2; xk3 = xk1 if xk1 + h is larger than xk2; where the constant h represents the gradient increment for poisoning degree. For simplicity, the gradient constant h = 0.001. Then the results under gradient constant for poisoning degree are plotted in Fig. 13. The results in Fig. 13 show that distinct spiral wave can be developed to occupy the network when a gradient poisoning is considered. Furthermore, we also investigate the development of spiral wave under ion channel noise, and the results are plotted in Fig. 14 (for xk = 0.4) and Fig. 15 (for xk = 0.5). The results in Fig. 14 confirm that spiral wave is also induced in the network in the presence of channel noise. Spiral waves emerge close to the top end of border lines of the poisoned area, stable rotating spiral waves will emerge in the network only when the border lines of poisoned area are fixed completely. Furthermore, the case when the poisoning ratio is selected as xk = 0.5 is also investigated, and the results are plotted in Fig. 15. The results in Fig. 15 show that spiral waves begin to emerge with increasing the size of the poisoned area, and stable rotating spiral wave is formed in the network in the presence of channel noise when the size of the poisoned area is increased to a fixed value, which a maximal number of neurons about 60,000 are included in this area. It is also interesting to investigate this problem with gradient poisoning being considered, and the results are plotted in Fig. 16. The results in Fig. 16 confirm that perfect spiral wave could also be induced to occupy the network when the effect of channel noise and gradient poisoning are considered. It is found that border lines of the poisoned area in Fig. 16 are not as clear as the ones in Fig. 15 under gradient poisoning, the generated spiral wave begins to grow up when the size of the poisoned area is increased to a fixed value. Compared the results in Fig. 13, Fig. 16, it is found that the gradient poisoning plays an important role in developing perfect spiral wave. In a realistic neuronal systems, the poisoned degree in the target points (area), where the drug is injected, is often higher than the adjacent points to be infected or poisoned, thus a gradient poisoning factor is reasonable to be considered. In summary, ordered wave could be induced by external forcing currents with diversity, artificial defects, and poisoned area. Spiral wave can be induced when the target wave interacts with the traveling wave (induced by the poisoned area), no matter the border lines of the poisoned area is regular or not. Perfect spiral wave emerges in the network under gradient poisoning even if the channel noise is considered. The development and growth of spiral wave

Fig. 16. The development of spiral wave due to the interaction between target wave and the traveling wave induced by the poisoned area with irregular border lines. The maximal poisoning ratio xk = 0.5, gradient poisoning h = 0.001, I1 = 80 for the nodes 191 6 i 6 200, 191 6 j 6 200 and I2 = 40 for the other nodes in the network; coupling intensity D = 4, for t = 250 time units(a), t = 550 time units(b), t = 750 time units(c), t = 950 time units(d), t = 1350 time units(e), t = 1650 time units(f), the maximal number of poisoned neuron is 50,000 and the number of open(working) ion channels is N0 = 1000.

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depends on the competition between the target wave and the traveling wave (induced by the defects or poisoned area). In fact, the target wave is blocked by the border lines of defects of poisoned area, and the broken waves can develop to form a perfect spiral wave, the formation of spiral wave in the network of neurons due to ion channels poisoning or blocking with irregular border lines could be suitable to understand the formation mechanism for the emergence of spiral wave in the cortex of brain. There are some other additional questions that should be clarified in a brief way. (1) Neurons in a realistic system could be connected in different type. For example, long-range connection with certain probability for some neurons could be important to change the collective behaviors of neurons in the network, and some researchers claimed that small-world connection type could be better to study the electric activities of neurons. In this case, the effect of long-range connection could be more important than a simple regular connection, where neurons are coupled with a nearest-neighbor connection. In the case of spiral wave, it is found that local regular connections are in supporting the spiral wave, while the long-range connection often destroys the spiral wave in a homogenous media. Spiral wave breakup occurs when the probability of long-range connection exceeds certain threshold. As reported in Ref. [81], long-range connection in the small-world can decrease the local heterogeneity in the small-world network. (2) In a reaction–diffusion or spatial network [82], the wave front often depends on the coupling intensity, and the profile of the wave is often associated to the coupling intensity, it was found that the wave becomes sparse when a higher coupling intensity is used. For the emergence of spiral wave, no spiral wave can be generated when the coupling intensity exceeds certain threshold. (3) Formation of target wave. In this paper, target wave is generated by imposing external forcing currents with diversity. In fact, this diversity could be associated to local heterogeneity in the network. In fact, neurons in different area of the network respond to external forcing in different degree due to local heterogeneity, thus different transmembrane currents are generated. Therefore, local heterogeneity accounts for the emergence of target wave in the network or media. (4) In the discussion presented above, it just investigates the collective behavior of excitatory neurons, and the effect of inhibitory neurons is left out. That is to say, the model corresponds to a simple scenario of excitatory neurons, and that inhibition is ignored. In the brain and in most of the neuronal tissues there are excitatory and inhibitory neurons. Although excitatory neurons drive activity, inhibitory neurons play an important role in shaping the dynamics of the propagating front. For example, the effect of clustered excitatory connections on the dynamics of neuronal networks that exhibited high spike time variability owing to a balance between excitation and inhibition [83]. Neural circuit process information through the temporal and spatial patterns of their spikes, and these patterns of spikes are surprisingly variable because the firing rates and spike timing are changed due to a balanced excitation and inhibition [84]. It was also confirmed that inhibitory connection offsets excitatory input therefore reduces firing rate and synchrony [85]. Interestingly, in contrast to the conventional belief that inhibition suppresses firing, Ref. [86] reported that a facilitatory role of inhibitory synaptic input that can enhance a neuron’s firing rate. In short, the effect of inhibition on the neurons is multifarious and its role in regulating the collective behaviors of neurons in network deserves further detailed investigation.

4. Conclusions In this paper, formation of ordered wave such as target wave and spiral wave in the network of Morris–Lecar neuron is investigated, respectively. (1) Target wave is induced to occupy the network when external forcing currents with diversity are imposed on the network; the potential mechanism is that transmembrane currents emerge some diversity when the same external forcing current is imposed on the neurons in the network due to heterogeneity in realistic neuronal systems. (2) Target wave could be induced by artificial defects, which the membrane of potentials of neurons in a local area are selected with fixed value. (3) Traveling wave could be induced by the poisoned area in the network of neurons. (4) Target wave is broken by the traveling wave induced by the poisoned area, and thus spiral wave could be induced even if the channel noise is considered. (5) The border lines of the poisoned area are irregular when the size of the poisoned area is increased to a fixed value in a random way, thus the problem of stochastic poisoning in channels is simplified and detected. Compared our results in this paper with the results for network of a simplified neuron model [87,88], we focus on the problem when the poisoned area for ion channels in neurons is selected with irregular border lines, the effect of channel noise, and dynamical diffusive poisoning in ion channels in neurons are considered. These results could be helpful to understand the formation and development of spiral wave in the realistic neuronal systems.

Acknowledgement This work is supported by the National Nature Science of Foundation of China under the Grant No. 11265008.

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