Stability of target waves in excitable media under electromagnetic induction and radiation

Physica A 521 (2019) 519–530

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Physica A journal homepage: www.elsevier.com/locate/physa

Stability of target waves in excitable media under electromagnetic induction and radiation Yin Zhang a , Fuqiang Wu a , Chunni Wang a , Jun Ma a,b , a b

∗

Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China School of Science, Chongqing University of Posts and Telecommunications, Chongqing 430065, China

highlights • • • •

Biological neuron model is improved to consider the effect of electromagnetic induction. Heterogeneity formation is explained on conductance and excitability diversity. Electromagnetic radiation can suppress propagation of target wave. Breakup of target wave can be induced by electromagnetic radiation.

article

info

Article history: Received 24 September 2018 Received in revised form 21 December 2018 Available online 30 January 2019 Keywords: Bifurcation Target wave Electromagnetic induction Memristor Ion channel

a b s t r a c t Local periodic stimulus can trigger stable pulse and target wave in the excitable media, the potential mechanism is that each cell is activated by continuous driving to keep pace with the external forcing. Indeed, local diversity in excitability and heterogeneity are induced when external forcing with diversity is applied on the media. When target wave is formed and propagated, the media will be modulated, e.g. it can emit electric signal from sinoatrial node and develops target wave in the cardiac tissue, and then the heartbeat is controlled completely. In fact, the distribution of electromagnetic field in the media is changed when external electric stimulus is applied on the media; therefore, the effect of electromagnetic induction becomes important during the information encoding and signal propagation. In this paper, the Morris–Lecar model is used to describe the local kinetic of the media and an induction current is considered thus the effect of electromagnetic induction can be described. Then the artificial heterogeneity is triggered to investigate the formation and development of target waves in the excitable media with electromagnetic induction and even radiation, which is described by using Gaussian white noise. Diversity in excitability and conductance of ion channels is considered, respectively, thus different kinds of artificial heterogeneity are developed to find the stability of target waves. It is found that the target wave encounters breakup with increasing of the intensity of electromagnetic radiation. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Target waves and spiral waves can be observed in chemical and biological systems, which show distinct spatial distribution under multistability. For example, the cardiac tissue [1], animal retina [2], chemical reactions [3] can support the propagation of standing target wave and spiral waves, which play an important role in regulating the signal propagation ∗ Corresponding author at: Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China. E-mail address: [email protected] (J. Ma). https://doi.org/10.1016/j.physa.2019.01.098 0378-4371/© 2019 Elsevier B.V. All rights reserved.

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as a pacemaker when the media and networks. These traveling waves can also be detected in the chaotic oscillators [4] of network under coupling. Readers can find relevant evidences in the review and references therein [5]. Some evidences have confirmed that spiral waves in cardiac tissue may block the normal wave propagation emitted from sinoatrial node, as a result, cardiac arrhythmias and even fibrillation [6,7] can be induced to cause potential heartbeat breakdown. The self-sustained spiral wave, which accounts for a topological defect, is often found in other spatiotemporal systems such as Rayleigh–Benard convection [8], the Belousov–Zhabotinsky reaction [9], nonlinear optics [10], and gas discharge system [11]. To decrease and suppress the harmful effect of spiral waves, many schemes have been proposed to prevent the emergence and breakup of spiral waves in the reaction–diffusion systems and network of coupled oscillators [12–17]. Spiral wave and target wave are also reproduced in the neuronal network, and some researchers built different types of networks to discuss the stability and formation mechanism of spiral waves thus the potential biological function of spiral wave in cortex can be understood [18–23]. The synchronization approach and pattern formation on the network can provide an easy insight to understand the collective behaviors of neuronal activities when different types of connection are considered. As is well known, synapse plays an important role in receiving and encoding information, and it also gives an effective way to bridge connection between neurons [24–27]. The complex electrophysiological activities have been detected in cardiac tissue and nervous systems, and nonlinear analysis on these sampled time series for membrane potential can provide insights to understand the occurrence mechanism of neuronal disease [28–34]. Indeed, phase transition and selection for spatial patterns are characterized in electrical activities, density of species concentration of products and reactants. For example, linear stability analysis is applied on an exponential discrete Lotka–Volterra system by [35] to obtain the conditions for the Turing instability and the emergence of spiral patterns is demonstrated as well. A mechanism for the generation of spiral waves in the photosensitive Belousov–Zhabotinsky reaction is explained, and spiral waves appear in the wake of a propagating wave following a local perturbation in illumination intensity [36]. The pattern transition from target waves to spiral waves was also confirmed in chemical reaction systems, and the results show that control behavior is up to the bifurcation parameter from observation [37]. Both spiral waves and target waves can control the collective behaviors of excitable media and network completely by behaving a stable pacemaker. Regular electrical activation waves in cardiac tissue lead to the rhythmic contraction and expansion of the heart that ensures enough blood supply to the whole body. Irregularities in the propagation of these activation waves, which are induced by spiral waves and spiral segments, can result in cardiac arrhythmias, like ventricular tachycardia (VT) and ventricular fibrillation (VF). The occurrence of spiral wave often causes harm on normal signal propagation in nervous system and cardiac tissue. Therefore, many schemes [38–42] have been proposed to prevent emergence and instability of spiral wave in cardiac tissue so that the rate of cardiopathy may be kept at normal level. For example, Shajahan et al. [38] developed a low-amplitude defibrillation scheme for the elimination of VT and VF, especially in the presence of heterogeneity that occurs commonly in cardiac tissue. Panfilov et al. [39] suggested that multiple electrical shocks can be applied to suppress spiral waves in cardiac tissue. Ma et al. [40] confirmed that discontinuous electric shock was also effective to suppress spiral wave in the cardiac tissue. It is found that continuous target wave from sinus node of the heart can spread through the heart conduction system into the two atria and then two ventricles [41]. On the other hand, pinned spiral wave often binds to the heterogeneity and it takes great difficult to remove these harmful waves [42,43]. In fact, it is important to understand potential mechanism of formation and instability [44–47] in spiral wave thus appropriate schemes can be applied to control the spiral wave and prevent its breakup as well [48,49]. As confirmed in Ref. [44], the heart rate can be suppressed under electromagnetic radiation, and target waves develop into spiral waves in local media when the effect of electromagnetic induction is considered in cardiac tissue model. Indeed, cardiac tissue models with heterogeneity would be of great interest to investigate the formation and development of spatial patterns in excitable media thus the wave propagation and electrical response can be understood from dynamical view. In fact, heterogeneity in nature affects collective behaviors and pattern formation is spatiotemporal systems. For example, spiral wave unpinning [49] is found in the heterogeneity of a cardiac tissue by setting the boundary condition, and this heterogeneity can initiate or suppress spatiotemporal patterns under different circumstances. Therefore, it is interesting to discuss the pattern formation in excitable media with heterogeneity when the effect of electromagnetic induction is considered. On the other hand, time-varying electromagnetic field can be triggered in cell when the spatial distribution of charged ions is changed during the flow of charges and fluctuation of membrane potential. As presented in Refs. [50–52], induction current is used to describe the effect of electromagnetic field when magnetic flux is coupled with membrane potential via memristor. Furthermore, Ge et al. [53] discussed the mode selection of neuronal activities when different frequencies of electromagnetic radiation are applied on an isolate neuron. Diversity in conductance of ion channels under poisoning can modulate the electrical activities of neurons, and then Xu et al. [54] investigated the effect of channel blocking on stochastic Hodgkin–Huxley neuron with electromagnetic radiation. In this paper, the biological Morris– Lecar model [55,56] is improved to investigate the effect of electromagnetic induction and then electromagnetic radiation is applied to study the stability of target waves, which are induced by heterogeneity resulting from diversity in excitability and conductance. 2. Model setting and scheme approach The FitzHugh–Nagumo model and its revised version are often used to describe the electrical activities in cardiac tissue. Considering the effect of ion channels and benefits for fast calculation, the Morris–Lecar (ML) model is used for improvement

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so that electromagnetic induction can be considered. The generic ML model is described as follows

⎧ ∂V ⎪ = −Iion + Iext + D∇ 2 V ⎨C m ∂t ⎪ ⎩ ∂w = 0.03 w∞ (V ) − w ∂t τ∞ (V )

(1)

where V is a transmembrane potential, w represents the ionic conductivity as the recovery of variable in excitable media, and ∇ 2 is a Laplace operator and used to calculate the diffusion in the media. Iext is the external forcing current, D measures the coefficient of diffusion as 1 cm2 /ms, and Cm denotes the membrane capacitance and it is often selected as 1 µF/cm2 . Iion is current flowed through the axon membrane with capacitor Cm , it is calculated by Iion = gCa m∞ (V )(V − VCa ) + gK w∞ (V − VK ) + gL (V − VL )

(2)

where gCa , gK , and gL denotes the conductance for calcium ion, potassium ion, and leakage current, respectively. VCa , VK , and VL are resting potentials connected with different variables. m∞ (V ), w∞ (V ), and τ∞ (V ) are important functions depended on membrane potential, and these time-varying parameters are approached by m∞ (V ) = 0.5(1 + tanh

V − V1 V2

); w∞ (V ) = 0.5(1 + tanh

V − V3 V4

); τ∞ (V ) = (cosh

V − V3 V4

)−1

(3)

The parameters for conductance functions and resting potentials are fixed at V1 = −1.2 mV, V2 = 18 mV, V3 =12 mV, V4 = 17.4 mV, VL = −60 mV, VCa = −120 mV, VK = −80 mV. The values for ionic conductance is selected as gCa = 4.0 mS/cm2 , gK = 8.0 mS/cm2 , and gL = 2.0 mS/cm2 , respectively. According the law of electromagnetic induction, a timevarying electromagnetic field may be triggered when variant ionic currents pass through ion channels embedded into the membrane. The magnetic flux presents the time-varying electromagnetic field as far as possible. Inspired by Refs. [44,52], which the spatiotemporal dynamics is investigated in cardiac tissue under electromagnetic induction, the same effect of electromagnetic induction can be considered on the biological ML model with ion channel. That is, the important electric device, the memristor [57,58] is used to bridge the magnetic flux and membrane potential and induction current is effective to calculate the effect of electromagnetic induction in the cell. The nonlinear memductance for memristor is described by

ρ (ϕ ) =

dq(ϕ ) dϕ

= α + 3βϕ 2

(4)

where q(ϕ ) is the memristor constitutive relation and ϕ denotes magnetic across memristor, the parameters α , β are often selected as α = 0.2, β = 0.3 in some previous works. As a result, the improved ML neuron model for cardiac tissue is defined as follows ⎧ ∂V ⎪ = −Iion + Iext + k0 ρ (ϕ )V + D∇ 2 V Cm ⎪ ⎪ ⎪ ∂ t ⎪ ⎨ ∂w w∞ (V ) − w = 0.03 (5) ⎪ τ∞ (V ) ⎪ ∂t

⎪ ⎪ ⎪ ⎩ ∂ϕ = k V − k ϕ 1 2 ∂t

where the nonlinear term k0 ρ (ϕ )V denotes induction current across the flux-controlled memristor, and it describes the coupling between membrane potential and magnetic flux in media. When the Laplace operator is considered on onedimensional space, it describes the case on cable-like neuron and the signal propagation is estimated on chain network. On the other hand, the diffusion effect is calculated on two-dimensional space such as square array which signal can be propagated in the form of target wave, spiral wave or plane wave. In numerical study, the media is often cut on the array with N × N nodes uniformly. A statistical synchronization factor [5,19] R is defined by using the mean field theory thus the stability and synchronization degree can be estimated as follows R=

⟨F 2 ⟩ − ⟨ F ⟩2 1 N2

∑N

i=1,j=1

2

⟨Vi,j ⟩ − ⟨Vi,j ⟩

2

;F =

1 N2

N ∑ (

Vi,j

)

(6)

i=1,j=1

where the N 2 denotes the number of nodes in network, the symbol ⟨∗⟩ represents the average value over time. The synchronization factor R∼1 means that perfect synchronization is reached in two-dimensional array spaces and the network tends to become homogeneous. While the synchronization factor R∼0 denotes that this network presents non-perfect synchronization and smaller synchronization factor is helpful to support pattern formation and selection. As is well known, the excitability of media is dependent on the external stimulus. As a result, diversity in external forcing can cause gradient excitability and thus target-like wave can be developed to occupy the media. In fact, this kind of diversity in excitability just generates local heterogeneity in the media. On the other hand, diversity in conductance of ion channels can also induce local heterogeneity and continuous wave fronts can emit from this local heterogeneity. For simplicity, channel blocking and poisoning in ion channels can be applied to generate diversity in channel conductance. A diagram about heterogeneous area is plotted as Fig. 1.

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Fig. 1. The blue area represents the size of heterogeneity. Wi is the width and length of rectangle Area I (i[90:95], j[90:95]), respectively.

Fig. 2. Target pattern is developed by applying gradient external stimulus at t = 200, 400, 600, 800, 1000, 2000 time units. The external forcing Iext = 45 mA is imposed on local nodes of square area [i(90:95), j(90:95)], while Iext = 40 mA is applied on other nodes. The snapshots are shown color scale.

As mentioned above, gion represents the ion channel conductance, such as gCa , gK , and gL respectively. A heterogeneity is developed in local area when the diversity in ion channel conductance between Area I and Area II occurs. The heterogeneity is estimated by calculating ∆gion = gion (Area I) − gion (Area II), and it describes the outward (gion > 0) or inward (gion < 0) direction in heterogeneity, respectively. 3. Numerical results and discussion The membrane potential series for all nodes are calculated by using the Euler forward algorithm with time step h = 0.01, and no-flux boundary condition is used when the media is discretized on the two-dimensional space with 200 × 200 nodes. The initial values for all nodes in the network are selected as (u, v, ϕ ) = (1.0 + Random, 0.1 + Random, 0.01 + Random), and it sets Random in the scope [0, 1]. At first, the development of spatiotemporal pattern is detected when the parameters are selected as k0 = −0.1, k1 = 0.01, k2 = 0.1, α = 0.2, β = 0.07. Gradient excitability is generated by applying external stimulus with diversity, and the development of spatial patterns in the media is plotted in Fig. 2.

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Fig. 3. Sampled time series of membrane potential on the node (100,100) by applying different feedback gains k0 . (a) k0 = −1.0; (b) k0 = −0.1; (c) k0 = 0.0; (d) k0 = 0.01. Parameters are fixed as k1 = 0.01, k2 = 0.1, α = 0.2, β = 0.07.

Fig. 4. Bifurcation diagram of ISI (interspike interval) via varying the feedback gain k0 for the induction current. Parameters are set as k1 = 0.01, k2 = 0.1, α = 0.2, β = 0.07.

It is found in Fig. 2 that target wave is triggered and developed to occupy the media by applying gradient external forcing current in presence of electromagnetic induction. Continuous wave fronts are emitted from the heterogeneity and the modulated media shows distinct periodicity. Due to the diversity in the initials setting, the involvement of magnetic flux and induction current can enhance the diversity in spatial excitability of media, and the developed target cannot suppress spatial irregularity completely. With the increase of diffusion coefficient D, the diffusion can be enhanced and target wave can be propagated more quick thus the network can be occupied completely within shorter transient period. Furthermore, the sampled membrane potential series on node (100,100) are calculated by selecting appropriate coupling gain k0 , the results are presented in Fig. 3. It is found in Fig. 3 that negative feedback in the induction current (k0 < 0) can decrease the excitability thus periodic wave propagation can be supported. While the electrical activities are suppressed when positive feedback gain in induction current is applied which higher excitability blocks the wave propagation in the media. To observe the dynamical characteristic of cardiac tissue under electromagnetic induction, the bifurcation diagram is plotted with increase of feedback intensity k0 , and the results are show in Fig. 4. The bifurcation diagram in Fig. 4 confirms that multiple peaks can be found in the sampled time series for membrane potentials. And it can present different firing patterns in electrical activities of cardiac tissue under appropriate feedback

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Fig. 5. Developed spatial pattern is plotted at t = 800 time units under different gradient gap for calcium channel conductance. (a)∆gCa = 0.0 mS/cm2 ; (b) ∆gCa = −0.4 mS/cm2 ; (c) ∆gCa = 0.4 mS/cm2 ; (d) ∆gCa = 6.0 mS/cm2 . The external forcing current is set at Iext = 40 mA, calcium channel conductance is fixed as gCa = 4.0 mS/cm2 in Area II, and parameter are selected as gK = 8.0 mS/cm2 , gL = 2.0 mS/cm2 in the media. The snapshots are shown color scale. The right panel plots the distribution for synchronization factors. Parameters are fixed as k0 = −0.1, k1 = 0.01, k2 = 0.1, α = 0.2, β = 0.07.

intensity in induction current. Positive feedback gain in the induction current can increase the spatial diversity in excitability and the oscillation can be suppressed even though it can contribute to the oscillation in isolate neuron which can also be activated by the external stimulus. The potential mechanism is that the induction current is dependent on the distribution of magnetic flux and then the excitability of media is modulated as the external forcing current in the same way. The heterogeneity substantially affects the formation of spatiotemporal pattern. In the following studies, diversity in channel conductance is presented to induce local heterogeneity. Three types of heterogeneous area are designed by varying the intensity of calcium, potassium, and leakage ionic channel conductance, respectively. In the first case, collective behavior and dynamics are investigated by setting the heterogeneous area involving the diversity in conductance of calcium channel. Furthermore, the spatiotemporal pattern and synchronization factor are calculated by changing the gradient gap for calcium ionic channel conductance, and the results are shown in Fig. 5, respectively. From Fig. 5, it is demonstrated that random spatial pattern is developed in homogeneous tissue (∆gCa = 0.0 mS/cm2 ) and the media cannot support regular spatial pattern because of diversity in excitability induced by the random induction current. The wave cannot penetrate the heterogeneity and wave fronts are pinned to heterogeneity in local area when center area (Area I) are blocked to hold smaller conductance for calcium ion channels. With increasing the conductance for calcium in area I, distinct gradient conductance diversity in calcium channels can contribute to the formation and development of target waves in the medial. On the other hand, the factor of synchronization is decreased to support pattern formation and wave propagation when the center area begins to activate electrical activities with higher conductance than nodes in the Area II. Furthermore, it is also interesting to discuss the case when conductance diversity in potassium channels is considered, the evolution of spatiotemporal pattern and synchronization factor are plotted in Fig. 6, respectively. It is found in Fig. 6 that the synchronization factors always approach high values beyond 0.99, which means the media tends to present homogeneous distribution. The developed patterns confirmed that target seeds are compressed in a local area pinned to heterogeneity, the potential mechanism could be that the conductance of potassium channels seldom modulates the dynamical properties in electrical activities of cell completely even electromagnetic induction is considered. Furthermore, we also investigated the case for gCa = 3.5 mS/cm2 , which is associated with the middle turn point in curve for synchronization factor in Fig. 5, the pattern formation and distribution for synchronization factors are show in Fig. 7. Similar to the results in Fig. 6, target seed is compressed in a local area and synchronization factors decreased slightly but are kept with higher value, which means the homogeneous network blocked the growth of target wave when the gradient conductance in calcium channels is selected with higher value. In the third case, the formation of spatiotemporal pattern is discussed in heterogeneous tissue when conductance diversity is considered on leakage current channel. The evolution of spatial pattern and synchronization factor are calculated in Fig. 7, respectively. From Fig. 8, the synchronization factor tends to approach the maximal value and perfect synchronization is reached to support homogeneous state in the media with further increase of conductance value for leakage current. No target waves can be developed to occupy the media when the conductance for leakage current in the center (Area I) is much higher than the rest region (Area II). However, target wave is formed to control the media completely when conductance for leakage current in the center is less than the value setting for Area II. The mechanism could be that the occurrence of leakage current is associated with the chlorination (Cl− ), which holds active role in binding the Ca2+ and K+ . In fact, the transport of Ca2+ ,

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Fig. 6. Developed spatial pattern is plotted at t = 800 time units under different gradient gap for potassium channel conductance. (a) ∆gK = 0.0 mS/cm2 ; (b) ∆gK = −0.4 mS/cm2 ; (c) ∆gK = 0.4 mS/cm2 ; (d) ∆gK = 1.0 mS/cm2 . The external forcing current is set at Iext = 40 mA, potassium ionic channel conductance is fixed as gK = 8.0 mS/cm2 in Area II, and parameters are selected as gCa = 4.0 mS/cm2 , gL = 2.0 mS/cm2 in the media. The snapshots are shown color scale.The right panel plots the distribution for synchronization factors. Parameters are fixed as k0 = −0.1, k1 = 0.01, k2 = 0.1, α = 0.2, β = 0.07.

Fig. 7. Developed spatial pattern is plotted at t = 800 time units under different gradient gap for potassium channel conductance. (a) ∆gK = 0.0 mS/cm2 ; (b) ∆gK = −0.4 mS/cm2 ; (c) ∆gK = 0.4 mS/cm2 ; (d) ∆gK = 1.0 mS/cm2 . The external forcing current is set at Iext = 40 mA, potassium ionic channel conductance is fixed as gK = 8.0 mS/cm2 in Area II, and parameters are selected as gCa = 3.5 mS/cm2 , gL = 2.0 mS/cm2 in the media. The snapshots are shown color scale. The right panel plots the distribution for synchronization factors. Parameters are fixed as k0 = −0.1, k1 = 0.01, k2 = 0.1, α = 0.2, β = 0.07.

K+ can be suppressed when the area II holds higher conductance for leakage current and chlorination because chlorination ions are active to combine with the Calcium and potassium ions. As mentioned in Ref. [59], target waves can be developed by applying different kinds of forcing currents and feedback schemes. In fact, this target wave shows different stability and robustness under noise. Therefore, the collision and coexistence of target waves can be worthy of further discussion. Indeed, diversity in excitability and channel conductance can enhance the heterogeneity in the media. For simplicity, two separate local areas are controlled to adjust the conductance of calcium channels with gradient diversity. The evolution of spatiotemporal pattern is plotted in Fig. 9. As shown in Fig. 9, the two target waves can coexist and occupy the media completely because both of them are triggered in the same scheme and thus they can stand alive in the same media. That is, sole diversity in conductance of same ion channels can cause gradient excitability and then target waves can be induced to occupy the network completely. In fact, the the Morris–Lecar neuron has two types of ion channels, and the conductance diversity in different kinds of ion channels can generate different kinds of gradient excitability. As a result, the pattern selection and wave propagation will be dependent on the selection of conductance diversity in ion channels. It is interesting to explore the collision between two target fronts which are triggered by conductance diversity in calcium and potassium. The results are calculated in Fig. 10.

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Fig. 8. Developed spatial pattern is plotted at t = 800 time units under gradient gap for leakage ionic channel conductance. (a) ∆gL = 0.0 mS/cm2 ; (b) ∆gL = −0.4 mS/cm2 ; (c) ∆gL = 0.4 mS/cm2 ; (d) ∆gL = 8.0 mS/cm2 . The external forcing current is set at Iext = 40 mA, leakage ionic channel conductance is fixed as gL = 2.0 mS/cm2 in Area II, and others are selected as gCa = 4.0 mS/cm2 , gK = 8.0 mS/cm2 in whole media. The snapshots are shown color scale. The right panel plots the distribution for synchronization factors. Parameters are fixed as k0 = −0.1, k1 = 0.01, k2 = 0.1, α = 0.2, β = 0.07.

Fig. 9. Developed spatial pattern is plotted at t = 200, 400, 600, 800, 1000, 2000, 3000, 4000 time units. The calcium channel conductance gCa = 9.0 mS/cm2 is selected for nodes in the heterogeneous Area I (i[60:65], j[90:95]) and Area I′ (i[100:105], j[90:95]), while it sets gCa = 4.0 mS/cm2 in Area II. The external stimulus Iext = 40 mA, gL = 2.0 mS/cm2 , gK = 8.0 mS/cm2 are considered on the media. The snapshots are shown color scale. Parameters are fixed as k0 = −0.1, k1 = 0.01, k2 = 0.1, α = 0.2, β = 0.07.

It is confirmed in Fig. 10 that target waves cannot grow up to occupy the media. Conductance diversity in potassium channels just generates defects and blocks the wave propagation from the target waves generated by conductance diversity in calcium channels. Then target seeds and fronts are compressed to coexist in a local area in presence of electromagnetic induction. It is important to detect the stability of target waves when external electromagnetic radiation is considered. For simplicity, the electromagnetic radiation is described by Gaussian white noise, the statistical properties of noise and interplay on the excitable media are described by

⎧ ⎨ ∂ϕ = k V − k ϕ + ξ (t); 1 2 ∂t ⟨ ⟩ ⎩ ⟨ξ (t)⟩ = 0; ξ (t)ξ (t ′ ) = 2D0 δ (t − t ′ );

(7)

where ξ (t) denotes Gaussian white noise, D0 is the radiation intensity, δ denotes the Dirac function, ⟨∗⟩ represents statistical average over time. The results for pattern development are plotted in Fig. 11.

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Fig. 10. Developed spatial pattern is plotted at t = 200, 400, 600, 800, 1000, 2000, 3000, 4000 time units. It sets calcium channel conductance gCa = 9.0 mS/cm2 and potassium channel conductance gK = 10.0 mS/cm2 in heterogeneous Area I (i[60:65], j[90:95]) and Area I′ (i[100:105], j[90:95]), respectively. While it sets gCa = 4.0 mS/cm2 , gK = 8.0 mS/cm2 in Area II. The external stimulus Iext = 40 mA is applied uniformly and gL = 2.0 mS/cm2 is considered on the media. The snapshots are shown color scale. Parameters are fixed as k0 = −0.1, k1 = 0.01, k2 = 0.1, α = 0.2, β = 0.07.

Fig. 11. Developed spatial pattern is plotted at t = 800 time units when external electromagnetic radiation is applied on the media uniformly. Noise intensity is selected for (a) D0 = 0.0; (b) D0 = 2.5; (c) D0 = 7.5; (d) D0 = 20.0. The calcium channel conductance gCa = 8.0 mS/cm2 in heterogeneous Area I (i[90:95], j[90:95]), and it sets gCa = 4.0 mS/cm2 in Area II. The external stimulus Iext = 40 mA, gL = 2.0 mS/cm2 , gK = 8.0 mS/cm2 are considered on the media. The snapshots are shown color scale. Parameters are fixed as k0 = −0.1, k1 = 0.01, k2 = 0.1, α = 0.2, β = 0.07.

From Fig. 11, it is demonstrated that breakup of target wave occurs far away from the center and the target wave is compressed in a local area under electromagnetic radiation. With further increase of electromagnetic radiation intensity, target wave is destroyed and spatiotemporal turbulence is generated. The potential mechanism is that the diversity in induction current is enhanced and the media becomes anisotropic because of the excitability in spatial distribution is changed greatly under electromagnetic induction and radiation. The distribution for synchronization factors also indicates that electromagnetic radiation at appropriate intensity can enhance the regularity and suppress the propagation of target wave in the media. Furthermore, electromagnetic radiation is applied on a local area, and blocking of wave propagation is calculated in Fig. 12.

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Fig. 12. Developed spatial pattern is plotted at t = 200, 400, 600, 800 time units. The local external electromagnetic radiation sets as D0 = 10.0 in Area I(i[70:75], j[90:95]). The calcium channel conductance gCa = 8.0 mS/cm2 in heterogeneous Area I(i[90:95], j[90:95]), it sets gCa = 4.0 mS/cm2 for other nodes in Area II. The external stimulus Iext = 40 mA, gL = 2.0 mS/cm2 , gK = 8.0 mS/cm2 are considered on the media. The snapshots are shown color scale. Parameters are fixed as k0 = −0.1, k1 = 0.01, k2 = 0.1, α = 0.2, β = 0.07.

That is, local electromagnetic radiation can also generate defects in the media thus the propagation of target wave is blocked. Therefore, target wave can be suppressed completely when electromagnetic radiation is applied on the media uniformly. Developing target wave ever confirmed its effectiveness on removal of spiral wave in the excitable media [12]. In this way, it just provide important clue to suppress spiral wave and target wave. Therefore, electromagnetic radiation can be applied on the source area emitting regular wave fronts to destroy the regular propagation. For example, emergence of spiral wave in the cardiac tissue can block the regular wave propagation from atrionector, then electromagnetic radiation and electric shock can be applied on the defects area, which blocks the target wave front to form spiral fronts, thus the propagation of spiral wave can be terminated completely. In a summary, we discussed the wave formation and stability under electromagnetic radiation in the media based on reaction–diffusion system. However, most of the researchers prefer to explore the collective behaviors and synchronization consensus on neuronal networks and complex network with a variety of topological connection [60–62], e.g. sub-networks are included. Both chemical and electric synapses contribute to the signal processing and information encoding for neurons, while the author of this paper ever argued that field coupling [63–65] can provide another effective bridge to connect neurons effectively, and synchronization consensus of network can be modulated by field coupling. In this way, researchers can extend this problem on the network with synapse coupling and field coupling under consideration. It also confirmed some interesting results in Ref. [66] when electromagnetic radiation is imposed on myocardial cells and the mechanism for failure in heart is explained. On the other hand, the authors [67] investigated the spatiotemporal formation of patterns in a diffusive chain network with electromagnetic induction, and linear stability analysis was presented to explore the impact of magnetic flux when the generation of pulses and rhythmics behaviors like breathing are confirmed. In fact, the stability of target wave in the cardiac tissue and excitable media much depends on the formation mechanism while the mentioned target wave in this work is activated by using gradient diversity in excitability. Therefore, the propagation of pulses, target wave, spiral wave and stability of these waves are worthy of further investigation when electromagnetic radiation is considered. 4. Conclusions The excitability of media can be modulated when external electric stimulus is applied. On the other hand, electromagnetic radiation can adjust the transport process of ions because of polarization and magnetization on the media. The effect of electromagnetic induction becomes distinct and non-negligible when a lot of ions (Ca2+ , K+ , Na+ ) transport and pass though the channels embedded in the membrane of cells. Inspired by the physical law of electromagnetic induction, induction current is proposed to describe the effect of electromagnetic induction in the ML neuron with ion channels, and magnetic flux flow is used to map the effect of time-varying electric field. Based on the improved neuron model for cardiac tissue, heterogeneity induced formation of target waves is discussed. It is found that excitability diversity generated by gradient external stimulus can trigger stable target wave in the media. Blocking in channels of calcium in local area also can develop target wave in the media while conductance diversity in potassium shows slight modulation on wave propagation in the media. However, conductance diversity for leakage current can modulate the wave propagation by generating local defects and this heterogeneity contributes to the formation of target wave. Furthermore, propagation of target wave can be

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