Emergence and robustness of target waves in a neuronal network

International Journal of Modern Physics B Vol. 29, No. 23 (2015) 1550164 (13 pages) c World Scientific Publishing Company

DOI: 10.1142/S0217979215501647

Emergence and robustness of target waves in a neuronal network

Int. J. Mod. Phys. B 2015.29. Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 09/18/15. For personal use only.

Ying Xu, Wuyin Jin and Jun Ma∗ Department of Physics, Lanzhou University of Technology, Lanzhou 730050, P. R. China College of Mechano-Electronic Engineering, University of Technology, Lanzhou 730050, P. R. China ∗ [email protected]

Received 5 May 2015 Revised 10 June 2015 Accepted 16 June 2015 Published 5 August 2015 Target waves in excitable media such as neuronal network can regulate the spatial distribution and orderliness as a continuous pacemaker. Three different schemes are used to develop stable target wave in the network, and the potential mechanism for emergence of target waves in the excitable media is investigated. For example, a local pacing driven by external periodical forcing can generate stable target wave in the excitable media, furthermore, heterogeneity and local feedback under self-feedback coupling are also effective to generate continuous target wave as well. To discern the difference of these target waves, a statistical synchronization factor is defined by using mean field theory and artificial defects are introduced into the network to block the target wave, thus the robustness of these target waves could be detected. However, these target waves developed from the above mentioned schemes show different robustness to the blocking from artificial defects. A regular network of Hindmarsh–Rose neurons is designed in a two-dimensional square array, target waves are induced by using three different ways, and then some artificial defects, which are associated with anatomical defects, are set in the network to detect the effect of defects blocking on the travelling waves. It confirms that the robustness of target waves to defects blocking depends on the intrinsic properties (ways to generate target wave) of target waves. Keywords: Target wave; neuronal network; defects; factor of synchronization. PACS numbers: 05.45.–a, 47.54.–r, 87.19.lq, 87.18.Hf

1. Introduction The heart in animal often beats in rhythm and the sinus node used to emit continuous electrical waves to propagate among the cardiac tissue, thus target wave can be ∗ Corresponding

author. 1550164-1


Y. Xu, W. Jin & J. Ma

developed to regulate the shrinkage and relaxation of heat. However, spiral waves begin to emerge in the cardiac tissue when the target wave generated from the sinus node is blocked by anatomical defects such as heart ischemia in local area, as a result, the heartbeat could become inordinate which is often associated with arrhythmia disease.1 – 6 The scheme of local pacing7 – 10 by imposing periodical signal on local area shows great effectiveness to suppress the spiral wave and turbulence in the cardiac tissue by generating continuous target waves or plane waves, thus it could be practical and reliable to prevent the emergence of ventricular fibrillation. Indeed, continuous pacing in local square area in the media can generate stable and continuous target wave and thus the spiral wave or spatiotemporal chaos can be suppressed completely. The realistic neuronal system is made of a large number of neurons, and it is believed that the electrical activities of neuron could be described by some theoretical models.11 – 15 The discharge mode of electrical activity in neuron can be changed by applying appropriate external forcing current. The collective behaviors of neurons are often detected on neuronal networks, and this topic is associated with synchronization problem,16 – 19 pattern selection20 – 27 and control in network or coupled oscillators. As a result, complex network with different topologies such as regular network with nearest neighbor connection, small-world connection type, even scale free type is often designed to discern the collective behavior or transition of electrical activities in neurons.28 – 30 Indeed, pattern selection and recognition could be also important to understand the emergence of electrical activities in neuronal network because the distribution of membrane potentials of neurons in network could be observed in experimental way. As mentioned in the previous works, target wave, spiral wave can emerge in the neuronal network under appropriate condition and the developed target wave or spiral wave can regulate the collective behavior of neuronal network effectively as a powerful pacemaker.31,32 In a neuronal network, spiral wave can be induced by using specific initial values or breaking the travelling waves. For the emergence of target wave, multiple schemes can be used to induce and develop target wave so that the collective behaviors of network could be adjusted. For example, it is believed that target wave can be induced by external forcing with diversity or local pacing in the network.33,34 Heterogeneity35,36 is also effective to generate target waves in the media and/or network as well, for simplicity; parameter in diversity is used to reproduce the heterogeneity in media. Interestingly, feedback in local area37 is also reliable to develop continuous target wave in the network completely. Indeed, it is important to discern the difference and robustness of the target waves developed in different ways. In this paper, the robustness of target wave induced by different potential ways is detected, and then artificial defects are introduced into the network. The collision between the target waves and defects is investigated; it is found that spatial pattern selection in network blocked by artificial defects is dependent on the properties of target waves induced by different ways. The content is arranged as follows: In Sec. 2, the model is presented and three different schemes are proposed to generate 1550164-2


continuous target wave by selecting different local control terms in the network. Numerical studies and discussion are presented in Sec. 3 including three subsections as well. Each scheme is arranged in a subsection and artificial defects are introduced to block the target wave in each case so that the robustness of target wave could be detected. Furthermore, time series of sampled membrane potentials of nodes and factor of synchronization are also calculated for signal analysis and detection of collective behaviors of the network. Finally, conclusion is provided in the last section.


2. Model and Scheme The dynamical equations for a regular network of Hindmarsh–Rose neurons12,13 driven by local forcing or feedback are described by    dxij = yij − ax3ij + bx2ij − zij + Iext + D(xi+1j + xi−1j + xij+1 + xij−1 − 4xij )   dt     + f δiα δjβ ,    dyij  = c − dx2ij − yij ,   dt         dzij = r[s(xij − χ) − zij ] . dt (1) The network is connected with nearest-neighbor type in a two-dimensional square array; the local kinetics of each node is described by the three-variable Hindmarsh– Rose neuron model. Here the variable x, y and z denotes the membrane potential, recovery variable associated with slow current and adaption current, respectively. Iext often represents the external forcing current, D is the coupling intensity between adjacent nodes, the subscripts i, j, α and β are integers and f is additive forcing term with multiple forms and will be imposed on a local area in the network. δiα = 1 for i = α, otherwise, δiα = 0; δjβ = 1 for j = β, otherwise, δjβ = 0. For an isolate neuron, the Hindmarsh–Rose neuron model can show quiescent state, spiking, bursting even chaotic states by selecting appropriate intensity of external forcing current carefully at fixed parameters as a = 1.0, b = 3.0, c = 1.0, d = 5.0, s = 4.0, r = 0.006, χ = −1.56. To discern the statistical properties of collective behaviors, a factor of synchronization R is defined by using mean field theory, and the synchronization factor21,22 is defined as follows: R=

hF 2 i − hF i2 N N 1 XX 2 (hx i − hxij i2 ) N 2 j=1 i=1 ij

and F =

N X N X

xij /N 2 ,

(2)

j=1 i=1

where xij could be calculated from Eq. (1), the symbol h∗i represents the average calculation over time with certain transient period, N 2 is the number of nodes in the two-dimensional square array, perfect synchronization is realized at R ∼ 1 1550164-3



and nonperfect synchronization occurs at R ∼ 0. Indeed, the network holds high synchronization under homogeneous state, and smaller factor of synchronization is observed when the network of the media is dominated by distinct ordered wave such as target waves or spiral waves. The distribution for factor of synchronization could also be effective to give some clues to discern the difference of target waves developed in different ways. To further discern the robustness of target waves, artificial defects are introduced into the media. The artificial defects in the neuronal network could be reproduced by setting the membrane potentials of some neurons in a local area as certain constant value. There are multiple types of additive forcing that could be imposed on a local area of the network, it often reads f = A cos ωt + 1 ,

(3a)

f = k(V0 − xij ) ,

(3b)

f = k(xmn − xij ) .

(3c)

Equation (3a) indicates that each node in a local square array will be imposed the same periodical forcing current. Here V0 in Eq. (3b) is constant, ω is angular frequency, k is feedback gain; xmn is the sampled time series for any node in the network. The three types of additive forcing on neurons in a local area can generate continuous wavefront to regulate the collective behaviors of neurons as pacemaker because continuous feedback or forcing is introduced in a local area of the network, and the results are verified by numerical studies in the next section. 3. Numerical Results and Discussions In the numerical studies, the Euler forward algorithm is used to calculate the time series for membrane potentials of neurons of the network under no-flux boundary condition, time step h = 0.02, the initial values for each node of the network are selected as (−1.31742, −7.67799 and 1.13032) so that a homogeneous quiescent state begins for the network, and transient period for calculating is about 20,000 time units. Then the additive forcing f is selected with different forms to explore the potential ways for generating continuous target waves. For simplicity, 200 × 200 neurons are placed on nodes in a two-dimensional square array uniformly. In the following, each subsection gives a case for the development of target wave and its robustness to blocking by artificial defects will be investigated, respectively. 3.1. Local periodical pacing-induced target wave and defects blocking For simplicity, external periodical forcing is imposed on a square array with 6 × 6 nodes (80 ≤ i, j ≤ 85, α = 80, 81, 82, 83, 84, 85 and β = 80, 81, 82, 83, 84, 85), the membrane of potentials for neurons in the area (i = 100, 1 ≤ j ≤ 100) are set to constant V so that artificial defects could be reproduced in this area. The 1550164-4



Fig. 1. (Color online) Formation of target wave induced by local periodical forcing in the network at t = 20,000 time units. The external forcing is imposed on nodes (80 ≤ i, j ≤ 85), coupling intensity D = 1, external forcing current Iext = 1.0, for (a) ω = 0.001, (b) ω = 0.01, (c) ω = 0.1.

external forcing is set as f = 1 + 2 cos ωt and different angular frequencies are used to observe the wave emergence and pattern selection in the network, as shown in Fig. 1. It is found that local periodical forcing can develop continuous target wave in the neuronal network by applying appropriate angular frequency and smaller angular frequency is much helpful to develop a full target wave in the network and thus the network can be regulated by the target wave completely. To discern the robustness of this target wave induced by external local pacing, line defects are generated in the network by setting the membrane potentials of neurons in the area (i = 100, 1 ≤ j ≤ 100) as constant V . For simplicity, the external periodical forcing is selected as f = 1 + 2 cos 0.001t, and the robustness of target wave is observed in Fig. 2. The results in Fig. 2 confirm that the developed target wave could be blocked by the defects, but spiral waves are also induced when the target wave is broken by the defects. In two-dimensional excitable media, the velocity–curvature relationship of wavefronts can be written in the form of an eikonal equation as V (r) = Vn = Vp − D/r, where Vn is the normal velocity, Vp is the plane wave velocity, D is the diffusion coefficient, and 1/r is the wave curvature.38 – 40 For the

Fig. 2. (Color online) Collision between the target wave and line defects at t = 20,000 time units, external forcing current Iext = 1.0. The threshold value for the defects is selected for (a) V = −0.5, (b) V = 0.0 and (c) V = 0.5. The coupling intensity is set as D = 1, defects locate at the area (i = 100, 1 ≤ j ≤ 100) and additive forcing f = 1 + 2 cos 0.001t is imposed on nodes (80 ≤ i, j ≤ 85). 1550164-5



two-dimensional network, this result can still be verified by numerical results. That is to say, the spiral wave can propagate more quickly and occupy the network by further increasing the coupling intensity. Similar results are approached by selecting the coupling intensity D = 2, however, no target wave and spiral waves could be observed in the network by increasing coupling intensity to D = 3.0, the potential mechanism could be that neurons are coupled to reach complete synchronization thus no ordered wave could be induced. Furthermore, the time series for sampled membrane potentials of two different nodes are plotted in Fig. 3 and the distribution of synchronization factor is calculated in Fig. 4 by changing the threshold value V for defects area. It is found in Fig. 3 that intermittent spiking states emerge in the time series of the node (100, 150) when the wavefronts were not blocked by the defects and the propagation of wave in the left-hand side is modulated by the defects, while the time series for node (180, 150) decrease to stable state within certain transient period, the potential mechanism could be that the wavefronts were blocked by the line defects, the broken waves are anchored to the defects thus no continuous wave can pass through the defects and let alone to propagate in the right-hand side of the network. Interestingly, by further increasing the coupling intensity, the broken

Fig. 3. The time series for membrane potentials of neurons for node (100, 150) (a) without defects being considered; time series for node (100, 150) (b) and node (180, 150) (c) the threshold value for defects is set as V = 0 and the defects locate at the area (i = 100, 1 ≤ j ≤ 100), external forcing current Iext = 1.0.

Fig. 4. Distribution for factor of synchronization under different thresholds for artificial defects V for defects area (i = 100, 1 ≤ j ≤ 100). The transient period is about 20,000 time units and coupling intensity is set as D = 1, external forcing current Iext = 1.0. 1550164-6



waves can propagate with higher speed thus, more neurons on the right side can be activated. In addition, the threshold for artificial defects is changed to calculate the factor of synchronization, and the results are shown in Fig. 4. The results in Fig. 4 confirm that the factor of synchronization decrease slightly by increasing the threshold value for artificial defects (for the threshold value V for defect degree, it could be associated with poisoning ratio or depth in ion channels), it indicates that standing waves could exist in the network. Indeed, continuous pacing in a local area can generate stable target wave in the network though the target wave could be broken by the defects and converts into spiral wave, as a result, ordered wave still occupies the network and the sampled series shows intermittent spiking states. It is also interesting to investigate the formation of target wave by applying local feedback in the network. 3.2. Local track control-induced target wave formation and defects blocking In this case, the membrane potentials of neurons in a local area are detected and compared with a standard signal V0 in Eq. (3b) and then the errors between the sampled membrane potentials and the standard signal are feedbacked into the local area, thus quasi-periodical forcing could drive the network to generate target-like waves. In a practical way, the standard signal V0 could be constant introducing from external forcing, but also can be sampled from a certain node in the network as well. For simplicity, the case for V0 = −0.8 is investigated, and the results are shown in Figs. 5. It is found in Fig. 5 that symmetric spatial patterns can be developed in the network under track control in a local area of network; the network can reach

Fig. 5. (Color online) Developed pattern in the network at t = 20,000 time units. The forcing term is described by f = k (−0.8 − xij ), coupling intensity D = 1.0 and external forcing current Iext = 1.5. For (a) k = 0.2, (b) k = 0.25, (c) k = 0.3 and (d) k = 0.35. 1550164-7



Fig. 6. (Color online) Developed pattern in the network at t = 20,000 time units under different thresholds for defects. The forcing term is described by f = k (V0 − xij ), coupling intensity D = 1.0, external forcing current Iext = 1.5 and feedback gain k = 0.3. For (a) V0 = −0.9, (b) V0 = −0.8, (c) V0 = −0.7, (d) V0 = 0.1, (e) V0 = 0.3, (f) V0 = 1.2.

Fig. 7. Distribution for factor of synchronization under different thresholds for defects at k = 0.3. The transient period for calculating is about 20,000 time units, and coupling intensity is set as D = 1.0, external forcing current is selected as Iext = 1.5.

homogeneous state by further increasing the feedback gain k. In fact, the neurons in a local area will be imposed a feedback action only when the membrane potentials of these controlled nodes deviate the target value V0 (= −0.8), thus appropriate gradient forcing is generated to drive the network-like pacemaker. To discern the effect of target value V0 on the pattern selection, extensive results are calculated in Fig. 6 for snapshots, and the distribution for synchronization factor in Fig. 7. The results in Fig. 6 show that the developed spatial patterns are associated with the target value greatly and perfect symmetry could be observed in the spatial 1550164-8



patterns. It is interesting to explore the potential significance for this type of feedback on neuronal network, indeed, when the target value comes from the center nervous system, the neurons in a local area received this signal or information will adjust their electrical activities by tracking the target value and then generate continuous wave front to regulate other neurons, as a result, different spatial patterns are developed and corresponding collective behaviors are approached. To detect the synchronization degree, the factors of synchronization under different target values are calculated in Fig. 7. It is found that synchronization factors hold higher value under track control and perfect synchronization could be reached when the target value is much too negative, it also indicates that this type of feedback forcing could be effective to reach synchronization of the network. By further increasing the target value V0 , the factor of synchronization shows slight decreasing, interestingly, perfect symmetric spatial patterns are formed in the network. Furthermore, the defects blocking is also considered. For example, the local forcing term is selected as f = k (V0 − xij ) = 0.3(−0.8 − xij ), no ordered wave could be detected in the network and the time series for membrane potentials of some nodes become stable within certain transient period, and the results are shown in Fig. 8. The results in Fig. 8 show that the time series for sampled membrane potentials decrease to stable value within certain transient period by applying local track control on the network, and the distribution of membrane potentials show slight difference. Furthermore, the dependence of synchronization factor on the thresholds of defects is calculated, and the results are plotted in Fig. 9. It is found that the factor of synchronization is close to certain value about 0.71 and the network tends to synchronization but not reach complete or perfect synchronization and the threshold for defects show slight impact on the distribution for factor of synchronization. Now it is important to investigate the case that the target value is time varying, which could be much associated with a realistic neuronal system when center nervous system emits different signals to regulate other neurons.

Fig. 8. The time series for membrane potentials of neurons for node (100, 150) (a) without defects being considered; time series for node (100, 150) (b) and node (180, 150) (c) the threshold value for defects is set as V = 0 and the defects locate at the area (i = 100, 1 ≤ j ≤ 100). The forcing term is described by f = 0.3 (−0.8 − xij ), coupling intensity D = 1.0, external forcing current Iext = 1.5. 1550164-9



Fig. 9. Distribution for factor of synchronization under different thresholds for defects at k = 0.3. The transient period for calculating is about 20,000 time units and coupling intensity is set as D = 1.0, external forcing current Iext = 1.5, the forcing term is described by f = 0.3 (−0.8 − xij ).

3.3. Local self-feedback-induced target wave formation and defects blocking For simplicity, the node (85, 85) is used to generate target value V0 , and neurons in a local area are controlled by applying the self-feedback control as shown in Eq. (3c) and the results are shown in Fig. 10. It finds that stable target wave can be developed in the network by self-feedback in a local area when appropriate feedback gain is used. The network emerges homogeneous state in large area by applying stronger feedback gain in the external forcing term; otherwise, target wave just locates in a local area when smaller gain is used in the external forcing term. To detect the robustness of this target wave, defects are introduced into the network, and blocking effect is investigated in Fig. 11.

Fig. 10. (Color online) Developed patterns in the network at t = 20,000 time units. The forcing term is described by f = k(x85,85 −xij ), at coupling intensity D = 1.0 and external forcing current Iext = 1.5. For (a) k = 0.32, (b) k = 0.35, (c) k = 0.38 and (d) k = 0.4. 1550164-10



Fig. 11. (Color online) Developed pattern in the network at t = 20,000 time units under different thresholds for defects. The forcing term is described by f = k (x85,85 − xij ), coupling intensity D = 1.0, external forcing current Iext = 1.5 and feedback gain k = 0.38. For defects degree (a) V = −0.4, (b) V = 0.0, (c) V = 0.2.

Fig. 12. The time series for membrane potentials of neurons for node (100, 150) (a) without defects being considered; time series for node (100, 150) (b) and node (180, 150) (c), the threshold value for defects is set as V = 0 and the defects locate at the area (i = 100, 1 ≤ j ≤ 100). The forcing term is described by f = k (x85 85 − xij ), coupling intensity D = 1.0, external forcing current Iext = 1.5 and feedback gain k = 0.38.

The results in Fig. 11 confirm that this type of target wave cannot keep robustness to the defects and disordered spatial pattern emerge in the network though some local area could be occupied by target-like waves. The developed state is also dependent on the threshold for defects as well. To discern the periodicity of network, the time series for sampled membrane potentials are calculated in Fig. 12. The time series for membrane potentials show that self-feedback in local area can generate continuous spiking states; furthermore, the time series can be regulated to bursting when defects effect is considered. It finds that the thresholds for defects change the distribution for factors of synchronization, and the factors of synchronization reach a higher value with small diversity even if the thresholds for defects are changed. It indicates that no perfect ordered wave can occupy and regulate the collective behaviors of the network. It also means that no distinct ordered waves can be developed to occupy the network completely. In a summary, target wave can be formed in the network via three different ways, and these developed target waves shows different robustness to defects blocking. 1550164-11



Fig. 13. Distribution for factor of synchronization under different thresholds for defects at k = 0.38. The transient period for calculating is about 20,000 time units and coupling intensity is set as D = 1, external forcing current Iext = 1.5, the forcing term for self-feedback is described by f = 0.38(x85 85 − xij ).

Indeed, the local pacing due to external periodical forcing in local area can develop continuous target waves and can keep certain robust to the defects blocking, and target wave can also be induced when the target wave is blocked by defects. It differs from the formation of target wave by imposing external forcing with diversity. The track control and self-feedback on local area also can induce target waves in the network, but these target waves can be destroyed by defects though some sampled membrane potentials for neurons could show distinct periodicity. In the last case, the synchronization degree is enhanced. As a result, the first scheme that generating continuous target waves by imposing appropriate external periodical forcing in local area could be very effective to keep the orderliness of the media, and suppress the spatial irregularity of the network.

4. Conclusion The potential mechanism for formation of target waves is discussed in three different ways. These target waves show different robustness to defects blocking. The external pacing-induced target wave by imposing external periodical forcing in local area shows great robustness to defects blocking and could be used to suppress spatiotemporal chaos. The track control and self-feedback in local area also can induce target waves in the network, but these target waves are sensitive to defects blocking though this scheme could be helpful to realize synchronization of the network. As a result, normal wave propagation from center nervous system in the network could be terminated when the media or network suffers from abnormality in a local area, thus some nervous disease could be triggered. 1550164-12


Acknowledgment This project is partially supported by the National Nature Science of Foundation of China under Grant Nos. 11372122 and 11365014.


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Emergence and robustness of target waves in a neuronal network

Emergence and robustness of target waves in a neuronal network

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