Effect of Topology Structures on Synchronization Transition in Coupled Neuron Cells System

Transcription

1 Commun. Theor. Phys. 60 (2013) Vol. 60, No. 3, September 15, 2013 Effect of Topology Structures on Synchronization Transition in Coupled Neuron Cells System LIANG Li-Si ( Ò), ZHANG Ji-Qian ( ), XU Gui-Xia (Æ ), LIU Le-Zhu ( Ï ), and HUANG Shou-Fang (á Ǒ) College of Physics and Electronic Information, Anhui Normal University, Wuhu , China (Received February 4, 2013) Abstract In this paper, by the help of evolutionary algorithm and using Hindmarsh Rose (HR) neuron model, we investigate the effect of topology structures on synchronization transition between different states in coupled neuron cells system. First, we build different coupling structure with N cells, and found the effect of synchronized transition contact not only closely with the topology of the system, but also with whether there exist the ring structures in the system. In particular, both the size and the number of rings have greater effects on such transition behavior. Secondly, we introduce synchronization error to qualitative analyze the effect of the topology structure. Furthermore, by fitting the simulation results, we find that with the increment of the neurons number, there always exist the optimization structures which have the minimum number of connecting edges in the coupling systems. Above results show that the topology structures have a very crucial role on synchronization transition in coupled neuron system. Biological system may gradually acquire such efficient topology structures through the long-term evolution, thus the systems information process may be optimized by this scheme. PACS numbers: Sn, Bb, Rh, Aa Key words: neuron network, topology configuration, synchronization transition 1 Introduction It is well known that the cooperative behavior in complex neuron network has attracted much attention. Many researchers have studied the synchronization and coherence in completely local regular or completely random networks, they found that the dynamical process of the neuron is very complex, so some effective methods have been proposed to research such collective behaviors. [1 12] So far, in order to reveal the character and inherent law of the complex coupled system, especially the life system, many mathematical models of neurons have been established, [1 3] such as FitzHugh Nagumo, Hodgkin Huxley, HR etc. [4 5] These models and corresponding methods were used to investigate the dynamic process, and many interesting phenomena were found, for example, people researched the neuron firing rhythm and bifurcation regular pattern, and found that interspike intervals (ISI) could be used to reveal some firing rhythm patterns rules in biology, [6] as well as the principles and internal signal transfer interaction in biological systems. [5,7 8] Recently, some researchers found that network topology has a significant effect on the evolution of spatiotemporal dynamics and pattern formation in the neural network. Such as, spiral waves and spatiotemporal chaos could be observed in the heart by controlling the inhibition of calcium and potassium ions flow. [9] The channel disturbance may be conducive to the conversion between the neural network pattern formation and the spiral waves. [10] It is reported that complex spatiotemporal patterns could propagate towards both sides of the pacemaker in a ring neuronal network. [11] Neuronal synchronization on complex networks has been widely explored heoretically, such as, with certain coupled strength and the periodic stimulation, the complete synchronization or coherence between two neurons could appear. [4 5,7,12 13] The chaotic bursting is lag synchronization in HR system via a single controller. [14] Wang et al. have investigated the dependence of synchronization transitions of bursting oscillations on the information transmission delay over scale-free neuronal networks with attractive and repulsive coupling, and found that information transmission delays play a pivotal role in ensuring synchronized neuronal activity. [15] They also found that the typical patterns may emerge spatially, and firing synchronization may decrease with the increase of the noise level. [16] For multi-neuronal cell system, there are some different kinds of coupling modes, such as, regularity network, small-world networks, as well as scale-free networks. [17 18] For example, in 2D square lattice of HR neurons, it was found that signal lossless transmission can be achieved with appropriate coupling. In learning neuron networks with small world connectivity, the firing rate of neuron networks may decrease with the increment of synaptic learning rate, namely, synaptic learning can suppress the excitement in networks. [13] The generalized recurrent neuronal networks with multiple discrete delays and distributed delays may reach global asymptotical stability. [19] In some neuronal networks, the internal structure and its Supported by the National Natural Science, and Special Found for the Theoretical Physics of China under Grant Nos , , , the Special Foundation of Education of Anhui Province for Excellent Young Scientists under Grant No. 2011SQRL023 zhangcdc@mail.ahnu.edu.cn c 2013 Chinese Physical Society and IOP Publishing Ltd

2 No. 3 Communications in Theoretical Physics 381 space distribution have certain stability, and almost do not change with the increase of the connection units. [20 22] Recent studies about firing patterns transition and state to state transition also have proved that structures of the system play a significant role. [7,23 24] Above results imply that cooperative behavior is a major focus of study in complex neuron networks. In fact, nearly any complex system in nature could be viewed as a network in which vertices represent the dynamic elements of the system, and the edges denote the interactions or couplings between them. To our knowledge, neuron is one type of highly specialized cell; it is one of the basic structural and functional units in the nervous system. Its basic function is to finish the process of information exchange by acceptance, integration, transmission and output of information. [25] The electrical synapses among neurons can be connected by gap junctions, then they could response the stimulation and conduct the impulse, thus, the neuron system may finish the information process and realize response function for analysis and integration the environment stimulation which comes from the outside or inside of brain. However, the number of neurons in the biological coupled system is so large that the relationship between them is very complex, so there are some very interesting questions worth exploring: How do so large numbers of cells couple together? How does the coupling mode affect the conduction function in the information process? And how many cells are involved in such process? Of cause, to study how the interplay between the intrinsic dynamics among elements and their complex connectivity, it is very important to choose an efficient simulation method instead of the usual scheme. To this end, evolutionary algorithms may be one of the most potential methods, because the improved evolutionary algorithm has advantages of high precision, fast convergence speed and strong stability. [25 26] The evolutionary algorithm is combined with BP algorithm, and hybrid genetic algorithm, thus it is widely used to study the dynamics of networks. In this paper, by using the HR neuron model as a project, we construct the different complex networks with N HR cells as the units, and select I ext as the control parameter. We mainly study the regulative effect of topology structures on synchronization transition in N-cells coupled system. In the simulation, the time step is set to be dt = 0.001, and no flow boundary conditions are adopted. Equation (1) is numerical integrated by using the fourthorder Runge Kutta method. The best network structures with the optimal topology configurations are filtrated by improved evolutionary algorithm. The average ISI is obtained for each structure by averaging over 50 different realizations. Furthermore, we introduce the synchronous error to study the effects of topology structure on the collective behavior of the system. Finally, the inherent law and mechanisms of information process in neuron activities is also discussed. 2 Model In this paper, the coupled system is built by using N HR neuron cells, and the dynamical equation for each cell satisfies: dx i dt = y i x 3 i + bx 2 i z i + I ext + C N δ ij (x j x i ) j=1 dy i dt = c dz i dx2 i y i, dt = r[s(x i X 0 ) z i ], (1) where N is the total number of the cells, for the i-th cell, x i, y i and z i are the membrane potential, recovery variable and a slow adaptation current, respectively. X 0 is used to adjust the resting state, I ext is external current. The last term in the first equation in Eq. (1) is the coupling term. C is the coupling strength. δ ij is the coupling parameter between the two neurons i and j, if these two neurons are coupled to each other, δ ij = 1, other wise, δ ij = 0. Other parameters: b = 3.0, c = 1.0, d = 5.0, s = 4.0, r = 0.006, X 0 = Fig. 1 Two typical topological configurations for 5-cell coupled system and their state-state transition behavior. Left and right panels correspond to the results for the single chain and ring configurations, respectively; (b) and (e) are the transition results from periodic-1 to periodic-2 state, the ordinate ISI represents the interspike interval; (c) and (f) are the plots of synchronization error e t change with external stimulation intensity a.

3 382 Communications in Theoretical Physics Vol. 60 After studied the interspike interval (ISI) in a single HR neuron change with stimulating current I ext, Holden found that with the increase of current I ext, the HR neuron may undergo resting state, periodically bursting as well as chaotic state. [26 27] As shown Fig. 1 in Refs. [7]. The relevant parameters and their physical meaning in the model see as in Ref. [7]. 3 Results and Discussion In this paper, the numerical simulation is carried out by the scheme mentioned above according to the following steps. Firstly, a two-way coupled system is constructed with 5 HR cells. The coupling strength is set to the appropriate fixed value (C = 0.5), then the control parameters I ext of each cell are set near to the left side of the first bifurcation point (I ext0 = 1.3). Under this condition, the system will stay at the periodic-1 state, in order to detect the ability of the system to external weak signal, a feeble signal with certain frequency is switched on the stimulating current, i.e.: I ext = I ext0 (1 + a sin(2πft)), (2) where f = 2.4 Hz, and a is the amplitude of the external signal. For a certain type of topology configuration, we can find that when the signal strength a is set to be a proper value (a = 0.05), the system will undergo from periodic-1 state to periodic-2 (see as in Fig. 1(e)). One can notice that only a few configurations are helpful for such transition even though there are 19 different configurations in such coupled system, this indicates that, in such configuration, system may response to external low frequency weak signal. Such transition behavior is shown in Fig. 1(e), which induced by one typical configuration in Fig1d. Of cause, when the 5-cell neurons are coupled into the open-loop (chain Fig. 1(a)) structure, the state-state transition is unable to achieve (Fig. 1(b)). Obviously, such result implies that a coupled system with a closed ring structure is more conducive to achieve synchronous transition. Secondly, by the same scheme we have investigated the effect of topology on synchronization transition in coupled system for different cell numbers, such as, N = 10, 20, 50,100, 1000, respectively. And found similar phenomena. Above results show that, no matter how large the total number of cells in the coupled system, as long as there exist one or some numbers of ring structures with certain size, the transition behavior between the different states will appear easily, herein, the weak external stimulus signal could be tested by such mode. It will be helpful if we introduce a variable to represent the effect of synchronous transition, which is called synchronization error e and the definition can refer to Ref. [27]. The variable is as follows: e t = 1 N (xi x) N 2 + (y i ȳ) 2 + (z i z) 2, i=1 e = e t, (3) where e t is the average relative synchronous error for the three variables of the system at a certain time. e is the average over a period of time. The smaller the value of e, the better the effect of the synchronization transition. Thus, the state-state transition could successfully achieve. Other wise, such transition will not be able to realize. By using the formula 3, we have calculated the synchronization error of two typical topology, the results are shown in Figs. 1(c) and 1(f). One can see from Fig. 1(c), with the increment of the stimulus intensity, there are some scattered points whose values are higher than zero, and distribute in the parameter space. This implies the system could not turn to periodic-2 state synchronously (Fig. 1(b)). However, the synchronization error shown in Fig. 1(f) is near 0 (e = ), it is much lower than that of Fig. 1(c), so the system is able to achieve statestate transition successfully (Fig. 1(e)). Thirdly, due to there is close connection between the synchronous transition and system scale, so, in the following sections, we explore the effects of system size on the synchronous transition, and take the coupled system with 20 cells as an example, we select the number or the size of ring structure as a parameter, and set the coupling strength to be a fixed value (C = 0.5). In order to ensure that there not exists isolated cell in the coupled system, at first, all the cells are connected one by one to form a long chain, then we randomly select p cells to construct q rings. The size of the ring p can be adjusted from 3 to 19 for the coupled system with 20 cells, thus the number of rings will change with the ring size. For each ring structure, we calculate the synchronization error, and observe the state-state transition. For each same p and q, we do calculating the average synchronous error for each structure by averaging over 50 different realizations. Our simulation results are shown in Fig. 2(b). From this figure, one can see there are two obvious features: on one hand, for a certain size of the rings, with the increment of the number of rings, the synchronous error monotonously decreases. For example, when p = 3, there are C20 3 = 1140 types of possible rings. But not all of these rings are needed, i.e., to achieve synchronous transition only proper number of ring structures may be used. If q = 0, that is no ring existing in the system, the synchronization error is very high, even if outside stimulus intensity is strong, the system could not achieve synchronization transition (see the point at the beginning of black line in Figs. 1(a), 1(b), and Fig. 2(a)). If q > 0, there exist some rings in the coupled system, one can find the synchronization error decrease with the increase of the ring numbers rapidly. Of cause, if the size of the ring is small (p = 3), even if there are many ring structures in the coupled system (q = 19), the synchronous transition effect is not good.

4 No. 3 Communications in Theoretical Physics 383 On the other hand, when the size of the rings increase, such as, if the size of the ring is greater than 3, i.e., p = 9, 11 and 13, to obtain a better synchronous transition, the number of the rings in the system only need q = 8, 6 and 4 (see the lines below the black one shown in Fig. 2(b)). Similar results could be observed in the coupled systems with N = 10, 50 (see as in Figs. 2(a) and 2(c)). Comparing the results shown in Fig. 2, one can notice that for the fixed size p of the ring, the synchronization effect of the system will become better with the increment of the ring numbers in the system; As long as ring size p is large enough, the system could quickly achieve synchronization with proper numbers of ring in coupled system; At the same time, with the increase of the total number of coupling cell, for the better synchronization it does not need a lot of the number and the size of rings, such as if p = 7 15, the number q is about 10. However, if a small number of cells are in the isolated connective state, the synchronization effect will become very poor, even there exist rings in the coupled system. One can see this case shown in Fig. 3, this figure plots two typical topology configurations for N = 10 and p = 3. In Fig. 3(a), there not exist a larger number of rings and the connection edges among cells for effective synchronous behavior. But if there exist one or two relative isolation cells in the coupled system, the synchronous behavior cannot be achieved, even though there are both more connected edges and ring structures (see as in Fig. 3(b), the number 1 cell is isolated). This case may be common in some area of the tissues which have some impurities or dead cells in the life systems, thereby affect the information process in the system. Fig. 3 (Color online) Two typical topology configurations for N = 10 cells system, (a) e = 0.989, proper number of rings and few connection edges among cells for effective synchronous behavior; (b) e = , the presence of isolated cell makes the synchronous behavior cannot be achieved. Fig. 2 Average synchronization transition error change with the number and the size of the rings in the coupled systems for three typical size of the system. p and q represent the size of the ring and the number of the rings in the coupled system, respectively. Error bar represents the fluctuation deviation caused by the randomly realization. The results above indicate the topology structure of coupling neurons has a very important role on the synchronous transition, especially the ring structure strengthens the coordinated capability between cells in the system. To achieve better synchronous effect, a certain number of ring structures are required. This actually shows that a certain number of connections are required between the cells. Of course, it is not true that a larger number of connective edges are necessary for the better synchronous behavior. In fact, only the number of connections between the cells reaches an appropriate value. To verify this conjecture, in the following discussion, we use the C program in the method of evolutionary algorithm, and select coupling system with 10 cells as an object to find the optimal configurations as follows:

5 384 Communications in Theoretical Physics Vol. 60 (1) Setting the coupling strength and the bifurcation parameter according to the method mentioned above, and then randomly creating h coupled neuron cell systems. These systems are used as original structures, in which random connections are used to ensure all cells are contacted each other. (2) Reconstruct j substructures for each original structure. Randomly removing one of the connection edges in the original structure, and then retying with other cells. By this method, we may totally rebuild h j substructures. (3) Calculating the value of the synchronization error e for each new sub-configuration. Find out in those structures which synchronous errors are small, and take them as the next generation parent configurations. (4) Repeating steps 2 and 3, until the value of synchronous error becomes to be the minimum, at this time, these configurations are the best configurations of what we are looking for. Similarly, we can also find the optimal configurations by gradually removing one edge randomly, until the value of e reaches the minimum, thus the previous configuration is the optimal one. In the process of simulation, we found that: (1) Generally, the more connection edges, the lower e value, thus the better synchronization transition. (2) It is not necessary to have much more connection edges for the better effect. There is the minimum critical number of connection edges of the coupled system. Such as, for the coupled system with N = 10, the minimum critical value of edges is near about 7 10 (see as in Fig. 3(a)). From Fig. 3(a), if the information is transmitted from one cell to another one (such as from 3-rd cell to 9-th one), it usually needs to go through several edges, and each edge is called one step. This kind of connection may have more than one path. For a certain coupled system, its synchronous transition behavior may be closely associated with these parameters: the number of the steps a ij, the information transmits from i cell to j one; the number of the path b ij from i to j, as well as the number of connections c i of i cell. For example, if we want to have a very low e, the minimum number of steps should be 1 2. Therefore if the minimum number of steps for any pair of cells exceeds 3, the value of e will greatly increase. Meanwhile the more number of paths between two cells, the lower the value of e. As shown in Fig. 3(a), the minimum number of step from the 3rd cell to the 9-th one is 2, and there are three paths between them: 3-2-9, and So the connection between these two cells is very close. Similarly, so do the contact among other cells. Thus, the system with such configuration may easily achieve the synchronous transition behavior. However, compared from Fig. 3(b), even if the number of connection edges is very large, due to there are some isolate cells, so the minimum numbers of steps from 1st to others get larger, thus the synchronized effect of the system becomes very poor. To analyses the simulation results above by nonlinear fitting, after a lot of artificial tries and computer simulation test, we proposed the relationship between these three parameters and the synchronous error, following is the corresponding fitting formula for e g : ( ( (b ij 1) 1.7 e g = lg a 1.9 i =j k ij 5 N ( d ci )) 1.5) d cj 1, (4) where N is the total number of cells in the coupling system. k is the path ID for steps from i to j. d is the convergence constant, in order to make the results become easy to be convergence, d usually is set to be a large integer (such as d > 1000). Wherein the sum calculation result of a great magnitude scale range results, and the sum of the b ij in the bracket of formula increased exponentially, so, logarithm makes e g more corresponding to the value of e. If the number of connections for the i cell c i 1, it indicates this cell is a isolated unit, and the value of e will be large, so the synchronization transition effect is poor; On the contrary, if c i > 1, herein the value of e g is smaller, this could represent that there is no isolate unit in the system. Fig. 4 e g and e changes with the different configurations for three typical coupling systems. N is 10, 20 and 50. Other parameters see as in Fig. 2. Comparing the results from these two formulas Eqs. (3) and (4), which is shown in Fig. 4, we found that

6 No. 3 Communications in Theoretical Physics 385 both of these two variables could be used to characterize the synchronous transition behavior. For example, in Fig. 4(b), if e g < 3.2, of e is 0, which shows that the system could stay in the completely synchronization state; If the value e g is between 3.2 and 8, it represents that the synchronous effect is poor. While if e g > 8, the value of e is large and the system is the corresponding value in a completely non-synchronized state. Similarly, if we adjust the number of system N, the same phenomenon could be obtained (Figs. 4(a) and 4(c)). Fig. 5 (a) The minimum number n of connections for the optimal structure in the coupled system changes with the number N of cells. (b) One of the optimal structures when N = 10, the average number of connection edges is 3 4 for each cell. Due to the formula 4 could be debugged and controlled conveniently, especially, for a coupled system with very large number of neuron N, it spends much less time to calculate the synchronous error by using the formula 4 than 3. Thus, by using this formula, we can explore the relations of synchronization effect and topology structure in the system, and found that for each number of N, there are optimum connection structure and the least number of connection edges. The curve of the minimum number n of connections for the optimal structure in the coupled system as a function of the number N of cells is plotted in Fig. 5. One can find that with the increasing of N, the proper connective edges increase proportional rather than explosively. For example, when N = 10, the average number of connection edges is 3-4 for each cell. When N = 20, it is 4 to 6. Figure 5(b) shows one of the optimal structures for N = Conclusions In summarize, we have investigated the effects of topology structures on synchronization transition behavior. By using the evolutionary algorithm, we found that synchronous behavior is closely related not only with the topology of the system, but also with both the size and the number of rings. Furthermore, we introduce the synchronous error of qualitative describe the relationship between the topology and the synchronous transition. Finally, by nonlinear fitting, the relationship between the topology and parameters is also discussed, we found that, to obtain the best synchronous behavior in coupled system there exists a minimum number of edges and the optimization topology structure. Our simulation results show that, the topology structures may play an important role on synchronous transition in coupled neuron cells system. Biological systems may gradually acquire such efficient topology structures through the long-term evolution, thus the systems information process may be optimized by this scheme. References [1] P. Wang, J.Q. Zhang, and H.L. Ren, Chin. J. Chem. Phys. 23 (2010) 23. [2] J.Q. Zhang, C.D. Wang, M.S. Wang, and S.F. Huang, Neurocomputing. 74 (2011) [3] L. Li, H.G. Gu, M.H. Yang, Z.Q Liu, and W. Ren, Acta Biophysica. Sinica 20 (2004) 471. [4] D.Q. Wei, X.S. Luo, and Y.L. Zou, Commun. Theor. Phys. 50 (2008) 267. [5] J.D. Cao, K. Yuan, and H.X. Li, IEEE. T. Neural. Networ. 17 (2006) [6] W.H. Xia, L.Q. Shao, and Z.Y. Hong, Journal of Dynamics and Control. 7 (2009) 293. [7] J. Zhou, X.Q. Wu, and W.W. Yu, Chaos 18 (2008) [8] H.J. Yu and J. Peng, Chaos, Solitons and Fractals 29 (2006) 342. [9] Z. Min and T.G. Ning, J. Comput. Phys. 28 (2011) 119. [10] L.S. Bao and W. Ying, Journal of Dynamics and Control. 8 (2010) 284. [11] Q.Y. Wang, Q.S. Lu, and G.R. Chen, Physica A 374 (2007) 869. [12] Y. Wu, J.X. Xu, D.H. He, and W.Y. Jin, Acta Phys. Sin- Ch. Ed. 54 (2005) [13] F. Han, Q.S. Lu, M. Wiercigroch, J.A. Fang, and Z.J. Wang, Int. J. Nonlin. Mech. 47 (2012) [14] Z.L. Wang and X.R. Shi, Appl. Math. Comput. 215 (2009) 1091.

7 386 Communications in Theoretical Physics Vol. 60 [15] Q.Y. Wang, G.R. Chen, and M. Perc. Plos One 6 (2011) e [16] Q.Y. Wang, Z.S. Duan, L. Huang, G. R. Chen, and Q.S. Lu, New J. Phys. 9 (2007) 383. [17] S. Boccaletti, V. Latora, Y. Moreno, et al., Complex Systems And Complexity Science 3 (2006) 56. [18] D.H. Feng, L.S. Zhuo, W.F. Marcus, Y.Z. Shan, and Y.X. Song, Acta. Phys. Sin. -Ch. Ed. 56 (2007) [19] D.Q. Wei, X.S. Luo, and Y.H. Qin, Physica A 387 (2008) [20] S.F. Huang, J.Q. Zhang, and S.J. Ding, Chin. Phys. Lett. 26 (2009) [21] L.A. Muffley, S.C. Pan, A.N. Smith, M. Ga, A.M. Hocking, and N.S. Gibran, Exp. Cell. Res. 318 (2012) [22] S.J. Morrison, Curr. Opin. Cell. Biol. 13 (2011) 666. [23] S.N. Dorogovtsev and J.F.F. Mendes, Adv. Phys. 51 (2002) [24] A.V. Holden and Y.S. Fan, Chaos, Solitons & Fractals 2 (1992) 221. [25] A.V. Holden and Y.S. Fan, Chaos, Solitons & Fractals 3 (1993) 439. [26] Q.Y. Wang and H.H. Zhang, Advances in Mechanics. 43 (2013) 149. [27] L. Bergeron and J.Y. Yuan, Curr. Opin. Neurobiol. 8 (1998) 55.