Parameter Diversity Induced Multiple Spatial Coherence Resonances and Spiral Waves in Neuronal Network with and Without Noise

Commun. Theor. Phys. 57 (2012) 817 824 Vol. 57, No. 5, May 15, 2012 Parameter Diversity Induced Multiple Spatial Coherence Resonances and Spiral Waves in Neuronal Network with and Without Noise LI Yu-Ye (Ó ã), 1 JIA Bing ( ), 1,2 GU Hua-Guang ( Ù½), 1,2, and AN Shu-Cheng (Ë ) 1 1 College of Life Sciences, Shaanxi Normal University, Xi an 710062, China 2 School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China (Received December 2 2011; revised manuscript received February 10, 2012) Abstract Diversity in the neurons and noise are inevitable in the real neuronal network. In this paper, parameter diversity induced spiral waves and multiple spatial coherence resonances in a two-dimensional neuronal network without or with noise are simulated. The relationship between the multiple resonances and the multiple transitions between patterns of spiral waves are identified. The coherence degrees induced by the diversity are suppressed when noise is introduced and noise density is increased. The results suggest that natural nervous system might profit from both parameter diversity and noise, provided a possible approach to control formation and transition of spiral wave by the cooperation between the diversity and noise. PACS numbers: 05.45.Xt, 87.18.Tt, 87.18.Hf Key words: multiple spatial coherence resonance, spiral wave, diversity, neuronal network 1 Introduction It is well known that noise plays a constructive role in many nonlinear systems. Two prominent examples are stochastic resonance (SR), [1 2] where the response of nonlinear systems to a weak periodic forcing becomes maximally ordered at a resonant noise density, and coherence resonance (CR), [3 4] where the nonlinear systems exhibit a maximum with respect to noise density without external periodic forcing. Both SR and CR have been verified experimentally in many different fields, and examples were founded in physical, chemical, and biological systems. [5 7] Particularly in biological nervous systems, some systems may take advantage of the benefits of ambient noise to discover the feeble information rather than ignoring it. [6 7] In theoretical and computational studies, the focus of researchers about the constructive role of noise has shifted from the simple, low-dimension, dynamical systems to the higher dimensional extended systems composed of many coupled identical units. [8 20] For instance, spatiotemporal SR, spatiotemporal CR, spatial SR, and spatial CR have been investigated in excitable media and neuronal network with identical units. [8 11,13 16] In many observations of resonance, a great many of spatiotemporal patterns induced by noise have been observed, such as synchronization, spiral wave, spatial periodicity, and temporal order. [3,14 17,20 21] Many realistic systems are spatially extended, including the nervous system which is composed of billions of coupled cells. Therefore, the existence of parameter diversity between individuals is inevitable. Recently, diversity (also called variability or heterogeneity) has been proved to play a crucial role in the spatiotemporal dynamics of nonlinear system. [22 24] For instance, diversity can enhance synchronization in an array of Josephson junctions, [25 26] and induce spatial stochastic or coherence resonance in a lot of systems, including bistable or chaotic systems, [23,27] excitable media, [28] and neuronal network. [29] A series of nonlinear spatiotemporal patterns such as synchronization and spiral wave induced by diversity have been observed in the spatial extended excitable media and neuronal network. [14,28,30 33] Most of the previous articles about SR, CR and spatiotemporal patterns considered separately the effect of noise on parameter homogeneous network or the effect of parameter diversity on network in the absence of noise. Actually, the existence of both internal noise and diversity in many natural systems is inevitable, especially in biological systems. The realistic nervous system evolves in noisy environment and is composed of numerous cells with different functions and characteristics. Hence, it is necessary that the interplay of noise and parameter diversity have been considered in nervous systems. Recently, studied in network of coupled FitzHugh Nagumo (FHN) elements with noise, it has shown that parameter diversity can induce spatial stochastic resonance. When more noise is present, less variability is necessary to achieve op- Supported by National Natural Science Foundation of China under Grant Nos. 11072135 and 10772101, the Fundamental Research Funds for the Central Universities under Grant No. GK200902025 Corresponding author, E-mail: guhuaguang@263.net; guhuaguang@tongji.edu.cn c 2011 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

818 Communications in Theoretical Physics Vol. 57 timal signal enhancement. [22] Diversity in the parameter of the array, which is a chain of coupled FHN neurons, can enhance the coherence resonance induced by noise. [25] Parameter diversity can also induce a transition from oscillatory to excitable behavior in a coupled network with noise. [32] Among the above-mentioned investigations about various spatiotemporal resonances, most articles reported that the effect of resonance manifests as a single peak in noise-signal ratio (SNR), while there is a few reports on multiple resonances, [11 12,34 40] especially multiple spatial stochastic resonances or coherence resonances. In spatial extended systems, multiple stochastic resonances [38] or multiple temporal resonances [39] induced by delay, and multiple spatial coherence resonances induced by an artificial stochastic signal or noise [11 12,40] have been studied in the neuronal network. However, diversity-induced multiple spatial coherence resonances have not been clearly reported. In this paper, the diversity-induced multiple spatial coherence resonances and spatiotemporal patterns are studied in a neuronal network composed of Morris Lecar (ML) model. Considering that the real nervous systems evolve in noisy environment, we further investigate that multiple spatial coherence resonances induced by parameter diversity in the neuronal network with white noise. This paper is organized as follows. A two-dimensional ML neuronal network with a parameter diversity and white noise is introduced in Sec. 2. In Sec. 3, multiple spatial coherence resonances and spiral waves induced by parameter diversity in the neuronal network without noise are simulated. The influences of noise on multiple spatial coherence resonances induced by the diversity are introduced in Sec. 4. Finally, conclusion and discussion are given in Sec. 5. 2 Neuronal Network and Morris Lecar Model The two-dimensional neuronal network considered in this paper is composed of a set of coupled ML [41 42] neurons, presented as: dv i,j dt dw i,j dt = 1 C ( g l(v i,j v l ) g ca m (V i,j )(V i,j v ca ) g k W i,j (V i,j v k ) + g c εm,n,i,j (V m,n V i,j ) + I i,j ) + ξ i,j (t), = λ W (V i,j )(W (V i,j ) W i,j ), (1) here, i, j=1, 2,...,N specifies the neuron index, N is the size of network, which is the two-dimensional N N coupled ML neurons and a regular networks with nearest neighbor coupling. The last term of Eq. (1) runs over all lattice sites, if the site (m, n) is coupled to (i, j), ε m,n,i,j = 1, otherwise ε m,n,i,j = 0 (more details about scheme of network can be found in Refs. [11 12]). g c is the coupling strength. V i,j, W i,j, and I i,j are the membrane potential, the recovery variable and the background current of each neuron, respectively. C is the membrane capacitance, t is time. Moreover, the constants v k, v ca, and v l are the reversal potential of potassium, calcium, and leakage current, respectively, while g k, g ca, and g l are the corresponding maximal conductances. m, W, and λ W are functions of V i,j defined as: m = 0.5[1 + tanh((v i,j + 1.2)/18)], W = 0.5[1 + tanh((v i,j 12)/17.4)], λ W = 0.067 cosh[(v i,j 12)/(34.8)]. In the present contribution only noise in the membrane potential V i,j and diversity are considered. The noise term ξ i,j (t) is a spatiotemporal Gaussian white noise with a zero mean, white in space and time, i.e., ξ i,j (t) = 0, ξ i,j (t)ξ m,n (t ) = 2Dδ(t t )δ i,m δ j,n, (2) where D is the noise density and δ( ) is the Diract δ function. Diversity is applied in the background current I. The parameter value I i,j is Gaussian distributed numbers with a fixed mean I i,j = I 0, (I i,j I 0 )(I k,l I 0 ) = I w δ i,j,k,l, (3) I w is the diversity strength of I i,j. Throughout this paper, the parameters are taken as follows: C = 20 µf/cm 2, g k = 8 ms/cm 2, g l = 2 ms/cm 2, g ca = 4.0 ms/cm 2, v ca = 120 mv, v k = 84 mv, v l = 60 mv. The system size, the coupling strength and the mean background current are used N = 128, g c = 1, and I 0 = 39.7, respectively, while the noise density D, and the diversity strength I w are varied. The stochastic coupled ML model driven by a Gaussian white noise was numerically simulated by the stochastic Runge Kutta algorithms [43] with an integration time step 0.1 ms. The dynamic behavior of a single uncoupled ML neuron with bifurcation parameter I is described here briefly. ML model shows a saddle-node bifurcation on invariant cycle [44] at I 39.96, as shown in Fig. 1(b). When I < 39.96, an unstable focus, a saddle, and a stable node coexist, such as I = 39.7 shown in Fig. 1(a). Here, the ML model is at a resting state, a small perturbation away from a resting state can result in a large excursion of its potential (solid line) before returning to quiescence, as shown in Fig. 1(a). When I > 39.96, an unstable focus and a stable limit cycle coexist, which enables the ML model to oscillate (solid line), such as I = 41 shown in Fig. 1(c). The corresponding bifurcation diagram of the ML model with respect to I is shown in Fig. 1(d). In this paper, I 0 = 39.7 to make sure that all neurons in the network stay at a resting steady state without diversity or noise, i.e. I w = 0 and D = 0. The initial conditions of all neurons are chosen as (V, W) = (4.583 345, 0.298 910 89).

No. 5 Communications in Theoretical Physics 819 3 Multiple Spatial Coherence Resonances and Spiral Waves Induced by Diversity Without Noise In the section, multiple spatial coherence resonances and spiral waves induced by diversity are investigated in the neuronal network without noise. In the absence of noise, the system would remain quiescent forever without parameter diversity. When the parameter diversity is introduced, the ordered spatial pattern emerges. Characteristic snapshots of spatial pattern for different I w are presented, as shown in Fig. 2. It can be observed that the network shows ordered spiral waves in some regions of I w, as presented Fig. 2(b) 2(p). And with the increase of I w, the structure of the spiral wave pattern transits between simpler patterns that have main one tip (Figs. 2(b), 2(d), 2(f), 2(h), 2(j), 2(l), 2(n), 2(p)) and complex pattern patterns that have more than main one tip (Figs. 2(c), 2(e), 2(g), 2(i), 2(k), 2(m), 2(o)). But for small I w, the diversity is not able to evoke ordered spatial pattern, as shown in Fig. 2(a). While for large I w, disordered patterns emerges (Fig. 2(j)). The results implicate that there exist multiple transitions between patterns of spiral waves in the studied neuronal network, implying multiple resonances to be quantified. To measure quantitively the phenomenon of spatial coherence resonance, we may use the structure function P(k x, k y ) of the membrane potential V i,j and the signal-to-noise ratio (SNR) calculated by the structure function P(k x, k y ), which have been widely used in literatures. [3,11,45 46] In the following, the structure function P(k x, k y ) [3,45] of the membrane potential is defined by P(k x, k y ) = H 2 (k x, k y ), (4) where H(k x, k y )is the two-dimensional Fourier transform of V i,j field at a particular time t, and, is the ensemble average over the simulated data series. The spatial structure function P(k x, k y ) values corresponding to different I w are presented in Fig. 3. For intermediate levels of diversity (e.g., I w = 18.7 and I w = 19) there indeed exists a preferred spatial frequency (Fig. 3(b) and Fig. 3(c)), and spatial frequency for simpler pattern (Fig. 3(b)) emerges more remarkable than one complex pattern (Fig. 3(c)). While for smaller and very large I w do not induce particularly expressed spatial frequency shown in Figs. 3(a) and 3(d), respectively.

820 Communications in Theoretical Physics Vol. 57 Fig. 2 In the absence of noise (D = 0), the spatial structure of V ij at different diversity strength I w. (a) I w = 2; (b) I w = 4; (c) I w = 6; (d) I w = 7; (e) I w = 9.5; (f) I w = 10.7; (g) I w = 11; (h) I w = 13.1; (i) I w = 13.15; (j) I w = 15.2; (k) I w = 18.7; (l) I w = 19; (m) I w = 22.5; (n) I w = 24; (o) I w = 28; (p) I w = 30; (q) I w = 60.

No. 5 Communications in Theoretical Physics 821 Then, we calculate the circular average of the spatial structure function P(k x, k y ) [3,45] according to equation p(k) = P(k x, k y ), (5) Ω k where k = (k x, k y ), Ω k is the circular shell with radius k = k. The variation of p(k) for different I w are shown in Fig. 4(a). For intermediate diversity strength (e.g., I w = 18.9 and I w = 19), there indeed exists a particular spatial frequencies, marked with the vertical dashed line at k = k max (such as I w = 19 shown in Fig. 4(b)). Corresponding to the complex spiral wave (e.g., I w = 18.9), the particular spatial frequency exhibits a low and broad peak at k = k max, while corresponding to the simple spiral wave (e.g., I w = 19), one exhibits a high and narrow peak, as shown in Fig. 4(a). But, the fingerprint of spatial coherence was disappeared for I w = 2, while reduced for I w = 60. Fig. 3 Spatial structure functions obtained for different values of I w in the absence of noise. (a) I w = 2; (b) I w = 18.7; (c) I w = 19; (d) I w = 60. Fig. 4 Circular averages p(k) of structure functions P(k x, k y) presented in Fig. 3. In order to extract the particular spatial frequencies more precisely, the SNR [3,45] is calculated for each diversity strength by SNR = p(k max )/ p, (6) where p = (p(k a ) + p(k b ))/2 is an approximation for the level of background fluctuations, k a and k b mark the minimal value around k max, such as I w =19 shown in Fig. 4(b). Variations of SNR curve depend on I w values, as shown in Fig. 5. SNR curve shows multiple local maximal values with respect to increase of I w, showing that there exist multiple coherence resonances. Comparing the SNR curve (Fig. 5) with the pattern of spiral waves (Fig. 2), it can be found that the values of SNR become higher when the structures of spiral wave are simple and lower when the structures are complex. While SNR value becomes very small with breakdown spiral, and no spiral wave. The results show that the transitions between the simple and complex structures of spiral wave led to the emergence of multiple spatial coherence resonances.

822 Communications in Theoretical Physics Vol. 57 Fig. 5 The dependence of the SNR on diversity strength I w in the absence of noise. 4 Diversity-Induced Multiple Spatial Coherence Resonance in Network with Noise In this section, diversity-induced the spatiotemporal dynamics in network with noise is considered. Variations of the SNR curve depend on I w values under the different noise densityd, as shown in Fig. 6. The SNR curves exhibit multi-peaks with respect to the increase of I w, indicating that the multiple spatial coherence resonances can be induced by diversity in neuronal network with noise. Compared the SNR curves shown in Fig. 6, the maximal SNR value is reduced as D is increased, as shown in Fig. 7. It shows that the coherence degree induced by the diversity is suppressed when noise density is increased. Fig. 6 The dependence of the SNR on diverse strength I w under different noise density D. (a) D = 0.003; (b) D = 0.01; (c) D = 0.03; (d) D = 0.1; (e) D = 0.5; (f) D = 2.

No. 5 Communications in Theoretical Physics 823 Fig. 7 The dependence of the maximal SNR values on noise density D. 5 Conclusion and Discussion In summary, we have shown that the parameter diversity can induce the multiple spatial coherence resonances in a two-dimensional neuronal network with or without noise. In the absence of noise, multiple spatial coherence resonances and spiral waves are induced by parameter diversity. The multiple resonances are related to the transitions between simple patterns and complex patterns of spiral waves. Namely, the values of SNR become higher when the structures of spiral wave are simple and lower when the structures are complex. When noise is present in network, multiple spatial coherence resonances and spiral waves induced by the diversity are simulated. The resonance degree induced by diversity is reduced with respect to the increase of noise density. The result shows that the noise strongly influences the diversity-induced multiple spatial coherence resonances. The phenomenon of spatial stochastic or coherence resonances induced by noise have been extensively reported in nonlinear systems. And a single or multiple resonances have also existed. [8 12,14 16] Lately, spatial stochastic or coherence resonances induced by diversity have been investigated in the spatial extended systems. [26,32,37,40 41] However, these results often show that diversity-induced resonances exhibit a single peak in SNR curve. In this paper, multiple spatial coherence resonances can be induced by parameter diversity in a two-dimensional neuronal network with or without noise. Although the presented results offer persuasive evidence for the existence of multiple spatial coherence resonance in the studied neuronal network, the mechanisms are not clear and need further study. The realistic nervous system evolves in noisy environment and is composed of a large number of cells, which display a strong diversity in morphology and dynamical activity. At present, experiments have shown that synchronized states can occur in many special areas of the brain, such as the olfactory system or the hippocampal region, [47 48] and the spiral wave can be discovered in the visual cortex, cardiac tissue, and brain slices (Hippocampal Slice Cultures). [49 51] The synchronization and spiral wave were speculated to be a spatial framework to organize cortical oscillation, might contribute to seizure generation, serve as emergent population pacemakers, and be used for coordinating oscillation phase. [52] Theoretically, synchronization and spiral wave induced by noise have been extensively simulated in neuronal network. [14,53 54] Recently, it has been found that parameter diversity can enhance synchronization in an array of Josephson junctions, [25 26] and induce spiral wave in this paper and in two-dimensional networks of Hodgkin Huxley model. [14] It is sufficiently demonstrated that parameter diversity, similar to noise, plays radical roles in those complex procedures of pattern formation and transition. Our results have indicated that an appropriate level of diversity could improve the response of the system. Such a resonant effect renders us to speculate that diversity has an important function. Considering that the neurons in the network are different in parameters, the results of this paper also suggest that the spiral waves discovered in the experiment on real nervous system [52] might be induced by the parameter diversity. Thus, the presented results show that the interaction between noise and diversity might have a crucial impact on neuronal network, and help to understand the phenomenon in experimental study. References [1] R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A 14 (1981) L453. [2] J.K. Douglass, L. Wilkens, E. Pantazelou, and F. Moss, Nature (London) 365 (1993) 337. [3] M. Perc, Europhys. Lett. 72 (2005) 712. [4] G. Hu, T. Ditzinger, C. Ning, and H. Haken, Phys. Rev. Lett. 71 (1993) 807. [5] T. Wellens, V. Shatokhin, and A. Buchleitner, Rep. Prog. Phys. 67 (2004) 45. [6] E. Manjarrez, I. Mendez, L. Martinez, A. Flores, and C.R. Mirasso, Neurosci. Lett. 415 (2007) 231. [7] H.G. Gu, W. Ren, Q.S. Lu, S.G. Wu, and W.J. Chen, Phys. Lett. A 285 (2001) 63. [8] C.S. Zhou and J. Kurths, Phys. Rev. E 65 (2002) 040101. [9] M. Ozer, M. Perc, and M. Uzuntarla, Phys. Lett. A 373 (2009) 964.

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