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In a neuronal system, propagation speed of neuronal information is mainly determined by the length, the diameter, and the kind of the axons between the neurons. Thus, some communications between neurons are not instantaneous, and others are instantaneous or with some negligible delay. In the past years, effects of time delay on neuronal dynamics, such as synchronization, stochastic resonance, firing regularity, etc., have been investigated. For stochastic resonance, it has been reported recently that stochastic multi-resonance in a neuronal system can be induced by time delay. However, in these studies, time delay has been introduced to every connection of the neuronal system. As mentioned in the beginning, in a real neuronal system, communication between some neurons can be instantaneous or with some negligible delays. Thus, considering the effect of partial time delay (time delay is called as partial time delay if only part of connections are delayed) on neuronal dynamics could be more meaningful.In this paper, we focus on discussing effect of partial time delay on response amplitude of a Watts-Strogatz neuronal network which is locally modeled by Rulkov map. With the numerically obtained results, we can see that partial time delay can induce a stochastic multi-resonance which is indicated by the multi-peak characteristics in the variation of response amplitude with partial time delay. Namely, partial time delay could also induce stochastic multi-resonance in a neuronal system. Moreover, we also find that optimal response amplitude can be reached in much wider range of the partial time delay value when delayed connections are less (i.e., the partial time delay probability is small). This is different from the case in which all connections are delayed, where response amplitude become optimal only when time delay is nearly the multiples of external signal's period. But the range of the partial time delay value becomes narrower and narrower with the increasing of the partial time delay probability and when the partial time delay probability is large enough, response amplitude becomes optimal only when time delay is nearly the multiples of external signal period. It is similar to the case where all connections are delayed. Furthermore, effects of random rewiring probability and total link number in the neuronal network on partial time delay induced stochastic multi-resonance are also studied. It is found that partial time delay induced stochastic multi-resonance is robust to random rewiring probability but not robust to total link number. Stochastic resonance is a very important nonlinear phenomenon in neuroscience, thus, our obtained results could have some implications in this field.
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Keywords:
- stochastic multi-resonance /
- neuronal network /
- partial time delay
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[1] Benzi R, Sutera A, Vulpiani A 1981 J. Phys. A: Math. Gen. 14 L453
[2] Pikovsky A S, Kurths J 1997 Phys. Rev. Lett. 78 775
[3] Masoliver J, Robinson A, Weiss G H 1995 Phys. Rev. E 51 4021
[4] Porra J M 1997 Phys. Rev. E 55 6533
[5] Collins J J, Chow C C, Capela A C, Imhoff T T 1996 Phys. Rev. E 54 5575
[6] Collins J J, Chow C C, Imhoff T T 1995 Phys. Rev. E 52 R3321
[7] Heneghan C, Chow C C, Collins J J, Imhoff T T, Lowen S B, Teich M C 1996 Phys. Rev. E 54 R2228
[8] Vilar J M G, Rub J M 1997 Phys. Rev. Lett. 78 2882
[9] Longtin A, Bulsara A, Moss F 1991 Phys. Rev. Lett. 67 656
[10] Douglass J K, Wilkens L, Pantazelou E, Moss F 1993 Nature 365 337
[11] Sun Z K, Lu P J, Xu W 2014 Acta Phys. Sin. 63 220503 (in Chinese) [孙中奎, 鲁捧菊, 徐伟 2014 物理学报 63 220503]
[12] Jin Y F 2015 Chin. Phys. B 24 110501
[13] Xu Y, Wu J, Zhang H Q, Ma S J 2012 Nonlinear Dyn. 70 531
[14] Gammaitoni L, Hnggi P, Jung P 1998 Rev. Mod. Phys. 70 223
[15] Lindner B, Garcia-Ojalvo J, Neiman A, Schimansky-Geier L 2004 Phys. Rep. 392 321
[16] Perc M 2007 Phys. Rev. E 76 066203
[17] Sun X J, Perc M, Lu Q S, Kurths J 2008 Chaos 18 023102
[18] Sun X J, Lu Q S 2014 Chin. Phys. Lett. 31 020502
[19] Qin H X, Ma J, Wang C N, Wu Y 2014 PLoS One 9 e100849
[20] Gu H G, Jia B, Li Y Y, Chen G R 2013 Physica A 392 1361
[21] Yu H T, Guo X M, Wang J, Deng B, Wei X L 2015 Physica A 419 307
[22] Volkov E I, Ullner E, Kurths J 2005 Chaos 15 023105
[23] Liu Z Q, Zhang H M, Li Y Y, Hua C C, Gu H G, Ren W 2010 Physica A 389 2642
[24] Lin X, Gong Y B, Wang L 2011 Chaos 21 043109
[25] Jia Y B, Gu H G 2015 Chaos 25 123124
[26] Wang Q Y, Zhang H H, Perc M, Chen G R 2012 Commun. Nonlinear Sci. Numer. Simul. 17 3979
[27] Wang Q Y, Perc M, Duan Z S, Chen G R 2009 Chaos 19 023112
[28] Hao Y H, Gong Y B, Lin X 2011 Neurocomputing 74 1748
[29] Rulkov N F 2001 Phys. Rev. Lett. 86 183
[30] Ibarz B, Casado J M, Sanjuan M A F 2011 Phys. Rep. 501 1
[31] Hilborn R C 2004 Am. J. Phys. 72 528
[32] Rulkov N F, Timofeev I, Bazhenov M 2004 J. Comput. Neurosci. 17 203
[33] Rulkov N F, Bazhenov M 2008 J. Biol. Phys. 34 279
[34] Nowotny T, Huerta R, Abarbanel H D I, Rabinovich M I 2005 Biol. Cybern. 93 436
[35] Watts D J, Strogatz S H 1998 Nature 393 440
[36] Landa P S, McClintock P V E 2000 J. Phys. A: Math. Gen. 33 L433
[37] Zaikin U A, Garca-Ojalvo J, Bscones R, Kurths J 2003 Phys. Lett. A 312 348
[38] Rajasekar S, Used J, Wagemakers A, Sanjuan M A F 2012 Commun. Nonlinear Sci. Numer. Simul. 17 3435
[39] Zhao Z G, Gu H G 2015 Chaos, Solitions Fractals 80 96
[40] Gu H G 2015 PLoS One 10 e0138593
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