Unstable discrete modes in Hindmarsh-Rose neural networks under magnetic flow effect. (English) Zbl 1448.92023
Summary: The competitive effect between electric and magnetic flux couplings is used, in the context of modulational instability, to describe the collective dynamics in a modified Hindmarsh-Rose neural networks. The multiple-scale expansion is utilized to reduce the system to a nonlinear differential-difference equation, whose plane wave solutions are found to be unstable for some values of parameters. Particular interest is given to the influence of changing both the electric and magnetic coupling strengths, and confirmation of analytical results is given via numerical integration of the generic Hindmarsh-Rose model. The model presents a rich variety of spatiotemporal patterns propagating in the network, as the result of the interplay between nonlinear and dispersive effects. The electromagnetic induction appears to be responsible for regular bursting patterns and synchronous states in the network. With increasing the electric coupling, full synchronization is difficult to realize and irregular spatiotemporal patterns of action potentials are predominant.
MSC:
| 92B20 | Neural networks for/in biological studies, artificial life and related topics |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
References:
| [1] | Hodgkin, A. L.; Huxley, A. F.; Physiol, J., (London), 117, 500 (1952) |
| [2] | FitzHugh, R., Biophys J, 1, 445 (1961) |
| [3] | Hindmarsh, J. L.; Rose, R. M., Proc R Soc London Ser-B, 221, 87 (1984) |
| [4] | Binczak, S.; Eilbeck, J. C.; Scott, A. C., Physica D, 148, 159 (2001) · Zbl 0961.92007 |
| [5] | Grothe, B.; Carr, C.; Casseday, J.; Fritzsch, B.; Köppl, C., The Evolution of Central Pathways and their Neural Processing Patterns, (Manley, G. A.; Popper, A. N.; Fay, R. R. (2004), Springer: Springer New York) |
| [6] | Bose, A., SIAM J Appl Math, 55, 1650 (1995) · Zbl 0836.34055 |
| [7] | Maïna, I.; Tabi, C. B.; Mohamadou, A.; Ekobena, H. P.F.; Kofané, T. C., Chaos, 25, 043118 (2015) · Zbl 1374.92024 |
| [8] | Tang, J.; Ma, J., Sci China Technol Sci, 60, 1011 (2017) |
| [9] | Volterra, A.; Meldolesi, J., Nat Rev Neurosci, 6, 626 (2005) |
| [10] | Guo, S. L.; Tang, J.; Ma, J.; Wang, C., Complexity, 2017, ID4631602 (2017) |
| [11] | Etémé, A. S.; Tabi, C. B.; Mohamadou, A., Commun Nonl Sci Numer Simul, 43, 211 (2017) · Zbl 1467.92016 |
| [12] | Etémé, A. S.; Tabi, C. B.; Mohamadou, A., Chaos Solit Fract, 104, 813 (2017) · Zbl 1380.92007 |
| [13] | Bekkers, J. M., Curr Biol, 13, R433 (2003) |
| [14] | Bekkers, J. M., Curr Biol, 8, R52 (1998) |
| [15] | Wang, L.; Zeng, Y. J., Neurol Sci, 34, 1977 (2013) |
| [16] | Wang, C. N.; Guo, R. L.; Xu, Y.; Tang, J.; Alzahrani, F.; Hobiny, A., Complexity, 2017, ID5436737 (2017) · Zbl 1367.92025 |
| [17] | Guo, S. L.; Wang, C. N.; Ma, J.; Jin, W., Neurocomputing, 216, 627 (2016) |
| [18] | Majhi, S.; Perc, M.; Ghosh, D., Chaos, 27, 073109 (2017) |
| [19] | Bera, B. K.; Majhi, S.; Ghosh, D.; Perc, M., EPL, 118, 10001 (2017) |
| [20] | Rakshit, S.; Bera, B. K.; Perc, M.; Ghosh, D., Phys Rep, 7, 2412 (2017) |
| [21] | Majhi, S.; Perc, M.; Ghosh, D., Phys Rep, 6, 39033 (2016) |
| [22] | Lv, M.; Ma, J., Neurocomputing, 205, 375 (2016) |
| [23] | Lv, M.; Wang, C.; Ren, G.; Ma, J.; Song, X., Nonl Dyn, 85, 1479 (2016) |
| [24] | Chua, L. O., IEEE Trans Circ Theory, 18, 507 (1971) |
| [25] | Xu, Y.; Jia, Y.; Ma, J.; Alsaedi, A.; Ahmad, B., Chaos Solit Fract, 104, 435 (2017) |
| [26] | Li, Q. D.; Tang, S.; Zeng, H. Z.; Zhou, T., Nonl Dyn, 78, 1087 (2014) · Zbl 1331.94090 |
| [27] | Rostami, Z.; Jafari, S.; Perc, M.; Slavinec, M., Nonl Dyn, 94, 679 (2018) |
| [28] | Ma, J.; Wang, Y.; Wang, C. N.; Ren, G., Chaos Solit Fract, 99, 219 (2017) |
| [29] | Ozer, M.; Uzuntarla, M.; Kayikcioglu, T.; Graham, L. J., Phys Lett A, 372, 6498 (2008) · Zbl 1225.92009 |
| [30] | Uzun, R.; Ozer, M.; Perc, M., EPL, 105, 60002 (2014) |
| [31] | Perc, M., Phys Rev E, 72, 016207 (2005) |
| [32] | Perc, M., Chaos Solit Fract, 31, 64 (2007) |
| [33] | Sun, X.; Perc, M.; Lu, Q.; Kurths, J., Chaos, 18, 023102 (2008) |
| [34] | Yilmaz, E.; Ozer, M.; Baysal, V.; Perc, M., Phys Rep, 6, 30914 (2016) |
| [35] | Tabi, C. B.; Etémé, A. S.; Mohamadou, A., Physica A, 474, 186 (2017) · Zbl 1400.92026 |
| [36] | Panguetna, C. S.; Tabi, C. B.; Mohamadou, A., Commun Nonl Sci Numer Simul, 55, 326 (2018) |
| [37] | Panguetna, C. S.; Tabi, C. B.; Mohamadou, A., Phys Plasmas, 24, 092114 (2017) |
| [38] | Koko, A. D.; Tabi, C. B.; Ekobena, H. P.F.; Mohamadou, A.; Kofané, T. C., Chaos, 22, 043110 (2012) · Zbl 1319.81091 |
| [39] | Tabi, C. B.; Dang, A. K.; Doko, R. O.; Ekobena, H. P.F.; Kofané, T. C., Physica A, 442, 498 (2016) · Zbl 1400.92415 |
| [40] | Tabi, C. B.; Ondoua, R. Y.; Ekobena, H. P.; Mohamadou, A.; Kofané, T. C., Phys Lett A, 380, 2374 (2016) |
| [41] | Tabi, C. B.; Maïna, I.; Mohamadou, A.; Ekobena, H. P.F.; Kofané, T. C., Physica A, 435, 1 (2015) · Zbl 1400.92203 |
| [42] | Wang, Q. Y.; Duan, Z. S.; Perc, M.; Chen, G. R., EPL, 83, 50008 (2008) |
| [43] | Wang, Q. Y.; Perc, M.; Duan, Z. S.; Chen, G. R., Phys Rev E, 80, 026206 (2009) |
| [44] | Wang, Q. Y.; Chen, G. R.; Perc, M., PLoS ONE, 6, e15851 (2011) |
| [45] | Hao, Y. H.; Gong, Y. B.; Wang, L.; Ma, X. G.; Yang, C. L., Chaos Soliton Fract, 44, 260 (2011) |
| [46] | Guo, D. Q.; Wang, Q. Y.; Perc, M., Phys Rev E, 85, 061905 (2012) |
| [47] | Xu, B.; Gong, Y. B.; Wang, L.; Wu, Y. N., Nonlin Dyn, 72, 79 (2013) |
| [48] | Sun, X. J.; Lei, J. Z.; Perc, M.; Kurths, J., Chaos, 21, 016110 (2011) |
| [49] | Tabi, C. B.; Maïna, I.; Mohamadou, A.; Ekobena, H. P.F.; Kofané, T. C., EPL, 106, 18005 (2014) |
| [50] | Ma, J.; Lv, M.; Zhou, P.; Xu, Y.; Hayat, T., Appl Math Comput, 307, 321 (2017) · Zbl 1411.92049 |
| [51] | Ren, G.; Xu, Y.; Wang, C., Nonl Dyn, 88, 893 (2017) |
| [52] | Wang, Y.; Ma, J.; Xu, Y.; Wu, F.; Zhou, P., Int J Bifurcat Chaos, 27, 1750030 (2017) · Zbl 1362.34080 |
| [53] | Xu, Y.; Jia, Y.; Ma, J.; Hayat, T.; Alsaedi, A., Sci Rep, 8, 1349 (2018) |
| [54] | Weicker, L.; Erneux, T.; Keuninckx, L.; Danckaert, J., Phys Rev E, 89, 012908 (2014) |
| [55] | Wang, Q.; Zheng, Y.; Ma, J., Chaos Solit Fract, 56, 19 (2013) |
| [56] | Leon, J.; Manna, M., J Phys A: Math Gen, 32, 2845 (1999) · Zbl 0941.35110 |
| [57] | Leon, J.; Manna, M., Phys Rev Lett, 83, 2324 (1999) |
| [58] | Tabi, C. B.; Maïna, I.; Mohamadou, A.; Ekobena, H. P.F.; Kofané, T. C., Physica A, 435, 435 (2015) · Zbl 1400.92203 |
| [59] | Tabi, C. B.; Mohamadou, A.; Kofané, T. C., Eur Phys J E, 32, 327 (2010) |
| [60] | Tabi, C. B.; Mohamadou, A.; Kofané, T. C., Chaos, 19, 043101 (2009) · Zbl 1311.92148 |
| [61] | Rosenblum, M. G.; Pikovsky, A. S., Phys Rev Lett, 92, 074104 (2004) |
| [62] | Dhamala, M.; Jirsa, V. K.; Ding, M., Phys Rev Lett, 92, 028101 (2004) |
| [63] | He, D. H.; Hu, G.; Zhan, M.; Lu, H. P., Physica D, 156, 314 (2001) · Zbl 1026.92008 |
| [64] | Guo, S.; Xu, Y.; Wang, C.; Jin, W.; Hobiny, A.; Ma, J., Chaos Solit Fract, 105, 120 (2017) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
