Abstract
In this work, we study the collective dynamics and energy aspects of star-coupled Hindmarsh–Rose neuron model with memristor. In the presence of chemical coupling and field effects, the system exhibits desynchrony, synchrony and drum head mode states. The electrically coupled network with field effects shows desynchronized and synchronized regions. The parameter space has been plotted to explain the transition from desynchronized state to synchronized and drum head mode regions. The time evolution of membrane potential in the absence of synaptic coupling reveals that field coupling regulates the electrical modes of the system. Based on Helmholtz theorem, the Hamilton energy function associated with the system has been derived. The average energy variation of chemically coupled neurons shows two important regions. A fluctuating regime corresponding to the desynchronized state and a linearly increasing regime corresponding to the synchronized state with amplitude death have been observed. In electrically coupled star network, the average energy returns to its initial uncoupled value in the synchronized state. The study finds applications in identical and nonidentical networks of chaotic oscillators with different coupling topologies.
Similar content being viewed by others
References
Belykh, I., Hasler, M.: Mesoscale and clusters of synchrony in networks of bursting neurons. Chaos 21, 016106 (2011)
Pikovsky, A., Rosenblum, M.: Dynamics of globally coupled oscillators: progress and perspectives. Chaos 25, 097616 (2015)
Omelchenko, I., Omelchenko, O.E., Hovel, P., Scholl, E.: When nonlocal coupling between oscillators becomes stronger: patched synchrony or multi-chimera states. Phys. Rev. Lett. 110, 224101 (2013)
El-Nashar, H.F., Zhang, Y., Cerdeira, H.A., Ibiyinka, A.F.: Synchronization in a chain of nearest neighbors coupled oscillators with fixed ends. Chaos 13, 1216 (2003)
Pereda, A.E.: Electrical synapses and their functional interactions with chemical synapses. Nat. Rev. Neurosci. 15(4), 250–263 (2014)
Veenstra, R.D.: Cell Physiology, 4th edn. Academic Press, New York (2012)
Thottil, S.K., Ignatius, R.P.: Influence of memristor and noise on h–r neurons. Nonlinear Dyn. 95(1), 239–257 (2018)
Xu, Y., Ying, H., Jia, Y., Ma, J., Hayat, T.: Autaptic regulation of electrical activities in neuron under electromagnetic induction. Sci. Rep. 7, 43452 (2017)
Xu, Y., Jia, Y., Ma, J., Hayat, T., Alsaedi, A.: Collective responses in electrical activities of neurons under field coupling. Sci. Rep. 8, 1349 (2018)
Hrg, D.: Synchronization of two Hindmarsh–Rose neurons with unidirectional coupling. Neural Netw. 40, 73–79 (2013)
Ali, M.K.: Synchronization of a chaotic map in the presence of common noise. Phys. Rev. E 55, 4804 (1997)
Andrade, V., Davidchack, R.L., Lai, Y.C.: Noise scaling of phase synchronization of chaos. Phys. Rev. E 61, 3230 (2000)
Abrams, D.M., Strogatz, S.H.: Chimera states for coupled oscillators. Phys. Rev. Lett. 93, 174102 (2004)
Wang, Z., Shi, X.: Lag synchronization of multiple identical Hindmarsh–Rose neuron models coupled in a ring structure. Nonlinear Dyn. 60(3), 375–383 (2010)
Shi, X., Wang, Z.: Adaptive synchronization of time delay hindmarsh–rose neuron system via self-feedback. Nonlinear Dyn. 69(4), 2147–2153 (2012)
Pecora, L.M., Carroll, T.L.: Synchronization stability in coupled oscillator arrays: solution for arbitrary configurations. Int. J. Bifurcat. Chaos 10(2), 273–290 (1999)
Usha, K., Subha, P.A., Nayak, C.R.: The route to synchrony via drum head mode and mixed oscillatory state in star coupled Hindmarsh–Rose neural network. Chaos Solitons Fractals 108, 25–31 (2018)
Jonq, J., Yu-Hao, L.: Cluster synchronization in networks of neurons with chemical synapses. Chaos 24, 013110 (2014)
Xu, F., Zhang, J., Fang, T., Huang, S., Wang, M.: Synchronous dynamics in neural system coupled with memristive synapse. Nonlinear Dyn. 92(3), 13951402 (2018)
Hu, X., Liu, C., Liu, L., Ni, J., Yao, Y.: Chaotic dynamics in a neural network under electromagnetic radiation. Nonlinear Dyn. 91(3), 1541–1554 (2017)
Lv, M., Wang, C., Ren, G., Ma, J., Song, X.: Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn. 85(3), 1479–1490 (2016)
Xu, Y., Jia, Y., Kirunda, J.B., Shen, J., Ge, M., Lu, L., Pei, Q.: Dynamic behaviors in coupled neuron system with the excitatory and inhibitory autapse under electromagnetic induction. Complexity 2018, 3012743 (2018)
Rostami, Z., Jafari, S., Perc, M., Slavinec, M.: Elimination of spiral waves in excitable media by magnetic induction. Nonlinear Dyn. 94(1), 679692 (2018)
Xu, Y., Jia, Y., Ge, M., Lu, L., Yang, L., Zhan, X.: Effects of ion channel blocks on electrical activity of stochastic hodgkin-huxley neural network under electromagnetic induction. Neurocomputing 283, 196–204 (2018)
Ge, M., Jia, Y., Kirunda, J.B., Xu, Y., Shen, J., Lu, L., Liu, Y., Pei, Q., Zhan, X., Yang, L.: Propagation of firing rate by synchronization in a feed-forward multilayer hindmarsh–rose neural network. Neurocomputing 320, 60–68 (2018)
Lu, L., Jia, Y., Kirunda, J.B., Xu, Y., Ge, M., Pei, Q., Yang, L.: Effects of noise and synaptic weight on propagation of subthreshold excitatory postsynaptic current signal in a feed-forward neural network. Nonlinear Dyn. https://doi.org/10.1007/s11071-018-4652-9 (2019)
Xu, Y., Jia, Y., Wang, H., Liu, Y., Wang, P., Zhao, Y.: Spiking activities in chain neural network driven by channel noise with field coupling. Nonlinear Dyn. https://doi.org/10.1007/s11071-018-04752-2 (2019)
Torrealdea, F.J., Sarasola, C., d’Anjou, A.: Energy consumption and information transmission in model neurons. Chaos Solitons Fractals 40, 60–68 (2009)
Guan, W., Yi, S., Quan, Y.: Exponential synchronization of coupled memristive neural networks via pinning control. Chin. Phys. B 22(5), 050504 (2013)
Wang, Y., Wang, Z.D., Wang, W.: Dynamical behaviors of periodically forced hindmarsh-rose neural model: the role of excitability and ‘intrinsic’ stochastic resonance. J. Phys. Soc. Jpn. 69, 276–283 (2000)
Sarasola, C., Torrealdea, F.J., d’Anjou, A., Moujahid, A., Grana, M.: Energy balance in feedback synchronization of chaotic systems. Phys. Rev. 69, 011606 (2004)
Yu, L.C., Liu, L.W.: Optimal size of stochastic hodgkin-huxley neuronal systems for maximal energy efficiency in coding pulse signal. Phys. Rev. E 89, 032725 (2014)
Lu, L., Jia, Y., Xu, Y., Ge, M., Yang, L., Zhan, X.: Energy dependence on modes of electric activities of neuron driven by different external mixed signals under electromagnetic induction. Sci. China Technol. Sci. 62(3), 427–440 (2019)
Moujahid, A., d’Anjou, A., Torrealdea, F., Torrealdea, F.: Efficient synchronization of structurally adaptive coupled hindmarsh–rose neurons. Chaos Solitons Fractals 44, 929–933 (2011)
Ma, J., Wu, F., Jin, W., Zhou, P., Hayat, T.: Calculation of hamilton energy and control of dynamical systems with different types of attractors. Chaos 27, 053108 (2017)
Ma, J., Zhou, P., Ahmad, B., Ren, G., Wang, C.: Chaos and multi-scroll attractors in rcl-shunted junction coupled jerk circuit connected by memristor. PLoS ONE 13(1), e0191120 (2018)
Usha, K., Subha, P.A.: Energy feedback and synchronous dynamics of hindmarsh–rose neuron model with memristor. Chin. Phys. B 28(2), 020502 (2019)
Xin-lei, A., Li, Z.: Dynamics analysis and hamilton energy control of a generalized lorenz system with hidden attractor. Nonlinear Dyn. 94(4), 2995–3010 (2018)
Li, F., Yao, C.: The infinite-scroll attractor and energy transition in chaotic circuit. Nonlinear Dyn. 84(4), 2305–2315 (2016)
Hindmarsh, J.L., Rose, R.M.: A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B 221, 87–102 (1984)
Rose, R.M., Hindmarsh, J.L.: A model of a thalamic neuron. Proc. R. Soc. Lond. B 225, 161–193 (1985)
Storace, M., Linaro, D., de Lange, E.: The hindmarsh–rose neuron model: bifurcation analysis and piecewise-linear approximations. Chaos 18, 033128 (2008)
Usha, K., Subha, P.A.: Star-coupled hindmarsh–rose neural network with chemical synapses. Int. J. Mod. Phys. C 29, 1850023 (2018)
Thottil, S.K., Ignatius, R.P.: Nonlinear feedback coupling in hindmarsh–rose neurons. Nonlinear Dyn. 87(3), 1879–1899 (2017)
Belykh, V.N., Belykh, I.V., Mosekilde, E.: Cluster synchronization modes in an ensemble of coupled chaotic oscillators. Phys. Rev. E 63, 036216 (2001)
Somers, D., Kopell, N.: Rapid synchronization through fast threshold modulation. Biol. Cybern. 68(5), 393–407 (1993)
Usha, K., Subha, P.A.: Hindmarsh–rose neuron model with memristors. BioSystems 178, 1–9 (2019)
Monteiro, L.H.A., Filho, A.P., Chaui-Berlinck, J.G., Piqueira, J.R.C.: Oscillation death in a two neuron network with delay in a self connection. J. Biol. Syst. 15(1), 49–61 (2007)
Torrealdea, F.J., d’Anjou, A., Grana, M., Sarasola, C.: Energy aspects of the synchronization of model neurons. Phys. Rev. E 74, 011905 (2006)
Acknowledgements
UK would like to acknowledge University Grants Commission, India, for providing financial assistance through JRF scheme for doing the research work. PAS would like to acknowledge DST, India, for their financial assistance through FIST program.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest:
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Usha, K., Subha, P.A. Collective dynamics and energy aspects of star-coupled Hindmarsh–Rose neuron model with electrical, chemical and field couplings. Nonlinear Dyn 96, 2115–2124 (2019). https://doi.org/10.1007/s11071-019-04909-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-019-04909-7