Research Article
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Route to Chaos and Chimera States in a Network of Memristive Hindmarsh-Rose Neurons Model with External Excitation

Year 2022, Volume 4, Issue 3, 119 - 127, 30.11.2022
https://doi.org/10.51537/chaos.1144123

Abstract

In this paper we have introduced and investigated the collective behavior of a network of memristive Hindmarsh-Rose (HR) neurons. The proposed model was built considering the memristive autapse of the traditional 2D HR neuron. Using the one-parameter bifurcation diagram and its corresponding maximal Lyapunov exponent graph, we showed that the proposed model was able to exhibit a reverse period doubling route to chaos, phenomenon of interior and exterior crises. Three different configurations of the ring-star network of the memristive HR neuron model, including ring-star, ring, and star, have been considered. The study of those network configurations revealed incoherent, coherent , chimera and cluster state behaviors. Coherent behavior is characterized by synchronization of the neurons of the network, while incoherent behaviors are characterized by the absence of synchronization. Chimera states refer to a differet state where there is a coexistence of synchroniaed and asynchronized nodes of the network. One of the interesting result of the paper is the prevalence of double-well chimera states in both ring and ring-star network and has been first mentioned in the case of memrisitve HR neuron model.

References

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  • Guo, Y., Z. Zhu, C. Wang, and G. Ren, 2020 Coupling synchronization between photoelectric neurons by using memristive synapse. Optik 218: 164993.
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  • Hou, Z., J. Ma, X. Zhan, L. Yang, and Y. Jia, 2021 Estimate the electrical activity in a neuron under depolarization field. Chaos, Solitons & Fractals 142: 110522.
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  • Joshi, S. K., 2021 Synchronization of coupled hindmarsh-rose neuronal dynamics: Analysis and experiments. IEEE Transactions on Circuits and Systems II: Express Briefs 69: 1737–1741.
  • Li, K., H. Bao, H. Li, J. Ma, Z. Hua, et al., 2021a Memristive rulkov neuron model with magnetic induction effects. IEEE Transactions on Industrial Informatics 18: 1726–1736.
  • Li, Y., 2021 Simulation of memristive synapses and neuromorphic computing on a quantum computer. Physical Review Research 3: 023146.
  • Li, Z., H. Zhou, M. Wang, and M. Ma, 2021b Coexisting firing patterns and phase synchronization in locally active memristor coupled neurons with hr and fn models. Nonlinear Dynamics 104: 1455–1473.
  • Lin, H., C. Wang, Q. Deng, C. Xu, Z. Deng, et al., 2021 Review on chaotic dynamics of memristive neuron and neural network. Nonlinear Dynamics 106: 959–973.
  • Lin, H., C. Wang, Y. Sun, and W. Yao, 2020 Firing multistability in a locally active memristive neuron model. Nonlinear Dynamics 100: 3667–3683.
  • Liu, Y., W.-j. Xu, J. Ma, F. Alzahrani, and A. Hobiny, 2020 A new photosensitive neuron model and its dynamics. Frontiers of Information Technology & Electronic Engineering 21: 1387–1396.
  • Liu, Z., C. Wang, G. Zhang, and Y. Zhang, 2019 Synchronization between neural circuits connected by hybrid synapse. International Journal of Modern Physics B 33: 1950170.
  • Muni, S. S., H. O. Fatoyinbo, and I. Ghosh, 2022 Dynamical effects of electromagnetic flux on chialvo neuron map: nodal and network behaviors. arXiv preprint arXiv:2201.03219 .
  • Muni, S. S. and A. Provata, 2020 Chimera states in ring–star network of chua circuits. Nonlinear Dynamics 101: 2509–2521.
  • Njitacke, Z. T., J. Awrejcewicz, B. Ramakrishnan, K. Rajagopal, and J. Kengne, 2022a Hamiltonian energy computation and complex behavior of a small heterogeneous network of three neurons: circuit implementation. Nonlinear Dynamics 107: 2867–2886.
  • Njitacke, Z. T., I. S. Doubla, S. Mabekou, and J. Kengne, 2020 Hidden electrical activity of two neurons connected with an asymmetric electric coupling subject to electromagnetic induction: coexistence of patterns and its analog implementation. Chaos, Solitons & Fractals 137: 109785.
  • Njitacke, Z. T., S. D. Isaac, T. Nestor, and J. Kengne, 2021a Window of multistability and its control in a simple 3d hopfield neural network: application to biomedical image encryption. Neural Computing and Applications 33: 6733–6752.
  • Njitacke, Z. T., B. N. Koumetio, B. Ramakrishnan, G. D. Leutcho, T. F. Fozin, et al., 2021b Hamiltonian energy and coexistence of hidden firing patterns from bidirectional coupling between two different neurons. Cognitive Neurodynamics pp. 1–18.
  • Njitacke, Z. T., C. N. Takembo, J. Awrejcewicz, H. P. E. Fouda, and J. Kengne, 2022b Hamilton energy, complex dynamical analysis and information patterns of a new memristive fitzhugh-nagumo neural network. Chaos, Solitons & Fractals 160: 112211. Njitacke, Z. T., N. Tsafack, B. Ramakrishnan, K. Rajagopal,
  • J. Kengne, et al., 2021c Complex dynamics from heterogeneous coupling and electromagnetic effect on two neurons: Application in images encryption. Chaos, Solitons & Fractals 153: 111577.
  • Njitacke Tabekoueng, Z., S. Shankar Muni, T. Fonzin Fozin, G. Dolvis Leutcho, and J. Awrejcewicz, 2022 Coexistence of infinitely many patterns and their control in heterogeneous coupled neurons through a multistable memristive synapse. Chaos: An Interdisciplinary Journal of Nonlinear Science 32: 053114.
  • Qin, Y., T. Menara, D. S. Bassett, and F. Pasqualetti, 2021 Phaseamplitude coupling in neuronal oscillator networks. Physical Review Research 3: 023218.
  • Roberts, L. G. and B. D. Wessler, 1970 Computer network development to achieve resource sharing. In Proceedings of the May 5-7, 1970, spring joint computer conference, pp. 543–549.
  • Shaffer, A., A. L. Harris, R. Follmann, and E. Rosa Jr, 2016 Bifurcation transitions in gap-junction-coupled neurons. Physical Review E 94: 042301.
  • Shepelev, I., A. Bukh, S. Muni, and V. Anishchenko, 2020a Role of solitary states in forming spatiotemporal patterns in a 2d lattice of van der pol oscillators. Chaos, Solitons & Fractals 135: 109725.
  • Shepelev, I., S. Muni, and T. Vadivasova, 2021a Spatiotemporal patterns in a 2d lattice with linear repulsive and nonlinear attrac tive coupling. Chaos: An Interdisciplinary Journal of Nonlinear Science 31: 043136.
  • Shepelev, I. A., A. V. Bukh, S. S. Muni, and V. S. Anishchenko, 2020b Quantifying the transition from spiral waves to spiral wave chimeras in a lattice of self-sustained oscillators. Regular and Chaotic Dynamics 25: 597–615.
  • Shepelev, I. A., S. S. Muni, E. Schöll, and G. I. Strelkova, 2021b Repulsive inter-layer coupling induces anti-phase synchronization. Chaos: An Interdisciplinary Journal of Nonlinear Science 31: 063116.
  • Shepelev, I. A., S. S. Muni, and T. E. Vadivasova, 2021c Synchronization of wave structures in a heterogeneous multiplex network of 2d lattices with attractive and repulsive intra-layer coupling. Chaos: An Interdisciplinary Journal of Nonlinear Science 31: 021104.
  • Shu, H., J. Zhou, Q. Lian, H. Li, D. Zhao, et al., 2021 Modeling gene regulatory networks using neural network architectures. Nature Computational Science 1: 491–501.
  • Sysoeva, M. V., I. V. Sysoev, M. D. Prokhorov, V. I. Ponomarenko, and B. P. Bezruchko, 2021 Reconstruction of coupling structure in network of neuron-like oscillators based on a phase-locked loop. Chaos, Solitons & Fractals 142: 110513.
  • Tabekoueng Njitacke, Z., J. Kengne, and H. B. Fotsin, 2020a Coexistence of multiple stable states and bursting oscillations in a 4d hopfield neural network. Circuits, Systems, and Signal Processing 39: 3424–3444.
  • Tabekoueng Njitacke, Z., C. Laura Matze, M. Fouodji Tsotsop, and J. Kengne, 2020b Remerging feigenbaum trees, coexisting behaviors and bursting oscillations in a novel 3d generalized hopfield neural network. Neural Processing Letters 52: 267–289.
  • Tabekoueng Njitacke, Z., I. Sami Doubla, J. Kengne, and A. Cheukem, 2020c Coexistence of firing patterns and its control in two neurons coupled through an asymmetric electrical synapse. Chaos: An Interdisciplinary Journal of Nonlinear Science 30: 023101.
  • Takembo, C. N., H. P. E. Fouda, and T. C. Kofane, 2022 Modulational instability in chain diffusive neuronal networks under electric field. Indian Journal of Physics pp. 1–9.
  • Tsumoto, K., H. Kitajima, T. Yoshinaga, K. Aihara, and H. Kawakami, 2006 Bifurcations in morris–lecar neuron model. Neurocomputing 69: 293–316.
  • Wouapi, K., B. H. Fotsin, F. P. Louodop, K. F. Feudjio, Z. T. Njitacke, et al., 2020 Various firing activities and finite-time synchronization of an improved hindmarsh–rose neuron model under electric field effect. Cognitive Neurodynamics 14: 375–397.
  • Wouapi, M. K., B. H. Fotsin, E. B. M. Ngouonkadi, F. F. Kemwoue, and Z. T. Njitacke, 2021 Complex bifurcation analysis and synchronization optimal control for hindmarsh–rose neuron model under magnetic flow effect. Cognitive neurodynamics 15: 315–347.
  • Wu, F., J. Ma, and G. Zhang, 2020 Energy estimation and coupling synchronization between biophysical neurons. Science China Technological Sciences 63: 625–636.
  • Xu, Q., T. Liu, C.-T. Feng, H. Bao, H.-G. Wu, et al., 2021 Continuous non-autonomous memristive rulkov model with extreme multistability. Chinese Physics B 30: 128702.
  • Yao, Z., P. Zhou, Z. Zhu, and J. Ma, 2021 Phase synchronization between a light-dependent neuron and a thermosensitive neuron. Neurocomputing 423: 518–534.
  • Zhang, G., D. Guo, F. Wu, and J. Ma, 2020a Memristive autapse involving magnetic coupling and excitatory autapse enhance firing. Neurocomputing 379: 296–304.
  • Zhang, G., C. Wang, F. Alzahrani, F. Wu, and X. An, 2018 Investigation of dynamical behaviors of neurons driven by memristive synapse. Chaos, Solitons & Fractals 108: 15–24.
  • Zhang, Y., C. Wang, J. Tang, J. Ma, and G. Ren, 2020b Phase coupling synchronization of fhn neurons connected by a josephson junction. Science China Technological Sciences 63: 2328–2338.
  • Zhou, J.-F., E.-H. Jiang, B.-L. Xu, K. Xu, C. Zhou, et al., 2021a Synaptic changes modulate spontaneous transitions between tonic and bursting neural activities in coupled hindmarsh-rose neurons. Physical Review E 104: 054407.
  • Zhou, P., Z. Yao, J. Ma, and Z. Zhu, 2021b A piezoelectric sensing neuron and resonance synchronization between auditory neurons under stimulus. Chaos, Solitons & Fractals 145: 110751.

Year 2022, Volume 4, Issue 3, 119 - 127, 30.11.2022
https://doi.org/10.51537/chaos.1144123

Abstract

References

  • Bao, B., A. Hu, Q. Xu, H. Bao, H. Wu, et al., 2018 Ac-induced coexisting asymmetric bursters in the improved hindmarsh–rose model. Nonlinear Dynamics 92: 1695–1706.
  • Bao, H., A. Hu, W. Liu, and B. Bao, 2019 Hidden bursting firings and bifurcation mechanisms in memristive neuron model with threshold electromagnetic induction. IEEE transactions on neural networks and learning systems 31: 502–511.
  • Buri´c, N., K. Todorovi´c, and N. Vasovi´c, 2008 Synchronization of bursting neurons with delayed chemical synapses. Physical Review E 78: 036211.
  • Cai, J., H. Bao, Q. Xu, Z. Hua, and B. Bao, 2021 Smooth nonlinear fitting scheme for analog multiplierless implementation of hindmarsh–rose neuron model. Nonlinear Dynamics 104: 4379– 4389.
  • Chay, T. R., 1985 Chaos in a three-variable model of an excitable cell. Physica D: Nonlinear Phenomena 16: 233–242.
  • Doubla Isaac, S., Z. T. Njitacke, and J. Kengne, 2020 Effects of low and high neuron activation gradients on the dynamics of a simple 3d hopfield neural network. International Journal of Bifurcation and Chaos 30: 2050159.
  • Galinsky, V. L. and L. R. Frank, 2021 Collective synchronous spiking in a brain network of coupled nonlinear oscillators. Physical review letters 126: 158102.
  • Guo, Y., Z. Zhu, C. Wang, and G. Ren, 2020 Coupling synchronization between photoelectric neurons by using memristive synapse. Optik 218: 164993.
  • Hindmarsh, J. and R. Rose, 1982 A model of the nerve impulse using two first-order differential equations. Nature 296: 162–164.
  • Hindmarsh, J. L. and R. Rose, 1984 A model of neuronal bursting using three coupled first order differential equations. Proceedings of the Royal society of London. Series B. Biological sciences 221: 87–102.
  • Hodgkin, A. and A. Huxley, 1990 A quantitative description of membrane current and its application to conduction and excitation in nerve. Bulletin of mathematical biology 52: 25–71.
  • Hou, Z., J. Ma, X. Zhan, L. Yang, and Y. Jia, 2021 Estimate the electrical activity in a neuron under depolarization field. Chaos, Solitons & Fractals 142: 110522.
  • Izhikevich, E. M., 2003 Simple model of spiking neurons. IEEE Transactions on neural networks 14: 1569–1572. Izhikevich, E. M. and R. FitzHugh, 2006 Fitzhugh-nagumo model. Scholarpedia 1: 1349.
  • Joshi, S. K., 2021 Synchronization of coupled hindmarsh-rose neuronal dynamics: Analysis and experiments. IEEE Transactions on Circuits and Systems II: Express Briefs 69: 1737–1741.
  • Li, K., H. Bao, H. Li, J. Ma, Z. Hua, et al., 2021a Memristive rulkov neuron model with magnetic induction effects. IEEE Transactions on Industrial Informatics 18: 1726–1736.
  • Li, Y., 2021 Simulation of memristive synapses and neuromorphic computing on a quantum computer. Physical Review Research 3: 023146.
  • Li, Z., H. Zhou, M. Wang, and M. Ma, 2021b Coexisting firing patterns and phase synchronization in locally active memristor coupled neurons with hr and fn models. Nonlinear Dynamics 104: 1455–1473.
  • Lin, H., C. Wang, Q. Deng, C. Xu, Z. Deng, et al., 2021 Review on chaotic dynamics of memristive neuron and neural network. Nonlinear Dynamics 106: 959–973.
  • Lin, H., C. Wang, Y. Sun, and W. Yao, 2020 Firing multistability in a locally active memristive neuron model. Nonlinear Dynamics 100: 3667–3683.
  • Liu, Y., W.-j. Xu, J. Ma, F. Alzahrani, and A. Hobiny, 2020 A new photosensitive neuron model and its dynamics. Frontiers of Information Technology & Electronic Engineering 21: 1387–1396.
  • Liu, Z., C. Wang, G. Zhang, and Y. Zhang, 2019 Synchronization between neural circuits connected by hybrid synapse. International Journal of Modern Physics B 33: 1950170.
  • Muni, S. S., H. O. Fatoyinbo, and I. Ghosh, 2022 Dynamical effects of electromagnetic flux on chialvo neuron map: nodal and network behaviors. arXiv preprint arXiv:2201.03219 .
  • Muni, S. S. and A. Provata, 2020 Chimera states in ring–star network of chua circuits. Nonlinear Dynamics 101: 2509–2521.
  • Njitacke, Z. T., J. Awrejcewicz, B. Ramakrishnan, K. Rajagopal, and J. Kengne, 2022a Hamiltonian energy computation and complex behavior of a small heterogeneous network of three neurons: circuit implementation. Nonlinear Dynamics 107: 2867–2886.
  • Njitacke, Z. T., I. S. Doubla, S. Mabekou, and J. Kengne, 2020 Hidden electrical activity of two neurons connected with an asymmetric electric coupling subject to electromagnetic induction: coexistence of patterns and its analog implementation. Chaos, Solitons & Fractals 137: 109785.
  • Njitacke, Z. T., S. D. Isaac, T. Nestor, and J. Kengne, 2021a Window of multistability and its control in a simple 3d hopfield neural network: application to biomedical image encryption. Neural Computing and Applications 33: 6733–6752.
  • Njitacke, Z. T., B. N. Koumetio, B. Ramakrishnan, G. D. Leutcho, T. F. Fozin, et al., 2021b Hamiltonian energy and coexistence of hidden firing patterns from bidirectional coupling between two different neurons. Cognitive Neurodynamics pp. 1–18.
  • Njitacke, Z. T., C. N. Takembo, J. Awrejcewicz, H. P. E. Fouda, and J. Kengne, 2022b Hamilton energy, complex dynamical analysis and information patterns of a new memristive fitzhugh-nagumo neural network. Chaos, Solitons & Fractals 160: 112211. Njitacke, Z. T., N. Tsafack, B. Ramakrishnan, K. Rajagopal,
  • J. Kengne, et al., 2021c Complex dynamics from heterogeneous coupling and electromagnetic effect on two neurons: Application in images encryption. Chaos, Solitons & Fractals 153: 111577.
  • Njitacke Tabekoueng, Z., S. Shankar Muni, T. Fonzin Fozin, G. Dolvis Leutcho, and J. Awrejcewicz, 2022 Coexistence of infinitely many patterns and their control in heterogeneous coupled neurons through a multistable memristive synapse. Chaos: An Interdisciplinary Journal of Nonlinear Science 32: 053114.
  • Qin, Y., T. Menara, D. S. Bassett, and F. Pasqualetti, 2021 Phaseamplitude coupling in neuronal oscillator networks. Physical Review Research 3: 023218.
  • Roberts, L. G. and B. D. Wessler, 1970 Computer network development to achieve resource sharing. In Proceedings of the May 5-7, 1970, spring joint computer conference, pp. 543–549.
  • Shaffer, A., A. L. Harris, R. Follmann, and E. Rosa Jr, 2016 Bifurcation transitions in gap-junction-coupled neurons. Physical Review E 94: 042301.
  • Shepelev, I., A. Bukh, S. Muni, and V. Anishchenko, 2020a Role of solitary states in forming spatiotemporal patterns in a 2d lattice of van der pol oscillators. Chaos, Solitons & Fractals 135: 109725.
  • Shepelev, I., S. Muni, and T. Vadivasova, 2021a Spatiotemporal patterns in a 2d lattice with linear repulsive and nonlinear attrac tive coupling. Chaos: An Interdisciplinary Journal of Nonlinear Science 31: 043136.
  • Shepelev, I. A., A. V. Bukh, S. S. Muni, and V. S. Anishchenko, 2020b Quantifying the transition from spiral waves to spiral wave chimeras in a lattice of self-sustained oscillators. Regular and Chaotic Dynamics 25: 597–615.
  • Shepelev, I. A., S. S. Muni, E. Schöll, and G. I. Strelkova, 2021b Repulsive inter-layer coupling induces anti-phase synchronization. Chaos: An Interdisciplinary Journal of Nonlinear Science 31: 063116.
  • Shepelev, I. A., S. S. Muni, and T. E. Vadivasova, 2021c Synchronization of wave structures in a heterogeneous multiplex network of 2d lattices with attractive and repulsive intra-layer coupling. Chaos: An Interdisciplinary Journal of Nonlinear Science 31: 021104.
  • Shu, H., J. Zhou, Q. Lian, H. Li, D. Zhao, et al., 2021 Modeling gene regulatory networks using neural network architectures. Nature Computational Science 1: 491–501.
  • Sysoeva, M. V., I. V. Sysoev, M. D. Prokhorov, V. I. Ponomarenko, and B. P. Bezruchko, 2021 Reconstruction of coupling structure in network of neuron-like oscillators based on a phase-locked loop. Chaos, Solitons & Fractals 142: 110513.
  • Tabekoueng Njitacke, Z., J. Kengne, and H. B. Fotsin, 2020a Coexistence of multiple stable states and bursting oscillations in a 4d hopfield neural network. Circuits, Systems, and Signal Processing 39: 3424–3444.
  • Tabekoueng Njitacke, Z., C. Laura Matze, M. Fouodji Tsotsop, and J. Kengne, 2020b Remerging feigenbaum trees, coexisting behaviors and bursting oscillations in a novel 3d generalized hopfield neural network. Neural Processing Letters 52: 267–289.
  • Tabekoueng Njitacke, Z., I. Sami Doubla, J. Kengne, and A. Cheukem, 2020c Coexistence of firing patterns and its control in two neurons coupled through an asymmetric electrical synapse. Chaos: An Interdisciplinary Journal of Nonlinear Science 30: 023101.
  • Takembo, C. N., H. P. E. Fouda, and T. C. Kofane, 2022 Modulational instability in chain diffusive neuronal networks under electric field. Indian Journal of Physics pp. 1–9.
  • Tsumoto, K., H. Kitajima, T. Yoshinaga, K. Aihara, and H. Kawakami, 2006 Bifurcations in morris–lecar neuron model. Neurocomputing 69: 293–316.
  • Wouapi, K., B. H. Fotsin, F. P. Louodop, K. F. Feudjio, Z. T. Njitacke, et al., 2020 Various firing activities and finite-time synchronization of an improved hindmarsh–rose neuron model under electric field effect. Cognitive Neurodynamics 14: 375–397.
  • Wouapi, M. K., B. H. Fotsin, E. B. M. Ngouonkadi, F. F. Kemwoue, and Z. T. Njitacke, 2021 Complex bifurcation analysis and synchronization optimal control for hindmarsh–rose neuron model under magnetic flow effect. Cognitive neurodynamics 15: 315–347.
  • Wu, F., J. Ma, and G. Zhang, 2020 Energy estimation and coupling synchronization between biophysical neurons. Science China Technological Sciences 63: 625–636.
  • Xu, Q., T. Liu, C.-T. Feng, H. Bao, H.-G. Wu, et al., 2021 Continuous non-autonomous memristive rulkov model with extreme multistability. Chinese Physics B 30: 128702.
  • Yao, Z., P. Zhou, Z. Zhu, and J. Ma, 2021 Phase synchronization between a light-dependent neuron and a thermosensitive neuron. Neurocomputing 423: 518–534.
  • Zhang, G., D. Guo, F. Wu, and J. Ma, 2020a Memristive autapse involving magnetic coupling and excitatory autapse enhance firing. Neurocomputing 379: 296–304.
  • Zhang, G., C. Wang, F. Alzahrani, F. Wu, and X. An, 2018 Investigation of dynamical behaviors of neurons driven by memristive synapse. Chaos, Solitons & Fractals 108: 15–24.
  • Zhang, Y., C. Wang, J. Tang, J. Ma, and G. Ren, 2020b Phase coupling synchronization of fhn neurons connected by a josephson junction. Science China Technological Sciences 63: 2328–2338.
  • Zhou, J.-F., E.-H. Jiang, B.-L. Xu, K. Xu, C. Zhou, et al., 2021a Synaptic changes modulate spontaneous transitions between tonic and bursting neural activities in coupled hindmarsh-rose neurons. Physical Review E 104: 054407.
  • Zhou, P., Z. Yao, J. Ma, and Z. Zhu, 2021b A piezoelectric sensing neuron and resonance synchronization between auditory neurons under stimulus. Chaos, Solitons & Fractals 145: 110751.

Details

Primary Language English
Subjects Engineering, Multidisciplinary
Journal Section Research Articles
Authors

Sishu Shankar MUNİ>
Massey University
0000-0001-9545-8345
New Zealand


Zeric NJITACKE> (Primary Author)
University of Buea
0000-0001-7797-8929
Cameroon


Cyrille FEUDJİO>
University of Buea
0000-0002-9316-1860
Cameroon


Théophile FOZİN>
University of Buea
0000-0001-7385-5462
Cameroon


Jan AWREJCEWİCZ>
Lodz University of technology
0000-0003-0387-921X
Poland

Publication Date November 30, 2022
Published in Issue Year 2022, Volume 4, Issue 3

Cite

APA Muni, S. S. , Njıtacke, Z. , Feudjio, C. , Fozin, T. & Awrejcewicz, J. (2022). Route to Chaos and Chimera States in a Network of Memristive Hindmarsh-Rose Neurons Model with External Excitation . Chaos Theory and Applications , 4 (3) , 119-127 . DOI: 10.51537/chaos.1144123

Chaos Theory and Applications in Applied Sciences and Engineering: An interdisciplinary journal of nonlinear science 23830