Dynamic Analysis and Synchronization of Two Uncoupled Chaotic Hindmarsh-Rose Neurons
Main Article Content
Abstract
In this paper, the problem of synchronization of two uncoupled chaotic Hindmarsh-Rose (HR) neurons is addressed. First, the dynamic behaviors of a single HR neuron stimulated by an external applied current are studied. By using the concept of fast/slow dynamic analysis, the bursting mechanism of the HR neuron is investigated. Considering the applied current as a bifurcation parameter, the chaotic behavior as well as other dynamic behaviors is reported. Second, the author formulated a method for synchronization of two uncoupled chaotic HR neurons. By using a Lyapunov function, a nonlinear feedback control law is designed that guarantees that the two uncoupled neurons are globally asymptotically synchronized. Finally, in order to verify the effectiveness of the proposed method, numerical simulations are carried out, the results of which are provided herein.
Keywords
Chaos, Bursting mechanism, Hindmarsh-Rose neuron, Lyapunov function, Synchronization
Article Details
References
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[13] J. Wang, T. Zhang, and B. Deng; Synchronization of FitzHugh-Nagumo neurons in external electrical stimulation via nonlinear control; Chaos, Solitons & Fractals 31(1) (2007) 30-38.
[14] J. Wang, Z. Zhang, and H. Li; Synchronization of FitzHugh-Nagumo systems in EES via H-infinity variable universe adaptive fuzzy control; Chaos, Solitons & Fractals 36(5) (2008) 1332-1339.
[15] M. Rehan and K.-S. Hong; LMI-based robust adaptive synchronization of FitzHugh-Nagumo neurons with unknown parameters under uncertain external electrical stimulation; Physics Letters A 375(15) (2011) 1666-1670.
[16] J. Rinzel; A formal classification of bursting mechanisms in excitable systems; Proceedings of the International Congress of Mathematicians (1986) 1578-1593.
[17] R. Bertram and A. Sherman; Dynamical complexity and temporal plasticity in pancreatic beta-cells; Journal of Biosciences 25(2) (2000) 197-209.
[18] H. K. Khalil; Nonlinear systems; third edition, New Jersey, Prentice Hall, 2002.
[2] R. Fitzhugh; Thresholds and plateaus in the Hodgkin-Huxley nerve equations; The Journal of General Physiology 43(5) (1960) 867-896.
[3] C. Morris and H. Lecar; Voltage oscillations in the barnacle giant muscle fiber; Biophysical Journal 35(1) (1981) 193-213.
[4] J. L. Hindmarsh and R. M. Rose; A model of neuronal bursting using three coupled first order differential equations; Proceedings of the Royal Society of London B 221 (1984) 87-102.
[5] R. Eckhorn; Neural mechanisms of scene segmentation: recording from the visual cortex suggests basic circuits or linking field models; IEEE Transactions on Neural Networks 10(3) (1999) 464-479.
[6] A. B. Neiman and D. F. Russell; Synchronization of noise-induced bursts in noncoupled sensory neurons; Physical Review Letters 88(13) (2002) 138103.
[7] Y. Wu, J. Xu, W. Jin, and L. Hong; Detection of mechanism of noise-induced synchronization between two identical uncoupled neurons; Chinese Physics Letters 24(11) (2007) 3066-3069.
[8] J. Wang, B. Deng, and K. M. Tsang; Chaotic synchronization of neurons coupled with gap junction under external electrical stimulation; Chaos, Solitons & Fractals 22(2) (2004) 469-476.
[9] L. H. Nguyen and K.-S. Hong; Synchronization of coupled chaotic FitzHugh-Nagumo neurons via Lyapunov functions; Mathematics and Computers in Simulations 82(4) (2011) 590-603.
[10] L. H. Nguyen and K.-S. Hong; Adaptive synchronization of two coupled chaotic Hindmarsh-Rose neurons by controlling the membrane potential of a slave neuron; Applied Mathematic Modelling 37 (2013) 2460-2468.
[11] L. H. Nguyen and K.-S. Hong; Lyapunov-based synchronization of two coupled chaotic Hindmarsh-Rose neurons; Journal of Computer Science and Cybernetics 30(4) (2014) 337-348.
[12] O. Cornejo-Pérez and R. Femat; Unidirectional synchronization of Hodgkin-Huxley neurons; Chaos, Solitons & Fractals 25(1) (2005) 43-53.
[13] J. Wang, T. Zhang, and B. Deng; Synchronization of FitzHugh-Nagumo neurons in external electrical stimulation via nonlinear control; Chaos, Solitons & Fractals 31(1) (2007) 30-38.
[14] J. Wang, Z. Zhang, and H. Li; Synchronization of FitzHugh-Nagumo systems in EES via H-infinity variable universe adaptive fuzzy control; Chaos, Solitons & Fractals 36(5) (2008) 1332-1339.
[15] M. Rehan and K.-S. Hong; LMI-based robust adaptive synchronization of FitzHugh-Nagumo neurons with unknown parameters under uncertain external electrical stimulation; Physics Letters A 375(15) (2011) 1666-1670.
[16] J. Rinzel; A formal classification of bursting mechanisms in excitable systems; Proceedings of the International Congress of Mathematicians (1986) 1578-1593.
[17] R. Bertram and A. Sherman; Dynamical complexity and temporal plasticity in pancreatic beta-cells; Journal of Biosciences 25(2) (2000) 197-209.
[18] H. K. Khalil; Nonlinear systems; third edition, New Jersey, Prentice Hall, 2002.