Nonlinear dynamics, chaos and control of the Hindmarsh-Rose neuron model
DOI:
https://doi.org/10.5269/bspm.47770Abstract
Mathematics has changed over time to comprise interdisciplinary fields of research, and considering this, biomathematics has arisen as an interface study. In this work, we analyze the dynamical behavior of the Hindmarsh-Rose (HR) neuron model, which describes the neuronal bursting in a single neuron. A stability study through the Lyapynov exponents method is proposed and evidence of a chaotic dynamics is presented. Therefore, a control design based on the State-Dependent Ricatti Equation (SDRE) is proposed aiming to reduce the oscillation of the system to a desired orbit. The results show that the controller is efficient and robust as a method for preventing epileptic seizures.
References
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13. Mracek, Curtis P., and James R. Cloutier. Control designs for the nonlinear benchmark problem via the state-dependent Riccati equation method. International Journal of robust and nonlinear control 8.4-5 (1998): 401-433. https://doi.org/10.1002/(SICI)1099-1239(19980415/30)8:4/5<401::AID-RNC361>3.0.CO;2-U
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15. Palus, Milan, et al. Synchronization and information flow in EEGs of epileptic patients. IEEE Engineering in Medicine and Biology Magazine 20.5 (2001): 65-71. https://doi.org/10.1109/51.956821
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19. Zoski, Cynthia G., ed. Handbook of electrochemistry. Elsevier, 2006.
2. Bear, Mark F., Barry W. Connors, and Michael A. Paradiso. Neurociencias: desvendando o sistema nervoso. Artmed Editora, 2002.
3. Chavez, Mario, et al. Spatio-temporal dynamics prior to neocortical seizures: amplitude versus phase couplings. IEEE Transactions on Biomedical Engineering 50.5 (2003): 571-583. https://doi.org/10.1109/TBME.2003.810696
4. Cloutier, James R., Christopher N. D'Souza, and Curtis P. Mracek. Nonlinear regulation and nonlinear H∞ control via the state-dependent Riccati equation technique: Part 1, theory. Proceedings of the First International Conference on Nonlinear Problems in Aviation and Aerospace. Daytona Beach, FL: Embry-Riddle Aeronautical Univ. Press, 1996.
5. da Silva, Fernando H. Lopes, et al. Dynamical diseases of brain systems: different routes to epileptic seizures. IEEE transactions on biomedical engineering 50.5 (2003): 540-548. https://doi.org/10.1109/TBME.2003.810703
6. FitzHugh, Richard. Impulses and physiological states in theoretical models of nerve membrane. Biophysical journal 1.6 (1961): 445-466. https://doi.org/10.1016/S0006-3495(61)86902-6
7. Hindmarsh, James L., and R. M. Rose. A model of neuronal bursting using three coupled first order differential equations Proceedings of the Royal society of London. Series B. Biological sciences 221.1222 (1984): 87-102. https://doi.org/10.1098/rspb.1984.0024
8. Hindmarsh, James L., and R. M. Rose. A model of a thalamic neuron. Proceedings of the Royal society of London. Series B. Biological sciences 225.1239 (1985): 161-193. https://doi.org/10.1098/rspb.1985.0057
9. Hodgkin, Alan L., and Andrew F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of physiology 117.4 (1952): 500-544. https://doi.org/10.1113/jphysiol.1952.sp004764
10. Lopes, Vera Lucia da Rocha, and Marcia A. Gomes Ruggiero. Calculo numerico-Aspectos teoricos e computacionais. (1996).
11. Jun, Ma et al. Control Chaos in Hindmarsh-Rose Neuron by Using Intermittent Feedback with One Variable. Chinese Physics Letters, 25.10 (2008): 3582. https://doi.org/10.1088/0256-307X/25/10/017
12. Monteiro, Luiz Henrique Alves. Sistemas dinamicos. Editora Livraria da Fısica, 2006.
13. Mracek, Curtis P., and James R. Cloutier. Control designs for the nonlinear benchmark problem via the state-dependent Riccati equation method. International Journal of robust and nonlinear control 8.4-5 (1998): 401-433. https://doi.org/10.1002/(SICI)1099-1239(19980415/30)8:4/5<401::AID-RNC361>3.0.CO;2-U
14. Nagumo, Jinichi, Suguru Arimoto, and Shuji Yoshizawa. An active pulse transmission line simulating nerve axon. Proceedings of the IRE 50.10 (1962): 2061-2070. https://doi.org/10.1109/JRPROC.1962.288235
15. Palus, Milan, et al. Synchronization and information flow in EEGs of epileptic patients. IEEE Engineering in Medicine and Biology Magazine 20.5 (2001): 65-71. https://doi.org/10.1109/51.956821
16. Rossler, O. E. An equation for hyperchaos. Physics Letters A 71.2-3 (1979): 155-157. https://doi.org/10.1016/0375-9601(79)90150-6
17. Savi, Marcelo Amorin. Dinamica nËœao-linear e caos. Editora E-papers, 2006.
18. Wolf, Alan, et al. Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena 16.3 (1985): 285-317. https://doi.org/10.1016/0167-2789(85)90011-9
19. Zoski, Cynthia G., ed. Handbook of electrochemistry. Elsevier, 2006.
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2022-02-02
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