Synchronization of Different Coupled Chaotic Neurons under Controlled External Electrical Stimulation

Abstract

The swift evolution of biofeedback control has opened up new voyages in the field of biomedical engineering and provides powerful perspective to researchers for viewing many real problems such as tumor, surgeries of sensitive parts of body, control of glucose level of a patient, heart diseases and brain disorders associated with humans. Consequently, mysterious and intricate biological phenomena can now be studied and investigated by utilizing the knowledge of control and nonlinear dynamics of physical systems. The emerging theory of biofeedback control can be more fruitful for understanding the brain functioning in order to cope with various neural disorders. Neuron, being an innate sophisticated structural entity of nervous system, plays an imperative role owing to its chief biophysical features and key mechanism of operations, for effective transmission of neuronal signals to the brain and the muscles. The probe of neuron doctrine gives an insight into understanding of brain information processing and information transmittance among neurons which may further ii corroborate a close relevance between the synchronization of neural systems and the information of cerebral process. Thus, neuronal synchronization under deep brain stimulation has become a potential application in the study of clinical treatment mechanisms for neurodegenerative disorders. Moreover, the famous FitzHugh-Nagumo (FHN) model under external electrical stimulation (EES; e.g. deep brain stimulation), is extensively used as synchronization study tool for its utility in symbolizing the dynamical behavior of neurons. The embryonic impact of biofeedback control in improving external therapies for patients suffering cognitive disorders such as Parkinson’s disease, epilepsy and dystonia is the main motivation to this research work. “This thesis presents an efficient novel mechanism for synchronization of two different, chaotic and distant coupled neurons with unknown parameters subjected to external electrical stimulation and disturbances” This research investigates the chaotic behavior and synchronization of two different coupled chaotic FitzHugh-Nagumo (FHN) neurons with unknown parameters under external electrical stimulation (EES). The coupled FHN neurons iii of different parameters admit unidirectional and bidirectional gap junctions in the medium between them. Dynamical properties, such as increase in synchronization error as a consequence of the deviation of neuronal parameters for unlike neurons, the effect of difference in coupling strengths caused by the unidirectional gap junctions, and the impact of large time-delay due to separation of neurons, are studied in exploring the behavior of the coupled system. A novel integral-based nonlinear adaptive control scheme to cope with infeasibility of the recovery variable, for synchronization of two coupled delayed chaotic FHN neurons of different and unknown parameters under uncertain EES is derived. Further, to guarantee robust synchronization of different neurons against disturbances, the proposed control methodology is modified to achieve the uniformly ultimately bounded synchronization. The parametric estimation errors can be reduced by selecting suitable control parameters. The effectiveness of the proposed control scheme is illustrated via simulation results. Keywords: Chaos synchronization; FitzHugh-Nagumo model; External electrical stimulation; Robust adaptive control; Lyapunov function.