Synchronization Criteria for Delay Coupled Izhikevich Neurons
In this chapter, we investigate the chaotic synchronization of two coupled Izhikevich neurons via a gap junction. In the absence of a controller, the coupled neurons will achieve complete chaotic synchronization only when the degree of connectivity or the
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Abstract In this chapter, we investigate the chaotic synchronization of two coupled Izhikevich neurons via a gap junction. In the absence of a controller, the coupled neurons will achieve complete chaotic synchronization only when the degree of connectivity or the coupling strength exceeds a critical value. This transition to synchronization with varying connectivity strengths is analysed with conditional Lyapunov exponents. Synchronization of gap junction separated, coupled Izhikevich neurons using control laws has remained non-investigated to this date. As such, in this chapter we propose a nonlinear adaptive controller, in order to obtain complete chaotic synchronization for any value of coupling strength and delay, based on the Lyapunov stability theory. Effectiveness of the proposed nonlinear controller for synchronizing delayed-coupled Izhikevich neurons are shown through numerical simulations. Keywords Izhikevich model ⋅ Gap junction ⋅ Delay coupled ⋅ Synchronization ⋅ Nonlinear control
1 Introduction Chaotic synchronization is one of the most interesting dynamical behaviours that can arise in a neuronal network. Important brain functions such as working memory, selective attention, sensory perception, and multisensory integration which requires I.T. Hettiarachchi (✉) ⋅ L. Shanmugam ⋅ A. Bhatti ⋅ S. Nahavandi Institute for Intelligent Systems Research and Innovation, Deakin University, Geelong, Australia e-mail: [email protected] L. Shanmugam e-mail: [email protected] A. Bhatti e-mail: [email protected] S. Nahavandi e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2017 A. Bhatti et al. (eds.), Emerging Trends in Neuro Engineering and Neural Computation, Series in BioEngineering, DOI 10.1007/978-981-10-3957-7_7
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efficient sensory and cognitive processing are the result of synchronized firing activity of coupled neuronal networks. The information processing of the brain is a result of intensity keeping of the neuronal response during the propagation over the network, which occurs during neuronal synchronous states [1]. Further, disturbances in the synchronized network activity causing imbalance of the neurons are the cause of clinical disorders such as Parkinson’s disease, schizophrenia and epilepsy [2]. Therefore, studying the synchronization and de-synchronization of neuronal spike-burst behaviours can give us insight to the information processing in the brain and even the origins of certain mental disorders. In this regard, biophysical models of single neurons have been proposed in the literature, which are useful in understanding various neuronal behaviour such as spiking, bursting, and chaos [3–7]. These models can reproduce various spiking activity observed in neural systems by tuning the parameters, and also can be used to investigate the phenomenon of synchronization of neuronal networks [8]. The Hodgkin–Huxley (HH) neuron model [3] is known to be the most comprehensive, yet the most complex spiking neuron model for s
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