Control algorithms for feedback tracking in the small populations of Hodgkin-Huxley neurons

Abstract

The purpose of the thesis is to design powerful mathematical control algorithms for the tracking and modeling spiking and bursting behaviors of real biological neurons in 4-dimensional dynamical systems. For this aim, 4-dimensional Hodgkin-Huxley’s (HH) nonlinear dynamical system including differential equations preferred. Because HH model represents a realistic mathematical model for the real neurons and it analytically accepted. Applied external current as a control signal initiate stimulating of the neuron cells in the neuronal networks serve while the membrane action potentials are outputs. We applied two different control methods; speed gradient (SG) of Fradkov’s and target attractor (TA) of Kolesnikov’s feedbacks for the modeling and controlling spiking and bursting regime that axon membrane potential created by the control signal in HH neuron clusters. These algorithms show high effectiveness and robustness in the managed HH dynamical neuron system. This study provides generating arbitrary forms of single spikes, train of spikes and bursts for chosen cells in the various configurations of HH neuron clusters (linear chain and ring-type chain) with the control over a selected element of the network. In this study, developed algorithms applied to epileptiform collective bursting in a small cluster of HH neurons for make suppression. The scope of this thesis is to develop new control methods for mathematical modeling to control of real neurons and effectively can use in computational neuroscience and diagnosis or treatment of neural dysfunctions such as epileptiform or abnormal behavior in the HH neuron networks.