Novel Generation of Fixed Point Transformation for the Adaptive Control of a Nonlinear Neuron Model

The neurons as living cells work as essentially nonlinear oscillators or spike generators. In the case of a particular model the "ideal", the till acceptable i.e. "Healthy", and the impaired ("sick") operation of a neuron can be represented by appropriate parameter settings. A practically interesting control task may be forcing the motion of a "sick" neuron to trace the trajectory generated by an "ideal" one on the basis of an available approximate model. In both cases the existence of three different parameter settings is assumed. As is well known essentially nonlinear systems cannot be well controlled on the basis of linearized models and linear techniques. The general nonlinear technique uses Lyapunov's "direct" method that guarantees global stability of the solution that otherwise suffers from several deficiencies. An alternative approach that removes these deficiencies at the cost of giving up global stability uses a special iteration created by a particular fixed point transformation. In the present paper a systematic method is presented for the generation of whole families of fixed point transformations that can be used in nonlinear adaptive control of Single Input -- Single Output (SISO) systems. The applicability of the novel method is demonstrated by the adaptive control of the FitzHugh-Nagumo neuron model investigated by simulations.