Insights from control theory into deep brain stimulation for relief from Parkinson´s disease

Using ideas from control theory. i.e., the root locus method, Lyapunov´s theorem of the first approximation, the describing function, Nyquist stability theory and the concept of the equivalent nonlinearity associated with dither injection in a nonlinear feedback loop, the phenomenon of quenching of pathological neural oscillations by deep brain stimulation is explored. The model used contains a second order unstable, linear, dynamical system, in a negative feedback loop with a nonlinearity comprising a linear gain in parallel with a “signed square”. This mimics, what is referred to by Alim Louis Benabid, the great pioneer of deep brain stimulation as “excitation of inhibitory pathways that lead to functional inhibition”. Describing function analysis is used to give a very close estimate of the inherent, almost sinusoidal oscillation, which is quenched by deep brain stimulation. The relationship between the critical amplitude of deep brain stimulation (expressed either in volts or milliamps) and the fractional pulse width needed for quenching the oscillation is derived. This is fitted as closely as possible to experimental results by Benabid et al., by minimizing a sum of squared error index.