On modeling the neuronal activity in movement disorder patients by using the Ornstein Uhlenbeck Process

Mathematical models of the neuronal activity in the affected brain regions of Essential Tremor (ET) and Parkinson´s Disease (PD) patients could shed light into the underlying pathophysiology of these diseases, which in turn could help develop personalized treatments including adaptive Deep Brain Stimulation (DBS). In this paper, we use an Ornstein Uhlenbeck Process (OUP) to model the neuronal spiking activity recorded from the brain of ET and PD patients during DBS stereotactic surgery. The parameters of the OUP are estimated based on Inter Spike Interval (ISI) measurements, i.e., the time interval between two consecutive neuronal firings, by means of the Fortet Integral Equation (FIE). The OUP model parameters identified with the FIE method (OUP-FIE) are then used to simulate the ISI distribution resulting from the OUP. Other widely used neuronal activity models, such as the Poisson Process (PP), the Brownian Motion (BM), and the OUP whose parameters are extracted by matching the first two moments of the ISI (OUP-MOM), are also considered. To quantify how close the simulated ISI distribution is to the measured ISI distribution, the Integral Square Error (ISE) criterion is adopted. Amongst all considered stochastic processes, the ISI distribution generated by the OUP-FIE method is shown to produce the least ISE. Finally, a directional Wilcoxon signed rank test is used to show statistically significant reduction in the ISE value obtained from the OUP-FIE compared to the other stochastic processes.