  Deep brain stimulation (DBS) is a surgical treatment for Parkinson's Disease.
Static models based on quasi-static approximation are common approaches for DBS
modeling. While this simplification has been validated for bioelectric sources,
its application to rapid stimulation pulses, which contain more high-frequency
power, may not be appropriate, as DBS therapeutic results depend on stimulus
parameters such as frequency and pulse width, which are related to time
variations of the electric field. We propose an alternative hybrid approach
based on probabilistic models and differential equations, by using Gaussian
processes and wave equation. Our model avoids quasi-static approximation,
moreover, it is able to describe dynamic behavior of DBS. Therefore, the
proposed model may be used to obtain a more realistic phenomenon description.
The proposed model can also solve inverse problems, i.e. to recover the
corresponding source of excitation, given electric potential distribution. The
electric potential produced by a time-varying source was predicted using
proposed model. For static sources, the electric potential produced by
different electrode configurations were modeled. Four different sources of
excitation were recovered by solving the inverse problem. We compare our
outcomes with the electric potential obtained by solving Poisson's equation
using the Finite Element Method (FEM). Our approach is able to take into
account time variations of the source and the produced field. Also, inverse
problem can be addressed using the proposed model. The electric potential
calculated with the proposed model is close to the potential obtained by
solving Poisson's equation using FEM.
