A Three Spatial Dimension Wave Latent Force Model for Describing Excitation Sources and Electric Potentials Produced by Deep Brain Stimulation

TL;DR

This paper introduces a novel wave-based latent force model for deep brain stimulation that captures dynamic electric fields without relying on quasi-static assumptions, enabling more accurate modeling of stimulation effects and source recovery.

Contribution

It presents a hybrid probabilistic-differential equation model using Gaussian processes and wave equations, improving upon static models by accounting for time-varying sources and fields in DBS.

Findings

Model accurately predicts electric potentials for static and dynamic sources.

Inverse problem solution successfully recovers excitation sources.

Potential closely matches FEM-based solutions.

Abstract

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.