A mathematical model for coregistered data from electroencephalography and diffusive optical tomography

¹ LAGA UMR CNRS 7539, Université Sorbonne Paris Nord, France
² LMR UMR CNRS 9008, Université de Reims-Champagne Ardenne, Reims, France

^* Corresponding author; stephanie.lohrengel@univ-reims.fr

Received: 28 May 2024
Accepted: 9 January 2025

Abstract

A mathematical model for coregistered time-dependent electroencephalography (EEG) and diffusive optical tomography (DOT) is developed and analysed. Evolution with time is introduced by considering time-dependent dipolar sources in the EEG model and time-dependent optical parameters for DOT. Dimensional analysis shows that time-derivatives can be neglected. A non-linear system of differential equations from literature is used to model the postsynaptic current and hemodynamic parameters at the neuron level. A key point of the full model is to explain how these quantities provide, at the level of the whole head, the moment of the dipolar source term of the EEG problem and the behaviour in time of the optical parameters of the DOT model. The well-posedness of the timedependent EEG problem is proved by the subtraction approach for moments with L²-regularity in time and continuous source trajectories. For the time-dependent DOT model with continuous optical parameters in time, standard results of functional analysis apply. We explain the full pipeline from the stimulation current up to the simulated signals recorded at the electroptodes. Numerical results for a three-dimensional realistic head model illustrate the capacity of simultaneous EEG/DOT measurements to attest neurovascular coupling between the neural activity and changes in the hemodynamic parameters.