A hybrid analytical–numerical algorithm for determining the neuronal current via electroencephalography. Issue 163 (12th February 2020) - Hashemzadeh, Parham; Fokas, A. S.; Schönlieb, C. B.
Abstract : Specific mental processes are associated with brain activation of a unique form, which are, in turn, expressed via the generation of specific neuronal electric currents. Electroencephalography (EEG) is based on measurements on the scalp of the electric potential generated by the neuronal current flowing in the cortex. This specific form of EEG data has been employed for a plethora of medical applications, from sleep studies to diagnosing focal epilepsy. In recent years, there have been efforts to use EEG data for a more ambitious purpose, namely to determine the underlying neuronal current. Although it has been known since 1853, from the studies by Helmholtz, that the knowledge of the electric potential of the external surface of a conductor is insufficient for the determination of the electric current that gave rise to this potential, the important question of which part of the current can actually be determined from the knowledge of this potential remained open until work published in 1997, when it was shown that EEG provides information only about the irrotational part of the current, which will be denoted by Ψ ; moreover, an explicit formula was derived in the above work relating this part of the current, the measured electric potential, and a certain auxiliary function, v s, that depends on the geometry of the various compartments of the brain–head system and their conductivities. In the present paper: (i) Motivated by recent results which show that, in the case of ellipsoidal geometry, the assumption of the L 2 minimization of the current yields a unique solution, we derive an analogous analytic formula characterizing this minimization for arbitrary geometry. (ii) We show that the above auxiliary function can be computed numerically via a line integral from the values of a related function v s computed via OpenMEEG; moreover, we propose an alternative approach to computing the auxiliary function v s based on the construction of a certain surrogate model. (iii) By expanding Ψ in terms of an inverse multiquadric radial basis we implement the relevant formulae numerically. The above algorithm performs well for synthetic data; its implementation with real data only requires the knowledge of the coordinates of the positions where the given EEG data are obtained.