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In this study, the neuronal current in the brain is represented usingHelmholtz decomposition. It was shown in earlier work that data obtained viaelectroencephalography (EEG) are affected only by the irrotational component ofthe current. The irrotational component is denoted by $\Psi$ and has support inthe cerebrum. This inverse problem is severely ill-posed and requires thatadditional constraints are imposed. Here, we impose the requirement of theminimization of the $L_2$ norm of the current (energy). The function $\Psi$ isexpanded in terms of inverse multiquadric radial basis functions (RBF) on auniform Cartesian grid inside the cerebrum. The minimal energy constraint inconjunction with the RBF parametrization of $\Psi$ results in a Tikhonovregularized solution of $\Psi$. The RBF shape parameter (regularizationparameter), is computed by solving a 1-D non-linear maximization problem.Reconstructions are presented using synthetic data with a signal to noise ratio(SNR) of $20$ dB. The root mean square error (RMSE) between the exact and thereconstructed $\Psi$ is RMSE=$0.1122$. The proposed reconstruction algorithm iscomputationally efficient and can be vectorized in MATLAB.

A hybrid analytical-numerical algorithm for determining the neuronal current via EEG