Estimation of current source density CSD from the low-frequency part of extracellular electric potential recordings is an unstable linear inverse problem. To make the estimation possible in an experimental setting where recordings are contaminated with noise, it is necessary to stabilize the inversion. Here we present a unified framework for zero-and higher-order singular-value-decomposition SVD-based spectral regularization of 1D linear CSD estimation from local field potentials. The framework is based on two general approaches commonly employed for solving inverse problems: quadrature and basis function expansion. We first show that both inverse CSD iCSD and kernel CSD kCSD fall into the category of basis function expansion methods. We then use these general categories to introduce two new estimation methods, quadrature CSD qCSD, based on discretizing the CSD integral equation with a chosen quadrature rule, and representer CSD rCSD, an even-determined basis function expansion method that uses the problem's data kernels representers as basis functions. To determine the best candidate methods to use in the analysis of experimental data, we compared the different methods on simulations under three regularization schemes Tikhonov, tSVD, and dSVD, three regularization parameter selection methods NCP, L-curve, and GCV, and seven different a priori spatial smoothness constraints on the CSD distribution. This resulted in a comparison of 531 estimation schemes. We evaluated the estimation schemes according to their source reconstruction accuracy by testing them using different simulated noise levels, lateral source diameters, and CSD depth profiles. We found that ranking schemes according to the average error over all tested conditions results in a reproducible ranking, where the top schemes are found to perform well in the majority of tested conditions. However, there is no single best estimation scheme that outperforms all others under all tested conditions. The unified framework we propose expands the set of available estimation methods, provides increased flexibility for 1D CSD estimation in noisy experimental conditions, and allows for a meaningful comparison between estimation schemes.
['Pascal Kropf', 'Amir Shmuel']
1d current source density csd estimation in inverse theory: A unified framework for higher-order spectral regularization of quadrature and expansion-type csd methods