An image resolution perspective on functional activity mapping.

In this paper we apply techniques for numerical estimation of system resolution from imaging, to the regression problem of relating biological data to phenotypes. Our approach can be viewed as an extension of Backus-Gilbert theory, which attempts to find the most concentrated estimator that may be reliably computed in an inverse problem. Applied to a regression model, we estimate a minimal combination of collinear variables that may be found in a predictor, which gives a robust multivariable estimate of the network relationships in the data. Our extension of this approach incorporates a sparsity prior in order to adapt the concept to the high noise and small sample regime. The result is a compromise between the Backus-Gilbert and sparse regularized estimates, which may be adjusted to trade-off benefits of both and provide a result which we demonstrate to be more robust. This is applied to a dataset containing fMRI activity maps and SNP's for subjects with schizophrenia and related disorders. We find the resolution estimate identifies plausible modular behavior among neighboring variables and between regions. We further demonstrate the ability to find differences in these relationships using different response variables or additional data, providing a means to extract more specialized information.