The Human Genome project promised great advances in understanding genetic factors for disease, and genome-wide association studies (GWAS) were the main instrument to achieve that understanding. However they have mostly failed to deliver. Recently several papers have suggested that there may be many common variants contributing slightly to the risk of complex disease, and that GWAS may in fact be detecting these, but below the strict threshold of significance. I describe a novel method for estimating the distribution of small effects taking as data the distribution of small p-values; the crucial step is to construct a Fredholm integral equation of the first kind. I validate the method by simulation studies, and then provide a modern estimate of the genetic architecture of schizophrenia.
A major problem in the initial stages of neural signal analysis is the identification and compensation of artifacts, many of which reflect physiological processes, such as breathing, and pulse. I introduce a method of constructing synthetic controls to identify artifacts in fMRI or other neural time series data. Synthetic controls are differences of little biological significance, which however differ in their relation to anticipated (but unmeasured) sources of artifact. These differences turn out to have very strong systematic patterns summarized by a very few principal components. These factors in turn predict typically 50% of variance in the original data. Subtracting these estimates from the raw data seems to improve the S/N ratio of the data by almost a factor of two over the current `gold standard'.
Two unexpected applications of mathematics to biology in genetics and neuroimaging
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