Ryan Budney

Nicole Croteau, Farouk S. Nathoo, Jiguo Cao et al.

Brain decoding involves the determination of a subject's cognitive state or an associated stimulus from functional neuroimaging data measuring brain activity. In this setting the cognitive state is typically characterized by an element of a finite set, and the neuroimaging data comprise voluminous amounts of spatiotemporal data measuring some aspect of the neural signal. The associated statistical problem is one of classification from high-dimensional data. We explore the use of functional principal component analysis, mutual information networks, and persistent homology for examining the data through exploratory analysis and for constructing features characterizing the neural signal for brain decoding. We review each approach from this perspective, and we incorporate the features into a classifier based on symmetric multinomial logistic regression with elastic net regularization. The approaches are illustrated in an application where the task is to infer, from brain activity measured with magnetoencephalography (MEG), the type of video stimulus shown to a subject.

Ryan Budney, William Sethares

The musical realm is a promising area in which to expect to find nontrivial topological structures. This paper describes several kinds of metrics on musical data, and explores the implications of these metrics in two ways: via techniques of classical topology where the metric space of all-possible musical data can be described explicitly, and via modern data-driven ideas of persistent homology which calculates the Betti-number bar-codes of individual musical works. Both analyses are able to recover three well known topological structures in music: the circle of notes (octave-reduced scalar structures), the circle of fifths, and the rhythmic repetition of timelines. Applications to a variety of musical works (for example, folk music in the form of standard MIDI files) are presented, and the bar codes show many interesting features. Examples show that individual pieces may span the complete space (in which case the classical and the data-driven analyses agree), or they may span only part of the space.