Brain activation during anticipation of sound sequences

Music consists of sound sequences that require integration over time. As we become familiar with music, associations between notes, melodies, and entire symphonic movements become stronger and more complex. These associations can become so tight that, for example, hearing the end of one album track can elicit a robust image of the upcoming track while anticipating it in total silence. Here, we study this predictive “anticipatory imagery” at various stages throughout learning and investigate activity changes in corresponding neural structures using functional magnetic resonance imaging. Anticipatory imagery (in silence) for highly familiar naturalistic music was accompanied by pronounced activity in rostral prefrontal cortex (PFC) and premotor areas. Examining changes in the neural bases of anticipatory imagery during two stages of learning conditional associations between simple melodies, however, demonstrates the importance of fronto-striatal connections, consistent with a role of the basal ganglia in “training” frontal cortex (Pasupathy and Miller, 2005). Another striking change in neural resources during learning was a shift between caudal PFC earlier to rostral PFC later in learning. Our findings regarding musical anticipation and sound sequence learning are highly compatible with studies of motor sequence learning, suggesting common predictive mechanisms in both domains.

Thin Sequences and the Gram Matrix

We provide a new proof of Volberg's Theorem characterizing thin interpolating sequences as those for which the Gram matrix associated to the normalized reproducing kernels is a compact perturbation of the identity. In the same paper, Volberg characterized sequences for which the Gram matrix is a compact perturbation of a unitary as well as those for which the Gram matrix is a Schatten-2 class perturbation of a unitary operator. We extend this characterization from 2 to p, where 2 <= p <= infinity.