Discovering how the brain functions has been proved valuable for understanding the brain's behavioral control as well as guiding treatment of mental diseases. In response to stimuli, the brain generates a mix of brain waves that are dynamic and frequency-specified. Thus, time-frequency analysis has been widely used in analyzing brain signals. However, due to huge variation of the characteristics of brain signals, analysis measures providing the signal-invariant resolution cannot well reveal dynamic structure of various brain signals.

We introduce an adaptive S transform (AST), a new multi-resolution time-frequency representation whose resolution is adaptively adjusted to its analyzed signal. The proposed representation is built on the S transform with additional parameters to control its resolution. Given any specific signal, we implement a numerical procedure that automatically determines optimal parameters so that the resulting representation has the signal energy highly concentrated at the involved frequencies and time duration. It hence provides a time-frequency analysis tool offering good resolution to describe behaviors of various signals. We then use the AST to derive a number of measures for analyzing brain time series recorded by electroencephalography and magnetoencephalography. These measures include the AST-based power spectrum for revealing the characteristics of functional activity at a single brain area, and the AST-based coherence and phase-locking statistic for investigating the functional connectivity between multiple brain areas. Numerical simulations are presented to demonstrate performances of the AST and the corresponding brain time series measures. Finally, we apply the proposed AST-based analysis tools to investigate functional activity of motor cortices when subjects perform the multi-source interference task, a behavioral experiment involving tasks at multiple levels of difficulty.