Understanding Brain Dynamics Through Neural Koopman Operator with Structure-Function Coupling

The fundamental question in neuroscience is to understand the working mechanism of how anatomical structure supports brain function and how remarkable functional fluctuations emerge ubiquitous behaviors. We formulate this inverse problem in the realm of system identification, where we use a geometric scattering transform (GST) to model the structure-function coupling and a neural Koopman operator to uncover dynamic mechanism of the underlying complex system. First, GST is used to construct a collection of measurements by projecting the proxy signal of brain activity into a neural manifold constrained by the geometry of wiring patterns in the brain. Then, we seek to find a Koopman operator to elucidate the complex relationship between partial observations and behavior outcomes with a relatively simpler linear mapping, which allows us to understand functional dynamics in the cliché of control system. Furthermore, we integrate GST and Koopman operator into an end-to-end deep neural network, yielding an explainable model for brain dynamics with a mathematical guarantee. Through rigorous experiments conducted on the Human Connectome Project-Aging (HCP-A) dataset, our method demonstrates state-of-the-art performance in cognitive task classification, surpassing existing benchmarks. More importantly, our method shows great potential in uncovering novel insights of brain dynamics using machine learning approach.