A coupled node-network dynamics with Koopman and Krankheit-operators

The brain can be described as a network of nodes and links between them, the connectome, either anatomic or functional, containing information on the neural pathways and the signal exchanged across them. Recent studies proposed the Krankheit-operator (K-operator) acting on the connectome and modeling the effect of neurological disorders, while Koopman operators K describe neural dynamics through lifted linear representations. This is a position paper where we propose a coupled node-network model of brain dynamics. Structural changes on the functional connectome are induced by K, whose information is injected within the Koopman operators on each node. As a first, simple proof-of-concept, we propose a computational approximation of the model, imposing the network dynamics modeled by the K-operator on the local, node-level Koopman operators. We show an improvement in time-series reconstruction by using K as a nonlinear correction, rather than injecting it within linear Koopman frameworks. We also find a small, yet statistically significant effect of a Krankheit-Koopman coupling.