Relationships between structural and functional connectivity (SC/FC relationships) serve as a cohesive, unifying structure to the ITN. The driving force behind ESR2 is the goal to construct minimal models for many of the relationships between structural and functional connectivity (SC/FC relationships) observed in the diverse application scenarios. A minimal model is a mathematical representation (network + dynamical model) that is capable of displaying the desired behaviour and no simpler system can be envisioned. In addition to network architectures directly provided by the application scenarios, various models of random graphs (e.g., scale-free random graphs, small-world random graphs) will be employed. On the level of dynamics, we will resort to a set of generic models: excitable dynamics, random walks and diffusion, flow dynamics, avalanches on graphs. In a subsequent step, these generic models will be refined in a close dialogue with the application projects, in order to capture the most relevant properties of the dynamical processes within each specific application. In this way the project will contribute to a unified scientific framework that captures Connectivity Science and relates the theoretical structures and their properties to specific functions, methods and tools, which have been developed for diverse investigations across the various disciplines involved in the training network. By doing so, this project will provide a roadmap linking theory and methods as these support the diverse applications.
A minimal model to each application project capable of qualitatively reproducing the SC/FC correlations observed in the respective application nature, the transferability and the range of validity of each of these SC/FC correlations. The minimal models will be accompanied with a catalogue of structures, properties, functions, methods and tools related to connectivity. This universal framework will reveal those important theoretical structures, properties and functionalities related to connectivity. Under the light of these universal and strong theoretical results, important problems related to connectivity in various sciences and applications can now be re-investigated with better possibilities of success.