Averaging, either spatial or temporal, is a powerful technique in complex multi-scale systems.
However, in some situations it can be difficult to justify.
For example, many real-world networks in technology, engineering and biology have a function and exhibit dynamics that cannot always be adequately reproduced using network models given by the smooth dynamical systems and fixed network topology that typically result from averaging. Motivated by examples from neuroscience and engineering, we describe a model for what we call a "functional asynchronous network". The model allows for changes in network topology through decoupling of nodes and stopping and restarting of nodes, local times, adaptivity and control. Our long-term goal is to obtain an understanding of structure (why the network works) and how function is optimized (through bifurcation).
We describe a prototypical theorem that yields a functional decomposition for a large class of functional asynchronous networks. The result allows us to express the function of a dynamical network in terms of individual nodes and constituent subnetworks.