Geometric constraints on the dynamics of networks with refractory node states

Correspondence: [email protected] Department of Bioengineering, Department of Neurosciences, and Center for Engineered Natural Intelligence, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92037-0412, United States Full list of author information is available at the end of the article Abstract Understanding how local interactions among connected nodes in a network result in global network dynamics and behaviors remains a critical open problem in network theory. This is important both for understanding complex networks, including the brain, and for the controlled design of networks intended to achieve a specific function. Here, we describe the construction and theoretical analysis of a framework derived from canonical neurophysiological principles that models the competing dynamics of incident signals into nodes along directed edges in a network. The framework describes the dynamics between the offset in the latencies of propagating signals, which reflect the geometry of the edges and conduction velocities, and the internal refractory dynamics and processing times of the downstream node. One of the main theoretical results is the definition of a ratio between the speed of signaling or information flow, which is bounded by the spatial geometry, and the internal time it takes for individual nodes to process incoming signals. We show that an optimal ratio is one where the speed of information propagation between connected nodes does not exceed the internal dynamic time scale of the nodes. A mismatch of this ratio leads to sub-optimal signaling and information flows in a network, and even a breakdown in signaling all together.