Connecting structure of individual networks to dynamics in oscillator systems

New developments in connectomic reconstruction are rapidly increasing the ability to map connection patterns in neural systems, ranging from the microscopic synaptic connection patterns between individual neurons to the macroscopic connectivity patterns between cortical areas. The pace of these developments is increasing with time due to experimental support from programs such as the B.R.A.I.N. Initiative. At the same time, however, a challenge commonly arises: even if we knew the complete connectivity diagram for a single model organism, how could we understand anything about the resulting nonlinear dynamics? Here, we present a new mathematical approach to go from the connectivity of an individual network - for example the realization of a random graph on an individual trial or the precise connection patterns in an experimental reconstruction - to the spatiotemporal pattern of oscillations in a neural system. By introducing a complex-valued matrix formulation for the Kuramoto model, we develop an analytical approach to study the transient behavior arising from the precise topology of a single network. This approach allows us to analytically study the collective behavior of oscillator networks and offers a new, geometric perspective of synchronization phenomena in terms of the spectrum of the network's adjacency matrix. We then use this novel approach to study two important dynamical phenomena in neural systems: (1) the emergence of synchronous oscillations and complex spatiotemporal patterns like chimera states, and (2) patterns of neural activity important for working memory, a short-term store in the brain that is dynamic, flexible, and modifiable online. These results provide new analytical insight into how sophisticated spatiotemporal dynamics arise from specific connection patterns in neural systems.