New computational and mathematical approaches to large-scale recordings and networks in neuroscience, Part 1
New technologies are rapidly advancing large-scale recordings and network reconstructions in neuroscience. In this mini-course, I will introduce our research that has developed new computational and mathematical approaches in two domains: (1) spatiotemporal dynamics in cortex and (2) non-asymptotic, algebraic approaches to the topology of random graphs.
In the first section, I will introduce the computational approaches to spatiotemporal data we have developed over the past several years. With these methods, we have found that specific cortical activity patterns are organized into neural traveling waves, at the mesoscopic (single region) scale in wake and the macroscopic (whole-brain) scale in sleep. These waves have implications for the way we think about noise in the brain, and for theories of sensory processing. I will introduce approaches we have taken in recent work on understanding dynamics and computation in systems with time delay.
In the second section, I will introduce networks and graph theory in neuroscience. I will summarize the approaches taken to characterize structure in brain networks and recent technological advances that have led to network reconstructions at increasingly large scales. I will then describe a non-asymptotic analytical approach to random graphs. While theoretical investigations have generally applied asymptotic evaluations in the large-N limit or the continuum approximation, our approach allows to obtain analytical expressions for algebraically well-defined graph-theoretic measures, valid outside the large-N limit. These results allow rigorously assessing the agreement of structural models, such as the Erdős–Rényi random graph or the Watts-Strogatz small world graph, with data obtained from network reconstructions.
Taken together, these works represent new computational and mathematical approaches to the data that are currently revolutionizing neuroscience. In future work, we aim to use these methods to understand dynamics and computation in cortical networks during natural sensory processing.
