Differential Geometry

Functional connectivity has been the focus of many research groups aiming to study the interaction between cells and brain regions. A standard method for analyzing connectivity is to statistically compare pairwise interactions between cells or brain regions across behavioral states or conditions. This methodology ignores the intrinsic properties of functional connectivity as a multivariate and dynamic signal, expressing the correlational configuration of the network. In this talk, I will present a geometric approach, combining Graph Theory and Riemannian Geometry to build "a graph of graphs" and extract the latent dynamics of the overall correlational structure. Using this approach, we formulate the statistical relations between network dynamics and spontaneous behavior as a second-order Taylor’s expansion. Our analysis shows that fast fluctuations in functional connectivity of large-scale cortical networks are closely linked to variations in behavioral metrics related to the arousal state. We further expand this methodology to longer time scales to study the effect of dopamine on network dynamics in the primary motor cortex (M1) during learning. We developed a series of analysis methods indicating that as animals learn to perform a motor task, the network of pyramidal neurons in layer 2-3 gradually and monotonically reorganizes toward an "expert" configuration. Our results highlight the critical role of dopamine in driving synaptic plasticity: Blocking dopaminergic neurotransmission locally in M1 prevented motor learning at the behavioral level and concomitantly halted plasticity changes in network activity and in functional connectivity. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2025-01-29. Recording duration: 00:27:08.

Michael Hutchings introduces Reeb vector fields and explains why their periodic orbits are substantially better understood in three dimensions through Seiberg-Witten theory. The colloquium develops Irie's closing lemma, which creates a periodic orbit through a selected region after a small perturbation, and discusses a quantitative refinement connecting the perturbation size to an upper bound on the orbit period.

One of the main achievements of my last four years at Perimeter has been the development and formulation of intrinsic null geometry. This framework has served as a common denominator for the study of the (algebraic) quantization of gravity on null hypersurfaces; connections to quantum-gravity phenomenology for causal diamonds and the derivation of the Verlinde-Zurek fluctuation identity; a complete characterization of the null gravitational phase space; links to asymptotic null infinity; and the related classification of eBMS anomalies. After introducing the basic toolkit of null geometry, I will digress on Carrollian connections and on how the intrinsic geometry is related to an ambient manifold. I will then introduce the null Brown-York stress tensor, which allows the intrinsic gravitational constraints to be recast as conservation laws. Finally, I will briefly retrace the path through the aforementioned applications, with particular emphasis on new and ongoing results.