Cartography of low-dimensional dynamical systems and their spiking properties

Qualitative study of low-dimensional dynamical systems using bifurcation theory is key to understanding systems from biological clocks and neurons to physical phase transitions. However, phenomena that are distinguished by their geometric features, such as excitability, are invisible to these topological tools, and identifying such behaviors is often difficult analytically. To this end, we develop a contrastive learning technique to identify dynamical behaviors from trajectory data, and leverage transition path theory to characterize systems beyond topological features. These techniques allow us to explore a large ensemble of stochastic dynamical systems, revealing a nonlinear gain-sensitivity tradeoff for excitable systems.