First-Exit Statistics in Biased, Active and Cognitive Systems

A complete understanding of how biased, active, and cognitive systems process information and behave under fluctuations remains an open challenge. Although first-exit statistics are well known for processes like biased Brownian motion, analytic derivations are often tedious. For active systems, symmetries in first-exit time problems remain poorly understood. Meanwhile, cognitive neuroscience and stochastic thermodynamics have progressed largely independently, despite their shared focus on nonequilibrium dynamics. This Thesis aims to develop a comprehensive analysis for characterizing first-exit time statistics in biased, active, and cognitive systems. To this end, we employ stochastic processes theory, including short-cuts from the martingale framework, to derive exact analytical results for first-exit problems related to biased Brownian functionals and run-and-tumble models. Analytical expressions are validated through numerical simulations. In parallel, psychophysical experiments were conducted with 45 human participants, who made bias-direction judgments from visualized biased Brownian trajectories presented in short movies. We analyzed the behavioral data within the framework of first-exit phenomena and stochastic thermodynamics, and compared to the optimal Wald’s sequential probability ratio test (SPRT). The results yield exact analytical expressions, computed in just a few lines, for distributions and functionals in first-exit problems associated to biased Brownian trajectories. These include the mean area swept and the hitherto unexplored cross-correlation function between the first-exit time and the first-exit site. In active matter, we uncover asymmetry in first-exit time statistics for run-and-tumble models in symmetric barriers, which is quantified via the Kullback–Leibler divergence and signal-to-noise ratios, and identify conditions under which symmetry is restored. In human perceptual decision-making, we found that performance limits were consistent with recently established thermodynamic trade-offs (thermodynamic uncertainty relations) involving speed, accuracy, and dissipation; higher rates of entropy production lead to shorter decision times. Furthermore, decision times exceeded Wald’s SPRT predictions. We propose an alternative model that integrates evidence with a memory time constant that closely matches observed behavior. These results advance statistical physics by extending the first-exit theory in biased, active, and cognitive systems while providing cognitive neuroscience with a link to thermodynamic trade-offs.