Ergodicity shapes inference in biological reactions driven by a latent trajectory

Abstract: Many natural phenomena are quantified by counts of observable events, from the annihilation of quasiparticles in a lattice to predator-prey encounters on a landscape to spikes in a neural network. These events are triggered at random intervals, when an underlying, often unobserved and therefore latent, dynamical system occupies a set of reactive states within its phase space. We show how the ergodicity of this latent dynamical system, i.e. existence of a well-behaved limiting stationary distribution, constrains the statistics of the reaction counts. This formulation makes explicit the conditions under which the counting process approaches a limiting Poisson process, a subject of debate in the application of counting processes to different fields. We show that the overdispersal relative to this limit encodes properties of the latent trajectory through its hitting times. These results set bounds on how information about a latent process can be inferred from a local detector, which we explore for two biophysical scenarios. First, in estimating an animal's activity level by how often it crosses a detector, we show how the mean count can fail to give any information on movement parameters, which are encoded in higher order moments. Second, we show how the variance of the inter-reaction time sets a fundamental limit on how precisely the size of a population of trajectories can be inferred by a detector, vastly generalizing the Berg-Purcell limit for chemosensation. Overall, we develop a flexible theoretical framework to quantify inter-event time distributions in reaction-diffusion systems that clarifies existing debates in the literature and explicitly shows which properties of latent processes can be inferred from observed reactions.