How insects navigate complex odor plumes, where the location and timing of odor packets are uncertain, remains unclear. Here we imaged complex odor plumes simultaneously with freely-walking flies, quantifying how behavior is shaped by encounters with individual odor packets. We found that navigation was stochastic and did not rely on the continuous modulation of speed or orientation. Instead, flies turned stochastically with stereotyped saccades, whose direction was biased upwind by the timing of prior odor encounters, while the magnitude and rate of saccades remained constant. Further, flies used the timing of odor encounters to modulate the transition rates between walks and stops. In more regular environments, flies continuously modulate speed and orientation, even though encounters can still occur randomly due to animal motion. We find that in less predictable environments, where encounters are random in both space and time, walking flies navigate with random walks biased by encounter timing.

In statistical data assimilation one evaluates the conditional expected values, conditioned on measurements, of interesting quantities on the path of a model through observation and prediction windows. This often requires working with very high dimensional integrals in the discrete time descriptions of the observations and model dynamics, which become functional integrals in the continuous-time limit. Two familiar methods for performing these integrals include (1) Monte Carlo calculations and (2) variational approximations using the method of Laplace plus perturbative corrections to the dominant contributions. We attend here to aspects of the Laplace approximation and develop an annealing method for locating the variational path satisfying the Euler-Lagrange equations that comprises the major contribution to the integrals. This begins with the identification of the minimum action path starting with a situation where the model dynamics is totally unresolved in state space, and the consistent minimum of the variational problem is known. We then proceed to slowly increase the model resolution, seeking to remain in the basin of the minimum action path, until a path that gives the dominant contribution to the integral is identified. After a discussion of some general issues, we give examples of the assimilation process for some simple, instructive models from the geophysical literature. Then we explore a slightly richer model of the same type with two distinct time scales. This is followed by a model characterizing the biophysics of individual neurons.

We and others have shown that during odor plume navigation, walking Drosophila melanogaster bias their motion upwind in response to both the frequency of their encounters with the odor (Demir et al., 2020), and the intermittency of the odor signal, which we define to be the fraction of time the signal is above a detection threshold (Alvarez-Salvado et al., 2018). Here we combine and simplify previous mathematical models that recapitulated these data to investigate the benefits of sensing both of these temporal features, and how these benefits depend on the spatiotemporal statistics of the odor plume. Through agent-based simulations, we find that navigators that only use frequency or intermittency perform well in some environments - achieving maximal performance when gains are near those inferred from experiment - but fail in others. Robust performance across diverse environments requires both temporal modalities. However, we also find a steep tradeoff when using both sensors simultaneously, suggesting a strong benefit to modulating how much each sensor is weighted, rather than using both in a fixed combination across plumes. Finally, we show that the circuitry of the Drosophila olfactory periphery naturally enables simultaneous intermittency and frequency sensing, enhancing robust navigation through a diversity of odor environments. Together, our results suggest that the first stage of olfactory processing selects and encodes temporal features of odor signals critical to real-world navigation tasks.