Odor Landscapes in Turbulent Environments

The olfactory system of male moths is exquisitely sensitive to pheromones emitted by females and transported in the environment by atmospheric turbulence. Moths respond to minute amounts of pheromones, and their behavior is sensitive to the fine-scale structure of turbulent plumes where pheromone concentration is detectible. The signal of pheromone whiffs is qualitatively known to be intermittent, yet quantitative characterization of its statistical properties is lacking. This challenging fluid dynamics problem is also relevant for entomology, neurobiology, and the technological design of olfactory stimulators aimed at reproducing physiological odor signals in well-controlled laboratory conditions. Here, we develop a Lagrangian approach to the transport of pheromones by turbulent flows and exploit it to predict the statistics of odor detection during olfactory searches. The theory yields explicit probability distributions for the intensity and the duration of pheromone detections, as well as their spacing in time. Predictions are favorably tested by using numerical simulations, laboratory experiments, and field data for the atmospheric surface layer. The resulting signal of odor detections lends itself to implementation with state-of-the-art technologies and quantifies the amount and the type of information that male moths can exploit during olfactory searches.

Popular Summary

Sex pheromones are chemicals secreted by animals for the purpose of attracting mates. Female moths emit blends of pheromones, which act as a form of long-range communication, to entice male moths located hundreds of meters away. These pheromones are transported by atmospheric turbulence, which creates strong gradients of concentration. The resulting signal detected by male moths is known to be strongly intermittent and sporadic, yet a quantitative characterization of this signal is still lacking. Such characterization is important since the behavioral response of male moths is sensitive to the structure of the plumes of pheromones subject to distortion caused by atmospheric turbulence. Furthermore, the statistics of stimuli in physiological conditions is crucial to building olfactory stimulators aimed at reproducing those signals in well-controlled laboratory conditions. We investigate the intensity, temporal structure, and chemical composition of the pheromone signal at the receiver’s end.

We develop a theory that predicts the explicit form of the probability distribution of the pheromone concentration as a function of the distance from the source. We also investigate the durations of pheromone detections and their spacing in time. Our work hinges on Lagrangian methods and is based on the analysis of the trajectory of particles transported by the turbulent flow. We successfully test predictions by numerical simulations and experimental data, both in the laboratory and in the field.

Our work quantifies the amount and the type of information that male moths can exploit during olfactory searches and has applications in a variety of fields, including entomology, neurobiology, and technology. Olfactory signals dictate the flight patterns of male moths, shedding light on the neurobiology of how insects communicate. Our results also have potential applications to techniques that inhibit the mating behaviors of invasive pests. We expect that our findings will pave the way for new laboratory experiments using tethered insects.

Figure 1

The structure of a turbulent odor plume. (a) A two-dimensional section of a plume from the jet-flow experiment [3]. The shaded area is the projection of the conical average plume, i.e., the region outside of which crosswind transport is weak and the odor concentration decays rapidly. (b) A typical time series of the odor concentration at a given point in space [3]. Red triangles indicate the occurrence of whiffs, i.e., intervals when the local concentration is above the threshold $c_{\text{thr}}$ indicated by the red line. For olfactory searches, the threshold is comparable to the sensitivity of the pheromone receptors of the insects. The blue line indicates the average concentration $C$ in the regions where the signal is above the noise level. (c) A two-dimensional section of two blending plumes from the jet-flow experiment [4]. The two different chemicals mix as they progress downwind, and the resulting signal is a blend.

Figure 2

Scheme of the Lagrangian approach. The concentration $c$ at a given location $\mathit{x}$ and time $t$ is expressed in terms of the history of a Lagrangian puff, that is, an ensemble of particles transported by the turbulent flow, all starting at $\mathit{x}$ at time $t$ and dispersing backwards in time. The concentration $c$ is determined by the size of the Lagrangian puff when it hits the source (if it does): (a) Average values $c\simeq C$ correspond to the puff hitting the source with a typical value of the size; (b) intense concentrations $c$ correspond to the puff hitting the source with unusually small sizes; (c) the concentration $c$ vanishes if the puff never hits the source throughout its history. (d) The sketch of a time series. From left to right: blank, the concentration $c$ vanishes; whiff, the puff hits the source with a small size and $c$ passes the threshold of detection $c_{\text{thr}}$; blank, turbulent diffusion enlarges the size of the puff and $c$ decays below the threshold, then $c$ vanishes because of the puff losing contact with the source. The red strips indicate the regions of the puff overlapping with the source as the puff is swept by the turbulent flow.

Figure 3

The statistics of the concentration of odors for the Kraichnan flow [26] [panel (a)] and jet-flow experiments [3] (all other panels). (a) Moments of the concentration for the Kraichnan flow, as a function of the ratio $x/a$ between the distance $x$ to the source and its linear size $a$. Solid lines are the theoretical predictions in Eq. (8). (b) Same as in (a), for the jet-flow experimental data, compared to our predictions (9). (c) The probability density function (pdf) of the concentration (rescaled by its typical value $C$ within the whiffs) at various distances (shown in the inset) from the source, compared to our prediction (10). Data have been shifted vertically for viewing purposes. (d) Pdf for the duration $t_w$ of the whiffs (time intervals when the concentration remains above a threshold of detection $c_{\text{thr}}$) at various distances from the source. Solid lines are our predictions (5). Durations are rescaled by their most likely value $\tau$. (e) The pdf for the duration of the whiffs vs $c_{\text{thr}}$ (at $x/a=5$), compared to our prediction (5). (f) The pdf for the duration of the blanks (intervals without detections) at various distances from the source (shown in the inset), compared to our prediction (6).

Figure 4

Odor statistics in the atmospheric surface layer. (a, b) The intensity of the fluctuations of odor concentration as a function of the downwind distance $x$ and of the ratio between the crosswind distance $y$ and $x$, multiplied by the ratio between the mean $U$ and the turbulent component $v$ of the flow transporting the odors. Solid lines are our predictions, from Eq. (11). (c) The probability of detection, i.e., that the local concentration of odors is above a certain threshold, vs the value of the detection threshold. The solid line is the theoretical prediction from Eq. (12). The dashed line is the intermittency factor $\chi =\text{Prob}(c>0)$ in Eq. (11). (d, e) The probability density functions for the duration of the whiffs (time intervals when the concentration remains above the threshold of detection) and the upcrossing time intervals, defined as the time elapsed between the beginnings of two successive whiffs [$t_w+t_b$ in Fig. 1]. The power law $-3/2$ is the theoretical prediction derived here.