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Algorithmically Defining Olfactory Responses in Animals

For their survival, animals use olfaction (their sense of smell) to locate food, find mates, and avoid predators. While animals excel at these tasks, the algorithmic and mechanistic processes governing this behavior are not well understood. Our lack of understanding is made vivid by the pre-industrial nature of current odor source localization technology; we use dogs to search for contraband, pigs to find truffles, and pouch rats to detect land mines.

While animals exhibit a wide range of morphological and physiological characteristics, they use similar cues to locate odor sources: differences in odor concentration across space and over time, measurements of local concentration gradients, frequency of odor encounters, and environmental cues about flow direction, such as wind or water flow. Moreover, animals share many morphological and kinematic similarities in their olfactory systems, including bilateral sensors, dynamic sampling behaviors (sniffing, antenna movements), and common anatomical features (neural circuitry in the first stages of olfactory processing). These commonalities suggest that convergent evolution has identified robust strategies for locating odors in complex environments.

Finding odor sources is hard because odor environments are complex. Odors in natural environments travel along turbulent flow paths governed by wind, topography, and molecular diffusion. What exactly does the odor landscape look like? Figure 1a shows an odor plume imaged by our collaborator, John Crimaldi (University of Colorado). It’s clear that the concentration is far from being uniform or smooth in space. Locations near the source at some instants of time show nothing, while even distant places show bursts of high concentrations. Figure 1b displays the output from a photoionization detector (PID) at different distances from a line of odorant placed on a table in a room. Unlike the plume, the only airflow is the ambient turbulent flow arising from ventilation, movement, etc. in the room. Presumably, this is what a mouse perceives when trying to locate an odor trail. In the PID measurements, the baseline is higher near the source than away from it. But another common feature of these complex odor landscapes is that the rate of fluctuation increases with proximity to the source.

Figure 1a. Snapshot of a chemical plume in water. Image credit: John Crimaldi. 1b. Photoionization detector (PID) time series at 2 centimeters and 8 centimeters from an odor source. Image credit: Nathan Urban.

Our group is using the behavior of animals in constrained environments to infer the algorithms they utilize to locate odor sources and follow odor trails. For example, in videos created by our collaborator Kathy Nagel (New York University), we observe fruit flies walking as controlled odors are turned on and off. The flies transiently make more frequent turns when an odor is switched from on to off. Nathan Urban (University of Pittsburgh), another collaborator on this project, films mice as they follow odor trails associated with a food reward. His lab finds that the mice are less accurate at following trails when one nostril (naris) is occluded, suggesting that “stereo-olfaction” may be important in trail following. The subjects in both of these experimental setups exhibit frequent movement called casting: the mouse moves its head around while sniffing or the fly moves orthogonal to the direction of the wind.

We are currently using behavioral analysis to create models and test them on various odor landscapes. Testing the models in realistic settings is a challenge since we are just now beginning to understand what the landscape “looks” like to an animal. Approaches to modeling the odor concentration range from simple heuristic models to full Navier-Stokes simulations in controlled and closed environments. Any simulation of the environment should capture the intermittency of the odor, as seen in Figure 1. Furthermore, any navigation algorithm must be robust in the face of distortions of the plume, e.g., due to obstacles, changes in humidity and temperature, and sniffing or other movements of the animal itself. Once we have a good understanding of the algorithms, we want to then understand their neural and physiological basis. There is considerable knowledge of how odors are converted to electrical signals in the brain, but how these signals can be used for odor source localization remains a mystery. This article and a follow-up piece to appear in the October issue of SIAM News discuss a few of the simple algorithms that animals may use to find an odor source. Here I explore possible mechanisms for odor location that do not depend on the actual concentration.

Besides concentration, what other signals could be used to produce odor localization? Figure 1 shows rather clearly that the concentration fields are quite complex and chaotic. Figure 1b demonstrates that the frequency of fluctuations is much greater closer to the source; the plume in Figure 1a depicts similar behavior and a clear direction of the flow. These images suggest a method for finding the odfigor based on the frequency of odor encounters, and when possible, direction of the wind. For example, a very simple algorithm would move in a direction that shortens the intervals between odor encounters. An excellent algorithm, Infotaxis, is built with this premise in mind; the agent moves in a direction that increases the Shannon information [1]. At each time step, Infotaxis builds up an estimate of the probability that the source is at some location, $$\hat{x}$$. For instance, if the agent hasn’t yet found the source, it is certain that the source is not at the current position. With this estimated probability, the agent estimates the entropy in each of the four cardinal points and moves to the point where the entropy is minimized.

While Infotaxis is an optimal strategy for finding an odor source when the encounters are rare, it is very computationally intensive. It is also not clear how a biological organism could implement Infotaxis. Instead, we have developed algorithms based on the rate of encounters between “spikes” of odor that work reasonably well in simple models for the odors. Since many odors arrive in a windborne plume, it is possible to use additional information about the wind direction to follow the odor upwind, and move crosswind or backwards when the odor is lost to increase the chance of finding it again. An example algorithm that does not use the wind works by keeping track of the last three odor encounters. If the time between those encounters is decreasing or the same, a possible strategy would be to continue to move in the current direction; otherwise, move in a random direction that is near the current direction.

For a very simple algorithm, this approach does reasonably well. Figure 2a shows the results of a Monte Carlo simulation of this algorithm applied to a single Gaussian source, which indicate that the agent spends much more time at the center of the source. Nagel has made careful observations of fly behavior in the presence of attractive odors. By temporally varying the on and off rates of an odor, she found a few rules that flies implement when walking in a small arena. Figure 2b shows an example simulation of her rules in the presence of a particle model for a plume. In this simple plume model, particles from the source are emitted randomly and drift/diffuse. A slowly-varying wind direction has a mean in the y-direction. The figure only represents a snapshot of the “plume” – if the fly is close enough to a particle, it registers as a hit. The rules use the wind direction as well as the frequency of on and off hits with the particles, and flies have a number of cells that register responses both when the odor is present and when it disappears. As with the spatial comparison algorithms, both of these temporal algorithms include a stochastic component when the hits become infrequent enough.

Figure 2a. Fraction of time spent at different locations with a Gaussian odor source using the time between events. 2b. A more sophisticated algorithm for a fly locating an odor in a “particle”-based plume. This shows the fly’s track from the start (blue dot) and a snapshot of the “plume.” Image courtesy of Kathy Nagel.

We are just beginning this work, and many mathematical and biological questions remain. How can we quantify and statistically imitate real odor landscapes in order to test models? What are the best algorithms for locating an odor and how do their parameters change at different spatial and temporal scales? What are the best search strategies when the odor is lost? Where in the olfactory system can the algorithms be implemented?

In a follow-up article to appear in the next issue, the author will describe mathematical aspects of tracking odors based on concentration difference, along with some aspects of foraging.

References
[1] Vergassola, M., Villermaux, E., & Shraiman, B.I. (2007). Infotaxis as a strategy for searching without gradients. Nature, 445(7126), 406-409.

Bard Ermentrout is a professor of mathematics at the University of Pittsburgh. He works in many areas of mathematical biology, with a focus on neuroscience.