Measuring and controlling turbulent signals

Although it is relatively straightforward to reconstruct natural flight trajectories using videos, precise spatiotemporal patterns of a naturalistic odor signal are challenging to measure, since odor is usually advected by turbulent flow [24]. Instead one often measures the coarse-grained statistics of a turbulent signal, such as its time-averaged concentration, intermittency, or plume envelope, foregoing the ability to infer the odor concentration at precise timepoints, but allowing one to explore how trajectory features depend on these statistics [6, 7]. Alternatively one can present a stationary plume with a pre-measured profile, which allows one to infer the concentration experienced by a flying insect on a moment-by-moment basis, but in which case the true signal is less naturalistic. In this analysis we have examined flight trajectories generated in the latter context, since it allowed us to infer the moment-by-moment sensory experience of each insect and to look for correlations with subsequent behaviors. Further, from the perspective of the fly, which follows a highly variable path through the wind tunnel, its encounters with the plume may be a sufficient approximation to the encounters it would have with a turbulent plume [15]. Importantly, although the plume itself is stationary, the variation in each trajectory yields variation in the sensory experience of each fly. This was key in allowing us to explore how different features of plume encounters elicited different navigational decisions.

Informative features of plume encounters

Our first result that flies respond to differences in the concentration experienced during an in-flight odor encounter suggests that this feature may contain usable information about the fly’s current position relative to the plume source. Indeed, it has been shown theoretically that the distribution of nonzero instantaneous concentrations depends on displacement from the plume source [13], and our results propose that flies detect and act on this information. This suggests that theoretical trajectory-generating plume-tracking strategies, such as infotaxis [22], should incorporate non-binary odor detections in order to better recapitulate observed animal trajectories; whether such information significantly improves tracking efficiency, however, remains an open question. For example, it may be the case that higher instantaneous concentrations indicate to the fly that it is closer to the plume centerline, such that it would behoove it to make a stronger upwind turn, as we observe in the data. Our result also corroborates the fact that flies have neural mechanisms capable of detecting graded levels of odor concentration, which have been observed at the level of ORNs [25, 26], but which may also be present in the antennal lobe, one synapse downstream from the detector neurons.

Our second result, that the history prior to an odor encounter influences a fly’s response to that encounter, suggests that the number of encounters throughout a naturalistic trajectory also contains information about the fly’s displacement from the source. For example, our observations are consistent with the possibility that increased encounters might indicate that the fly has moved closer to the source, such that it should diminish its upwind turns so as not to overshoot it. Indeed, such a strategy would be consistent with moth plume tracking behavior, in which it was seen that moths tended to slow their flight as they approached a pheromone source [19]. Alternatively, as the flies accumulate odor encounters they might be more inclined to investigate the visual features of the surrounding wind tunnel, such as the floor pattern [15], which might slow their upwind progress.

More generally, the dependence of the flight trajectories on odor history is consistent with the flies’ acting on information obtained from multiple, temporally separated samples of the plume, an idea fundamental to strategies based on extended accumulation of information [22, 23]. The neural mechanisms enabling history-dependent behaviors in plume tracking are unknown, but a simple possibility is that they result from adaptation of the peripheral olfactory neurons (ORNs) in the insects’ antenna and maxillary palp, which typically decrease their sensitivity after extended odor exposure [27, 28]. ORN adaptation is typically studied, however, with much longer pulses than the sparse, intermittent puffs expected in a turbulent plume, so it is unclear how strongly this mechanism would influence natural tracking. An alternative is that history dependence arises from state changes in the antennal lobe (AL), the insect olfactory network immediately downstream of the initial receptors, analogous to the mammalian olfactory bulb. Indeed, many local neurons within the Drosophila AL substantially change their odor-evoked responses over the course of an approximately 10 second train of short pulses, with some exhibiting decreased, and others increased responses to later pulses in the train [29]. Interestingly, these AL response dynamics occur on about the same timescale as the history dependence we have observed from in the behavioral analysis. Determining the precise neural mechanisms underlying our results will be an exciting future task, and this suggests that exploring antennal lobe mechanisms may be a promising avenue for investigation.

An important factor in experiments involving flight trajectories through wind tunnels is the visual experience of the insect. Since wind tunnels are typically small due to technical constraints, i.e., between 1–2 m long and between 0.3–1 m wide, such that the insects can likely see the boundaries of the environment, it is likely the case that flight decisions are driven by a combination of both visual and olfactory stimuli. Indeed, the optomotor response of flying insects requires visual input in order for the flies to determine the wind direction relative to the ground [1], so that experiments cannot be performed in the absence of visual input. In our analyses we have attempted to control for visual effects in several ways. Since the main effect we expect is for the fly’s distance to the upwind or downwind walls to influence their propensity to turn upwind (for example, they may be less inclined to turn upwind if there is little space for them to turn into), in our analyses of the empirical data we excluded plume crossings occurring in the upwind-most and downwind-most 30 cm of the wind tunnel. As an additional control, however, in each of our data analyses we subtracted out the best linear prediction of the quantity of interest given the position x₀ of the fly along the upwind-downwind axis of the wind tunnel. Specifically, in quantifying the concentration dependence of plume-crossing-triggered heading, we first subtracted the best prediction of heading from x₀ and then calculated the correlation of the residual heading variation with plume-crossing concentration, with a similar analysis applied to the dependence of plume-crossing-triggered heading on the number of prior crossings in the trajectory. In the future, it will be useful to conduct plume-tracking experiments in larger wind tunnels, such that the visual experience does not substantially vary across different locations in the wind tunnel, but doing so at present remains a significant technical challenge.

Identifying evidence of theoretical search strategies in data

There has been much interest in elucidating theoretical principles of how insects track turbulent plumes [2]. However, due to the highly dynamical nature of flight trajectories, identifying evidence of specific search strategies in data is not straightforward. In particular, because the dynamical odor experience of each insect is controlled not by the experimenter but rather by the insect’s own movements, one cannot compare theoretical vs. empirically observed trajectories on a one-to-one basis. More precisely, one cannot force a simulated trajectory to have a data-matched odor experience without causing it to deviate potentially quite strongly from the algorithm that was supposed to have generated it.

Given these limitations, it is preferable to instead ask how the distributions over specific trajectory features vary between simulations and data. Since a key theoretical question asks about the extent to which insects accumulate and use information about the plume’s source location over extended time periods, we chose to ask how such information accumulation might affect the history dependence of navigational decisions observed in real trajectories vs. those simulated by theoretical algorithms. Consequently, we examined the joint distributions of plume-crossing-triggered heading responses and a variable indicating whether each plume crossing occurred early vs. late in the total sequence. Our findings were that although plume-crossing-triggered flight maneuvers were dominated by a brief upwind turn (Fig 1D), they indeed differed early or late in the trajectory (Fig 3A–3D, Fig 4). This suggests that although the real plume-tracking strategy used by the insects we explored likely involves a significant reactive component, it is also modulated by the insect’s history with the plume since it took to the air. This rules out plume-tracking strategies that are 100% reactive, such as the pure surge-cast strategy depicted in Fig 3E, and it suggests that including such history-dependent modulation may increase the efficiency of successful source localization.

Since the purely reactive surge-cast model (Fig 3E) did not reproduce the early-late distinction we observed in the empirical crossing distribution, we examined the history dependence predicted by two information-accumulation-based theoretical strategies: the centerline-inferring model, based on the model by Grunbaum and Willis [23], and infotaxis, in which movement of a tracking agent explicitly maximizes expected information about the location of the source in 3D space [22]. We found that the centerline-inferring model predicted a history dependence opposite to that seen in the data: later crossings generated stronger upwind turns. This inconsistency suggests that accumulated information about a plume’s centerline may not be directly converted into increased upwind advancement along the inferred centerline.

The failure of the centerline-inferring model motivates the consideration of infotaxis, an algorithm in which the tracking agent uses internal knowledge of turbulent plume statistics to interpret odor encounters and guide its search. Though the true plume used both in the original experiments and in our infotaxis simulations was cylindrical and contained within a laminar flow, we nonetheless allowed the simulated agent to assume an internal model of a more natural, plume governed by turbulent diffusion. Essentially, we wanted to ask how a search agent used to turbulence would behave when the plume was actually laminar and cylindrical; this, we felt, best approximated the experimental situation in which flies evolved to track natural plumes were introduced to an artificial one. Additionally, though it is possible the flies in the wind tunnel noticed and adapted to the real plume’s non-turbulent structure, given the high variability of their trajectories and the intermittency of their odor encounters, their dynamic flight and odor experience was likely reasonably similar to that one would expect in more naturalistic plume tracking. Performing similar experiments in the presence of turbulent, yet measurable plumes (measured, for example, using Schlieren imaging, a technique that could detect concentration changes through the optical distortions they cause over a patterned background [30, 31] will be an important step to confirming this hypothesis.

In infotaxis, though the magnitude substantially differed, the sign of the observed history dependence was consistent with infotaxis. One possible reason that infotaxis, and perhaps the empirical trajectories, exhibit weaker upwind turns for later encounters is because the physical scale at which information is gained is larger for earlier encounters and smaller for later encounters. That is, the first encounters in a trajectory confine the possible source location to a relatively large region of space upwind of where the encounters occurred, so that it advantages the tracking agent to fly upwind into this space to increase its chances of encountering the source or additional odor puffs. Later encounters, however, provide primarily fine-scale information about the source, since the large-scale location may already be known. Thus, it may behoove the tracking agent to pay more attention to the specific details of where it thinks the source might be, as opposed to flying indiscriminately upwind.

In the experiments we analyzed, the fruit flies tracked ethanol, an attractive odorant that emanates from fruit, and the mosquitoes tracked carbon dioxide, which is produced by mammalian hosts. While we did not investigate plume tracking history dependence in the presence of other odors, if the behaviors we observed were indeed evidence of efficient source localization strategies we would expect to see the same history dependence when flies tracked other odors arising from sources with positive valence. Indeed, for other attractive odors, such as Vector 960 (an odorant used in some fruit fly traps), the reflexive odor-triggered turns are quite similar to those triggered by ethanol [15]. Investigating more detailed plume tracking features as a function of odor identity would be a fruitful topic of future research. For aversive odors, on the other hand, we would not expect the same behavioral responses, as the insects’ goal would presumably no longer be to localize and approach the odor source, and so one would expect different navigation strategies.

On the role of infotaxis in biological source localization

Although infotaxis was proposed in the context of biological plume tracking, few studies have compared theory to data. In fact, to our knowledge the only study that has directly compared infotaxis to animal behavior was that of Calhoun, Chalasani, and Sharpee [32], in which the authors showed that the foraging behavior of C. elegans in the absence of a food source exhibited transitions from local to global search strategies similar to those predicted by infotaxis. In addition to the already discussed challenge of comparing theoretical to empirical flight trajectories, infotaxis trajectories bear the challenge of depending on the parameters the tracking agent uses in its turbulent plume model, such as the source emission rate or turbulent diffusivity coefficient. While our qualitative results were relatively invariant to small changes in these parameters, the observed infotaxis history dependence changed sign when the source emission rate assumed by the insect decreased too sharply (S3 Fig). D, the turbulent diffusivity coefficient assumed by an infotaxis agent in its internal plume model, also strongly affects the source position information gained from each odor detection: with a very small D, odor detection implies the source is almost directly upwind, since little spreading should have occurred and which should strongly bias movement in that direction; with a larger D, however, odors can be detected from sources with more variable positions. When we re-ran our infotaxis simulations we saw this was indeed the case: smaller D led to the predicted stronger upwind turns (S3 Fig). The sign of the history dependence, however, which was the key focus of our analysis, did not depend on D; for both small and large D early plume-crossing-triggered turns had a stronger upwind component than late turns (S3 Fig). This shows that while trajectories differ in some ways when the internal plume model changes, certain coarse-resolution features remain the same.

Since an infotaxis agent uses these parameters to infer the source location given a sequence of odor encounters, a key question is how such an agent would learn these parameters in the first place, which can vary over several orders of magnitude in natural environments [33], and incorrectly estimated values of which can affect the efficiency of its search trajectory [34]. One possibility is that by incorporating a reactive component to their behaviors, as observed in the data we analyzed, a tracking strategy might gain robustness to variations in the plume statistics that would otherwise confound a pure infotaxis strategy. That is, a moderately strong reactive component might stabilize behavioral responses that would otherwise be sensitive to the parameters of the plume. Indeed, when plume-tracking robots were programmed with different search strategies, under conditions of high odor concentration, reactive strategies outperformed infotaxis [35], perhaps because of mismatches in the true plume parameters vs. those assumed by the robot. Future work could test this hypothesis explicitly by building a full trajectory-generating model that was a hybrid of infotaxis and a reactive strategy, and calculating source-finding efficiency under varying plume statistics.

The role of vision in search and infotaxis

An important component of an insect’s search strategy is to explore visual features after encountering attractive odors [15]. Their choice to explore a visual feature likely depends on many factors, which may include the concentration and frequency and number of odor encounters. Thus, it is conceivable that our observed difference in behavior for early and late plume crossings is influenced by the insect’s choice to explore visual features (such as the checkerboard pattern on the floor of the wind tunnel) rather than solely being a function of plume interactions. These same visual features, however, could also help to guide the animal’s surge and cast behavior by providing helpful landmarks.

Of the models we tested, only the infotaxis model captured the correct qualitative change in behavior between early and late plume encounters. The infotaxis model, however, requires that the animal has access to its exact position in world coordinates, something that is quite challenging for an insect flying in the wind over large distances (separating distance, ground speed, and wind speed, from measurements of optic flow and air speed is not trivial [36]. Instead, it possible that insects could use visual features as relative landmarks in an infotaxis-like strategy. We hope that our results will inspire the development of new biologically plausible search theories similar to infotaxis, but that take into account the limited information available to insects, which includes visual landmarks, but not exact location information.

Dataset

We used datasets previously described by van Breugel and Dickinson [15] and van Breugel et al. [21]. See the referenced works for more details. Briefly, groups of either fruit flies or mosquitoes were released in a 1.5 m x 0.3 m x 0.3 m wind tunnel (with a 1.3 m x 0.3 m x 0.3 m section available for flight) containing a gentle, laminar wind flowing along the long axis (30–60 cm/s wind speed). When any insect took off, a camera-based tracking system began recording it, and the insect’s 3D trajectory was later reconstructed from the video. In the experimental conditions we have focused on, a laminar odor plume was also present in the wind tunnel (ethanol for fruit fly experiments and CO2 for mosquito experiments). The plume retained a stationary profile due to the laminar wind flow, and was approximately cylindrical in shape, with the highest concentration along the center of the cylinder. The odor concentration as a function of 3D position in the wind tunnel was measured prior to releasing any insects. For the ethanol plume in the fruit fly experiments, there was no discernible widening of the plume from the upwind to the downwind end of the wind tunnel, such that the plume’s cross section did not change along the wind axis. For the CO2 plume some diffusion occurred, with the plume widening slightly with increasing x, i.e. towards the downwind end of the wind tunnel. However, since our history dependence analysis accounted for x before comparing early vs. late plume crossings, it is unlikely that the trend observed in mosquitoes tracking CO2 was due to perceived changes in the plume cross section. With both plume types, however, the plume’s stationarity allowed the approximate concentration experienced by the insect to be inferred from its position. The resulting dataset that we used as our starting point thus contained registered time-series of experienced odor concentration (up to a multiplicative factor), 3D position and 3D velocity (Kalman filtered with a constant velocity Kalman filter) at every time point in every trajectory of every insect.

Identifying plume crossings

We defined a plume crossing as the interval during which the odor concentration experienced by the insect rose above a certain threshold. To determine an optimal threshold for each species we asked which threshold best separated potential plume crossings (in which the insect entered the vicinity of the plume but may or may not have crossed it) into below- and above-threshold crossing groups. Specifically, for a range of candidate thresholds, we asked when the difference between the post-crossing headings in the above-threshold group (averaged over crossings and time) was maximally different from that in the below-threshold group. We reasoned that if our chosen threshold were below its true value, the above-threshold group would get diluted with non-crossings, making it more closely resemble the below-threshold group; if our chosen threshold were above the true value, however, then the below-threshold group would get diluted by true crossings, making it more closely resemble the above-threshold group. An optimal threshold that ideally separates the below- and above-threshold groups should exist somewhere in between. However, upon performing this analysis a large uncertainty arose surrounding the calculated difference between the two groups’ heading time series (S1 Fig), such that the optimality of the chosen threshold could not be guaranteed. To keep our analysis straightforward, we chose an ethanol threshold approximately corresponding to the “elbows” in S1A Fig (for fruit flies) and S1D Fig (for mosquitoes), corresponding to 0.0009% ethanol and 430 ppm CO2, respectively, to define plume crossings.

Importantly, however, none of our results depend on the precise threshold used to define a plume crossing, since an additional threshold was either kept as a free parameter (Fig 2) or the chief comparison was made between groups defined by the same plume-crossing threshold but which differed along other dimensions, such as crossing history. To provide further evidence that the precise crossing threshold value did not affect our qualitative results, we re-ran the history dependence analysis from Fig 4 using the 0.3 m/s wind speed fly-tracking-ethanol experiments, but extracting crossing sets according to a range of thresholds that varied over almost two orders of magnitude (S5 Fig). We found, however, that regardless of which threshold we chose, the results followed the same trend: late crossings yielded turns with a weaker upwind component than earlier crossings, all else held constant. Additionally, we re-ran our hybrid model analysis using a wide range of thresholds (S7 Fig) but also found that this parameter did not substantially affect our results.

In all of our analyses we chose t = 0 to correspond to the peak odor concentration experienced during the plume crossing (e.g., as in Fig 1D). We defined the end of each plume crossing as the time when the insect either reentered the plume (the experienced concentration rose above threshold again) or landed.

Data exclusion criteria

In order to limit the influence of geometrical confounds on the results of our analysis, unless otherwise noted we excluded the following sets of plume crossings from all empirical and model-generated datasets when performing our analysis. First, we excluded all crossings occurring in the upwind- or downwind-most 30 cm of the wind tunnel. This was motivated by the possibility that flies already in the upwind end of the wind tunnel may exhibit more crosswind-oriented flight simply because they have less of an area to turn into upwind of them. Second, we excluded all crossings that did not occur at a crosswind angle, i.e., in which the initial crossing heading was not between 60° and 120° from upwind. We made this exclusion simply because we were seeking out upwind directed changes in heading, which would not be visible if flies were already heading upwind. Relaxing these criteria did not change the qualitative results of our analysis of the dataset, but they did introduce confounding factors into our simulated trajectories. For example, when we included the upwind-most 30 cm of the wind tunnel in the surge-cast model analysis (Fig 3E), late crossings indeed appeared more crosswind, simply because many late crossings occurred in this upwind-most portion, where the simulated agents’ contacting of the upwind wall often prevented them from turning as upwind as in the earlier crossings that occurred in the central portion of the wind tunnel.

Partial correlation analysis

To calculate the partial correlation between peak concentration experienced during a plume crossing and the fly’s heading at time t following the crossing (Fig 2), we first fit a linear model predicting the heading h(t) from the initial heading h₀ = h(0) and the position x₀ along the long axis of the wind tunnel. We then subtracted this prediction from h(t) to yield a residual heading h’(t), and we subsequently calculated the correlation between peak concentration c_peak and h’(t). Given this framework, any correlations between c_peak and h’(t) cannot be through the confounding factors of initial heading or position in the wind tunnel.

Threshold and threshold-linear model fitting

We defined the threshold model as: where h(t) is the heading t seconds after the plume crossing; h₀ is the heading at the time of the plume crossing, scaled by parameter a_h; x₀ is the position at the time of the plume crossing along the upwind/downwind axis, scaled by parameter a_x; h_< is the heading predicted when the peak crossing concentration c_peak is below the threshold c_th; h_> is the additional heading change predicted when the concentration was above c_th, and Θ is the Heaviside step function, which is 0 below 0 and 1 above 0.

We defined the threshold-linear model as: where a_c is an additional parameter specifying the linear dependence of heading on concentration when c_peak is greater than the threshold c_th.

In both models we fit a_h, a_x, h_<, h_>, and c_th. In the threshold-linear model we additionally fit a_c. The threshold model is thus “nested” within the threshold-linear model. For both models we chose as the best fit parameters those which minimized the squared difference between the true and the model-predicted headings. To determine whether the threshold-linear model fit the data significantly better, we used an F-test. When calculating the p-value, however, we replaced the number of crossings with the number of unique trajectories, since each trajectory yielded multiple crossings, thus providing a more conservative estimate of significance by not treating all crossings as independent occurrences.

History dependence analysis

To prevent geometrical factors from confounding the results of our comparison of early vs. late plume crossings, we first constructed a new variable h*(t) representing the change in heading at time t that remained after linear predictions of Δh(t) from x₀ and t_flight (the time spent flying since takeoff) were subtracted from Δh(t). In particular, we fit the predictor coefficients and calculated the predictions of Δh(t) using all non-excluded plume crossings, regardless of crossing number. Following the calculation of h*(t) for each crossing at each time point, we grouped crossings into early and late classes as before and compared h*(t) between the two groups using a t-test. Finally, since there were occasionally multiple non-excluded crossings from one trajectory in the same class, when calculating the p-values shown in Fig 4 we used as the number of data points in each class the number of unique trajectories, as opposed to the number of unique crossings.

In an additional related analysis, also aimed at uncovering a relationship between crossing number and post-crossing heading, instead of splitting crossings into early vs. late crossings we simply calculated the partial correlation between crossing number and Δh time-averaged from 350–450 ms post-crossing. This partial correlation was calculated by first subtracting out the best linear predictions of the time-averaged Δh from x₀ and t_flight (S4 Fig).

Base model for surge-cast and centerline-inferring tracking algorithms

We based both our surge-cast and centerline-inferring algorithms on a correlated random walker, described by: where v is the tracking agent’s velocity, with the dot indicating its time derivative, τ is the timescale of turning, η is a Gaussian white noise variable sampled independently at each timestep with zero-mean and diagonal covariance matrix with diagonal entries η, w_c is a unit vector in the crosswind plane pointing toward the centerline of the wind tunnel, and b is a positive scalar multiplying w_c. Before augmenting the base model to include either surging or centerline-inferring features, we fit τ, η, and b to approximately match the speed, angular velocity, and crosswind position distributions of a simulated agent to those of the empirical data (S6 Fig). This yielded values of τ = 0.42 s, η = 1.9 m/s and b = 0.25 m/s. Trajectories generated by the base model alone thus followed a correlated random walk structure with a 0.42 s timescale and with a moderate “casting” bias of oscillating around the wind tunnel centerline.

In both the surge-cast and centerline inferring models, we modeled the wind tunnel to have the same geometry and the plume to have the same concentration profile as in the empirical data. We then analyzed the resulting plume crossings using the same exclusion criteria as we did with the real data.

Surge-cast model

To add a surging component to the base model just described we augmented it with an additional upwind bias term such that where w_u is a unit vector pointing upwind, and a(t) is the convolution of an alpha function with timescale 0.07 s and maximum amplitude 0.8 m/s and a train of delta functions placed at the time points corresponding to the peak odor experienced in each crossing. The timescale and amplitude of the alpha function were chosen so that the simulated tracking agent’s plume-crossing triggered responses exhibited approximately the same dynamics as observed in the empirical plume-crossing data.

Centerline-inferring agent simulation

To generate trajectories for the centerline-inferring algorithm we augmented the base model with a different upwind bias term:

Here k is a positive-valued scalar that increases as the uncertainty of the plume’s centerline location decreases. In particular, we let k = k*/|K|, where k* is the upwind bias and K is the covariance matrix of a posterior belief probability over the 2D centerline location (y*, z*), whose determination is given shortly. Furthermore, in the centerline-inferring agent, we let w_c be a vector that points not to the centerline of the wind tunnel, but rather to the maximum a posteriori estimate of the centerline of the plume. As with the surge-cast model, we chose the scaling factor for the upwind bias such that the plume-crossing triggered responses showed approximately the same dynamics as in the data.

The 2D belief distribution over the plume centerline location (y*, z*) was updated in a Bayesian way each time the simulated insect crossed the plume at location (x_n, y_n, z_n) (with n indexing the plume crossing number):

We assumed that the likelihood and prior (the two terms on the right-hand side of the equation, respectively) were both Gaussian with diagonal covariances. Thus, the posterior (the term on the left-hand side of the equation), the likelihood, and the prior could each be parameterized by a mean and covariance matrix. Note that in this model the posterior after one crossing acts as the prior at the next crossing. We chose the likelihood covariance K_s and the covariance K₀ of the prior over (y*, z*) before any crossings had occurred to have diagonal entries (2m)² and (5m)², respectively. Finally, to model short-term memory decay over several seconds, we allowed the mean of the posterior to decay exponentially to (0, 0) over τ_m = 10 seconds, with the posterior covariance K decaying to K₀ over the same timescale.

Infotaxis simulation

The infotaxis algorithm we used was developed by and detailed in [22]. In this algorithm a tracking agent moves along a rectangular lattice in order to localize the source of an odor plume. To model turbulence, the odor plume is not assumed to have a concentration gradient, but rather to yield random “hits”, with hit probability maximized immediately downwind of the source. The tracking agent maintains a belief distribution over the possible source locations that is updated at each time step, depending on whether the agent receives a hit or a miss. This update is Bayesian and thus depends on an explicit formula for the probability of receiving a hit a function of displacement from the source, that is, the likelihood. The principle difference between the Bayesian update in the centerline-inferring model vs. in infotaxis is that in the centerline-inferring model the belief distribution is a 2D Gaussian over the centerline location, whereas in infotaxis the distribution is over the 3D source position coordinate and has no parametric form. Since hit probability depends on the agent’s position, and hits and misses update the source position distribution in different ways, the tracking agent moves so as to maximize the expected decrease in uncertainty of the source position distribution, as measured by its Shannon entropy.

The likelihood formula is derived in [22] from the time-averaged hit rate one would expect at different displacements from a source in a turbulent medium. Importantly, this hit rate depends on D, the effective diffusivity of the turbulent medium [24]. When D is small, the plume is confined to a narrow band downwind of the source, and when D is large the plume becomes much more spread out, and can even yield hits upwind of the source.

We ran our simulations on a 3D lattice that simulated positions accessible within the wind tunnel, with lattice points 2 cm apart from one another. A discretized version of the non-turbulent cylindrical plume used in the experiments was simulated in the center of the lattice, and the tracking agent received a hit at every time step it was inside the plume. Note, however, that even though the simulated plume was cylindrical and non-turbulent, the actions of the tracking agent in our simulation were driven by its assumption that it was interacting with a naturalistic, turbulent plume. For each experimental flight trajectory we generated one corresponding infotaxis trajectory that started at the same (but discretized) location in the wind tunnel as the experimental trajectory and which lasted for the same number of timesteps as the number of lattice points that the experimental trajectory traversed.