Navigating a diversity of turbulent plumes is enhanced by sensing complementary temporal features of odor signals

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, i.e. 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.


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The complexity of natural odor plumes makes olfactory navigation a difficult task. Turbulent flows 32 produce rapid changes in the local concentrations of the odor and instantaneous local gradients 33 of the odor often do not point toward the source (Celani et al., 2014;Crimaldi and Koseff, 2001). 34 Encounters between the animal and odorized packets of air are intermittent, with durations and  To navigate plumes exhibiting this degree of temporal complexity, animals must be able to detect what conditions these two temporal features -odor intermittency and encounter frequency -can 103 enhance the navigation of turbulent odor plumes. To examine the contribution to navigation from 104 these two temporal features alone, we ignore all other sensory modalities, such as concentration 105 sensing, concentration gradient sensing, bilateral sensing, and vision. We first demonstrate 106 analytically that the dynamical model proposed in the first study above picks out (in appropriate 107 limits) odor signal intermittency, while the model in the second study responds to the frequency 108 of odor hits. These two temporal features are complementary and can be varied independently, 109 forming a natural basis of temporal sensing. We devised a simple combined model that 110 incorporates intermittency sensing and frequency sensing in a minimal way, and uses these two 111 "sensors" to drive upwind orientations. Using agent-based simulations, we first show that this 112 combined model requires both sensors to successfully navigate both measured plumes used in 113 the two studies. We then applied the navigational model to simulated plumes, leveraging an 114 advecting-diffusing packet framework that mimics odor motion in turbulent flows (Farrell et al.,115 2002). We find that to robustly navigate across a variety of plumes, agents should leverage both 116 intermittency and frequency sensing. However, there is a steep tradeoff in performance when 117 using both temporal features simultaneously, which persists across a variety of plumes. This we refer to this model as the intermittency model.

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In the second model ( Figure 1A) (Demir et al., 2020), a detection threshold is used to detect when 158 the odor arrives. This results in a binary time series ( ), which spikes as a -function each time 159 the odor concentration crosses the threshold from below, and is 0 otherwise. The frequency of 160 odor encounters is then estimated by filtering ( ) with an exponential: .
(4) To illustrate how each of these two sensory modalities respond to the temporal features of odor 173 signals, we plotted the output of each filter in response to square-wave odor pulses of given 174 frequency and intermittency ( Figure 1B). These two features can be independently tuned -an 175 odor signal can be high frequency and high intermittency if the whiffs (periods above threshold) 176 are interrupted frequently with blank periods that are very short (region 1 in Figure 1B), while it 177 can have high intermittency but low whiff frequency if whiffs are interrupted with short blank 178 periods occurring more sparsely (region 2 in Figure 1B). In the first 2 regions of the signal, where 179 intermittency is high, the response of the ON( ) model approaches a high value after an initial 180 transient, while it drops to a lower steady state in region 3 where the signal intermittency is lower.

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The steady state response of ON( ) model is sensitive to the signal intermittency, but is 182 independent of the whiff frequency, as indicated by the average response asymptote /(1 + ), 183 which monotonically increases with intermittency (Methods). In contrast, the frequency model 184 responds strongly in regions 1 and 3 where whiff frequency is high, consistent with its asymptotic 185 response • (Methods). This happens irrespective of the disparity in signal intermittency 186 between these regions ( Figure 1B, bottom trace). Note that both models are sensitive to the 187 temporal characteristics of the signal, but not absolute concentration. plot) into a binary representation ( ), which is then passed through an exponential filter with timescale ' to generate 197 ( ). B. Response of each of the models (bottom two plots) to a binary odor signal (top plot) of high intermittency, high 198 frequency (Region 1), high intermittency, low frequency (Region 2) and low intermittency, high frequency (Region 3). 199 The intermittency model is sensitive to the intermittency of the signal -in regions 1 and 2, it approaches a high value where I is signal intermittency (Methods for derivation).

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Conversely, the frequency model exhibits sensitivity to the frequency of encounters, tending asymptotically towards 203 ' where f is the signal frequency (dashed line). The frequencies in the three regions are 2Hz, 0.5Hz, and 2 Hz, the 204 encounter durations are 0.45s, 1.8s, 0.1s, and the intermittencies are thus 0.9, 0.9, and 0.1.

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Though these two models were extracted from the same model organism with the same were much shorter (~100ms) ( Figure 2D). The two navigational models these experiments 218 informed were clearly shaped by the plumes' natural features: in the first, odor intermittency 219 reached as high as 100% and whiff frequencies rarely surpassed 1 Hz ( Figure 2E), whereas in 220 the latter, the signal had intermittency mostly below 30% but whiff frequencies of several Hz 221 ( Figure 2F). Together, these two experiments and corresponding models suggest that flies use  to C for the high frequency plume. Data is clustered within a higher range of frequencies but low intermittencies. To address this question, we incorporated the two sensing capabilities, intermittency sensing and in which the adaptive compression from the original ( ) sensor has been replaced with a 249 thresholded odor signal. Θ is the Heaviside step function and K is the odor detection threshold. 250 We kept the frequency sensor ( ) the same as above (Eq. 4), and used the same filtering 251 timescales = ,-= * for both the ( ) and ( ) sensors.

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It is known that odor signals influence many behavioral actions, including speeding, turning, Thus, the likelihood that a turn is directed upwind (versus downwind) increases sigmoidally with 265 a linear combination of ( ) and ( ). In the absence of signal, upwind and downwind turns are 266 equally likely: (upwind|turn) = 0.5 (further specifics of parameters choices in Methods). For 267 now, the "sensor gains" 4 and 5 were set to 1.9 and 0.2, respectively, by comparing to 268 experimental data (Methods). We refer to these values as the "base gains" and denote them as performance was relatively robust to initial angle and position ( Figure 3D). However, when either 282 frequency sensing ( 5 = 0) or intermittency sensing ( 4 = 0), were removed, performance 283 degraded ( Figure 3D) in one of the plumes, and became more sensitive to initial conditions.

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Though not wholly surprising that removing sensors degrades performance, this suggests that a 285 simple linear combination robustly navigates two disparate odor plumes, without exhibiting any 286 obvious failure modes due to interference between sensors.

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We expect that the two sensors do not contribute equally at all times to the navigation and that 289 the relative contribution of either sensor may depend on location within the plume -for example, 290 in the high frequency plume, the intermittency sensor might be more active near the plume 291 centerline, where the signal is more likely to be present. To quantify this, we measured the relative  crosswind position y (2 nd column) and orientation (3 rd column) for successful agents, for the high intermittency (top row) 313 and high frequency (bottom row) plumes. Colors correspond to same models as in D. Upwind heading is 180° and 314 shaded regions represent SEMs obtained from bootstrapping (Methods) E. Relative filter weight ≔ different points in the two plumes.  Figure 4C). Increasing the wind speed to = 300mm/s and 335 decreasing effective diffusivity to = 10mm = /s resulted instead in a higher frequency plume with 336 much shorter whiffs ( Figure 4D). In each plume, we simulated 10,000 agents with uniformly 337 distributed initial position and heading angle, where each agent navigated with a fixed set of gains 338 4 and 5 . We investigated various choices of 4 and 5 , from 0 to 50X the base gains (Methods).

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The 4 and 5 maximizing performance in our simulated high intermittency plume was reasonably   To get a better understanding of how navigational performance in these two simulated plumes 376 depends on the sensor weights, we did a tighter sweep of gains near the performance maxima  Figure 4G). The resulting scatterplot quantifies the performance in the two plumes 381 for different navigational models, where each model is parameterized by its sensor weights 4 and 5 . In general, the scatter plot fills out a region near the origin, bounded by a curve that forms 383 a "Pareto front" of navigational performance. This Pareto front reveals a performance tradeoff for 384 the different models: combinations of 4 and 5 that are weighted toward I do better in the high 385 intermittency plume, while combinations weighted toward F outperform in the high frequency 386 plume ( Figure 4G). There was no fixed set of gains that performs optimally in both plumes.

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Importantly, the apparent concavity of the Pareto front illustrates a somewhat steep tradeoff, and 388 suggests that flies might be better off modulating gains and switching between using intermittency 389 and frequency sensors to bias upwind motion, as opposed to using both simultaneously.

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Finally, we wondered how this tradeoff manifests across a more diverse spectrum of plumes. The In this work, we used numerical simulations to explore the value of two temporal features of the 420 signal -odor intermittency and encounter frequency -in navigating naturalistic odor plumes 421 spanning a range of spatial and temporal complexity. These two features are a natural set in that 422 they can be varied independently to create a variety of odor signals (Figure 1). Other

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We emphasize that our work explores normative strategies, so our results have no bearing on 435 whether such adaptation actually occurs. There is, however, evidence that such adaptation may In the latter half of this study, we simulated odor plumes using a simple drift-diffusion model  There are several aspects of olfactory navigation not considered in this work. In particular, we    The authors declare that no competing interests exist. One can thus see that (14) 592 593 Once the number of pulses is much greater than " , i.e ≫ " , we get 594 595 (15) 596 597 598 599 Since this is the value of ( ) at the end of a pulse, it will be the maximum value of ( ) over one 600 period. Ultimately, however, we are interested in computing ( ), which obeys the equation To understand the response of we can consider three different signal time scales. If the signal 608 fluctuates quickly with respect to " i.e. and D E − ≪ " , then for ≫ " one can approximate 609 ( ) with its average value over one period, which is given by Hence 653 654 heading angle was uniformly distributed from 0 to 360 degrees. The simulation was run for the 676 length of the video, (240s) and the discrete time step was set to be the reciprocal of the frame rate (1/15s).

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The second plume recording we used was taken from (Demir et al., 2020). We call this the high 680 frequency plume. The odor detection threshold of each agent was set the same way it was in 681 (Demir et al., 2020). Again 10,000 agents were initialized with uniformly distributed initial position 682 and heading. The initial x-position was between 38.45mm and 288.45mm and the initial y position 683 was between -74mm and 86mm. Initial heading was uniformly distributed from 0 to 360 degrees.

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The simulation was run for 123.3s, starting from the 600 th frame of the video to the last frame, at 685 89.94 frames per second, corresponding to the frame rate used in (Demir et al., 2020). The first 686 600 frames were dropped so that the plume had expanded to full size when the simulations began.

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In both simulations, odor signal was computed by averaging over an elliptical antenna-sensing 689 region in front of the agent, as in (Demir et al., 2020). The length of the region's major axis was 690 1.5mm and the length of the minor axis was 0.5mm. The ellipse was centered 1mm in front of the 691 agent. In both simulations, if agents went outside the frame region then they were allowed to 692 continue but received zero signal in those regions. Thus there were no walls in these simulations.

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For these simulations, was computed as for the square-wave pulses, with a detection threshold 695 as described above, but we also enforced that the whiff time series ( ) could not register two The base gains, 4) and 5) , which were used for the simulations in Figure 3, and in multiples of in the high intermittency and high frequency plumes respectively, we have 4) ) = 5) ) . We thus 709 determined a 4) of 1.936.

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Regarding the remaining parameters, the turn-rate was set to 1.3/s, walking speed set to 10.1 712 mm/s, and filter decay timescale was set to 2s, all in accordance with the findings of (Demir et 713 al., 2020). Note that the same timescale was used for the and filters.

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Error bars for success rates ( Figure 3C) were computed by bootstrapping data from a simulation 718 of 10,000 flies-1000 resamples were used with each resample size being equal to 10,000.

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Similarly, for the histograms of successful initial conditions, the data was resampled 1000 times,  In general, parameters were chosen to be physically realistic and also give concentration time-751 series and odor plumes that were qualitatively similar to those in the videos. To set ) , we defined 752 the detection threshold to be 1 and enforced that an agent more than 1.6 standard deviations 753 away from an initial packet would not be able to detect its presence. See Table 2  The order of magnitude for was set by the fact that attractive odorants for Drosophila 759 melanogaster tend to have molecular diffusivities of around 10mm = /s, eg. Ethyl acetate. The eddy 760 diffusivity was set in accordance with (Drivas et al., 1996). The release rate and initial size were 761 chosen to be similar to those in (Farrell et al., 2002). The wind speed was chosen to be similar to To define the antenna-sensing region, space was discretized into "pixels" with 0.154mm as the 777 pixel width, matching the spatial resolution of the high frequency plume. The concentration was 778 then computed by averaging over the pixels in an elliptical region, with the region defined as in 779 the previous section.

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To set the level of noise added to the and filters, we first computed a characteristic value in 782 the simulated high intermittency plume, ) , by averaging values over a region 192mm < < 783 205mm and 0mm < < 9mm and then averaging over the length of the simulation. We did the 784 same for values in the simulated high frequency plume to obtain ) . The values we obtained 785 were ) = 0.776 and ) = 3.14. We then used 5% of these values as the standard deviation for 786 Gaussian white noise to be added to the output of the and filters respectively at each time 787 step. We also used ) and ) as representative and values in order to assign a single relative 788 filter weight with which to color each set of gains in Figure 4G.