Most macroscopic organisms detect odors in intermittent bursts, that may be separated by extended regions with no odor. Organisms leverage this complex dynamics efficiently for diverse tasks, including locating and identifying an odor source Murlis et al., 1992; Mafra-Neto and Cardé, 1994; Vickers, 2000; Riffell et al., 2014; Ache et al., 2016; Ackels et al., 2021. However, what are the most informative features of intermittent odor cues remains largely unclear. There are two broad classes of measures that quantify the dynamics of olfactory cues: those that depend on odor intensity including e.g. odor gradients in space or time, and those that do not depend on odor intensity but only on its timing, i.e. on whether the odor is on or off regardless of its concentration. To compute quantities that depend on odor intensity, an accurate representation of the odor is needed. In contrast, measuring the timing of odor detection simply requires to mark at all times whether the odor is on or off, thus a binary switch is sufficient.

Behavioral evidence suggests that animals use both intensity and timing of odor encounters for olfactory navigation Baker et al., 2018. At close range, mammals appear to compare odor intensity either across nostrils or across sniffs Catania, 2013; Gire et al., 2016; Findley et al., 2021. On the other hand, mounting evidence suggests timing of odor detection also plays a key role for olfactory navigation Ache et al., 2016: moths respond to odor pulsed at specific frequencies Riffell et al., 2014; Vickers et al., 2001; fruit flies respond to timing since last odor detection van Breugel and Dickinson, 2014; Demir et al., 2020; lobsters and sharks compare odor arrival time across their paired olfactory organs and orient toward the side that detected the odor first Basil and Atema, 1994; Gardiner and Atema, 2010; many organisms will move upwind upon detection of an odor Kennedy and Marsh, 1974; Murlis et al., 1992; Steck et al., 2012.

Neural recordings upon stimulation with intermittent odor cues confirm that the brain of many animals is able to record information both about intensity (and its derivatives) as well as timing of odor encounters (most information comes from work on arthropods Nagel and Wilson, 2011; Vickers et al., 2001; Brown et al., 2005; Gorur-Shandilya et al., 2017; Jacob et al., 2017; Riffell et al., 2014, but see also Parabucki et al., 2019; Lewis et al., 2021). For example, when insects are presented with intermittent odor cues, information about intensity and timing is recorded in their antennal lobe (see e.g. Vickers et al., 2001; Brown et al., 2005). Odors that mimic natural intermittency elicit a response that preserves an accurate measure of timing in fruit flies and moths Gorur-Shandilya et al., 2017; Jacob et al., 2017. In lobsters, bursting olfactory neurons encode specifically for the time between successive odor encounters, see Park et al., 2014; Park et al., 2016 and references therein. Interestingly, the neural activity varies considerably with the dynamics of the odor cues Nagel and Wilson, 2011; Vickers et al., 2001; Lewis et al., 2021, but how intermittency of an odor affects its neural representation is not well understood.

This evidence suggests animals are able to identify when they detect an odor as well as how intense it is; but whether they record and rely on both kinds of information is not understood. From a physical perspective, these two measures clearly provide information about source location. Indeed, we know from theoretical Shraiman and Siggia, 2000; Falkovich et al., 2001; Celani et al., 2014 and experimental Justus et al., 2002; Moore and Crimaldi, 2004 work that turbulence causes the odor to be distributed in highly intermittent patches separated by blanks with no odor. Both intensity and timing of these intermittent bursts vary depending on the location of the source Celani et al., 2014, as early recognized by Atema, 1996, thus can be used to infer source location or navigate to it Vergassola et al., 2007; Schmuker et al., 2016; Boie et al., 2018; Leathers et al., 2020; Michaelis et al., 2020.

Here, we ask what salient features of turbulent odor signals best predict the location of the odor source and specifically compare quantities related to intensity vs timing of odor encounters. We first compose a dataset of realistic odor fields at scales of several meters using accurate state-of-the-art fluid dynamics simulations. We then develop machine learning algorithms that predict source location based on these synthetic odor fields.

We find that measures of odor temporal dynamics based on a short memory span (down to about 1 s) hold information about source location. Close to the source or close to the substrate, measures of intensity predict distance better than measures of timing; but this ranking is reversed at further distance from the source or from the substrate. Pairing the two kinds of measure improves dramatically the quality of the prediction robustly across all datasets, whereas pairing two measures of intensity or two measures of timing is either useless or detrimental.

Our results demonstrate that timing and intensity are complementary attributes of odor dynamics and are most effective in more dilute and concentrated conditions respectively. These different conditions exist in different portions of space because odor gets transported, mixed and diluted by the fluid. As a result, the spatial range of operation of a living organism constrains the solutions it may evolve to make predictions with turbulent odors.

Odor cues at several meters from the source are often turbulent. Figure 1a–c and show snapshots of the velocity field and odor cues in space, resulting from direct numerical simulations of the turbulent flow in a channel of length L, width W and height H (also see Figure 1—video 1). Air flows from left to right at a mean speed Ub and hits a cylindrical obstacle that generates turbulence. The height of the obstacle is H/4 and tunes the intensity of turbulent fluctuations relative to the mean velocity. To characterize the flow we show in Figure 1d that the mean velocity profile, for z+≳ 30, follows the law of the wall ⟨U⟩uτ=1κ⁢ln⁡z++B, where κ and B are constants and z+=zδν, recovering classical statistics for channel turbulence Pope, 1984. The odor field is emitted from a concentrated source downstream from the obstacle; it develops as a meandering filament that fluctuates as it travels downstream and soon breaks into discrete pockets of odor (whiffs) separated by odor-less stretches (blanks) (Figure 1b–c and f). The Kolomogorov scaling for the spectra of odor fluctuations k-5/3 holds for k⁢η≲0.1 (see Figure 1e), consistent with previous experimental results in channel flow (see e.g. Saddoughi and Veeravalli, 1994). Typical time courses of the odor are shown in Figure 1f. Note that depending on the sampling location, odor may be more or less sparse (compare for example Figure 1f left and right). All parameters and methods are summarized respectively in Table 1 and in Materials and Methods.

Do odor cues bear information about source location meters away from the source? To answer this question, we develop supervised machine learning algorithms that learn the relationship between the input (odor) and the distance from the source (output) from a large dataset of examples. In order to dissect what are the best predictors of source location and how ranking depends on the statistics of the odor, we need to detail more specifically the input and output of the algorithm.

To design the input we start with the odor concentration field c(z,t) which varies stochastically in space and time as a result of turbulent transport. Here, z=(z1,z2,z3) is a location in the three dimensional space and t is time. We focus on a plane at a fixed height, and consider the conical region where odor can be detected, the ‘cone of detection’ (Figure 2a). We first compose time series of the odor field; each time series is indicated with ci and consists of the odor sampled at M equally spaced times with frequency ω at a discrete location zi within the cone of detection. Thus, each time series is a vector ci=(c⁢(zi,ti),…,c⁢(zi,ti+M)), where ti+M-ti=M/ω is the temporal span of the time series, or memory. From each time series, ci we calculate five features xi1,…,xi5, where xi1 is the temporal average of the concentration during whiffs in the time series ci; $x_i^2$ is its average slope (time derivative of odor upon detection, averaged across whiffs within ci); xi3 is the average duration of blanks (stretches of time when odor is below detection within ci); xi4 is the average duration of whiffs (stretches of time when odor is above threshold within ci); and xi5 is the intermittency factor (the fraction of time the time series ci is above threshold). The detection threshold is defined adaptively as discussed in Materials and methods and Figure 2—figure supplement 1. Features x1 and x2 depend explicitly on odor concentration, whereas features x3, x4 and x5 only depend on when the odor is on or off, but not on its intensity. To remark this difference, we refer to x1 and x2 as intensity features, and x3, x4 and x5 as timing features. Our input xi=(xi1,…,xid) is composed of d-dimensional vectors of features and we will focus on d=1,2,5. We seek to infer distance from the source, thus our output y is the coordinate of the sampling point z in the downwind direction, i.e. y=z1, with the source placed at the origin (see sketch in Figure 2a). We refer to the figure supplements for results in the crosswind direction, y=z2. We train the algorithm by providing N examples of input-output pairs (xi,yi) selected randomly from the full simulation, and obtain the function that connects input and output: y≈f⁢(x).

We propose a machine learning approach where the different odor features are ranked based on their predictive power, rather than their fitting properties. Different data-sets of odor/distance pairs are defined. The data-sets differ in the way odor measurements are represented in terms of feature vectors. For each data-set we learn a function to predict the distance to target given the corresponding odor features. The predictive power of each function, and corresponding set of features, is then assessed. More precisely, each data-set is split in a training and a test set, as custom in machine learning. Training sets are used to learn functions connecting odor to target location, whereas test sets are used to assess their prediction properties. The training/test split is crucial since the goal is to make good predictions on new, unseen points, that are not within the training sets. From a modeling perspective, a flexible nonlinear/nonparametric approach based on kernel methods is contrasted and shown to be superior to a simpler linear model (Figure 2—figure supplement 2). A careful protocol based on hold-out cross-validation is used to select the hyper-parameters of the considered learning models (we refer to Materials and methods and Figure 2—figure supplement 3 for more details).

To illustrate the results we pick the two-dimensional plane at height H/4 that contains the source. The first result is that individual features (d=1) bear useful information for two-dimensional source localization even at several meters from the source. Performance is quantified by the normalized squared error averaged over the Nt points in the test set χ=∑i=1Nt[yi-f⁢(xi)]2/∑i=1Nt[yi-y¯]2. For this dataset, intensity features rank higher than timing features (Figure 2b–c), consistent with previous work Atema, 1996 and predictions are more accurate in the crosswind than in the downwind direction (compare with Figure 2—figure supplement 4). For reference, a random guess with flat probability within the correct lower and upper bounds yields χrandom=2, whereas a target function ftrivial⁢(x)=⟨y⟩test that learns the average of the output over the test set yields χtrivial=1.

In order to prove that the algorithm captures the dynamics of the scalar and not of the underlying velocity field, we realize three computational fluid dynamics simulations with the odor source at different locations downstream of the obstacle. Each source is placed at a different distance from the obstacle and thus feels different velocity fields, because the flow is inhomogeneous in the downstream direction. To perform a fair comparison we train the algorithm on points sampled from a conical domain based on the most downstream source; for the other two sources, we use an identical domain shifted upstream so that the vertex of the cone is located at the respective source location (see Figure 2—figure supplement 5 left). Performance at source height varies little over the three locations, demonstrating that the algorithm is learning the dynamics of the scalar and not of the carrying flow (see Figure 2—figure supplement 5 center). Additionally we demonstrate that, irrespective of source position, timing features acquire predictive power with height, whereas the opposite is observed for intensity features (see Figure 2—figure supplement 5 right). We discuss this property in further detail in the following. . Finally, we train the algorithm with the odor fields emanating from one source location, and test its performance over odor fields generated by the two other sources. Figure 2—figure supplement 6 shows that performance of individual features varies little when test and training are performed over different dataset. Learning from pairs of features appears somewhat more sensitive to the details of the dataset. In the aggregate, the analysis corroborates that the algorithm captures odor dynamics and is rather insensitive to details of the underlying flow. In the rest of the manuscript we focus on the leftmost source, so as to exploit the full spatial range available from the simulations.

Next we analyze whether and how the sampling strategy affects performance and ranking of the features. Most results are shown for a memory of 100⁢τη≈15⁢s. Performance improves with longer memory (Figure 3a), because this allows to better average out noise and obtain more stable estimates of the features. But improvement follows a slow power law so that waiting for example 20 times longer yields predictions only about twice as precise. On the other hand, waiting as little as 10⁢τη≈1.5 seconds still allows to make predictions, albeit less precise. We then verify whether performance may improve with a larger training set. Because we infer distance from an individual (scalar) feature, the problem is one dimensional and we find that a small number of training points, which we indicate with N, is sufficient to reach a plateau in prediction performance (Figure 3b). We choose N=5000 training points, which is also robust to the case with more than one feature (Figure 3—figure supplement 1). Finally, sampling more frequently than once per Kolmogorov time does not essentially affect the results nor ranking (Figure 3c). Similar results hold for the crosswind direction (Figure 3—figure supplement 2).

Pairing two observables improves performance in some cases, but not always. In fact, pairing two features of the same category results in little to no improvement (Figure 4 and similarly for the crosswind direction, Figure 4—figure supplement 1). In contrast, combining one intensity and one timing feature improves performance considerably, up to 65%. This result can be understood by mapping the error done by individual features in space (Figure 5—figure supplement 1), showing that intensity and timing features are complementary, that is intensity features perform well in locations where timing features perform poorly.

We next seek to clarify whether the results depend on space. To this end we compose five different dataset, a to e, obtained by extracting odor snapshots from horizontal planes at source height (b), above the source (c to e), and below the source (a) (Figure 5a). From a to e, sparsity increases and intensity decreases (Figure 5b) simply because closer to the boundary, the air slows down and the odor accumulates. By analyzing performance across these dataset, we find that ranking of individual features shifts considerably. The two intensity features outperform all timing features when the dataset is not very sparse (dataset a-b, Figure 5c and d left). In contrast, two timing features (intermittency factor and blank duration) outperform all others for the more sparse and less intense dataset d-e (Figure 5c and d right). Whiff duration performs poorly in d-e because intermittency is too severe and whiffs are short in duration thus bear little information (the average whiff duration is 1–7 time steps in over 90% of the time series). Although the ranking of individual features shifts with height, pairing one intensity and one timing feature remains the most successful strategy across all heights (Figure 5c and d). In contrast, combining all five features contributes little improvement (Figure 5—figure supplement 2).

Let us now focus on the plane at source height and separate locations based on their distance from the source. We assemble a distal dataset and a proximal dataset, composed of points that are further and closer than 2330η from the source respectively (Figure 6a). The odor is more intense and more sparse closer to the source and it becomes more dilute and less sparse with distance from the source (Figure 6b). Performance of individual features degrades with distance (Figure 6d). Intensity features clearly outperform timing features at close range, as seen both from various percentiles of the test error (Figure 6d, left) as well as the full distribution (Figure 6c, left). The disparity between timing and intensity features disappears in the distal problem: the error distribution for all individual features is essentially superimposed except for the tails (Figure 6c, right and inset), which cause small differences in the median and other percentiles of the error (Figure 6d, right). Remarkably, mixed pairs outperform all individual features in both the distal and proximal problems (Figure 6c–d). In the aggregate, results demonstrate that, even within a single turbulent flow, ranking shifts considerably. Namely, measuring timing of odor encounters is most useful in regions where the odor is dilute, that is far from the source and from the substrate, whereas measuring intensity is most useful in concentrated conditions, that is close to the source or the substrate.

Thus, the predictive power of different features varies greatly in space and time. Next, we show how this spatial variation is dictated by the statistical properties of the odor plume. To this end, we provide an analytical characterization of the test error of individual features, χ, that connects directly to the physics of the problem. Such a characterization is consistent with the observed variation in performance. Note that for large enough samples, the test error χ approximates the following expected error(1)∫0∞dx⁢∫0Rdy⁢(y-f*⁢(x))2⁢p⁢(x,y)∫0R(y-y¯)2⁢p⁢(y)⁢dy.

The latter is called expected error and is the ideal version of the test error. It can be interpreted as the error summed over all possible input-output pairs, weighted by their corresponding joint probability to be sampled. Here, f*⁢(x)=⟨y|x⟩=∫0Ry⁢p⁢(y|x)⁢dy is the so called regression function, which minimizes the expected error among all possible functions. The regression function can be approximated using Kernel ridge regression and sufficiently rich kernels. Indeed, kernel ridge regression is known to be a so called universal estimator Hastie et al., 2001; Steinwart, 2002. In the above expression, R is the length of the cone, p⁢(y|x) is the conditional probability distribution of the output given the input, p⁢(x,y) is the joint probability distribution of the input output pair (x,y) is the prior distribution on the output y. The idea is to relate f*, hence the expected error, to the distribution p⁢(x|y) of observing feature x depending on distance y. Indeed, the latter is dictated by the fluid dynamics of odor plumes from a concentrated source, and hence provides a more direct connection between the expected error and the physics of the problem. Since the considered features are sample averages, in the limit of large samples, their distribution is well approximated by a Gaussian, hence fully characterized by the mean and standard deviation. Then simple estimates can be computed empirically from our data. Equation (1), provided with these estimates, reproduces the predictive power of individual features showed in Figure 5c (see Figure 5—figure supplement 3). Note that whiffs deviate considerably from a normal distribution, hence the argument needs to be revised for this feature. To move beyond empirical estimates and extend the above reasoning to other flows and generic combinations of (possibly non Gaussian) features, a generalization of the asymptotic arguments proposed in Celani et al., 2014 is needed.

Our results demonstrate that within the cone of detection, the time course of an odor bears useful information for source localization even at meters from the source. We find that the concentration and the slope of a turbulent odor signal, averaged over a memory lag, are particularly useful to predict source location at close range or near the boundary. These features quantify the intensity of the odor and its variation. The primacy of the intensity features wanes in more challenging conditions, for example moving away from the source or away from the boundary. In these portions of space, where the odor is scarcer, features that quantify timing of odor detection become as effective as intensity features, or more effective. One of the best studied example of olfactory search in dilute conditions is arguably the case of insects.

Interestingly, olfactory receptor neurons in insects appear to encode efficiently information about timing across a wide range of intensities Gorur-Shandilya et al., 2017; Martelli et al., 2013.

Note that while the statistics of an odor plume clearly depends on all details of the flow and the source, see e.g. Justus et al., 2002; Celani et al., 2014; Fackrell and Robins, 1982, here we keep all of these parameters constant and demonstrate that even within a single flow, odor dynamics and the best predictors vary considerably in space. This begs the next question: do organisms switch between different modalities depending on attributes of odor dynamics, which will vary in space? This could be the case for mice, where the neural activity in the first relay of olfactory processing does in fact depend on how sparse is the odor Lewis et al., 2021. Specifically, sparse odor cues elicit individual responses that follow closely the ups and downs of the odor in time. In contrast, continuous signals elicit intense responses which are however uncorrelated to the temporal dynamics of the odor itself Lewis et al., 2021.

We find that features within the same class are redundant whereas features from different classes are complementary. Indeed, features of the same class have similar patterns of performance in space, but each class has a distinct pattern. As a consequence, measuring both timing and intensity is beneficial, but using more than one feature to quantify either timing or intensity provides no advantage. Combining all features does not improve over the performance of mixed pairs, consistent with redundancy within each class. Note that there is no fundamental reason to expect features from the same class to be redundant, and further work with a larger library of features is needed to prove or disprove this notion.

Importantly, mixed time/intensity pairs of features outrank individual features robustly, that is in all portions of space, regardless of distance from the source and from the ground. This is in contrast with individual features and suggests relying on simultaneous timing and intensity features is advantageous when odors are sensed at various distances from the source and from the substrate. Interestingly, the coexistence of bursting olfactory neurons and canonical olfactory neurons in lobsters suggests these animals are in fact able to measure simultaneously timing and intensity Park et al., 2014; Ache et al., 2016, which is consistent with the increased predictive power of the mixed pairs of features. Similarly, in mammals, optogenetic activation of the olfactory bulb Smear et al., 2013 demonstrates that both kinds of measures guide behavior (lick vs no lick).

In this work, we have investigated the problem of predicting the location of a target from measures of the time course of a turbulent odor. Previous work explored a related question, that is how to best represent instantaneous snapshots of the odor to encode maximum information about source location Victor et al., 2019. The two approaches are not immediately comparable: first, Victor et al., 2019 consider few snapshots of the odor, rather than measures of its time course. Second, maximizing information does not guarantee good predictions (to make predictions information needs to be extracted and processed, and importantly the focus is on new data that were not previously seen). We provide two comments that are relevant if information is the limiting factor for prediction accuracy: (i) binary representations were suboptimal in all conditions considered in Victor et al., 2019; Boie et al., 2018, that is at few tens of cm from the source. This is consistent with our results in concentrated conditions, where timing features -accessible through binary representations- are suboptimal. Our evidences suggest, however, that the result may not hold in more dilute conditions, where the gap between binary and more accurate representations should become increasingly small. (ii) Individual snapshots of odor from Victor et al., 2019; Boie et al., 2018 contained 1–2 bits of information about source location, but allocating more resources to represent how the odor varies in time was found informative Victor et al., 2019; Boie et al., 2018. Our mixed pairs of features at close range achieve precisions of 5–6%, corresponding to coding for position with words of 4–4.3 bits. Our results thus confirm that memory is indeed useful, but the gain does not increase indefinitely with further memory.

The literature on olfactory navigation is vast. Although a complete review of available algorithms is beyond the scope of the present work, we remark that recent results investigated gradient descent algorithms using either concentration alone Gire et al., 2016, or various measures of timing and intensity Park et al., 2016; Michaelis et al., 2020; Leathers et al., 2020. Overall, both intensity and timing appear to have a potential to lead to an odor source, consistent with our results on individual features. A combination of the two kinds of features was found beneficial in Leathers et al., 2020, consistent with our results on mixed pairs. Note that predicting source location and navigating to reach it are distinct tasks. Although they are often assumed to be intimately connected, whether good predictors may be good variables for navigation in more general contexts remains to be understood.

Here, we have analyzed the features that enable the most accurate prediction of source location. We add a few observations about the significance of the results for animal behavior. First: whether animals rely on features from either class will depend on what features best support behavior. It is often implicitly assumed that features that bear reliable information on source location are also the most useful for navigation. However, this connection between prediction and navigation is far from straightforward and more work is needed to establish whether accurate predictions imply efficient navigation. Second: animals are unlikely to have prior information on the details of the odor source, for example its intensity. Timing features are more robust than intensity features with respect to the intensity of the source and may thus be favored regardless of their performance, which was argued in Schmuker et al., 2016. In our work, timing features are precisely invariant with source intensity because we define the detection threshold adaptively (see Materials and methods). More realistic conditions will need to be evaluated, where dependence on source intensity emerges as a result of non-linearities that we did not model in this work. These effects emerge for example, close to a boundary which partially absorbs the odor Gorur-Shandilya et al., 2019, or in the case of fixed thresholds, although this dependence is weak in the far field where timing features are most useful Celani et al., 2014. Third: we have focused on predicting source location from within the cone of detection, where an agent will detect the odor quite often. However, a crucial difficulty of turbulent navigation is to find the cone itself. We cannot address the problem of predicting source location from outside the cone because detections are so rare that we lack statistics. The distinction between inside and outside the cone of detection is key for navigation with sparse cues (see Reddy et al., 2021) and deserves further attention.