Our study is motivated by two models recently extracted from experimental observations of walking Drosophila navigating odor plumes (Álvarez-Salvado et al., 2018; Demir et al., 2020). Here, we examine how they each respond to distinct temporal features of the odor concentration. We focus on temporal changes in odor concentration rather than odor flux (which depends also on air speed) as Drosophila melanogaster ORN responses are invariant to air speed (Zhou and Wilson, 2012). In the first model (Figure 1A; Álvarez-Salvado et al., 2018), the instantaneous odor concentration odor(t) is first compressed into the range 0–1 using an adaptive Hill function:(1)C(t)=odor(t)odor(t)+kd+A(t).

The half-max is set by At , a low-pass-filtered sliding average of the instantaneous odor concentration(2)τAdAdt=odor(t)−A(t).

This mimics the gain adaptation of ORNs to the mean signal (Cao et al., 2016; Gorur-Shandilya et al., 2017). At the onset of a sudden increase in odor concentration, the compressed signal Ct increases instantaneously before relaxing back to ~0.5 with timescale τA=9.8 s. The compressed signal Ct is then exponentially filtered into an ‘ON’ function,(3)ON(t)=∫0t1τON⋅et′−tτON⋅C(t′)dt′,

which drives odor-elicited behavioral actions. When ON(t) is high, the fly accelerates and biases its heading upwind; when ON(t) is low, the fly’s orientation randomizes and drifts downwind and its walking speed reduces (Álvarez-Salvado et al., 2018). We show analytically (Materials and methods) that the value of ON(t) – and therefore the navigational actions – is largely determined by the intermittency of the odor signal, defined as the percentage of time an odor signal is present. Thus, we refer to this model as the intermittency model.

In the second model (Figure 1A; Demir et al., 2020), a detection threshold is used to detect when the odor arrives. This results in a binary time series wt , which spikes as a δ-function each time the odor concentration crosses the threshold from below, and is 0 otherwise. The frequency of odor encounters is then estimated by filtering wt with an exponential:(4)F(t)=∫0tet′−tτFw(t′)dt′.

Thus, Ft rises by 1 at each threshold crossing, before decaying exponentially with timescale tw until the next odor hit. In this model, Ft plays a similar role as ONt in the previous model, in that it drives behavioral response to odors. When Ft increases, flies increase their bias upwind and stop less frequently and for a shorter time (Demir et al., 2020). Since Ft is effectively a running average of the frequency of odor hits, we refer to this model as the frequency model.

To illustrate how each of these two sensory modalities respond to the temporal features of odor signals, we plotted the output of each filter in response to square-wave odor pulses of given frequency and intermittency (Figure 1B). These two features can be independently tuned – an odor signal can be high frequency and high intermittency if the whiffs (periods above threshold) are interrupted frequently with blank periods that are very short (region 1 in Figure 1B), while it can have high intermittency but low whiff frequency if whiffs are interrupted with short blank periods occurring more sparsely (region 2 in Figure 1B). In the first two regions of the signal, where intermittency is high, the response of the ON(t) model approaches a high value after an initial transient, while it drops to a lower steady state in region 3 where the signal intermittency is lower. The steady-state response of ON(t) is sensitive to the signal intermittency, but is independent of the whiff frequency, as indicated by the average response asymptote I1+I, which monotonically increases with intermittency (Materials and methods). In contrast, the frequency model responds strongly in regions 1 and 3, where whiff frequency is high, consistent with its asymptotic response f∙τF (Materials and methods). This happens irrespective of the disparity in signal intermittency between these regions (Figure 1B, bottom trace). Note that both models are sensitive to the temporal characteristics of the signal, but not absolute concentration.

Though these two models were extracted from the same model organism with the same locomotive repertoire – fruit files walking in a 2D arena – the experiments were performed in very different odor and flow conditions. The intermittency model was first extracted from flies navigating a uniformly odorized region of odor within a laminar airflow (Álvarez-Salvado et al., 2018). Using simulations, the model was then shown to qualitatively recapitulate navigational behavior in a measured near-bed turbulent plume (Connor et al., 2018; Figure 2A), which we call the high-intermittency plume, in which the odor signal was ever-present and varied on relatively long timescales of several seconds or more (Figure 2B). In contrast, the frequency model was fit to trajectories of flies navigating a plume with a high degree of spatial complexity (Figure 2D) generated by perturbing a fast laminar flow with stochastic lateral jets, which we call the high-frequency plume. In that experiment, odor whiffs occurred frequently (2–5 Hz) (Figure 2E and F) and were much shorter (~100 ms) (Figure 2E). The two navigational models these experiments informed were clearly shaped by the plumes’ natural features: in the first, odor intermittency reached as high as 100% and whiff frequencies rarely surpassed 1 Hz (Figure 2C), whereas in the latter, the signal had intermittency mostly below 30% but whiff frequencies of several Hz (Figure 2F). Together, these two experiments and corresponding models suggest that flies use both odor frequency and intermittency to navigate upwind in different environments. This prompted us to ask how this dual-sensing capability might enhance the efficacy and robustness of navigation in different conditions.

To next investigate how these dual-sensing capabilities – odor intermittency sensing and frequency sensing – shape navigational performance in distinct odor landscapes, we incorporated them into a combined navigational model. It is known that odor signals influence many behavioral actions, including accelerating, turning, and stopping (Álvarez-Salvado et al., 2018; Baker and Vickers, 1997; Demir et al., 2020; Mafra-Neto and Cardé, 1994; Vickers and Baker, 1994). Given the near-universal response of insects to turn upwind or bias their turns upwind in the presence of odor (Baker et al., 2018), here we assumed agents walk at a constant speed unless they are turning and focused on signal-driven changes in orientation. Turns occur randomly at a Poisson rate λturn, and turn magnitudes are sampled from a normal distribution N30o,8o as found before (Demir et al., 2020). Turn directions (sign of the orientation change) are modeled as(5)p(turnupwind|turning)=11+e−gION−gFF.

Thus, the likelihood that a turn is directed upwind (versus downwind) increases sigmoidally with a linear combination of Ft and ONt. In the absence of signal, upwind and downwind turns are equally likely: Pupwindturn=0.5. To allow frequency sensing to be adaptive, we set the detection threshold for Ft to be variable and equal to 12At, where At is defined in Equation 2. The ‘sensor gains’ gI and gF were set to 3.9 and 0.2, respectively, by comparing to experimental data (Materials and methods). For now, we hold the gains fixed at these ‘base’ values gI0=3.9 and gF0=0.2; below, we investigate the performance of different gI and gF. Finally, we define intermittency-only and frequency-only sensing models by setting gF and gI to 0, respectively.

To examine how frequency and intermittency contribute to navigational performance in this combined model, we simulated N agents navigating both the high-intermittency and high-frequency plumes. The initial position and orientations of the agents were randomized uniformly. Performance was quantified as the fraction of agents that reach within 15 mm of the source in the presence of an odor signal, NsN , minus the fraction of agents, NcN , that reach the source by chance, that is, when no signal is present. Individual trajectories of successful flies in either plume look similar: when oriented away from the source, agents are quickly able to reorient within the plume region and navigate to the source with relatively straight trajectories combined with occasional corrective kinks (Figure 3B). Overall, agents navigated successfully in both plumes (Figure 3C), and performance was relatively robust to initial angle and position (Figure 3D). However, when either frequency sensing gF=0 or intermittency sensing gI=0 was removed, performance degraded (Figure 3D) in one of the plumes and became more sensitive to initial conditions. Though not wholly surprising that removing sensors degrades performance, this suggests that a simple linear combination robustly navigates two disparate odor plumes, without exhibiting any obvious failure modes due to interference between sensors.

Our upwind bias function (Equation 5), though phenomenological, is a natural choice in that it allows an increased upwind response to both the ON and F filters. In fact, it very closely approximates a logical OR gate for the two filters (Materials and methods; Equations 34-35). This raises the question of whether this particular logical operation is ideal. We similarly investigated an AND gate implementation, finding clear failure modes (Materials and methods).

We expect that the two sensors do not contribute equally at all times to the navigation and that the relative contribution of either sensor may depend on plume statistics or on the location within a plume (Rigolli et al., 2021). For example, in the high-frequency plume, the intermittency sensor is likely to also be active near the plume centerline, where the signal is more likely to be present, while in the high-intermittency plume the frequency sensor is likely to be active on the edges where the presence of odor is less certain. To quantify this, we measured the relative weight of each sensor gIONt-gFFtgIONt+gFFt , which interpolates between pure intermittency sensing (+1) and pure frequency sensing (–1). As expected, the intermittency sensor dominates in the high-intermittency plume, whereas the frequency sensor dominates in the high-frequency plume (Figure 3E). Still, this dominance is not absolute. For example, frequency sensing plays a role near the conical boundary of the high-intermittency plume. Likewise, intermittency contributes along the centerline of the high-frequency plume.

These modest but significant contributions led us to next wonder how the sensors might be relatively weighted to optimize navigational performance and how this weighting might change in different plumes. Therefore, for tractability, we constructed a simpler model that eliminated some parameters. Firstly, we retained the frequency sensor Ft (Equation 4), but used a fixed odor detection threshold K rather than an adaptive threshold as before. Secondly, we replaced the ONt function with:(6)I(t)=12⋅∫0t1τI⋅et′−tτI⋅Θ(odor(t′)−K)dt′.

where Θ is the Heaviside step function. The primary change from ONt is the replacement of adaptive odor compression with a fixed binarizing odor threshold. The factor of ½ is kept for ease of comparison between It and ON(t), so that both filters asymptotically approach ½ in the presence of continuous odor (Materials and methods). Filtering timescales were set at τI=τF=2s for both It and Ft . While these changes do affect some quantities, like the relative filter weight in the two environments, the overall effect on navigational success is minimal (Figure 3—figure supplement 1). Thus, to study the effect of various model parameters in detail, we used this simplified model for all further investigations.

Upwind bias, and therefore navigation performance, depends on the sensor gains (Equation 5), which up to now we have fixed to experimentally informed values (the ‘base’ gains). To investigate the influence of relative sensor weight in navigation, we quantified navigational performance as a function of both the sensor weights gI and gF and the plume’s spatiotemporal complexity. To remove constraints due to the limited spatial and temporal resolution of the recorded plume videos, and to easily investigate a wide range of environments, we switched to simulated plumes using a simple dispersion model (Farrell et al., 2002). Gaussian packets of odor are released from a source at a fixed Poisson rate λ and advected by a velocity field composed of a uniform downwind velocity U. Normally distributed random perturbations ηx and ηy are added to the packet positions in the crosswind and downwind directions, respectively, at each time step, to account for the effects of turbulent diffusivity. The turbulent diffusivity models the effects of turbulent eddies as a diffusive process, but with diffusion constant κ that can greatly exceed molecular diffusivity. In addition, the Gaussian packets grow in size with an effective diffusivity D to account for the combined effects of molecular diffusion and smaller eddies in the wind flow (Figure 4A and B). Varying U and D allowed us to generate plumes with diverse temporal statistics. U=36mm/s and D=52mm2/s resulted in a plume with longer whiff durations and high intermittency (Figure 4C and E). Increasing the wind speed to U=300 mm/s and decreasing effective diffusivity to D=10mm2/s resulted instead in a high-frequency plume with much shorter whiffs (Figure 4D and F). In each plume, we simulated 10,000 agents with uniformly distributed initial position and heading angle, where each agent navigated with a fixed set of gains gI and gF . We investigated various choices of gI and gF , from 0 to 50× the base gains.

The gI and gF maximizing performance in our simulated high-intermittency plume was reasonably constrained, with a clear maximum occurring around the experimentally derived base gain (Figure 4—figure supplement 1). However, in the simulated high-frequency plume, a variety of gains led to similarly maximal performance (Figure 4—figure supplement 1), including some with values an order of magnitude larger than the base gains. Performance was largely independent of τI over nearly two orders of magnitude (unchanged even for a null algorithm that drives upwind orientation whenever odor is present, i.e., τI=0) and scaled with τF in a way that could be absorbed into the gF, (Figure 4—figure supplement 2; Materials and methods), so these trends were fundamentally due to the sensor gains rather than other model features. On the other hand, models with extreme gain factors could compound the effects of noise, leading to a lack of robustness in natural conditions. We therefore added Gaussian noise to the I and F filters – noise amplitude was 5% of the average value of I (F) in the center of the simulated high-intermittency (high-frequency) plume. This removed maxima at high gains but retained clear maxima at lower gains (Figure 4G and H). Interestingly, the unique maxima sat fairly close to the base gain values (values of 1 in Figure 4G and H), suggesting a degree of tuning within the biological fly olfactory circuit. Finally, the optimal gains for the simulated high-intermittency and high-frequency plumes had gF=0 and gI=0, respectively, indicating that optimal performance in either plume requires silencing the nonrelevant sensors. This inherent trade-off illustrates that simply augmenting the sensory capability can at times degrade performance. This suggests a benefit for sensor specialization in distinct environments.

To get a better understanding of how navigational performance in these two simulated plumes depends on the sensor weights, we did a tighter sweep of gains near the performance maxima (Figure 4G and H) for each plume. For each set of gains, we then plotted performance in the high-intermittency plume against that in the high-frequency plume. For comparison, we also plotted the set of gains (gI∗,gF∗) that maximized the geometric mean of normalized success in both plumes (indicated in Figure 4I). The resulting scatterplot quantifies the performance in the two plumes for different navigational models, where each model is parameterized by its sensor weights gI and gF . In general, the scatterplot fills out a region near the origin, bounded by a curve that forms a ‘Pareto front’ of navigational performance. This Pareto front reveals a performance trade-off for the different models: combinations of gI and gF that are weighted toward I do better in the high-intermittency plume, while combinations weighted toward F outperform in the high-frequency plume (Figure 4I). There was no fixed set of gains that performs optimally in both plumes. Importantly, the apparent concavity of the Pareto front illustrates a somewhat steep trade-off and suggests that flies might be better off modulating gains and switching between using intermittency and frequency sensors to bias upwind motion, as opposed to using both simultaneously.

We then wondered how this trade-off manifests across a more diverse spectrum of plumes. The computational simplicity of the turbulent plume model allowed us to study a wide array of turbulent plumes differing in their temporal statistics. We fixed the gains to the values that optimized the geometric mean between the high-intermittency and high-frequency plumes, (gI∗,gF∗), and then varied the environmental parameters U and D to smoothly interpolate between the high-frequency and high-intermittency plumes investigated above. Success was roughly uniform in the different environments (Figure 5A). However, removing the frequency sensor (gF=0) significantly improved performance in the slowly advecting and highly diffusive plumes (low U; high D), which tend to be smoother in their concentration profiles. The reverse was true when we removed intermittency sensing (gI=0), exemplifying a trade-off in navigational performance that persists across this wide range of odor environments. Together with the results presented above (Figure 3), this suggests that while a naïve summation of temporal sensors may be beneficial in some cases, in general, navigation can always be improved by some degree of specialization.

Our results so far suggest that dual sensing of two complementary odor signal features, intermittency and frequency, aids navigation across a diversity of odor plumes, albeit with a trade-off. To what extent is this dual-sensing capability enabled by the Drosophila olfactory circuit? Prior experimental and modeling work has shown that synaptic and circuit mechanisms in the olfactory periphery allow for accurate signal transmission across a range of frequencies (Martelli and Fiala, 2019; Nagel et al., 2015), while fast ORN adaptation allows signals to be encoded without saturation (Gorur-Shandilya et al., 2017). These various mechanisms suggest that the natural structure of the fly olfactory circuit may be well-primed for robust encoding of multiple temporal features of the odor signal.

We thus combined prior models (Gorur-Shandilya et al., 2017; Nagel et al., 2015) into a single model of odor binding, ORN firing, and PN response, and fed this naively into a behavioral module to investigate navigational performance. At the first stage of processing, odors bind an olfactory receptor/co-receptor (Or/Orco) complex, which can be active (ion channel open) or inactive (closed). Assuming fast binding dynamics, the average activity a of the complex is(7)a=1+eϵ⋅1+CKoff1+CKon-1

where C is the odor concentration, ϵ is the free energy difference between the active and inactive states when unbound, and where the dissociation constant between odorant and the complex in the inactive state, Koff , is much higher than that for the active state, Kon . To model adaptation, receptor activity feeds back into ϵ via(8)dϵdt=βa-a0

where β is an adaptation rate and a0 is a baseline activity. ORN firing rate is then obtained by passing a through a linear filter and static nonlinearity (Figure 6A; see Materials and methods). Finally, ϵ is bounded from below (ϵ§amp;gt;ϵL) so that ORNs shut off with sufficiently weak odor.

ORN firing rate is converted into a PN membrane potential through a postsynaptic conductance with two timescales (Nagel et al., 2015). Conductances are weakened over time via synaptic depression, also with two timescales (Figure 6A). This depression is modeled by a scaling factor of the conductance, Afastt (analogously for Aslowt):(9)dAfastdt=-rfaststAfastt+1-AfasttτAfast

where s is the ORN firing rate, rfast is the rate that A decays with increased firing rate, and τAfast is the timescale it takes for Afast to relax back to 1. This scaling factor then affects the synaptic conductance via(10)dqfastdt=kfasts(t)⋅Afast(t)−qfast(t)τgfast

where qfast is the fast conductance (analogous for the slow conductance). The fast and slow conductances are summed to give a total synaptic conductance qsyn . The PN membrane potential Vt then obeys(11)dVdt=-Vt-Eleak+qsyntRmVt-Esynτm

where Eleak,Esyn are the reversal potential for leak and synaptic currents, respectively, Rm is the resistance of the membrane, and τm is the timescale of the membrane. For parameter values, see Materials and methods and Nagel et al., 2015.

We first looked to see how the PN membrane potential responds to environments of different temporal statistics. As in Figure 1, we simulated the potential in square-wave environments of varying frequencies and intermittencies. We find that average membrane potential increases with frequency and intermittency independently (Figure 6B). This suggested that this membrane potential could be used to navigate environments where only one of intermittency or frequency is high. To test this, we considered a navigator that used the difference between the membrane potential and its resting potential (i.e., Eleak) to generate an upwind bias:(12)p(turnupwind|turning)=11+e−gPN(V−Eleak)

where gPN is the base gain for this model chosen analogously to the other base gains (see Materials and methods). While the circuit-inspired model was outperformed by the single-sensor models when these were used in matching environments (i.e., the F model in the high-frequency plume and the I model in the high-intermittency plume) (Figure 6C and D), it performed better than the individual F and I models when those were used in suboptimal environments. Thus, the dual-sensing capability of the ORN-PN circuit translates directly to more effective navigation across diverse plumes. Of course, as our results above showed, some degree of modulation of the gains could further enhance performance (Figures 4I and 6C and D, purple) – say by amplifying frequency sensing in certain plumes. It would be interesting to investigate whether any such modulation is enacted by the insect olfactory circuit.

The frequency response function is defined as the convolution between the whiff onset time series wt and an exponential filter with decay timescale τF where the whiff time series is a sum of delta functions occurring at the onset of each whiff. Thus, we have(13)F(t)=∫−∞tw(t−s)e−sτFds=∑k∫−∞tδ(t−tk−s)e−sτFds=∑ke−t−tkτF

where k enumerates the whiffs. Note that Ft+Δt=Fte-ΔtτF . Therefore, in discrete time steps we have wt+Δt=1 if odort§amp;lt;K and odort+Δt≥K and 0 otherwise and Ft+Δt=Ft⋅e-ΔtτF if wt+Δt=0 and Ft+Δt=Ft⋅e-ΔtτF+1 if wt+Δt=1.

For ONt , we use Euler’s method to numerically integrate Equation 2 to obtain At and then similarly integrate the following equation:(14)dONdt=1τON(C(t)−ON(t))

where Ct is defined in Equation 1, and the above equation is equivalent to Equation 3. τF was set to 2 s (Demir et al., 2020) while τA and τON were set to 9.8 s and 0.72 s, respectively (Álvarez-Salvado et al., 2018). The detection threshold was assumed to be below the signal amplitude, and kd was set to be 1% of the signal amplitude.

To illustrate how the ON and F filters respond to the frequency and duration of odor signals, we consider their response to square-wave odor pulses of given frequency f, duration D, and amplitude S0 . We first consider the ON response. To understand the ON response, we first calculate At . From Equation (2), we have(15)dAdt=1τA⋅(odor−A)

Let An denote the value of A at the offset of the nth pulse of signal and An∗ denote the value of A at the onset of the nth pulse. We wish to obtain a recursive relation for An, which will allow us to solve for An and from there obtain the value of A at all times. At the offset of a pulse, odor=0 and A will exponentially decay with time scale τA until the onset of the next pulse. This time of decay is given by 1f-D. Hence at the onset of the next pulse, An+1∗=An⋅e−(1τA⋅(1f−D)) . At this point, for a time period D, that is, until the offset of the (n+1)th pulse, A obeys the equation(16)dAdt=1τA⋅(S0−A)

with initial value An+1∗ . Hence,(17)∫An+1∗An+1dAS0−A=DτA(18)An+1=Ane−1fτA+S0(1−e−DτA)(19)An=A0e−nfτA+S0(1−e−DτA)∑k=0n−1e−kfτA(20)=A0e−nfτA+S0(1−e−DτA)⋅1−e−nfτA1−e−1fτA.(21)An≈S0(1−e−DτA)1−e−1fτA.

Since this is the value of At at the end of a pulse, it will be the maximum value of At over one period. Ultimately, however, we are interested in computing ONt, which obeys the equation(22)dONdt=1τON⋅(odorodor+kd+A(t)−ON).

To understand the response of ON, we can consider three different signal timescales. If the signal fluctuates quickly with respect to τA, that is, D and 1f-D«τA , then for t≫τA one can approximate At with its average value over one period, which is given by(23)f⋅⟮∫01f−DS0(1−e−DτA)1−e−1fτA⋅e−tτAdt+∫0DS0(1−e−DτA)1−e−1fτAe−(1f−D)τA⋅e−tτAS0⋅(1−e−tτA)dt⟯(24)=S0⋅f⋅D

Notice f∙D=I, the intermittency of the signal. Hence in this limit, and assuming S0≫kd, when the signal is present, we have(25)dONdt=1τON⋅(11+I−ON)

Thus, ONt obeys the same dynamics as At , except that it adapts to a square wave of amplitude 11+I instead of S0 and with a different timescale. Thus by the same reasoning as for At , the maximum value of ONt over one period (once t≫τA,τON) is approximately 11+I∙1-e-DτON1-e-1fτON, and the average value over one period is I∙11+I .

If instead τA≈D or τA≪D, then At≈odort, and we get(26)dONdt=1τON⋅(12−ON)

Finally, we can consider the case where τA≫D and τA≪1f-D . In this case, At≈0 and ONt adapts to a square wave with amplitude ≈ 1. The average value of ONt is I (and the maximum value would be 1-e-DτON1-e-1fτON).

In summary, we see that in all these cases the average value of ON depends only on the intermittency and increases monotonically with intermittency.

For F, it is easiest to consider Fn as the value of F just after the onset of the nth pulse. Since F increases by 1 at the onset of each pulse and then decays exponentially with timescale τF until the onset of the next pulse, one has(27)Fn+1=Fn⋅e−1fτF+1.

Hence,(28)Fn=F0⋅e−(n−1)fτF+1−e−nfτF1−e−1fτF

where F0 is the value of F right before the onset of the first pulse. For t≫τF, we have n≫fτF and Fn≈11-e-1fτF . Since F jumps at the onset of a pulse and then decays, this is the maximum value of F. The average value of F over one period is thus(29)11−e−1fτw⋅f⋅∫01fe−tτwdt=f⋅τw

Hence, the average value of F is linearly proportional to the frequency of the signal.

In a square wave, the It filter obeys the exact same dynamics as At , except with a pre-factor of 1/2 (assuming the amplitude of the wave is above the detection threshold) and thus has an asymptotic average response of I/2 .

We claim that Equation 5 is very similar to an OR gate in the variables gION and gFF. To see this, let us first define what we mean by an OR gate. Normally, an OR gate in two binary variables A and B returns a 1 if any one of A, B is nonzero. This results in the following ‘truth table’:

The first plume recording we used is the same as used in Álvarez-Salvado et al., 2018. We call this plume the high-intermittency plume. The odor detection threshold of the agents was set by analyzing the signal in a region outside the plume. In this region, pixel values of 0 were removed and nonzero values were fit to a Gaussian. The detection threshold was then set to be the 3 standard deviations above the mean of this fit. 10,000 agents were initialized with uniformly distributed starting position, where the x-position was between 50 mm and 300 mm from the source and the y-position went from 80 mm below the source to 80 mm above the source. The initial heading angle was uniformly distributed from 0 to 360°. The simulation was run for the length of the video (240 s), and the discrete time step was set to be the reciprocal of the frame rate (1/15 s).

The second plume recording we used was taken from Demir et al., 2020. We call this the high-frequency plume. The odor detection threshold of each agent was set the same way it was in Demir et al., 2020. Again 10,000 agents were initialized with uniformly distributed initial position and heading. The initial x-position was between 38.45 mm and 288.45 mm, and the initial y-position was between –74 mm and 86 mm. Initial heading was uniformly distributed from 0 to 360°. The simulation was run for 123.3 s, starting from the 600th frame of the video to the last frame, at 89.94 frames/s, corresponding to the frame rate used in Demir et al., 2020. The first 600 frames were dropped so that the plume had expanded to full size when the simulations began.

In both simulations, odor signal was computed by averaging over an elliptical antenna-sensing region in front of the agent, as in Demir et al., 2020. The length of the region’s major axis was 1.5 mm, and the length of the minor axis was 0.5 mm. The ellipse was centered 1 mm in front of the agent. For all models, odor values below the detection threshold described above were set to 0 to minimize the effect of camera shot noise. When computing the ON filter, the kd value was also set at this detection threshold value. If agents went outside the frame region, then they were allowed to continue but received zero signal in those regions. Thus, there were no walls in these simulations.

For these simulations, F was computed as for the square-wave pulses, with a detection threshold as described above, but we also enforced that the whiff time series wt could not register two whiffs less than 40 ms apart to capture the idea that the time resolution of individual whiffs is not arbitrarily precise and to avoid spurious detections due to the random fluctuations in the signal, as suggested by Demir et al., 2020.

The base gains, gI0 and gF0 , which were used for the simulations in Figure 3, and in multiples of which the gains in Figures 4 and 5 are reported, were determined the following way. Demir et al., 2020 experimentally extracted a sigmoidal turning bias, as in Equation 6, except only using the F filter and reported a gain of 0.242. We thus set gF0=0.242. gI0 was set so that the contribution from I in the high- intermittency plume would be roughly the same size as the contribution from F in the high-frequency plume. So defining I0 and F0 to be typical I and F values in the high-intermittency and high-frequency plumes, respectively, we have gI0I0=gF0F0 . We thus determined a gI0 of 1.936. For the PN model, we considered V0 to be the average value of the membrane potential in a high-intermittency environment and then set gPNV0-Eleak=gF0F0 , where Eleak was set to –70 mV (see below). We thus determined gPN to be 0.057 /mV. Finally, for the parameters dictating the navigational actions, the turn rate was set to 1.3 /s, walking speed to 10.1 mm/s, and filter decay timescale τ to 2 s, all in accordance with the findings of Demir et al., 2020. Note that the same timescale was used for the I and F filters.

Error bars for success rates (Figure 3C) were computed by bootstrapping data from a simulation of 10,000 flies – 1000 resamples were used with each resample size being equal to 10,000. Similarly, for the histograms of successful initial conditions, the data was resampled 1000 times, where each resample size was the size of the original data and means and standard deviations were computed and used for each histogram bin.

The simulated odor plumes were created using the strategy laid out by Farrell et al., 2002. Plumes consisted of growing Gaussian packets of odor concentration, released as a Poisson process with rate λ, that were advected by a uniform mean wind velocity and perturbed by turbulent diffusivity. The concentration at a point x,y due to a packet centered at xi,yi was computed as(37)odori(x,y)=C0π(R02+4Dti)exp(−ri2(R02+4Dti)),

where r2=x-xi2+y-yi2 , R0 is the initial packet radius, ti is the time since the release of this particular packet, D is a diffusivity that governs the packet growth, meant to account for molecular diffusivity and the effects of small eddies ,and C0 sets the initial concentration amplitude. The total odorx,y,t is then the sum over all packets that have been released up to time t. The packet center was computed the following way:(38)xi(t+Δt)=xi(t)+UΔt+η1(39)yi(t+Δt)=yi(t)+η2,

where U denotes the mean wind velocity, and η1 and η2 are Gaussian white noise perturbations with mean 0 and standard deviation 2κΔt , representing the effects of turbulent dispersion with eddy diffusivity κ.

In general, parameters were chosen to be physically realistic and also give concentration time series and odor plumes that were qualitatively similar to those in the videos. To set C0 , we defined the detection threshold to be 1 and enforced that an agent more than 1.6 standard deviations away from an initial packet would not be able to detect its presence. See the following table:

The order of magnitude for D was set by the fact that attractive odorants for D. melanogaster tend to have molecular diffusivities of around 10mm2/s , for example, ethyl acetate. The eddy diffusivity κ was set in accordance with Drivas et al., 1996. The release rate and initial size were chosen to be similar to those in Farrell et al., 2002. The wind speed was chosen to be similar to those used experimentally in Demir et al., 2020 and (Álvarez-Salvado et al., 2018).

Additionally, to improve computational efficiency, packets were no longer tracked once their x position was so large that even if all released packets were at that position, the sum of their contributions would still be less than the detection threshold.

10,000 agents were initialized with uniformly distributed initial position and angle, with x between 50 mm and 400 mm, y between –110 mm and 110 mm, and 0°§amp;lt;θ§amp;lt;360°, where x and y positions are defined relative to the source location, as in Figure 3. Plumes were simulated for enough time steps so that the expected x position of a packet released at time 0 would be equal to the maximum initial x for navigating agents, before navigating agents were introduced and simulated for 120 s. Once again, a trajectory’s success was defined by whether it got within 15 mm of the source location.

To define the antenna-sensing region, space was discretized into ‘pixels’ with 0.154 mm as the pixel width, matching the spatial resolution of the high-frequency plume. The concentration was then computed by averaging over the pixels in an elliptical region, with the region defined as in the previous section.

To set the level of noise added to the I and F filters, we first computed a characteristic I value in the simulated high-intermittency plume, I0 , by averaging I values over a region 192mm<x<205mm and 0mm<y<9mm and then averaging over the length of the simulation. We did the same for F values in the simulated high-frequency plume to obtain F0 . The values we obtained were I0=0.388 and F0=3.14. We then used 5% of these values as the standard deviation for Gaussian white noise to be added to the output of the I and F filters, respectively, at each time step. We also used I0 and F0 as representative I and F values in order to assign a single relative filter weight with which to color each set of gains in Figure 4G.

To understand how performance depended on the filter timescales τI and τF , we varied the two timescales independently, and for each pair of timescales simulated 10,000 flies in the two simulated plumes explored thus far. No noise was added to the sensor outputs, and gains were set at the base gains. Given that the average response of the intermittency filter is independent of the filtering timescale, it is unsurprising that for the fixed τF performance does not change significantly for values of τI nearly two orders of magnitude apart and only starts to degrade once the timescale gets on the order of 10 s (Figure 4—figure supplement 2A and B). This degradation is also expected: at very long timescales, it requires significant time for the I filter to reach an appreciable value, even in the case of constant odor. There was also no significant difference in performance in either plume between an I-only model with an infinitely fast (τI=0) timescale (and thus flat response power spectrum) and an I-only model with a 2 s timescale. This is to be expected as even with an infinitely fast timescale such a model has an upwind bias if and only if the signal is present and thus is only responding to the intermittency of the signal. We also see that performance is impacted by varying τF (Figure 4—figure supplement 2A and B) but that this is largely equivalent to fixing τF but varying gF instead (Figure 4—figure supplement 2C), as predicted by Equation 29.

ORN firing rates were computed from Equations 7 and 8. Once odor activity a was obtained, it was convolved with a normalized sum of two gamma distributions, N⋅Γ1-0.5⋅Γ2 , where the timescales for the two gamma distributions were 6 ms and 8 ms, respectively (Gorur-Shandilya et al., 2017), and the shape parameters 2 and 3, respectively, giving the shape seen in Figure 6A. This convolution was then multiplied by 300 Hz to get a firing rate. Since the model is only valid in regions where Kon§amp;lt;odor§amp;lt;Koff , we set any odor less than Kon to 0. In the simulated plumes, Kon was set to 1 and Koff was set to 400. a0 was set to 0.15 in order to get a baseline firing rate of about 40 Hz in the presence of continuous odor. In order to ensure the activity would go to 0 once there was no signal, ϵ was bounded below by ϵL and ϵL was set to be greater than the steady-state ϵ when no signal is present, which is given by ln⁡(1a0−1)≈1.73. Thus, ϵL was set to 2.5 and activity less than 11+eϵL was set to 0. β was set to 0.8 /s, in accordance with Gorur-Shandilya et al., 2017.

Once the ORN firing activity was obtained, PN membrane voltages were obtained using Equations 9–11. All parameters in Equations 9–11 were taken from Nagel et al., 2015. Since the fastest timescales were around 5 ms, responses were calculated through Euler integration with a timescale of 0.5 ms.