A simplifying assumption.

Because the out-of-phase correlations of our data are typically very close to zero, in what follows we will assume that these correlations are zero. In that case, and Eq 19 simplifies to (21)

To evaluate the agreement of the asymmetric Laplacian in Eq 21 with the observed distribution of correlations we compare their cumulative distribution functions (CDFs). In Fig 7A–7C we make the comparison for the 5 Hz correlation data at three different intersource distances. We quantified the agreement between the CDFs as 1 minus the largest absolute difference between them. A value of 1 would indicate perfect agreement, while, a value of 0, the smallest possible would indicate non-overlapping distributions. A heatmap of the match over the full range of intersource distances and frequencies (Fig 7G) reveals a match of ∼0.7 and higher over most of this range.

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Fig 7. Modeling the distribution of observed correlations.

(A–C) Examples of observed (F_data, thick pink) and predicted cumulative distribution function for the 5 Hz component of correlations, for three different intersource distances (indicated above each panel). Predictions were for the asymmetric Laplacian distributions (F_alap, pink) in Eq 21, asymmetric Laplacian with intermittency (F_int, light blue), and the best-fitting model when testing both intermittent and non-intermittent versions of the asymmetric Laplacian, Gamma, and generalized inverse Gaussian (F_best, navy). The parameters of F_alap were derived from Gaussian fits to the distributions of Fourier coefficients, the parameters for the remaining fits were found by directly fitting the correlation data. (D–F) Absolute difference between the predicted and observed cumulative distribution functions in the corresponding panels in the first row. Data in A-F may be optimistic because fits were computed using some (though not all) of the data they are being qualitatively evaluated against in those panels. See panel H for performance on held-out data. (G) Fit quality for the asymmetric Laplacian model, measured as 1 minus the largest absolute difference between the predicted and observed CDFs, computed for all distances and frequencies listed. Higher values indicated better fits. Values may be optimistic because parameters were both trained and evaluated on the same (full) dataset. Coloured dots correspond to data points shown in the left three panels. (H) As in panel G, but for the best fitting model when testing both intermittent and non-intermittent versions of the asymmetric Laplacian, Gamma, and generalized inverse Gaussian, fitted directly to the correlations, and evaluated on unseen data. See Methods Sec 5.2.4.

https://doi.org/10.1371/journal.pone.0297754.g007

Although there is good qualitative agreement between the data distributions of correlations and the corresponding asymmetric Laplacians of Eq 21, the plots in Fig 7A–7C also suggest that there are systematic differences between the two. In Fig 7D–7F we have plotted the differences between the data distribution and the asymmetric Laplacian fits. These plots have similar shapes, consisting of a narrow positive lobe followed by a broader negative lobe, eventually decaying to zero as both CDFs approach 1. This reveals that the data had more correlation values concentrated near zero, and correspondingly fewer large correlation values, than the asymmetric Laplacians.

To achieve a better fit ot the observed distribution of correlations, we elaborated the asymmetric Laplacian model in two ways to better capture small correlations (see Methods Sec 5.3.4). First, turbulent flows produce intermittent signals [50] so some very small correlations may be noise, not actual correlations. Therefore, similar to [43] for single-source concentration models, we incorporated intermittency using a binary random variable z that indicated whether an observation y_n was of a correlation r_n drawn from our correlation model p(r_n|s), or noise w: (22) Second, we generalized the exponential decays in our correlation model of Eq 21 to the generalized inverse Gaussian p(r) ∝ r^k−1 e^{−α(r/λ+λ/r)/2} which also includes the Gamma distribution as a special case.

We fit both intermittent and non-intermittent versions of these models to the correlations at each frequency and intersource separation separately. Examples of such fits are shown in Fig 7, showing improved agreement with the observed correlations. The performance of the best-fitting models on unseen data over the full range of frequencies an intersource distances are shown Fig 7H, demonstrating a very good fit over the entire range.

2.3.1 Which frequencies are more informative?

The Fisher information heatmaps for our CFD plumes suggest that high frequencies are more informative when sources are close together. To see what such heatmaps would look like when the ground-truth distribution of information over frequencies is known, we generated surrogate plume datasets with similar power spectra to our CFD plumes, but with different distributions of information across frequencies (see Methods Sec 5.2.7).

To do so, we used the fact that the Fisher information depends on the in-phase correlations ρ_n(s) according to Eq 23, and that these in turn are determined by the correlation (equivalently, covariance, since coefficients have zero mean) the in-phase Fourier coefficients by Eq 14b. We therefore generated surrogate data for R odour sources at a given harmonic by sampling from a zero-mean R-dimensional Gaussian and adjusted the correlation of its dimensions, representing odour sources, to decay with intersource distance to produce the desired Fisher information at that harmonic.

In the first such dataset, all frequencies were set to be equally informative. To achieve this, we set the correlation between two surrogate sources to decay at the same exponential rate with their intersource distance at each harmonic (see Table 2). In Fig 11B we have plotted the Fisher information heatmap for this dataset. The information content varies with intersource distance, but is homogeneous with frequency, as expected, and unlike that of our CFD plumes (reproduced in Fig 11A).

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Fig 11. Fisher information for simulations and surrogate data.

(A) Fisher information heatmap for our simulated plumes (reproduced from Fig 10B). (B) Heatmap for surrogate data with power spectrum similar to our simulated data and where all frequencies are equally informative. (C) Heatmap for surrogate data with power spectrum similar to our simulated data and where frequencies >12.5 Hz have the same, higher value of Fisher information than the lower frequencies. (D) Fisher information for our simulated plumes when using a 1-second boxcar window.

https://doi.org/10.1371/journal.pone.0297754.g011

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Table 2. Summary of surrogate datasets and the power and correlation functions used.

The kernel for each dataset was the product of the power and correlation functions: k(i, j, n) = S(n)G(|i − j|, n).

https://doi.org/10.1371/journal.pone.0297754.t002

Next, we generated a second surrogate dataset in which correlations decayed with intersource distance six times faster for the higher half of the frequency range than the lower half. This should result in the high frequencies being more informative at small intersource distances. At large intersource distances, high-frequency correlations will have decayed to zero due to their fast decay rate, yielding little information. Low frequencies will not have decayed completely to zero and will remain informative. Thus we expected to see high frequencies being more informative at small intersource distances, and low frequencies more informative at large intersource distances. This is indeed what we saw when we plotted the Fisher information heatmap for this dataset in Fig 11C.

The similarity of the information heatmap for our CFD plumes (Fig 11A) to that of the surrogate dataset where high-frequencies were a priori more informative (Fig 11C), and its dissimilarity to the heatmap of the dataset where all frequencies were a priori equally informative, supports the conclusion that for our CFD plumes high frequencies are more informative when sources are close together.

To emphasize the importance of windowing to these results, in Fig 11D we plot the Fisher information heatmap for our CFD plumes, but when analyzed with a 1-second boxcar window. The boxcar window, also known as a rectangular window, weights all samples equally. The poor spectral leakage properties of this window coupled with the power-law power spectrum of our data (S14 Fig) results in the low-frequency information masking that of the higher frequencies. This then results in an heatmap where information appears to be homogeneously distributed across frequencies, similar to Fig 11B.

From Eq 23, Fisher information depends on the rate at which correlations change with distance. The fact that correlations decay faster for high frequencies than for low frequencies means that when sources are closer together, the correlations at high frequencies will be more informative. This is shown in the left-most panel of Fig 12A, where we have plotted the information available at each frequency for an intersource distance of ∼0.1 pitch. At that intersource distance information shows a positive trend with frequency. As sources move farther apart, high-frequency correlations will have decayed, eventually changing at the same rate with distance as the slower decaying low-frequency correlations. This will result in all frequencies being similarly informative, as shown in the middle panel of Fig 12A for an intersource separation of ∼1 pitch. As sources move even farther apart, the lower frequencies will become more informative, as shown in the right panel of Fig 12A for an intersource separation of ∼2 pitches. The overall change in the trend as intersource distance increases is shown in Fig 12C and shows a distinctive elbow shape, so in what follows we will frequently refer to such data as ‘elbow plots.’ For surrogate data in which all frequencies are equally informative, trends in information with frequency are minimal at all intersource separations; see Fig 12B and 12C. Note that the exponential decay in correlations with frequency means that regardless of which frequencies are more informative, the absolute amount of information available will decrease with intersource distance. This is because information is derived from the rate of change of correlation with intersource distance, and this decreases as sources move farther apart. This effect is shown by the decrease in the down-scaling applied to the data in the panels of Fig 12A and 12B.

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Fig 12. Fisher information vs. frequency.

(A) Base 10 logarithm of the Fisher information at each frequency component (dots) and linear fits (dashed lines), for intersource separations of ∼0.1, 1, and 2 pitches (left, middle and right panel, respectively), for the simulated plumes using a 1-second Hann window. Coefficients β of the linear fits are stated in the legends. Data have been scaled by the amount shown in each legend to ease comparison. (B) As in panel A but for surrogate data where all frequencies are equally informative. (C) Bootstrap median (dots) and 5th-95th percentiles (bands) of the coefficients β of linear fits to the logarithm of Fisher information regressed on frequency for the CFD plumes (blue) and 10 surrogate datasets where all frequencies are equally informative (gray). Positive coefficients indicate information increases with frequency, negative coefficients that it decreases. Centres of orange markers indicate the integral length scale (L_U, see Methods Sec 5.2.8) in the vertical direction (orthogonal to the mean flow) of the vertical velocity field, computed at the midline where the odour sources were located (‘origin’, L_U = 0.85ϕ) and at the probe location (‘probe’, L_U = 2.6ϕ). Linear fits were computed using robust regression, see Methods Sec 5.2.5.

https://doi.org/10.1371/journal.pone.0297754.g012

The analyses above relied on the use of windows that reduced spectral leakage when the amount of power in different frequency bands is different, as is the case in real plumes and in our simulations. By using the Hann window to reduce leakage we were able to distinguish the different rates at which correlations decay in different frequency bands, and therefore the different amounts of information that they contain. As we show in S7 and S8 Figs these effects were not limited to the 1-second Hann window used above, and we also observed them when we used other windows that similarly reduced spectral leakage.

2.3.2 Testing other probe locations and source geometries.

The results we have presented above were computed at a single probe location, located on the midline of the flow downstream of the odour sources. Because animals will find themselves oriented at a variety of locations relative to odour sources of interest, it is important to determine the extent to which our results generalize to other locations within the plume. Therefore, we selected 8 additional probe locations within the plume, chosen to be dispersed enough to test the generalizability of our results, while also staying within the plume and avoiding boundary effects (see S1 Fig). These locations are indicated with coloured ‘×’s in Fig 13A. The blue ‘×’ marks the probe location for the results we’ve presented so far, while the remaining ‘×’s mark the new probe locations within the plume.

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Fig 13. Results for other probe locations and source geometries.

(A) Plume snapshot and probe locations for the cross-stream source geometry. Sources are equally spaced at x = 0 along a line transverse to the mean flow. The plumes (colours) are generated by the most distal pair of sources (white ‘o’s). Probe locations are marked with coloured ‘×’s. The cross-stream source geometry is the one analyzed in most of the Main Text, using the data measured at the blue probe location. (B) The decay of correlations with intersource distance for a few example frequency bands, at the nine di.erent probe locations in panel A, indicated by the background colour of each panel. The data in the blue panel is for the principal probe location analyzed in the Main Text and was previously shown in Fig 9A. (C) Elbow plots for each probe location in panel A, and coloured accordingly. The blue curve is for the principal probe location analyzed in the Main Text and is the same as in Fig 12C. Only the bootstrap medians, and not the confidence intervals, are shown, for clarity. (D,E,F) As in panels A-C, but for the oblique source configuration, where sources are equally spaced along a line at 45-degrees to the mean flow. (F,I) As in panels A-C, but for the streamwise source configuration, where sources are equally spaced along a line parallel to the mean flow.

https://doi.org/10.1371/journal.pone.0297754.g013

We performed the analysis describe in the previous section for each new probe location, culminating in ‘elbow’ plots like those of Fig 12C indicating whether high or low frequences were more informative at each intersource distance. The panels in Fig 13B show how correlations decay with intersource distance at each probe location. These panels reveal that at every probe location high-frequency correlations decay faster than lower frequency correlations when sources are close together. In Fig 13C we have overlayed the resulting elbow plots for all probe locations, showing only the medians for clarity, and coloured according to the probe locations in Fig 13A. The blue curve is the mean data previously shown in Fig 12C, while the remaining curves show the results for the new probe locations. All of the curves show the same qualitative behaviour: each starts at a positive value for β at low intersource distance, indicating that high frequencies are more informative than low frequencies when sources are close together. As intersource distance increases, all the curves trend downwards, and most eventually reach negative values of β, indicating that low frequencies become more informative when sources are sufficiently far apart.

Our results thus far are for sources located perfectly ‘cross-stream’, transverse to the mean flow. Natural odour sources will assume many other orientations relative to the mean flow. It is therefore important to determine whether our results generalize to other source orientations. To do so, we performed simulations using two new source orientations.

In the first, odour sources were equally spaced on a line at 45 degrees to the mean flow. In Fig 13D we have plotted a snapshot of the plumes generated by two most distant sources in this configuration. The panels in Fig 13E show how correlations decay at each probe location for this source configuration, again revealing that high frequencies decay faster than low frequencies when sources are close together. The resulting elbow plots of Fig 13F reflect this and show that at all probe locations in this source configuration, high frequences are more informative than low frequencies when sources are close together.

In the second new source configuration, we placed our sources parallel to the flow (streamwise). Fig 13G shows the plumes generated by the most distant sources in this configuration. The correlation decay results in Fig 13H reveal the same qualitative trends as in the corresponding panels for the previous two configurations, with high-frequency correlations decaying faster than those at low frequencies when sources are close together. This is reflected in the the elbow plots of Fig 13I which show that high frequencies are more informative than low frequencies when intersource distances are low.

In summary, our analysis of 8 new probe locations and two new source configurations all reveal the same same qualitative trends that we observed in our original probe location and source configuration, namely that high-frequency correlations decay faster than low-frequency correlations when sources are close together (compare Figs 9A, 13B, 13E and 13H), resulting in high frequencies being the most informative when sources are closer together, with the balance shifting towards low frequencies as intersource distance increases (compare Figs 12C, 13C, 13F and 13I). We saw the same qualitative trends when we performed our analyses using longer windows (2-second Hann, S21 Fig), shorter windows (0.5-second Hann, S22 Fig) and windows with different shape (1-second Kaiser-16, S23 Fig).

Finally, we repeated our entire procedure for a supplementary set of simulations, and observed the same qualitative effects (S24 Fig), though there were exceptions for some probe locations and source configurations. For example, high frequencies decay more slowly than low frequencies for small intersource distances at the red and brown probe locations in the oblique plume (S24E Fig), which may contribute to low frequencies being more informative for nearby sources than high frequencies at the red probe location (note the inverted elbow in S24F Fig). Another prominent example comes from the same source configuration but analyzed using 500 msec. window, for which the elbows from a few additional probe locations are inverted (S26F Fig). However, all of these exceptions are for probe locations that are at the edge of the plume. This is partially due to the fact that odour sources are also closer to the probes in the supplementary simulations. Both of these properties may contribute to the behaviour we observe at those locations.

When computing the Fisher information for the new source geometries above, we continued to use the in-phase component of the correlations. Initially, we had expected that the out-of-phase component of the correlations would play a larger role in these new geometries. For example, in the setting sources are arranged parallel to the flow direction, the velocity of the flow and the spacing between any pair of sources imply an initial phase difference between them, which may survive until the flow reaches the probe locations and manifest as strong out-of-phase correlations. However, we found that the magnitude of the out-of-phase correlations for the two new source configurations (S18D and S19D Figs) was small and similar to that for the original, transverse source configuration (S17D Fig), justifying our approximation of the Fisher information to use only the in-phase correlations for those source geometries as well.

3.1.1 Implications of statistical assumptions.

To determine the Fisher information contained in correlations about source separation, we required a probabilistic model of these correlations and how they vary with source separation. In our initial approach, we made simplifying assumptions about the Gaussianity of the plumes and their interactions and derived the predicted correlation distributions to be an asymmetric Laplacian (Eq 19). While this distribution fit the data well (Fig 7), we found that it deviated from the observations systematically due to its inability to describe the high probabilities of low correlation events (i.e. intermittency). This is perhaps unsurprising given that instantaneous plume distributions are known to be non-Gaussian [43] with the exception of large distances from the source, exhibiting transitions from exponential-like behaviors near the source through a log-normal transition to the far-field Gaussian regime [24]. To address these deficiencies in the correlation models, we then modeled the correlations directly using the asymmetric Laplacian as a template. We firstly extended it to incorporate intermittency by attributing very small correlations to noise. Secondly, in addition to the asymmetrical Laplacian, we also evaluated the ability of the Gamma and generalized inverse Gaussian distributions to describe the distribution of observations flagged as true correlations. We found that the extended models were able to capture the observed correlation distributions quite well across all source separations and frequencies (Fig 7H). The improved fits also indicate that the joint distribution of coefficients is not mulitvariate Gaussian. An interesting avenue for future investigation would be to determine the joint distribution of Fourier coefficients that would predict the same parameteric form for the distribution of correlations as our model when we fitted the correlations directly.

Ideally, we would have used these more sophisticated correlation models to compute the Fisher information; however, this computation would require describing how additional parameters depend on intersource distance, greatly increasing the difficulty of our analysis. We therefore used the simpler Gaussian assumption to compute the Fisher information. Would our findings hold if we used more accurate assumptions to derive Fisher information? Our simplified analysis suggested that an important quantity that determines Fisher information is the speed with which correlations decrease with intersource distance. A robust observation in our data, independent of any statistical assumptions, was that high-frequency correlations decreased more rapidly than low-frequency correlations. Because Fisher information is a measure of how rapidly the distribution of an observed quantity (like correlations) changes with respect to variations in estimated parameter (like intersource distance), the faster changes in high-frequency correlations with intersource distance that we observe will likely mean that more sophisticated computations of Fisher information using more accurate statistical models will still yield our core observation that high frequencies are more informative than low frequencies at small source separations.

3.1.2 Other considerations.

We aimed to determine whether high frequencies contained more spatial information than low frequencies. We found that this was the case when sources were close together. At intermediate source separations, all frequencies were equally informative. Low frequencies were more informative when odour sources were far apart. However, the absolute amount of information in these latter cases was much lower, and presumably more easily obscured by noise. Thus, when correlations contain enough spatial information to rise above the noise floor, it may be contained mainly in high frequencies. Therefore the sensitivity to high-frequency correlations recently observed in mice [20] may endow animals with fine spatial resolution when locating odour sources in the environment. We note, however, that diffusion acts to smear out high-frequency information more quickly than low-frequency information, given the sharper concentration gradients associated with these fluctuations. The distance that high-frequency information can persist in a given olfactory context is therefore set by the mean wind speed, the strength of turbulent straining, and the odourant molecular diffusivity.

To test the generalizability of our results we additionally analyzed plumes generated by sources arranged oblique and streamwise to the mean flow. An interesting result of this analysis was that the roll-off of the elbows shifted to larger intersource distances as the source separations transitioned from cross-stream to streamwise configurations (compare Fig 13C, 13F and 13I). In our developing grid turbulence flows, local integral length scales grow with streamwise distance (see S1A Fig), and these length scales relative to the source size set the plume dispersion characteristics. Therefore, sources separated in the cross-stream direction are introduced into similar length scales in the flow and thus disperse similarly, while sources separated in the streamwise direction disperse under the influence of different length scales in the flow. Furthermore, for the array of fixed probe locations considered here, sources separated in the cross-stream, oblique, and streamwise directions are located at different relative streamwise and lateral distances from the probe locations. These two effects likely drive the different roll-off behaviors observed in the elbow plots for different source configurations. Future work could investigate these connections.

We observed that correlations dropped approximately exponentially with intersource distance, see e.g. Fig 3C. This exponential decrease with intersource distance has been previously observed in the fluid dynamics literature [52]. These correlations are composed of the component correlations at each frequency component. Therefore, there is considerable latitude in the way the component correlations may decay: they are bounded by the variance at each frequency and must sum at each intersource separation to the overall correlation at that separation. Nevertheless, we observed that the component correlations all had the same, nearly exponential form (Fig 9A). Explaining the reason why the component correlations decay exponentially would be an interesting topic of future work.

Our work highlights the importance of window shape when analyzing signals where information is distributed across the frequency spectrum and where spectral power spans a wide range. Our information measures were computed for 1-second windows. Longer windows would reduce leakage effects, but would lengthen the time needed for behvioural decisions. An interesting question is how the distribution of spectral changes depends on the time windows used—or in ethological terms, which frequencies the olfactory system should attend to if odour source localizations have to be made quickly. We leave the investigation of these important questions to future work.

Fisher information is a ‘local’ measure about how a given signal can distinguish a given parameter value from neigbouring values. Our expression for Fisher information Eq 23 is accordingly local and depends on the value of component correlations ρ_n(s) at a given intersource separation and how they change around that point. To actually evaluate the Fisher information we used parametric fits to these component correlations (Eq 24). In contrast to the local nature of Fisher information, these parameteric fits are ‘global’ in that the fitted value at each intersource separation is influenced by all the data, not just those local to the separation. For example, deviations at large intersource separations can affect the length constant γ of the fit, which will in turn affect the value of Fisher information reported for small intersource distances (and all others). It will therefore be useful to investigate whether fitting approaches that are more local, such as splines, produce better estimates of the Fisher information.

We chose to compare the two plumes by computing their Pearson correlations. We chose the Pearson correlation because it is insensitive to the mean and scale of the signals being compared and only registers their covariation, and it is this covariation that reflects the relative separation of the odour sources. By our choice of Pearson correlation we do not mean to imply that it is the optimal measure for decoding intersource distance. It is, rather, a simple measure whose information content provides a lower bound for the total information present in the interacting plumes about source separation. Other measures, for example those based on event timings (see e.g. [53]) may be more effective for odour source localization.

We used Fisher information as our information measure. An obvious alternative is mutual information between the intersource distances and correlations. One advantage of Fisher information is that it is based on the likelihood function p(r_n|s) only and does not require the specification of a prior distribution on intersource distances. Mutual information requires the joint distribution p(r_n, s) of intersource distances and correlations and thus does require a prior on intersource distances—or alternatively a prior p(r_n) and likelihood p(s|r_n) for correlations, which seem harder to specify. However, this is not a major shortcoming as a prior on intersource distances would not be hard to motivate or to determine empirically for a given environment. The type of information provided by Fisher and mutual information are also different. Fisher information is ‘local’ in the unknown parameter (intersource distance) and indicates how discriminable a given value of that parameter is from another value infinitesimally close. Hence it is a function of the unknown parameter, as seen in Fig 10. It also bounds the variance of unbiased estimates of intersource distance from correlations, and would thus be particularly relevant if the animal needs to accurately localize the relative locations of odour sources. Mutual information is a ‘global’ measure in that its computations involves averaging over the full joint distribution of intersource distances and correlations. We leave the task of determining which of these two measures of information is most relevant for quantifying the spatial information in plumes to future work.

5.1.1 Model domain, simulations, and meshing.

The model domain (Fig 2) consists of a wide rectangular wind tunnel (0.62 m x 0.4225 m, x by y) with a mixing grid cylinder array distributed across the flow development inlet section (0.12 m in streamwise length). An array of 16 odour sources (8 source pairs mirrored about the domain centerline at y=0) of width L_f was distributed spatially across a transect a short distance downstream of the mixing grid at x = 0, covering a range of source separations s for analysis (Table 1). Source y-locations were constrained to the central 1/4 of the total inlet width to minimize walls effects, i.e. any impacts on cross-stream plume spread due to growth of the velocity boundary layer along the lateral walls. A uniform inlet velocity upstream of the mixing grid produces a chaotic flow field downstream through the nonlinear interactions of wake structures shed by the mixing grid. Simulations spanned a dimensional duration 70 s consisting of i. a 10 s start up period to establish fully chaotic flow conditions throughout the model domain and ii a subsequent 60 s analysis period containing dynamic multispecies plumes generated by the source array issuing into the chaotic flow environment.

Numerical simulations were performed via finite element discretization of the Navier-Stokes and continuity equations (Eqs S1 and S2 in S1 File) governing fluid flow, and the coupled, non-reactive advection-diffusion equation (Eq S4 in S1 File) governing odour transport and diffusion (see Sec S2 in S1 File). The COMSOL Multiphysics package (ver. 6.0) was used to generate the mesh in the model domain described below and to solve the system of equations resulting from the weak-form discretization of the governing equations, subjected to prescribed initial and boundary conditions described below. Geometric, flow, fluid, and odourant properties for model runs in the domain depicted in Fig 2 are summarized in Table 1.

The unstructured finite-element mesh contained 289,607 total triangular and quadrilateral elements (110 elements/cm² on average), locally refined in spatial regions with strong fluid velocity or odour concentration gradients, notably along the solid surfaces of the mixing grid cylinders and lateral walls and near the finite-width odour sources where strong initial odour gradients persist. The level of mesh refinement (increasing model degrees of freedom) was iterated to a final spatial resolution sufficient to resolve the smallest anticipated fluid velocity and odour concentration gradients. The final mesh implemented for all model runs had typical maximum and minimum element sizes (spatial resolution) of approximately 2 mm and 0.2 mm, respectively, with approximately 35 and 10 mesh elements per pitch ϕ and per source diameter L_f, respectively.

5.1.2 Initial and boundary conditions.

Initial conditions were u* = 0, p* = 0 and c* = 0 everywhere. The mixing grid and lateral walls were no-slip conditions, u* = 0, with zero total odour flux normal to the surfaces, −n ⋅ (J + u*c*) = 0, where the diffusive flux is J = −n ⋅ D∇c* and u*c* is the advective flux (n is the unit vector normal). The velocity inlet condition was normal, uniform flow with speed U_o, u = U_on and the outlet condition on the downstream end of the domain was zero pressure p* = 0. For solver stability we ramped the inlet velocity from zero to U_o over short interval in time. Odour sources of strength c* = 1 were introduced as constant concentration constraints from finite-width locations (Table 1), where the source profile at the origins was a smoothed top hat. The inlet and outlet odour boundary conditions were no diffusive flux normal to the boundary −n ⋅ J = −n ⋅ D∇c* = 0.

5.1.3 Discretization, solvers, and convergence.

Lagrangian shape functions were used for weak-form discretization of the fluid velocity, pressure, and odour concentration fields. The order of the integration scheme was matched to the element order for all dependent variable (first-order here). The time-dependent solver employs an implicit backward differentiation formula (BDF) method with maximum second order schemes, balancing numerical stability and damping tendencies for our smoothly varying velocity and scalar concentration gradients. BDF methods use variable-order, variable step-size backward differentiation and are known for their stability [61, 62]. The variable step size taken by the solver was informed by a prescribed absolute tolerance for the nonlinear solver and an implicit formulation of the mesh Courant-Friedrichs-Lewy (CFL) number. The solution sequence for the resulting systems of equations was fully-coupled in all dependent variables (fluid velocity and pressure, scalar concentration) for each solver iteration using an affine invariant form of the damped Newton method [63]. The nonlinear systems of equations were solved iteratively with specified convergence criteria using the direct PARDISO solver optimized for parallelized solutions of sparse systems of equations [64, 65]. The discretization schemes and solvers detailed above on the described mesh yielded good numerical stability and solution convergence with acceptable memory and computation requirements.

5.1.4 Lagrangian coherent structures.

To investigate the dynamics of the Lagrangian flow structures responsible for odour mixing we computed backwards finite-time Lyaponov exponent (FTLE) fields per [55]. The computation followed the methods described in [56], and the resulting backwards time FTLE fields provided a proxy for the attracting LCS in the flow. The integration time T_LCS was set to 0.6 seconds based on preliminary investigations of the spatially-varying integral time scales. The temporal resolution of the LCS computation matched the resolution of the underlying velocity data (20 ms), and the initial spatial resolution the Lagrangian tracer grid was 250 μm. These integration parameters yielded well-defined FTLE fields with strong ridge lines that were in good qualitative agreement with spatial regions odour pair coalescence, suggestive of convergence in the FTLE computation.

5.2.1 Computing plume correlations.

Fourier decomposition and windowing of plumes. To compute the correlations between a pair of plumes, a short-time Fourier transform (STFT) was applied to each plume using the scipy.signal.stft [66]. The STFT was computed for a specified window length using rectangular (boxcar) windows, with an overlap of half the window length, and using no padding or boundary. The application of different window shapes was implemented through the detrending function supplied to the STFT. The detrender first applied the desired window before z-scoring the result through mean-subtraction followed by scaling by its standard deviation. This detrending was performed so that the coefficients of the decomposition of two signal into trigonometric coefficients could be easily combined as in Eq 59 to yield the components of the Pearson correlation contributed by each harmonic. The conversion of Fourier coefficients to their trigonometric equivalents is completely standard so we have provided the derivation in the Supporting Information. The derivation shows that a signal x[n] of length L with discrete Fourier transform coefficients X[k] = u_k + jv_k can be expressed in terms of sines and cosines as (27a) (27b) (27c) where ⌊⋅⌋ is the floor function.

Computing correlations from Fourier coefficients. The component of the correlation between two plumes for a given time window in a given harmonic was computed by combining the sine and cosine coefficients from each source according to Eq 3. To compute the full Pearson correlation we summed the harmonic correlations over all harmonics as in Eq 2. To compute the distribution of correlations at a given intersource distance we pooled all correlations for all time windows computed for all pairs of sources separated by the given intersource distance. Cumulative distribution functions for some of this data are plotted in the top panels of Fig 7.

We fitt the distribution of correlations.

5.2.2 Modeling the dependence of Fourier coefficients on intersource distance.

To compute the empirical distribution of the Fourier coefficients from two sources at a fixed intersource distance s and for a given harmonic n, we first listed all pairs of sources that were the desired distance apart. Our source locations only differed in the y-coordinate, and we used a signed distance, so dist(y_i, y_j) ≜ y_i − y_j. Therefore, a desired intersource distance d yielded a list of ordered pairs of sources (y_i, y_j) such that y_i − y_j = d. We then concatenated the sine and cosine coefficients in the given harmonic for all time windows across all sources in the first coordinate of each pairing. This yielded a vector of length (28) where the factor of 2 is because of the concatenation of both sine and cosine coefficients. The length is a function of intersource distance s because the number of pairs a given distance apart was distance-dependent. We then stacked this vector on top of the same concatenation of data applied to the sources in the second coordinate of each pairing. This yielded a 2 × N(s) matrix Data_n(s), (29) where each element of the first row was one Fourier coefficient computed in one time window from one source, and the corresponding element in the second row was the corresponding coefficient for another source a distance s apart. We plotted some of this data in Fig 6.

The columns of Data_n(s) represent samples from the joint distribution p(a_n, c_n|s) of Fourier coefficients for the n’th harmonic, from sources a distance s apart. To fit a zero-mean bivariate Gaussian to Data_n(s) as in Eq 14b we computed the principle variances λ_n(s) and μ_n(s) along the major (y = x) and minor (y = −x) axes, respectively. To determine these we computed the variances of the projections of the data on the unit vectors and , respectively, (30a) (30b) We computed the variance as the mean of the principle variances at s = 0, (31) This is equivalent to computing the pooled variance of all coefficients from all sources for the given harmonic.

5.2.3 Parametric fits to ρ_n(s) as a function of distance.

We next fit each ρ_n(s) to the parametric form in Eq 24 by non-linear least squares using curve_fit from the scipy.optimize package [66]. The parameters to be learned were the length scale ι and the offset b. We constrained the parameters to be non-negative, and initialized ι to 1 and b to 0. All other parameters to curve_fit were kept at their default values. Because the data sometimes exhibited changes in length scale with intersource distance (see e.g. Fig 8B) we only used the data for intersource distances up to one pitch to compute fit parameters. Some example fits and parameters learned are in Fig 8.

5.2.4 Fitting the distribution of correlations.

We fit intermittent and non-intermittent versions of three different models to the correlation data. The models differed in the probability distributions they used to fit the correlations. Each model was based on a probability distribution on non-negative values. To extend the domain to cover negative values, the same base distribution was used but applied to the absolute value of the negative values, and with new set of parameters to cover these values, and the overall distribution normalized to one. That is, for a base distribution on non-negative values with parameters θ₊, (32) the extended distribution was defined as (33) For example, the asymmetric Laplacian distribution in Eq 21 corresponds to a base Exponential distribution, with parameters λ = (1 + ρ_n(s))/2 covering the positive correlations, and μ = (1 − ρ_n(s))/2 covering the negative correlations.

Note: In what follows we will frequently refer to the extended models by their base distributions, so e.g. Exponential when referring to the asymmetric Laplacian distribution.

Evaluating models by comparing CDFs.

We evaluated correlation models by comparing their predicted cumulative distribution functions (CDFs) to those we observed.

To compute the empirical CDF for a set of N observed correlations, we first sorted the values in ascending order, to yield We then computed the empirical CDF evaluated at each data point r_i as (34) The predicted CDF can be derived using the probability density function of Eq 95, which is (35) where ρ and η are the correlation and noise distributions, respectively (see Eq 99 for a particular case). The CDF is then a linear combination of the CDFs for the correlation and noise distributions: (36)

When comparing CDFs the R² value can be overly optimistic since both CDFs start at zero and increase monotonically to 1. Therefore, we compared CDFs more stringently by computing the largest absolute difference in their values, which is bounded below by 0 and above by 1. That is, we computed (37)

When numerically computing the predicted CDFs we sometimes encountered values outside of the valid [0, 1] range. This was particularly the case when using scipy.stats.geninvgauss to compute the CDF of the generalized inverse Gaussian. Therefore, when computing fit quality we only used the subset of points for which the CDF values were valid.

Fitting models using expectation-maximization.

To fit models with intermittency we had to determine which of a set of observed correlations {y₁, …, y_N} at a given harmonic and intersource distance were attributed to noise. From Eq 95 this requires determining the values of the binary latent variables {z₁, …, z_N} corresponding to each of the observations. To fit the parameters θ of a model while accounting for these latent variables, we used the Expectation-Maximization algorithm [67]. Initialized with an initial guess of the model parameters, this algorithm consists of repeatedly estimating the values of the latent variables (the ‘E’-step), then using those estimates to update the estimates of the model parameters (the ‘M’-step). Below, we summarize the parameter updates. Derivations are supplied in Sec S5 in S1 File.

The E-step update for all models at the t+ 1’st iteration has the simple form (38) Here ρ(y_i|θ_t) and η(y_i, θ_t) are the probabilities of the observation y_i under the correlation, and noise, distributions, respectively, θ_t is the current estimate of the model parameters, and ι_t is the current estimate of the intermittency parameter. The E-step therefore declares an observation y_i to have been a correlation rather than noise if its probability according to the correlation distribution is greater than its probability according to the noise distribution by a threshold that depends on the current estimated intermittency level. For the non-intermittent models, all observations were marked as being correlations.

During the M-step model parameters were updated using the current estimate of the latent variables {z_1,t, z_2,t, …, z_N,t}. All models had two parameters that related to intermittency: the intermittency level ι, and the standard deviation σ of the noise distribution. These parameters were updated the same way for all models. For the non-intermittent models, ι was fixed at 1, and σ at 0. Otherwise, these parameters were updated as described below.

To update the intermittency level ι, it is helpful to define the data intermittency at time t as (39) The intermittency level is updated by balancing the data intermittency against a prior on intermittency, , (40) This can be interpreted as weighting Nα observations with an intermittency of against N observations at the data intermittency . The hyperparameter was fitted using grid search (see below). The hyperparameter α, which sets the relative strength of the prior, was fixed at 1.

We updated the noise level by setting its variance to be the mean sum of squares of the observations attributed to noise in the E-step. Intuitively, this is because under our assumption that the noise distribution has mean zero, the mean sum of squares of the noise observations is an estimate of its variance. If the E-step did not flag any observations as noise we set the estimated noise variance to a small fraction of the variance of the observed correlations. We did this because setting the noise level to exactly zero would mean that the estimated noise variance would remain at zero for all future iterations. This is because the probability of any non-zero observations would be 0 under this degenerate noise distribution, so all observations in the next E-step would also be flagged as correlations and not noise, leaving the noise variance stuck at zero.

Letting be the number of observations attributed to noise after the current E-step, (41) where is the variance of the observed correlations.

The rest of the parameters varied by model so their updates will be described for each model separately.

Exponential model. In addition to the intermittency parameters, the Exponential model had two parameters, λ and μ, describing the decay rate of the positive, and negative, correlations, respectively. The M-step updates for these parameters had the closed form (42a) (42b) Here is the number of observations that were designated as correlations in the E-step. These can be split into positive and negative correlations, and and are the averages of the absolute values of the corresponding observations. The maximum operations prevents μ_t from being set to zero when the E-step did not designate any observations as noise. The added factor of 10⁻⁸ in the denominators prevents division by zero when N_t = 0 i.e. all observations were estimated to be noise.

Gamma model. The Gamma model has all the parameters of the Exponential model, plus two shape parameters, k and m, to describe the positive and negative correlations. To update the parameters during the M-step we minimized the negative log likelihood of the observations that were marked as correlations in the E-step: (43a) (43b) The minimization was performed using Nelder-Mead method as implemented by the scipy.minimize [66] function. The bounds of the search were [10⁻⁶, 10] for λ and μ, and [0, 10] for k and m. The search was initialized at the parameter values from the previous iteration, clipped to lie within these bounds.

Generalized inverse Gaussian. The Generalized inverse Gaussian model has all the parameters of the Gamma model, plus two additional shape parameters, α and β, for the positive and negative correlations, respectively. During the M-step we performed a damped update of the parameters towards the minimum of the negative log likelihood of the observations that marked as correlations in the E-step: (44a) where K_v(a, b) is the modified Bessel function of the second kind of real order a evaluated at b. Damping was used to aid convergence, with the damping factor δ fixed at 0.5. The minimization was performed the same way as for the Gamma model, with the new parameters α and β bounded to [10⁻⁶, 10]. The search was initialized at the parameter values from the previous iteration, clipped to lie within these bounds.

Ancestral initialization.

The Gamma distribution subsumes the Exponential distribution, because the former reduces to the latter when the shape parameter is set to 1. In turn, the generalized inverse Gaussian distribution subsumes the Gamma distribution, because the former reduces to the latter when its β parameter (the coefficient of 1/x in the argument of the exponential) is set to zero. Because models with fewer parameters can be easier to fit, to aid the fitting process we fit our models ancestrally. That is, to fit a model using the Gamma distribution we first fit the data using an Exponential distribution, and then initialized the Gamma fit with the parameters of the Exponential, initializing the shape parameter of the Gamma to zero. Similarly, when fitting a generalized inverse Gaussian, we initialized its β parameter at zero, and initialized the remaining parameters from those of the best fit Gamma. The Gamma in turn was initialized with the parameters of the best fit Exponential.

Evaluating models using nested cross-validation.

We fit a range of models to the correlation data by testing all combinations of the following hyperparameters:

• Whether the model was intermittent or not;
• If intermittent, the mean of the prior Beta distribution on the intermittency parameter, for the values {0.1, 0.2, …, 0.9}.
• The base probability for the correlations, whether Exponential, Gamma, or Generalized Inverse Gamma.

The remaining hyperparameters, listed below, were held constant at the stated values

• The minimum (0) and maximum (10) shape parameters for the Gamma and Generalized Inverse Gaussian distributions.
• The strength (1) of the Beta prior on intermittency.

We characterized these models in two ways. First, we estimated the performance of each model in fitting the correlation data. To do so, we repeatedly split the data into training (67%) and test (33%) sets. Each model was fitted to the training set and evaluated on the test set. Its performance was estimated as its average performance over three such random splits.

Secondly, we estimated the rank of each model, so that we could say e.g. which model was best overall, or which was the best among those with intermittency. One way to do this would be simply by ranking the performances we computed above. However, doing so would risk overfitting. To see why, note that by ranking the models we are in effect fitting the hyperparameters. Ranking the models using the same data used to evaluate their performance would then mean estimating performance with the same data used to fit the (hyper)parameters, risking overfitting.

To estimate the rank of each model while avoiding making this estimation on the same data that was used to estimate performance, we performed an inner loop of cross validation for each iteration of the outer loop. In each iteration of the inner loop, the training set provided by the outer loop was further split at random into a training subset (67%) and a validation set (33%). Models for every setting of the hyperparameters were fit on the training subset, and evaluated on the validation set. Models were ranked by their mean validation performance over three such random splits. The resulting model ranks were then averaged over iterations of the outer loop.

To see why this nested cross-validation procedure avoids overfitting, observe that each iteration of the outer loop yields, first, an estimate of model performance as computed on the test data, and second, model rank as determined from the training data by the inner cross-validation loop. Therefore, in each outer loop iteration, model rankings and model performance are estimated using different data. We then reduce the noise in these estimates by averaging over iterations of the outer loop.

5.2.5 Regressing information on frequency.

To produce the results in Fig 12 we regressed, for each value of intersource distance, Fisher information on frequency. The Fisher information at some intersource distances exhibited outliers that overly influenced the results of a standard linear regression, as judged by visual inspection. In addition to filtering out data points for which the computed Fisher information had NaN, infinite, or negative values, we took two measures to make the results more robust to these outliers. First, we limited the range of frequencies considered to be 1 Hz to 15 Hz. This was because some of the data, particularly in the supporting simulations, showed sudden drops in Fisher information above ∼15 Hz. Secondly, we replaced ordinary linear regression with robust regression. We used Huber regression, as implemented in scikit-learn’s linear_model.HuberRegressor. We used the default parameters, except for max_iter which we set to 10,000.

5.2.6 Bootstrapping procedure.

To estimate the variability of the various statistics used in our approached we computed their bootstrap distributions. These statistics were computed from the trigonometric form of the Fourier decompositions computed for each plume in each time window. Therefore we first computed these coefficients for all time windows, and then created each bootstrap dataset by sampling time windows with replacement until we had as many time windows as the original dataset. We generated 50 bootstrap datasets in this way. We then computed our various statistics using the data for the time windows chosen for each bootstrap dataset. This then gave us as many point estimates of each statistics as bootstrap datasets, from which we computed e.g. the 5th, 50th and 95th percentiles of Fisher information shown in Fig 10A.

5.2.7 Generating surrogate data.

In brief, we generated surrogate data consisting of artificial plumes with covariance structure similar to real plumes but for which we could adjust the amount of spatial information. To generate these plumes we randomly generated coefficients according to the bivariate normal model of Eq 14b using covariance kernel K(n, s) to relate the coefficients from plumes a distance s apart. Specifically, (45) where as usual, a_m and c_n are the cosine coefficients for the m and n’th frequency components, generated by two plumes a distance s apart, b_m and d_n are the corresponding sine coefficients, and expectations are taken over time windows, and the δ function δ(m, n) ensures that coefficients for different harmonics are uncorrelated. We combined these coefficients with their corresponding sine and cosine waveforms to generate the surrogate signals.

In detail, we generated signals for M sources, each of length 2⌊L/2⌋ + 1, by first creating the kernel relating the trigonometric coefficients of their Fourier decompositions. The signals had zero-mean, therefore for each source, we needed to specify ⌊L/2⌋ sine and cosine coefficients, corresponding to the normalized frequencies , covering all positive harmonics of the fundamental up to the Shannon limit. The kernel needed to specify how each harmonic at every source was correlated with each harmonic at every other source, a total of M²⌊L/2⌋² values. We organized these values into a symmetric kernel matrix of size M⌊L/2⌋ × M⌊L/2⌋. The elements of this matrix were determined as (46) where k(i, j, n) was the desired covariance of the trigonometric coefficients for the n’th harmonic for sources i and j (see below). We assumed coefficients at different harmonics were uncorrelated.

To compute cosine coefficients with covariance K, we first computed the Cholesky decomposition K = LL^T. We then generated M⌊L/2⌋ i.i.d. samples from a standard normal, u, and determined the cosine coefficients as c = Lu. We generated a second set of samples, v, from the standard normal in the same way, and used those to produce the sine coefficients s = Lv. Indexing these coefficients by source m and harmonic n, we produced the signals at each source by combining the coefficients with their corresponding trigonometric waveforms, (47)

Kernel functions.

We used several different kernel functions k(i, j, n) to generate the surrogate data in the text. Each of these kernel functions was the product of a term S(n) determining the power at the n’th harmonic, and a function G(i, j, n) determining the correlation between sources i and j, that in some cases depended on the harmonic. In every case, this correlation function was a function only of the absolute difference s = |i − j| of the sources, so it simplified to G(s, n). Thus our kernels were of the form (48)

Our kernels differed by whether they used a flat power spectrum or one similar to that of the simulations. We captured both cases by using a 1/f power spectrum: (49) where f_s was the sample rate in Hz. The maximum operation provides a frequency cutoff at 1 Hz below which the power is set to 1. To achieve a ‘white’ spectrum, we set α = 0. To achieve a ‘pink’ spectrum similar to our CFD simulations, we set α = 4.

Our surrogate data also differed in the correlation functions used. For the data where all frequencies were equally informative, we set (50) For the data where the high frequencies were more informative for sources close together than the lower frequencies, we set (51) Thus the length scale of decay was 6 times shorter for high harmonics (those in the upper half of the range) than for low harmonics.

The table below summarizes the surrogate datasets and the power and correlation functions used in each.

5.2.8 Computing integral length scales.

We computed integral length scales from velocity autocorrelation functions as defined in [68]. For example, to compute the integral length scale in the y-direction at a location (x, y) of interest, we first computed the y-velocity autocorrelation function in the y direction at that location by computing the time average of the product of the y-velocity at the location of interest, u_y(t;x, y) and that at a fixed positive displacement r in the y-direction, u_y(t;x, y + r). That is, we computed (52) We also computed the symmetric average in the negative y-direction, (53) We then averaged these two to arrive at the autocorrelation for a single, positive value of r (54) If e.g. (x, y + r) was not in the domain, then would not be computed and Q_yy took the value of , and similarly if (x, y − r) was not in the domain.

We then computed the longitudinal velocity correlation function by normalizing the velocity correlation function by its value at r = 0 (55) Finally, we computed the integral length scale by integrating this function from 0 to ∞: (56) The r values we used were discrete and spaced Δr = 0.02ϕ apart, so we approximated the integral above as (57) where N was the number of r values used.

In S20 Fig we have plotted the longitudinal velocity autocorrelation functions in the x and y directions, and the corresponding integral length scales, for two locations of interest.

5.3.1 Decomposing Pearson correlation.

We can express two concentration profiles x(t) and y(t) observed over a window of width T in terms of their Fourier decompositions, (58a) (58b)

Using the orthogonality of different harmonics of the fundamental frequency we can express the Pearson correlation in terms of the Fourier components as (59)

5.3.2 Relating the coefficients at one source to those at another.

To determine the uncertainty about the coefficients c_n, d_n of a component waveform at a second source, (60) given the coefficients a_n, b_n of a first, (61) we express the former as the sum of a deterministic component formed of a scaled and phase-shifted version of the component waveform from the first source, plus noisy residual. That is, (62) where β_n and θ_n are the best-fit scaling and phase-shift of the first source. These can be found to satisfy (see below) (63a) (63b) where as usual expectations are over time windows.

The residual ε(t) therefore captures the uncertainty remaining about the second component waveform, given knowledge of the first. To determine its statistical properties we express it by subtracting Eq 62 from Eq 61 as (64) This is a linear combination of sinusoidal waveforms at the same frequency ω, so we can write it as (65) By performing trigonometric expansions, the coefficients are found to be (66a) (66b) For n > 0, a_n, b_n, c_n, and d_n have mean zero (Eq 8). Therefore the coefficients above, and in turn ε(t), as linear combinations of zero-mean random variables, are also zero mean.

To determine the variance of the residuals, we first show in Supporting Information Sec S6.2 in S1 File that the coefficients and are uncorrelated. Therefore, from Eq 65, the residual variance is the scaled sum of the variances of the coefficients. To compute these variances, we have (67) (68) where we’ve used the relations in Eq 66 to express coefficient correlations in terms of β_n and θ_n, and the final equality follows by temporal stationarity. We then arrive at (69) Knowledge of the mean and variance would completely specify the probability distribution of the residual if it were a Gaussian random variable. Although it can be expressed as a linear combination of the Gaussian random variables a_n, b_n, c_n and d_n (Eq 64), these latter variables are not necessarily independent. Therefore, their combination is not necessarily Gaussian. To enforce this requirement we made the assumption 2.2.

To determine the scaling β_n and phase-shift θ_n, we equate Eq 62 an Eq 62 and match coefficients of cos(ωt) and sin(ωt), to get (70a) (70b) Correlating these equations against a_n and b_n, respectively, and using Eq S29 in S1 File and the definition Eq 10 of the coefficient variances , we get (71a) (71b) Correlating against b_n and a_n, respectively, and using Eq S30 in S1 File, we get (72a) (72b)

5.3.3 Deriving the distribution of correlations.

Here we derive the distribution p(r_n|s) of the component correlations r_n for the n’th harmonic for sources separated by a distance s, given our assumptions about the distribution of Fourier coefficients, a_n and b_n, from the first source and c_n and d_n from the second source.

Accounting for location dependence.

We must first account for the fact that the distribution of correlations is formed by the contribution of all pairs of sources separated by the given distance. To do so, we first assign integer indices to the sources. A pair of sources is then (i, j), the distance between them, D_ij, and the number of pairs a distance s apart, N_s. We can then use the rules of probability to express the distribution of correlations in terms of the contributions from each pair (73) The first term in the summand is the probability of the pair (i, j) being selected given the sources are a distance s apart. Because pairs at this distance will be selected uniformly and all others will not, (74) Our decomposition then simplifies to (75)

The summand is the distribution of correlations, given the pair of sources generating them and the distance between those sources. However, once the pair generating the correlations is specified the distance between the pair is redundant and we can remove it from the conditioners, yielding (76)

The new summand is the distribution of correlations generated by a given pair of sources. This in turn is related to the Fourier coefficients from each source. Let a_n, b_n be the cosine and sine coefficients from the first source in the pair, and c_n, d_n the corresponding coefficients from the second source. Then (77) The distribution of correlations for a given intersource distance is then (78)

This expression depends explicitly on the identity of the sources, because the joint distribution of coefficients generated by each pair of sources can be different for each pair. To emphasize the relationship of coefficients from one source to those from the other we decompose the joint distribution as, (79) To quantify effects that depend only on the relative location of sources, rather than the specific locations of specific sources we made location our independence assumptions. The first was that the distribution of coefficients from a source was the same for all sources, so (80) The second was that the distribution of coefficients c_n, d_n from source j given the coefficients a_n, b_n from source i depends only on the distance between them, (81) Substituting these two assumptions into Eq 79, and that result into Eq 79 yields Eq 5.

Deriving the distribution of correlations.

Dropping the n subscripts and the conditioning on s for clarity, (82) From Eq 12 We have that (83) By defining z = (a, b), x = (c, d) we can write Eq 83 as (84) Combining this with Eq 10 to get p(z) we have (85) where we’ve used Eq 10 to specify p(z). The integrand in Eq 85 is rotationally invariant in z since any rotation in z is matched by x through the δ function. So we can compute Eq 85 by computing it for one radial slice of z, say z = z e₁ = (z, 0), and multiplying the result by 2π. Switching z to polar coordinates z = (z, θ), for which (86) we have (87) To compute the latter integral, we switch x from (c, d) to (u, t) coordinates, defined for z = (z, 0) as (88) Then the area element dcdd becomes 2z⁻¹dudt and we have for the latter integral in Eq 87, (89) (90) (91) Now for our slice z = (z, 0), , so (92) Hence Eq 91 evaluates to (93) Substituting this in for the latter integral in Eq 87, we have Finally, to arrive at Eq 19, we simplify the length constant of the exponential by relating η² in Eq 69 to Z in Eq 20 as η² = −σ⁻²Z² − σ²β²cos(ϕ)², and use the definition of ρ in Eq 15 to get (94)

5.3.4 Improving the model of correlations.

Our correlation model captures intermittency by introducing a binary random variable z that determined whether the observation y_n was of a correlation r_n, or noise w. That is, (95) so that when the masking variable z = 1, we observe a ‘true’ correlation r_n, and when z = 0, we observe an instance of noise instead. We assume that the masking variable z obeys a Bernoulli distribution, (96) The intermittency parameter ι_n(s) ∈ [0, 1] determines the fraction of observations we deem to be true correlations, and can depend both on the harmonic n and the intersource distance s. The true correlations r_n are drawn from the asymmetric Laplacian distribution of correlations in Eq 21, (97) while the noisy observations w come from a zero-mean Gaussian (98) Like the intermittency parameter, the variance of the noise can depend on both the harmonic and intersource distance. The predicted distribution of correlations for our ‘intermittent asymmetric Laplacian’ model is then a linear combination of the ‘true’ correlation distribution p(r_n|s) and the noise distribution determined by the intermittency parameter ι_n(s) (99) Setting ι_n(s) to 1 corresponds to our initial asymmetric Laplacian model, while setting it zero represents a fully intermittent case where all correlations are in fact noise. The amount of intermittency and the variance of the noise were determined from the data (see Methods, Sec 5.2.4).

To demonstrate the effect of including intermittency in our model we have added the CDFs for the distributions in Eq 99 fitted to the data in Fig 7A–7C to the corresponding panels in those plots (light blue traces), revealing an improved qualitative agreement. In Fig 7D–7F we have added the deviations between the data CDFs and those of the new model that includes intermittency. The panels show that both the positive and negative deviations have been reduced, as expected, but not eliminated.

The fact that both positive and negative deviations remain in Fig 7D–7F indicates that the data have more small values than even augmentation with intermittency can capture. Therefore, we extended our model by replacing the Exponential distributions in Eq 21 that constitute the positive and negative halves of the asymmetric Laplacian with distributions that had higher densities near zero, but that included the Exponential as special cases. One candidate is the Gamma distribution, p(r) ∝ r^k−1e^r/λ. Just like the Exponential distribution, the Gamma distribution has a scale parameter, λ for the positive correlations above. In addition, it introduces a shape parameter, k > 0. When k = 1, the Gamma distribution reduces to the Exponential distribution. When k < 1, the density approaches infinity as r → 0. This singularity at 0 aids in modeling the large number of low correlation values we observe. Analogously to Eq 21, we ‘sandwich’ two Gamma distributions together so that we can also cover negative values (see also Methods Sec 5.2.4), and arrive at our correlation distribution (100) Here Z is the normalizing function of the Gamma distribution, k and λ are the shape and scale parameters for the positive correlations, and m and μ are the corresponding parameters for the negative correlations. We have omitted the dependence of these parameters on the harmonic n and the intersource distance s for clarity.

Another probability distribution, which includes the Gamma (and therefore the Exponential) as a special case, is the generalized inverse Gaussian. The parameterization of this distribution that we use gives the probability density function p(r) ∝ r^k−1e^{−α(r/λ+λ/r)/2}. This distribution introduces a further shape parameter α > 0 and a term proportional to 1/r in the argument of the exponential that provide additional flexibility in modeling small observations. The scale of the distribution can be taken to be the length constant of the r term in the exponential, 2λ/α. Keeping this constant while reducing αλ towards zero, one approaches the Gamma distribution. As before, we sandwich two such distributions together to arrive at our distribution of correlations, (101) Here Z is the normalizing function of the generalized inverse Gaussian distribution, k, λ, and α are the parameters for the positive correlations, and k and μ and β are the corresponding parameters for the negative correlations. As before, we have omitted the dependence of these parameters on the harmonic n and the intersource distance s for clarity.

Our elaborated models of the correlations were thus of the same form as Eq 99, with p(r_n|s) set to either asymmetric Laplacian, or sandwiched versions of the Gamma, or generalized inverse Gaussian distributions.

5.3.5 Fisher information.

To compute Fisher information for the distribution in Eq 19, we need (dropping subscripts for clarity), Then, defining λ = σ²ρ (102) Now defining λ_⊥ = σ²ρ_⊥, we have so (103) Then the square is (104) We need to compute the expectation of this with respect to r under p(r|s). We can split this into the contributions from each component.

Constant term.

The expectation of the constant term is just (105)

Linear term.

Defining we have (106)

Quadratic term.

Defining we have (107) (108)

Fisher Information.

Collecting terms, we arrive at (109) This expression simplifies significantly if we assume that λ_⊥(s)≈0, since then Z ≈ σ², Z′ ≈ 0, and (110) which is Eq 23. For ρ = (1 − b)e^−s/γ + b as in Eq 24 we get, in terms of normalized distances , (111) which is Eq 26.