Noise enhances odor source localization

TL;DR

This work shows that in turbulent odor plumes, precise proprioception is not strictly necessary for accurate target localization: a calibrated level of proprioceptive noise can improve Bayesian inference by exploiting the anisotropic plume geometry. The authors derive an intuitive two-detection geometry and an asymptotic theory for Bernoulli sensing, linking the optimal perceived size $\sigma^*$ to the optimal pair distance $a^*$, and demonstrate, via CFD-based simulations, that turbulence enhances the benefit of noise while isotropic plumes negate it. They further introduce empirical noise tuning $\hat{\eta}$ that estimates the optimal noise from observed detection statistics and explore multiple noise modalities, finding robust improvements across realistic conditions. The findings have potential applications in octopus-inspired robotics and in understanding sensory processing in biological systems under uncertain, correlated environments.

Abstract

We address the problem of inferring the location of a target that releases odor in the presence of turbulence. Input for the inference is provided by many sensors scattered within the odor plume. Drawing inspiration from distributed chemosensation in biology, we ask whether the accuracy of the inference is affected by proprioceptive noise, i.e., noise on the perceived location of the sensors. Surprisingly, in the presence of a net fluid flow, proprioceptive noise improves Bayesian inference, rather than degrading it. An optimal noise exists that efficiently leverages additional information hidden within the geometry of the odor plume. Empirical tuning of noise functions well across a range of distances and may be implemented in practice. Other sources of noise also improve accuracy, owing to their ability to break the spatiotemporal correlations of the turbulent plume. These counterintuitive benefits of noise may be leveraged to improve sensory processing in biology and robotics.

Figures (18)

• Figure 1: (a) A snapshot of the odor field $c(\boldsymbol{\phi})$ emitted from an olfactory target (crab), obtained through direct numerical simulations (top) and its binarized version $m(\boldsymbol{\phi})$ (bottom). The multisensor agent is represented as a circle of radius $R$ centered in $(x,y)$; full and empty dots represent the $N$ sensors, either detecting or non-detecting odor. Our goal is to infer the coordinates $(x,y)$ of the center from binary odor detections. (b) Empirical likelihood $\ell(\boldsymbol{\phi})=\langle m(\boldsymbol{\phi})\rangle$. The brown rectangle delimits a uniform prior. (c) Mean Squared Error in the $x$ coordinate (main plot) and in the $y$ coordinate (inset), averaged over all 52 points $(x,y)$: $\langle \text{MSE}_x\rangle = 1/52 \sum_{i=1}^{52} \text{MSE}_x^i$, for inference with a single sensor and $N_T=20$ time points, sampled at regular intervals $\Delta t$ (blue line). For $\Delta t \gtrsim 15 \tau$, $\langle {\text{MSE}_x} \rangle$ plateaus to $3.3 \text{Var}_x$, matching predictive accuracy with the same number of time points sampled randomly in the whole simulation (red line), confirming measures are uncorrelated. (d) Same as (c) for a single sensor and an increasing number of time points $N_T$ sampled at an equal interval $\Delta t = 15 \tau$.
• Figure 2: Instantaneous inference with many sensors and perfect proprioception is challenging due to correlations in the odor signal. (a) Sketch of test points within the domain. (b) Maximum a posteriori estimate $\hat{x}$vs ground truth $x$, average and standard deviation across 200 realizations. Colors represent different locations $y$ relative to the centerline, sketched in (a). Grey line: ideal estimate $\hat{x}=x$. Accuracy degrades with $x$ as detections become rare, and improves with $R$, as the agent collects increasingly independent odor measures. (c) Performance degrades as the agent moves away from the source. Bias $b_x=\sum_{i=1}^{N_r}(\hat{x}_i-x)/N_r$, where $N_r$ is the number of realizations, normalized with the standard deviation of $x$, Std$(x)$, as a function of the number of sensors $N$ for $x/L=0.23,0.58,0.93$ from top to bottom. Mean (solid lines) and standard deviations (shades) computed across realizations and averaged for all $4$ values of $y$. Light, medium, and dark gray represent results with $R=10, 25, 40$ (in units of the grid spacing $\Delta x$), respectively. Bias for single sensor $b_0=-0.327, -1.904, -3.479$, calculated analytically (see Materials and Methods).
• Figure 3: Proprioceptive noise improves inference far from the target. (a) Sketch of an agent with perfect proprioception (left) and noisy proprioception (right). (b) Box plot of Square error $\text{SE}=(\hat{x}_i-x_j)^2$ for $i\in(1, N_r)$ and $N_r=100$ realizations, for 55 test locations $x_j$ close to the source (top), 50 locations mid distance (center) and 50 locations far from the source (bottom). Perfect proprioception (blue), noisy proprioception with $\eta = 7.4R$ (green). Perfect proprioception outperforms noisy proprioception only close to the source and noise improves the accuracy of Bayesian estimates far from the source. A random estimate within the prior $\hat{x}_{\text{random}}\sim P(x)$ is shown for comparison (gray). (c) Box plot of mean square error $\text{MSE}_x=\frac{1}{N_r}\sum_{i=1}^{N_r}(\hat{x}_i-x_j)^2$ for the ensemble of $j$ locations close, mid, far (top to bottom), as a function of the intensity of noise ($\eta$) relative to the size of the agent ($R$) corroborating that larger values of noise are useful as the agent moves further away from the target.
• Figure 4: An optimal noise exists that depends on $x$ and tuning noise to this value greatly improves inference. (a) $\text{MSE}_x$ has a marked minimum at a specific value of $\eta^*$ which increases with distance $x$. (b) Perceived size, at the optimal noise as a function of $x$, for different values of $y$ (pink, blue, red and orange lines), as well as their average (black). The optimal perceived size compares well with the optimal distance between detection pair, $a^*_{\text{pair}}$ (black line). (c) Maximum a posteriori estimate $\hat{x}$vs ground truth $x$, average (dots) and standard deviations (errorbars) are computed over 100 realizations and 5 values of $y$. Comparison between perfect proprioception $\eta=0$ (blue), noisy proprioception with noise tuning $\eta^*(x)$ (green) and noisy proprioception with empirical noise $\hat{\eta}(\hat{\theta})$ (gray). (d) Aggregate statistics of square error across all points, color code as in panel (c). Random (darker gray): inferred position is a random point with flat probability within the prior. For these results, we excluded realizations with fewer than 2 detections, which are entirely dominated by the prior.
• Figure 5: The two detection limit illustrates how the perceived sensor position tunes Bayesian estimates of source location. (a) Top: Likelihood in spatial coordinates $(\phi_x,\phi_y)$ (green maps), with two detections perceived to be located at $\hat{\bm\xi}_{\pm}=(x,y\pm a)$ with $a$ small (left), intermediate (center) and large (right). The contour line of the likelihood $\phi_y=c_a(\phi_x)$ is defined as the one whose width matches the perceived pair distance: $\max c_a = a$. Bottom: posterior distribution $p(x,y|1,1)$ after measuring two detections at $\hat{\bm\xi}_{\pm}$ obtained as the product of the likelihood shifted of $\pm a$. $p(x,y|1,1)$ is maximum at the location where the contour $c_a$ is maximally wide $\hat{x}_a=\arg\max c_a$. The distance between the two detections dictates the estimated position $\hat{x}_a$ and an optimal perceived distance $a^*$ exists such that $\hat{x}_{a^*}=x$. (b) 1D maximum a posteriori estimates on the centerline of the anisotropic turbulent plume from numerical simulations with two detections separated by $a^*$ are indeed nearly perfect.