Zero-information limit of a collective olfactory search model

Abstract

We address the problem of how individuals can integrate efficiently their private behavior with information provided by others within a group. To this end, we consider the model of collective search introduced in [https://doi.org/10.1103/PhysRevE.102.012402], under a minimal setting with no olfactory information. Agents combine a private exploratory behavior and a social imitation consisting in aligning to their neighbors, and weigh the two contributions with a single ``trust" parameter that controls their relative influence. We find that an optimal trust parameter exists even in the absence of olfactory information, as was observed in the original model. Optimality is dictated by the need to explore the minimal region of space that contains the target. An optimal trust parameter emerges from this constraint because it it tunes imitation, which induces a collective mechanism of inertia affecting the size and path of the swarm. We predict the optimal trust parameter for cohesive groups where all agents interact with one another. We show how optimality depends on the initialization of the agents and the unknown location of the target, in close agreement with numerical simulations. Our results may be leveraged to optimize the design of swarm robotics or to understand information integration in organisms with decentralized nervous systems such as cephalopods.

Figures (8)

• Figure 1: Sketch of the model. (Top) Schematic representation of the geometry of the search task. $\text{R}_s$ denotes the swarm radius while $L$ and $H$ indicate the position of the target relative to the swarm center. The initial heading angles of the agents are drawn from a Gaussian distribution with mean $\mu$ and standard deviation $\sigma$. (Bottom) Illustration of the two behavioral components of motion: the private velocity $\boldsymbol{v}_i^{\text{priv}}(t)$ is a cast-and-surge exploratory strategy (left), and the public velocity $\boldsymbol{v}_i^{\text{pub}}(t)$ results from Vicsek-like alignment with neighbors within the interaction radius $\text{R}_{\text{v}}$ (right). At time 0, agents are assigned a random clock $\gamma_i$ with uniform probability between $0$ and $t_{\text{clock}}$, so that $\boldsymbol{v}_i^{\text{priv}}(t)=\boldsymbol{v}^{\pm}_{CS}(t+\gamma_i )$, where $\boldsymbol{v}^{+}_{CS}$ represent the cast and surge program depicted in the sketch, and $\boldsymbol{v}^{-}_{CS}$ is its mirror reflected with respect to the x axis. The suffix $\pm$ reflects that each agent follows either the cast and surge as depicted, or its mirror with equal probability.
• Figure 2: Sketch illustrating the effect of the trust parameter $\beta$ on the trajectory of a casting agent interacting with another casting agent. Each agent combines its private velocity $\boldsymbol{v}_i^{\text{priv}}(t)$, prescribing a sharp turn, with the public velocity $\boldsymbol{v}_i^{\text{pub}}(t)$, which corresponds to the delayed velocity of its neighbor at time $t-t_{\text{mem}}$. As $\beta$ increases, imitation dominates over sharp turning, resulting in smoothed, more elongated casting trajectories.
• Figure 3: Representative trajectories of pairs of agents for different values of the trust parameter $\beta$ illustrating the effect of social alignment. Solid lines: trajectory of one of the two agents (colors correspond to different values of $\beta$, see legend); gray lines: trajectory of the second agent. Inset: zoom in to visualize $\beta=0$. At $\beta=0$, the dynamics are purely governed by private behavior, resulting in stereotypical casting dynamics. At $\beta=1$, agents follow exclusively public behavior, leading to ballistic motion in the initial heading direction defined by the average initial condition, here oriented at $45\deg$ relative to the centerline, which is marked with a black arrow. For $0<\beta<1$, trajectories smoothly interpolate between the casting and ballistic regimes. Note: for simplicity we choose $t_{\text{mem}}=\delta t$ where $\delta t$ is the numerical time step. However results depend weakly on $\delta t$ as long as $\delta t<t_{\text{mem}}$.
• Figure 4: Performance measures for finite interaction range $\text{R}_{\text{v}}=1$ under different initial conditions: (a) varying the shift $H$ of the target from the centerline, with $\mu, \sigma = 0$, (b) varying the mean initial angle $\mu$, with $H, \sigma = 0$ and (c) varying the angular dispersion $\sigma$, with $H, \mu = 0$. Left column: normalized first-passage time $\tau$ (averaged over 50 simulations) as a function of $\beta$. Right column: corresponding success rate $\rho$, defined as the fraction of successful trials out of 50 total simulations. Crosses in left column mark $\beta^*_\tau$.
• Figure 5: Performance measures for infinite interaction range $\text{R}_{\text{v}}$ under the same conditions as in \ref{['fig:perf_finite']}. Missing points in the $\tau$ plots correspond to cases where $\rho=0$, i.e. no agents reached the target. Crosses in right column mark $\beta^*$.