Chemotaxis emerges as the optimal solution to cooperative search games

Alberto Pezzotta, Matteo Adorisio, and Antonio Celani

Phys. Rev. E 98, 042401 – Published 1 October 2018

Abstract

Cooperative search games are collective tasks where all agents share the same goal of reaching a target in the shortest time while limiting energy expenditure and avoiding collisions. Here we show that the equations that characterize the optimal strategy are identical to a long-known phenomenological model of chemotaxis, the directed motion of microorganisms guided by chemical cues. Within this analogy, the substance to which searchers respond acts as the memory over which agents share information about the environment. The actions of writing, erasing, and forgetting are equivalent to production, consumption, and degradation of chemoattractant. The rates at which these biochemical processes take place are tightly related to the parameters that characterize the decision-making problem, such as learning rate, costs for time, control, collisions and their tradeoffs, as well as the attitude of agents toward risk. We establish a dictionary that maps notions from decision-making theory to biophysical observables in chemotaxis, and vice versa. Our results offer a fundamental explanation of why search algorithms that mimic microbial chemotaxis can be very effective and suggest how to optimize their performance.

Received 10 January 2018
Revised 12 September 2018

DOI:https://doi.org/10.1103/PhysRevE.98.042401

Physics Subject Headings (PhySH)

Chemotaxis Collective behavior Optimization problems Stochastic processes

Physics of Living SystemsInterdisciplinary PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Alberto Pezzotta¹, Matteo Adorisio¹, and Antonio Celani²

¹International School for Advanced Studies (SISSA), via Bonomea 265, 34136 Trieste (Italy)
²The Abdus Salam International Centre for Theoretical Physics (ICTP), strada Costiera 11, 34151 Trieste (Italy)

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Vol. 98, Iss. 4 — October 2018

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Figure 1
Scheme of the cooperative search game for $N = 2$ agents. The agents are driven by the controls $u_{1}$ and $u_{2}$ , which depend on both their positions $x_{1}$ and $x_{2}$ . They pay individual costs associated with control and time expenditure, respectively with a rate $γ u^{2} / 2$ and $q$ (possibly position dependent), and a pairwise cost associated with the interaction—collision—with a rate proportional to a Dirac- $δ$ function. Optimal controls minimize the average cost (or an exponential average in the risk-sensitive case).
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Figure 2
Optimal cooperative predation. Comparison between the uncontrolled (a) and controlled dynamics (b), in the noninteracting case ( $g = 0$ ). The agents are initially localized in a small region of space and are required to reach the target (grey disk). They initially undergo unbiased diffusion during the scouting phase and when some reach the target, the recruiting phase begins. The chemical cue is emitted from the target and degraded at constant rate, resulting in a gradient (grey contour lines, in logarithmic scale) which elicits a drift toward the target in all other agents. In these simulations the parameters are $γ = 1$ , $q = 10$ , $D = 1$ , $g = 0$ , $ε = 0.1$ . (c) Average cost rate for time (uncontrolled: dashed orange line; controlled: solid blue line) and for control (dash-dotted blue line). The scouting phase (S, shaded) and the recruiting (R) phase are dominated by time cost and by control cost, respectively. (d) Probability density function of the time to reach the target [color code as in (c)]. For small times the distributions are similar, while at large times controlled agents display an exponential decay against a $- 3 / 2$ power law for uncontrolled ones.
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Figure 3
Optimal crowd evacuation. Agents are injected at a constant rate at the center of the maze and have to find the exit (on the right side of the maze) as quickly as possible while minimizing collisions. The panels show a numerical simulation of Eqs. (6), (8), and (10). The desirability (= concentration; see Table 1) is shown in the top panels, while the flux of agents $J_{ρ}$ is displayed in the bottom panels. During scouting (left column), the population consumes the chemical, leading to an outward-driven scouting process, faster than pure diffusion. Upon reaching the target, agents lay the beacon signal and recruit those who lag behind to the target (middle column). Eventually, since agents are continuously injected in this case, a stationary state is reached where agents track the optimal path from the center to the exit (right column). The parameters are $D = 1$ , $γ = 1$ , $ε = 10^{- 1}$ , $q = 10$ , and $g (N - 1) = 100$ .
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Figure 4
Risk-sensitive cooperative predation. Probability density functions of the total cost per agent under the optimal risk-neutral control (solid blue), risk averse ( $α = 0.4$ , dashed red), and risk seeking ( $α = - 0.4$ , dashed-dotted green). Other parameters are set as in Fig. 2. The same distributions are plotted on linear scale in the inset. Here we simulate the dynamics under the optimal control drift (no learning), which can be computed exactly in this case (see Appendix pp1-s4). Positive values of $α$ (optimal risk-aversive behavior) lead to steeper exponential decay for large values of the cost while negative values (optimal risk-seeking behavior) enhance fluctuations with lower-than-average cost, at the price of increasing the frequency of higher-than-average fluctuations. The thin solid lines are the analytical estimates for the right tails, which can be calculated in the one-dimensional case (see Appendix pp1-s3).
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Figure 5
Robustness of the optimal solution. The risk-sensitive cost function $F_{α}$ is plotted as a function of the control amplitude $U$ , in a risk-neutral (solid blue), risk-averse (dashed red), and risk-seeking (dashed-dotted green) situation. The parabolic approximation around the minimum is shown for the risk-neutral case (dotted blue line). A deviation of the control from the optimum by a quantity $δ U$ corresponds to an increase in the cost $δ F$ (in the parabolic approximation). These two quantities are related by Eq. (A34). Curves have been obtained with the same parameters as in Figs. 2 and 4 of the main text.
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Figure 6
Risk neutral vs risk sensitive. Two-dimensional histograms of time expenditure and control costs, from simulations of Eq. (A1) with the mean-field control in Eq. (A37), for the risk-averse, risk-neutral, and risk-seeking situation. The costs for time expenditure and control are positively and almost linearly correlated. The optimal control for the risk-neutral problem is such that control and time expenditure costs are very similar. Instead, the risk-averse optimal controller tends to pay more on control (reducing possible dangerous fluctuations towards high values of the cost), whereas the risk-seeking controller allows for large time-expenditure cost while reducing the control.
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