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Ehud agents -seeking of groups in emerging behavior efficient and socialtaxis Information naethsaottelcto fasuc n hwdta it that showed and www.pnas.org/cgi/doi/10.1073/pnas.1618055114 source a of location that the information about the presented has maximizing (14) agent on al. an based et Vergassola algorithm about (13). foraging patches source a these the from of information location use the and strategy read different to a needed gradient, is continuous disconnected a and Without sparse, scale (12). a random, patches macroscopic from into coming broken a signal typically on the is flows, source or turbulent sources, by characterized weak is chemotaxis that for bacterial However, of example, 11). for studies (1, demonstrated, experimental as and scale, theoretical environ- microscopic by nonturbulent a in on source or strong ments a finding for strategy ing general exploitation, a as and (9) (10). exploration learning learning making, and optimization, machine decision and eco- for in 8), framework pivotal (7, been for- have physics of foraging nature nomics, of the models characterizing Theoretical on aging. depends and critically functions behav- (1–6) social utility ior and the patterns movement and underlie that navigate organ- computations and how space Understanding represent environments. isms complex in cues sory F exploitation foraging settings. general in agents, dynam- computation social altruistic collective study to vs. and extended ics selfish naturally be of can it groups how dis- reflecting in predict to couplings socialtaxis optimal use We tinct noise. robust behavioral Moreover, highly and is conspecifics. sensory and to tuning their parameter require infer of not to does behavior computations socialtaxis simple the on from rely share and information information agents no where the or settings, to little socialtaxis comparable plausible efficiency biologically high is the of which show we Importantly, agents, behavior. group opportunistic optimal of emerges behavior groups collective in efficient increase diversity, to information aim information individuals their social when that a show and we Surprisingly, simultaneously information term. group sensory own the its in social maximizes and individual infotaxis forag- each unifies group where that for interactions, “socialtaxis,” groups algorithm term such we an which in present ing, We processing unknown. information mostly extensively, of are studied principles been design have of the animals source role between and valuable interactions nature a the of Whereas be environment. the can about as conspecifics information well to settings, as leads nature, theoretical In gain in foraging. information individual ethological showed expected and algorithm the efficient infotaxis maximizing the locally humans, chemotaxis, that gradient- of the to by nature Inspired bacteria behavior. climbing individual from reward on studying gathering, focusing for mostly information setting canonical and a seeking exploita- been and has exploration balancing Foraging 2016) of 31, tion. terms October in review science, described for computer (received often 2017 and is 17, economics, April biology, approved in and behavior, NJ, Individual Princeton, University, Princeton Jr., Callan G. 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Email: addressed. be should correspondence whom To ore sa piiainpicpeta a ietindividual direct can behavior. group that social and principle and optimization an sensory as This from sources behavior. maximization group information collective suggests optimal biologically nearly robust, overlap maximize and efficient, highly informational plausible, to a is its aims result The minimize agent others. and with model each information a which own present in its We foraging, searches. group efficient for in about results information in source maximize but the signal, to a of moving gradient environments, the noisy following environ- by nonturbulent done a be in can ment source a exploration Finding between trade-off exploitation. a and of terms experimentally in key theoretically, analyzed a and often been behavior, has complex foraging of individual example humans, to bacteria From Significance epeetamdlo oaigi ru htcombines that group a in foraging of model a present We (16), amoebae (15), bacteria including species, many nature, In www.pnas.org/lookup/suppl/doi:10. 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ECOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY in which agents share little or no information, and its robust- by Gi , or how individuals affect one another, and β, the cou- ness to various forms of noise. Finally, we show socialtactic opti- pling strength or inverse “temperature,” which balances individ- mal behaviors in -like settings of competing and ual information and social interactions: β = 0 reduces socialtaxis opportunistic agents, suggesting a wide range of applications for to the standard individual infotaxis behavior of each agent, and this model. β → ∞ implies pure socially driven behavior. Following many models of , a natural choice From Individual Infotaxis to Socialtaxis for the relation between individuals would be one that is deter- The infotaxis algorithm (14) describes a strategy for individual mined by the physical distance between them. We consider here foraging in space, where an agent (or animal) moves to maxi- the case where the free energy of agent i is given by mize the information it has about the location of a stationary dist α source that emits signals (or particles) in random directions at Fi = S[Pi ] + βΣj dij , [2] a low rate. In its discretized version, the search space is divided into “cells,” and the agent estimates at each time t a probability where dij is the Euclidean distance between agents i and j , α > 1 map of the possible locations (r) of the source Pt (r). The is a tunable parameter that determines the scaling of the dis- of the map at time t, given by S [Pt (r)] = −ΣrPt (r) log2Pt (r) tance between agents, and the summation goes over all group (32), measures the uncertainty or missing information about the members. We explored the behavior of groups of 2–20 agents location of the source, in bits. At each time point, the agent may searching for a single sparse source in a 2D arena and found (i) find the source, (ii) detect a signal from the source (“hit”), or that for each choice of α, there was an optimal β that was always (iii) detect nothing. The agent then updates its probability map positive and minimized the time needed for the first member of over the space based on one of these three events and uses the the group to find the source (Distance-Based Socialtaxis). How- new map to decide whether to move to a neighboring cell or stay ever, even for the optimal values, the improvement in search time still. This decision is based on the expected information it would compared with independent agents was very small (more below). gain from each possible move, which is given by the expected Fig. 1B shows an example of the cohesive trajectories of two cou- change in entropy of each of the possible events in that location pled agents with their optimal βopt (Movie S1). (finding the source, getting a hit, or not getting a hit), weighted We then considered a socialtaxis model in which social inter- by the probability of occurrence of each event (Materials and actions were based on the information that agents have and not Methods). Fig. 1A shows the trajectories of two agents who are their physical locations. Here each agent aims to minimize the independently searching the source using infotaxis in a 2D arena, combination of its entropy and the overlap in the information it completely ignoring one another (Movie S1). The paths of these has in common with the other agents, independent agents are typical of individual infotaxis searches info (14), resembling trajectories of insects in nature. Fi = S[Pi ] − βΣj DKL [Pi ||Pj ], [3] If there are other agents (animals) around, then in addition to its own sensory information about the source, each agent may where Pi and Pj are the probability maps of agents i and j , and use information it receives from conspecifics or infers from their DKL is the Kullback–Leibler divergence (32) between probabil- behavior. We therefore extended the infotaxis algorithm to con- ity maps, which measures in bits the dissimilarity between the sider the behavior of groups of agents. Here, each agent moved distributions. For positive β this social term would drive agents to minimize a combination of its own entropy (as in infotaxis) to make their probability maps different from one another (con- and some function of its relation to other agents. Thus, rather strained by the individual infotaxis term), whereas for negative than minimizing entropy, each agent minimized a “free energy,” β the social term would drive agents to merge their knowledge given by and make their individual maps more similar. Fig. 1C shows the trajectories of two agents under this model with strong cou- F = S[P ] + βG ( ), [1] i i i group pling β = 10, reflecting their seemingly diverging yet highly time- where S[Pi ] is the entropy of the probability map Pi of agent i, efficient paths toward the source (Movie S1). The agents’ tra- as in the single-agent infotaxis above, Gi is a social interaction jectories are qualitatively different from the independent or term between agent i and the rest of the group, and β weighs distance-based algorithm and are similar to the trajectories of the balance between the individual’s information and its cou- fully cooperating agents who use a joint probability map based pling to others. The nature of this socialtaxis model is determined on their joint hits, and individual movements are determined by

AIndependent agents B Distance socialtaxis C Information socialtaxis D Full cooperation

10 steps

Fig. 1. Characteristic trajectories of pairs of agents searching for a source under different social interaction regimes. Shown are examples of pairs of agents, searching for a source located in the middle of the arena (purple square). Agents begin their search at the top left corner (purple triangles), and their trajectories are marked by blue and green paths, which are becoming lighter with time along the search; the circles on top of the trajectories indicate positions where hits were obtained (main text). The grayscale background represents the mean detection rate, decaying exponentially at large distances. (A) Independent agents, each performing infotaxis and ignoring one another. (B) Distance-dependent socialtaxis agents (Eq. 2) remain close to each other and are slightly more efficient than independent agents. (C) Information-based socialtaxis agents (Eq. 3) start off in different directions, seemingly dividing the search space and then quickly converging, resulting in highly efficient searches. (D) Fully cooperating pairs, who share their hits and coordinate their movements using a joint infotaxis algorithm (Materials and Methods), are highly efficient at finding the source.

2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1618055114 Karpas et al. Downloaded by guest on October 6, 2021 Downloaded by guest on October 6, 2021 ac ewe h oreadteaetta nsi s ie eut in Results time. vs. it finds that agent the and and repre- source the bars strengths between coupling error different tance and for time regions vs. shaded agents between source; Negative the shown SEM. and Results posi- initial sent 1. agents the to the of configurations of is chosen tions agents randomly 3,000 such independent over rescaled two averaged times of are search time with search (blue), the agents that cooperating fully and distance- (gray) a or (purple, (red) one strength socialtaxis based coupling information-based either vs. using time agents search aging Mean (A) coupling. social 2. Fig. Coupling on of negative Times Search tuning Notably, the any of Strength). Dependence require Functional Analy- the (Analytical not of analytically sis show did we which behavior parameters, model optimal This agents. as β monotonically of that improved with agents 2 compared time (Fig. search agents in independent decrease abrupt and all. at clear source the find to failure nonzero in a resulted typically toward negative ues, whereas them results, similar drove qualitatively gave that distance agents between slower optimal interaction much than an with were faster even they of 2A), little (Fig. choice but a agents agents, cooperating source fully independent the the of than find case to social- the distance-based agents in the the reflect, enabled 1 the taxis Fig. As 33). in ref. trajectories also any see example avoid configuration, (to of source choice the specific of of location effects the and agents config- the initial of chosen urations randomly 3,000 with each values, parameter 1D, (Fig. Methods). strategy infotaxis joint a apse al. et Karpas 1. Fig. in is as agent are socialtaxis details the other of arrow; information direction an same the by the shown direction; with different agent a socialtaxis choose a cor- would where with the points overlaid of are the one agents of to infotactic marking close independent another, two one of Trajectories near (D) starting ners. agents and center the in A Distance to source [ ] CD Search time [normalized] a showed socialtaxis information-based the contrast, clear In different under models, socialtaxis different the simulated We hsmdlwso a ihtecs fflycooperating fully of case the with par on was model this 0.2 0.4 0.6 0.8 0.7 0.8 0.9 C 1 1 r vrgdoe ,0 iuain ftoaet,wt h source the with agents, two of simulations 1,000 over averaged are 0 200 100 0 102030 oilai fcec eoe pia ihsrn information strong with optimal becomes efficiency Socialtaxis Coupling strength β opt Information socialtaxis Time [steps] Distance socialtaxis Independent agents hc ehdt n ueial.Aspring-like A numerically. find to had we which , Full cooperation α β =1 = ausldt aldsace.( searches. failed to led values ) erhtmsaesonfrtoindependent two for shown are times Search 4). =0 .Ipraty h efrac of performance the Importantly, A). 0 β aus hc nutvl might intuitively which values, B

Distance between agents [ ] and S1, Movie β 0.2 0.4 0.6 0 nrae,adwt large with and increased, 0 200 100 0 Time [steps] =0 =1 vrg distance Average B) β (C . aeil and Materials β vrg dis- Average ) o w for- two for β val- B ie oprdwt hs fidpnetaet,ee nthe in S4 even (Fig. search ters agents, in independent (N improvement of case small best those with a compared only times showed socialtaxis (Fig. based scaling power-law in a of improvement possibility 3B, given the monotonic at a hinting showed For time, size search 3B). group (Fig. on source dependency the of finding faster of trajectories size 3A the (Fig. study groups larger we in agents when apparent information-based becomes the socialtaxis in behavior collective spatial emerging An Gain Information Their Maximizing Socialtaxis Agents Opportunistic of Behavior Group Emergent with compared space ( 3D agents a independent of in case times the search reducing in efficient ar faet,w n htfragvnculn au fagent of value coupling given on a 1, for Focusing agent that experience. to find 2 from we learned agents, or of cou- individ- these pairs innate conspecifics—as each be its may where to differently plings case, coupled realistic be may con- biologically ual we more First, groups? a animal real sidered for socialtaxis is relevant How Agents Altruistic or of Opportunistic, Behavior Selfish, Optimal and Socialtaxis model. Plausible the Biologically in move- term spatial coherent explicit an and of lack structure” the despite “social ments, spatial cou- “Finder” a the First as demonstrates the faster Beyond it S5 Group, find (Fig. the agent still stronger first would the becomes the they that of pling know source, rest not the the did found for agents benefit had other clear the is if there Even agent, group: resid- first obvious the no of is source there indepen- whereas source of Thus, case the ual the 3E). in finds (Fig. than agent it agents to first dent closer the much all when are indepen- yet they (faster), than 3D), spatially (Fig. clustered agents socialtaxis also dent less information have consistently agents The agents other information. all keeps more source, the much finds Consequently, gathered agent 3C). first of (Fig. initial rate the on the the when later boost in interactions significantly agents social gain independent the information of but search, that their to of similar parts is that rate a cases all almost in filter source the and find information to their failure (compare average in to resulted agents noise, enable to seem is S2 (Figs. noise Environ- a computational the to or try, of Robust Modeling Is Wrong (Socialtaxis ment and source Environments the decay of from and source, Multitude signal source, the from the of strength distances of arena, starting length the of the shape and as size such the variations, different (Fig. of to directions robust kinds is different socialtaxis in infotactic information-based Notably, move where 2D). would path, agents the socialtactic the along and locations from individual away the them paring take that trajec- the infotaxis (t of source standard part in source. initial seen the the tories the in toward “overshoots” reduced long agents convergence characteristic between inter- accurate coupling social more information the results Accordingly, and which search, knowledge, the faster individual in the in by Later constrained space. is search action the of sion stro- more on, social early as the other ngly each of “repel” Because agents efficiency: term, information its delineates socialtaxis based h oilatcaet anifrainaottesuc at source the about information gain agents socialtactic The h aueo rjcoiso h gnsi h information- the in agents the of trajectories of nature The .Aan ag ruso gnsta eido distance- on relied that agents of groups large Again, Inset). N N and nraigtewih ftesca term social the of weight the increasing , eedneo nomto hrn ntrso ieto time of terms in sharing information of dependence β ∈ ,wihlast nefciedivi- effective an to leads which 2B), (Fig. increased is ,a ela ifrn om fsnoy odome- sensory, of forms different as well as S1), Fig. oi S2 Movie [70, ). ≈ β 2→1 100] ,adrqie n uigo oe parame- model of tuning fine required and 15), h entm ofidtesuc decreased source the find to time mean the , for .Ti sfrhrrflce ycom- by reflected further is This 2C). Fig. in > β 0 and and i.S1E). Fig. and oi S3 Movie .Fragvngroup given a For S4). Movie .Cneunl,tegroup the Consequently, ). and oilai sEfiin for Efficient Is Socialtaxis NSEryEdition Early PNAS S3 for .I a lohighly also was It ). < β β eutdi a in resulted 0). β au,the value, | f6 of 3

ECOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY A time = 25 time = 80 time = 130 would benefit an individual’s chances of finding the source are different from what would be good for the group: For a given =0 value of β2→1 the optimal β1→2 that would raise the probabil- ity for agent 1 to find the source first has a peak at β1→2 ≈ 0.5, regardless of the value of β2→1 (Fig. 4B). Thus, optimizing group success or individual success requires very different coupling val- ues. Still, even the most opportunistic choice of coupling values was more efficient for the group than the case of independent 10 steps agents and would benefit the group. The critical question of relevance of the socialtaxis model to real animals is the feasibility of information sharing between =10 agents (34). Surprisingly, we found that even a very limited form of information sharing gives a highly efficient socialtaxis: If instead of sharing their full probability map at each time point (which is obviously biologically unrealistic), agents shared only the location of the peak of a Gaussian that approximates their current probability map, the search time of the group was com- parable or even better than the case where the whole map was shared (Fig. 4C). Interestingly, the difference between the shar- B C ing of just the peak and that of their full maps was most appar- 300 14 ent for weak coupling between agents; for strong couplings the 300 =0 =0 200 0.6 12

[steps] =10

=10 [bits] 250 S [bits] 0.4 10 100 =10 200 21030 0 75 150 A B N Time 250 0.8

N=5 - S 0.2 21 150 N=10 =0

N=20 S

Search time [steps] 0 0.6 200 21=10 N=30 =0.25 100 21 = 012 0 50 100 150 21 12 =10 0.4 Coupling strength Time [steps] 150 2 1 =0.25 21 2 1 0.2 D 1 E Search time [steps] 100 =0 0.15 =10 0510 0123 In cluster 1 2 1 2 0.1 0.5 =10

p(R) C =0 Probability 0.05 1 Isolated

0 0 0.9 0 50 100 050100 Time [steps] Dist. to source R [steps] 0.8 Fig. 3. Information socialtaxis in large groups and the of collec- tive behavior. (A) Examples of 30 agents performing infotaxis independently 0.7 (Top) and information-based socialtaxis with β = 10 (Bottom). Each panel shows a snapshot in time of the agents. Starting area of agents is shown by a purple triangle and the source by a purple square. For presentation clarity, 0123 source is shown to be larger than its actual size, which is a single lattice site. (B) Mean search time for different group sizes N, performing information- based socialtaxis. Results are for 250 different groups and error bars show Fig. 4. Biological relevance and extensions of socialtaxis: diverse coupling SEM. B, Inset shows mean search time as a function of group size for two strength, limited information sharing, and inference. (A) Mean search time values of β.(C) Difference in information gain between agents in informa- of pairs of agents for different combinations of individually different cou- tion socialtaxis and independent agents as a function of time. C, Inset shows plings between them. Each line shows for a given coupling strength between the average entropy of agents in the two cases as a function of time. Gray 2 and 1, β2→1, the search time as a function of the coupling from 1 to 2, β1→2. line shows the mean residual entropy when the source is found. (D) Aver- (B) The probability that agent 1 would be the first one to find the source is age probability of an agent to be in a close group (cluster) with other agents shown as a function of the coupling strength from 1 to 2, β1→2, whereas β2→1 (dotted lines) or alone (solid lines) vs. time for infotaxis and socialtaxis (see is held fixed at different values. Results for A and B are averaged over 3,000 Materials and Methods for details). (E) Distribution of distances of agents random starting configurations. (C) Mean search time vs. coupling strength from the source when the first agent has found it, for independent info- for two socialtaxis agents that share their respective maps with one another taxis agents and information socialtaxis ones. Groups started from the same as above (dark red) and the search time when they share just the peak of initial conditions in both cases, and distributions are averaged over 1,000 a Gaussian fit with a predetermined variance to their map (dark blue). For random simulations. the normalized mean search time for agents that do not share information, but infer their respective maps from the movement of other agents, infer- ence of the full map based on estimated hits (pink) gives very small improve- monotonically as a function of the reciprocal coupling β1→2 (Fig. ment compared with the case of independent agents (gray), whereas infer- 4A). Thus, even when each agent can have an individually cho- ring only the peak location using GLM gives a significant improvement (light sen coupling to the others, strong couplings are beneficial for blue). Search times are normalized by the search times of two independent the group as a whole. However, the optimal coupling value that agents (set to 1). Error bars shown are SEM.

4 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1618055114 Karpas et al. 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isolated trtvl sdiscretmdlo h a of map the of model current its used iteratively R · ∇ Z −R(r t 1 hc iistesz ftegopta ol use could that group the of size the limits which , r isolated (r λ a 0 = p si e.1,w elc h aeof case the neglect we 14, ref. in As . 0  nietenihoho fa gn,which agent, an of neighborhood the inside t sh ) D i +ρ p exp (t ( (r) t )|r = = (r e λ 1 0 · 0 a enda sltd eew used we Here isolated. as defined was −R(rt n nopesbeflow incompressible and 0 e (r  ,btdfeetcocso parameters of choices different but 3, ) )[−S olwn asne l 3) econsider we (37), al. et Masson Following stecreainlnt ftesource. the of length correlation the is R(r d −R(r − 0 )∆S p rudi.I lse a endas defined was cluster In it. around (y i i (t )|r t + ] i sindahtto hit a assigned rmec ftepsil moves possible the of each from (r) (t y )|r − 1 0 )|r 2D e ] ) ieto)i h two-dimensional the in direction) , R(rt 0 −R(rt i (1 0 y ,if ), ), bann ntotiig hit a obtaining) (not obtaining 0 = nlzn eairo large of behavior Analyzing )V − ) hc a ensonto shown been has which 0), )|r )|r p  if NSEryEdition Early PNAS t 0 0 K (r ,if ), ), k k 0 0 i i ))[ρ  = = r i |r if eoe h position the denotes 1 0 0 tec iepoint, time each At j (r − , k k λ ihaprobability a with 0 = = )∆ r r 0 0 · ∇ | (“exploitation 1 0 S  , 0 , ρ r ~ v k j k by (r = a the was > | 0 gives 0 sthe is ) h(r) due 1 f6 of 5 [5] [4] [6] [7] sh d ≡ ,

ECOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY ˆj memory of past behavior of the other agents gave a small improvement of Pi (t), was a Gaussian fit to the estimated peak position at t, with variance performance, but at an exponential computational cost. In the peak infer- predetermined as in the regular Gaussian case described in the main text. To ence case, each agent tried to infer the location of the peak of the prob- make this algorithm more stable and computationally compact, we used a ability map of other agents, using a GLM. The x and y coordinates of the limited temporal window for the social term, t ∈ [50, 150], and the estimate peak were estimated independently of each other with different models, of the location of the peak was updated only every five steps. where the past positions of the other agent were weighted and fed to a ACKNOWLEDGMENTS. We thank Roy Harpaz, Ori Maoz, Oren Forkosh, Aia nonlinear function, to estimate the x or y location of the peak. We learned Haruvi, Rachel Ludmer, Amir Bar, Ofer Feinerman, and Gasper Tkacik for a separate model for a different starting region of the agents (space was discussions. This work was supported by a Human Frontier Science Program, split into 64 different 7×7 starting regions), and each model was trained European Research Council Grant 311238 and an Israel Science Foundation 5 using 10 simulated trajectories. This procedure was performed separately Grant 1629/12, as well as research support from Martin Kushner Schnur and for every coupling strength β. In that case, the inferred map that i has of j, Mr. and Mrs. Lawrence Feis.

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