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Positional choice and robust collective behavior in fish schools: biohybrid experiments and modeling
Vahagn Grigoryan, Donato Romano, Cesare Stefanini, Giulia De Masi
TL;DR
The paper asks whether individual motion rules assumed by fish-school models reflect real fish behavior. It compares classical models with positional observations from a live fish interacting with robotic fish, then adds a positional-preference term. Classical models matched observations only with precise parameters, while the new model was robust to small parameter errors and produced collective patterns in multi-agent simulations.
Problem
Group-level validation does not establish whether the individual rules assumed by collective-motion models match fish behavior.
Source: §1 · p. 3Method
The authors compared simulated agents following classical models with measured positional choices of a fish interacting with four robotic fish, then added a positional force to a generalized cohesion model.
Source: §3.3.1 · p. 19, §5 · p. 29Results
Classical models closely matched experimental observations only with precise parameter choices and failed under minor perturbations, whereas the generalized cohesion model was robust to small parameter errors and produced parallel and swarming behavior in multi-agent simulations.
Source: §5 · pp. 29–30Takeaways & Limitations
The generalized cohesion model increased robustness while preserving cohesion, swarming, and highly polarized collective motion in multiple agents.
Source: §5 · p. 30, §6 · p. 32Scaling positional tendencies to collective behavior remains a mechanistic hypothesis; the model does not establish identical rules across fish or independent positional decisions, and it assumes no correlation between positions.
Source: §5 · p. 31, §6 · p. 32
Abstract
from arXiv · showhide
Collective behavior of fish schools is usually modeled on the assumption that each individual follows specific rules of motion that depend on its position and velocity relative to its neighbors. Although these models reproduce many schooling patterns observed in nature, it remains unclear whether the assumed rules are realistic at the individual level. To address this question, we first analyzed a set of experiments in which a live fish interacted with four moving robotic fish in a tank, and measured the time it spent in each position relative to the robots. We then simulated this experiment, replacing the fish with an agent, and assessed the extent to which classical models agree with the experimental observations. An extensive exploration of the parameter space showed that these models closely matched the experimental observations, with a very specific choice of parameters. However, they were highly sensitive to the parameter values: a minor perturbation caused them to fail completely. To resolve this, we propose a new model that incorporates an additional exploratory term characterizing the agent's positional preference at every moment. This model proved robust to small errors in the parameters while successfully replicating the experiments. Furthermore, when generalised to multiple fish, the model reproduced schooling behaviour and common schooling patterns, while also exhibiting exploratory behaviour at both the individual and the school level. This shows that social cohesion coexists with individual exploratory behaviour, and that realistic models of collective behaviour should account for both social interactions and this exploratory component.
1 Introduction
Fish-school models can reproduce group patterns, but agreement at the group level does not establish that their assumed individual rules match real fish behavior. This study tests individual positional choices in a controlled fish–robot setting and asks how those choices relate to collective behavior and stimuli.
1 Introduction
- Collective-motion models prescribe individual rules so group patterns emerge, and existing approaches include force-based, behavioral-zone, leader-formation, and data-inferred rules.These models reproduce group dynamics including schooling, swarming, milling, and predator avoidance.Source: §1 · p. 2
- Group-level agreement leaves open whether modeled individual rules match the behavior fish actually express, because distinct rules can generate similar patterns and conspecifics complicate individual observation.The position an individual chooses within the group, and how external context affects that choice, has consequently been overlooked.Source: §1 · pp. 2–3
- Schooling is a context-dependent social strategy involving decisions about movement, spacing, and orientation, with both fitness benefits and costs.Benefits include reduced predation risk and improved foraging; costs include competition, conspicuousness, and exposure to inaccurate social information.Source: §1 · pp. 2–3
- The study isolates one live fish interacting with four unresponsive robotic fish to identify individual behaviors associated with biological schooling.The setup examines behavior in the presence of a repulsive stimulus, specifically a predator.Source: §1 · p. 3
- The paper asks whether a single fish follows assumed schooling rules, whether applying its behavior to multiple agents produces schooling, and how a repulsive stimulus changes behavior.Source: §1 · p. 3
2 State of Art
Prior work models collective behavior through force rules, behavioral zones, prescribed formations, data-inferred responses, and continuum descriptions. Other strands incorporate fluid dynamics, predators, obstacles, or robotics constraints, with varied collective outcomes.
Force-based models
- Force-based models combine attraction, repulsion, and alignment, sometimes adding stochasticity, predator forces, or environmental details.Examples include predator-avoidance and survival-probability modeling, predator and obstacle forces, and the fish-inspired robotic algorithm’s wandering, memory, and delay compensation.Source: §2 · pp. 5–6
Range-based models
- Range-based models prioritize repulsion over alignment or attraction and have produced swarms, milling, and parallel-group states.Related zone-based models use randomly selected neighbors and variable speeds, and have been used to simulate predator–prey systems.Source: §2 · pp. 6–7
Formation-based models
- A leader-following robotics approach adjusts a predefined formation around the leader and group barycenter to study coordinated movement.The work investigated which formation patterns facilitate successful coordinated movement.Source: §2 · p. 7
Data-driven models
- Data-driven approaches infer behavioral response from trajectories or observations, including angular modulation over Voronoi neighbors and a proportional-derivative law for zebrafish.The trajectory-inferred model reproduces schooling, swarming, milling, and winding states; bioPD reproduced zebrafish social response and scaled across species, groups, and multi-robot tasks.Source: §2 · pp. 7–8
Macroscopic and control-theoretic models
- Macroscopic and control-theoretic approaches analyze collective dynamics and convergence, with formal algorithms addressing cohesion, velocity matching, collisions, and obstacles.A control-theoretic variant without navigational feedback leads to regular fragmentation, while another adds feedback to keep groups cohesive.Source: §2 · pp. 6–7
Fluid-coupled models
- A hybrid particle–fluid model feeds behavioral targets to a simulated fish body and reproduces schooling and milling in groups of 30–50 individuals.It reports speed fluctuations across collective modes and faster, lower-cost swimming than an equivalent solitary swimmer.Source: §2 · p. 8
Antiflocking
- Antiflocking work repurposes force-like rules for dispersion and distributed area coverage around unknown obstacles.Its agents share maps through neighbor-to-neighbor communication and simulations showed reduced overlap in non-coverable regions.Source: §2 · p. 8
3 Materials and Methods
The methods are presented as a procedural overview.
3 Materials and Methods
- Figure 2 is identified as a flowchart of the experiment and model design, and the methodology procedure is illustrated there.Source: Figure 2 · p. 9, §3 · p. 9
3.1 Experimental Design
The experimental design paired one live fish with a moving four-robot school, with predator and no-stimulus conditions. Researchers recorded the fish’s time across five positions relative to the robots.
3.1 Experimental Design
- Four robotic fish were arranged 90° apart on a 32 mm-radius circle and moved along an elliptical path inside the tank.The ellipse had 195 mm × 80 mm semi-axes and the school moved at 10 mm/s.Source: §3.1.1 · p. 10
- A removable opaque partition separated the school compartment from a stimulus compartment containing either no cue or a robotic predator.The tank measured 600×600×400 mm, with a water depth of 300 mm.Source: §3.1.1 · p. 10
- The predator replica followed a 25 mm-radius semicircle and was revealed by removing the partition; the no-stimulus condition used the same procedure without a cue.Source: §3.1.1 · p. 10
- Researchers defined five positions relative to the robotic school and recorded how long fish occupied each over 20 minutes after stimulus presentation.Each condition was repeated 20 times with 20 different fish, after a 30-minute acclimation period with the partition in place; position-occupancy probabilities were inferred from the measured times.Source: §3.1.1 · p. 10
- The study simulated the robotic-school experiments by replacing the live fish with an agent following candidate models.Source: §3.1.1 · p. 11
3.2 Models
This section presents force- and range-based schooling models and an extension of the force-based model that adds experimentally informed positional preference alongside social interactions, including its consideration for multiple fish. It also defines generalized cohesion through attraction points based on neighbors’ positions and average direction; for Central Position, attraction is directed to their barycenter, recovering the original definition for four robotic fish.
3.2.1 Generalized force-based model
- The generalized force-based model moves fish at constant speed, combining neighbor alignment and attraction–repulsion with stochastic noise.The model uses a discrete time step of τ = 0.1 s and defines the neighborhood as the four robotic fish.Source: §3.2.1 · pp. 11–12
3.2.2 Range-based model
- The Couzin range-based model uses repulsion, alignment, and attraction zones, prioritizes repulsion, and adds Gaussian turning noise subject to a turning-rate limit.Neighbors in the alignment and attraction zones count only when within the fish’s field of perception.Source: §3.2.2 · pp. 15–16
3.2.3 New generalized cohesion model
- The new cohesion model adds a position force toward a randomly selected preferred position drawn from the experimental distribution, introducing exploration alongside social interactions.The fish selects a new preferred position after remaining at its current selection for Δt = 10 seconds; setting cp = 0 recovers the generalized force-based model.Source: §3.2.3 · pp. 17, 19
- For multiple fish, the proposed model assigns each individual preferred positions within its neighborhood, which the authors define as metric.The supplied passage does not specify further details of the neighborhood.Source: §3.2.3 · p. 19
- The generalized model defines attraction points from neighbors within distance R, partitioning them around the average-direction axes; its Central Position attraction point is their barycenter, and the definition reduces to the original four-robot version.The passage also describes attraction points for other positions as reflections of the overall neighbor barycenter relative to subgroup barycenters.Source: §3.2.3 · p. 19
3.3 Statistical Analysis
The analysis screens model parameters by comparing simulated and experimental distributions of time spent in each position, then tests local robustness and summarizes school structure with statistical measures.
3.3.1 Simulation-Based Parameter Screening
- An ABC-inspired screen accepts parameter combinations compatible with observed positional data, rather than estimating a posterior distribution.The error compares distributions of time spent in each position, not matched experiment pairs.Source: §3.3.1 · pp. 19–20
- 20 observations across five positions are scaled to 1,200 per row, and 105 sampled parameter combinations are screened against the experimental data.For the no-stimuli case, runs with a Frontal Position mean of zero are excluded from the reported tables because they show no frontal exploration.Source: §3.3.1 · pp. 19–20
3.3.2 Local Sensitivity Analysis
- Local sensitivity analysis tests robustness by comparing an accepted reference with its five nearest neighbors in normalized parameter space.Parameters with different units, ranges, or sampling scales are mapped onto the unit interval before distances are computed.Source: §3.3.2 · pp. 20–21
3.3.3 Statistical Measures
- School statistics include polarization over time and length and width measured from a minimum-area rectangle aligned with the school's average direction.Polarization approaches zero for swarming and one for parallel motion, assuming the individuals form a single school.Source: §3.3.3 · p. 22
4 Results
The generalized cohesion model matched the experimental positional distributions across a broader parameter neighborhood than the classical models. In multiple-fish simulations, school formation and polarization depended on group size and model weights, while individual exploration persisted.
4.1.1 Generalized force-based model: Constant Alignment
- Only 62 combinations fell below tolerance; nine also had nonzero frontal means, but three low-σ cases spun rapidly and nearby fits were scarce.Four of the five nearest neighbors failed the tolerance; the remaining run had a zero frontal-position mean.Source: §4.1.1 · pp. 22–23, Figure 5 · p. 23, Table 2 · p. 24
4.1.2 Generalized force-based model: Positive Lennard-Jones Alignment
- Only 41 combinations fell below tolerance; four passed the nonzero-frontal guard, but three had low frontal means, while all five neighbors to case 3 exceeded tolerance.Four neighbors had errors more than ten times that of case 3.Source: §4.1.2 · pp. 25–26
4.1.3 The range-based model
- 98 combinations fell below tolerance, but the five neighbors of the selected case all exceeded it, with frontal means ranging from 19.38 to 781.87.The selected case was the lowest-error case among those investigated.Source: §4.1.3 · p. 26
4.1.4 Generalized cohesion model
- 14,892 combinations fell below tolerance; 14,588 also had nonzero frontal means, and all five nearby combinations remained acceptable.The five neighbors’ frontal means stayed close to the experimental value.Source: §4.1.4 · p. 26
4.2 Generalized cohesion model for multiple fish
- Polarization was mainly near 1 for N ≤30, exclusively parallel for 40≤N≤70, and sometimes swarming, near 0.1, at larger sizes.Moderate polarization for N ≤30 was attributed as likely due to multiple schools.Source: Figure 7 · p. 27, §4.2 · p. 29
- With cp fixed, low cw produced swarming, while high cw and relatively low cf produced parallel alignment; the cohesion model’s polarization curve was shifted versus the force-based model.Mean and maximum polarization were both close to 1 in the parallel regime, indicating sustained high polarization.Source: §4.2 · pp. 27–28
- Over 50,000 seconds, the fish extensively explored; frequent desired-position changes indicated it eventually reached each selected position.Desired position was updated only when the fish reached it.Source: Figure 8 · p. 28, §4.2 · p. 29
- The fish formed one school in 100% of cases for N ≥40, while high length and width outliers suggested occasional difficulty merging when N ≤30.School dimensions were evaluated after 5,000 time steps.Source: §4.2 · pp. 28–29
5 Discussion
Classical models could match the fish’s observed positions, but only with narrowly tuned parameters; the generalized cohesion model was more robust to parameter changes. In multiple-agent simulations, positional exploration coexisted with cohesion and coordinated motion, while the study’s controlled setup and scaling assumptions limit biological interpretation.
5 Discussion
- Fewer than 0.2% of classical-model runs met the tolerance, versus more than 10% for the generalized cohesion model, which also tolerated small parameter errors.Classical models could reproduce observations with precise parameter choices, but minor perturbations could make them fail.Source: §5 · p. 29
- For N ≥40, simulations formed a single school that remained together for 5000 seconds; larger groups could also switch between parallel and swarming behavior.At N = 100, varying weights reproduced both patterns; polarization stayed near 1 while average direction varied, consistent with school-level reorientation.Source: §5 · p. 30
- The position force represents a latent tendency to seek a position relative to the group, rather than necessarily a physical force perceived by fish.Positional choices may reflect context-dependent trade-offs involving safety, information, and resource access.Source: §5 · p. 30
- In long simulations, fish repeatedly changed desired and occupied positions while cohesion persisted, showing that individual exploration need not destabilize collective organization.The authors suggest flexible movement among spatial roles may support information acquisition and responses to changing ecological conditions.Source: §5 · p. 31
- The fish–robot experiment isolates a focal fish responding to an unresponsive, standardized social configuration, not the feedback dynamics of a fully biological school.In natural schools, one fish’s response can alter neighbors’ sensory environments and subsequent decisions.Source: §5 · p. 31
- Scaling observed positional tendencies to multiple agents remains a mechanistic hypothesis; the simulations do not establish that real fish share one rule or make independent positional decisions.The authors identify individual heterogeneity and neighbor identity, movement, and past choices as factors to incorporate.Source: §5 · p. 31
6 Conclusions
The study used a controlled fish–robot setup to examine whether individual fish behavior can account for schooling patterns. Classical models matched observed position distributions only in restricted parameter regions, whereas positional tendencies broadened compatibility and supported cohesive, exploratory multi-agent behavior.
6 Conclusions
- A single fish interacting with an unresponsive robotic school allowed its behavior during schooling and external stimuli to be isolated within a standardized social configuration.The authors describe the experimental setup and modeling as unique in the literature to their knowledge.Source: §6 · p. 31
- Classical force- and range-based models matched observed distributions only in restricted parameter regions, while the generalized cohesion model worked across a substantially broader range.The classical models also matched predator-context observations, but only in restricted parameter regions.Source: §6 · p. 32
- Positional tendencies generated stable collective organization while allowing continued spatial exploration, suggesting behavioral variability and group-level coherence can coexist.The authors note trade-offs involving predation risk, resources, and social information.Source: §6 · p. 32
- The model is a mechanistic hypothesis, and its tested classical alternatives were limited to force-based models and one range-based model.Future work would quantify positional transitions, residence times, and individual heterogeneity, and test reciprocal interactions in responsive groups.Source: §6 · p. 32
Supplementary Materials
The supplementary materials show error and local-sensitivity figures for the generalized force-based model in the predator context, alongside tables listing parameter combinations that meet specified criteria.
Supplementary Materials
- Figure S1 plots the error distribution across 100,000 parameter combinations for the generalized force-based model with constant alignment in the predator context.The figure description does not state the distribution’s outcome.Source: Supplementary Materials · p. 33, Figure S1 · p. 33
- Figure S2 examines local sensitivity around run 11442 for that model in the predator context.The figure description identifies the model variant and context but does not state the sensitivity result.Source: Supplementary Materials · p. 33, Figure S2 · p. 34
- Tables S1–S3 list parameter combinations whose error falls below tolerance and whose mean frontal position is non-zero for the respective classical models.Table S3 reports noise amplitude σ in units of 10^-2; each table’s final row gives experimental reference values.Source: Table S1 · p. 34, Table S2 · p. 35, Table S3 · p. 35