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Emergence of collective criticality from future-anticipation maximisation

This paper proposes that the Future Anticipation Maximisation (FAM) principle, where individuals locally maximize the diversity of future perceptual states through mutual anticipation, spontaneously generates and maintains critical-like collective dynamics and morphological transitions in animal groups without requiring fine-tuning or explicit alignment rules.

Original authors: Takayuki Niizato

Published 2026-09-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Takayuki Niizato

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

In the natural world, from the flickering patterns of a school of fish to the shifting formations of a bird flock, life often seems to hover on a razor's edge between order and chaos. Scientists call this state "criticality." It is a condition where a system is so sensitive to small changes that a single individual's movement can ripple through the entire group, allowing for rapid information transfer and incredible adaptability. For decades, researchers have wondered how living things maintain this delicate balance. The prevailing idea was that nature must constantly fine-tune itself to stay at a specific, fixed point of criticality, much like a tightrope walker adjusting their balance to stay on a single wire. However, this view struggles to explain how animal groups can spontaneously change their entire shape—swarming, schooling, or spinning in circles—while remaining adaptable. If the rules of the group are constantly changing, how does the system stay balanced?

A new study by Takayuki Niizato at the University of Tsukuba offers a different perspective, suggesting that criticality is not a fixed destination but a continuous process of keeping options open. By simulating the behavior of one hundred to five hundred virtual agents, the researcher discovered that a simple rule based on anticipating the future can generate complex, critical-like dynamics without any need for fine-tuning. The agents in the simulation were not programmed to align with their neighbors or avoid collisions in the traditional sense. Instead, each agent was driven by a single goal: to choose the action that would give it the most diverse range of possible future views of its surroundings. This principle, called "Future Anticipation Maximisation," relies on the agents mutually guessing what their neighbors might do next, rather than predicting their own distant future alone.

The results of these simulations were striking. Without any explicit instructions to form specific shapes, the groups spontaneously transitioned between three distinct morphologies: a disorganized swarm, a coordinated school moving in one direction, and a milling formation where the group rotates like a vortex. These transitions happened naturally across a wide range of conditions, creating a "broad critical region" rather than a narrow, fragile point. The study showed that as the groups switched between these states, they exhibited a phenomenon known as "critical slowing down." This is a hallmark of critical systems where the time it takes for the group to recover from a disturbance increases, and the fluctuations in their movement become larger and more complex. The researchers found that these transitions were not random; they followed specific statistical patterns, including movements that resembled a "Lévy walk," a type of search pattern characterized by many short steps mixed with occasional long jumps, which is often observed in real animals searching for food.

What makes this finding particularly significant is how it challenges the idea that biological criticality requires a pre-set, perfect balance. In the simulation, the agents did not know they were creating a critical system. They simply acted to maximize the variety of future scenarios they could perceive through their neighbors' potential actions. This mutual anticipation meant that the future possibilities for each individual were constantly reshaped by the group itself. The study suggests that the "critical" state emerges naturally from this local interaction, rather than being imposed from the outside. The researchers observed that in some parameter settings, the groups switched rapidly between states, driven by temporal fluctuations, while in others, the switching was slower and driven by spatial fluctuations, yet both regimes maintained the hallmarks of criticality.

The research also delved into how these group-level changes connect to the behavior of individuals. By analyzing the movement of single agents within the different group shapes, the study found that the statistical properties of their motion depended heavily on the collective state. In a schooling formation, individual movement showed a clear pattern of long, straight runs mixed with shorter turns. In a milling formation, this pattern changed; movement along the circle remained similar to the schooling pattern, but movement toward or away from the center became more restricted. This indicates that the famous "Lévy-like" movement patterns seen in animal groups are not just a result of the group switching shapes, but are deeply rooted in how individuals fluctuate within those specific shapes. The geometry of the group itself shapes the individual's freedom of movement.

Crucially, the study argues against the notion that these complex behaviors require agents to have a deep, internal model of the world or to plan far into the future. Unlike other theories that suggest animals calculate the consequences of actions many steps ahead, this model showed that a simple, one-step mutual anticipation was sufficient to generate the entire spectrum of complex dynamics. The agents did not need to know the rules of the group or the concept of criticality; they only needed to evaluate the immediate combinatorial possibilities of their neighbors' next moves. This approach avoids the computational burden of predicting long-term futures, suggesting that nature may rely on these simple, local interactions to maintain adaptability.

The implications of this work extend beyond understanding fish or birds. It proposes a fundamental shift in how we view biological adaptability. Instead of seeing criticality as a static state that organisms must strive to maintain, the study presents it as a dynamic process that is continuously regenerated through local interactions. The "critical" nature of the system is not a fixed point on a map but a consequence of the system's ability to keep its future possibilities open. By maximizing the diversity of what they can perceive next, individuals inadvertently create a group that is highly sensitive to change, capable of rapid reorganization, and robust enough to survive in a changing environment. The study concludes that biological criticality is not a state of being, but a process of becoming, driven by the simple, relentless drive to keep options open.

Technical Summary: Emergence of Collective Criticality from Future-Anticipation Maximisation

Problem Statement
Biological criticality is widely recognized as a fundamental principle underlying adaptability in living systems, from neural networks to animal collectives. However, the mechanisms by which critical-like dynamics are generated and maintained under dynamically changing system constraints remain unresolved. This is particularly pertinent to animal collectives (e.g., bird flocks, fish schools), which spontaneously transition between distinct morphologies (swarming, schooling, milling), each characterized by different interaction structures. Conventional theoretical frameworks often assume a fixed critical point or stable morphology, failing to explain how critical-like dynamics persist when the collective's own constraints are continuously reshaping. Furthermore, existing future-oriented models (e.g., causal entropic force, free-energy principle) typically rely on extended individual prediction horizons or explicit internal generative models, which may not scale efficiently or explain how local decisions generate collective criticality without fine-tuning.

Methodology: Future Anticipation Maximisation (FAM)
The author introduces the Future Anticipation Maximisation (FAM) principle to address these gaps. Unlike previous approaches where agents independently predict future trajectories over extended horizons, FAM operates on a one-step combinatorial mutual anticipation framework.

Agent Dynamics: A system of $N$ self-propelled agents moves in a 2D continuous domain. At each time step, agents select a turning angle from a set of candidate actions.
Mutual Anticipation: To evaluate a candidate action, an agent does not predict a single future path. Instead, it considers the set of possible one-step actions of its Delaunay neighbours. It constructs a combinatorial space of future configurations based on the joint possibilities of its own action and the anticipated actions of its neighbours.
Perceptual Diversity Maximisation: For each candidate action, the agent calculates the number of unique future perceptual states (binary occupancy patterns of neighbours in discretized angular sectors) that could result from the combinations of neighbour actions. The agent selects the action that maximizes this diversity ( $U_{i,k}$ ).
Key Distinction: Future possibilities are not estimated independently from empirical distributions but are jointly constructed through reciprocal anticipation. The "context" is an emergent property of the interaction structure, not an externally specified condition.
Parameters: The study explores the parameter space defined by perceptual heterogeneity ( $\kappa_{per}$ ) and option variability ( $\kappa_{op}$ ) across group sizes ( $N=100$ to $500$).

Key Results

Spontaneous Morphological Transitions:
Without explicit alignment rules or tuning to a predefined critical point, FAM generates spontaneous transitions among swarming, schooling, and milling states. The phase diagram reveals distinct regions:
- Milling-dominant: High rotational order.
- Schooling-dominant: High alignment.
- Coexistence Regions (SMS/MS): Where the system stochastically switches between states.
- Switching Dynamics: Transitions occur via "flip" events where a stable structure (e.g., milling) becomes unstable, passes through a schooling state, and reconstructs with inverted orientation. These switching intervals exhibit heavy-tailed distributions, lacking a characteristic timescale.
Emergence of Critical-like Slowing Down:
The system exhibits critical-like slowing down near state transitions, characterized by the scaling relationship between relaxation time ( $\tau$ ) and susceptibility ( $\chi$ ): $\tau \propto \chi^\alpha$ .
- Two Distinct Regimes: The study identifies qualitatively different balances between temporal and spatial fluctuations:
  - SMS Phase (Swarm-Milling-Schooling): Dominated by temporal fluctuations (frequent transitions). The critical exponent converges with group size, suggesting finite-size effects.
  - MS/Milling-Dominant Phases: Dominated by spatial fluctuations (stable milling interrupted by schooling). The critical exponent remains invariant across group sizes ( $\alpha \approx 0.5-0.6$ ).
- Broad Critical Region: Unlike classical criticality confined to a narrow point, FAM produces a broad critical-like region where these dynamics emerge.
Lévy-like Collective and Individual Dynamics:
- Group Level: The centre-of-mass trajectories exhibit Lévy-like statistical behaviour (heavy-tailed step-length distributions) across all phases, though the underlying mechanisms differ. In the SMS phase, this emerges only at large group sizes; in MS and milling phases, it is stable across sizes.
- Individual Level: The study decomposes individual motion relative to the centre of mass.
  - Schooling: Exhibits clear Lévy-like statistics in flight-length distribution.
  - Milling: Exhibits anisotropic fluctuations. Tangential motion remains Lévy-like, while radial motion is constrained by the annular geometry (higher exponents), though it retains heavy tails.
- Link: Morphological switching (stop-and-go dynamics) connects state-dependent individual fluctuations to group-level Lévy dynamics.

Significance and Claims
The paper claims that biological criticality need not arise from fine-tuning to a fixed critical point or from fixed disorder (as in Griffiths phases). Instead, it can be continuously regenerated through local interactions under dynamically changing collective constraints.

Mechanism: Criticality emerges from the local principle of maximizing future perceptual diversity via mutual anticipation. This principle maintains "future possibilities" without requiring agents to optimize for a specific state or possess complex internal generative models.
Scalability: By relying on one-step combinatorial anticipation rather than extended planning horizons, FAM avoids the computational explosion associated with predicting future states in large populations.
Theoretical Shift: The findings suggest that biological criticality is better understood as a dynamic process of reorganizing interaction structures in response to changing contexts, rather than a static state. The "keeping options open" philosophy is realized not through independent prediction, but through the collective construction of future possibilities.

The study concludes that the FAM principle provides a mechanistic explanation for how critical-like dynamics, morphological transitions, and Lévy-like movements can coexist and emerge naturally in animal collectives without predefined critical states.

Technical Summary: Emergence of Collective Criticality from Future-Anticipation Maximisation

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