The emergence of predictability in an unpredictable system

Researchers discover a fully deterministic, nonchaotic model that is initially unpredictable yet gives rise to predictability that gradually self-organizes through the model’s dynamics.

Predicting a system’s final outcome from its initial state is the ultimate goal for many physicists. In certain complex systems, however, this goal is thwarted by chaos, where even the subtlest tweaks of the initial state can lead to completely different fates, making the system almost impossible to predict.

In research published in the journal Nature Communications on Sept. 11, 2026, Illinois physicists have developed a model showing that unpredictability can also arise in nonchaotic systems. Despite being fully deterministic, the team’s model resists computational attempts to predict its final state based on its initial configuration. Remarkably, however, the scientists find the model’s dynamics gives rise to topological structure that can eventually be used as a reliable predictor of final fate, demonstrating that predictability itself can emerge over time.

Unpredictability in deterministic systems

Push a block of known mass with a known force. As any physics student knows, the block’s initial position allows us to predict its final position at some specified time. In deterministic systems such as this one, the final outcome is predetermined by the initial configuration and the rules governing its evolution through time.

In practice, however, things aren’t always so simple. In chaotic deterministic systems, for instance, although the governing rules may be known exactly, minuscule deviations in the systems’ initial configurations can become magnified, leading to different final states than one might have expected.

But it seems chaos isn’t the only source of unpredictability, an idea Illinois Physics Professor Hyun Youk stumbled across some years back.

“When I started my group 11 years ago,” he said, “we started working on models of living systems to understand how complex dynamics can arise from simple deterministic rules, specifically systems of living cells that interact with each other to form spatial patterns.

“In 2020, we computationally searched for ways that cells could communicate by secreting molecules and sensing those from other cells. We found communication modes that matched how cells in nature form the very same types of spatial patterns.”

The model Youk’s team focused on was a cellular automaton (CA), a discrete grid of individual cells, each of which is in one of a finite number of possible states and can change its state according to fixed update rules. Although these rules are often very simple, CAs can develop remarkably sophisticated patterns that appear to move, interact, and even self-replicate, behaviors that have inspired their use as models for pattern formation within the fields of physics, computer science, and biology.

In the 2020 work, Youk’s team defined their own CA, one whose cells resemble those found in living tissues. Each cell takes on one of four states, visualized as a color, and after every timestep, changes its state according to a special gene circuit, mimicking how real cells secrete and sense molecules around them. For ease of simulation, the CA also possesses periodic boundary conditions (PBCs), so that each lattice boundary matches up with its opposite boundary, similar to Pac-Man, who can reappear on the left edge of a screen if he goes past the right.

When simulated, the CA starts in a randomly chosen initial configuration before terminating in exactly one of three final configurations, or fates: a static fate composed of same-state cells; a moving rectilinear wave; or a moving spiral wave.

As it turns out, because the number of possible configurations is finite—albeit astronomically large—this CA is not chaotic. Nevertheless, fate is difficult to predict from the initial configuration alone, at least visually. Indeed, a difference of just a single cell state between two initial configurations often leads to different fates with little to suggest what causes the difference.

In the current study, to test if this complexity was genuine unpredictability, the researchers first turned to machine learning (ML). They gave various ML algorithms a simple binary-classification task: based on their initial configurations, forecast whether the CA will land in either a static or moving-wave fate. Surprisingly, half of the time the algorithms got the fates right, and the other half of the time they got them completely wrong—no better than random guessing! This deterministic system was genuinely unpredictable.

Illinois Physics graduate student and co-author of the study Elinor Kay emphasized, “Despite the simplicity of our system, the cells self-organized in a way that no human or machine could initially predict.”

Reimagining states reveals hidden vortices

To understand where this unpredictability was coming from, Youk’s team changed their perspective. Rather than using colors, they reimagined the cells’ states as arrows pointing in one of four directions—up, down, left, or right. What emerged was striking: “cores” of oppositely pointing arrows completely surrounded by closed “loops” of arrows tracing out circular paths, forming “vortices.” Based on its degree of circular tracing and orientation, each vortex was labelled as positive, negative, or neutral.

Youk and his former student, lead author Lars Koopmans, found that the vortices were highly dynamic.

Youk recalled, “Lars and I were looking at a lot of simulation movies, and watching them eventually trained our eyes to see these vortices. At first, they were oddities, but then we noticed that their abundances always decreased over time.”

In fact, in each simulation, after a short period of rapid proliferation, the vortices moved about like Brownian particles, merging and annihilating each other. Interestingly, if all vortices disappeared, the fate was either static or a rectilinear wave, whereas if at least one vortex survived, the fate was a spiral wave. They also noticed that every positive vortex was connected to a negative one, and vice versa, by an unbroken string of same-state cells.

Additionally, positive and negative vortices always appeared and disappeared together, so that the system’s total “charge” was conserved, remaining zero throughout the course of every simulation—an unexpected conservation law, as it wasn’t encoded into the CA’s update rules at all.

Periodic boundary conditions give rise to topological structure 

The researchers soon realized that some of the vortices’ behaviors could be traced back to the PBCs. For example, whenever any vortex loop crosses a lattice boundary, it reenters the lattice to generate a new vortex core of opposite orientation and charge, explaining why charged vortices come in pairs and why charge is conserved.

Youk mentioned, “We originally imposed periodic boundary conditions because they made the simulations easier to implement. But it turns out this seemingly trivial decision is actually very important. Without these boundary conditions, we wouldn’t have unending waves or pairing of oppositely charged vortices.”

But that wasn’t all. In fact, something deeper was going on. The PBCs gave rise to new structures, ones that become apparent if we examine the underlying lattice topology.

According to topology, a two-dimensional grid having PBCs is topologically identical to a torus, or donut shape. This can be visualized by “gluing” the lattice’s opposite edges together.

Because of this connectivity, some strings wrap around the donut and can’t be untangled from the donut’s hole, earning them the name noncontractible loops (NCLs). Could NCLs, with their origins in topology, be used as reliable predictors of fate?

Recalling how the annihilation of the final vortex pair signaled either a static or rectilinear-wave fate whereas vortex survival signaled a spiral wave, the researchers tested the NCLs’ predictive potential.

In simulations leading to static or rectilinear-wave fates, the average NCL count fluctuated near zero before annihilation, dropping to zero more frequently as it neared annihilation. In principle, then, observing rapid drops to zero signal a static or rectilinear-wave fate prior to the actual deciding event.

On the other hand, in simulations bound for spiral-wave fates, the average NCL count exhibited no other behaviors besides fluctuating near one, a situation that still allows for a static or rectilinear-wave fate to result. So although NCLs are predictors for non-spiral-wave-bound simulations, they’re not good predictors for spiral-wave-bound ones.

The winding field is a reliable predictor of fate

The researchers explored other promising predictor candidates. When these candidates were fed to the ML algorithms, their accuracies increased markedly as each simulation proceeded, pointing to predictive structures that emerged over time. Yet these structures emerged too late to be reliable.

Notably, the best-performing algorithm, a convolutional neural network (CNN), was sensitive to the lattice’s full spatial configuration. Following this lead, Youk’s team introduced a generalization of NCLs called a winding field, which captures this full configuration. Instead of looking at only wrapped strings, this topological quantity looks at entire connected regions of same-state cells that wrap around the torus. It assigns a pair of numbers, or winding vector, to each cell quantifying the number of times its connected region wraps around the torus.

 

The researchers then ran multiple simulations to see how the winding field evolved. At first, nearly every initial configuration had a zero winding field. But soon, within the first one percent of each simulation’s runtime, the field “turned on” as vortices materialized. As the dynamics unfolded, the field correspondingly self-organized, expanding as the vortices moved and annihilated each other.

Crucially, depending on the simulation, the field then changed in one of two ways. It either abruptly engulfed the whole lattice, resulting in a static or rectilinear fate, or gradually expanded so that regions of zero winding field persisted, resulting in a spiral wave—a difference that enables one to reliably discriminate among all three types of fate, making the winding field a reliable predictor!

What also stuck out was the way in which this predictability appeared. When the researchers fed the winding field to the CNN, its accuracy grew increasingly as the simulations ran on, performing no better than random guessing at their start to almost perfectly at their end. This result shows that, despite being initially unpredictable, the CA develops predictive structure that emerges over time.

As Kay summed up, “This shows that information is always present but slowly becomes accessible, which is very exciting because it implies that there’s a greater order just below our grasp.”

At the intersection of determinism and uncertainty

That unpredictability can still arise in systems having predetermined outcomes and also exhibit emergent predictive structure is challenging traditional notions of predictability, notions lying at the intersection of both determinism and uncertainty. Youk’s lab is among the first to probe these new ideas.

“We didn’t expect to find a new source of unpredictability at all,” Youk admitted. “It was an accidental discovery that arose from my lab’s general interest in self-organizing systems—a good example of how research can often take unexpected turns.”

Kay added, “What’s most exciting is the way our results exemplify the rich dynamics and layers of order that can form out of only local rules.”

In future studies, the team hopes to fully explore their system’s topological details and pin down the right definition of predictability.

Youk concluded, “So far, we haven’t come up with a deep answer to why topology matters so much in our simulations. And while we have an operational definition of predictability based on the ability of a human observer or machine-learning model to predict fate better than chance, this definition hasn’t been mathematically formalized yet, so rigorously defining predictability and examining its properties are our next goals.”

 

This research was funded by the National Institutes of Health through an NIH-NIGMS R35 grant under Grant No. GM147508 and the National Science Foundation’s Science and Technology Center for Quantitative Cell Biology under Grant No. 2243257.