For centuries, one of the deepest ambitions of physics has been the ability to read a system’s initial conditions and deduce its destiny. Push a block of known mass with a known force, and any student can calculate exactly where it will end up. But nature is rarely so cooperative. In chaotic systems, the tiniest perturbation of a starting configuration can snowball into radically different outcomes, and prediction collapses into guesswork. Now, a team of physicists at the University of Illinois Urbana-Champaign has reported something stranger still: a fully deterministic system, governed by simple and completely known rules, that is nonetheless genuinely unpredictable at the outset—and yet, remarkably, builds its own predictability as it evolves. The research, published in Nature Communications on September 11, 2026, suggests that predictability is not merely a property we impose on the world, but something that can self-organize within it.
The story began more than a decade ago, when Hyun Youk, a professor of physics at Illinois, started his laboratory with a broad interest in how complex dynamics can arise from simple deterministic rules, particularly in systems of living cells that interact to form spatial patterns. In 2020, his group computationally explored how cells might communicate by secreting molecules and sensing those released by their neighbors, and they discovered communication modes that matched the ways real cells in nature construct the very same kinds of spatial patterns. That work led the team to a cellular automaton, a discrete grid of individual cells, each occupying one of a finite number of possible states and updating according to fixed rules. Cellular automata are famous for generating astonishingly sophisticated behavior—patterns that appear to move, interact, and even self-replicate—from deceptively simple local rules, which is why they have long served as models of pattern formation in physics, computer science, and biology.
The automaton Youk’s team studied was designed to mimic living tissue. Each cell takes on one of four states, and after every timestep it changes state according to a gene-circuit-like rule that imitates how real cells secrete and sense signaling molecules. The grid also carries periodic boundary conditions, meaning each edge wraps around to meet its opposite edge, much like the classic video game character Pac-Man reappearing on the left side of the screen after exiting the right. When simulated, the automaton starts from a randomly chosen initial configuration and always terminates in exactly one of three final fates: a static configuration of same-state cells, a moving rectilinear wave, or a moving spiral wave. Because the number of possible configurations is finite—though astronomically large—the system is, by the strict mathematical definition, not chaotic. Yet a difference of just a single cell’s state between two initial configurations often produced entirely different fates, with nothing visually obvious to explain why.
To test whether this apparent complexity amounted to genuine unpredictability, the researchers turned to machine learning. They gave a suite of algorithms a deceptively simple binary-classification task: given an initial configuration, forecast whether the automaton would end in a static fate or a moving-wave fate. The results were startling. Half the time the algorithms guessed correctly, and half the time they were completely wrong—performance indistinguishable from a coin flip. Here was a deterministic system whose future could not be extracted from its past by any human or machine. “Despite the simplicity of our system, the cells self-organized in a way that no human or machine could initially predict,” said Elinor Kay, an Illinois Physics graduate student and co-author of the study. The finding pointed to a source of unpredictability fundamentally distinct from chaos, one that had gone largely unnoticed in discussions of determinism and forecasting.
The breakthrough came when the team changed how they looked at the system. Instead of visualizing the four cell states as colors, they reimagined them as arrows pointing up, down, left, or right. What emerged was striking: cores of oppositely pointing arrows, completely surrounded by closed loops of arrows tracing circular paths, forming structures the researchers called vortices. Each vortex could be labeled positive, negative, or neutral depending on its orientation and degree of circular tracing. Youk and his former student, lead author Lars Koopmans, found the vortices to be highly dynamic entities. “Lars and I were looking at a lot of simulation movies, and watching them eventually trained our eyes to see these vortices,” Youk recalled. “At first, they were oddities, but then we noticed that their abundances always decreased over time.”
The vortices behaved like Brownian particles, drifting across the lattice, merging, and annihilating one another after an initial period of rapid proliferation. Their fate carried decisive information: if every vortex vanished, the system settled into a static configuration or a rectilinear wave, whereas the survival of even a single vortex guaranteed a spiral wave. The team also noticed that every positive vortex was tethered to a negative one, and vice versa, by an unbroken string of same-state cells, and that positive and negative vortices always appeared and disappeared together. The system’s total charge, therefore, remained exactly zero throughout every simulation—an unexpected conservation law that was nowhere encoded in the automaton’s update rules. Some of this behavior traced back to the periodic boundary conditions: whenever a vortex loop crossed a lattice boundary, it reentered the grid and generated a new vortex core of opposite orientation and charge, explaining both the pairing of charged vortices and the conservation of total charge. “We originally imposed periodic boundary conditions because they made the simulations easier to implement,” Youk said. “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.”
Something deeper was at work, rooted in topology. A two-dimensional grid with periodic boundary conditions is topologically equivalent to a torus—the surface of a donut—which can be visualized by gluing the lattice’s opposite edges together. Because of this connectivity, certain strings of same-state cells wrap around the donut and cannot be untangled from its hole; these are known as noncontractible loops. The researchers tested whether these loops could serve as predictors of fate. In simulations destined for static or rectilinear-wave outcomes, the average count of noncontractible loops fluctuated near zero before the final vortex annihilation, dipping to zero with increasing frequency as the deciding moment approached. In principle, observing such rapid drops could signal an impending static or rectilinear fate before it actually occurred. But in simulations bound for spiral-wave fates, the loop count simply fluctuated near one, a condition that still permits either non-spiral outcome. The loops, in other words, could predict some fates but not all.
The decisive predictor emerged from a generalization of these loops that the team calls the winding field, a topological quantity capturing the lattice’s full spatial configuration—precisely the kind of information to which the best-performing machine-learning model, a convolutional neural network, had proven sensitive. Rather than examining only wrapped strings, the winding field considers entire connected regions of same-state cells that wrap around the torus, assigning to each cell a pair of numbers, a winding vector, that quantifies how many times its region winds around the donut. At the start of nearly every simulation, the field was zero. But within the first one percent of each run’s runtime, the field switched on as vortices materialized, then self-organized as the dynamics unfolded, expanding as vortices moved and annihilated. Crucially, the field then changed in one of two characteristic ways: it either abruptly engulfed the entire lattice, producing a static or rectilinear fate, or it expanded gradually, leaving persistent regions of zero winding field and producing a spiral wave. That distinction allowed reliable discrimination among all three fates. When the winding field was fed to the convolutional neural network, its accuracy climbed steadily over the course of each simulation, from no better than random guessing at the start to near perfection at the end. “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,” Kay said.
The discovery challenges traditional notions of predictability that sit at the intersection of determinism and uncertainty, and Youk’s laboratory is among the first to probe these ideas. “We didn’t expect to find a new source of unpredictability at all,” Youk admitted, describing the finding as an accidental discovery born of the lab’s general interest in self-organizing systems. “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,” Kay added. Much remains to be understood. The team hopes to explore the system’s topological details more fully and to pin down a rigorous definition of predictability, which they currently define operationally as the ability of a human observer or machine-learning model to forecast fate better than chance. “So far, we haven’t come up with a deep answer to why topology matters so much in our simulations,” Youk concluded. “Rigorously defining predictability and examining its properties are our next goals.” The work was funded by the National Institutes of Health through an NIH-NIGMS R35 grant under Grant No. GM147508 and by the National Science Foundation’s Science and Technology Center for Quantitative Cell Biology under Grant No. 2243257.
Subject of Research: Emergence of predictability in a deterministic, nonchaotic cellular automaton modeling living cell communication
Article Title: The emergence of predictability in an unpredictable system
Article References: The emergence of predictability in an unpredictable system. (n.d.). Original publication
Image Credits: AI Generated
DOI: Not provided
Keywords: cellular automata, predictability, chaos, topology, emergence, self-organization, machine learning, winding field, vortices, spatial patterns, determinism, Nature Communications
