A Simple Grid Model Is Fully Deterministic Yet Unpredictable, Until Topology Reveals Its Fate

Predicting a system's final outcome from its starting state is the ultimate goal of many physicists. In chaotic systems that goal fails, because the smallest tweak to the initial state can lead to a completely different fate. Physicists at the University of Illinois Urbana-Champaign now report that unpredictability can also arise in systems that are not chaotic at all, and that predictability can emerge on its own as such a system evolves.

The team, led by physics professor Hyun Youk, built a model that is fully deterministic: the final state is set entirely by the starting configuration and the rules. Yet the model resists computational attempts to forecast its outcome. Even so, the researchers found that its dynamics generate a topological structure that can eventually be used as a reliable predictor of its fate.

Youk's group works on models of living systems, trying to understand how complex dynamics arise from simple rules, in particular how cells that interact form spatial patterns. In earlier work in 2020 they searched computationally for ways cells could communicate by secreting and sensing molecules, and found modes that matched how cells in nature form the same kinds of patterns. The model in the new study is a cellular automaton, a grid of cells each in one of a finite number of states that update by fixed rules. Each cell takes one of four states, shown as colors, and changes state according to a rule mimicking a gene circuit. The grid has periodic boundary conditions, so opposite edges are joined, like Pac-Man reappearing on the left after leaving the right.

Starting from a random configuration, the system always ends in one of three fates: a static pattern 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 not chaotic. Still, changing a single cell in the starting grid often produces a different fate with little to show why.

To test whether the difficulty was real, the team gave machine learning algorithms a simple task: look at the initial configuration and say whether the system ends static or in a moving wave. The algorithms were right half the time and wrong half the time, no better than guessing. "Despite the simplicity of our system, the cells self-organized in a way that no human or machine could initially predict," said graduate student Elinor Kay.

The breakthrough came from changing perspective. The researchers treated each cell's state as an arrow pointing up, down, left or right. They saw cores of oppositely pointing arrows surrounded by closed loops, which they called vortices, labeled positive, negative or neutral. The vortices moved like Brownian particles and annihilated in pairs, and the system's total charge stayed at zero, a conservation law that was not built into the rules. If every vortex vanished, the fate was static or a rectilinear wave. If at least one survived, the fate was a spiral wave.

The wraparound edges explained the pairing. A grid with those boundaries is topologically a torus, a doughnut, and some strings of cells wrap around it and cannot be untangled, which the team calls noncontractible loops. In runs headed for static or rectilinear fates, the loop count dropped to zero more often as annihilation neared, a warning sign before the deciding event. Youk said the boundary conditions, first chosen only to make simulations easier, turned out to matter deeply, because without them there would be no unending waves or vortex pairing.