  Much of dynamical algebraic combinatorics focuses on global dynamical systems
defined via maps that are compositions of local toggle operators. The second
author and Roby studied such maps that result from toggling independent sets of
a path graph. We investigate a "toric" analogue of this work by analyzing the
dynamics arising from toggling independent sets of a cycle graph. Each orbit in
the dynamical system can be encoded via a grid of 0s and 1s; two commuting
bijections on the set of 1s in this grid produce torsors for what we call the
infinite snake group and the finite ouroboros groups. By studying related
covering maps, we deduce precise combinatorial properties of the orbits.
Because the snake and ouroboros groups are abelian, they define tilings of
cylinders and tori by parallelograms, which we also characterize. Many of the
ideas developed here should be adaptable both to other toggle actions in
combinatorics and to other cellular automata.
