Chaos is defined as extreme (exponential) sensitivity to initial conditions. Phenomenologically it is sometimes difficult to distinguish chaos from randomness, especially at the Heisenberg level where the underlying drivers become truly random. Nevertheless chaos has a distinct and precise mathematical formulation.

In dynamic systems chaos is usually described by a "chaotic attractor" that allows us to visualize the general shape of the topology in phase space. Chaotic attractors are known for a wide variety of systems, however it is possible to arbitarily synthesize them by function approximation if the network can be controlled that way.

The library of primitives for this purpose is not as simple as a collection of Gaussians with a spectrum of means and variances. Nevertheless there are machine learning models that can closely match the Kuramoto parameters to the resulting dynamic behavior. It is not entirely clear yet what this actually tells us, since neural oscillators are usually far from sinusoidal, but it's a good place to start.

Measuring Chaos

How do you determine when a system is in a chaotic state ? One way is to do it theoretically, using Lyapunov exponents. However for this we must know and understand the network structure in advance, and it would be a whole lot more convenient to be able to put an electrode in the brain and just watch for a while. Such asymptotic tests do exist, like the 0-1 chaos test, but there are pretty stringent conditions under which the results can be reasonably evaluated. Tests like this are good for quick hit-or-miss guesstimates, but rigorous study probably requires a better treatment.