Dynamical systems exhibiting Heteroclinic and Heteroclinic-like dynamics offer a new natural computation framework in which symbolic dynamics seem to arise from non-symbolic systems, in continuous time (as opposed to my Neural Turing Computation with R-ANNs project). Heteroclinic dynamics is hypothesized to be one of the key tools in the computational toolbox used by the brain, but its relevance goes beyond the lab: some of these heteroclinic systems can be built in hardware, others can be implemented in existing platforms for Neuromorphic Computing.
Importantly, very simple systems endowed with Heteroclinic dynamics can encode surprisingly complex information and produce meaningful computations!
As an illustration, I include a video of a simple visualizer I made to show the heteroclinic switching for a simple system of 5 interconnected LIF oscillators, as described in Neves, F.S. & Timme, M. (2012). In the video, every node in the graph represents a synchronized state within the oscillators. For example, in node ababc, the first and the third oscillators are synchronized (the a’s are in the first and third place), the second and the fourth oscillators are synchronized (the b’s are in the second and fourth place), and the fifth oscillator is a singleton.
The oscillators de-synchronize and re-synchronize continuously but predictably, and the sequence of de- and re-synchronizations (i.e. the state switching) depends on the input. In this example system, it is possible to retrieve a partial ordering of the input currents simply by observing the switching sequence: it discriminates the smallest 2 currents from the greatest 3.
A Heteroclinic system with as little as 5 oscillators and three synchronization clusters (a, b and c) can thus perform a categorization task distinguishing meaningfully between 30 percepts! And due to some factorial considerations, the number of percepts which can be categorized by systems with more and more synchronization clusters grows really fast.