2023/09/09 by Peter Ashwin, Ashwin, Peter, Andrea Ceni +1 · 1 citation
Computer Science · Engineering · Neuroscience · #Advanced Memory and Neural Computing #Artificial Intelligence (cs.AI) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Neural Networks and Reservoir Computing #Neural dynamics and brain function
paper · pdf · doi:10.48550/arxiv.2309.04728
openalex publication_date 2023/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The echo index counts the number of simultaneously stable asymptotic responses of a nonautonomous (i.e. input-driven) dynamical system. It generalizes the well-known echo state property for recurrent neural networks - this corresponds to the echo index being equal to one. In this paper, we investigate how the echo index depends on parameters that govern typical responses to a finite-state ergodic external input that forces the dynamics. We consider the echo index for a nonautonomous system that switches between a finite set of maps, where we assume that each map possesses a finite set of hyperbolic equilibrium attractors. We find the minimum and maximum repetitions of each map are crucial for the resulting echo index. Casting our theoretical findings in the RNN computing framework, we obtain that for small amplitude forcing the echo index corresponds to the number of attractors for the input-free system, while for large amplitude forcing, the echo index reduces to one. The intermediate regime is the most interesting; in this region the echo index depends not just on the amplitude of forcing but also on more subtle properties of the input.