Neuronal circuits for robust online fixed-point detection

A fundamental problem in systems neuroscience is understanding how the brain learns the non-linear dynamics of the complex world and identifies the environment's state. Data-driven learning in such high-dimensional spaces requires lifting the curse of dimensionality. One way of reducing dimensionality relies on extracting from observed trajectories the underlying topological skeleton, i.e., fixed points connected by invariant manifolds. Thus, online extraction of fixed points from input trajectories is an important task. Whereas Dynamic Mode Decomposition (DMD) provides a framework to calculate fixed points offline, efficiently extracting fixed points online remains an unsolved problem. Moreover, implementing the online algorithm in a biological neuronal circuit requires it to satisfy more constraints, such as local update rules and no reliance on external memory.

In this work, we propose two biologically plausible neural networks with multi-compartment neurons for online fixed-point detection. The first neuronal circuit (circuit A) employs mostly local learning rules to update synaptic weights and estimate the linearized forward dynamics with high accuracy, and then utilizes the learned recurrent circuit to infer nearby fixed points. The second algorithm (circuit B) is a simpler recurrent neural network with synaptic weights learned by anti-Hebbian plasticity. Experiments show that circuit A can efficiently and robustly detect stable and unstable fixed points and all saddle points in switch-linear and non-linear systems. Though circuit B satisfies biological constraints more strictly, it converges slowly in practice. Our circuits are potential building blocks for a larger neuronal circuit for systems identification and model-based control.