Low-Dimensional Manifolds Support Multiplexed Integrations in Recurrent Neural Networks

We study the learning dynamics and the representations emerging in recurrent neural networks (RNNs) trained to integrate one or multiple temporal signals. Combining analytical and numerical investigations, we characterize the conditions under which an RNN with $n$ neurons learns to integrate $D(≪n)$ scalar signals of arbitrary duration. We show, for linear, ReLU, and sigmoidal neurons, that the internal state lives close to a $D$-dimensional manifold, whose shape is related to the activation function. Each neuron therefore carries, to various degrees, information about the value of all integrals. We discuss the deep analogy between our results and the concept of mixed selectivity forged by computational neuroscientists to interpret cortical recordings.