  We study the learning dynamics and the representations emerging in Recurrent
Neural Networks trained to integrate one or multiple temporal signals.
Combining analytical and numerical investigations, we characterize the
conditions under which a RNN with n neurons learns to integrate D(n) scalar
signals of arbitrary duration. We show, both for linear and ReLU neurons, that
its 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.
