Theory of gating in recurrent neural networks

Recurrent neural networks (RNNs) are powerful dynamical models, widely used in machine learning (ML) for processing sequential data, and in neuroscience, to understand the emergent properties of networks of real neurons. Prior theoretical work in understanding the properties of RNNs has focused on networks with additive interactions. However, gating -- i.e. multiplicative -- interactions are ubiquitous in real neurons, and gating is also the central feature of the best-performing RNNs in ML. Here, we study the consequences of gating for the dynamical behavior of RNNs. We show that gating leads to slow modes and a novel, marginally-stable state. The network in this marginally-stable state can function as a robust integrator, and unlike previous approaches, gating permits this function without parameter fine-tuning or special symmetries. We study the long-time behavior of the gated network using its Lyapunov spectrum, and provide a novel relation between the maximum Lyapunov exponent and the relaxation time of the dynamics. Gating is also shown to give rise to a novel, discontinuous transition to chaos, where the proliferation of critical points (topological complexity) is decoupled from the appearance of chaotic dynamics (dynamical complexity), in contrast to a seminal result for additive RNNs. The rich dynamical behavior is summarized in a phase diagram indicating critical surfaces and regions of marginal stability -- thus, providing a map for principled parameter choices to ML practitioners. Finally, we develop a field theory for gradients that arise in training, by combining the adjoint formalism from control theory with the dynamical mean-field theory. This paves the way for the use of powerful field theoretic techniques to study training and gradients in large RNNs.