Operative dimensions in unconstrained connectivity of recurrent neural networks

Recurrent Neural Networks (RNNs) are commonly used models to study neural computation. However, a comprehensive understanding of how dynamics in RNNs emerge from the underlying connectivity is largely lacking. Previous work derived such an understanding for RNNs fulfilling very specific constraints on their connectivity, but it is unclear whether the resulting insights apply more generally. Here we study how network dynamics are related to network connectivity in RNNs trained without any specific constraints on several tasks previously employed in neuroscience. Despite the apparent high-dimensional connectivity of these RNNs, we show that a low-dimensional, functionally relevant subspace of the weight matrix can be found through the identification of operative dimensions, which we define as components of the connectivity whose removal has a large influence on local RNN dynamics. We find that a weight matrix built from only a few operative dimensions is sufficient for the RNNs to operate with the original performance, implying that much of the high-dimensional structure of the trained connectivity is functionally irrelevant. The existence of a low-dimensional, operative subspace in the weight matrix simplifies the challenge of linking connectivity to network dynamics and suggests that independent network functions may be placed in specific, separate subspaces of the weight matrix to avoid catastrophic forgetting in continual learning.