arXiv:2506.07919v1 Announce Type: cross Abstract: Memory and long-range temporal processing are core requirements for sequence modeling tasks across natural language processing, time-series forecasting, speech recognition, and control. While nonlinear recurrence has long been viewed as essential for enabling such mechanisms, recent work suggests that linear dynamics may often suffice. In this study, we go beyond performance comparisons to systematically dissect the functional role of nonlinearity in recurrent networks--identifying both when it is computationally necessary, and what mechanisms it enables. We use Almost Linear Recurrent Neural Networks (AL-RNNs), which allow fine-grained control over nonlinearity, as both a flexible modeling tool and a probe into the internal mechanisms of memory. Across a range of classic sequence modeling tasks and a real-world stimulus selection task, we find that minimal nonlinearity is not only sufficient but often optimal, yielding models that are simpler, more robust, and more interpretable than their fully nonlinear or linear counterparts. Our results provide a principled framework for selectively introducing nonlinearity, bridging dynamical systems theory with the functional demands of long-range memory and structured computation in recurrent neural networks, with implications for both artificial and biological neural systems.