We establish a mathematical correspondence between state space models, a state-of-the-art architecture for capturing long-range dependencies in data, and an exactly solvable nonlinear oscillator network. As a specific example of this general correspondence, we analyze the diagonal linear time-invariant implementation of the Structured State Space Sequence model (S4). The correspondence embeds S4D, a specific implementation of S4, into a ring network topology, in which recent inputs are encoded, as waves of activity traveling over the one-dimensional spatial layout of the network. We then derive an exact operator expression for the full forward pass of S4D, yielding an analytical characterization of its complete input-output map. This expression reveals that the nonlinear decoder in the system induces interactions between these information-carrying waves that enable classifying real-world sequences. These results generalize across modern SSM architectures, and show that they admit an exact mathematical description with a clear physical interpretation. These insights enable a new level of interpretability for these systems in terms of nonlinear oscillator networks.