Prospective Coding Improves Learning in Deep Continuous-Time Recurrent Networks

Temporal integration gives continuous-time recurrent networks memory, but in deep stacks it also delays bottom-up signals and attenuates top-down errors. We ask whether this failure mode can be addressed by making the bottom-up input to each layer prospective, replacing instantaneous layer inputs with local look-ahead signals that compensate for integration-induced lag. We develop Recursive Quadrature Filters (RQFs), a biologically motivated class of complex-valued temporal filters that remain equivalent to single-channel diagonal state-space models (SSMs). This equivalence makes RQF layers trainable with the same parallel scan and convolutional algorithms used for diagonal SSMs. At the implementation level, this prospective input amounts to a lightweight two-tap input update, making it a drop-in modification for RQFs and SSMs. For the resulting discrete-time network, we prove that spatial-only backpropagation in a deep RQF network with instantaneous bottom-up inputs yields error signals that decay geometrically with depth, whereas prospective-input coding restores order-one gradient flow. We evaluate prospective-input coding on the Speech Commands dataset using standalone RQF recurrent stacks across two feature representations, mel-frequency cepstral coefficients (MFCCs) and raw audio, and local and non-local credit-assignment strategies. Under spatial-only backpropagation, prospective-input coding improves validation accuracy from 65.7% to 84.9% on MFCC features and from 46.4% to 61.0% on raw audio. Under full backpropagation through time, the gains persist, improving accuracy from 89.4% to 93.2% on MFCC features and from 80.1% to 82.7% on raw audio. Taken together, our results identify prospective-input coding as a local, broadly applicable mechanism for improving credit assignment in multi-layer recurrent networks with a continuous-time substrate.