Trajectory-Induced Modes: Decomposing Layerwise Representation Flow in Deep Neural Networks

Deep neural networks progressively transform internal representations across depth, yet how representational structure persists through this process remains poorly understood. Existing analyses characterize layerwise content, cross-layer similarity, or output decodability, but do not directly characterize persistence as a property of representation flow. We introduce Trajectory-Induced Modes (TIMs), which models stimulus-specific hidden-state trajectories through a reduced spectral operator and uses eigenvalue magnitude as an operational measure of cross-layer persistence, distinguishing attenuated, preserved, and amplified components. Across 108 frozen pretrained models spanning vision, audio, and language, TIMs reveals prevalent persistent modes with systematic modality-dependent profiles. Language models are dominated by highly persistent components, vision models exhibit broader mixtures of attenuation and amplification, and audio models show an intermediate pattern. Crucially, TIM persistence is not merely a spectral property. Higher-persistence components retain substantially more information aligned with terminal representations, and removing them from intermediate activations produces larger task-loss increases than removing transient or random TIM-subspace components. Higher persistence is also associated with stronger cortical alignment in audio and language, with more limited gains in vision. These findings remain consistent across perturbation, layer subsampling, rank variation, and residual-structure controls. Together, these results establish cross-layer persistence as a measurable and functionally meaningful property of deep representations, offering a new perspective on how information is transformed across network depth.