Manifold-constrained plasticity enables stable learning in recurrent neural circuits

The activity of large neuronal populations is often confined to low-dimensional manifolds that can drift over time, posing a challenge for learning rules that assume stable, full-rank representations. Here, we introduce SPLiT (Synaptic Projection Learning with intrinsic Tracking), a synaptic plasticity rule for recurrent neural networks that combines unsupervised manifold tracking with supervised learning. SPLiT uses an online Oja rule to continuously estimate the intrinsic low-dimensional activity subspace and performs normalized least-mean-squares learning in manifold coordinates. In this way, SPLiT keeps synaptic weights aligned with evolving population dynamics. Using a rate-based recurrent network, we show that SPLiT reliably learns time-varying target signals under both constrained dynamics, where activity is restricted to a fixed low-dimensional manifold, and unconstrained dynamics exhibiting changes in the dominant activity subspace. We show that manifold-constrained learning with SPLiT yields faster convergence, less sensitivity to recurrent gain, smaller weight updates, and is more robust to noisy teaching signals. Analytical results show that SPLiT learns the optimal decoder projected onto the instantaneous principal subspace and maintains bounded error under drift. Together, these findings provide a mechanistic account of how synaptic plasticity can leverage the structure of neural manifolds to enable stable and efficient supervised learning despite representational drift.Author summary: Neural activity in the brain can often be described by a small number of coordinated patterns that are far fewer than the total number of active neurons. These patterns, known as neural “manifolds,” can gradually change over time. This creates a fundamental challenge for learning: how can downstream circuits maintain stable behavior if the internal representations they depend on are drifting? In this work, we introduce a learning rule designed to address this problem by continuously tracking the dominant patterns of population activity and restricting learning to those patterns. Our model combines two complementary components: an unsupervised mechanism that estimates the current low-dimensional structure of neural activity, and a supervised mechanism that adjusts output connections to minimize task error within that structure. We show that this approach enables stable and efficient learning even as internal representations evolve. Across several tasks, including periodic signals, handwritten digit trajectories, and chaotic dynamics, our method outperforms existing learning rules when activity is confined to low-dimensional subspaces. It further predicts that learning within an intrinsic manifold should be faster and require smaller synaptic adjustments than learning that depends on activity outside that manifold. Together, these results suggest that tracking low-dimensional population structure provides a general strategy that allows neural circuits to function reliably despite ongoing representational drift.