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Manifold transform by recurrent cortical circuit enhances robust encoding of familiar stimuli
Fig 4
A Locally Linear Dynamic Strategy for Manifold Learning.
(A) This schematic illustrates how a recurrent circuit can perform a manifold transform. On the left, the red and blue curves represent the variants θ manifold and the concept γ manifold, respectively. Similar to Fig 3B, the black dot is the representation of stimulus xi, and the red dot is the positive sample, the blue dot is the negative sample. The displacement between two attractors (
) on the manifold decomposes into locally linear (within the vicinity of the attractor, denoted by the dashed circle) and globally nonlinear components, and the network can reshape the manifold by modulating either. The local linear transform is the strategy that manipulates the local linear component to drive global geometric changes. The objective of the local linear strategy is to anisotropically adjust network’s recurrent gain, thereby stretching the signal geometry while compressing the noise (pre-training vs post-training). This can be achieved via two possible mechanisms (middle): selective spectrum modulation (top), which increases the spectrum of signal-oriented modes and decreases the spectrum of noise-oriented modes, or alignment modulation (bottom), which rotates the modes to be more signal-oriented. The blue and red arrows represent signal and noise directions in the input 
space (
and 
, respectively). The grey, solid arrow represents pre-training collective modes, and the black, dashed arrow represents post-training collective modes. (B) Evolution of the linearized 
and the 
across three different noise levels. For 10–30% noise, R shows an early drop (maximal linear compression, epoch 30) that parallels the full metrics; at low noise, R rebounds in the late training stage while the full metric remains reduced, indicating additional nonlinear contributions. At 50% noise, R increases, marking a regime not well captured by local linearization. (C) Corresponding evolution of linearized 
, 
and 
used to calculate R in panel B, which also mirrors full distances at low-to-mid noise levels in the early stage (compare to Fig 3F). D–F are shown at epoch 30, the time of maximal linear compression for 10–30% noise to isolate the locally linear mechanism. (D) Density distributions of the normalized change in modes’ alignment (
) to signal versus noise direction. For 10-30% noise, where the Hebbian network employs a local linear transform, learning selectively increases signal direction alignment (blue) while simultaneously decreasing noise direction alignment (orange). The noise alignment here represents the average of the level and residual alignments. (E) Density distributions of the normalized change in the collective mode spectrum (
) for signal-oriented modes versus noise-oriented modes. In contrast to alignment modulation, the change in the spectrum is largely non-selective. 
exhibits no significant change for both signal-oriented (blue) and noise-oriented (orange) modes. The signal-oriented modes are those that align more with the signal direction in the input 
space pre-training, and similarly for the noise-oriented modes. The noise alignment here represents the average of the level and residual alignments. (F) Large plot on the left: Joint density plot showing the relationship between the normalized sensitivity magnitude (
) and the normalized change magnitude (
) for the spectrum. The two small plots on the right show similar relations for signal and noise alignment (
and 
). The sensitivity quantifies the contribution of the change in mode alignment or spectrum to the linear relative distance. The three density plots reveal a consistent inverse relationship: learning primarily modifies modes that were initially insensitive (low sensitivity, high change, top-left cluster), while leaving highly sensitive modes largely unchanged (high sensitivity, low change, bottom-right cluster).
doi: https://doi.org/10.1371/journal.pcbi.1013587.g004