Fig 1.
Recurrent circuit model of the primary visual cortex.
(A) The network architecture. An excitatory sub-circuit, spanning multiple hypercolumns, encodes a global image. Different local subcircuits encode different global images. Neurons compete via two inhibitory mechanisms: (i) suppression among neurons within the same feature channel across different hypercolumns, and (ii) divisive normalization among neurons representing different features within the same hypercolumn. The network input, , is the sparse code representation of an image [17], and its output,
, is the resulting steady-state response of the excitatory-inhibitory population. (B) Computational function. The circuit operates as an attractor network, transforming the sparse code of an input stimulus
that is generated via convolution by a dictionary of filters (lower left: three example filters) into neural representations (
). Right: trajectories of population activity of excitatory neurons to two stimuli (triangle = start, circle = end). Both converge to distinct fixed points, providing stable stimulus representations. The example input image is publicly available here, also see [21]. (C) The excitatory neighborhood. For a given target excitatory neuron (orange circle), its neighborhood consists of all excitatory neurons (gray circles) within an
square region, spanning all feature channels. (D) The inhibitory neighborhood. For a given target inhibitory neuron (blue circle), its presynaptic neighborhood of excitatory neurons (gray circles) mediates two functions: surround suppression via connections from an
spatial region and divisive normalization via connections from the same hypercolumn. (E) The neuronal activation function, which maps total synaptic input to a non-linear firing rate response. (F) Connection types and plasticity. The model includes excitatory-to-excitatory connections (E-E, initial weight wee), excitatory-to-inhibitory connections (E-I, weight wie), and inhibitory-to-excitatory connections (I-E, weight wei = −1). Only E-E connections are plastic.
Fig 2.
Tuning curve sharpening and familiarity suppression.
(A) Suppression of the average population response following familiarity training with both BCM and Hebbian rules. The curve represents the average response across all excitatory neurons and all stimuli, normalized according to r = r/rmax, where rmax is the peak response over time. (B) Population histograms of SI of individual neurons. Both learning rules result in a statistically significant decrease in the SI. The inset shows the convergence of the distribution’s mean over training epochs. Triangle markers indicate the mean of each distribution. (***: p<0.001, one-sided t-test against a mean of 0). (C) Stimulus-averaged change in population tuning curve. For each stimulus, neuronal responses were sorted in descending order and organized in a log scale. Both rules lead to a sharpening of the population tuning: responses of the most selective neurons are enhanced, while responses of moderately selective neurons are suppressed. Both rules produce comparable profiles. (D) Example of a single neuron’s tuning curve across 25 stimuli before (blue) and after (orange) familiarity training. The tuning curve sharpens, characterized by an enhanced response to the most preferred stimulus and suppressed responses to non-preferred stimuli. (E) Population histograms of the relative change in the lifetime sparsity. Both learning rules produce a statistically significant increase in lifetime sparsity, indicating that neuronal tuning becomes more selective. The inset shows the convergence of the distribution’s mean over training epochs. (*** : p<0.001, one-sided t-test against a mean of 0). (F) Scatter plot of maximum values of tuning curves pre- and post-training. Both rules result in an increase in peak response in the responsive neuron.
Fig 3.
Manifold transformation in the familiarity association experiment.
(A) Conceptual illustration of a visual feature manifold. Consider a smooth surface of local visual features on which a sparse coding dictionary provides discrete samples () [13]. Nuisance transformations of a visual stimulus, such as noise or changing viewpoints, correspond to flows along this manifold (blue dotted lines). The circles and triangles represent distinct sets of activated dictionary elements for two perceptually similar stimuli generated by the nuisance transformation (xi and x(n(i))), with arrows indicating response amplitudes. Because the dictionary is unordered, a smooth manifold flow can result in dissimilar sparse code activations (top). (B) Schematic of the manifold transform performed by the learned circuit. The model considers two types of relationships: a ’concept manifold’ (blue curve) representing the neural codes of distinct stimuli such as a target image xi (black dot) and a dissimilar stimulus xj (blue dot); and a ’variants manifold’ (red curve) representing the neural codes of variations of the same concept, such as xi and a related similar stimulus xn(i) (red dot). The learned circuit encoder,
, maps the sparse codes to a new representation, r. This transformation compresses the variants manifold, reducing the distance between xi and xn(i) to better reflect their geometric relationship. For visual simplicity, the concept manifold is depicted as unchanged by the encoder. (C) Example images from the stimulus set (CIFAR100, publicly available here, also see [21]), corrupted with 10%, 30%, and 50% salt-and-pepper occlusion noise. (D) Schematic of the neural manifold geometry in the familiarity association experiment. The orange rings represent the signal variance for stimuli at different noise levels, forming a “signal cone.” Each target image and its corrupted samples would form an opposite “noise cone”. The blue rings represent the noise variances inside the noise cone at varying noise levels. From a specific noise sample (the red dot), the level distance is marked by the light blue arrow, the residual distance is marked by the deep blue arrow, and the signal distance is marked by the orange arrow. (E) Across noise levels,
and
over the first 200 epochs show a two-phase trajectory: a minimum at epoch 30 (dashed line), and an overall decrease. This indicates a compression of the neural manifold in both level and residual directions. Ribbons represent the standard deviation across different target images. (F)
,
and
over first 200 epoch at all noise levels. The net decrease in both relative distances is primarily due to the larger increase in
. (G, H) The relative distances exhibit a reverse correlation with neuronal tuning selectivity and the magnitude of SI. The darkness of the scatter indicates the number of epochs, with deeper colors corresponding to earlier epochs. The solid lines represent the fitted regression lines, with the corresponding Pearson correlation coefficient noted aside.
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).