Fig 1.
Calculation of shared dimensionality and percent shared variance.
(A) Factor analysis partitions the spike count covariance matrix into shared and independent components. (B) Shared dimensionality (dshared) was defined as the number of eigenvectors (modes) required to explain 95% of shared variance. (C) The percent shared variance for an individual neuron is defined as the neuron’s shared variance divided by its total variance. We then averaged this across all neurons to obtain an overall percent shared variance.
Fig 2.
Neural populations and network models.
(A) Neural activity was recorded using a Utah array implanted in V1 of macaque monkeys during presentation of an isoluminant gray screen. (B) Clustered network consisted of 4000 excitatory neurons grouped into 50 clusters of 80 neurons. Triangles represent excitatory neurons and circles represent inhibitory neurons. Clusters had high within-cluster connection probability relative to between-cluster connection probability. Connection probabilities between excitatory and inhibitory neurons indicated above corresponding arrow. (C) Non-clustered network consisted of 4000 excitatory neurons with homogeneous connection probabilities.
Fig 3.
Scaling properties of shared dimensionality and percent shared variance with neuron and trial count in V1 recordings.
The dshared and percent shared variance over a range of (A) neuron counts and (B) trial counts from population activity recorded in V1. Each triangle represents the mean across single samples from each of three arrays. Error bars represent one standard error across the three arrays.
Fig 4.
Modes of shared variability in V1 recordings.
(A) Left: Modes of in vivo recordings. Each column of the heatmap is an eigenvector of the shared covariance matrix computed from a set of 80 neurons and 1200 trials. Columns are ordered with modes explaining the most shared variance on the left. Neurons (rows) are ordered with highest mean firing rate at the top to lowest mean firing rate at the bottom. Right: Percent of shared variance explained by each mode. Plot shows mean across three arrays. Trends were similar in each of the three arrays. (B) Principal angles between modes in in vivo recordings for 20- (black), 40- (blue), or 60-neuron (red) analyses and corresponding neurons from 80-neuron analyses. Modes were identified by computing the eigenvectors of the shared covariances corresponding to neurons from the 20-neuron set. Triangles and error bars represent mean and standard error across the three arrays, respectively. Grey triangles represent principal angles (mean ± one standard deviation) between random 20-dimensional vectors. (C) Percent shared variance along each mode in the clustered network for 20-neuron analyses (blue) and 80-neuron analyses (black) used in (B). Note that the maximum number of modes (across the three arrays) in the 20-neuron sets was 9 and the maximum number of modes in the 80-neuron sets was 22. The recordings from each array had at least 5 modes. Curves and error bars represent mean percent shared variance and standard error for each mode across single samples from each of three arrays.
Fig 5.
Scaling properties of shared dimensionality and percent shared variance with neuron and trial count in spiking network models.
The dshared and percent shared variance over a range of (A) neuron counts and (B) trial counts from clustered (filled circles) and non-clustered (open circles) networks. Circles represent mean across the five non-overlapping sets of neurons and five non-overlapping sets of trials (25 total sets) and error bars represent standard error across all sets. Standard error was generally very small and therefore error bars are not visible for most data points.
Fig 6.
Scaling properties of shared dimensionality and percent shared variance with large neuron and trial counts in spiking network models.
The dshared and percent shared variance over a range of (A) neuron counts and (B) trial counts from clustered (filled circles) and non-clustered (open circles) networks. Insets zoom in on range of neurons used in in vivo recordings in Fig 3. Circles represent mean across the five non-overlapping sets of neurons and five non-overlapping sets of trials (25 total sets) and error bars represent standard error across all sets. Standard error was generally very small and therefore error bars are not visible for most data points.
Fig 7.
Influence of cluster representation on shared dimensionality and percent shared variance in the clustered network.
Dependence of (A) dshared and (B) percent shared variance on cluster representation in the set of sampled neurons. Analyses were performed for 50 neurons with 10,000 trials. ‘Rand’ indicates random sampling over all excitatory neurons. Circles represent mean across five non-overlapping sets of neurons and five non-overlapping sets of trials (25 total sets) of a single network with clustered structure. Error bars represent standard error across all sets.
Fig 8.
Modes of shared variability for spiking network models.
(A) Left: Modes of clustered network. Each column of the heatmap is an eigenvector of the shared covariance matrix computed from a set of 500 neurons and 10,000 trials. Columns are ordered with modes explaining the most shared variance on the left. Neurons (rows) are ordered by cluster (black lines indicate cluster boundaries), sorted with the highest mean firing rate clusters at the top. Note that due to random sampling there are an unequal number of neurons in each cluster. (B) Modes of non-clustered network. Same conventions as in (A), except rows are ordered by firing rate of individual neurons, with the highest mean firing rate at the top. The number of dimensions that maximized the cross-validated data likelihood was 100 in (A) and 110 in (B). (C) Percent of shared variance explained by each mode in (A). (D) Percent of shared variance explained by each mode in (B).
Fig 9.
Stability of modes of shared variability in network models.
(A) Principal angles between top five modes in clustered network for 20- (blue), 40- (black), or 80-neuron (red) analyses and corresponding neurons from 500-neuron analyses. Modes were identified by computing the eigenvectors of the shared covariances corresponding to neurons from the 20-neuron set. Plots show mean and standard error across 25 sets of 500 neurons and 10,000 trials. Grey circles represent principal angles (mean ± one standard deviation) between random 20-dimensional vectors. (B) Principal angles between modes in the non-clustered network. Same conventions as in (A). (C) Percent shared variance along each mode in the clustered network for 80-neuron analyses (red) and 500-neuron analyses (black) shown in (A). The maximum number of modes across the 25 sets was 75 for the 80-neuron analysis and 130 for the 500-neuron analysis. The two curves were nearly identical between modes 50 and 75 and therefore only the first 100 modes are shown. Curves represent mean percent shared variance for each mode across 25 sets. Error bars show standard error computed across the 25 sets. (D) Percent shared variance along each mode in the non-clustered network for the 80-neuron analyses (red) and the 500-neuron analyses (black) used in (B). Same conventions as in (C). The maximum number of modes across the 25 sets was 45 in the 80-neuron analysis and 130 for the 500-neuron analysis. Inset shows zoomed in vertical axis.