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The authors have declared that no competing interests exist.

Recent studies have applied dimensionality reduction methods to understand how the multi-dimensional structure of neural population activity gives rise to brain function. It is unclear, however, how the results obtained from dimensionality reduction generalize to recordings with larger numbers of neurons and trials or how these results relate to the underlying network structure. We address these questions by applying factor analysis to recordings in the visual cortex of non-human primates and to spiking network models that self-generate irregular activity through a balance of excitation and inhibition. We compared the scaling trends of two key outputs of dimensionality reduction—shared dimensionality and percent shared variance—with neuron and trial count. We found that the scaling properties of networks with non-clustered and clustered connectivity differed, and that the

We seek to understand how billions of neurons in the brain work together to give rise to everyday brain function. In most current experimental settings, we can only record from tens of neurons for a few hours at a time. A major question in systems neuroscience is whether our interpretation of how neurons interact would change if we monitor orders of magnitude more neurons and for substantially more time. In this study, we use realistic networks of model neurons, which allow us to analyze the activity from as many model neurons as we want for as long as we want. For these models, we found that we can identify the salient interactions among neurons and interpret their activity meaningfully within the range of neurons and recording time available in current experiments. Furthermore, we studied how the neural activity from the models reflects how the neurons are connected. These results help to guide the interpretation of analyses using populations of neurons in the context of the larger network to understand brain function.

Dimensionality reduction methods (for review, see [

While our ultimate goal is to understand the population activity structure of

Spiking network models that balance excitation and inhibition have been widely studied to understand the mechanisms underlying spike timing variability and correlated variability across neurons (e.g., [

To study single-trial population activity, we used factor analysis (FA), a linear dimensionality reduction method [

In this work, we leveraged this separation of variability into shared and independent components to quantify two aspects of the population activity structure: (1) shared dimensionality, which is a measure of the complexity of the shared activity co-fluctuations, and (2) percent shared variance, which measures the prominence of the shared component in the spiking activity. These measures generalize the ideas behind spike count correlation [

We studied the scaling trends of shared dimensionality and percent shared variance with increasing numbers of neurons and trials. To perform this analysis we used spontaneous activity recorded in the primary visual cortex (V1) of anesthetized macaque monkeys and activity generated from non-clustered and clustered spiking network models. In addition, we assessed the effects of network structure on these metrics and found substantial differences in the scaling properties of the clustered and non-clustered networks, with the clustered network showing many similarities to the

A standard approach to studying pairwise relationships in populations of simultaneously recorded neurons over many trials is to compute the spike count covariance of the population (_{shared} (_{shared} measures the complexity of the shared activity co-fluctuations, or the number of modes, among the neurons. For example, if _{shared} equals one, then all of the shared variance in the population can be attributed to a single mode, whereas larger _{shared} indicates the presence of multiple modes of shared variability. Percent shared variance measures the prominence of the shared component in the spiking activity, and is computed based on how much of each neuron’s activity co-varies with the activity of at least one other recorded neuron.

(A) Factor analysis partitions the spike count covariance matrix into shared and independent components. (B) Shared dimensionality (_{shared}) 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.

Before studying how _{shared} and percent shared variance scale with the number of neurons and trials, it is important to recognize that these two measures are distinct. To see this, consider a population of neurons modulated by a common multiplicative gain factor that accounts for a large portion of the variance [_{shared} would be one, and the percent shared variance would be high. On the other hand, suppose that a population is grouped into pairs of neurons, where each pair is modulated by a distinct multiplicative gain factor that accounts for a small portion of total variance. In this scenario, _{shared} would be high (roughly equal to half the number of neurons in the population), and the percent shared variance would be low. Similar scenarios can be imagined that result in low _{shared} and low percent shared variance or high _{shared} and high percent shared variance. These scenarios show that _{shared} and percent shared variance do not necessarily change together.

Below, we first assess the _{shared} and percent shared variance of

(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.

We first studied how _{shared} and percent shared variance scale with neuron count for _{shared} and percent shared variance for each neuron or trial count. We expected _{shared} and percent shared variance to either saturate or to increase with increasing neuron or trial count. Saturating _{shared} would suggest that we have identified all of the modes for the network (or networks) sampled by the recording electrodes and increasing _{shared} would suggest that additional modes are being revealed by monitoring additional neurons or trials. We found that _{shared} increased with neuron count (_{shared} and stable percent shared variance (_{shared} and percent shared variance remained the same for spike count bins ranging from 200 ms to 1 second (

The _{shared} 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.

Recent studies have explored how different modes of population activity are used during different task epochs [

We first examined the modes for the

(A) Left: Modes of

To measure how modes of shared variability from

In the previous sections, we identified trends in _{shared} and percent shared variance using

We consider recurrent spiking network models with distinct excitatory and inhibitory populations whose synaptic interactions are dynamically balanced [

In the particular clustered network studied here, each cluster resembles a bistable unit with low and high activity states that lead neurons in the same cluster to change their activity together. We expected to identify dimensions that reflected these co-fluctuations within clusters, resulting in _{shared} bounded by the number of clusters (i.e., 50 dimensions) and high percent shared variance. In contrast, the non-clustered network lacks the highly correlated activity seen in the clustered network [_{shared} being zero. Small amounts of shared variance relative to total variance would result in low percent shared variance and either low or high _{shared} depending on the multi-dimensional structure of the shared variance.

To test how clustered connectivity affects the population activity structure and to understand how the population-level metrics scale with the number of neurons and trials, we performed the following analysis. We applied FA to spike counts, from non-clustered and clustered network simulations. Each spike count was taken in a 1-second bin of simulation time, which we refer to as a ‘trial’ in analogy to physiological recordings. We then increased the neuron count, as we did in _{shared} with neuron count in the clustered network and a _{shared} of zero for all neuron counts in the non-clustered network (_{shared} and high percent shared variance in the clustered network and zero _{shared} and percent shared variance in the non-clustered network.

The _{shared} 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.

We next investigated how _{shared} and percent shared variance change for an increasing number of trials, with the number of model neurons fixed at 80 to match the analyses shown in _{shared} and percent shared variance would increase to a saturation point after which enough trials would be available to reliably identify all of the modes of shared variability. In the clustered network, we observed that _{shared} (_{shared} and percent shared variance for 80 neurons sampled from the clustered network. In the non-clustered network, we observed zero _{shared} and percent shared variance for all trial counts. Therefore, of the two networks studied, only the clustered network demonstrated population-level shared structure within the range of trials obtained in the

Comparing the model network results (_{shared} and saturating percent shared variance with neuron and trial count. Note that we did not tune network parameters (e.g., firing rates, number of clusters, etc.) in the clustered network to match the _{shared} or percent shared variance to match in the two cases. However, the trends of increasing _{shared} with neuron and trial count accompanied by stable percent shared variance suggest that, in both cases, the population activity is largely governed by a few dominant modes that are well characterized within the range of neurons and trials obtainable with current recording technology.

To better understand how the outputs of dimensionality reduction for limited sampling reflect larger portions of the network, we investigated how the trends from _{shared} in the clustered network saturated with roughly 100 neurons, whereas _{shared} in the non-clustered network continued to increase with neuron count (_{shared} and percent shared variance for both networks (_{shared} and percent shared variance with few neurons relative to the network size. This likely stemmed from the fact that neurons from the same cluster varied together. Therefore, we were able to identify the majority of shared variance once multiple neurons were sampled from most clusters. That result contrasts with our observation of increasing dimensionality and low shared variance in the non-clustered network, which lacks modes describing activity from groups of co-varying neurons. It is therefore likely that we identified many modes that each explain small amounts of variability. We investigate this in greater detail below.

The _{shared} 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

To study the effects of large trial count on population-level metrics, we computed _{shared} and percent shared variance for 80 neurons while varying the trial count up to 20,000 (_{shared}. It is clear from this result that trial counts within the experimental regime were insufficient to identify shared dimensions, but that additional trials revealed shared dimensions. Percent shared variance followed a similar trend, with zero percent shared variance below 5,000 trials, as expected given zero _{shared}. These results show that many more trials were required to identify the small amounts of shared variability in the non-clustered network compared to the clustered network.

The above analyses showed substantial differences between the two model networks. In the clustered network, the shared population activity structure was salient (approximately 90% of the raw spike count variability was shared among neurons) and defined by a small number of modes (approximately 20 modes), all of which could be identified using a modest number of neurons and trials. In contrast, for the non-clustered network, the shared population activity was more subtle (approximately 20% of the raw spike count variability was shared among neurons), distributed across many modes, and required large numbers of trials to identify.

So far we have sampled neurons at random from the model networks. However, in our _{shared} and percent shared variance. To investigate the effects of non-random sampling procedures, we varied the number of clusters represented in a 50-neuron set. We found that _{shared} generally increased with cluster representation (_{shared} exceeded cluster representation for low cluster counts, likely representing less dominant modes that are washed out when more clusters are represented. For the 50-cluster case, _{shared} was 22.5 ± 0.17 (mean ± standard error), roughly equal to the saturation value for the clustered network of 21.8 ± 0.25 (mean ± standard error, 500-neuron, filled data point in _{shared}, percent shared variance remained stable across a wide range of cluster representation (

Dependence of (A) _{shared} 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.

In our analyses, the distribution of samples across the network influenced the observed _{shared}, with broader distributions (i.e., with more clusters being represented in the sample) better characterizing the overall network. With only 50 carefully chosen neurons, we obtained the saturation _{shared} and percent shared variance shown in

As we did with the ^{th} mode did not reflect the cluster identities of the neurons, but instead described more subtle interactions between neurons both within and across clusters. Additionally, neurons or clusters of neurons with higher mean firing rates tended to be involved in more dominant modes in both model networks (

(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).

Comparing the modes for the model networks (

Each of the modes in _{shared}, defined as the number of modes needed to explain 95% of the shared variance (see _{shared} to equal the number of clusters (50) in the clustered network, we found that _{shared} was 20 because the top 20 modes were sufficient for explaining 95% of the shared variance.

We then assessed how the modes of shared variability changed direction in the multi-dimensional population activity space with increasing neuron count, using the same procedure as with the

(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.

Analyzing the modes of shared variability allows us to better understand trends observed in _{shared} increases because each dimension explains some amount of (positive) shared variance. However, for the non-clustered network, we found that _{shared} increased without an associated increase in percent shared variance. This can be understood by the fact that the dominant modes changed as more neurons were added to the analysis (_{shared} to increase without an associated increase in percent shared variance.

For the _{shared} increases without an associated increase in percent shared variance (

In summary, for the clustered network, the dominant modes of shared variability among the original neurons remained stable as neurons were added to the analysis. In contrast, the non-clustered network modes changed as neurons were added to the analysis and tended to become less dominant (i.e., the percent shared variance along those modes decreased). The results shown here for the clustered network are largely consistent with the results for

In this study, we used V1 recordings and spiking network models to understand how the results obtained using dimensionality reduction methods generalize to recordings with larger numbers of neurons and trials, as well as how these results relate to the underlying network structure. We found that recordings of tens of neurons and hundreds of trials were sufficient to identify the dominant modes of shared variability in both

We focused on variability that is shared among simultaneously-recorded neurons. Shared variability has been widely studied due to its implications for the amount of information that is encoded by a population of neurons [_{shared}) would remain at zero because independent neurons have no shared variance.

We used FA to partition the raw covariance matrix into shared and independent components and measured the dimensionality of the shared component [

In this work, we studied spontaneous activity during

We focused on trends in shared dimensionality and percent shared variance. Specific shared dimensionality and percent shared variance values obtained for the model networks likely depend on model parameters, including the number of clusters, the synaptic weights, and the probability of synaptic connection. We used the parameters described in [

While many existing models reflect various aspects of neural activity, we studied two balanced spiking network models, which can be viewed as representing the two ends of a connectivity spectrum. At one end is the classic balanced network with homogeneous connectivity which has been studied for decades (i.e., the non-clustered network) [

Comparisons between network models and

Recent developments in neural recording technology are making it feasible to record from orders of magnitude more neurons simultaneously than what is currently possible (e.g., [

All experimental procedures followed guidelines approved by the Institutional Animal Care and Use Committees of the Albert Einstein College of Medicine at Yeshiva University and New York University, and were in full compliance with the guidelines set forth in the US Public Health Service Guide for the Care and Use of Laboratory Animals.

Details of the neural recordings were reported previously [

We implanted multi-electrode arrays in primary visual cortex (V1) in three hemispheres of two anesthetized macaque monkeys. We recorded spontaneous activity for 20–30 minutes while a uniform gray screen was displayed on a computer monitor in front of the animal. Recorded waveform segments were sorted off-line using a competitive mixture decomposition method [

Network simulations were performed using the same parameters as described in [_{syn} is the total synaptic input to the neuron. The membrane time constant _{th} = 1, after which the neuron was reset to _{re} = 0 with an absolute refractory period of 5 ms. Here we have normalized the voltages to range between 0 and 1, with a value of 0 corresponding to roughly -65 mV and a value of 1 corresponding to roughly -50 mV, as in biological networks.

For each neuron, total synaptic input current, ^{y}, is denoted by *. The filter, ^{y} is described by the equation
_{2} = 3 ms for excitatory synapses and 2 ms for inhibitory synapses and _{1} = 1 ms for all synapses, consistent with fast glutamatergic and GABAergic synaptic transmission. These values were selected to reproduce the effects of fast-acting excitatory and inhibitory neurotransmitters. One trial was defined as one second of time according to the simulation.

We simulated two network structures, one with homogeneous connection probability across excitatory neurons (non-clustered network) and one with clusters of high within-cluster connection probability (clustered network). In the non-clustered network, synaptic strengths were ^{EE} = 0.024, ^{EI} = −0.045, ^{IE} = 0.014, and ^{II} = −0.057. The probability of synaptic connection between excitatory neurons projecting onto other excitatory neurons occurred with probability ^{EE} = 0.2. All other types of synaptic connections occurred with probability ^{EI} = ^{IE} = ^{II} = 0.5. These connection probabilities are similar to the connection probabilities seen in cortex [

In the clustered network, the probability of connection between excitatory neurons depended on whether two neurons were in the same cluster or in different clusters, with ^{EE} = 0.2). This ratio of

We used factor analysis (FA) to characterize the population activity structure [

FA is defined by:

As shown in ^{T}, and an independent component, ^{T} indicates the number of latent variables needed to describe the shared covariance. If

In this study, we used two key metrics derived from the shared covariance matrix to describe population activity: shared dimensionality (_{shared}) and percent shared variance. First, we sought to measure the number of dimensions in the shared covariance as a metric for the complexity of the population activity. We followed a two step procedure to obtain this metric. We first found the _{shared} as the number of dimensions that were needed to explain 95% of the shared covariance, ^{T}. We did this for the following reasons. In simulations, we found that, when training data were abundant, there was not a strong effect of overfitting and the cross-validated data likelihood curve saturated at large dimensionalities. As a result, the peak data-likelihood appeared at widely varying dimensionalities along the flat portion of the curve, leading to variability in the value of _{shared} as described above provided a more reliable estimate of dimensionality across analyses, even if it may have been slightly smaller than the true dimensionality.

Second, we measured the amount of each neuron’s variance that was shared with at least one other neuron in the recorded population (^{th} neuron was computed as:
_{k} is the ^{th} row of the factor loading matrix and Ψ_{k} is the independent variance for the ^{th} neuron. The values reported in this paper (see Figs

For Figs ^{th} mode was computed as:
_{i} is the eigenvalue of ^{T} corresponding to the ^{th} mode and

In Figs ^{th} mode for the ^{th} neuron as:
_{ik} is the ^{th} entry in the ^{th} eigenvector of ^{T}. We then averaged this value across all neurons. This allowed us to break down the contribution of each mode to percent shared variance and illustrated a contrast between the clustered and non-clustered network models. Note that

A central goal of this study was to determine how _{shared} and percent shared variance vary with neuron and trial counts for _{shared} and percent shared variance. To increase the neuron counts, we augmented the next smaller sample of neurons with additional randomly selected neurons. For example, we first randomly selected 10 neurons, computed _{shared} and percent shared variance for this set, and then added 10 additional randomly-selected neurons to obtain a new sample of neurons. For the

For the model networks, we repeated this procedure 25 times at each neuron count using 5 non-overlapping sets of neurons and 5 non-overlapping sets of trials, using either 1200 or 10,000 trials for each neuron count. All neuron samples were obtained exclusively from the excitatory populations in the two networks. Since inhibitory neurons in both model networks did not have clustering structure, exclusion of this population allowed us to isolate the impact of clustering on the observed trends.

We studied how _{shared} and percent shared variance change with trial count by performing the same procedure as described above, except that trials were increased rather than neurons. For

We sought to assess how the identified modes of shared variance change with increasing neuron count. We could do this by measuring the angles between corresponding modes; however as neuron count increases, modes can change order, causing direct angle measurements between modes with the same index to overestimate the difference between the two sets of modes.

To overcome this, we measured principal angles between sets of modes [^{T}. We then computed the principal angles between modes from two different conditions (i.e., different numbers of neurons). Since the vector defining each mode had length equal to the number of neurons in the sample, we could not directly measure the principal angle between the eigenvectors of conditions with different neuron counts. To overcome this, we first computed ^{T} in each condition using only the rows of ^{T} in each condition, we then measured the angle between the modes as described above. Additionally, to restrict the analysis to the most dominant modes, only the five modes explaining the largest amount of shared variance were included in all principal angle measurements.

To study how the choice of sampled neurons influences _{shared} and percent shared variance, we sampled excitatory neurons from the clustered network by varying the number of clusters represented in a set of 50 neurons sampled. This was done by first randomly selecting

To assess the effects of spike count bin size on the observed results, we repeated the analyses in _{shared} and percent shared variance over a range of (A) neuron counts and (B) trial counts from population activity recorded in V1 with spike counts taken in 1000 ms (black), 500 ms (blue), and 200 ms (red) bins. Each triangle represents the mean across single samples from each of three arrays. Error bars represent one standard error across the three arrays. We observed trends of increasing _{shared} and stable percent shared variance with neuron and trial counts for all bin sizes, implying that the observed trends are consistent across a range of timescales. Furthermore we found that _{shared} and percent shared variance were lower for smaller bin sizes, consistent with previous work showing that noise correlations scale with bin size [

(EPS)

Factor analysis does not take into account sequential relationships between time bins. In other words, it assumes that the spike counts in consecutive time bins are independent of one another. We found that this assumption is valid for 1-second time bins using the following analyses. (A) We computed the autocorrelation of the spike counts in 1-second bins and found near-zero auto-correlation at all non-zero lags. Black line represents mean across 80 neurons from 3 arrays (_{shared} of 4.3, mean percent shared variance of 51.8%, labeled here as ‘Random’) using three different trial sampling methods: (1) sampling of adjacent trials (‘Skip 0s’), (2) sampling trials separated by 1 second (‘Skip 1s’), and (3) sampling trials separated by 2 seconds (‘Skip 2s’). We used 400 trials in each case. Triangles represent mean across 3 arrays and error bars represent one standard error across the three arrays. All of these measures produced qualitatively similar results across these sampling methods and no sampling method was significantly different from the random set (

(EPS)

(A) We applied the k-means algorithm to the rows of the modes matrix in

(EPS)

(A) Principal angles between top five modes in clustered network for 20- (blue), 40- (black), or 60-neuron (red) analysis and corresponding neurons from 80-neuron analysis. 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 80 neurons and 1200 trials. Grey circles represent principal angles (mean ± one standard deviation) between random 20-dimensional vectors. (B) Percent shared variance along each mode in the clustered network for the 20-neuron analyses (blue) and the 80-neuron analyses (black) shown in (A).

(EPS)

Some of the neural data were collected in the laboratory of Tony Movshon at New York University. We are grateful to Tony Movshon and his laboratory, as well as Tai Sing Lee and his laboratory, for research support.