## Abstract

The amount of information encoded by cortical circuits depends critically on the capacity of nearby neurons to exhibit trial-to-trial (noise) correlations in their responses. Depending on their sign and relationship to signal correlations, noise correlations can either increase or decrease the population code accuracy relative to uncorrelated neuronal firing. Whereas positive noise correlations have been extensively studied using experimental and theoretical tools, the functional role of negative correlations in cortical circuits has remained elusive. We addressed this issue by performing multiple-electrode recording in the superficial layers of the primary visual cortex (V1) of alert monkey. Despite the fact that positive noise correlations decayed exponentially with the difference in the orientation preference between cells, negative correlations were uniformly distributed across the population. Using a statistical model for Fisher Information estimation, we found that a mild increase in negative correlations causes a sharp increase in network accuracy even when mean correlations were held constant. To examine the variables controlling the strength of negative correlations, we implemented a recurrent spiking network model of V1. We found that increasing local inhibition and reducing excitation causes a decrease in the firing rates of neurons while increasing the negative noise correlations, which in turn increase the population signal-to-noise ratio and network accuracy. Altogether, these results contribute to our understanding of the neuronal mechanism involved in the generation of negative correlations and their beneficial impact on cortical circuit function.

## Introduction

During the past decade, it has become increasingly understood that the trial-by-trial variability in neuronal responses, or “noise,” is not independent but exhibits correlations (Shadlen and Newsome 1998; Bair et al. 2001; Kohn and Smith 2005; Averbeck et al. 2006; Gutnisky and Dragoi 2008; Hansen et al. 2012). Estimation of correlated neuronal firing is fundamental for understanding how populations of neurons encode sensory inputs. Indeed, the structure of noise correlations across the network has been shown to influence the available information in the responses of a population of cells (Abbott and Dayan 1999; Sompolinsky et al. 2001) and possibly limit behavioral performance (Abbott and Dayan 1999; Palmer et al. 2007; Chen et al. 2008). In addition, correlations between neurons can serve to constrain the possible schemes employed by the cortex to code and decode sensory stimuli. For instance, Fisher information (FI), which bounds the variance with which a stimulus feature encoded by a neuronal population can be estimated (Cox and Hinckley 1974), has been shown to be strongly influenced by the distribution of noise correlations across the population of cells (Abbott and Dayan 1999; Averbeck et al. 2006; Gutnisky and Dragoi 2008; Josić et al. 2009). Stimulus-triggered responses of sensory cortical neurons are often characterized by bell-shaped tuning curves (Hubel and Wiesel 1962; Miller et al. 1991; Ringach et al. 2002). For such population-coding schemes, it was shown that positive noise correlations could decrease coding accuracy estimated with FI (Abbott and Dayan 1999; Sompolinsky et al. 2001). In general, the reduction in the correlated variability of neural responses has been found to be beneficial for coding accuracy (Abbott and Dayan 1999; Gutnisky and Dragoi 2008; Ecker et al. 2010).

Most studies in sensory cortex have reported noise correlations that are positive in mean with average values between 0.1 and 0.3 (Zohary et al. 1994; Bair et al. 2001; Kohn and Smith 2005; Gutnisky and Dragoi 2008; Smith and Kohn 2008), but see Hansen et al. 2012. However, a positive mean correlation coefficient only reflects the fact that the distribution of correlation coefficients is biased toward positive correlation values, hence masking the fact that many neurons may exhibit negative correlations in their responses. Although negative correlation coefficients have been occasionally reported in electrophysiological studies (Lee et al. 1988; Zohary et al. 1994; Gutnisky and Dragoi 2008; Smith and Kohn 2008; Ecker et al. 2010), their impact on population coding was not adequately examined, but see Abbott and Dayan 1999 and Sompolinsky et al. 2001. That is, the distribution of negative correlation coefficients across a population of cells and their functional significance for network coding are still not fully understood.

We addressed this issue by performing multi-electrode recordings in the superficial layers of V1 of alert macaque and computed the distribution of positive and negative noise correlations and their impact on population-coding accuracy (Abbott and Dayan 1999; Sompolinsky et al. 2001). We found that network-orientation-discrimination performance was improved when the strength of negative correlations was increased even though the mean correlation coefficient was unchanged. To explore the possible network mechanism responsible for the emergence of negative correlations, we used a conductance-based integrate-and-fire model of orientation selectivity (Somers et al. 1995; Chelaru and Dragoi 2008). We found that the magnitude of negative correlations depends on the balance between the strength of the excitatory and inhibitory synaptic currents, that is, strong negative correlations are caused by strong inhibition and weak excitation. Despite decreasing neuronal firing rates, the increase in negative correlations causes an increase in network accuracy and the population signal-to-noise ratio.

## Materials and Methods

### Electrophysiological Recordings

All experiments were performed in accordance with protocols approved by the U.S. National Institutes of Health Guidelines for the Care and Use of Animals for Experimental Procedures and were approved by the Institutional Animal Care and Use Committee at the University of Texas-Houston Medical School. Two male rhesus monkeys (Macaca mulatta) were trained to fixate on a centrally located fixation point (0.2° in size) within a 1° fixation window. We used standard methods for single-unit extracellular recordings as described previously (Dragoi et al., 2002; Gutnisky and Dragoi 2008). Microelectrodes (tungsten/glass, 1–2 MΩ at 1 kHz, FHC, Inc.) were advanced transdurally through stainless steel guide tubes into V1. We recorded up to 8 units simultaneously in each session at depths between 200 and 400 μm (recording sites were located between 1 and 2 mm of each other; the majority of the neurons (>80%) were recorded on different electrodes. More than 80% of cells were complex. Single-unit isolation was assessed offline using waveform clustering based on parameters such as spike amplitude, width, valley, and peak. When a unit was isolated, its receptive field was mapped using an automatic procedure while the animal maintained fixation. Receptive field eccentricities ranged between 2° and 6° from the center of gaze (receptive field positions were reconfirmed at the end of the experiments). Monkeys were trained to fixate on a small spot (0.1°) presented on a video monitor placed 57 cm in front of the monkey. Once the animal achieved stable fixation for 300 ms, the visual stimulus was presented for 500 ms followed by another fixation period of 300 ms (Fig. 1A). Monkeys were required to hold fixation throughout stimulus presentation to earn a juice reward; the trial was automatically aborted if fixation instability exceeded 0.25° at any time during stimulus presentation and fixation period. Eye position was continuously monitored using an infrared eye-tracking system operating at 500 Hz (EyeLink II). The stimulus presentation, behavioral trials, and eye position control and recording were done using Psychophysics Toolbox and a programable behavioral control module (ECM, FHC, Inc.). Stimuli were oriented gratings flashed for 500 ms within the receptive fields of simultaneously recorded V1 neurons (8 orientations × 40 repeats; spatial frequency 2 cpd; 75% contrast; binocular presentation). In order to remove slow-wave fluctuations in responses across trials, all neurons underwent a detrending procedure (Bair et al. 2001) in which the spike counts for each trial were high-pass-filtered using a linear-phase Finite Impulse Response filter having a 0.1-normalized cutoff frequency.

Figure 1.

Structure of positive and negative noise correlations in V1 cell populations. (A) Schematic description of the fixation task protocol. After 300 ms of fixation, a sine–wave grating of fixed orientation was presented for 500 ms. Following stimulus presentation, an additional 300 ms of fixation was required for the monkey to complete the trial and receive a reward. (B) Peristimulus time histograms for 2 pairs of neurons (cell 1—red; cell 2—blue) exhibiting positive (top) and negative (bottom) correlated variability. (C) Mean correlations decayed exponentially as a function of the orientation difference between the cells in a pair (Δθ). The mean correlation coefficient was 0.13 ± 0.03 (mean ± S.E.M). (D) Mean-positive and -negative correlation coefficients as a function of Δθ. Positive correlation coefficients were large (mean 0.21) and decayed exponentially as a function of Δθ; negative correlations were smaller (mean −0.098) and remained relatively constant as a function of Δθ. Error bars represent SEM. (E) In comparison with positive correlations, only 10.6% of cell pairs exhibited statistically significant negative correlations.

Figure 1.

Structure of positive and negative noise correlations in V1 cell populations. (A) Schematic description of the fixation task protocol. After 300 ms of fixation, a sine–wave grating of fixed orientation was presented for 500 ms. Following stimulus presentation, an additional 300 ms of fixation was required for the monkey to complete the trial and receive a reward. (B) Peristimulus time histograms for 2 pairs of neurons (cell 1—red; cell 2—blue) exhibiting positive (top) and negative (bottom) correlated variability. (C) Mean correlations decayed exponentially as a function of the orientation difference between the cells in a pair (Δθ). The mean correlation coefficient was 0.13 ± 0.03 (mean ± S.E.M). (D) Mean-positive and -negative correlation coefficients as a function of Δθ. Positive correlation coefficients were large (mean 0.21) and decayed exponentially as a function of Δθ; negative correlations were smaller (mean −0.098) and remained relatively constant as a function of Δθ. Error bars represent SEM. (E) In comparison with positive correlations, only 10.6% of cell pairs exhibited statistically significant negative correlations.

### Fisher Information Estimation

We built a statistical model by using Gaussian functions to model the orientation tuning curves (OTCs) of a population of 180 neurons, uniformly spanning preferred orientations between 0° and 90° (Fig. 2A). Mean values of the tuning curve parameters were: baseline 7 Hz, peak 35 Hz, and width 50°. These parameters were consistent with the average tuning curve parameters of the population of V1 cells from the experiment. For each neuron, the parameters of the tuning curve were defined as sums between mp, the mean value of the respective parameter (e.g., mp = 35 Hz for peaks), and noise (a normal process with zero mean and STD = 0.05 × mp).

Figure 2.

Negative correlations influence network accuracy. To compute network accuracy using FI (FI), we performed a Monte Carlo simulation. For each simulation case, we generated at random a set of OTCs, (A) a correlation matrix with positive and negative correlation coefficients (C) and then analytically estimated FI (see Materials and Methods). (B) We generated correlation matrices such that the mean-positive correlation (mean rsc+, blue line) decreased exponentially with the orientation difference of the cells in a pair whereas the mean-negative correlation (mean rsc, red line) was held constant. The mean profile of positive and negative correlations was changed such that the mean spike count correlations (mean rsc, black line) remained constant in each bin of orientation difference Δθ. (D) For each decrement (−0.01) of negative correlations, we averaged FI computed 10 000 times (see Materials and Methods). Fisher information increases almost linearly with the increase in negative correlation strength.

Figure 2.

Negative correlations influence network accuracy. To compute network accuracy using FI (FI), we performed a Monte Carlo simulation. For each simulation case, we generated at random a set of OTCs, (A) a correlation matrix with positive and negative correlation coefficients (C) and then analytically estimated FI (see Materials and Methods). (B) We generated correlation matrices such that the mean-positive correlation (mean rsc+, blue line) decreased exponentially with the orientation difference of the cells in a pair whereas the mean-negative correlation (mean rsc, red line) was held constant. The mean profile of positive and negative correlations was changed such that the mean spike count correlations (mean rsc, black line) remained constant in each bin of orientation difference Δθ. (D) For each decrement (−0.01) of negative correlations, we averaged FI computed 10 000 times (see Materials and Methods). Fisher information increases almost linearly with the increase in negative correlation strength.

We estimated FI for the population of 180 neurons by using the OTCs of the population of cells (Fig. 2A) together with the correlation matrix (Fig. 2C) approximating the distribution of noise correlation found in the experiment (Fig. 1C and D). To explore the influence of negative correlations on population-coding accuracy, we gradually changed the amplitude of mean-negative and mean-positive correlations such that their sum remained constant. Specifically, we started from zero negative correlations and gradually incremented their amplitude (from 0 to −0.1) whereas positive correlations were appropriately increased such as to keep the mean correlation coefficient constant for each Δθ bin (Fig. 2B). In agreement with experimental data (Fig. 1D), the change in the mean-positive correlation coefficient with respect to Δθ was modeled as an exponential function r = a e−bΔθ (e.g., a = 0.305, b = 0.016 in Fig. 2B), whereas mean-negative correlations were held constant. For each mean correlation profile, we generated, at random, 10 000 correlation matrices (Fig. 2C). For each correlation matrix, we generated a set of OTCs for the neuronal population (Fig. 2A). For each case, FI was calculated as the product between the inverse of covariance matrix computed from the generated correlation matrix, Q1, and the orientation tuning derivatives, f′, corresponding to a 90° orientation stimulus, with the equation: FI =f′(90°)TQ1f′(90°) (Abbott and Dayan 1999, Supplementary Material). The FI curve shown in Figure 2D is the average of the FI computed for 10 000 cases corresponding to each negative correlation decrement of −0.01.

### Noise Correlations

For each pair of neurons (i,j), we computed the noise correlation for a stimulus with orientation θ, rsc(θ), using the Pearson correlation coefficient

$rsc(θ)=∑k=1N(rik−ri¯)(rjk−rj¯)σiσj.$
where N is the number of trials, $rik$ is the firing rate of neuron i in trial k, $r¯i$ is the mean firing rate, and σi is the standard deviation of the responses for neuron i. Then, we computed the noise correlations of neuronal pairs by averaging the Pearson correlations across all M stimulus orientations between 0° and 180°,
$rsc=1M∑i=1Mrsc(θi),$
with M = 8 for experiment and M = 6 for the noise correlation model (Gutnisky and Dragoi 2008; Hansen et al. 2012). For each stimulus orientation, we computed the noise correlation rsc(θ) for N = 50 trials of 500 ms length each.

### Recurrent Model

The model had 3 interconnected stages: retina, LGN, and V1 (see Supplementary Material for a detailed description). Retina contained 2 layers of ON-center surround cells, and respectively OFF-center surround cells, driven by image inputs. Inputs consisted of 4 × 1° oriented bars presented for 500 ms. Retinal cells were modeled by a difference of Gaussian filters. Retinal activity was modulated by a saturating nonlinearity to account for the stimulus contrast sensitivity and sent to the LGN cells with a random delay. LGN cells were organized in 2 ON/OFF layers and were connected one-to-one with the retinal cells. LGN cells output modulated Poisson spiking generators that fed the V1 network. We ensured that the LGN input to a cortical cell was broadly tuned for orientation in agreement with experimental studies (Tanaka 1983; Reid and Alonso 1995; Ferster et al. 1996; Ferster and Miller 2000). The layer of cortical cells simulated a hypercolumn of layer 4 that consisted of simple cells. The network had 1008 excitatory regular spiking cells and 252 inhibitory fast-spiking cells. The network had 252 preferred orientations in the interval (0°, 180o), with 4 excitatory neurons and 1 inhibitory neuron per orientation. All neurons were modeled as conductance-based integrate-and-fire neurons. The thalamocortical afferents were established at random from ON and OFF subfields that were defined over the ON and OFF LGN layers by using Gabor functions. The recurrent connections between V1 cells were established at random, following a Gaussian probability distribution centered on the cell's preferred orientation (Fig. 3A inset). The response of LGN afferents was broadly tuned to stimulus orientation, and then intracortical connections within V1 sharpened orientation selectivity (Fig. 3A). We studied the dependence of noise correlation structure on the synaptic conductances by changing slightly the afferent excitatory synaptic conductances: The excitatory conductances $g¯EE$ around 1.2 nS and the inhibitory conductances $g¯EI$ around 4.9 nS, both in the relative range of ±16% (Fig. 5 and 6).

Figure 3.

Exploring the mechanism of negative correlations using a spiking network model. (A) Gray diamonds: Mean pooled response of LGN afferents to V1 excitatory neurons to a vertically orientated bar of 4° length and 1° width, presented for 500 ms in 1008 trials. Each point represents the sum of firing rates of LGN spikes converging to a V1 neuron, having the preferred orientation indicated on x-axis. Black diamonds: Mean V1 population response to a bar stimulus of 90o orientation. Intracortical connections increase the orientation selectivity of the weakly selective responses of the LGN inputs. (A inset) Neurons are connected randomly according to a normal distribution centered at the 0° difference in the preferred orientation of the cells in a pair (Δθ). The y-axis represents the probability that a postsynaptic neuron receives excitatory/inhibitory input from a cell oriented Δθ degrees away. The inhibitory inputs to a cortical neuron originate from a broader range of orientations than the excitatory inputs. We exemplify the connection probabilities corresponding to narrowly tuned excitation (σE = 12.5°), and broadly tuned inhibition (σI = 40°). (B) Distribution of binned mean-positive and -negative noise correlation. Noise correlation was computed as an average of noise correlations corresponding to 6 stimuli separated by 30°. For each stimulus orientation, we calculated the noise correlation for 50 trials of 500 ms each.

Figure 3.

Exploring the mechanism of negative correlations using a spiking network model. (A) Gray diamonds: Mean pooled response of LGN afferents to V1 excitatory neurons to a vertically orientated bar of 4° length and 1° width, presented for 500 ms in 1008 trials. Each point represents the sum of firing rates of LGN spikes converging to a V1 neuron, having the preferred orientation indicated on x-axis. Black diamonds: Mean V1 population response to a bar stimulus of 90o orientation. Intracortical connections increase the orientation selectivity of the weakly selective responses of the LGN inputs. (A inset) Neurons are connected randomly according to a normal distribution centered at the 0° difference in the preferred orientation of the cells in a pair (Δθ). The y-axis represents the probability that a postsynaptic neuron receives excitatory/inhibitory input from a cell oriented Δθ degrees away. The inhibitory inputs to a cortical neuron originate from a broader range of orientations than the excitatory inputs. We exemplify the connection probabilities corresponding to narrowly tuned excitation (σE = 12.5°), and broadly tuned inhibition (σI = 40°). (B) Distribution of binned mean-positive and -negative noise correlation. Noise correlation was computed as an average of noise correlations corresponding to 6 stimuli separated by 30°. For each stimulus orientation, we calculated the noise correlation for 50 trials of 500 ms each.

For each excitatory neuron of the recurrent network, we computed the sum of afferent synaptic currents, referred as the synaptic current in the following, every millisecond (see Supplementary Material). For a pair of neurons (i,j), we computed synaptic current cross-correlation as $C(τ)=∑k=−NNci(k)cj(k+τ),$ where ci(k) is the mean over trials of the synaptic current i at time t k, τ is the time lag, and N = 250. Then, we computed the area under the current correlation curve, denoted as CT, by summing the correlations corresponding to 500 lags between τ = −250 and τ = 250. We computed CT for 50 trials of 500 ms length each. We restricted the cross-correlation computation to neuronal pairs of preferred orientations within 30° around the 90° stimulus orientation (the noise correlations of these neuronal pairs were essential for the computation of FI). For each neuron, the afferent synaptic currents are combinations of excitatory and inhibitory neurons. Therefore, for a pair of neurons, CT = CEE+ CII+ CEI + CIE, where CEE is the cross-correlation area of excitatory currents afferent to the neuronal pair, CII is the cross-correlation area of the afferent inhibitory currents, and CEI is the cross-correlation area of excitatory and inhibitory afferent currents. We computed all the cross-correlation areas CT, CEE, CII, CEI, and CIE in order to study their efficacy in predicting the sign of the noise correlation rsc (Fig. 4, Supplementary Fig. 1).

Figure 4.

Current cross-correlation areas for neuronal pairs with positive (dark blue line) and negative (red line) noise correlation. We averaged the synaptic currents over trials and then computed the current cross-correlation of neuronal pairs for 500 lags of 1 ms each. We used 50 trials of 500 ms each and a stimulus with orientation 90°. Probability distribution function (pdf) of the area under the cross-correlation curve for (A) excitatory–excitatory currents (CEE), (B) inhibitory–inhibitory currents (CII), and (C) excitatory–inhibitory currents (CEI+ CIE). (D) Performance of SVM decoders that predicted the sign of noise correlations from the current cross-correlations areas CEE, CII, CEI + CIE, CT, and CALL (see Materials and Methods). Asterisks indicate that the average performance of each decoder is significantly different from chance. In addition, the performance of each decoder is statistically significant from the other decoders (ANOVA one-way test, F4 = 1636, P = 9.24 × 10−284, followed by post hoc multiple comparison with Bonferroni correction for the 0.05 significance level).

Figure 4.

Current cross-correlation areas for neuronal pairs with positive (dark blue line) and negative (red line) noise correlation. We averaged the synaptic currents over trials and then computed the current cross-correlation of neuronal pairs for 500 lags of 1 ms each. We used 50 trials of 500 ms each and a stimulus with orientation 90°. Probability distribution function (pdf) of the area under the cross-correlation curve for (A) excitatory–excitatory currents (CEE), (B) inhibitory–inhibitory currents (CII), and (C) excitatory–inhibitory currents (CEI+ CIE). (D) Performance of SVM decoders that predicted the sign of noise correlations from the current cross-correlations areas CEE, CII, CEI + CIE, CT, and CALL (see Materials and Methods). Asterisks indicate that the average performance of each decoder is significantly different from chance. In addition, the performance of each decoder is statistically significant from the other decoders (ANOVA one-way test, F4 = 1636, P = 9.24 × 10−284, followed by post hoc multiple comparison with Bonferroni correction for the 0.05 significance level).

To compute the population signal-to-noise ratio (SNR), we used 6 stimuli, uniformly distributed over the entire orientation range (0°, 180°), and for each stimulus, we used 50 trials of 500 ms length each. For each stimulus with orientation θi, we computed $SNR(θi)=Mr¯/∑i=1M∑j=1MCov(ri,rj),$ where M represents the number of neurons with preferred orientations within [−30°, 30°] around stimulus orientation, $r¯$ is their mean firing rate, and Cov(ri,rj) is the covariance between the responses of neurons i and j (Zohary et al. 1994). We computed the SNR for all the stimuli as $SNR=(1/6)∑i=16SNR(θi)$ (Fig. 6B).

### Support Vector Machine Decoder

We used a support vector machine (SVM) decoder, based on a radial basis function kernel (Scholkopf and Smola 2002), to predict the sign of noise correlation (+/−) from the synaptic currents correlations. We used different cross-correlation areas as inputs for the SVM decoder: CEE, CII, CEI + CIE, CT, and CALL = {CEE, CII, CEI + CIE} (Fig. 4D). For each type of input, we randomly selected 5000 cross-correlation areas of neuronal pairs of positive noise correlations and 5000 cross-correlation areas for neuronal pairs of negative noise correlations. We divided the input data equally between a training set and a test set. Each set contained 2500 input data corresponding to positive or negative noise correlations. We trained the SVM decoder with the training set and computed the decoder performance (number of correct responses/5000) for the test set. We repeated this procedure 100 times. Then, we compared the average performance of the SVM decoders for different cross-correlation-area inputs and tested the statistical significance using the ANOVA one-way followed by multiple post hoc group comparisons (Fig. 4D).

## Results

### Distribution of Negative Noise Correlations in V1 Networks

Two non-human primates (M. mulatta) performed a fixation task (Fig. 1A) whereas we used electrode arrays with electrode spacing within 2 mm targeting the superficial layers of V1 (200–400 μm depth, n = 601 pairs of cells). Monkeys were required to hold fixation within a 1° window throughout stimulus presentation to earn a juice reward; the trial was automatically aborted if fixation instability exceeded 0.25° at any time during stimulus presentation. While monkeys fixated a white dot in the center of a computer screen, a single oriented grating stimulus was flashed for 500 ms in the center of the neurons' receptive field (5° circular sine–wave gratings with a spatial frequency of 1.4 cycles per degree and a 50% contrast level presented binocularly). The range of stimulus orientation was 0–180° in steps of 22.5° (8 orientations in total) with each orientation randomly presented 50 times across trials (400 trials in total). After the stimulus was extinguished, an additional 300 ms of fixation was required before the monkey was rewarded for maintaining fixation throughout the entire trial.

We examined the fluctuations in neuronal responses, or “noise,” by measuring spike count correlations (rsc) between pairs of neurons. Spike counts were measured during stimulus presentation (500 ms in our experiment, see Materials and Methods). The main analysis focused on the differences between positive and negative noise correlations. For instance, Figure 1B shows 2 examples of cell pairs (peristimulus time histograms) exhibiting positive (top) and negative (bottom) correlated variability. We first confirmed previous reports (Lee et al. 1988; Zohary et al. 1994; Gutnisky and Dragoi 2008; Nauhaus et al. 2009; Hansen et al. 2012) that mean correlation strength decays exponentially (Fig. 1C) as a function of the preferred orientation difference between the cells in a pair (Δθ). Across the population of cell pairs, we found that the mean correlation coefficient was 0.13 ± 0.03 (mean ± SEM) and that correlations were generally independent of stimulus orientation (only 9% of the pairs exhibited a significant relationship between the correlation coefficient and stimulus orientation, Wilcoxon signed-rank test). We also found that a large number of correlation coefficients (58%) were significantly different from 0 (Fig. 1E, α = 0.05, two-tailed t-test) and that the percentage of significant correlation coefficients decreased with the difference (Δθ) between the cells' preferred orientations (Perrett et al. 1985; Bair et al. 2001; Reich et al. 2001; Kohn and Smith 2005). However, in contrast to positive correlation coefficients, 1) a much smaller percentage of cell pairs (10.6%) exhibited statistically significant negative correlations (α = 0.05, two-tailed t-test) and 2) negative correlations did not appear to depend on Δθ (Fig. 1D and E). To assess the specific contribution of positive and negative correlations to the mean correlation coefficient, we computed the mean-positive and mean-negative correlation coefficient as a function of Δθ. Interestingly, we found that whereas the mean-positive correlation coefficient was large (mean 0.21) and decayed exponentially as a function of Δθ, the negative correlations were smaller in magnitude (mean −0.098) and remained relatively constant as a function of Δθ (Fig. 1D separately averaging positive and negative correlation coefficients for each Δθ bin). This result indicates that positive and negative correlation coefficients have highly distinct strengths and distributions across the population of V1 cells.

### Negative Correlations Improve the Accuracy of the Population Code

What is the contribution of negative correlations to the accuracy of population coding? Although previous studies have examined the impact of correlations on network coding (Zohary et al. 1994; Abbott and Dayan 1999; Sompolinsky et al. 2001; Kohn and Smith 2005; Averbeck et al. 2006; Gutnisky and Dragoi 2008; Hansen et al. 2012), only the effect of mean correlations was considered. However, it is conceivable that positive and negative correlation coefficients, which both contribute to the mean correlation, may have distinct functional effects on network coding. Indeed, although negative correlation coefficients have been occasionally reported in electrophysiological studies (Lee et al. 1988; Zohary et al. 1994; Gutnisky and Dragoi 2008; Smith and Kohn 2008; Ecker et al. 2010), their functional significance for network coding is still not fully understood.

To address this issue, we estimated FI, which is the upper limit with which any decoding mechanism can extract information about stimulus orientation. We assumed that the joint neuronal responses to stimulus orientation can be described by a multivariate Gaussian defined by the mean firing rate and covariance matrix of trial-by-trial correlated variability (Abbott and Dayan 1999; Sompolinsky et al. 2001; Gutnisky and Dragoi 2008). Thus, we generated populations of cells with idealized Gaussian tuning curves uniformly spanning preferred orientations between 0° and 180° (parameters were consistent with the average tuning curve parameters of our recorded population of V1 cells). For each population of cells that was generated, we constructed a correlation matrix with positive and negative correlation coefficients (cf. Fig. 1D) such that mean-positive correlations' strength decreased exponentially as a function of the difference in cells' preferred orientation, whereas mean-negative correlations remained constant (cf. Fig. 2B, see Materials and Methods and Supplementary Material). We subsequently computed FI for our population of 180 neurons while varying the mean amplitude of positive and negative correlation coefficients, however without changing the overall mean correlation coefficient. Fisher information was computed using the slopes of the neurons' tuning curves and a positive definite approximation of the generated correlation matrix by performing Monte Carlo simulations for 10 000 neuronal populations for each decrement in mean-negative correlation (see Materials and Methods and Supplementary Material).

The main question that we examined is whether and how FI changes when the mean-negative correlations are decreased and the mean-positive correlations are increased such as to maintain the overall mean correlation coefficient unchanged. A somewhat surprising result is that that FI increases greatly even for a slight increase in negative correlations (Fig. 2D). Indeed, when negative correlations were increased above −0.02, FI increased almost linearly with the strength of negative correlations. As described in Supplementary Material, for higher negative correlations, the determinant of the positive definite matrix approximating the generated correlation matrix is smaller, and this leads to a larger inverse covariance matrix which is associated with higher FI values for the same derivatives of the tuning curves. It is noteworthy that increasing the amplitude of both positive and negative correlations should increase the range, and thus, variability of correlation coefficients. If anything, a higher variability of correlation coefficients would decrease FI (Abbott and Dayan 1999; Gutnisky and Dragoi 2008). Therefore, it is remarkable that we found a strong increase in FI when negative correlations were increased in amplitude despite the increase in the variability of correlations and the lack of change in the overall mean correlation coefficient. Taken together, these analyses clearly indicate that negative correlations increase the amount of information in the experimentally measured V1 population response, hence supporting the idea that negative correlations are beneficial for network coding accuracy.

### Recurrent Model of Noise Correlations

We next used an integrate-and-fire recurrent model to examine the mechanism by which negative correlations emerge in cortical circuits. To this end, we implemented a recurrent network consisting of 2 populations of excitatory and inhibitory spiking neurons, both receiving excitatory feedforward projections (Somers et al. 1995; Chelaru and Dragoi 2008). The model consists of 1008 excitatory and 252 inhibitory neurons receiving weakly selective excitatory feedforward projections from LGN (Tanaka 1983; Reid and Alonso 1995; Ferster et al. 1996; Ferster and Miller 2000), which are sharpened by intracortical connections (Fig. 3A). The functional connection probability varies with the difference between the neurons' preferred orientation. In agreement with experimental data (Roerig and Chen 2002; Roerig et al. 2003), the inhibitory inputs to a cortical neuron originated from a broader range of orientations than excitatory inputs (this difference could be attributed in part to the broader tuning of feedforward excitation to fast-spiking interneurons relative to the tuning of feedforward excitation to excitatory neurons [Fig. 3A inset, Hirsch et al. 2003; Swadlow 2003; Hansen et al. 2012]).

The model confirmed the distribution of noise correlations found experimentally: Positive correlations were large and decreased exponentially as a function of orientation difference (Δθ), whereas negative correlations were smaller and relatively constant as a function of Δθ (Fig. 3B), except for negative correlations at nearby preferred orientations, found only in small numbers in the experiment (Fig. 1C). This figure complements the experimental data to show that 1) for the cells preferring similar orientations (small Δθ), negative correlations do not contribute much to the mean correlation coefficient since the positive correlations are large, whereas 2) for the cells of intermediate and large orientation difference, negative correlations either greatly reduce or cancel out positive correlations to render the V1 neurons more independent.

We further examined the emergence of negative spike count correlations by focusing on the cross-correlation between synaptic currents. Indeed, the synaptic current to each cell consists of an excitatory and an inhibitory component, and current cross-correlation across cell pairs can be decomposed into cross-correlations between excitatory–excitatory currents (CEE), inhibitory–inhibitory currents (CII), and excitatory–inhibitory currents (CEI and CIE). We examined separately the pairs of cells exhibiting positive spike count correlations and those exhibiting negative correlations by computing the area of cross-correlation between the synaptic currents for individual pairs. The area under the current cross-correlation curve is related to the noise (spike count) correlation resulting from trial-by-trial variability (cf. Renart et al. 2010; see also Hansen et al. 2012); hence, examining the distribution of current correlation may provide insight into the mechanism of spike count correlations.

We examined the distribution of current cross-correlation area, referred as current correlations, for pairs of neurons with preferred orientations within 30° of the stimulus orientation (see Materials and Methods). We observed clear differences between the current correlation distributions corresponding to positive and negative spike count correlations (Fig. 4A–C). Specifically, negative noise correlations typically emerged for pairs of cells that were more weakly coupled, whereas positive noise correlations were more likely to emerge for cells that were more tightly coupled. This result was applicable to each current correlation that we investigated (Fig. 4A–C and Supplementary Fig. 1). We further asked whether we can identify the key current correlations that can be used to predict the sign of spike count correlations for our population of model cells. Thus, we compared the performance of SVM decoders trained with specific current correlations: CEE, CII, and CEI+ CIE (see Materials and Methods) associated with positive and negative noise correlations. Interestingly, we found that the decoder based on excitatory–excitatory and excitatory–inhibitory current correlations (CEE and CEI+ CIE) had a stronger mean performance relative to the decoder based on the inhibitory–inhibitory current correlations (CII, Fig. 4D). Furthermore, decoding the total current correlations (CT) or using each current correlation as a separate input to the decoder (CALL) improved the decoder performance only marginally relative to the decoder in which only CEE is considered (since many samples, n = 5000, were used for training and testing the decoders, the variability of the decoder performance was small, see Materials and Methods). Each type of SVM decoder, based on different types of current correlation inputs, had a mean performance that was significantly different from chance (P < 10−20) and was significantly different from the mean performance of the other decoders (one-way ANOVA test, F4 = 1636, P = 9.24 × 10−284 followed by multiple comparison with Bonferroni correction, α = 0.05). Taken together, these analyses indicate that negative noise correlations emerge in cortical networks when synaptic current correlations are weakened.

We further used the recurrent model to examine the variables controlling the strength of negative correlations. We focused on the balance between local excitation and inhibition as the main variable controlling the gain and selectivity of neuronal responses by varying the model excitatory and inhibitory afferent synaptic conductances, gEE and gEI. Not surprisingly, increasing inhibitory synaptic conductances (gEI) caused an increase in total inhibition, and subsequently, a decrease in the mean response amplitude across the population of cells (Fig. 5A–C). We parametrically changed the strength of excitatory and inhibitory afferent synaptic conductances, gEE and gEI, to examine the distribution of negative correlations across the population of cells. As shown in Figure 6A, increasing inhibitory synaptic conductances, gEI, while decreasing the excitatory synaptic conductances, gEE, caused an increase in the amplitude of negative correlation coefficients, which, according to Figure 2D, should cause an increase in network discrimination performance. Also, the orientation selectivity index (Wörgötter et al. 1991), which measures the strength of orientation tuning of individual neurons, was reduced with the increase in inhibition and reduction in excitation (Fig. 5D).

Figure 5.

The changes in synaptic conductances modulates the network performance. (A) Network response decreases with the increase of the afferent inhibitory synaptic conductance gEI. Each dot represents the mean firing rate of the neuron of preferred orientation θ (represented on the x-axis) for various values, low and high, of the afferent inhibitory synaptic conductance gEI. (B) Orientation tuning curves for 6 neurons selected from the recurrent network of orientation selectivity. Orientation tuning curves were computed for 2 typical values of the synaptic conductance between excitatory and inhibitory neurons gEI = 4.9 nS (dark blue line) and gEI = 5.7 nS (red line). For high inhibition (gEI = 5.7 nS), OTCs have peaks smaller than those for the control network (gEI = 4.9 nS). OTCs were computed for 64 stimuli, uniformly distributed in the interval [0°, 180°] and averaged for 16 trials of 500 ms length each. (C and D) Model simulations were performed using different combinations of excitatory (gEE) and inhibitory (gEI) conductances varying around control values (gEEC = 1.2 nS and gEIC = 4.9 nS). All simulations were run for a constant synaptic profile of excitatory and inhibitory connections (excitatory: orientation spread σE = 12.5°; inhibitory: orientation spread σI = 40°). gEE and gEI values were varied in 2% steps in the (−16%, 16%) range (ΔgEE = 100(gEEgEEC)/gEEC; ΔgEI = 100(gEIgEIC)/gEIC). For each point, we used 50 trials and 6 stimulus orientations evenly distributed across the 0–180° range. (C) Mean response amplitude computed for the neurons with preferred orientations within 30° of the stimulus orientation. (D) Mean orientation selectivity index (see Supplementary material) of the network OTCs.

Figure 5.

The changes in synaptic conductances modulates the network performance. (A) Network response decreases with the increase of the afferent inhibitory synaptic conductance gEI. Each dot represents the mean firing rate of the neuron of preferred orientation θ (represented on the x-axis) for various values, low and high, of the afferent inhibitory synaptic conductance gEI. (B) Orientation tuning curves for 6 neurons selected from the recurrent network of orientation selectivity. Orientation tuning curves were computed for 2 typical values of the synaptic conductance between excitatory and inhibitory neurons gEI = 4.9 nS (dark blue line) and gEI = 5.7 nS (red line). For high inhibition (gEI = 5.7 nS), OTCs have peaks smaller than those for the control network (gEI = 4.9 nS). OTCs were computed for 64 stimuli, uniformly distributed in the interval [0°, 180°] and averaged for 16 trials of 500 ms length each. (C and D) Model simulations were performed using different combinations of excitatory (gEE) and inhibitory (gEI) conductances varying around control values (gEEC = 1.2 nS and gEIC = 4.9 nS). All simulations were run for a constant synaptic profile of excitatory and inhibitory connections (excitatory: orientation spread σE = 12.5°; inhibitory: orientation spread σI = 40°). gEE and gEI values were varied in 2% steps in the (−16%, 16%) range (ΔgEE = 100(gEEgEEC)/gEEC; ΔgEI = 100(gEIgEIC)/gEIC). For each point, we used 50 trials and 6 stimulus orientations evenly distributed across the 0–180° range. (C) Mean response amplitude computed for the neurons with preferred orientations within 30° of the stimulus orientation. (D) Mean orientation selectivity index (see Supplementary material) of the network OTCs.

Figure 6.

Negative correlations improve network performance. (A–C) Model simulations were performed as in Figure 5. (A) Peak negative correlations computed in an interval of 15° around stimulus orientation. (B) Population SNR computed for an interval of 30° around stimulus orientation. (C) Fisher Information computed through a Monte Carlo simulation for each combination of synaptic conductances gEE and gEI. First, we generated a statistical model of 180 neurons having the OTCs defined by the mean peak values and bandwidth of the network OTCs, corresponding to a specific pair of synaptic conductances. Then, for the same pair of synaptic conductances, we generated correlation matrixes having the corresponding mean-negative correlation of the network (see Fig. 2C) and averaged the FI for 10 000 simulation cases.

Figure 6.

Negative correlations improve network performance. (A–C) Model simulations were performed as in Figure 5. (A) Peak negative correlations computed in an interval of 15° around stimulus orientation. (B) Population SNR computed for an interval of 30° around stimulus orientation. (C) Fisher Information computed through a Monte Carlo simulation for each combination of synaptic conductances gEE and gEI. First, we generated a statistical model of 180 neurons having the OTCs defined by the mean peak values and bandwidth of the network OTCs, corresponding to a specific pair of synaptic conductances. Then, for the same pair of synaptic conductances, we generated correlation matrixes having the corresponding mean-negative correlation of the network (see Fig. 2C) and averaged the FI for 10 000 simulation cases.

We further confirmed the beneficial effects of negative correlations for network coding by computing the population SNR. This was done by using 6 oriented bars presented for 500 ms each, uniformly distributed over the entire orientation range (0°, 180°), and presented for 50 trials each. For the calculation of the population SNR, we took into account the mean responses of the cells in the population and the covariance matrix (see Materials and Methods, Zohary et al. 1994). Our analysis shows that negative noise correlations, which are consistent with weak local excitatory connections and strong inhibitory connections (Fig. 6A), cause a strong increase in the population SNR. Indeed, when the strength of inhibitory conductances is increased and the strength of excitatory conductances is decreased, negative correlations increase in magnitude to boost the population SNR (Fig. 6B). The fact that the population SNR is maximized in the region of parameter space where negative correlations are high is not necessarily an expected result. Indeed, the population SNR is inversely related to the mean covariance; hence, lower mean correlations are equivalent to higher SNR. However, the decrease in the mean firing rates when noise correlations are decreased should, in principle, contribute to a decrease in the population SNR (Zohary et al. 1994). In agreement with our previous studies (Gutnisky and Dragoi 2008; Chelaru and Dragoi 2008), these results (Fig. 6B) demonstrate that noise correlations have more weight than signal correlations when information is extracted from the population activity.

Finally, we used the recurrent model to examine whether and how negative correlations influence network accuracy, computed using FI, when the network was stimulated with 4 × 1° oriented bars presented for 500 ms. We addressed this issue by varying the model excitatory and inhibitory afferent synaptic conductances, gEE and gEI to control the strength of negative correlations. In principle, FI should increase with the reduction in correlations, and thus, we expected that increasing inhibition and decreasing excitation, which would cause an increase in negative correlations, would contribute to higher values of FI (see Fig. 2). However, FI is also proportional to the slope of the OTCs, and since the increase in inhibition and decrease in excitation are associated with a decrease in orientation tuning (Fig. 5D), one can reasonably argue that the decrease in tuning (seen in the model only for a large increase in gEI) would cause a decrease in FI. In other words, what is the variable that critically controls network accuracy (tuning curves or correlations) when the relative balance between excitation and inhibition is varied? As Figure 6C shows, we found an increase in FI when negative correlations were increased despite the decrease in neuronal responses and orientation tuning caused by the increase in local inhibition and decrease in local excitation.

## Discussion

Although negative noise correlations have been observed in many cortical areas, such as V1 (Perrett et al. 1985; Gawne et al. 1996; Reich et al. 2001; Kohn and Smith 2005; Gutnisky and Dragoi 2008; Smith and Kohn 2008), MT (Zohary et al. 1994; Bair et al. 2001), or IT (Gawne and Richmond 1993), their functional role has remained elusive. Indeed, the structure of negative correlations across a population of cells and its impact on the network's capacity to encode information has been unclear. We demonstrate here that unlike positive correlations which decrease exponentially as a function of signal correlations (Mastronarde 1983; Zohary et al. 1994; Gutnisky and Dragoi 2008), negative correlations are uniformly distributed across a population of cells. Importantly, we found that even a small increase in the strength of negative correlations leads to a large increase in population-coding accuracy. However, despite their beneficial effect on sensory coding, the balanced excitation and inhibition regime in which biological networks operate (Mastronarde 1983; Haider et al. 2006; Vogels and Abbott 2009; Stimberg et al. 2009; Renart et al. 2010) ensures that negative correlations are not excessively increased. Indeed, strong negative noise correlations are obtained in networks with strong inhibitory connections, but strong inhibitory connections may cause a reduction in neuronal firing rates and a decrease in tuning (see also Ly et al. 2012).

The degree of correlated variability in cortical networks has been typically assessed based on the strength of positive noise correlations (Gawne et al. 1996; Bair et al. 2001; Kohn and Smith 2005; Reich et al. 2001; de la Rocha et al. 2007; Smith and Kohn 2008). However, some studies (Lee et al. 1988; Zohary et al. 1994; Gutnisky and Dragoi 2008; Ecker et al. 2010; Renart et al. 2010) have explicitly revealed negative correlation coefficients in the reported dataset, but their impact on network coding was not examined. For instance, under specific conditions, it has been shown that negative correlations could cancel out positive correlations to generate a mean correlation coefficient close to 0 (Ecker et al. 2010; Renart et al. 2010). We found that although the mean correlation coefficient in V1 superficial layers (cf. Hansen et al. 2012) is greater than 0, a substantial number of cell pairs exhibited negative correlations in the (−0.05, −0.1) range (Fig. 1B–D). Nonetheless, unlike positive correlations, negative correlations were independent on the orientation difference between the cells in a pair while the more dominant, positive correlations decreased exponentially (Mastronarde 1983; Lee et al. 1988; Zohary et al. 1994; Gutnisky and Dragoi 2008).

We used a network model to explore the source of negative correlations in cortical circuits. Importantly, our recurrent model shows that the strength of negative noise correlations depends on network connectivity. That is, the negative correlation strength is modulated by the ratio between the excitatory and inhibitory synaptic conductances, that is, both the strength of negative correlations and the number of cell pairs exhibiting negative correlations increase when inhibition is made stronger (Fig. 6A). Furthermore, even though the increase in local inhibition causes a strong reduction in the mean firing rates (Fig. 5A–C), the increase in negative correlations corresponds to an increase in population SNR and population-coding accuracy (Fig. 6B and C). This has a beneficial effect on network performance, potentially causing an enhancement in the accuracy of the behavioral response.

Theoretically, it has been previously suggested that negative correlations would improve the accuracy of population coding (e.g., Sompolinsky et al. 2001). However, previous studies have assumed that all correlation coefficients in the network were negative and small, and the results were compared with the case of uncorrelated neurons, thus ignoring the actual distribution of positive and negative correlations found in experimental studies. In reality, exactly how the distribution of positive and negative correlations influences coding accuracy has remained unknown. Our study has examined the impact on coding accuracy of the distribution of positive and negative correlation coefficients as measured experimentally (Fig. 1). We found that for fixed mean correlations, even a small increase in the strength of negative correlations leads to a substantial increase in FI (Fig. 2D). This finding suggests that negative correlations have an essential contribution to population coding by balancing the decrease in FI, and consequently network accuracy, due to strong positive correlations (Sompolinsky et al. 2001).

Our result that increasing negative correlations causes an increase in coding accuracy is consistent with most previous studies examining the impact of correlations on cortical function. Indeed, increasing negative noise correlations decreases the mean correlation coefficient across a population of cells, and a reduction in correlations has been associated with an increase in the theoretical limit with which a population of cells can encode information (Zohary et al 1994; Abbott and Dayan 1999; Averbeck et al. 2006; Gutnisky and Dragoi 2008; Chelaru and Dragoi 2008; Ecker et al. 2010; Hansen et al. 2012). However, correlations can be beneficial or detrimental for coding depending on the distribution of noise correlations across the entire network (Abbott and Dayan 1999) as well as the relationship to signal correlations (Averbeck et al. 2006). In the context of the network examined in our paper, correlated noise tends to occur along the same dimension as the signals that distinguish between input stimuli (Dayan and Abbott 2001; Seriès et al. 2004; Gutnisky and Dragoi 2008) and is therefore detrimental to decoding performance. Importantly, noise correlations were found to be lower after adaptation (Gutnisky and Dragoi 2008), in networks of heterogeneous neurons (Chelaru and Dragoi 2008; Padmanabhan and Urban 2010,), and in the granular layers of the primary visual cortex (Hansen et al. 2012), and in each case, the corresponding population-coding accuracy was found to be increased. Nonetheless, the idea that correlations are detrimental for sensory coding has been challenged recently by Ecker et al. (2011) who reported that in very large model, networks of heterogeneous neurons (>10 000 cells) coding accuracy (measured using FI) can be substantially increased by increasing the mean correlation. However, this conclusion was only supported for large correlation values (> 0.3) above those found in experimental studies and was drawn without distinguishing between positive and negative correlation coefficients. Importantly, one of our key findings is that negative correlations improve coding accuracy even when positive correlations are increased as long as the mean correlation coefficient is unchanged.

Although our study focuses on negative correlations, it would be incorrect to treat negative correlations separately from positive correlations. Indeed, as our experimental and modeling data clearly indicate, for a given neuronal population, the majority of the pairs exhibit positive correlated variability, whereas only less than 10% are negatively correlated. Clearly, in order for negative correlations to improve coding accuracy, positive correlations must also be significantly different from 0. Both the experimental data and the model show that negative correlations improve population coding by canceling out positive correlations to reduce the overall strength of noise correlations. In addition, our statistical model of correlations (see Fig. 2) cannot operate if the strength of positive correlations is smaller than that of negative correlations—in that case, it would be impossible to approximate the generated covariance matrices with positive definite matrices and hence FI cannot be computed (Rebonato 1999; Higham 2002).

Altogether, our results provide evidence that negative correlations are beneficial for network coding in that they cause an increase in the SNR and the amount of information in the population code. This raises the issue of whether negatively correlated neuronal responses always constitute an advantageous coding strategy in visual cortex. While this conclusion holds for the correlation structure that we were able to measure in the superficial layers of V1, different correlation distributions in different cortical layers or areas may possibly lead to the conclusion that negative correlations are in fact detrimental for network coding. For instance, very low positive and negative correlations such as those recently observed in the middle layers of V1 (Hansen et al. 2012; Smith et al. 2013) or negative correlations that are not uniformly distributed as a function of signal correlations may have a different impact on network coding.

## Authors Contributions

V.D. designed the study, performed the electrophysiological experiments, and analyzed the data. M.I.C designed and implemented the computational models and analyzed the data emerging from computer simulations. M.I.C. and V.D. wrote the paper.

## Funding

This work was supported by grants to V.D. from NEI grant 1R01EY016715, the Pew Scholars Program, the James S. McDonnell Foundation, NIH EUREKA Program grant 1R01MH086919, and Texas ARP/ATP.

## Notes

We are grateful to Diego Gutnisky, Kresimir Josić, and Charles Beaman for helpful comments on the manuscript. Conflict of Interest: None declared.

## References

Abbott
LF
Dayan
P
.
1999
.
The effect of correlated variability on the accuracy of a population code
.
Neural Comput
.
11
:
91
101
.
Averbeck
BB
Latham
PE
Pouget
A
.
2006
.
Neural correlations, population coding and computation
.
Nat Rev Neurosci
.
7
:
358
366
.
Bair
W
Zohary
E
Newsome
WT
.
2001
.
Correlated firing in macaque visual area MT: time scales and relationship to behavior
.
J Neurosci
.
21
:
1676
1697
.
Chelaru
MI
Dragoi
V
.
2008
.
Efficient coding in heterogeneous neuronal populations
.
PNAS
.
105
:
16344
16349
.
Chen
Y
Geisler
WS
Seidemann
E
.
2008
.
Optimal temporal decoding of neural population responses in a reaction-time visual detection task
.
J Neurophysiol
.
99
:
1366
1379
.
Cox
DR
Hinckley
DV
.
1974
.
Theoretical Statistics
.
London, UK
:
Chapman & Hall
.
Dayan
P
Abbott
LF
.
2001
.
Theoretical Neuroscience
.
Cambridge, MA
:
The MIT Press
.
de la Rocha
J
Doiron
B
Shea-Brown
E
Josić
K
Reyes
A
.
2007
.
Correlation between neural spike trains increases with firing rate
.
Nature
.
448
:
802
806
.
Dragoi
V
Sharma
J
Miller
EK
Sur
M
.
2002
.
Dynamics of neuronal sensitivity in visual cortex and local feature discrimination
.
Nat Neurosci
.
5
:
883
891
.
Ecker
AS
Berens
P
Keliris
GA
Bethge
M
Logothetis
NK
Tolias
AS
.
2010
.
Decorrelated neuronal firing in cortical microcircuits
.
Science
.
327
:
584
587
.
Ecker
AS
Berens
P
Tolias
AS
Bethge
M
.
2011
.
The effect of noise correlations in populations of diversely tuned neurons
.
J Neurosci
.
31
:
14272
14283
.
Ferster
D
Chung
S
Wheat
H
.
1996
.
Orientation selectivity of thalamic input to simple cells of cat visual cortex
.
Nature
.
380
:
249
252
.
Ferster
D
Miller
KD
.
2000
.
Neural mechanisms of orientation selectivity in the visual cortex
.
Annu Rev Neurosci
.
23
:
441
471
.
Gawne
TJ
Kjaer
TW
Hertz
JA
Richmond
BJ
.
1996
.
Adjacent visual cortical complex cells share about 20% of their stimulus-related information
.
Cereb Cortex
.
6
:
482
489
.
Gawne
TJ
Richmond
BJ
.
1993
.
How independent are the messages carried by adjacent inferior temporal cortical neurons?
.
J Neurosci
.
13
:
2758
2771
.
Gutnisky
DA
Dragoi
V
.
2008
.
Adaptive coding of visual information in neural populations
.
Nature
.
452
:
220
224
.
Haider
B
Duque
A
Hasenstaub
AR
McCormick
DA
.
2006
.
Neocortical network activity in vivo is generated through a dynamic balance of excitation and inhibition
.
J Neurosci
.
26
:
4535
4545
.
Hansen
BJ
Chelaru
MI
Dragoi
V
.
2012
.
Correlated variability in laminar cortical circuits
.
Neuron
.
76
:
590
602
.
Higham
NJ
.
2002
.
Computing the nearest correlation matrix—a problem from finance
.
IMA J Num Analysis
.
22
:
329
343
.
Hirsch
JA
Martinez
LM
Pillai
C
Alonso
JM
Wang
Q
Sommer
FT
.
2003
.
Functionally distinct inhibitory neurons at the first stage of visual cortical processing
.
Nat Neurosci
.
6
:
1300
1308
.
Hubel
DH
Wiesel
TN
.
1962
.
Receptive fields, binocular interaction and functional architecture in the cat's visual cortex
.
J Physiol
.
160
:
106
154
.
Josić
K
Shea-Brown
E
Doiron
B
de la Rocha
J
.
2009
.
Stimulus-dependent correlations and population codes
.
Neural Comput
.
21
:
2774
2804
.
Kohn
A
Smith
MA
.
2005
.
Stimulus dependence of neuronal correlation in primary visual cortex of the macaque
.
J Neurosci
.
25
:
3661
3673
.
Ly
C
Middleton
JW
Doiron
B
.
2012
.
Cellular and circuit mechanisms maintain low spike covariability and enhance population coding in somatosensory cortex
.
Front Comput Neurosci
.
6
:
7
. .
Lee
C
Rohrer
WH
Sparks
DL
.
1988
.
Population coding of saccadic eye movements by neurons in the superior colliculus
.
Nature
.
332
:
357
360
.
Mastronarde
DN
.
1983
.
Correlated firing of cat retinal ganglion cells. I. Spontaneously active inputs to X- and Y-cells
.
J Neurophysiol
.
49
:
303
324
.
Miller
JP
Jacobs
GA
Theunissen
FE
.
1991
.
Representation of sensory information in the cricket cercal sensory system. I. Response properties of the primary interneurons
.
J Neurophysiol
.
66
:
1680
1689
.
Nauhaus
I
Busse
L
Carandini
M
Ringach
DL
.
2009
.
Stimulus contrast modulates functional connectivity in visual cortex
.
Nat Neurosci
.
12
:
70
76
.
K
Urban
NN
.
2010
.
Intrinsic biophysical diversity decorrelates neuronal firing while increasing information content
.
Nat Neurosci
.
13
:
1276
1282
.
Palmer
C
Cheng
SY
Seidemann
E
.
2007
.
Linking neuronal and behavioral performance in a reaction-time visual detection task
.
J Neurosci
.
27
:
8122
8137
.
Perrett
DI
Smith
PA
Potter
DD
Mistlin
AJ
AS
Milner
Jeeves
MA
.
1985
.
Visual cells in the temporal cortex sensitive to face view and gaze direction
.
Proc R Soc Lond B Biol Sci
.
223
:
293
317
.
Rebonato
R
.
1999
.
Volatility and Correlation
.
Hoboken, NJ
:
John Wiley & Sons
.
Reich
DS
Mechler
F
Victor
JD
.
2001
.
Independent and redundant information in nearby cortical neurons
.
Science
.
294
:
2566
2568
.
Reid
RC
Alonso
JM
.
1995
.
Specificity of monosynaptic connections from thalamus to visual cortex
.
Nature
.
378
:
281
284
.
Renart
A
de la Rocha
J
Bartho
P
Hollender
L
Parga
N
Reyes
A
Harris
KD
.
2010
.
The asynchronous state in cortical circuits
.
Science
.
327
:
587
590
.
Ringach
DL
Shapley
RM
Hawken
MJ
.
2002
.
Orientation selectivity in macaque V1: diversity and laminar dependence
.
J Neurosci
.
22
:
5639
5651
.
Roerig
B
Chen
B
.
2002
.
Relationships of local inhibitory and excitatory circuits to orientation preference maps in ferret visual cortex
.
Cereb Cortex
.
12
:
187
198
.
Roerig
B
Chen
B
Kao
JP
.
2003
.
Different inhibitory synaptic input patterns in excitatory and inhibitory layer 4 neurons of ferret visual cortex
.
Cereb Cortex
.
13
:
350
363
.
Scholkopf
B
Smola
AJ
.
2002
.
Learning with Kernels
.
Cambridge, MA
:
MIT Press
.
Seriès
P
Latham
PE
Pouget
A
.
2004
.
Tuning curve sharpening for orientation selectivity: coding efficiency and the impact of correlations
.
Nat Neurosci
.
7
:
1129
1135
.
MN
Newsome
WT
.
1998
.
The variable discharge of cortical neurons: implications for connectivity, computation, and information coding
.
J Neurosci
.
18
:
3870
3896
.
Smith
MA
Jia
X
Zandvakili
A
Kohn
A
.
2013
.
Laminar dependence of neuronal correlations in visual cortex
.
J Neurophysiol
.
109
:
940
947
.
Smith
MA
Kohn
A
.
2008
.
Spatial and temporal scales of neuronal correlation in primary visual cortex
.
J Neurosci
.
28
:
12591
12603
.
Somers
DC
Nelson
SB
Sur
M
.
1995
.
An emergent model of orientation selectivity in cat visual cortical simple cells
.
J Neurosci
.
15
:
5448
5465
.
Sompolinsky
H
Yoon
H
Kang
K
Shamir
M
.
2001
.
Population coding in neuronal systems with correlated noise
.
Phys Rev E Stat Nonlin Soft Matter Phys
.
64
(5 Pt 1)
:
051904
.
Stimberg
M
Wimmer
K
Martin
R
Schwabe
L
Mariño
J
Schummers
J
Lyon
DC
Sur
M
Obermayer
K
.
2009
.
The operating regime of local computations in primary visual cortex
.
Cereb Cortex
.
19
:
2166
2180
.
HA
.
2003
.
Fast-spike interneurons and feedforward inhibition in awake sensory neocortex
.
Cereb Cortex
.
13
:
25
32
.
Tanaka
K
.
1983
.
Cross-correlation analysis of geniculostriate neuronal relationships in cats
.
J Neurophysiol
.
49
:
1303
1318
.
Vogels
TP
Abbott
LF
.
2009
.
Gating multiple signals through detailed balance of excitation and inhibition in spiking networks
.
Nat Neurosci
.
12
:
483
491
.
Wörgötter
F
Muche
T
Eysel
UT
.
1991
.
Correlations between directional and orientational tuning of cells in cat striate cortex
.
Exp Brain Res
.
83
:
665
669
.
Zohary
E
MN
Newsome
WT
.
1994
.
Correlated neuronal discharge rate and its implications for psychophysical performance
.
Nature
.
370
:
140
143
.