Noise correlations in area MSTd
Monkeys were presented with two types of heading stimuli while maintaining fixation on a head-fixed target: inertial motion delivered by a motion platform in the absence of optic flow (vestibular condition) and optic flow stimuli presented while the animal was stationary (visual condition, see Methods for details). Consistent with previous findings (Gu et al., 2006
; Takahashi et al., 2007
), many MSTd neurons were tuned to heading direction, and their responses mainly followed the Gaussian velocity profile of the stimulus (). We analyzed responses obtained during the middle 1 second of the stimulus period, during which neuronal activity was robust. Tuning curves of two simultaneously recorded cells are shown in . The similarity of heading tuning between pairs of neurons was quantified as the Pearson correlation coefficient of mean responses across all stimulus directions (‘signal correlation’, rsignal
). For this example pair of neurons, rsignal
= 0.83 and 0.79 for the visual and vestibular conditions, respectively.
Figure 1 Measuring noise correlation (rnoise) between pairs of single neurons in area MSTd. (A) Response time courses for populations of neurons with significant tuning (p<0.05, ANOVA) in the visual (solid curve, n=231) and vestibular (dashed curve, n=118) (more ...)
As in other cortical areas, the spike counts of MSTd neurons in response to an identical stimulus vary from trial to trial, as illustrated in (visual condition) and (vestibular condition). Each datum in these plots represents the spike counts of the two neurons from a single trial. Because heading direction varied across trials, spike counts from individual trials have been z-scored to remove the stimulus effect and allow pooling of data across directions (see Methods). ‘Noise correlation’ is then computed as the Pearson correlation coefficient of the normalized trial-by-trial spike counts, and reflects the degree of correlated variability across trials. For this example pair of cells, there was a weak positive correlation, such that when one neuron fired more spikes, the other neuron did as well (visual condition: rnoise = 0.29, p=0.04, ; vestibular condition: R=0.14, p=0.3, ).
We first examined whether correlated noise in MSTd depends on stimulus modality (). Noise correlations computed from visual and vestibular responses were significantly correlated across 179 pairs of neurons (R=0.38, p
0.001, Spearman rank correlation), and their means were not significantly different (vestibular: 0.035±0.014 s.e.m, visual: 0.039±0.015, p>0.8, paired t-test). Thus, to gain statistical power, we recomputed rnoise
by pooling z-scored responses across stimulus conditions, thereby obtaining a single value of rnoise
for each pair of neurons.
As observed in other visual areas (Huang and Lisberger, 2009
; Smith and Kohn, 2008
), noise correlations depended on the distance between two simultaneously recorded MSTd neurons, as illustrated in Suppl. Fig. 1
, which shows distributions of rnoise
for three distance groups: <0.05mm, 0.05 – 1mm and >1mm. Average noise correlations were significantly greater than zero for the first two groups (<0.05mm: 0.042±0.021 s,e,m., p=0.049, t-test; 0.05–1mm: 0.062±0.024, p=0.011), but not for the group of distant pairs (>1mm: 0±0.15, p=0.9). Thus, the following analyses were focused on 127 neuronal pairs separated by <1mm (results were similar for the whole dataset).
Comparison of noise correlations in trained and naïve animals
Our main goal was to examine whether training modifies interneuronal correlations. Five animals were previously trained to perform a heading discrimination task, in which they reported whether their heading was leftward or rightward relative to straight ahead (Gu et al., 2008a
; Gu et al., 2007
). These monkeys’ heading discrimination thresholds (corresponding to 84% correct) were high (>10°) at early stages of training, and gradually decreased to a plateau of only a few degrees (1~3°), as illustrated in (Fetsch et al., 2009
; Gu et al., 2008a
; Gu et al., 2007
). We measured noise correlations after these ‘trained’ animals had reached asymptotic performance, and we compared them to correlations measured in three ‘naïve’ animals that had never been trained to perform any task other than visual fixation.
Figure 2 Training effects on behavior and interneuronal correlations. (A) Vestibular psychophysical thresholds from 5 monkeys (denoted by different symbol shapes) decreased gradually during training on a heading discrimination task. Solid curves: best fitting (more ...)
Our most conspicuous finding was a difference in mean rnoise between ‘trained’ and ‘naïve’ animals (). Correlations in trained animals were shifted toward zero, as compared to those in naïve animals. The mean noise correlation in the trained group (0.023±0.017 s.e.m, n=89) was significantly smaller than that for ‘naïve’ animals(0.116±0.031, n=38, p=0.006, t-test). Note that, for both groups of animals, rnoise was measured during an identical passive fixation task (see Methods).
Because the stimulus was dynamic (, gray curve), we examined the time course of noise correlation in trained and naïve animals by computing rnoise
in 500ms sliding windows (with 50ms steps). As illustrated in , rnoise
was significantly greater in naïve than trained animals throughout the time course of the neural response (p=0.002, permutation test, see Methods). The difference in rnoise
between naïve and trained animals was largest at the beginning of the trial and gradually decreased with time (R = −0.9, p
0.001, Spearman rank correlation, Suppl. Fig. 2A
). Importantly, these observations held true when correlations were examined for individual animals (Suppl. Fig. 2B
). Thus, the overall reduction in correlated noise among MSTd neurons was a robust finding in trained animals.
Effects of training on tuning curves, variability, and sensitivity of single neurons
It is possible that the difference in correlated noise between naïve and trained animals could be an indirect effect of training on the response properties of individual neurons. Moreover, training-related changes in correlated noise might emerge in parallel with changes in the heading sensitivity of single neurons. To address these issues, we examined the effect of training on the time courses of firing rates and response variability. As illustrated in , the time course of the population-average response to the preferred heading was indistinguishable between trained and naive animals (p=0.8, permutation test, see Methods). There was also no significant effect (p=0.5, permutation test) of training on the time course of the Fano factor, which measures the ratio of response variance to mean response (, see also Methods and Suppl. Fig. 3
). This finding contrasts with a previous report that Fano factor in area V4 was significantly reduced after animals were trained to discriminate orientation (Raiguel et al., 2006
). In MSTd, the difference in noise correlation between naïve and trained animals does not appear to be linked to changes in firing rates or Fano factors.
Figure 3 Training does not affect the time courses of mean responses and response variability during visual (top) and vestibular (bottom) stimulation. (A, C) Time course of the average response to stimuli presented at each cell’s preferred heading in ‘trained’(red, (more ...)
We further explored whether training shaped the tuning properties of individual MSTd neurons. For this analysis, we only included neurons with significant heading tuning in the horizontal plane (p<0.05, one-way ANOVA). To gain statistical power, we exploited a much larger database of single-unit responses from naïve and trained animals, recorded with a single electrode (Vestibular: n=556; Visual: n=992). As shown in , distributions of tuning width (full width at half-height) were very similar for naïve and trained animals. There was no significant difference in median tuning width for the visual condition (naïve: 124.5° vs. trained: 126°, p=0.21, Wilcoxon rank-sum test). The difference in median tuning width was significant for the vestibular condition (naïve: 121° vs. trained: 131°, p=0.045). However, this effect was weak and, notably, training slightly increased
tuning width in the vestibular condition, an effect opposite to that expected if training increases discriminability (e.g. Yang and Maunsell, 2004
). Similarly, as shown in , training did not have any significant effect on the distribution of tuning curve amplitudes in either the visual condition (naïve: 35.4° vs. trained: 31.8°, p=0.24, Wilcoxon rank-sum test) or the vestibular condition (naïve: 17.4° vs. trained: 17.2°, p=0.36). Thus, training animals to perform a fine heading discrimination task did not significantly shape the heading tuning of individual MSTd neurons.
Figure 4 Effects of training on heading tuning in MSTd during visual (top) and vestibular (bottom) stimulation. (A, E) Distributions of tuning width (full width at half-maximum response) for naïve (blue) and trained (red) animals. (B, F) Distributions (more ...)
However, it remains possible that training only shaped the tuning of a subset of neurons that were most informative for heading discrimination around the straightforward reference used in training (e.g. Raiguel et al., 2006
; Schoups et al., 2001
). If so, then effects might only be seen for neurons most sensitive to heading variations around straight forward, and may have been missed in the above analysis. To examine this further, we interpolated tuning curves and used Fisher information analysis (Gu et al., 2010
, see Methods) to compute the sensitivity of each neuron for discriminating heading around straight forward. As shown in , the most sensitive neurons (lowest thresholds) are generally those that prefer lateral headings, such that their tuning curves have a steep slope around straight-ahead. For quantitative analysis, neurons were divided into two groups by heading preference: ’fore-aft’ neurons with heading preferences within 45° of forward (0°) or backward (±180°) motion, and ‘lateral’ neurons with heading preferences within 45° of leftward (−90°) or rightward (90°) movements. Consistent with previous findings (Gu et al., 2008a
; Gu et al., 2010
), lateral neurons were significantly more sensitive than fore-aft neurons for heading discrimination around straight ahead (p
0.001, Factorial ANOVA, ). However, neuronal sensitivity was not significantly different between naïve and trained animals (p>0.5, Factorial ANOVA) for either group of neurons, with no significant interaction effect (p>0.3). In summary, whereas heading discrimination training clearly reduced correlated noise among MSTd neurons, we find no clear evidence that training altered the basic tuning properties or sensitivity of individual neurons, including those neurons that are most informative for performing the task. This result also generalizes to neuronal discrimination of heading about any arbitrary reference (Supplementary Fig. 4
Training effects on the noise-signal correlation structure
It is well established that rnoise
is related to rsignal
(Cohen and Maunsell, 2009
; Cohen and Newsome, 2008
; Gutnisky and Dragoi, 2008
; Huang and Lisberger, 2009
; Kohn and Smith, 2005
; Smith and Kohn, 2008
; Zohary et al., 1994b
), so it is important to evaluate whether training alters this relationship. shows the relationship between rnoise
, with each datum corresponding to a pair of MSTd neurons. This relationship was quantified using general linear models (analysis of covariance, ANCOVA), with rsignal
in each stimulus condition (visual or vestibular) as a continuous variable and training group (trained or naïve) as a categorical factor. There was a significant positive correlation between rnoise
in both stimulus conditions (vestibular: p=0.0001; visual: p=0.0004, ANCOVA), reflecting the fact that noise correlations tended to be positive for pairs of neurons with similar tuning (rsignal
>0) and near zero or negative for pairs with opposite tuning (rsignal
Figure 5 Relationship between noise correlation (rnoise) and signal correlation (rsignal) in MSTd. (A, B) Noise correlations depend significantly on rsignal computed from visual (A) or vestibular (B) tuning curves. Lines represent regression fits (ANCOVA). Red: (more ...)
Importantly, the slope of the relationship between rnoise and rsignal ( )was not significantly affected by training (vestibular: p=0.9; visual: p=0.9, ANCOVA interaction effect), as also indicated by overlap of the 95% confidence intervals around the regression slopes (, nearly identical slopes were obtained by Type II regression). In contrast, training had a significant main effect on rnoise (vestibular: p=0.02; visual; p=0.008 ANCOVA), and the 95% confidence intervals around the regression intercepts were non-overlapping for naïve and trained animals (). Thus, training reduced noise correlations uniformly across all signal correlations, such that the dependency of rnoise on rsignal remained unchanged.
Multisensory MSTd neurons can have matched visual and vestibular heading preferences (‘congruent’ cells) or mismatched preferences (‘opposite’ cells) (Gu et al., 2008a
; Gu et al., 2006
). Thus, we also tested whether rnoise
depends on congruency. Specifically, the two units in each pair could be (1) both congruent, (2) both opposite or (3) a mixture of congruent and opposite cells. As illustrated in Suppl. Fig. 5
was not substantially affected by congruency. Next, we incorporate these results into an information analysis to investigate how the fidelity of population activity changes between naïve and trained animals.
Computation of covariance matrix
Although neurons were recorded pair-wise, our goal is to determine whether population activity in MSTd can account for the effect of training on behavioral sensitivity. For this purpose, we need to estimate the covariance matrix that characterizes correlations among the MSTd population in naïve and trained animals. This was done by assigning each value of the covariance matrix according to the measured noise and signal correlation structures in our data set. Because rnoise
depended on rsignal
in both the vestibular and visual conditions (), both relationships were taken into account when constructing the covariance matrices. For simplicity, all neurons in the simulations discussed below were assumed to have congruent visual and vestibular heading preferences. Results were similar when variable congruency was introduced into the simulation, consistent with the observation that noise correlations were not strongly influenced by congruency (Suppl. Fig. 5
We constructed covariance matrices with two different correlation structures (see Methods): (1) rnoise
depended on rsignal
with regression slopes and intercept specified according to data from naïve animals: rnoise
+0.072, and (2) rnoise
depended on rsignal
with slopes and intercept derived from trained animals: rnoise
+0.005. Note that the slopes were common across the two correlation structures, since no significant difference in slopes was found (). We then used these covariance matrices to compute the precision with which a population of MSTd neurons in naïve or trained animals could discriminate heading, as described below. Importantly, noise correlations did not depend on whether trained monkeys performed a passive fixation task or the heading discrimination task (p=0.3, t-test), as shown in Suppl. Fig. 6
for a subset of neuronal pairs recorded in both tasks. Thus, we are justified in predicting heading discrimination performance from population activity measured during the fixation task for both trained and naïve animals.
Effect of training on population coding efficiency
We computed population discrimination thresholds from the inverse of Fisher information (If
), an upper bound on information capacity that can be extracted by any unbiased estimator (Abbott and Dayan, 1999
; Seung and Sompolinsky, 1993
). Predicted thresholds from If
define the performance that an ideal observer could achieve, based on MSTd population activity, in a fine heading discrimination task. For a simulated population of neurons with independent noise, predicted thresholds decreased steadily with population size (, dashed black curve). As expected from previous findings (Bair et al., 2001
; Cohen and Maunsell, 2009
; Shadlen et al., 1996
; Smith and Kohn, 2008
; Zohary et al., 1994b
), correlated noise similar to that seen in our naïve animals degraded population coding efficiency (, blue curve). For a simulated population of 2000 neurons, the predicted heading discrimination threshold was ~5-fold larger compared to the case of independent noise. Surprisingly, the uniform reduction in rnoise
that we observed in trained animals () had little effect on predicted discrimination thresholds, as compared to naïve animals (, red curve).
Figure 6 Impact of noise correlations on population coding efficiency. (A) Population heading discrimination thresholds as a function of population size. Each simulated population contained neurons with wrapped-Gaussian tuning curves (bandwidth=135°) and (more ...)
Why doesn’t the reduction in mean noise correlation seen in trained animals improve the sensitivity of the population code? We simulated performance of a population of neurons using many covariance matrices that were constructed by systematically varying both the slope and intercept of the relationship between rnoise
. As shown in , predicted thresholds were very sensitive to changes in the slope of the relationship between rnoise
. In contrast, changes in the intercept of the rnoise
relationship had weak effects on predicted thresholds. Counterintuitively, a uniform increase in rnoise
(across all values of rsignal
) produced a mild decrease in population thresholds, improving performance slightly (barely visible in , see also Abbott and Dayan, 1999
; Wilke and Eurich, 2002
). These simulations suggest that a uniform reduction of noise correlations in trained animals is expected to have little impact on discrimination performance.
This conclusion is based on the assumption that all neurons contribute to discrimination performance. We can infer from the simulations of that a change in noise correlation produces different effects for neurons with positive and negative signal correlations. To illustrate this, consider a population consisting of a single pair of neurons, having rsignal
that could range from −1 (opposite heading preferences) to +1 (matched preferences). As illustrated in , reducing the noise correlation between this pair of neurons results in a lower population threshold (red curve below blue curve) when the pair of neurons has positive rsignal
. In contrast, reducing noise correlation increases the predicted threshold for negative rsignal
(see also Supplementary Fig. 7A
). This simple prediction was confirmed when decoding responses of pairs of MSTd neurons. For each pair of neurons, we compute a discrimination threshold under the assumption of correlated noise, as well as the assumption of independent noise. As shown in , pairs of neurons with positive rsignal
yield discrimination thresholds that increase with rnoise
, whereas pairs with negative rsignal
have discrimination thresholds that decrease with rnoise
0.001, Spearman rank correlation). Thus, in a population of neurons with an even mixture of positive and negative signal correlations, the opposite effects of correlated noise will counteract each other.
Figure 7 Reduced noise correlations improve coding efficiency for neurons with similar tuning and reduce coding efficiency for neurons with dissimilar tuning. (A) Heading discrimination thresholds a pair of neurons with various signal correlations. One neuron (more ...)
With this intuition in hand, we consider larger pool sizes (e.g., n=256 in ). If the direction preferences of neurons in the population are broadly distributed, roughly equal numbers of cell pairs will have positive and negative rsignal
(, left inset) and population thresholds for naïve and trained animals will be similar. If we narrow the distribution of direction preferences to generate more cell pairs with positive rsignal
, the weaker noise correlations in trained animals substantially enhance coding efficiency (, middle and right insets, see also Supplementary Fig. 7B
). The more similar the heading tuning among neurons in the population, the greater the benefit of reducing noise correlations. At best, however, the predicted population discrimination threshold for trained animals is ~8% lower than for naïve animals (, right inset, see also Supplementary Fig. 7B
). Clearly, the effect of interneuronal correlations on population coding depends critically on the structure of the correlations, which involves both the relationship between rnoise
and the distribution of tuning similarity among neurons.
Possible rsignal distributions in area MSTd
Might heading be decoded from a subpopulation of MSTd neurons with similar tuning properties (positive rsignal
), such that the uniform reduction of rnoise
in trained animals might improve discrimination performance? Although we cannot firmly exclude this possibility, two observations suggest that it is unlikely. First, electrical microstimulation of multi-unit clusters with either leftward or rightward heading preferences can bias choices during a heading discrimination task (Britten and van Wezel, 1998
; Gu et al., 2008b
). Second, significant choice probabilities, which may reflect the contribution of single cortical neurons to behavior (Britten et al., 1996
; Gu et al., 2007
; Purushothaman and Bradley, 2005
) (but also see Nienborg and Cumming, 2010
), were reported for MSTd neurons preferring both rightward and leftward headings (Gu et al., 2008a
; Gu et al., 2007
). Thus, we further examined the dependence of choice probability and noise correlation on heading preference.
Compared to neurons with lateral heading preferences, neurons with a preference for fore-aft movement show significantly smaller choice probabilities (p=0.019, t-test, ). This result is consistent with the notion that neurons with direction preferences deviated away from straight ahead are more sensitive to small heading variations and thus contribute more to perception (Gu et al., 2007
; Purushothaman and Bradley, 2005
). Importantly, there was no significant difference in average choice probability between neurons preferring leftward and rightward headings (p=0.11, t-test), suggesting that the population of neurons that contributes to heading perception includes cells with both positive and negative signal correlations (inset to ).
Figure 8 Relationships between choice probability, noise correlation, and heading preferences in MSTd. (A) Choice probability tends to be more deviated away from the chance level of 0.5 for neurons with lateral heading preferences. Filled symbols denote choice (more ...)
Interestingly, a similar dependence on heading preference was not observed for noise correlations in trained animals. As shown in , there was no significant dependence of noise correlation on the heading preferences of MSTd neurons (p=0.2, t-test). Indeed, the average noise correlation for lateral neurons is a bit smaller than that for the fore-aft neurons. This finding suggests that the variation in choice probability with heading preference () is not driven just by correlated noise, but also depends on other factors such as how the signals are read out by decision circuitry.