Task onset reduces neural variability and correlations across spiking populations in NHPs

We estimated the spiking variability and spike count correlations of cortical populations in NHPs following task cue onset (task periods) and during the inter-trial intervals (ITI) (rest periods). We found that across trials, global spiking variability and spike count correlations (r_sc) decreased during task as compared to rest (exploratory subject, variance diff = -3.12, t₃₀₃ = −10.91, p<10e-22, r_sc diff = -0.04, t₃₀₃ = −, p<10e-06; Fig 2C and 2D; replication subject, variance diff = -5.37, t₈₀₇ = −13.45, p<10e-35; r_sc diff = -0.04; t₈₀₇ = −9.08, p<10e-17; S1 Fig). Variability reductions were also observed using fano factor (rather than variance) at both the mean-field (averaged across neurons; S14 Fig) and for the majority of individual neurons in each cortical area (S12 and S13 Figs). Correlated variability reductions were also observed using spike count covariance (rather than correlations) (Fig 2H). In addition, we demonstrated that variability and correlation decreased within trial (across time points within a trial, after removing the mean task-evoked response), demonstrating that task state quenching also occurs on a moment-to-moment basis, rather than only on a slower trial-to-trial timescale (S2 Fig). We also measured the spiking variability for each cortical area separately, finding that 5/6 cortical areas reduced their spiking variability during task states in the exploratory subject (all areas except for MT, FDR-corrected p<0.05). In the replication subject, all cortical areas, including MT, reduced their spiking variability (FDR-corrected p < 0.0001). Similarly, we found that during task states, the spike count correlation significantly decreased between a majority of cortical areas (FDR-corrected p<0.05; Fig 2D–2G).

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Fig 2. Neural variability and correlations decrease during task states relative to rest in spiking data.

Results for the replication subject are reported in S1 Fig. a) We measured mean-field spike recordings from six different cortical areas during a motion-color categorization task. b) We calculated the average spike rate across all recordings during the rest period (ITI) and task period (task cue), across trials. Each data point reflects the firing rate across 25 consecutive trials. c) We calculated the cross-trial spiking variance for each region during task and rest states, and then averaged across all regions. Each data point reflects the spiking variance across 25 consecutive trials. d) We calculated the average cross-trial neural correlation for task and rest states between all pairs of recorded brain regions. (Spike rates were averaged within each cortical area.) Each data point reflects the correlation across 25 consecutive trials. e-g) For each pair of brain regions, we visualize the correlation matrices between each recording site for the averaged rest period, task period, and the differences between task versus rest state spike count correlations. h) We also observed no increases in covariance (non-normalized correlation) [31–33]. For panels e-h, plots were thresholded and tested for multiple comparisons using an FDR-corrected p<0.05 threshold. Boxplots indicate the interquartile range of the distribution, dotted black line indicates the mean, grey line indicates the median, and the distribution is visualized using a swarm plot. Scatter plot visualizations of b-d can be found in S15 Fig.

https://doi.org/10.1371/journal.pcbi.1007983.g002

We did not identify any pairwise correlation and covariance increases in our exploratory NHP (Fig 2G). However, in our replication NHP we found correlation increases between visual and frontal areas (i.e., MT/IT and PFC/FEF) (S1 Fig). When analyzed with covariance (rather than correlation), we found these covariance increases to be weak relative to the observed covariance decreases. (Moreover, the baseline correlation strength between these areas was very low during the ITI period.) Though these correlation increases were only observed in 1 of 2 NHPs, they were generally consistent with our fMRI data (below), which showed that though there were few correlation increases, variability and correlations across cortex were dominated by decreases during task states

To ensure that correlation and variability decreases were associated with increases in the mean activity (rather than just the task period), we estimated the mean spike rate across all regions during the task cue interval and the preceding ITI. Indeed, we found that the mean spike rate during task states was significantly greater than the mean spike rate during rest (exploratory subject, task vs rest firing rate difference = 0.41 Hz, t₃₀₃ = 5.77, p<10e-06, replication subject, rate difference = 0.50 Hz, t₈₀₇ = 3.93, p<10e-04). These findings suggest that task states increase neural activity while quenching spiking variability and spike count correlations across large cortical areas.

Importantly, to accurately dissociate first order statistical effects (mean) from second order effects (variance and covariance/correlation), we removed the cross-trial, mean-evoked response for each task condition. This essential step, which removes the main effect of task, is standard procedure in the spike count (noise) correlation literature [25]. This procedure isolated the underlying spontaneous/background neural activity during task states, which was subsequently used to infer neural interaction through spike count correlation analysis [11]. To ensure consistency between our spiking and fMRI analysis, it was critical that we also carefully removed the mean-evoked response associated with task blocks in our fMRI data (i.e., the main effect of task; see Methods) [27]. To maintain additional consistency between task and rest states in both data sets, we applied the same statistical procedure to our rest data (for both spiking and fMRI data) to control for the possibility that our findings were associated with artifacts related to this procedure (see Methods). (However, we note that the “mean effects” removed as a result from this step during rest periods were negligible.)

Task-state variability is globally quenched across a wide battery of tasks in human fMRI data

Consistent with the spiking literature, previous work in the fMRI literature has demonstrated that increased activity associated with task-evoked states quenches neural variability [1,2,4]. We extended those findings to evaluate variability quenching across seven additional cognitive tasks in humans using data from the Human Connectome Project (HCP) [34]. We calculated the variability (estimated using time series variance) during task blocks, averaged across tasks and across regions. Consistent with previous reports, we found that the global variability during task blocks was significantly lower than the variability during equivalent periods of resting-state activity (exploratory cohort variance difference = -0.019, t₁₇₅ = −23.89, p<10e-56; replication cohort variance difference = -0.019, t₁₇₅ = −20.72, p<10e-48; Fig 3A). These findings suggest that task states are associated with task-evoked variability reductions.

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Fig 3. Variability and correlations decrease during task states in human fMRI data.

Figures for the replication cohort are in S4 Fig. Figures for each task separately are shown in S8 and S9 Figs. a) We first compared the global variability during task and rest states, which is averaged across all brain regions, and then b) computed the task- versus rest-state variability for each brain region. c) Scatter plot depicting the variance of each parcel during task states (y-axis) and rest states (x-axis). Dotted grey line denotes no change between rest and task states. d) We next compared the correlation matrices for resting state blocks with (e) task state blocks, and (f) computed the task- versus rest-state correlation matrix difference. g) We found that the average FC between all pairs of brain regions is significantly reduced during task state. h) We found that the average correlation for each brain region, decreased for each brain region during task state. i) Scatter plot depicting the FC (correlation values) of each pair of parcels during task states (y-axis) and rest states (x-axis). Dotted grey line denotes no change between rest and task states. For panels b-f, and h, plots were tested for multiple comparisons using an FDR-corrected p<0.05 threshold. Boxplots indicate the interquartile range of the distribution, dotted black line indicates the mean, grey line indicates the median, and the distribution is visualized using a swarm plot.

https://doi.org/10.1371/journal.pcbi.1007983.g003

To better understand how global this phenomenon was, we plotted the change in variability from rest to task for each brain region separately. We found that almost all brain regions significantly reduced their variability from rest to task, suggesting that variability reduction occurs across most brain regions (cortical maps are thresholded using an FDR-corrected threshold of p < 0.05; Fig 3). This finding extends the work of a previous study in human fMRI data during a finger tapping task [1,8], suggesting that task-induced variability reduction is a general phenomenon consistent across most cortical regions, and across a wide variety of cognitive tasks.

Lastly, we evaluated whether variability quenching occurred during task blocks relative to inter-block intervals (rather than comparing task runs to resting-state runs). Since we z-normalized each task run with unit variance, we could evaluate the degree to which variability was quenched during task blocks relative to inter-block intervals by computing the average variance during task blocks relative to 1. (Note that z-normalization of the task time series was performed after removing the mean task-evoked response via a task GLM, such that reduced variability was not an artifact of preprocessing/z-normalizing the time series.) Indeed, we found that the variance during task blocks was reduced relative to the inter-block intervals (exploratory cohort variance −1 = -0.019, t₁₇₅ = −36.58, p<10e-83; replication cohort variance difference = -0.018, t₁₇₅ = −33.01, p<10e-76). Our findings demonstrate that task-evoked periods quench neural variability relative to both resting-state activity and inter-block intervals.

Task-state FC is globally quenched across a wide battery of tasks in human fMRI data

Despite multiple studies describing task-evoked FC changes [15–17], the precise mechanisms of how FC can change remain unclear. Our current findings illustrate that mean-field spike count correlations decrease during task-evoked states, consistent with previous literature that focused on local circuits [4,11]. Consistent with the spiking literature’s perspective on spike count correlations, and the theoretical evidence suggesting that the correlation of ongoing spontaneous activity should be suppressed during task to facilitate information coding [23], we hypothesized that FC would also be globally reduced during task states. To ensure consistency in the statistical analysis across spiking and fMRI data, we removed the mean task-evoked response using a finite impulse response (FIR) model. This approach is statistically equivalent to removing the cross-trial mean response of a task condition, and is a critical step when calculating noise correlations in the spiking literature [11]. This step characterizes the correlation of the background spontaneous neural activity (i.e., background connectivity in fMRI), dissociating task-to-neural interactions (main effect of task) from neural-to-neural interactions (FC) [35].

We first calculated the mean FC across all pairwise correlations across all cortical regions for both task and rest states (Fig 3D–3F). We found that during task states, the global FC was significantly reduced relative to resting-state fMRI (exploratory cohort FC diff = -0.05, t₁₇₅ = −13.83, p<10e-29; replication cohort FC diff = -0.046, t₁₇₅ = −14.00, p<10e-29; Fig 3G). Recent studies have suggested that the use of correlation provides an ambiguous description of how shared variability (relative to unshared variability) change between brain areas [31,32]. Thus, to generalize these results, we also calculated FC using covariance rather than correlation, finding that covariance also globally decreases (covariance diff = -192.96, t₃₅₁ = −27.30, p<10e-88; S5 Fig). Task-evoked global FC was also reduced in each of the 7 HCP tasks separately (all tasks FDR-corrected p<0.0001; S9 Fig). To identify exactly how global this phenomenon was, we plotted the average task versus rest FC change for each brain region (Fig 3H and 3I). We found that nearly all cortical regions significantly reduced their correlation with the rest of cortex during task states. To ensure that correlation differences between rest and task states were not associated with in-scanner head motion, we calculated the average number of motion spikes during rest and task scans using a relative root mean squared displacement threshold of 0.25mm [36]. For both the exploratory and replication cohorts, we found no significant differences in the percentage of motion spikes between rest and task states (exploratory set, average task = 0.91% of frames, average rest = 0.81% of frames, t₁₇₅ = 1.08, p = 0.28; replication set, average task = 0.009% of frames, rest = 0.008% of frames, t₁₇₅ = 1.53, p = 0.12).

While we primarily observed global decreases in FC, a small portion of connections increased their FC during task states (exploratory cohort, 7.59% of all connections; replication cohort, 9.07% of all connections; FDR-corrected p<0.05) (Fig 3I, S4 Fig). However, FC increases were typically limited to cross-network correlations between networks with different functions, where baseline resting-state FC is already quite low (e.g., cingulo-opercular network with the default mode network, or the frontoparietal network with the visual network) (Fig 3D–3F).

Task state variability and correlation are quenched independently of removing the mean task-evoked response in fMRI data

The above fMRI results employ the use of FIR modeling to remove the mean task-evoked response to compare task- and rest-state correlations/variability. Here we sought to demonstrate that neural variability and correlations are quenched in fMRI data in the absence of any task regression (e.g., FIR modeling). We used an approach that has been previously used to demonstrate variability quenching following task onset, by measuring the cross-trial variance at each time point [2,4]. We employ the same general approach, measuring the variance and correlation across blocks for each time point within the block. Moreover, to obtain statistically comparable estimates of resting state variability/correlations, we measured the cross-block variance/correlation during sham blocks during resting state by applying the identical task block structure to resting-state fMRI data. Critically, the removal of the mean task-evoked response was excluded from preprocessing for this analysis, and the time series were not z-normalized.

We found that cross-block variance for time points during task state were significantly reduced relative to resting state (var diff = -1009.56, t₁₇₅ = −37.34; p<10e-84; Fig 4A and 4B). We also found consistent results for correlations, finding that the cross-block correlation for time points during task state were significantly reduced relative to resting state (r diff = -0.04, t₁₇₅ = −10.91, p<10e-20; Fig 4C and 4D). These results demonstrate that the quenching of correlations and variability during task states are independent of any potential statistical artifacts that result from removing the mean task-evoked response using FIR task regression.

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Fig 4. Task variability/correlations decrease independently of mean task activity removal step in fMRI data.

Instead of computing variance/correlations across time points within task blocks (and removing mean task effects), variance/correlations can be calculated across task blocks (for each time point within a block). This approach isolates ongoing neural activity that is not task-locked, and has been used in both spiking and fMRI data [2,4]. a) To isolate ongoing spontaneous activity that is not time-locked to the task, we estimated the variance at each time point across task blocks. The variance at each time point was calculated for each ROI and task condition separately, but then averaged across ROIs and task conditions. Note that to obtain an equivalent variance estimate during resting state, we applied an identical block structure to rest data to accurately compare rest to task state variability. Variability across block time points was averaged across brain regions and task conditions. Error bars denote standard deviation across subjects. b) Variance across task block time points was significantly reduced during task blocks relative to identical control blocks during resting-state data. c) We performed a similar procedure for task functional connectivity estimates, correlating across blocks for all pairs of brain regions. Correlations across block time points were averaged for all pairs of brain regions and task conditions. d) Correlations during task state blocks were significantly reduced relative to identical control blocks during resting state. Boxplots indicate the interquartile range of the distribution, dotted black line indicates the mean, grey line indicates the median, and the distribution is visualized using a swarm plot.

https://doi.org/10.1371/journal.pcbi.1007983.g004

Task-evoked activations are negatively correlated with neural variability and correlations in human fMRI data

We showed that task states widely reduced neural variability and correlations. We sought to extend this work to directly demonstrate that decreases in neural variability and correlations are associated with changes in task-evoked activation levels. To provide evidence for this hypothesis, we computed the mean task-evoked activation (averaging across all regions). We found that the global activation was significantly greater than baseline across different task states (exploratory cohort, t₁₇₅ = 6.46, p<10e-9; replication cohort, t₁₇₅ = 12.63, p<10e-25), demonstrating that decreases in neural variability and FC were accompanied by global increases in task-evoked activations.

Previous work has shown that regions that have strong task activations (i.e., the magnitude of the task-induced activation, positive or negative) tend to have greater variability reductions [2]. (Task activation magnitudes reflect the deflection of the BOLD activity relative to baseline, or the inter-block interval.) We sought to replicate this effect in the current data set, while extending those results to demonstrate that more task-active regions also tend to reduce their FC during task states. We first correlated regional task-evoked activation magnitude with task-evoked variability reduction (task variance minus rest variance) across regions at the group-level. We found that regions with greater task-evoked activation magnitudes (averaged across tasks) exhibited greater variability reductions during task states, confirming previous findings in a finger tapping task (exploratory cohort rho = -0.32, p<10e-9; replication cohort rho = -0.49, p<10e-22; Supplementary Fig 3A and 3C) [2]. This negative relationship was also observed in 6/7 of the HCP tasks when analyzed separately (FDR-corrected p<0.01; S8 Fig). To link regional task activation magnitudes with FC decreases, we tested for a correlation between regional task-evoked activation magnitude and the average FC change during task states for each region. Consistent with our hypothesis, we found that regions with greater task-evoked activation magnitudes (averaged across tasks) reduced their average FC more during task states (exploratory cohort rho = -0.25, p<10e-05; replication cohort rho = -0.20, p = 0.0002; S3 Fig). When tasks were analyzed separately, this negative correlation was observed in 4/7 of the HCP tasks (FDR-corrected p<0.05; S9 Fig). Thus, brain areas with higher levels of task-evoked activation magnitudes (i.e., changes in activity relative to baseline) tend to reduce both their task-evoked variability and FC.

The information-theoretic relevance of task state reduction of neural correlations

Results from our empirical data converged across imaging modalities and species, illustrating that task states increased mean activity while reducing neural variability and correlations. However, the theoretical implication of a decreased correlated task state remains unclear. Here we sought to better characterize the information-theoretic implication of a global reduction in neural correlations. In particular, consistent with previous large-scale computational models that have predicted increased dimensionality with stimulus-driven activity [37,38], we hypothesized that reductions in neural correlations increase the effective dimensionality across units by suppressing background spontaneous activity/noise. While this increased dimensionality may potentially supports more robust information representations, we acknowledge that a change in neural dimensionality does not necessitate an improvement (or change) in cognitive information representation [23], and that future studies will need to evaluate the relationship between neural dimensionality and cognitive content. Further, we note that an increase in dimensionality is not trivially implied by decreased global correlations. Because we found that regional time series variance also decreases during task states, the neural data dimensionality would increase only if inter-region covariance decreases more than local regional variance (i.e., off-diagonal is reduced more than the diagonal of the variance-covariance matrix).

We measured the dimensionality using the ‘participation ratio’ [37,39] of the neural activity (for human fMRI and NHP spiking data) during rest and task states (see Methods). Consistent with our hypothesis, we found that task states increased their overall dimensionality relative to rest states (fMRI task versus rest, exploratory cohort difference = 16.13, t₁₇₅ = 19.31, p<10e-44, replication cohort difference = 15.78, t₁₇₅ = 21.66, p<10e-50; NHP task versus rest, exploratory subject difference = 0.13, t₃₀₃ = 5.77, p<10e-07, replication subject difference = 0.26, t₈₀₇ = 13.00, p<10e-34) (Fig 5). We also found that when analyzing each of the 7 HCP tasks separately, dimensionality increased in all 7 tasks relative to resting state (FDR-corrected p<0.0001; S10 Fig). The present results suggest that task states were associated with a decrease in neural variability and correlations, reflecting a suppression of shared and private spontaneous activity, which increased the dimensionality of neural activity.

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Fig 5. Dimensionality increases during task periods relative to resting-state activity.

a) For each subject, we calculated the dimensionality using the participation ratio [37,39] during task and rest states and found that during task states, dimensionality significantly increased. b) We calculated the dimensionality of spiking activity across trials and found that during task states, dimensionality significantly increased. These findings provide a potential information-theoretic interpretation of neural correlation and variability reduction during task states. Boxplots indicate the interquartile range of the distribution, dotted black line indicates the mean, grey line indicates the median, and the distribution is visualized using a swarm plot.

https://doi.org/10.1371/journal.pcbi.1007983.g005

From neurons to neural masses: Modeling neural dynamics of cortical areas

In the previous sections, we provided empirical evidence that task states reduce mean-field inter-area correlations and variability in spike rate and fMRI data. In this section, we construct a biologically plausible model that provides a parsimonious explanation of correlation and variability reductions in mean field spiking networks and cortical BOLD dynamics.

Neurophysiologically, functional brain areas are composed of local circuits with balanced excitatory and inhibitory neural activity (Fig 6A). In previous work, local circuits have been demonstrated to have clustered excitatory connections [40], leading to slow dynamics and high variability in spiking networks simulated in silico [7]. Using this previously established model, we systematically perturbed this balanced network under a distribution of inputs (both excitatory and/or inhibitory inputs) to estimate the excitatory output (i.e., mean-field transfer function) of a cortical population. Though most long-range cortical connections are excitatory, we incorporated excitatory and inhibitory stimulation effects on a local population (Fig 6B). This is because long-range excitatory afferents may target local inhibitory neurons, producing a net inhibitory effect. Under the presence of inputs, we found that the population transfer function approximated a sigmoid activation function (Fig 6B). We note that the upper bound on the sigmoid transfer function (Fig 6D) is likely due to inhibitory feedback on excitatory activity rather than the true saturating spiking regime in neurons. This is because excitatory neurons in a local population typically do not reach a saturating spiking regime even for strong visual stimuli [41], and instead reach an upper bound due to strong inhibitory stabilization preventing runaway excitation [5]. Importantly, simplifying the mean-field transfer function of a cortical area allowed us to focus our modeling efforts on simplified networks across large cortical areas [42].

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Fig 6. Inferring the mean-field transfer function of a neural population with a balanced spiking model with clustered excitatory connectivity.

a) Schematic illustration of the balanced spiking model with clustered excitatory connections. Network architecture and parameters are identical to those reported in [7]. Red triangles indicate excitatory cells, blue circles indicate inhibitory cells. b) The population spike rate (excitatory cells only) subject to inhibitory regulation. We systematically stimulated a subset of the neural population and measured the corresponding mean excitatory spike rate. Spike rates were normalized between 0 and 1. Excitatory stimulation was implemented by stimulating 400 excitatory neurons, and inhibitory stimulation was implemented by stimulating 400 inhibitory neurons. Spiking statistics were calculated across 30 trials, with each point in the scatter plot indicating a different 50ms time bin. c) Population neural variability (excitatory cells only), as a function of input stimulation. d) Based on panel b, we approximated the mean field neural transfer function as a sigmoid. A sigmoid transfer function produces optimal input-output dynamics for a narrow range of inputs (gray). The same input distribution mean shifted by some excitatory/inhibitory stimulation produces a quenched dynamic range.

https://doi.org/10.1371/journal.pcbi.1007983.g006

In this balanced spiking network, any evoked stimulation, excitatory or inhibitory, would result in reduced variability (Fig 6C). Specifically, the magnitude of stimulation was negatively correlated with spiking variability in the balanced spiking model (rho = -0.92; p<0.0001). While previous studies have suggested that the mean and variance of the spike rate may be independent of each other, those studies focused on mean-matching the spike rate of individual neurons within the same local population [4,7]. However, in this study, we focus exclusively on the mean-field level rather than individual neurons. We found a highly negative association between mean and variance under experimental perturbation, suggesting that at the mean-field level, mean and variance cannot be mechanistically dissociated. Based on these considerations, we hypothesized that during periods in which global neural activity levels are elevated, such as task states, both neural variability and correlations would be globally quenched.

Neural variability is quenched during task-evoked states in a neural mass model

Here we rigorously ground the intuition that task-evoked activity reduces output variability using neural mass modeling and dynamical systems theory. A recent study provided evidence that an evoked stimulus drives neural populations in sensory cortex around a stable fixed point attractor [5]. We first extended these findings using a simplified neural mass model, which allows for a comprehensive dynamical systems analysis that is mathematically difficult in higher dimensions. Additionally, this enabled a simpler theoretical approach to investigating changes in neural dynamics that are generalizable across mean-field neural cortical areas (i.e., populations with sigmoidal transfer functions).

We first characterized the relationship between task-evoked and spontaneous activity in a large neural population using a single neural mass unit. We simulated the neural population’s dynamics across a range of fixed input strengths (Fig 7A), finding a nonlinear relationship between stimulus strength and the observed variability of the neural population (Fig 7C). We found that variability was highest when there was no stimulation, while variability decreased for any type of evoked stimulation (e.g., negative or positive input amplitudes). Despite the model’s simplicity, these findings are consistent with our (and others’) empirical and model results demonstrating that task states quench time series variability in both human and animal data [1,4,5]. We also generalized the findings from our minimal (single region) model to large-scale firing rate models (with 300 regions), where we found variability decreases during task-evoked states in both network models with random structural connections and clustered structural connections (S11 Fig). We demonstrated this for network models with excitatory connections only, as well as networks with both excitatory and inhibitory connections. However, due to the large number of possible network models when scaling to n-dimensions, we constrained our analyses to only four network architectures, leaving a more complete analysis to future studies.

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Fig 7. Task-evoked activity induces changes in neural variability and the underlying attractor dynamics.

Our minimal modeling approach directly links descriptive statistics (e.g., time series variability) with rigorous dynamical systems analysis (e.g., attractor dynamics). a) During different evoked states (i.e., fixed inputs), there is a reduction in the observed time series variability (measured by variance across time). This is directly related to how input-output responses change due to the changing slope in the sigmoid transfer function. b) We visualized the phase space for each of the neural populations according to state by plotting the derivative of X₁ denoted by . For each state, we estimated the fixed point attractor (plotted as a star), denoting the level of mean activity the system is drawn to given some fixed input (or absence thereof). Arrows denote the direction/vector toward each fixed point, which specify the characteristic time scale (i.e., the speed) the system approaches the fixed point. c) We ran simulations across a range of stimulation amplitudes, calculating the variance across time at each amplitude. d) We characterized the shifting attractor dynamics for each stimulus by computing the characteristic time scale at the fixed point for each stimulation amplitude. The characteristic time scale across all fixed points are nearly perfectly correlated with the neural variability of the simulated time series across all fixed inputs (rank correlation = 0.9996).

https://doi.org/10.1371/journal.pcbi.1007983.g007

We sought to leverage the model’s simplicity to characterize dynamical systems properties governing the observed neural variability. This would provide rigorous evidence that shifting the underlying attractor dynamics alters the observed neural signals. We first performed a state space analysis in one dimension to identify the stable fixed point attractor (i.e., the equilibrium level of activity the system is drawn to during a particular state) for the intrinsic and evoked states (Fig 7B). The state space view enabled visualization of the system’s full dynamics across different evoked states (Fig 7B). For example, dynamics around the fixed point attractor in the intrinsic baseline (rest) state appeared to approach equilibrium slowly. This can be identified by observing the angle where the curve intersects 0 on the y-axis (i.e., when ; Fig 7B). The angle of this curve corresponds to the characteristic time scale, a dynamical property characterizing the speed with which the system approaches the attractor (a higher value reflects slower dynamics; see Methods) [43].

To quantify this more rigorously, we performed a linear stability analysis around the fixed point attractor of the system across the same range of stimulation amplitudes. For each input, we analytically calculated the characteristic time scale at each fixed point. Again, we found a nonlinear relationship between the amplitude of the stimulus and the characteristic time scale of the neural population (Fig 7D), and found that the characteristic time scale explained nearly 100% of the variance (rho = 0.9996) of the simulated stimulus-evoked variability (Fig 7C). These results demonstrate that changes in observed neural variability can be directly attributed to changes in the underlying attractor dynamics.

To ensure that the model explanation would generalize to data obtained on a slower time scale (e.g., fMRI BOLD data), we transformed the simulated neural activity into fMRI BOLD activity using the Balloon-Windkessel model [44]. The Balloon-Windkessel is a nonlinear transformation of neural activity to model the BOLD signal that takes into account the normalized blood volume, blood inflow, resting oxygen extraction fraction, and the normalized deoxyhemoglobin content. Consistent with previous accounts [2], we found that the characteristic time scale around the fixed point attractor was still strongly correlated with BOLD variability (rho = 0.97; p<0.0001; S6 Fig).

Neural correlations are quenched during task states in a network model

We generalized the dynamical systems analysis in one dimension to two dimensions, allowing us to focus on correlations across cortical areas. We show illustrations of the state space for intrinsic and task-evoked states (Fig 8B and 8D), as well as the corresponding time series (Fig 8A and 8C) for our model. Induced negative activity produced qualitatively similar results to the activated state (Fig 8D) due to subthreshold levels of activity rather than saturating levels of activity.

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Fig 8. Task-evoked activity quenches neural correlations by altering the underlying attractor dynamics.

We used a two unit network model, the minimal model necessary to study dynamic changes in neural correlations. a) At baseline, we observed slow, high amplitude fluctuations and high neural correlations. b) To characterize the underlying attractor dynamics, we visualized the two-dimensional state space, visualizing the flow field and the nullclines (blue and red curves, where the rate of change is 0) for each unit. The intersection of the two nullclines denote the fixed point attractor. We overlaid the simulated scatter plot (cyan dots) to illustrate the correspondence between the attractor dynamics and simulation. c) We injected a fixed input stimulation, shifting the network to an ‘evoked’ state, which caused a decrease in neural variability and correlation. d) The external input transiently moved the fixed point, altering the attractor dynamics and the corresponding scatter plot. e) We systematically injected a range of fixed inputs into the network. We found that neural correlations were optimal with no external stimulation, and decreased with any external stimulation. f) Across stimulation strengths, we found that the generalized characteristic time scale (see Methods) near the fixed point explained 98% of the neural correlation variance, providing a direct association between the network’s attractor dynamics and observed neural correlations.

https://doi.org/10.1371/journal.pcbi.1007983.g008

The state space analysis (Fig 8B and 8D) allowed us to track the simultaneous evolution of the two neural masses, providing a geometric interpretation of the system. We observed qualitatively that shifts in the attractor dynamics (i.e., changes to the flow field) due to stimulation were directly associated with changes to the correlation between the two neural masses. Specifically, we observed that intrinsic state dynamics supported slower, elongated trajectories along a diagonal axis, consistent with correlated neural activity between the two masses (Fig 8B). This was due to a slower characteristic timescale near the fixed point attractor, which corresponds mathematically to eigenvalues with smaller magnitudes. In contrast, during evoked states, the system approached the fixed point attractor at a faster speed, quenching trajectories in state space that supported correlated variability (Fig 8D). Thus, the visualization of the state space demonstrated that changes in neural correlations were associated with changes to the flow field around the fixed point attractor.

To more carefully test the relationship between state-dependent neural correlations, we simulated our network model across a range of fixed input amplitudes. Despite no changes to the network’s connectivity structure, we found that neural correlations systematically changed (decreased) as a function of evoked stimulation (Fig 8E). Further, using dynamical systems analysis, we found that a generalization of the characteristic time scale in higher dimensions accounted for changes in neural correlations (rho = 0.99; p<0.0001; Fig 8F). In other words, we analytically determined that evoked stimulation shifted the attractor dynamics, changing the neural correlations in a network model with fixed synaptic connections. We found consistent results after transforming the neural activity to fMRI BOLD activity using the Balloon-Windkessel model [44], finding that changes to the characteristic time scale accounted for changes in BOLD correlations (i.e., FC) (rho = 0.97; p<0.0001; S7 Fig). These results were reproduced using mutual information (rho = 0.94; p<0.0001), a nonlinear measure of statistical dependence [45], and non-parametric rank correlation (rho = 0.99; p<0.0001). This suggests that the quenching of shared variance encompasses both parametric and non-parametric linear and nonlinear measures of statistical dependencies.

To ensure that our findings in simplified two node networks would scale to large-scale network models, we simulated large-scale firing rate models (with 300 regions). We found correlation decreases during task-evoked states in both network models with random structural connections and clustered structural connections (S11 Fig), suggesting that the mechanisms we identified in these minimal models likely scale to the larger networks. We demonstrated this for network models with excitatory connections only, as well as networks with both E and I connections.

Spiking data: Data collection

The behavioral paradigm for each monkey was a motion-color categorization task (Fig 1B). Experimental methods for electrophysiology data collected for NHP was previously reported in [29] and [59]. Data was collected in vivo from two (one female) behaving adult rhesus macaques (Macaca mulatta) across 55 sessions. Data from six distinct cortical regions were recorded simultaneously from acutely inserted electrodes. Cortical regions included: MT, V4, PIT, LIP, FEF, and LPFC (Fig 1A). Spikes from each region were sorted offline into isolated neurons. However, given our interest in inter-region neural correlations across large scale neural systems, we pooled spikes from each functional area into a single spike rate time series. For each trial, spikes were sorted for a 5s period, beginning 2.5s prior to stimulus onset, and until 3.5s after stimulus onset. Further details regarding electrophysiological data collection can be found here:

http://www.sciencemag.org/content/348/6241/1352/suppl/DC1 and here:

http://www.pnas.org/content/pnas/suppl/2018/07/09/1717075115.DCSupplemental/pnas.1717075115.sapp.pdf

All statistical analyses in the main article (detailed below) were performed on a single monkey. Independent replication was performed on the second monkey, and is reported in S1 Fig.

Spiking data: Task versus rest variability analysis

Neural variability analysis was analyzed using an analogous approach to both the computational model and fMRI data. However, since we had no true ‘resting-state’ activity for the monkey data set, we used the inter-trial interval (ITI; 0.5s - 1s variable duration, see Fig 1B) as “resting-state activity”. We used the 0.5s - 1s interval immediately preceding the trial’s fixation period to avoid any reward/feedback signals from the previous trial. (Reward/feedback from the previous trial was provided more than 1.5s prior to the fixation period.) Spike counts were calculated by taking a 50ms sliding window with 10ms increments, consistent with previous studies [4]. The mean-evoked response across all trials for a given task rule (e.g., motion rule versus color rule) was calculated and removed from each trial, as is common in the spike count literature [11] and the fMRI literature [27]. (Statistically this is equivalent to performing the task activation regression in the fMRI data, described below.) The mean task-evoked response of the ITI period associated with each task condition was also removed. This was to control for any artifacts that might be induced due to removal of the mean-evoked response. Trials with less than 500ms (or 50 time points) worth of spiking data for either the ITI and/or task cue presentation were excluded. This was done to reduce variability of the estimated spike count correlations, since correlations with few observations are highly variable.

We computed the variance across 25 consecutive trials using the spike rate from each cortical recording during either the ITI or the task cue period. This was repeated for all trials for each subject. We used across-trial variance to calculate variability rather than Fano factor [4]. This choice was due to the insight from our model illustrating that the mean-evoked activity and the corresponding variance interact in a nonlinear manner, and that the Fano factor is computed as the variance over the mean. Cross-trial variance was computed as (1) Where n = 25 trials, r_trial reflected the spike rate of each trial, and is the cross-trial mean firing rate for the task condition (i.e., either the cross-trial mean firing rate during the color or motion task cue period).

The statistical difference in task versus rest neural variability was computed by using a two-way, paired t-test across all bins of 25 consecutive trials. The global neural variability change was computed by averaging the variance across all recording areas for each bin. Statistics for the regional neural variability change were corrected for multiple comparisons using an FDR-corrected threshold of p<0.05.

In addition, we computed the variance during the ITI and task cue period within trial (across time points) (S2 Fig). This analysis demonstrated that moment-to-moment variability (rather than trial-to-trial variability) was also quenched from rest to task periods, suggesting that variability quenching also occurs at faster timescales. The statistical difference in task versus rest neural variability was computed by using a two-way, paired t-test (paired by trial) across all trials for each monkey separately. The global neural variability change was computed by averaging the variance across all recording areas for each trial (S2 Fig). Statistics for the regional neural variability change were corrected for multiple comparisons using an FDR-corrected threshold of p<0.05.

Spiking data: Task versus rest state correlation analysis

Neural correlations for spiking data using the same preprocessing steps mentioned above for spike rate variability analysis. Specifically, the mean-evoked response across all trials for each task condition was removed from each trial. Spike count correlations were then computed across trials, using groups of 25 trials as described above (Fig 2E and 2F).

The difference in task versus rest neural correlations was calculated using a two-way, paired t-test (paired by each bin of 25 trials) for each subject separately using Fisher’s z-transformed correlation values. The global neural correlation change was computed by averaging the Fisher’s z-transformed correlation values between all pairs of cortical regions, and comparing the averaged task versus rest correlation for each bin (Fig 2D). Statistics for the pairwise neural correlation change (Fig 2E–2G) were corrected for multiple comparisons using an FDR-corrected threshold of p<0.05.

In addition, we computed the spike count correlation during the ITI and task cue period separately within trial (across time points) (S2 Fig). This analysis demonstrated that moment-to-moment correlations (rather than trial-to-trial correlations) were also quenched from rest to task periods, suggesting that correlation quenching also occurs at faster timescales. The statistical difference in task versus rest neural correlations was computed by using a two-way, paired t-test (paired by trial) across all trials for each monkey separately. The global neural correlation change was computed by averaging the correlation across all pairs of recording areas for each trial (S2 Fig).

fMRI: Data and paradigm

The present study was approved by the Rutgers University institutional review board. Data were collected as part of the Washington University-Minnesota Consortium of the Human Connectome Project (HCP) [34]. A subset of data (n = 352) from the HCP 1200 release was used for empirical analyses. Specific details and procedures of subject recruitment can be found in [34]. The subset of 352 participants was selected based on: quality control assessments (i.e., any participants with any quality control flags were excluded, including 1) focal anatomical anomaly found in T1w and/or T2w scans, 2) focal segmentation or surface errors, as output from the HCP structural pipeline, 3) data collected during periods of known problems with the head coil, 4) data in which some of the FIX-ICA components were manually reclassified; low-motion participants (i.e., exclusion of participants that had any fMRI run in which more than 50% of TRs had greater than 0.25mm framewise displacement); removal according to family relations (unrelated participants were selected only, and those with no genotype testing were excluded). A full list of the 352 participants used in this study will be included as part of the code release.

All participants were recruited from Washington University in St. Louis and the surrounding area. We split the 352 subjects into two cohorts of 176 subjects: an exploratory cohort (99 females) and a replication cohort (84 females). The exploratory cohort had a mean age of 29 years of age (range = 22–36 years of age), and the replication cohort had a mean age of 28 years of age (range = 22–36 years of age). All subjects gave signed, informed consent in accordance with the protocol approved by the Washington University institutional review board. Whole-brain multiband echo-planar imaging acquisitions were collected on a 32-channel head coil on a modified 3T Siemens Skyra with TR = 720 ms, TE = 33.1 ms, flip angle = 52º, Bandwidth = 2,290 Hz/Px, in-plane FOV = 208x180 mm, 72 slices, 2.0 mm isotropic voxels, with a multiband acceleration factor of 8. Data for each subject were collected over the span of two days. On the first day, anatomical scans were collected (including T1-weighted and T2-weighted images acquired at 0.7 mm isotropic voxels) followed by two resting-state fMRI scans (each lasting 14.4 minutes), and ending with a task fMRI component. The second day consisted with first collecting a diffusion imaging scan, followed by a second set of two resting-state fMRI scans (each lasting 14.4 minutes), and again ending with a task fMRI session.

Each of the seven tasks was collected over two consecutive fMRI runs. The seven tasks consisted of an emotion cognition task, a gambling reward task, a language task, a motor task, a relational reasoning task, a social cognition task, and a working memory task. Briefly, the emotion cognition task required making valence judgements on negative (fearful and angry) and neutral faces. The gambling reward task consisted of a card guessing game, where subjects were asked to guess the number on the card to win or lose money. The language processing task consisted of interleaving a language condition, which involved answering questions related to a story presented aurally, and a math condition, which involved basic arithmetic questions presented aurally. The motor task involved asking subjects to either tap their left/right fingers, squeeze their left/right toes, or move their tongue. The reasoning task involved asking subjects to determine whether two sets of objects differed from each other in the same dimension (e.g., shape or texture). The social cognition task was a theory of mind task, where objects (squares, circles, triangles) interacted with each other in a video clip, and subjects were subsequently asked whether the objects interacted in a social manner. Lastly, the working memory task was a variant of the N-back task.

Further details on the resting-state fMRI portion can be found in [60], and additional details on the task fMRI components can be found in [30]. All fMRI results reported in the main article reflect results found with the first cohort of subjects. Independent replication of these effects are reported in S4 Fig with the replication cohort.

fMRI: Preprocessing

Minimally preprocessed data for both resting-state and task fMRI were obtained from the publicly available HCP data. Minimally preprocessed surface data was then parcellated into 360 brain regions using the [61] atlas. We performed additional standard preprocessing steps on the parcellated data for resting-state fMRI and task state fMRI to conduct neural variability and FC analyses. This included removing the first five frames of each run, de-meaning and de-trending the time series, and performing nuisance regression on the minimally preprocessed data [36]. Nuisance regression removed motion parameters and physiological noise. Specifically, six primary motion parameters were removed, along with their derivatives, and the quadratics of all regressors (24 motion regressors in total). Physiological noise was modeled using aCompCor on time series extracted from the white matter and ventricles [62]. For aCompCor, the first 5 principal components from the white matter and ventricles were extracted separately and included in the nuisance regression. In addition, we included the derivatives of each of those components, and the quadratics of all physiological noise regressors (40 physiological noise regressors in total). The nuisance regression model contained a total of 64 nuisance parameters. This was a variant of previously benchmarked nuisance regression models reported in [36].

We excluded global signal regression (GSR), given that GSR artificially induces negative correlations [57,63], which would bias analyses of the difference of the magnitude of correlations between rest and task. We included aCompCor as a preprocessing step here given that aCompCor does not include the circularity of GSR (regressing out some global gray matter signal of interest) while including some of the benefits of GSR (some extracted components are highly similar to the global signal) [64]. This logic is similar to a recently-developed temporal-ICA-based artifact removal procedure that seeks to remove global artifact without removing global neural signals, which contains behaviorally relevant information such as vigilance [65,66]. We extended aCompCor to include the derivatives and quadratics of each of the component time series to further reduce artifacts. Code to perform this regression is publicly available online using python code (version 2.7.15) (https://github.com/ito-takuya/fmriNuisanceRegression).

Task data for task FC analyses were additionally preprocessed using a standard general linear model (GLM) for fMRI analysis. For each task paradigm, we removed the mean evoked task-related activity for each task condition by fitting the task timing (block design) for each condition using a finite impulse response (FIR) model [27]. (There were 24 task conditions across seven cognitive tasks.) We used an FIR model instead of a canonical hemodynamic response function given recent evidence suggesting that the FIR model reduces both false positives and false negatives in the identification of FC estimates [27]. This is due to the FIR model’s ability to flexibly fit the mean task-evoked response across all blocks. Removing the mean-evoked response of a task condition (i.e., main effect of task) is critical to isolate the spontaneous neural activity (and similarly the background connectivity [35]). Importantly, this procedure is standard when performing in spike count correlations [11,27]. Analogous statistical preprocessing steps were critical when comparing neural correlation measures across human fMRI data and NHP spiking data.

FIR modeled task blocks were modeled separately for task conditions within each of the seven tasks. Thus, the mean task-evoked activation was differentially accounted for according to each specific task condition. In particular, two conditions were fit for the emotion cognition task, where coefficients were fit to either the face condition or shape condition. For the gambling reward task, one condition was fit to trials with the punishment condition, and the other condition was fit to trials with the reward condition. For the language task, one condition was fit for the story condition, and the other condition was fit to the math condition. For the motor task, six conditions were fit: (1) cue; (2) right hand trials; (3) left hand trials; (4) right foot trials; (5) left foot trials; (6) tongue trials. For the relational reasoning task, one condition was fit to trials when the sets of objects were matched, and the other condition was fit to trials when the objects were not matched. For the social cognition task, one condition was fit if the objects were interacting socially (theory of mind), and the other condition was fit to trials where objects were moving randomly. Lastly, for the working memory task, 8 conditions were fit: (1) 2-back body trials; (2) 2-back face trials; (3) 2-back tool trials; (4) 2-back place trials; (5) 0-back body trials; (6) 0-back face trials; (7) 0-back tool trials; (8) 0-back place trials. Since all tasks were block designs, each time point for each block was modeled separately for each task condition (i.e., FIR model), with a lag extending up to 25 TRs after task block offset.

fMRI: Task state activation analysis

We performed a task GLM analysis on fMRI task data to evaluate the task-evoked activity. The task timing for each of the 24 task conditions was convolved with the SPM canonical hemodynamic response function to obtain task-evoked activity estimates for each task condition separately [67]. FIR modeling was not used when modeling task-evoked activity. Coefficients were obtained for each parcel in the Glasser et al. (2016) cortical atlas for each of the 24 task conditions.

fMRI: Task state versus resting-state variability analysis

To compare task-state versus resting-state variability, we regressed out the exact same task design matrix used on task-state regression on resting-state data. This was possible given that the number of timepoints of the combined resting-state scans in the HCP data set exceeded the number of timepoints of the combined task-state scans (4800 resting-state TRs > 3880 task-state TRs). This step was to ensure that any spurious change induced through the removal of the mean task-evoked response would also induce spurious changes in the resting-state data. However, results were qualitatively identical without the regression of the task design matrix on resting-state data.

After task regression, we obtained the residual time series for both resting-state and task state fMRI data. We then z-normalized each task run with zero-mean and unit variance such that we could appropriately compare the neural variability of task blocks across different runs. We emphasize that task activation regression (removal of the mean task-evoked response) was removed prior to z-scoring the time series. (Additionally, S5 Fig shows results without z-normalization, and the results are qualitatively identical.) This enabled us to evaluate whether the variability during task blocks significantly decreased relative to inter-block intervals by evaluating the variance of task blocks relative to 1. We then extracted the time series variance during task blocks, and then averaged the variance across all task conditions to obtain our statistic of task-evoked neural variability. To identify the resting-state neural variability, we applied the same exact procedure to resting-state time series using the task-state design matrix. A sanity check for our analysis was that the ‘intrinsic-state’ neural variability is close to 1 (given that the time series was normalized to have unit variance), while the task-state neural variability is significantly less than 1 (Fig 3A). This ensured that variability measures were not biased by the normalization step.

We compared the neural variability of the entire brain during task state periods versus resting-state periods. For each subject, we computed the variance during task and resting state separately, and then averaged across all brain regions. This resulted in two values per subject, representing task state and resting-state variability. We then performed a two-way group paired t-test across subjects to assess statistical significance (Fig 3A). We also computed the task state versus resting-state difference in neural variability for each brain region separately (Fig 3B). We corrected for multiple comparisons using an FDR-corrected threshold of p<0.05 (Fig 3B and 3C). Cortical surface visualizations were constructed using Connectome Workbench (version 1.2.3) [34].

fMRI: Task state versus resting-state correlation analysis

We compared task-state versus resting-state FC (i.e., neural correlations), after performing the exact same preprocessing steps as mentioned above. Results without z-normalization (and using covariance rather than correlations) on the task and rest residual time series are reported in S5 Fig.

We computed the correlation between all pairs of brain regions for each task condition during task block periods. We then averaged the Fisher’s z-transformed correlation values across all task conditions to obtain a general task state FC matrix (Fig 3E). We repeated the same procedure (i.e., using the same task-timed blocks) on resting-state FC to obtain an equivalent resting-state FC matrix for each subject (Fig 3D). We directly compared task-state FC to resting-state FC by performing two-way group paired t-tests for every pair of brain regions using the Fisher’s z-transformed correlation values. Statistical significance was assessed using an FDR-corrected threshold of p<0.05 (Fig 3F). To compare the average global correlation during task state and resting state, we computed the average correlation between all pairs of brain regions during task and resting-state, performing a group paired t-test (Fig 3G). To compare the average global connectivity profile of every brain region [68], we computed the average Fisher z-transformed correlation of a single region to all other brain regions during task and rest and performed a two-way group paired t-test between task and rest (Fig 3H and 3I). Statistical significance was assessed using an FDR-corrected threshold of p<0.05.

fMRI: Task state versus resting-state variability/correlation analysis without task regression

To compare task-state versus resting-state variability/correlations without regressing out task effects using FIR [27], we calculated the variance/correlations for each time point across blocks. This approach is similar to previous studies that measured variability changes after task/stimulus onset [2,4]. Importantly, because variance/correlations explicitly account for the mean across a sample, and variance/correlations are computed for each time point separately, this approach accurately accounts for task-locked effects.

Statistics (i.e., variance/correlations) were calculated across blocks at each time point, for each condition separately. To accurately compare task-state to resting-state statistics, we computed cross-block statistics for rest data using the same task block design (i.e., sham/control blocks). This controlled for the number of task blocks and temporal spacing between blocks. We included the first 15 time points following block onset for both the rest and task data. Thus, any task blocks that contained fewer than 15 time points were excluded. This was performed for all ROIs for every subject. Summary statistics were aggregated across ROIs, task conditions and time points (within rest or task states) and visualized in Fig 4.

(For this analysis, we used minimally preprocessed data (from the HCP). Additional nuisance regression was performed for both rest and task data as described above, excluding task activation regression.)

Information-theoretic analysis

We evaluated the information-theoretic relevance of rest and task states by characterizing the dimensionality of neural activity. To estimate the statistical dimensionality of neural data, we used the ‘participation ratio’, as previously described in [39]. We first obtain the covariance matrix W of activity for rest and task states separately. We then calculated (2) Where dim_W corresponds to the statistical dimensionality of W, and λ_i corresponds to the eigenvalues of the covariance matrix W. Intuitively, this is related to finding the number of components needed to explain variance greater than some fixed threshold, with more needed components reflecting a higher dimensionality of the data.

For human fMRI data, we estimated the task-state and resting-state dimensionality by calculating the whole-brain covariance matrix for each state. For task state this was done by estimating the covariance matrix using task block periods. For resting state this was done by calculating the covariance matrix across the equivalently lengthed resting-state periods (using the same data in the FC analysis above). We applied Eq 2 to the task-state and resting-state covariance matrix for each subject. Finally, we applied a two-way, group paired t-test comparing the dimensionality of task-state activity to resting-state activity (Fig 5A). We replicated this finding in the replication cohort. In addition, we performed this analysis for each fMRI task separately (S10 Fig).

For NHP spiking data, we estimated the task (task cue period) and rest (ITI) dimensionality by calculating the covariance matrix between all pairs of population recordings. We then applied Eq 2 to task and rest periods for each covariance matrix. (Each covariance matrix was calculated using bins of 25 consecutive trials.) Finally, we applied a two-way group paired t-test (across bins) comparing the dimensionality of task activity to rest activity (Fig 5B). We replicated this effect in the held-out second monkey.

Spiking model: Estimating the transfer function of a neural population with a balanced spiking model

Our goal was to evaluate the effects of evoked activity across large neural populations, rather than within populations. Thus, we first estimated the transfer function of a neural population using a previously established balanced neural spiking model, with 4000 excitatory and 1000 inhibitory units [7]. All parameters are taken directly from [7] with the description paraphrased below. Units within the network were modeled as leaky integrate-and-fire neurons whose membrane voltages obeyed the equation (3) where τ indicates the membrane time constant, μ is the bias term, and I_syn is the synaptic input. When neurons reached V_th = 1, a spike was emitted, and voltages were reset to V_re = 0 for an absolute refractory period of 5ms. τ was 15ms and 10ms for excitatory and inhibitory neurons, respectively. For excitatory neurons, μ was randomly sampled from a uniform distribution between 1.1 and 1.2. For inhibitory neurons, μ was randomly sampled from a uniform distribution between 1 and 1.05.

Synapses to a neuron were modeled as the sum of excitatory and inhibitory synaptic trains x_E and x_I, respectively, and was calculated as the normalized difference of exponentials describing the synaptic rise and decay times caused by each presynaptic event. This effectively captured the weighted effect of all presynaptic neurons to a target neuron, and specifically obeyed the equations (4) (5) where the synaptic rise and decay of x_E and x_I was modeled as the first order differential equations (6) (7) J_ij refers to the synaptic weight from neuron j to i, s_j indicates whether neuron j emitted a spike. Synaptic rise times were the same for excitatory and inhibitory neurons, with τ_E,rise = τ_I,rise = 1ms, while τ_E,decay = 3ms and τ_I,decay = 2ms. Connection probabilities p^xy from neurons in population y to x were p^EI = p^IE = p^II = 0.5, and on average, p^EE = 0.2. However, if two neurons were both excitatory and belonged to the same cluster, the connection strength was multiplied by 1.9. (We employed only the homogenous clustered networks, as described by [7]; J^EE = 0.024, J^EI = −0.045, J^IE = 0.014, and J^II = −0.057.) Excitatory stimulation was performed by increasing μ to the first 400 excitatory neurons from 0.05 to 0.5 in 0.05 increments. Inhibitory stimulation was performed by decreasing μ by 0.05 to 0.5 in 0.05 increments to 400 inhibitory neurons.

To estimate the population transfer function, we simulated 30 trials lasting 2s each at each stimulation amplitude. Spike train statistics were estimated across trials in 50ms sliding windows with 10ms shifts. Only excitatory neurons were included when calculating the population spike train statistics (i.e., mean and variance at each stimulation amplitude).

Model code was originally adapted from [7], and was simulated with Julia (version 1.1.1).

Model: One-dimensional minimal network model

We use the simplest model to mathematically characterize the relationship between evoked activity and neural variability: a one-dimensional mean-field model. We used Wilson-Cowan-type firing rate dynamics to simulate neural population activity [69]. Specifically, our population’s activity obeyed the equation (8) where x_i denotes the firing rate (or a measure of activity), τ_i = 0.1 denotes the time constant, w_ii = 1 refers to local coupling (auto-correlation), b_i = −0.5 refers to the input threshold for optimal activity (or a bias term), s_i refers to the evoked stimulation (s_i = 0 for intrinsic activity), I refers to background spontaneous activity sampled from a Gaussian distribution with mean 0 and standard deviation 0.25, and f is a sigmoid input-output activation function, which is defined as (9) where k = 1. Numerical simulations were computed using a Runge-Kutta second order method with a time step of dt = 10ms [70]. We simulated neural population activity injecting a fixed input (boxcar input) with amplitudes ranging from s_i ∈ [−5,5] in 0.01 increments (Fig 7C). Neural variability for each input strength was calculated using the standard deviation of the time series following the input onset and preceding input offset. Each trial was run for 20 seconds. Fig 7C was generated using input amplitudes of s_i ∈ {−3,0,3}.

To visualize the full dynamics of our single neural population, we visualized the one-dimensional phase space (i.e., flow field on a line) [43]. In particular, we calculated the flow field by plotting (i.e., ) as a function of x_i (Eq 8). Notably, fixed point attractors (equilibrium states) are defined where (Fig 6B).

Model: Two-dimensional minimal network model

To characterize the effects of evoked activity on neural correlations, we use a two-dimensional neural population model. We extended the one-dimensional network model to include two neural populations. The network dynamics obeyed the equations (10) (11) where x₁ and x₂ describe the activity of each population, and all other variables are as described above. Inter-regional coupling was set to be greater than local coupling, given evidence from previous studies that global coupling is greater than local coupling [48,71,72]. Specific network parameters for this network model were: w₁₁ = w₂₂ = 2, w₁₂ = w₂₁ = 4, b₁ = b₂= −3, τ₁ = τ₂ = 0.1. I₁ and I₂ were sampled from a Gaussian distribution with mean 0 and standard deviation 1. For this network model, we decreased the slope of the sigmoid k = 0.5 to allow for a larger dynamic, linear response range. The full model code is publicly available: https://github.com/ito-takuya/corrquench.

To quantify the relationship between evoked activity and neural correlations, we systematically simulated the network under different stimulation states (input strengths). Using the same methods as above, we simulated network activity for 50 seconds. We injected fixed input into both neural populations with amplitudes ranging from s_i ∈ [−5,5] in 0.01 increments (Fig 7E). Notably, given that the injected stimulation is uncorrelated (due to 0-variance in a fixed input), it is non-trivial that the FC between two nodes would change in response to different inputs. Neural correlations were calculated using a Pearson correlation of the two time series following input onset and preceding input offset.

The use of a minimal model constrained our network to two dimensions. This allowed us to leverage dynamical systems tools to visualize the flow field in the two-dimensional phase plane. To identify the fixed point attractors, we first calculated the nullclines for x₁ and x₂. Nullclines are defined as the values of x₁ and x₂ such that and , respectively. The fixed points lie at the intersection of the nullclines. For our particular system of equations, the nullclines of x₁ and x₂ were defined, respectively, as (12) (13) where parameters are identical to those used in Eqs 10 and 11. However, the background noise, parameter I, was removed when calculating the nullclines. Fixed point attractors (equilibrium states) are defined where (Fig 8B). The full flow field was obtained by applying the system of equations (Eqs 10 and 11) to every point in the phase space (e.g., all values of x₁ and x₂).

Model: Evaluating fixed point attractor dynamics and the characteristic time scale

Our models accurately demonstrated that evoked activity decreased the neural variability and correlations from a stochastic dynamical network model. Since our network model was governed by firing rate equations which provided us full access to the system’s dynamics, we sought to link dynamical mechanisms (in the absence of spontaneous activity) with changes in the descriptive statistics. Such an analysis would provide us with a mechanistic understanding between descriptive neural statistics used in empirical data analysis and the governing neural dynamics.

To understand how attractor dynamics influenced simulated activity in a network model, we characterized the dynamics around the network’s fixed point attractor. Specifically, we performed a linear stability analysis around the fixed point (i.e., the equilibrium level of activity the system is drawn to during a particular state or input, e.g., Fig 7B) in both the one-dimensional and two-dimensional network models. In the one-dimensional case, this analysis is equivalent to evaluating the first derivative of Eq 8 at the fixed point (e.g., the slope of the line at the starred locations in Fig 7B). We then calculate the characteristic time scale T at the fixed point x* (in one-dimension) with the equation (14) where f represents Eq 8 [43]. The characteristic time scale captures the speed with which the system approaches the fixed point attractor. We calculated the characteristic time scale across the same range of evoked stimulation strengths as in the neural variability analysis. Fixed points were computed numerically by running the network model until it reached a steady state in the absence of noise/spontaneous activity.

The characteristic time scale is an established measure for one-dimensional systems. However, we sought to extend the characteristic time scale beyond a single dimension to evaluate shifting attractor dynamics in higher dimensions. We first performed a linear stability analysis in two dimensions by evaluating the Jacobian matrix for our two-dimensional system at the fixed point (x*₁,x*₂) (15) Where f₁ and f₂ refer to the equations governing neural populations 1 and 2 (Eqs 10 and 11, respectively). For our particular system of equations, the Jacobian was calculated as (16) For each input strength (i.e., differing evoked states), we evaluated the Jacobian at the fixed point attractor. We then calculated the two eigenvalues (denoted λ₁ and λ₂) and eigenvectors (denoted v₁ and v₂) of the Jacobian using an eigendecomposition. To calculate the generalized characteristic time scale in two dimensions, we first calculated the linear combination of the eigenvectors weighted by the real eigenvalues, and computed the magnitude of the vector, such that (17) We then define the two dimensional characteristic time scale T as the reciprocal of the magnitude of v_sum(x,y), such that (18) We calculated T for a range of values s₁, s₂ ∈ [−5,5] in 0.01 increments, and correlated T across all values of s₁ and s₂ with the corresponding neural correlations [43].

Model: 300 unit firing rate model

To verify the findings observed in our minimal models would scale to larger networks (one and two-dimensional models), we included a 300 region mean-field firing rate model. We chose 300 regions given that most whole-brain human atlases contain 200–400 cortically defined parcels [13,61,73]. Our model followed the same equations as in our minimal model, though inter-area weights were appropriately scaled relative regional self-coupling parameters. Specifically, the network dynamics obeyed the equations (19) where x_i describes the activity of each population, and all other variables are as described above. Inter-regional coupling was set to be greater than local coupling (2:1 ratio), given evidence from previous studies that global coupling is greater than local coupling [48,71,72]. Specific parameters for this network model were specified such that: w_ii = 1, the mean of inter-region coupling parameters was b_i = −2, τ_i = 0.1. I_i was sampled from a Gaussian distribution with mean 0 and standard deviation 1. s_i = 0 during rest state and s_i = 1 during task state.

In total, we ran simulations for two classes of network models: a network with random connections and a network with clustered communities (S11 Fig). For the random network, we randomly sampled connections with 20% probability rate between all pairs of regions. For the clustered network model, we generated 10 communities of 30 nodes each. Regions within the community had a 20% probability rate for establishing a connection. Between-community connections had a 3% probability rate for establishing a connection.

For each class of network (random or clustered), we weighted connections with either positive weights only (i.e., only E connections) or both positive and negative weights (i.e., both E and I connections. For the E-only network, weights were sampled from a normal distribution with parameters μ = 1, σ = 0.2. For the network with both E and I weights (80% E, 20% I), weights were sampled from a normal distribution with parameters μ = 1, σ = 1.2.

Both the rest and task state simulation was run for 10 seconds each and sampled at 100ms. For each group analysis (S11 Fig), we simulated 20 subjects worth of data. The full model code is publicly available, and was written in python 3.7.3.

Model: Simulating fMRI BOLD activity

We used the above model to simulate fMRI BOLD activity to demonstrate that changes in neural variability and correlations would extend to fMRI BOLD dynamics (S6 and S7 Figs). Neural activity generated from our model simulations was transformed to fMRI BOLD activity using the Balloon-Windkessel model, a nonlinear transformation from neural activity to the fMRI BOLD signal [44,74]. Notably, the transformation assumes a nonlinear transformation of the normalized deoxyhemoglobin content, normalized blood inflow, resting oxygen extraction fraction, and the normalized blood volume. All state equations and biophysical parameters were taken directly from [44] (Eqs 4 and 5). The Balloon-Windkessel model was implemented in Python (version 2.7.13), and the implementation code has been made publicly available on GitHub (https://github.com/ito-takuya/HemodynamicResponseModeling).

Ethics statement

All procedures performed on the two rhesus monkeys (one male, one female) followed the guidelines of the Massachusetts Institute of Technology Committee on Animal Care and the National Institutes of Health.

All human participants included in this study gave signed, informed consent in accordance with the protocol approved by the Washington University institutional review board.