Abstract
Correlated activity fluctuations in the neocortex influence sensory responses and behavior. Neural correlations reflect anatomical connectivity but also change dynamically with cognitive states such as attention. Yet, the network mechanisms defining the population structure of correlations remain unknown. We measured correlations within columns in the visual cortex. We show that the magnitude of correlations, their attentional modulation, and dependence on lateral distance are explained by columnar OnOff dynamics, which are synchronous activity fluctuations reflecting cortical state. We developed a network model in which the OnOff dynamics propagate across nearby columns generating spatial correlations with the extent controlled by attentional inputs. This mechanism, unlike previous proposals, predicts spatially nonuniform changes in correlations during attention. We confirm this prediction in our columnar recordings by showing that in superficial layers the largest changes in correlations occur at intermediate lateral distances. Our results reveal how spatially structured patterns of correlated variability emerge through interactions of cortical state dynamics, anatomical connectivity, and attention.
Introduction
Neocortical circuits spontaneously generate varying patterns of neural activity, which profoundly influence sensory responses and behavior^{1,2,3,4,5}. These endogenous activity fluctuations are correlated across neural populations and are often quantified by correlations between pairs of neurons, called noise correlations^{6}. Noise correlations are thought to reflect the anatomical circuit connectivity, but they are also dynamically influenced by behavioral and cognitive states^{5,7,8,9}, in particular, during spatial attention^{10,11,12,13,14}. Implications of noise correlations for population coding and behavior have been studied extensively^{15,16,17,18,19}. Yet, how anatomical connectivity and cognitive states interact to define the structure of noise correlations across populations is not well understood.
Spatial selective attention offers a rich experimental domain for studying the combined influence of anatomical connectivity and cognitive factors on the population structure of noise correlations. Changes in noise correlations during attention have been measured across different anatomical dimensions, yielding heterogeneous results. Many studies of noise correlations involved recordings from neurons in different cortical columns, e.g., using rectangular Utah arrays which preferentially sample from laterally separated neurons in more superficial cortical layers^{10,12} (Fig. 1a). These studies found that noise correlations substantially decreased when attention was directed to the receptive fields (RFs) of recorded neurons^{10,11,12}. More recent studies used linear multielectrode arrays to measure attentional modulation of noise correlations within cortical columns (Fig. 1a) and found effects that varied with layer and area. In V4, noise correlations decreased during attention only in input layers during stimulusevoked but not spontaneous activity, and no significant changes were observed in superficial and deep layers^{13}. In V1, noise correlations decreased only in supragranular layers with no significant changes in granular and infragranular layers^{14}. In both areas, the magnitude of changes in noise correlations within columns appeared an order of magnitude smaller compared to a sizable reduction of correlations across columns. These data suggest that attentional modulation in correlated variability is not uniform across anatomical dimensions, but depends on lateral distance and cortical layer. The network mechanisms underlying these heterogeneous modulations are unknown.
We hypothesized that heterogeneous changes in noise correlations arise from the modulation of OnOff dynamics propagating locally across columns. The OnOff dynamics are spontaneous transitions between phases of vigorous (On) and faint (Off) spiking that occur synchronously across layers of neocortex^{20,21} and are observed in visual cortex of behaving monkeys^{22,23} (Fig. 1b). The On and Off phases of population activity persist on the timescale of ~100 ms with exponentially distributed durations^{22,23} and correlate with large fluctuations in neurons’ membrane potentials^{24,25}, which are signatures of bistable dynamics^{26}. The OnOff dynamics reflect the global cortical state associated with arousal and are also modulated locally within retinotopic maps during selective attention^{22,23}. We analyzed spiking activity recorded within cortical columns in V4^{27} and found that the scale of OnOff dynamics predicted the magnitude of noise correlations and their dependence on lateral distance.
To explain the spatial patterns of noise correlations, we developed a network model of interacting columns with spatially structured connectivity. The key mechanism in our model is OnOff dynamics that propagate across columns to form spatiotemporal population activity which shapes the structure of noise correlations. Cortical activity propagates laterally as waves across different spatial scales, brain states and behavioral conditions^{28,29}, and wavelike propagation of spontaneous activity fluctuations is observed in the visual cortex of behaving primates^{30}. In our model, attentional inputs shift the stability of local OnOff dynamics, which effectively regulates the efficacy of lateral interactions among columns and affects the spatial activity propagation. As a result, attentional inputs reduce the spatial extent of OnOff dynamics, leading to spatially nonuniform changes in noise correlations. This mechanism predicts that, during attention, noise correlations can change substantially at intermediate lateral distances (across columns) even when average changes within columns are very small. To test this prediction, we analyzed how changes in noise correlations depend on lateral distance in columnar recordings in V4. While average changes were small, when sorted by the lateral distance, the changes in noise correlations were near zero at zero distance and gradually increased with distance in superficial layers, consistent with predictions of our model.
The mechanism based on bistable OnOff dynamics differs from previous models of cortical variability operating in a balanced excitatoryinhibitory regime, where population activity fluctuates around a single global fixed point^{31,32}. Balanced networks with a global fixed point do not capture the slow timescale and bistable characteristics of cortical fluctuations that we observed in our data. In addition, the previous model predicts that attentional changes of noise correlations are spatially uniform^{31}, in opposition to our experimental observations. Our results indicate that visual cortex operates in a regime of local bistable dynamics in single columns that interact via structured anatomical connectivity. Our work provides a unifying framework that explains how heterogeneous patterns of correlated variability emerge within neocortex through interactions of network dynamics and cognitive state.
Results
OnOff dynamics predict the magnitude of noise correlations
We measured spiking activity from all layers within columns of the visual cortical area V4^{27}. Spiking activity was recorded with 16channel linear array microelectrodes (Fig. 1a) arranged so that receptive fields (RFs) on all channels largely overlapped^{22,33}. During recordings, monkeys performed a spatial attention task, which required detecting changes in the orientation of a visual stimulus in the presence of distractor stimuli (Methods section and Supplementary Fig. 1). On each trial, an attention cue indicated the stimulus that was most likely to change. In the attention condition, the cue directed animal’s attention to the RF stimulus. In the control condition, the cue directed attention to a location outside the RFs of recorded neurons.
In our columnar recordings, we examined the relationship between the scale of ongoing OnOff dynamics and the magnitude of noise correlations. We quantified the OnOff dynamics by fitting a twophase Hidden Markov Model (HMM) to the population spiking activity^{22,23} (Fig. 1b, Methods section). The HMM models the dynamics of a latent population state that switches between two phases, On and Off, to capture synchronized changes in firing rates across neurons. Spikes on recorded channels were modeled as inhomogeneous Poisson processes with different mean rates during the On and Off phases. The variance explained by a twophase HMM (R^{2}) varied across recording sessions and this variation was tightly correlated with the average noise correlation (Fig. 1c). For most recording sessions (31 total, 67%), the twophase HMM was the most parsimonious model among HMMs with 1 or up to 8 possible phases^{22} (Methods section). For the remaining 15 (33%) sessions, a onephase HMM (i.e. constant firing rates without OnOff transitions) was the most parsimonious model. These onephase recordings consistently showed lower average noise correlations of multiunit (MU) activity (mean 0.13) than twophase recordings (mean 0.32), with a pronounced 59% difference on average (Fig. 1c). Trialtotrial variability of MU activity, quantified by the Fano factor (FF, the ratio of the spikecount variance to the mean), was also lower in onephase (mean FF = 1.5) compared to twophase (mean FF = 2.3) recordings (35% difference). On the other hand, the mean firing rates of MUs were similar between the onephase (108 Hz) and twophase (114 Hz) recordings (5% difference). Thus the scale of OnOff dynamics predicted the overall magnitude of correlated variability in our data, which implicates OnOff dynamics as a major source of noise correlations in visual cortex.
Attentional modulation of noise correlations within columns
We quantified attentionrelated changes in noise correlations within columns, separately in superficial (which included granular and supragranular) and deep (infragranular) cortical layers. In each session, data from each of the recording channels were assigned laminar depth, relative to a common current source density marker^{33}. We combined the granular and supragranular layers because they showed similar changes in noise correlations (Supplementary Fig. 2). We found that noise correlations were slightly reduced in superficial and enhanced in deep layers in the attention relative to control conditions (Fig. 1d). To quantify these changes, we calculated a standard modulation index MI_{corr}, which was the difference between noise correlations in the attention and control conditions divided by the sum. In twophase recordings, the mean \({{{{{{{{\rm{MI}}}}}}}}}_{{{{{{{{\rm{corr}}}}}}}}}^{{{{{{{{\rm{MU}}}}}}}}}\) for MU was −0.029 in superficial layers (p < 10^{−5}, Wilcoxon signedrank test, n = 5088) and 0.022 in deep layers (p = 0.004, Wilcoxon signedrank test, n = 6128) (see Supplementary Tables 1 and 2 and Supplementary Fig. 3 for a full summary of results). The magnitude and laminar profile (Supplementary Fig. 2) of these noisecorrelation changes are consistent with other laminar recordings in V4^{13}.
These average changes of noisecorrelations within columns were much smaller than the robust and sizable reduction of noise correlations previously reported for neurons in different columns^{10,11,12,31}. For comparison, a previous study in V4 using rectangular Utah arrays^{10} found that the mean \({{{{{{{{\rm{MI}}}}}}}}}_{{{{{{{{\rm{corr}}}}}}}}}^{{{{{{{{\rm{MU}}}}}}}}}\) was −0.29, which is an order of magnitude larger than in our columnar recordings. Despite this striking difference in the modulation of noise correlations, the attentional modulation of firing rates and trialtotrial variability of individual neurons was similar in our data and the previous study. In twophase recordings, the mean \({{{{{{{{\rm{MI}}}}}}}}}_{{{{{{{{\rm{rate}}}}}}}}}^{{{{{{{{\rm{MU}}}}}}}}}\) was 0.023 in superficial layers (p < 10^{−10}, Wilcoxon signedrank test, n = 1752) and 0.018 in deep layers (p < 10^{−10}, Wilcoxon signedrank test, n = 2216), which is more comparable to the previous study^{10}. Similarly, in twophase recordings, the mean modulation index of Fano factor \({{{{{{{{\rm{MI}}}}}}}}}_{{{{{{{{\rm{FF}}}}}}}}}^{{{{{{{{\rm{MU}}}}}}}}}\) was − 0.010 in superficial layers (p < 10^{−10}, Wilcoxon signedrank test, n = 1752) and −0.007 in deep layers (p < 10^{−10}, Wilcoxon signedrank test, n = 2216), comparable to the previous study^{10}. The results were similar in onephase recordings (Supplementary Tables 1 and 2). These results suggest that attentionrelated changes in noisecorrelations depend on the relative positions of neurons in the cortex, with sizable changes across columns and minute, layerdependent changes within columns. Since the striking difference in the modulation of noise correlations is not accounted for by differences in the activity of individual neurons, it likely reflects the spatial structure of population dynamics across the cortex.
Network model of interacting cortical columns
We hypothesized that heterogeneous modulations of noisecorrelations across layers and columns arise from the OnOff population dynamics and spatial structure of anatomical connectivity in the cortex. To test this hypothesis, we developed a network model of interacting columns with spatially structured connectivity (Fig. 2). The model consists of units interconnected in two parallel twodimensional lattices, corresponding to the superficial and deep cortical layers (Fig. 2a). Each unit represents a local population of neurons within one layer—superficial or deep—of a single column. Each unit is connected to its four neighboring units in the same layer, mimicking the local structure of horizontal connectivity in the cortex. Visual stimuli and attention are modeled by external inputs to local groups of units. Since attentional modulation differs between superficial and deeper layers, we modeled each layer as a separate network. Each network receives different attentional inputs to account for differential changes in noise correlations in superficial versus deep layers.
The key mechanism generating correlated variability in our model is the stochastic OnOff dynamics of the population activity in single columns. In visual cortex of behaving monkeys, the durations of On and Off episodes are distributed exponentially with a timescale of ~100 ms^{22,23} (Supplementary Fig. 4), which indicates that the On and Off phases are metastable with transitions driven by noise^{26}. Accordingly, we model the dynamics of each unit in the network by a twodimensional dynamical system with two stable fixed points, corresponding to the On and Off phases (Fig. 2b). This dynamical system is a phenomenological meanfield description of a population of excitatory neurons coupled by the vertical recurrent connectivity within the column. The first dynamical variable r(t) represents the mean firing rate of the population. It receives a recurrent selfcoupling F(r) and a negative feedback from the second dynamical variable a(t) representing firingrate adaptation^{21,26}. The dynamical system is driven by white noise ξ(t), which causes stochastic transitions between the On and Off fixed points. Each unit also receives external currents \({I}_{{{{{{{{\rm{stim}}}}}}}}}(t)\) and I_{att}(t), which model the bottomup inputs from visual stimuli and topdown inputs during attention, respectively.
The dynamics of individual units reproduce the OnOff transitions in single columns and their modulation during attention. As in the data from visual cortex^{22,23} (Supplementary Fig. 4), the duration of On and Off episodes in the model are irregular and exponentially distributed (Fig. 2c). In this regime, the dynamics of each unit can be reduced to a twostate Markov process switching between the On and Off phases (Supplementary Note 2.1). The OfftoOn (α_{1}) and OntoOff (α_{2}) transition rates of the Markov process set the average duration of the On and Off episodes: τ_{on} = 1/α_{2} and τ_{off} = 1/α_{1}. Consistent with this description, a twostate HMM provides the most parsimonious fit of the OnOff dynamics in our twophase recordings. Further, our model captures the increase of Onepisode durations during attention as observed in the data^{22,23} (Supplementary Fig. 5). During attention, a local group of units representing the attended RFs receives a small excitatory input I_{att}. This attentional input slightly shifts the rnullcline (Fig. 2b) elevating the threshold for transitioning from the On to Off fixed point, which reduces the OntoOff transition rate of the Markov process and results in longer average Onepisode durations (Fig. 2c).
The horizontal connectivity in the network correlates the OnOff dynamics across units in the lateral dimension. Each unit’s firingrate variable r(t) receives a recurrent excitatory input I_{rec} from its four neighbors on the lattice. As a result, the OnOff dynamics of each unit are influenced by the activity of its neighbors. The more neighbors in the On phase, the larger is the excitatory input I_{rec}, which elevates the threshold for OntoOff and lowers the threshold for OfftoOn transitions. In the description of a twostate Markov process, this is equivalent to a dependence of the OfftoOn and OntoOff transition rates on the On/Off phases of the neighbors: α_{1} + βS_{±} and α_{2} − βS_{±} (Supplementary Note 2.3). Here, the variable S_{±} indicates the number of neighbors in the On phase at each time, and β is the effective coupling strength that depends on the parameters of the dynamical system as well as on external inputs (Supplementary Note 2.3). The reduced network of coupled binary OnOff units follows Glauber dynamics^{34} (Supplementary Note 2.4), allowing us to calculate noise correlations analytically in our model. In simulations, both the dynamicalsystem and binaryunit versions of the network exhibit similar spatiotemporal dynamics, where the On and Off phases form local spatial clusters (Fig. 2d), which propagate across columns in a pattern of local irregular waves^{29} (Supplementary Movie 1).
We model two sources of spiking variability: the OnOff fluctuations of the population activity and stochasticity of spike generation in individual neurons^{21,35}. We simulate spikes of individual neurons as inhomogeneous Poisson processes with different mean rates during the On and Off phases generated by the network (Fig. 2e). This doubly stochastic description coincides with the assumptions of the HMM used to fit the experimental data. We match the model parameters to the experimental data by fitting the data with the HMM, which provides us with the estimates of the OnOff transition rates (α_{1} and α_{2}) and the On and Off firing rates (r_{on} and r_{off}) for all MUs and single units (SUs) in each recording session and task condition (Fig. 2f). We then use these parameters to predict noise correlations and compare these predictions with the values measured for the same neuron pairs in the data.
The model accounts for correlated variability within columns
We used the twophase recordings to test how accurately the model predicts changes of correlated variability in single columns during attention. In the model, the OnOff dynamics are the source of correlations between responses of individual neurons. All neurons in the population represented by a single unit (column) follow the same shared sequence of On and Off episodes. The spiking responses differ across neurons because of the independent Poisson noises as well as differences in their On and Off firing rates. We derived analytical formulas for the Fano factor and noise correlations, measured over an arbitrary timewindow T, as functions of the model parameters: the OnOff transition rates and the On and Off firing rates of each neuron (Methods section and Supplementary Note 2.1). The model analytically predicts the dependence of Fano factor and noise correlations on the measurement timewindow^{6}, indicating that this dependence is determined by the timescales of OnOff dynamics (Supplementary Note 2.1). We used these analytical formulas with the parameter values estimated from the data by the HMM to predict the FF and noise correlations for, respectively, each neuron and neuron pair in our dataset. We compared these model predictions with direct measurements from the experimental data.
Our model makes a specific prediction that the key factor determining the magnitude of FF and noise correlations within columns is the OnOff firingrate difference Δr = r_{on} − r_{off}. Specifically, FF is directly proportional to Δr, and the noise correlation between neurons i and j is proportional to the product Δr_{i}Δr_{j} (Methods section and Supplementary Note 2.1). This dependence on Δr is intuitive because the source of correlations within a column is the shared OnOff switching, hence the stronger a neuron is modulated by the OnOff dynamics (the greater is Δr), the stronger it will be correlated with other neurons in the same column. The dependence of FF and noise correlations on Δr is evident in an example recording (Fig. 3a, b), where different MUs exhibit a variety of OnOff firingrate differences Δr. The FF ranges broadly across MUs, from ~1 up to ~9, and this variation is very well predicted by Δr (Fig. 3a). Although a few units exhibited very high Δr and hence unusually high FF values, the median FF of MUA was 1.8 similar to previous studies (Supplementary Fig. 6). Similarly, MU pairs with the largest product Δr_{i}Δr_{j} also exhibit the largest noise correlations (Fig. 3b). While both FF and noise correlations also weakly depend on the OnOff transition rates, the OnOff firingrate difference is the main factor defining the broad distributions of these quantities in single columns (Supplementary Fig. 7).
As a consequence, the model also predicts that the changes in FF and noise correlations during attention are proportional to the changes in Δr and Δr_{i}Δr_{j}, respectively. This prediction was clearly borne out by the data: the measured change in FF had a strong trend as a function of the change in Δr (yaxis vs. coloraxis in Fig. 3c), and the measured change in noise correlations had a strong trend as a function of the change in Δr_{i}Δr_{j} (yaxis vs. coloraxis in Fig. 3d). Moreover, changes in the FF and noise correlations measured from the data were accurately matched by the model predictions (y vs. xaxis in Fig. 3c, d). The attentionrelated changes in noise correlations range widely across the population, with noise correlations substantially reduced and enhanced in many pairs. This entire broad distribution is accurately matched by the analytical predictions of the OnOff dynamics model in single columns. Despite substantial changes in many pairs, the average change in noise correlations within columns is near zero (Supplementary Fig. 8b), since the changes of Δr_{i}Δr_{j} are broadly distributed but close to zero on average. Thus, our model of OnOff dynamics explains the observed changes in correlated variability during attention within columns.
Decay of noise correlations with lateral distance
Next, we analyzed the dependence of noise correlations on the lateral distance in our laminar recordings and in the network model. Previous studies in the visual cortex found that noise correlations decrease with the lateral distance^{31,36,37}. These studies used multielectrode arrays with lateral spacing between electrodes ranging from ~0.35 to 4 mm, i.e. sampling distant neuron pairs in different columns. With the laminar recordings, we tested how noise correlations depend on the lateral distance over a much shorter range of distances within single or nearby columns. We leveraged the fact that laminar recordings generally exhibit slight horizontal shifts due to variability in the penetration angle (Fig. 4a). As a surrogate for horizontal displacements between pairs of channels, we used distances between centers of their RFs. To estimate the range of physical distances in the cortex spanned by our laminar recordings, we converted the RFcenter distances to cortical distances using the cortical magnification factor for each eccentricity^{38}. The range of distances spanned by our recordings was ~4−6 dva or ~1.5 mm (Methods section and Supplementary Fig. 9).
We found that noise correlations decreased with lateral distance in twophase but not in onephase recordings. In twophase recordings, noise correlations monotonically decreased with the RFcenter distance both in superficial and deep layers (Fig. 4b, linear regression, onesided ttest, slope − 0.09 ± 0.01, p < 10^{−8} in superficial layers, slope −0.04 ± 0.01, p < 10^{−3} in deep layers). With the conversion to cortical distances^{38}, noise correlations also decreased continuously with the lateral cortical distance over ≲1 mm range (Supplementary Fig. 9). Note that most pairs in our data were at very short distances, with the median estimated cortical distance 0.72 mm. Thus the decay of noise correlations with lateral distance spans all distances within nearby and across distant columns. In contrast, noise correlations did not decrease with the RFcenter distance in the onephase recordings (Fig. 4c, linear regression, onesided ttest, slope −0.010 ± 0.008, p = 0.1 in superficial layers, slope 0.014 ± 0.004, p = 0.997 in deep layers). These results suggest that OnOff dynamics give rise to the lateral distancedependence of noise correlations.
In our network model, the dependence of noise correlations on the lateral distance arises from the spatiotemporal OnOff dynamics. Whereas all neurons represented by a single unit in the network follow the same shared sequence of OnOff phases, neurons represented by different units follow their respective OnOff sequences. Consistent with this assumption, in recording sessions with large lateral shifts between receptive fields, we can observe OnOff phases that occur synchronously only on a subset of adjacent channels and propagate across channels over time (Supplementary Fig. 10). The OnOff dynamics are correlated across nearby units in the lateral dimension due to horizontal connectivity in the network. Since the horizontal connections are spatially local and relatively weak, the synchrony of OnOff dynamics is not global across the entire network, but localized to a finite range of lateral distances. Thus the OnOff phases form spatial clusters with a characteristic spatial length scale, and beyond this spatial scale the OnOff phases are uncorrelated. This network mechanism leads to a continuous decrease of noise correlations with the lateral distance in the model (Fig. 4d).
Using the binaryunit reduced network model, we derived an analytical formula for the dependence of noise correlations on the lateral distance d (Methods section). Our calculations show that noise correlations decay with the distance exponentially as \({{{{{{{\mathcal{A}}}}}}}}\exp (d/L)\). This formula describes noise correlations both within and across columns. At zero distance (d = 0), the formula reduces to the prefactor \({{{{{{{\mathcal{A}}}}}}}}={{{{{{{\mathcal{A}}}}}}}}({\alpha }_{1},{\alpha }_{2},{{\Delta }}{r}_{i},{{\Delta }}{r}_{j})\), which accounts for the dependence of noise correlations on the OnOff transition rates and the OnOff firingrate difference, as described in the previous section. Across columns (finite d > 0), the formula accounts for the spatial structure of noise correlations with the exponential discount factor \(\exp (d/L)\). The spaceconstant L of this exponential decay, termed correlation length, depends on the OnOff transition rates and on the effective coupling strengths β between units in the network: \(L=\sqrt{\beta /({\alpha }_{1}+{\alpha }_{2})}\) (in dimensionless units of the lattice constant, see Methods section and Supplementary Note 2.3). This analytical result agrees well with simulations of the full dynamicalsystem network model (Fig. 4d).
The exponential decay of noise correlations with distance in our model, characterized by the correlation length L, is consistent with the decrease in noise correlations observed over a wide range of cortical distances spanned by our laminar (Fig. 4b) and previous lateral recordings^{31,36,37} from the primate visual cortex. Our model can also reconcile the decay of noise correlations with distance in lateral recordings^{31,36,37} with the lack of distance dependence in the onephase recordings. With some heterogeneity, if a random fraction of units in the model does not exhibit OnOff transitions (due to a more stable fixed point), the activity of these onephase units is not correlated with other units at all distances. Thus lateral sampling from a mixture of onephase and twophase phase units would uniformly lower the average of noise correlations without affecting their distance dependence.
Differential changes in noise correlations arise from attentional modulation of the correlation length
In our network model, attentional inputs restructure the spatiotemporal OnOff dynamics, leading to differential changes in noise correlations within versus across columns. In the network simulations, a local group of units receives an attentional input I_{att}, while other units without this input (I_{att} = 0) are in the unattended control condition (Fig. 2d). With an excitatory attentional input (I_{att} > 0), average noise correlations between neurons in the same column change very little relative to control (MI_{corr} = −0.05), while noise correlations between neurons in different columns are substantially reduced (MI_{corr} = −0.21, Fig. 5a). These results replicate the orderofmagnitude difference of MI_{corr} observed between laminar versus lateral recordings from the visual cortex. We repeated simulations for a range of excitatory (I_{att} > 0) and inhibitory (I_{att} < 0) attentional inputs. The excitatory inputs reduced noise correlations, whereas inhibitory inputs increased noise correlations, but in all cases, the average changes of noise correlations within columns were smaller compared to sizable changes across columns of the network (Fig. 5b).
To reveal the mechanism leading to differential changes of noisecorrelations within versus across columns, we examined how attentional inputs affect the dependence of noise correlations on the lateral distance in our network. In simulations, excitatory attentional inputs produce a faster decay of noise correlations with lateral distance, which corresponds to a shorter correlation length (L_{att} < L_{ctl}, Fig. 5c and Supplementary Fig. 11). Due to this faster spatial decay, noise correlations at intermediate lateral distances (finite d > 0) are considerably lower in attention relative to control conditions, even when changes of noisecorrelations within columns (d = 0) are small. Inhibitory attentional inputs, on the other hand, produce a slower decay of noise correlations with lateral distance, which corresponds to a longer correlation length (L_{att} > L_{ctl}) and results in increase of noise correlations at intermediate lateral distances. Thus changes of the correlation length L produce sizable changes of noise correlations at intermediate lateral distances (across columns) even when noise correlations within columns do not change.
To understand the network mechanism by which attentional inputs modulate the correlation length, we leveraged the analytical formula \(L=\sqrt{\beta /({\alpha }_{1}+{\alpha }_{2})}\). In the dynamicalsystem model, an excitatory attentional input shifts the rnullcline, which increases OfftoOn (α_{1}) and decreases OntoOff (α_{2}) transition rates (Fig. 2b, c). Since α_{1} and α_{2} change only moderately and in opposite directions, their sum remains nearly constant (Fig. 5d). Therefore changes of the correlation length L are mainly driven by changes in the effective coupling strength β, which decreases steeply with an increasing attentional input (Fig. 5e). The effective coupling strength β decreases because an excitatory attentional input stabilizes the On fixed point, thereby effectively reducing the efficacy of the lateral recurrent inputs to drive the OnOff transitions. Vice versa, an inhibitory attentional input makes the On fixed point less stable, thereby enhancing the relative efficacy of the lateral recurrent inputs and hence extending the spatial correlation length in the network (Supplementary Note 3.2). Thus the attentional input modulates the correlation length by regulating the relative efficacy of lateral interactions between columns^{39}, which leads to differential changes in noise correlations within versus across columns.
The model predicts distancedependent changes of noise correlations
The major changes in noise correlations in our model are driven by changes in the correlation length L. The model, therefore, makes a specific prediction that changes in noise correlations during attention are not uniform across space. Noise correlations decay exponentially with the lateral distance, with different decay rates in attention and control conditions. Hence the spatial profile of noisecorrelation changes is defined by the difference of two exponential decays: \(\exp (d/{L}_{{{{{{{{\rm{att}}}}}}}}})\) and \(\exp (d/{L}_{{{{{{{{\rm{ctl}}}}}}}}})\). At very short lateral distances (within columns), average changes of noisecorrelations are small (Fig. 1d). At very long lateral distances, the average changes are negligible, because the overall magnitude of noise correlations vanishes. Sizable changes in noise correlations are predicted to occur at intermediate lateral distances, where the difference between two exponential decays dominates. Thus the network mechanism in our model predicts that the magnitude of noisecorrelation changes depends on lateral distance (Fig. 6a, b). This prediction contrasts with the alternative balanced network model, where population activity fluctuates around a single global fixed point, which instead predicts that attentionrelated changes of noise correlations are spatially uniform^{31} (Fig. 6c and Supplementary Note 3). Therefore examining the spatial profile of noisecorrelation changes in the data could distinguish between network mechanisms these alternative models postulate.
We analyzed how changes in noise correlations during attention depend on the lateral distance (estimated by the RFcenter distance) in our laminar recordings. Although average noisecorrelation changes in our columnar recordings were small, when sorted by the lateral distance, the data revealed spatial patterns with substantial changes in noise correlations at longer distances. As predicted by our model, changes of noise correlations in the twophase recordings were not uniform across space. In the superficial layers, changes of noise correlations were smallest at very short lateral distances and became progressively larger at longer distances (Fig. 6d). Noise correlations decreased during attention, with greater reduction at longer distances (linear regression, onesided ttest, slope −0.017 ± 0.004, p < 10^{−3}). The extrapolation of this trend to intermediate lateral distances (1 ⩽ d ⩽ 4 mm) is consistent with a robust reduction of noise correlations during attention observed in Utaharray recordings, which also sample from superficial layers^{10,12}. In the deep layers, the attentional effects on noise correlations were reversed from that in superficial layers, showing a moderate increasing trend with a borderline statistical effect (Fig. 6e, linear regression, onesided ttest, slope 0.006 ± 0.004, p = 0.06; ttest for slopes superficial versus deep: p = 0.6 × 10^{−3}). In contrast, changes of noise correlations did not depend on lateral distance in the onephase recordings (Fig. 6f, linear regression, ttest, slope −0.0005 ± 0.0061, p = 0.93 in superficial layers and slope −0.01 ± 0.01, p = 0.35 in deep layers). These results indicate that OnOff dynamics give rise to distancedependence of noisecorrelation changes during attention.
The spatial profile of noisecorrelation changes in twophase recordings is consistent with our network mechanism but inconsistent with the previously proposed model, which predicts spatially uniform changes^{31}. The observed spatial profiles of noisecorrelation changes indicate that the correlation length decreases in superficial but not in deep layers, which suggests that superficial and deep layers receive different modulatory inputs during attention^{40}.
Discussion
Our results show that OnOff dynamics are a major source of correlated variability in the visual cortex. OnOff dynamics in the awake cortex resemble some features of UpDown transitions prominent during slowwave sleep and anesthesia. Up and Down states were originally used to describe the bimodal distribution of membrane potentials, and now are also used to refer to spiking (Up) and silent (Down) phases of population activity during slowwave sleep and anesthesia^{5}. UpDown transitions are a major source of noise correlations during anesthesia^{21,41}. Similarly, OnOff dynamics account for a dominant share of noise correlations in behaving animals.
We found that OnOff dynamics explained the magnitude of noise correlations, their attentional modulation and dependence on lateral distance. The average changes of noise correlations during attention were very small within columns of the visual area V4. Noise correlations slightly decreased in superficial and increased in deep layers, but the changes were an order of magnitude smaller than a robust and sizable reduction of noise correlations between neurons in different columns. A reduction of noise correlations was suggested to be a major contributor to the improved behavioral performance during spatial attention^{10}. Our results show, however, that changes of noise correlations are not uniform: their magnitude and sign depend on the relative anatomical positions of neurons within layers and columns of the visual cortex. These heterogeneous changes of noise correlations may reflect unique contributions to behavioral improvements from different functional groups of neurons defined by their anatomical positions within the circuit.
To explain differences in attentionrelated changes of noise correlations within versus across columns, we developed a network model of interacting cortical columns. The key mechanism generating correlated variability in the model is OnOff dynamics, metastable transitions between the high and low firingrate fixed points in single columns. OnOff dynamics propagate laterally across columns via spatially structured network connectivity to form activity clusters traveling as local irregular waves (Supplementary Movie 1). Our model, therefore, integrates experimental findings that spontaneous fluctuations of cortical activity follow bistable OnOff dynamics in single columns^{22} and propagate laterally across columns as waves^{30}. Due to the stochasticity of dynamics, the activity clusters do not propagate coherently across the entire network but travel only locally until they fade or merge with other clusters. Local irregular waves differ from global traveling waves, in which a wave can propagate coherently across the entire network and most neurons equally participate in each wave^{29}. The spatial scale of activity clusters defines the exponential decay constant of noise correlations with lateral distance, i.e. the correlation length. Attentional inputs restructure the spatiotemporal OnOff dynamics and modulate the correlation length, which results in distancedependent changes of noise correlations. The model qualitatively captures attentionrelated changes of noise correlations in our data. Moreover, it makes a testable prediction that the sizable changes of noise correlations occur at intermediate lateral distances. Consistent with this prediction, we found that in our laminar recordings, the magnitude of noisecorrelation changes gradually increased with lateral distance in superficial layers (a moderate trend with borderline statistical effect in deep layers). These results show that changes in noise correlations depend on lateral distance.
OnOff dynamics accounted for the dominant share of noise correlations in the majority of our recordings (twophase recordings), while in some recordings (onephase recordings) HMM did not detect OnOff transitions. Although noise correlations in onephase recordings were substantially smaller than in twophase recordings, they were nonzero, indicating that other sources of variability contribute to noise correlations besides OnOff dynamics. Since in onephase recordings, noise correlations did not depend on distance (Fig. 4c), these other variability sources may be more global and uniform within a cortical area, such as fluctuations in neural excitability related to arousal^{2,5,22}. Moreover, Fano factor and noise correlations in onephase recordings were modulated during attention, suggesting that the additional variability sources also interact with attentional mechanisms, producing spatially uniform changes in correlated variability (Fig. 6f).
Noise correlations can limit stimulus information encoded in the population, meaning that information saturates with increasing population size^{15,16,18,19}. Information saturation is caused by a specific pattern of correlations, known as differential correlations, which are proportional to the product of the derivatives of the tuning curves^{17,42}. In our model, assuming that stimulus does not change the statistics of OnOff dynamics and that changes in the OnOff dynamics do not affect stimulus tuning, we found that the OnOff dynamics influence the strength of differential correlations and thus affect the saturation level of information (Supplementary Note 4). Specifically, the linear Fisher information is monotonically decreasing with the correlation length. Hence, a reduction of the correlation length leads to an increase in stimulus information, as we observed in superficial cortical layers during attention. However, the OnOff dynamics and stimulus tuning are likely interdependent in the cortical circuitry, where both arise from the same structured connectivity. Deeper understanding of how OnOff fluctuations impact sensory coding will be possible using models with connectivity that supports stimulus tuning in addition to spatial receptive fields^{43} in future work.
Several mechanisms were proposed to explain how correlated fluctuations arise in cortical networks. There are two general classes of models: one relies on external shared variability and another generates shared variability via intrinsic network dynamics. In models with external shared variability, the source of correlated fluctuations is assumed to be outside the network, and the network merely filters the correlated input noise^{32,44,45}. In most of these models, the mechanism is based on a spatial connectivity structure that locally breaks the excitationinhibition balance. The classical balanced network model with random connectivity^{46,47} operates in an asynchronous regime, where the tight excitationinhibition balance cancels any input correlations resulting in zero average noise correlations^{48}. The spatial connectivity structure, where recurrent inhibition is broader than feedforward excitation, breaks the balance locally, hence the input correlations cannot be canceled resulting in positive average noise correlations^{44,45}. However, to match the experimentally observed temporal and spatial scales of correlations, all of these models have to assume ad hoc spatiotemporal structure of the input noise^{32,44,45}.
The second class of models can generate shared variability internally. One mechanism is based on breaking stability in some spatial Fourier modes in a spatially organized balanced network. For example, in a twodimensional balanced network with slow inhibitory kinetics, shared fluctuations arise from instability at some spatial frequency that generates rate chaos^{31}. Similarly, in a onedimensional balanced ring model, strong correlations arise from a feedforward structure in some Fourier modes of connectivity^{49}. In these models, correlations arise from fluctuations around a global fixed point with a timescale defined by the mismatch between excitatory and inhibitory synaptic timeconstants, i.e. just a few milliseconds. This fast timescale is inconsistent with experimental data as fluctuations of cortical activity occur on a timescale of about a hundred milliseconds^{50} and exhibit signatures of metastable dynamics^{22,23,24}.
An alternative mechanism that can account for the slow timescale of cortical fluctuations is based on multistability. In this case, slow correlated fluctuations arise from stochastic transitions between multiple fixed points in the network. Multistability can result from clustered excitatory connectivity, where each cluster corresponds to a fixedpoint attractor^{51}. Further, bistability between high and low firingrate attractors can arise in unstructured networks with strong recurrent excitation and slower negative feedback such as firingrate adaptation^{21,26,52,53} or shortterm synaptic depression^{54}. Our model of OnOff dynamics is based on bistability, which is consistent with exponential distributions of On and Off episode durations in behaving monkeys^{22}. Similar models with slow negative feedback were used to reproduce UpDown dynamics^{26,55}. This mechanism can generate slow alternations between high and low firing rates via several dynamical regimes. In particular, UpDown dynamics were found to be bistable during anesthesia^{26} and excitable during slowwave sleep^{55} (a single stable fixed point from which suprathreshold fluctuations induce large transient events). The models with multistability capture the slow timescale of cortical fluctuations and produce realistic noise correlations^{21,26,53}. However, the multistable networks studied previously were not endowed with a spatial connectivity layout akin to organization of cortical networks, hence they do not produce any spatial structure of noise correlations.
To account for both the slow timescale and spatial structure of noise correlations in the visual cortex, our network model combines local bistable OnOff dynamics with spatially organized connectivity. In our model, correlated variability arises from metastable transitions between the On and Off fixed points, and not from fluctuations around a single global attractor^{32}. Our results suggest that a theory of noise correlations in the visual cortex should incorporate both the anatomical connectivity structure of visual areas as well as the local bistability of population dynamics in single columns.
Recurrent network models were previously developed to suggest possible circuit mechanisms that produce a reduction of noise correlations during attention^{31,32,56}. These models are based on a dynamical mechanism, where the network operates around a global fixed point and attentional inputs increase the stability of this fixed point leading to suppression of correlated fluctuations. Specifically, in the network with intrinsically generated shared variability, the stability of the operating point can be increased by upregulating activities of inhibitory neurons^{31}. However, elevated inhibition reduces firing rates of excitatory neurons, which contradicts attentional enhancement of firing rates in experiments. In the network filtering external noise, the stability of the global fixed point can be increased by excitatory inputs when the network operates in inhibition dominated regime^{32}. In this scenario, an excitatory input increases effective lateral connectivity, which suppresses the transmission of the correlated input noise (Supplementary Note 3.4).
The mechanism we propose differs from these previous models. First, we show that a reduction of noise correlations during attention is not universal. Therefore, a network mechanism should account for heterogeneous changes of noise correlations across different anatomical dimensions. Second, the mechanism in our model is based on the local bistability of OnOff dynamics in single columns. Attentional inputs change the stability of the On and Off fixed points, which effectively modulates the efficacy of lateral interactions across the network leading to changes of the correlation length. This mechanism is fundamentally distancedependent, as the major changes of noise correlations in our model are driven by changes of the correlation length (Supplementary Note 2.3, 3.2). As a consequence, we find that in Fourier space the lower spatial frequency modes contribute most to noisecorrelation changes (Supplementary Note 3.2). This result partially agrees with the previous model^{31}, where the dominant part of noisecorrelation changes arises from zero spatial frequency mode, which, however, predicts a spatially uniform modulation of noise correlations. In contrast, contributions from higher spatial frequency modes are not negligible in our model. A combination of all spatial frequency modes generates a nonmonotonic profile of noisecorrelation changes in lateral dimension, a prediction that was confirmed in our data.
Several biophysical substrates could mediate the network mechanism of attentional modulation in our model. Topdown projections from frontal cortical areas, especially Frontal Eye Fields (FEF)^{57,58} can provide temporally and spatially precise inputs to drive fast and local modulation of OnOff dynamics in the visual cortex. Most FEF projections to V4 target pyramidal neurons^{40}, in agreement with our model where reduction of noise correlations in superficial layers is driven by external excitatory inputs, and unlike models where reduction of noise correlations is mediated by inputs to inhibitory neurons^{31}. Neuromodulatory inputs can also mediate effects of attention^{59} and can influence OnOff dynamics by modulating neural excitability and firingrate adaptation^{60}. The level of Acetylcholine (ACh) can modify the efficacy of synaptic interactions during attention in a selective manner^{61}. An increase in ACh strengthens the thalamocortical synaptic efficacy by affecting nicotinic receptors and reduces the efficacy of horizontal recurrent interactions by affecting muscarinic receptors. A decrease in the efficacy of horizontal interactions leads to a reduction of correlation length in our model. Further, laminar distribution of topdown inputs^{40} and of neuromodulation, combined with layerspecific horizontal connectivity could account for the differential modulation of noise correlations in superficial and deep layers that we observed. Identifying precise mechanisms by which these multiple biophysical components interact within a columnar microcircuit is an important direction for future work.
Methods
The research complies with all relevant ethical regulations. Experimental procedures were in accordance with NIH Guide for the Care and Use of Laboratory Animals, the Society for Neuroscience Guidelines and Policies, and Stanford University Animal Care and Use Committee.
Behavior and electrophysiology
Two male monkeys (Macaca mulatta, 8−12 kg, between 6 and 9 years of age) were used in experiments. The monkeys were trained on a cued changedetection task^{22,62}. The monkey was required to make a difficult visual discrimination at a peripheral location with a central cue indicating which location would contain the change. The monkey was rewarded for reporting a successful detection with a saccade to the diametrically opposite peripheral location (antisaccade response). On each trial, a small central cue indicated the stimulus that was most likely to change. The cued stimulus was therefore a target of covert attention. The monkeys reported stimulus changes with an antisaccade to the location opposite to the change, which was therefore a target of overt attention due to anticipation of antisaccadic response^{22,62} (Supplementary Fig. 1). Modulations of neural responses in V4 were highly similar during the covert and overt attention, including changes in firing rates, spiking variability and noise correlations^{22,62}, and therefore we combined the covert and overt attention conditions in our analyses. The monkey initiated each trial by fixating a central fixation dot on the screen. Within several hundred milliseconds, four peripheral stimuli appeared (static Gabor patches: oriented black and white gratings in a circular Gaussian aperture). After a short delay, the attention cue appeared: a short line originating at the fixation dot and extending in the direction of one of the four stimuli, randomly chosen on each trial with equal probability. The cue indicated with ~90% validity which of the four stimuli, if any, would change on each trial. After a postcue period of 600−2300 ms, all stimuli synchronously disappeared for a brief interval and then reappeared. On half of the trials, one of the four stimuli reappeared with a changed orientation (i.e. rotated in place), and the monkey was rewarded for performing a saccadic eye movement to the location opposite to the changed stimulus. On the other half of the trials, all stimuli reappeared with the same orientations as they had before disappearing, and the monkey was rewarded for maintaining fixation on the central dot.
While monkeys performed the attention task, recordings were made in the visual area V4 with a 16channel linear array microelectrodes^{22,62}. The total length of array is 2.25 mm, and the spacing between electrical contacts is 150 μm. Recordings were targeted with MRI to be as perpendicular to cortical layers as possible so as to maximize the overlap of receptive fields (RFs) of recorded neurons. Each of the recording channels was assigned laminar depth relative to a common current source density marker as described previously^{33}.
Data analysis
Data were analyzed with a custom code written in Matlab. We measured Fano factor and noise correlations in our recordings using spikecounts N of MUA and SUA in 200 ms bins (400–600 ms window after the attention cue onset). The Fano factor is the ratio of the spikecount variance to its mean across trials: Var[N]/E[N]. The noise correlation r_{sc} is the Pearson correlation coefficient between spikecounts N_{i} and N_{j} of two neurons:
We estimated parameters of the OnOff dynamics in single columns by fitting population spiking activity in our recordings with a twostate Hidden Markov Model (HMM) as described previously^{22}. HMM has a latent variable representing an unobserved population state that stochastically switches between the On and Off phases following Markov dynamics. Spikes on 16 simultaneously recorded channels are assumed to be generated by inhomogeneous Poisson processes, with different mean rates during the On and Off phases. The latent OnOff state is shared by the population, but the On and Off firing rates can differ across neurons. HMM was fitted to MUA spikecounts in 10 ms bins, during a timewindow starting at 400 ms after the attention cue onset and until the end of the postcue period. The duration of this timewindow ranges between 200 and 1900 ms across trials. HMM was fitted separately for each of the 32 task conditions (4 attention conditions × 8 grating orientations). The HMM parameters were optimized with the ExpectationMaximization algorithm^{22}. The HMM had 34 parameters: firing rates in the On (r_{on}) and Off (r_{off}) phases for each of 16 channels and transition probabilities p_{on} and p_{off} for the entire population.
We selected the optimal number of HMM states using the same crossvalidation procedure as in our previous work^{22}. We computed average crossvalidation error for HMMs with n phases (n = 1, . . . 8), normalized by the average crossvalidation error of the HMM with 1 phase. For each recording session, the 4fold crossvalidation error was computed in 200 ms windows for each condition (4 attention conditions × 8 stimulus orientations), and then averaged across all channels, conditions and crossvalidation folds. For most recordings, addition of the second phase greatly reduced crossvalidation error compared to the singlephase HMM, whereas adding more phases resulted in only marginal improvements: the error curves display an elbow at n = 2, suggesting that 2phase HMM is the most parsimonious model for our data. For some recording sessions, HMMs with n > 1 phases did not perform better than the 1phase HMM and the error curves did not exhibit a kink for these recordings. We defined these to be onephase recordings.
We estimated lateral shifts between channels in our laminar recordings by distances between centers of their RFs. The RF mapping procedure was described previously^{22}. RFs were measured by recording spiking responses to briefly flashed stimuli on an evenly spaced 6 × 6 grid covering the lower left visual field. Spikes in the window 0–200 ms relative to stimulus onset were averaged across all presentations of each stimulus. The RF center was defined as the center of mass of the response map. The lateral cortical distance d_{cortical} (mm) was estimated from the RFcenter distance d_{RF} (d.v.a) using the cortical magnification factor M for each eccentricity E^{38}:
Network model of interacting columns
The model describes spatiotemporal dynamics of neural population activity across the cortical surface. The network consists of two twodimensional square lattices of units, representing superficial and deep cortical layers. Each unit represents a local population of neurons within one layer of a single column. The dynamical variable r(x, t) represents the mean firing rate of this population. The twodimensional lateral coordinates are denoted as x. The dynamics of the network model are given by
Here a(x, t) is the adaptation variable, ξ is a white Gaussian noise of unit intensity, and we omit the spatial indices of variables r and a for clarity. ϵ ≪ 1 is a constant that separates the timescales of the fast dynamical variable r, and slow adaption variable a (Supplementary Note 2.2.3 and Supplementary Fig. 12).
The noise term ξ in the adaptation variable drives stochastic transitions between the On and Off phases in single columns. This phenomenological noise term models fluctuations in population activity due to internal spiking noise and/or external stochastic inputs. Spiking noise can arise from finitesize fluctuations^{63} as well as from biophysical sources, such as stochasticity of ion channels or synaptic release. External inputs correlated on the spatial scale of a column could also probabilistically drive OnOff transitions, for example, inputs related to microsaccades^{22}. The term ξ(t) models the net effect of various biological noise sources. Including the noise in the equation for adaptation variable enables analytical reduction of the dynamical system to the binaryunit model^{64} (Supplementary Note 2.2.4). Adding the noise term in the firingrate variable produces qualitatively similar results (Supplementary Fig. 13).
The function F(r) is given by
This piecewise linear function approximates the inverted Nshaped rnullcline, typically used in ratemodels with adaptation^{21}, which allows us to analytically reduce the dynamical system to a binaryunit model^{64}. The term W ∇^{2} r represents lateral interactions between neighboring units, where \({\nabla }^{2}r={\partial }_{x}^{2}r+{\partial }_{y}^{2}r\) implements a diffusive coupling and W is the interaction strength parameter. The external currents I_{stim} and I_{attn} are applied to local groups of units to model stimulus and attentional inputs, respectively. A constant ϵ ≪ 1 separates the timescales of the fast firingrate variable r and slow adaptation variable a. The parameters g, f, Q are chosen so that the system is bistable^{64}, where the population rate r stochastically switches between two stable fixed points, corresponding to the On and Off phases.
We match the model to experimental data using the fitted HMM parameters. Specifically, the HMM transition matrix P (p_{11} = p_{off}, p_{12} = 1 − p_{off}, p_{22} = p_{on}, p_{21} = 1 − p_{on}) provides an estimate of the OnOff transition rates: α_{1} = (1 − p_{off})/Δt and α_{2} = (1 − p_{on})/Δt, where Δt = 10ms is the bin size used for HMM fitting. HMM also estimates the On and Off firing rates r_{on} and r_{off} for each MUA and SUA, which we use to generate spikes of the model neurons. To this end, for each network unit we segment the simulated timeseries r(t) into the On (S = 1) and Off (S = 0) phases as S(t) = Θ[r(t)], where Θ is the Heaviside step function. The spike counts are then generated from inhomogeneous Poisson processes with rates r_{i}(t), where the firing rate for neuron i is
Simulations
We simulated the network model Eq. (3) on a 256 × 256 discrete square lattice with a time step of 0.005 s. The unit activities are initialized randomly. We compute noise correlations from 100 simulated trials for each set of parameters. On each trial, we simulated the period of spontaneous activity, stimulus period and attentioncue period, as in the experimental data. During stimulus period, external inputs \({I}_{{{{{{{{\rm{stim}}}}}}}}}\) were applied to two local groups of units with the size 50 × 50. During the attentioncue period, one of these two groups also received attentional inputs I_{att}. To calculate noise correlations, we either assigned fixed values of r_{on} and r_{off} or sampled them from distributions of r_{on} and r_{off} extracted from experimental data by HMM.
Reduction to a binaryunit network
When the dynamicalsystem network operates in the bistable regime, the activity of each unit i can be approximated by a binary variable S_{i}^{64}, where S_{i} = 1 refers to On phase, and S_{i} = 0 to Off phase. We derived a reduced network model, where the dynamical equations describe the state transition probabilities of binary units. Using the meanfield approximation, we derived an approximate form for transition rates of binary units (Supplementary Note 2.3). In the leading approximation order, we have
Here S_{i±1} are the sum of activities of neighboring units that are connected to a given unit S_{i}. α_{1}, α_{2}, and β_{1}, β_{2} are functions of parameters in the dynamicalsystem model: f, g, Q, I_{stim}, and I_{attn}. This reduced model allows us to derive analytical formulas for correlations between units in the network.
The reduced network model of binary units
The binaryunit network operates on a twodimensional square lattice. The network consists of N units. Each unit can be in a discrete On (S_{i} = 1) or Off (S_{i} = 0) state, represented by a binary variable S_{i} = {0, 1}, (i = 1, . . . , N). At time t, the probability of the system to be in a certain configuration {S} = {S_{1}, S_{2}, . . . , S_{N}} is denoted as P({S}, t). The rate of change of P({S}, t) is described by the master equation:
Here {S}^{i*} = {S_{1}, S_{2}, . . . , 1 − S_{i}, . . . , S_{N}}, and w(S_{i}) is the transition rate. When S_{i} = 0, the transition rate of S_{i} from 0 to 1 is
When S_{i} = 1, the transition rate of S_{i} from 1 to 0 is
Here α_{1} and α_{2} represent the baseline transition rates of each unit without interactions with other units, and β_{1,2} describe how the transition rates are influenced by nearby units S_{i±1}. The diffusive coupling between units is described by the discrete Laplacian:
For simplicity, we use a single index i to represent indices in arbitrary dimension. For example, in two dimensions i = (x, y), and we have
The dynamics of the binaryunit network resemble Glauber dynamics of the 2D Ising model. However, in general, the detailed balance condition does not hold in the binaryunit model, so its dynamics are different from the 2D Ising model (Supplementary Note 2.4).
Based on the master equation, the dynamics of the first and second moments are given by
We studied the dynamics of the binaryunit network analytically and in simulations. In simulations, the states of all units were updated based on their transition rates in 10 ms time bins.
Theoretical prediction of noise correlations
Assuming the network evolved to the equilibrium state, we derived in the continuum limit the steadystate solution for the averaged first moment S(∞) and quadratic moments G(d; ∞):
Here the dimensionless correlation length L is given by
and d is the dimensionless lateral distance measured in units of the lattice constant Δd.
Using these expressions for the first moment S(∞) and quadratic moments G(d; ∞), we derived an analytical formula for the noise correlations. Consider a pair of neurons (x, i) and (y, j) that are indexed by the lateral positions x, y of units to which they belong, and by their indices i, j within these units. Spikecounts N(x, i) and N(y, j) of these two neurons are measured in a timewindow of duration T. The theoretical prediction of noise correlation r_{sc}[N(x, i), N(y, j)] is given by
This equation shows that noise correlations decay exponentially with the lateral distance d = ∣x − y∣, with the decayrate characterized by the correlation length L. The amplitude \({{{{{{{\mathcal{A}}}}}}}}({\alpha }_{1},{\alpha }_{2})\) depends on the OnOff transition rates α_{1}, α_{2}, and on the On/Off firing rates r_{off}(x, i), r_{off}(y, j), Δr(x, i), Δr(y, j). Specifically,
where
The amplitude \({{{{{{{\mathcal{A}}}}}}}}({\alpha }_{1},{\alpha }_{2})\) is the theoretical prediction for noise correlations within single columns (in the limit where d = ∣x − y∣ → 0) used in Fig. 3. In Figs. 4d and 5c, we compute the noise correlation at d = 0 (i.e. \({{{{{{{\mathcal{A}}}}}}}}({\alpha }_{1},{\alpha }_{2})\)) with realistic On and Off firing rates. We sampled 1,000,000 pairs of On and Off firing rates from the distributions estimated by HMM in the V4 data and averaged the noise correlation over all sampled pairs. At distance d = 0, the difference between simulations and analytical prediction due to sampling was less than 1%. At distances d > 0, we calculate the analytical prediction of noise correlations as the product between the analytical noise correlation at d = 0 and the exponential spatial decay factor.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
Data availability
All behavioral and electrophysiological data used in this study are available as downloadable files at https://doi.org/10.6084/m9.figshare.16934326.v3. Source data are provided with this paper.
Code availability
The source code written in Matlab to reproduce results of this study is freely available on GitHub (https://github.com/engellab/NetworkmodelsofspatiotemporalOnOffdynamics).
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Acknowledgements
This work was supported by the Swartz Foundation (Y.S.), the NIH grant R01 EB026949 (T.A.E.), the Pershing Square Foundation (T.A.E.), Alfred P. Sloan Foundation Research Fellowship (T.A.E.), and the NIH grant EY014924 (T.M.).
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Y.S., N.A.S., K.B., T.M., and T.A.E. designed the study. N.A.S. and T.M. designed the experiments. N.A.S. performed experiments, spike sorting, and RF measurements. Y.S. and T.A.E. developed data analysis methods and mathematical models. Y.S. analyzed the data, performed analytical calculations and model simulations. Y.S., N.A.S., K.B., T.M., and T.A.E. discussed the findings and wrote the paper.
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Shi, YL., Steinmetz, N.A., Moore, T. et al. Cortical state dynamics and selective attention define the spatial pattern of correlated variability in neocortex. Nat Commun 13, 44 (2022). https://doi.org/10.1038/s41467021277244
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DOI: https://doi.org/10.1038/s41467021277244
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