Shared neural variability is ubiquitous in cortical populations. While this variability is presumed to arise from overlapping synaptic input, its precise relationship to local circuit architecture remains unclear. We combine computational models and in vivo recordings to study the relationship between the spatial structure of connectivity and correlated variability in neural circuits. Extending the theory of networks with balanced excitation and inhibition, we find that spatially localized lateral projections promote weakly correlated spiking, but broader lateral projections produce a distinctive spatial correlation structure: nearby neuron pairs are positively correlated, pairs at intermediate distances are negatively correlated and distant pairs are weakly correlated. This non-monotonic dependence of correlation on distance is revealed in a new analysis of recordings from superficial layers of macaque primary visual cortex. Our findings show that incorporating distance-dependent connectivity improves the extent to which balanced network theory can explain correlated neural variability.
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Cortical state dynamics and selective attention define the spatial pattern of correlated variability in neocortex
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We are grateful to T.S. Lee and A. Movshon for research support. This work was supported by National Science Foundation grants NSF-DMS-1517828 (R.R.), NSF-DMS-1313225 (B.D.), NSF-DMS-1517082 (B.D.), NSF-DMS-1612913 (J.E.R.), NSF-DMS-1516288 (J.E.R.) and NSF-DMS-1312508 (J.E.R.); National Institute of Health grants R01NS070865 (B.D., J.E.R.), CRCNS-R01DC015139 (B.D.), R01EY016774 (A.K.), R01EY022928 (M.A.S.) and P30EY008098 (M.A.S.); two grants from the Simons Foundation collaboration on the global brain (SCGB#325293MC;BD, B.D. and 364994AK, A.K.); by the Eye and Ear Foundation of Pittsburgh (M.A.S.); and by Research to Prevent Blindness (A.K., M.A.S.).
The authors declare no competing financial interests.
Integrated supplementary information
a) Firing rate as a function of time-averaged total synaptic input from the simulation in Figure 4. Blue dots are from 400 randomly selected excitatory neurons, the black curve is the best fit thresholded quadratic and the red dot is placed at the population-average firing rates, at which the gain can be computed as the derivative of the black curve. b) Same as (a), but for inhibitory neurons.
Supplementary Figure 2 Correlation as a function of distance for various connection probability profiles.
a) Bottom: Connection probability from feedforward layer (black) and connection probability within the recurrent layer (purple) as a function of neuron distance. Each curve is normalized by its peak. Top: Mean spike count correlation between excitatory neurons as a function of neuron distance from simulations (black) and from the theoretical calculation using Eq. (S.49) (red). b-h) Same as (a) except in (e) where recurrent excitatory (blue) and inhibitory (red) connection probabilities have different profiles.
a) The cross-spectrum, <Se,Se>(n), between excitatory neurons as a function of the magnitude of the spatial Fourier mode, |n|, computed using Eq. (S.49) with parameters from the example in Figure 4. The curve is normalized by its peak. b) The approximation, h(n), to the cross-spectrum computed using Eq. (S.53) and plotted as a function of spatial Fourier mode, |n|, with parameters from the simulation in Figure 4.
Supplementary Figure 4 Spike count correlations, firing rates, and input currents at increasing network size.
a) Average excitatory (bottom) and inhibitory (top) firing rates from simulations with increasing numbers of neurons (solid curves) and from from the theoretical predictions in Eqs. (S.5) (dotted gray lines). Blue (red) curves are for simulations identical to the one in Figure 3 (Figure 4), except that the number of neurons was varied, with connection probability held fixed, and simulation time was increased from 22 s to 42 s. b) Average excitatory (top), inhibitory (bottom) and total (middle) synaptic input current to 200 randomly sampled excitatory neurons from the simulations in (a). Synaptic input currents were normalized by capacitance and are therefore reported in units V/s. c) Mean spike count correlation between neuron pairs whose distance is between 0 and 0.1 (purple); between pairs with distance between 0.2 and 0.3 (green); and between pairs chosen randomly at all distances (black). Computed for simulations from the blue curves in (a). d) Same as (c), but on a log-log scale. Dashed lines are best fit lines of slope -1. e,f) Same as (c,d), but for the simulations from the red curves in (a,b).
a,b) Same as Figures 5b and 7b respectively, but distances are binned more finely. Correlations decreased from the first to the second bin, from the first to the third bin and from the second to the third bin (all three with p<10^−5; one-sided unpaired t-test). Correlations increased from the fifth to the seventh bin (p=0.008; one-sided un- paired t-test; t-value=2.4; df=1,704). The increase of correlations over the last three bins is significant under a Bonferroni correction for multiple comparisons over the last three bins (corrected p-value = 0.024). The apparent increase in correlation from the third to fourth bin is not significant (p=0.18; one-sided unpaired t-test; t-value=0.9; df=2,249). Distances 3.5-5mm are not shown since there were few such pairs and distances 5+mm are not shown because the precise distances for such pairs are not known, so they cannot be resolved into 0.5 mm bins (see Experimental Procedures and Figure 7b). Including these data does not alter significance (p-values remain <0.025). c) Histogram of all residual correlations. d,e,f) Solid black curves are the same as Figures 5b, 5c and 7b respectively, except we only included data from the two recording sessions in which linear electrodes were present. The increase from the third to the fifth bin was still significant when only including data from these two recording sessions (p=10^−4; one-sided, unpaired t-test; t-value=3.7; df=1,064). The dashed gray curves are the same as Figures 5b, 5c and 7b respectively.
a) Mean spike count correlation and b) residual correlation between putative L2/3 neurons in macaque primary cortex as a function of tuning similarity. Same as Figures 5b and 7b respectively, but correlations are partitioned by tuning similarity instead of distance. c,d) Same as (a,b), but from a computational model where αffwd = 20 degrees and for different values of αrec (see legend). e) Residual correlation as a function of the difference between preferred orientations in the model. All plots show mean +/- SEM. The non-monotonicity in (b) is not significant (p = 0.051; one-sided unpaired t-test between first and third bins; t-value=1.6; df=872).
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Rosenbaum, R., Smith, M., Kohn, A. et al. The spatial structure of correlated neuronal variability. Nat Neurosci 20, 107–114 (2017). https://doi.org/10.1038/nn.4433
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