# Noise correlation in V1: 1D-dynamics explains differences between anesthetized and awake

We discussed state dependence of noise correlations in macaque primary visual cortex [1] today. Noise correlation quantifies the covariability in spike counts between neurons (it’s called noise correlation because the signal (stimulus) drive component has been subtracted out). In a 2010 science paper [2], noise correlation was shown to be much smaller than previously reported; in the range of 0.01 compared to the usual 0.1-0.2 range and stirred up the field (see [3] for a list of values). In this paper, they argue that this difference in noise correlation magnitude is due to population level covariations during anesthesia (they used sufentanil).

They showed that a factor model with 1-dimensional latent process captures the excess noise correlations observed during anesthesia. They extended GPFA [4] to include a linear stimulus drive component, and fit the trials corresponding to the same stimulus. This model assumes that the square-root transformed spike counts in 100 ms bins $y(t)$ follows,

$y(t) \sim \mathcal{N}(f(s(t)) + cx(t) + d, R)$

$x(t) \sim \mathcal{N}(0, K)$

where $s(t)$ is the stimulus, $f(\cdot)$ is the tuning curve drive, $d$ is the mean, $R$ represents the individual (diagonal) covariance, $c$ is the factor loadings vector, and $x(t)$ is the 1-dimensional population wide latent process with assumed smoothness given by the kernel $K(t_1, t_2) = \exp(-(t_1 - t_2)^2 / 2 / \tau^2 )$. The square-root transformation is for the variance stabilization, making Poisson-like spike count observations more like Gaussian distributed.

They show that the most of the extra noise correlation observed during anesthesia is explained by the 1D latent process, and the remaining noise correlation is in par with awake condition, while noise correlations in the awake data didn’t get explained away by it (figure below).

A remarkable result! (red) anesthesia, (blue) awake (solid) raw noise correlation (dashed) residual noise correlation unexplained by the 1D-latent variable

Note that this nonlinear relation between the mean rate and noise correlation is not a natural consequence of the generative model they assumed (equation above), since the noise is additive while firing rate is modulated by stimulus. That’s why they had to fit different factor loadings vector $c$ for different stimulus conditions. However, they show that a multiplicative LNP model where a 1D latent process multiplicatively interacts with the stimulus drive in the log firing rate does reproduce this quality (Fig. 6). (It’s a shame that they didn’t actually fit this more elegant generative model directly to their data.)

Furthermore, they showed that local field potentials (LFP) in 0.5-2Hz range is correlated with the inferred latent variable. Hence, low-pass filtered LFP averaged over electrodes is a good proxy for L4 population correlation under anesthesia.

1D-latent process, LFP, and LNP results are all consistent with Matteo Carandini’s soloist vs chorister story except his preparations were awake rodent as far as I recall. This story also bears similarity with [5-8]. This is certainly an exciting type of model and there are many similar models being developed (e.g. [9, 10] and references therein). I hope to see more developments in these kinds of latent variable models; they have the advantage of scaling well in terms of number of parameters, but the disadvantage of being non-convex optimization. Also, I would like to see similar analysis for awake but spontaneous (non-task performing) state.

Their data and code are all available online (awesome!).

1. Ecker, A. S., Berens, P., Cotton, R. J., Subramaniyan, M., Denfield, G. H., Cadwell, C. R., Smirnakis, S. M., Bethge, M., and Tolias, A. S. (2014). State dependence of noise correlations in macaque primary visual cortexNeuron, 82(1):235-248.
2. Ecker, A. S., Berens, P., Keliris, G. A., Bethge, M., Logothetis, N. K., and Tolias, A. S. (2010). Decorrelated neuronal firing in cortical microcircuitsScience, 327(5965):584-587.
3. Cohen, M. R. and Kohn, A. (2011). Measuring and interpreting neuronal correlationsNat Neurosci, 14(7):811-819.
4. Yu, B. M., Cunningham, J. P., Santhanam, G., Ryu, S. I., Shenoy, K. V., and Sahani, M. (2009). Gaussian-Process factor analysis for Low-Dimensional Single-Trial analysis of neural population activityJournal of Neurophysiology, 102(1):614-635.
5. Luczak, A., Bartho, P., and Harris, K. D. (2013). Gating of sensory input by spontaneous cortical activityThe Journal of Neuroscience, 33(4):1684-1695.
6. Goris, R. L. T., Movshon, J. A., and Simoncelli, E. P. (2014). Partitioning neuronal variabilityNat Neurosci, 17(6):858-865. [last week’s blog post]
7. Tkačik, G., Marre, O., Mora, T., Amodei, D., Berry, M. J., and Bialek, W. (2013). The simplest maximum entropy model for collective behavior in a neural networkJournal of Statistical Mechanics: Theory and Experiment, 2013(03):P03011+.
8. Okun, M., Yger, P., Marguet, S. L., Gerard-Mercier, F., Benucci, A., Katzner, S., Busse, L., Carandini, M., and Harris, K. D. (2012). Population rate dynamics and multineuron firing patterns in sensory cortexThe Journal of neuroscience : the official journal of the Society for Neuroscience, 32(48):17108-17119.
9. Pfau, D., Pnevmatikakis, E. A., and Paninski, L. (2013). Robust learning of low-dimensional dynamics from large neural ensembles. NIPS 26