Correlations are small when based on few spikes

Correlations in pairs of neurons that fire few spikes per trial are weaker than in pairs that respond more strongly^{7, 8, 28, 34}. Mathematically, correlations between discrete variables such as spike counts need not depend on their magnitude. Binary variables can be perfectly correlated, uncorrelated, or correlated to any intermediate degree. The dependence of r_{sc} on spike count is therefore not a mathematical given but a biological and experimental phenomenon, as described below.

The number of spikes produced by a neuron is determined by its underlying firing rate and the time window over which responses are measured, and both factors vary considerably across studies. Firing rates depend on the stimulus, the animal's cognitive state, and the neurons’ tuning. In some studies, the stimulus is tailored to the tuning preferences of the neurons under study^{6, 12, 29}. In others, a common stimulus is used to drive a large number of simultaneously recorded cells^{7, 26, 28}, yielding weaker responses on average. In addition, neurons in some cortical areas, such as V1 and MT, are easier to drive than those in areas where stimulus preferences are less understood. The time window used to count spikes is purely an experimental decision, and thus also varies across studies. Below we show that both low firing rates and brief measurement windows can lead to lower measured values of r_{SC}.

The spike threshold can reduce spike count correlations

The relationship between r_{SC} and firing rate depends largely on the proportion of subthreshold events that are masked by the spiking threshold^{34}. Correlations in spiking responses arise because of cofluctuations in synaptic input, which give rise to correlated membrane potential fluctuations in pairs of neurons^{16, 33, 35-37}. The degree to which shared membrane potential fluctuations are observable in spiking responses depends on the firing rates of the cells: when the mean membrane potential is far below threshold, responses are weak and many of the shared membrane potential fluctuations are unobservable in the spiking responses^{34, 37, 38}.

This relationship is illustrated with a simple simulation shown in (similar to ^{34}). We simulated correlated membrane potentials by picking values for each of two neurons from a bivariate Gaussian distribution. The membrane potential produces a spike response defined by the non-linearity shown in the middle panel. The shape of the nonlinearity is not critical; ours was chosen so that the variance of the spiking response is nearly the same as the mean, for both weak and strong responses^{39}.

We set the membrane potential correlation to 0.2 in all of our simulations, but the measured spike count correlation depended on response strength. When the mean membrane potential is above threshold (), the correlation in spiking responses is similar to that of the subthreshold response. However, when the mean membrane potential is far below threshold (), r_{SC} is markedly lower than between the subthreshold responses. This masking of correlated activity cannot be overcome by making more observations, which reduces the variance of correlation measurements but does not alter the mean. If membrane potential correlations are the same for stimuli that drive weak and strong responses, the spiking responses for the latter will thus be more correlated (), reaching asymptote at the strength of the underlying membrane potential correlation.

The dependence of correlation on response strength is typically assessed by comparing r_{SC} to the geometric mean response of the two neurons^{7, 8, 28, 34}. However, r_{SC} will be reduced if either neuron responds weakly. For example, a pair in which one neuron has a mean response of 0.01 sp/s and the other 100 sp/s has the same geometric mean response as a pair whose mean responses are both 1 sp/s, but the measured r_{SC} of first pair is only 15% of the underlying correlation compared with 85% in the second pair. shows r_{SC} as a function of the response strength of the two neurons. A dependence of r_{SC} on the geometric mean response would appear as diagonal stripes from the top-left to bottom right. Instead, has vertical and horizontal bands, indicating that the magnitude of r_{SC} depends more on the minimum response of the two neurons than their mean.

Counting spikes over short windows can lead to weaker correlation

The number of spikes a neuron fires also depends on the time window over which responses are measured. The studies in use windows that range from tens of milliseconds to multiple seconds. Counting spikes over short epochs can lead to weaker observed values of r_{sc,} even if both neurons are sufficiently responsive to avoid the effect of thresholding described in .

We ran additional simulations to illustrate the dependence of r_{SC} on measurement window (). To do so, it was necessary to use a framework in which we could control the timescale of correlation. Since the simulations of do not specify a timescale, we instead imposed correlations by adding a small number of common spikes to the otherwise independent (Poisson) spike trains of two simulated neurons (; see also ^{40}).

This simulation illustrates that correlations are systematically underestimated if the counting window is shorter than the jitter in the timing of the common spikes. If the common spikes occur at the same instant (), the resulting synchrony will be evident in spike count correlations based on responses measured over arbitrarily small windows (line marked ‘no jitter’ in ). If the times of these common spikes are jittered (), however, the resulting spike count correlation will only be fully evident when it is calculated from responses during longer response epochs (lines labeled 5-80 ms in ). For instance, when spike times are jittered using a Gaussian distribution with a standard deviation of 80 ms, a window of several hundred milliseconds is needed to capture the full strength of correlation. A similar dependence on measurement window is observed for Poisson distributed spikes conditioned on correlated underlying firing rates (see

^{22} and

Supplementary Material for analytical description). Note that in this scenario correlations do not arise from nearly synchronous spikes, yet the observed correlations are still smaller for brief measurement windows.

Several studies have measured the timescale of correlation in cortex, providing estimates ranging from tens^{22} to a few hundred milliseconds^{8, 12, 23}. Measurements using response windows briefer than these timescales are thus almost certain to yield weaker correlations than those using longer response windows.

Effect of spike sorting errors on measured r_{SC}

Issues of recording quality or spike sorting can artifactually increase or decrease measurements of r_{SC}. In general, errors that add independent variability to the responses of one neuron will bias estimates of correlations toward zero, while errors that involve combining the responses of multiple cells will increase the magnitude of measured r_{SC}.

One spike sorting error that can lead to consistent overestimation of correlations is mistaking multiunit activity as spikes from a single neuron. Combining several units into one effectively averages out variability that is independent to each cell, so the correlation between two clusters of multiunit activity will be larger than between pairings of the constituent neurons. To quantify this influence, we simulated multiunit activity by grouping the responses of individual neurons produced from the simulation in . We computed r_{SC} between increasingly large groupings of neurons, made by simply summing the responses of individual units whose pairwise correlations ranged from very small (0.001) to more typical (0.1) values ().

These simulations show that when pairwise correlations are weak, the measured value of r

_{SC} grows slowly with the number of units grouped together. Specifically, the value of r

_{SC} between multiunit clusters (r

_{SC-measured}) is given by:

where

*n* is the number of units grouped together and r

_{sc-pair} is the pairwise correlation, which in this simulation was the same between all units, whether they were in the same or different groupings

^{41, 42}. For nr

_{sc-pair}<<1, r

_{SC} thus grows linearly with the number of units

*n* contributing to the multiunit clusters (i.e. as nr

_{sc-pair}). If, for example, the underlying pairwise correlation were 0.01, one would need to record simultaneously from clusters of nearly 20 cells to obtain the r

_{SC} of 0.2 that has been reported in many studies. Recordings from such large groups of cells would be evident by a proportional increase in firing rate and thus easy to distinguish from single unit activity.

It is important to note that excessively restrictive criteria in spike sorting can lead to an *underestimation* of r_{SC}. Sorting spike waveforms in extracellular recordings is essentially a decision about when noisy voltage traces are similar enough that they are likely to have come from a single neuron. We considered a simple scenario in which the waveforms from two neurons were recorded on separate electrodes and clearly distinct, but each waveform was corrupted by noise (). We simulated increasingly stringent spike sorting by discarding a proportion of waveforms from each neuron. In this simulation, no spikes are mistakenly assigned to the other unit; changing the threshold simply alters the proportion of events that are accepted as valid spikes. As fewer spikes are accepted (meaning that the criterion for acceptance becomes more stringent), the measured value of r_{SC} decreases (). Such oversorting (discarding valid waveforms) decreases r_{SC} because whether a spike is accepted as valid or not is a random event—dependent strictly on noise—and variability that is independent for the two cells weakens measured correlation.

The relationship between the measured r

_{SC} (r

_{SC-oversort}) and the proportion of spikes discarded is given by

where

*p*_{1},

*p*_{2} are the probabilities that a spike is discarded from neuron 1 and 2, respectively, and

*n1*,

*n2* are the spike counts of those cells (see

Supplementary Material for derivation). In our simulation, the probability of deletion,

*p*, is the same for the two cells (

*p*_{1}=

*p*_{2}), the two cells have equal rates and the Fano factor equals 1, so

*<n*_{1}>=

*<n*_{2}>=var(

*n*_{1}) =var(

*n*_{2}). In this case,

Therefore, when half the spikes are accepted, measured r

_{SC} decreases twofold (from 0.20 to 0.10). When the probability of discarding waveforms is different for the two cells (p1≠p2) or when the Fano factors are different from 1, the decay with sort stringency will be different from that depicted in . Because the denominator in

Equation 2 is always greater than 1, however, r

_{SC-oversort} will always underestimate r

_{SC-original}.

A second scenario in which spike sorting can reduce measured correlations is when waveforms belonging to a single neuron are mistakenly assigned to multiple neurons (). We simulated this scenario by randomly dividing the spikes from a single neuron into two units (different shades of red in ). We then assessed the correlations between these units and a single unit () measured on another electrode. This manipulation reduced r_{SC} by roughly 30% compared to the true underlying correlation (, open symbols compared to filled). When more than two units are created from a single unit, the measured value of r_{SC} falls further ().

Variability in internal states can affect r_{SC}

Measurements of r_{SC} are based on sets of trials in which the stimulus and behavioral conditions are held as constant as possible. Despite experimenters’ best efforts, however, internal factors such as arousal, attention, or motivation are bound to vary. Similarly, in experiments using anesthetics, the depth of anesthesia may vary over time. Such fluctuations could in principle comodulate the responses of groups of cells and thus directly contribute to measurements of r_{SC}.

It is impossible to determine experimentally the degree to which this is the case, as internal variables, by definition, are not under experimental control. However, comparing the timescale of fluctuations in internal states with that of correlations can provide important constraints on their contribution. In the absence of salient changes in the visual scene, animals can only shift their attention approximately once every 400 ms^{43-45}. Even shifts in exogenous attention take 100-200 ms following an abrupt stimulus change^{43, 44, 46-49}. Fluctuations in other cognitive states like arousal or motivation or in anesthetic state likely occur even more slowly.

In contrast, correlations are dominated by fluctuations on shorter timescales. The timescale of correlation can be estimated within a trial by comparing r_{SC} computed in time windows of different sizes or by using the spike train crosscorrelogram^{8, 12, 22}. The contribution of fluctuations that occur across trials can be estimated by calculating r_{SC} between responses measured on different but nearby trials (e.g. by shifting the trials of one of the two neurons)^{22, 28}. These measures reveal that correlations are typically dominated by fluctuations on the timescale of tens to a few hundred milliseconds; correlations between responses measured in different trials are usually near zero (^{22, 28}; Kohn and Smith, Cohen and Maunsell, unpublished observations). Thus, although variations in cognitive factors or anesthesia states may well contribute to measured correlations, these correlations likely arise in large part from fluctuations on faster timescales.