Abstract
A large population of neurons can, in principle, produce an astronomical number of distinct firing patterns. In cortex, however, these patterns lie in a space of lower dimension1,2,3,4, as if individual neurons were “obedient members of a huge orchestra”5. Here we use recordings from the visual cortex of mouse (Mus musculus) and monkey (Macaca mulatta) to investigate the relationship between individual neurons and the population, and to establish the underlying circuit mechanisms. We show that neighbouring neurons can differ in their coupling to the overall firing of the population, ranging from strongly coupled ‘choristers’ to weakly coupled ‘soloists’. Population coupling is largely independent of sensory preferences, and it is a fixed cellular attribute, invariant to stimulus conditions. Neurons with high population coupling are more strongly affected by non-sensory behavioural variables such as motor intention. Population coupling reflects a causal relationship, predicting the response of a neuron to optogenetically driven increases in local activity. Moreover, population coupling indicates synaptic connectivity; the population coupling of a neuron, measured in vivo, predicted subsequent in vitro estimates of the number of synapses received from its neighbours. Finally, population coupling provides a compact summary of population activity; knowledge of the population couplings of n neurons predicts a substantial portion of their n2 pairwise correlations. Population coupling therefore represents a novel, simple measure that characterizes the relationship of each neuron to a larger population, explaining seemingly complex network firing patterns in terms of basic circuit variables.
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Acknowledgements
We thank L. Buesing for advice on latent variable analysis, C. Reddy and T. Sato for technical assistance, and M. Häusser for advice on the manuscript. This work was supported by the Wellcome Trust (S.B.H., T.D.M.-F., M.C., K.D.H.), Engineering and Physical Sciences Research Council (K.D.H.), the European Research Council (T.D.M.-F.), the Medical Research Council (L.C.), National Institutes of Health (EY014924, N.A.S. and T.M.) and the Simons Foundation (M.C. and K.D.H.). M.C. holds the GlaxoSmithKline/Fight for Sight Chair in Visual Neuroscience.
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M.O. conceived the study and performed the in vivo electrophysiology experiments in mouse V1. M.O., N.A.S. and L.C. performed the analyses. N.A.S. and T.M. performed the experiments in primate V4. L.C., M.F.I., H.K., S.B.H. and T.D.M.-F. performed the imaging and in vitro experiments and contributed to data analyses. P.B. performed the experiments in rat A1. K.D.H. constructed the mathematical model. M.O., M.C. and K.D.H. designed the study and wrote the paper.
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Extended data figures and tables
Extended Data Figure 1 Pearson correlation between spike trains of individual units and the population rate.
To estimate the relation of a neuron to the population, an alternative to spike-triggered population rate (stPR) would have been to compute the Pearson correlation coefficient of the neuron’s spike train with the summed population rate of all other recorded cells (a measure we term ‘Pearson coupling’). This measure, however, is biased by firing rate. a, Pearson coupling and stPR were computed for a set of individual units in an example experiment. Pearson coupling is related to the stPR, but not identical to it. b, The numerical value of the Pearson coupling depends strongly on the bin size used, but the correlations measured with different bin sizes are tightly related. c, Pearson correlation is biased by firing rate41,42. The spike train of a single cell was ‘thinned’ to different firing rates by keeping only a random subset of its spikes; Pearson correlation with the population was recalculated for different values of firing rate. A strong effect of firing rate is seen. d, Performing the same analysis for population coupling (measured by stPR) demonstrates that this measure does not suffer from rate bias. For this reason, we chose to quantify population coupling with stPR in this work.
Extended Data Figure 2 Neighbouring neurons differ markedly in population coupling during spontaneous activity.
a, Dividing the data into two halves shows that population coupling, measured as the height of stPR at 0 time lag, is highly consistent over time (n = 431 neurons from 13 experiments; ρ = 0.76, P < 10−100, rank correlation). Coloured dots represent the four example cells. b, As in a for peak spike-triggered local field potential (stLFP) (ρ = 0.58, P < 10−100). c, Differences in stLFP disappear after shuffling spikes in a manner that preserves each neuron’s mean firing rate and the population rate (compare with Figure 1g). Inset, stLFPs in the actual spike trains (red) and after shuffling (grey), for neurons from all experiments (compare with Figure 1g). d, stPR size of V4 neurons is consistent over time (n = 262 neurons from 43 experiments; ρ = 0.95, P < 10−100, rank correlation).
Extended Data Figure 3 Neighbouring neurons in auditory cortex differ markedly in population coupling.
a, Spike-triggered population rate (stPR) for four example neurons recorded on the same electrode shank, during spontaneous activity in rat primary auditory cortex. b, Differences in population coupling disappear after shuffling spikes in a manner that preserves each neuron’s mean firing rate and the population rate distribution. c, d, As in a, b for the spike-triggered local field potential (stLFP). e, Dividing the data into two halves shows that population coupling, measured as the height of stPR at 0 time lag, is highly consistent over time (n = 76 neurons from 3 experiments; ρ = 0.92, P < 10−100, rank correlation). Coloured dots represent the four example cells. f, As in e for stLFP (ρ = 0.81, P < 10−100, rank correlation).
Extended Data Figure 4 Firing rate, burstiness and population coupling.
a, Similarly to other studies20,43 our recordings allow separation of narrow spiking (putative Pvalb+ inhibitory) and wide spiking (primarily excitatory pyramidal) neurons. Here, we used a trough-to-peak time of 0.66 ms as the separation criterion. b, There is a negative correlation between burstiness (the ratio between the peak and baseline of a neuron’s autocorrelogram) and mean firing rate, which is also the case individually for wide spiking (n = 384, ρ = −0.60, P < 10−9, rank correlation) and narrow spiking (n = 47, ρ = −0.82, P < 10−9, rank correlation) neurons. a.u., arbitrary units. c, There is a positive correlation between burstiness and population coupling, which is also the case individually for wide spiking (ρ = 0.46, P < 10−9, rank correlation) and narrow spiking (ρ = 0.50, P = 4 × 10−4, rank correlation) neurons. d, There is a negative correlation between firing rate and population coupling, which is also the case individually for wide spiking (ρ = −0.27, P = 10−7, rank correlation) and narrow spiking (ρ = −0.37, P = 0.01, rank correlation) neurons. The correlation between population coupling and firing rate can be predicted from the correlations between burstiness and firing rate and between population coupling and burstiness; the partial rank sum correlation between population coupling and firing rate, once burstiness is taken into account, is insignificant (ρ = 0.06, P = 0.25). This is also the case for wide spiking (ρ = 0.01, P = 0.78) and narrow spiking (ρ = 0.07, P = 0.65) neurons individually.
Extended Data Figure 5 Latent variable analysis.
a, Population coupling of each neuron is highly correlated with its loading in a single-factor latent variable model (see Methods). The similarity of each cell’s population coupling and loading indicates that the low-dimensional structure found by the latent variable model is homologous to that found by the coupling model. b, Percent of pairwise correlation structure explained by a latent variable model with 1–5 factors (black), and by the coupling model introduced in the present study (dashed purple line). Error bars show standard error. While the coupling model outperforms latent variable models with less than four degrees of freedom, this difference may arise primarily from the assumption of a Gaussian distribution for the latent variables. Indeed, if the population rate distribution generated by the latent variable model is substituted into the coupling model instead of the (correct) populate rate distribution, extremely poor performance results (dashed grey line).
Extended Data Figure 6 Population coupling and visual stimulation in mouse V1.
a, stLFPs computed for the four example neurons of Fig. 1a–f, from intervals of natural movie presentation (inverted for ease of comparison, see Figure 1f). b, Comparison of stLFP size during spontaneous and evoked activity across all experiments (ρ = 0.72, P < 10−100, rank correlation). c–e, Population coupling is plotted versus the f1/f0 ratio, preferred spatial frequency and orientation selectivity index (OSI) for neurons recorded in the infragranular layers of V1. All correlations are statistically insignificant. f, Similar to movie presentations (Figure 3e), the mean change in the activity of a cell in response to grating presentations (relative to baseline, averaged across contrasts and orientations) correlates with population coupling measured during spontaneous activity (ρ = 0.32, P = 2 × 10−6, n = 217, rank correlation). Black diamonds, running median. g, In the two-photon imaging data (of ∼10,000 cells) only a very weak correlation between OSI and population coupling was found (ρ = 0.066, P < 10−9, rank correlation).
Extended Data Figure 7 stLFP reflects the correlation between membrane potential (Vm) and LFP.
a, Example of a silicon probe population recording performed simultaneously with a whole-cell recording (in an anaesthetized animal). Four neurons shown in colour were recorded on the same shank of the silicon probe. b, Comparison of stLFP and Vm–LFP cross-correlation (VmLFPcc, appropriately scaled along the ordinate axis) for the intracellularly recorded cell. c, stLFP for the four neurons from a and the intracellularly recorded neuron, exhibiting diversity in the strength of coupling to LFP.
Extended Data Figure 8 Population coupling in two-photon data is not correlated with location and intrinsic properties of the neurons.
a, For each neuron in the central region of the imaging field (defined as a square quarter of the total imaging area), we compared its coupling to the population of all other neurons in the central region, with its coupling to population of all neurons outside of the central region. The two were highly similar; this was the case because the two population rate signals were themselves highly correlated (on average across experiments the Pearson correlation was 0.77). Thus, differences in population coupling measured between cells do not reflect differences in the fraction of nearby neurons imaged. b–d, No significant correlation was observed between population coupling (measured in vivo) and resting potential, input resistance and spike threshold (subsequently measured in vitro).
Extended Data Figure 9 Correlation between input connectivity and population coupling.
a, Cumulative distribution of population coupling of a target pyramidal neuron when an input connection was present (red) and when it was absent (blue). The medians (arrows) are significantly different (P = 0.008, rank sum test, n = 854 pairs). b, As in a for population coupling of the source pyramidal cells. The distributions shown were used for the logistic regression analysis in Fig. 4. c–f, To estimate what strength of correlation between input connectivity and population coupling would give rise to these observations, we constructed random directed graphs of 1,000 nodes (each node representing a L2/3 pyramidal cell) with the probability of connection from node 
to node 
given by
, where the propensities to receive and provide connections (
and 
, correspondingly) were randomly and independently chosen for each node from a Gaussian distribution. The resulting distribution of the number of input connections in a typical network is shown in c; the number of output connections was (by construction) similarly distributed. In addition, each node was assigned a population coupling value, highly correlated to the number of its input connections (on average ρ = 0.65); this correlation in a typical network is shown in d. e, f, We next asked how the relationship between measured connectivity and population coupling would look if we sample from 33 such randomly generated networks (equal to the number of animals used in our experimental data), the same amount of data empirically available in our in vitro recordings (that is, the connections between 2–3 randomly selected groups of 2–6 nodes). Results very similar to those of Fig. 4 were typically obtained (e, f; compare with a and Fig. 4d; error bars in f indicate standard error for binned data). In particular, when the entire procedure was repeated 1,000 times, in over 30% of the cases the P value of the difference between the medians (presented in a, e) was higher (that is, less statistically significant) than the value of 0.008 obtained in the actual data. Thus, the results shown in Fig. 4 and in a are consistent with a strong correlation between connection probability and population coupling.
Extended Data Figure 10 Mathematical model for the relationship between nonspecific connectivity, specific connectivity, and correlations.
a, A recurrent network where excitatory cells (triangles) send synaptic connections (arrows) to each other and to inhibitory cells (circles). Weakly coupled neurons (bottom) receive only connections from neurons with similar sensory preference (for example, for stimulus orientation, indicated in blue versus red). Strongly coupled neurons (top) also receive nonspecific connections from neurons of different sensory preference. b, The effect of nonspecific drive, such as caused by non-sensory top-down inputs, or occurring due to artificial optogenetic stimulation, is amplified through recurrent connections, leading to stronger activation of neurons with greater mean local input (darker shading). c, d, Correlations predicted by the model (analytically derived in Supplementary Information). c, Population coupling versus nonspecific connectivity 
, for all simulated excitatory neurons. d, Pseudocolour plot of predicted pairwise correlations for a random subset of excitatory neurons, ordered by population coupling. e–h, Dependence of correlations on specific and nonspecific connectivity. e, Predicted correlations based on nonspecific connections versus total observed correlations. f, Predicted correlation based on nonspecific connectivity versus difference in preferred orientation. As in the experimental data (Fig. 2e), no relation is observed. g, Observed correlation versus difference in preferred orientation. As has been widely reported, observed correlations are largest for neurons of similar orientation preference. h, Residual correlation (after removing prediction from nonspecific connectivity) versus difference in preferred orientation. Again as in our experimental data (Fig. 2f), the residual correlation is largest for neurons of similar orientation preference, indicating an additive relationship between correlations generated by specific connections and correlations generated by nonspecific connections.
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Okun, M., Steinmetz, N., Cossell, L. et al. Diverse coupling of neurons to populations in sensory cortex. Nature 521, 511–515 (2015). https://doi.org/10.1038/nature14273
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DOI: https://doi.org/10.1038/nature14273
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