Partitioning neuronal variability

Journal name:
Nature Neuroscience
Volume:
17,
Pages:
858–865
Year published:
DOI:
doi:10.1038/nn.3711
Received
Accepted
Published online

Abstract

Responses of sensory neurons differ across repeated measurements. This variability is usually treated as stochasticity arising within neurons or neural circuits. However, some portion of the variability arises from fluctuations in excitability due to factors that are not purely sensory, such as arousal, attention and adaptation. To isolate these fluctuations, we developed a model in which spikes are generated by a Poisson process whose rate is the product of a drive that is sensory in origin and a gain summarizing stimulus-independent modulatory influences on excitability. This model provides an accurate account of response distributions of visual neurons in macaque lateral geniculate nucleus and cortical areas V1, V2 and MT, revealing that variability originates in large part from excitability fluctuations that are correlated over time and between neurons, and that increase in strength along the visual pathway. The model provides a parsimonious explanation for observed systematic dependencies of response variability and covariability on firing rate.

At a glance

Figures

  1. The modulated Poisson model accounts for spike count variability.
    Figure 1: The modulated Poisson model accounts for spike count variability.

    (a) Model diagram. Spikes are generated by a Poisson process whose rate is the product of two signals: a stimulus-dependent drive, f(S), and a stimulus-independent gain, G, that is assumed to fluctuate slowly relative to the duration of experimental trials. (b) Variance-to-mean relation of the neural responses of a single V1 neuron stimulated with gratings drifting in different directions (gray dots), compared with predictions of the Poisson model (red line) and the modulated Poisson model (blue line). Responses were computed by counting spikes in a 1,000-ms window following response onset. Means and variances were calculated over 125 repetitions of each stimulus. The inset shows this relation measured over variable-duration windows for three drift directions (green, black and orange). Each data point is obtained from a randomly selected epoch of the corresponding raster with duration drawn uniformly from the range 1–1,000 ms (the orange data are taken from the inset raster). (c) Spike count distributions (gray histograms) measured for different stimulus drift directions compared to the best-fitting probability densities of the Poisson (red) and gamma-modulated Poisson (blue) models. (d) Log-probability of the cell responses under the Poisson model (red triangle) and the modulated Poisson model (blue triangle). Histograms illustrate the expected range of the log-probability statistic (computed with a 1,000 run parametric bootstrap) for the Poisson model (red) and the modulated Poisson model (blue). (e) Variance-to-mean relations predicted by the modulated Poisson model and an additive model for weak (light gray) to strong (black) fluctuations in gain.

  2. Comparison of neural response variability for cells in different visual areas.
    Figure 2: Comparison of neural response variability for cells in different visual areas.

    (a) Variance-to-mean relation for 63 LGN cells (orange), 396 V1 cells (dark green), 189 V2 cells (blue) and 137 MT cells (violet). Each data point illustrates the mean and variance of the spike count in a 1,000-ms window of one cell for one stimulus condition. (b) Comparison of the predictive accuracy of the Poisson and modulated Poisson models. Models are fit to a subset of data, and log-likelihood is computed on the remaining data and expressed per spike (Online Methods). (c) Distribution of stimulus-independent fluctuations in gain, summarized by the coefficient of variation of the gain. Triangles indicate the median value for each area. (d) Fraction of within-condition variance explained by gain fluctuations. ***P < 0.0001.

  3. Response correlation analysis for three example pairs of simultaneously recorded V1 neurons.
    Figure 3: Response correlation analysis for three example pairs of simultaneously recorded V1 neurons.

    (ac) Mean response to drifting sinusoidal gratings as a function of direction (72 stimulus conditions, 50 repeats, 1,280-ms count window). (df) Spike-count correlation as a function of the geometric mean of the mean spike counts of the two neurons. Each data point corresponds to a different stimulus condition. The blue line shows the correlations predicted by the best-fitting modulated Poisson model, and the surrounding light blue region indicates ± 1 s.d. of the distribution of estimates computed from 50 repeats. (gi) Spike-count correlation as a function of the mean response of the two neurons, as predicted by the modulated Poisson model. Color indicates correlation and points indicate response means for different stimulus conditions, as depicted in the two tuning curves shown in ac.

  4. Model-based decomposition of measured spike-count correlations into gain and point-process correlations.
    Figure 4: Model-based decomposition of measured spike-count correlations into gain and point-process correlations.

    (a,b) Measured spike-count correlation (a) and inferred point-process and gain correlations (b) as a function of electrode distance. (c,d) Measured and inferred correlations plotted as a function of the correlation in mean responses (tuning curves) of the two neurons. Thickness of bands indicates the 95% confidence interval.

  5. Gain fluctuations are correlated over time.
    Figure 5: Gain fluctuations are correlated over time.

    (a) Normalized responses as a function of time for three simultaneously recorded V1 neurons. (b) The autocorrelation function of the inferred gain for the example neurons. (c) The autocorrelation function of the gain, averaged across units for each data set. (d) The cross-correlation function of the gain, averaged across pairs for each data set.

  6. Analysis of spike-count variance for a population of MT neurons recorded in awake, behaving macaques.
    Figure 6: Analysis of spike-count variance for a population of MT neurons recorded in awake, behaving macaques18, 22.

    (a) Variance-to-mean relation for 307 MT cells. Each data point illustrates the mean and variance of the spike count in a 2,000-ms window of one cell for one stimulus condition. (b) Distribution of stimulus-independent fluctuations in gain, summarized by the coefficient of variation of the gain (top) and fraction of within-condition variance explained by gain fluctuations (bottom). (c) The autocorrelation function of the gain, averaged across units (trials are assumed to be separated by 5 s).

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Affiliations

  1. Center for Neural Science, New York University, New York, New York, USA.

    • Robbe L T Goris,
    • J Anthony Movshon &
    • Eero P Simoncelli
  2. Howard Hughes Medical Institute, New York University, New York, New York, USA.

    • Eero P Simoncelli

Contributions

R.L.T.G., J.A.M. and E.P.S. designed research; R.L.T.G. analyzed data; and R.L.T.G., J.A.M. and E.P.S. wrote the paper.

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The authors declare no competing financial interests.

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