On July 7, we discussed Partitioning Neural Variability by Gorris et al. In this paper, the authors seek to isolate the portion of the variability of sensory neurons that comes from non-sensory sources such as arousal or attention. In order to partition the variability in a principled way, the authors propose a “modulated Poisson framework” for spiking neurons, in which a neuron produces spikes according to a Poisson process whose mean rate is the product of a stimulus-driven component , and a stimulus-independent ‘gain’ term (G).

Applying the law of total variance, the variance for the number of spikes over a time period can then be written as . Moreover, with a gamma prior over , the distribution for has a negative binomial distribution, which can be fit to neural data.

The authors analyzed quite a large data set, from neurons in macaque LGN, V1, V2, and MT stimulated with drifting gratings. Their decomposition of the variance of (above) motivates the corresponding decomposition of within-condition sum-of-squares into components arising “from the point process” (i.e. corresponding to the term above) and from the gain signal. Using this decomposition they can estimate of the fraction of the within-condition variance that comes from the fluctuations of . They find that the modulated Poisson model fits the model better than the Poisson model, and that the gain share of variance is quite high (47.5%, compared to 47% due to stimulus drive and only 5.5% from the Poisson noise), and that it increases along the visual pathway.

The authors also write down an analogous decomposition for the covariance of two neurons into a “point-process” and gain components. A strength of this form for the covariance is that it allows for spike-count correlations to vary dramatically with stimulus drive. This decomposition of pairwise tuning correlation into “point-process” and gain components is then used for an analysis of correlations with respect to stimulus similarity, cortical distance, and temporal structure.

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My main issue with this paper was how they extended the Poisson-Gamma form to multivariate form. Their equation (5) doesn’t seem to have a good generative model. To me ‘point process’ component of the variability is independent by definition, but their covariance is both explained by population co-variability in the point process variability in addition to a slowly changing latent process.