Models neural activity by combining stimulus information with factor analysis of noise correlation. Discovery of latent variables governing noise correlation.

Problem

Model $p(\br \mid \bx)$ of population activity $\br \in ℝ^n$ to an arbitrary stimulus $\bx$. Two main sources of variance:

• Stimulus-driven activity: $\bf_θ(\bx) = 𝔼[\br \mid \bx]$, responses dependent on an arbitrary stimulus. Assumes conditional independence among neurons.
• Stimulus-conditioned variability: $\mathrm{Var}[\br \mid \bx]$, responses to repeated presentations of an identical stimulus.
  • Variability to the repeated presentations of identical stimuli, called noise correlations.
  • Dependent on stimulus, behavioral task, attention, brain state.
  • Others fit $p(\br\mid\bx)$ separately for each unique stimulus and require repeated presentations of said stimulus.

This paper introduces a model that accounts for both.

Model

• Spiking data is transformed using a trained normalizing flow $T_φ(\br)$
  $$T_φ(\br) = \mathrm{Affine} ∘ \exp ∘ \mathrm{Affine} ∘ \mathrm{ELU} ∘ \mathrm{Affine} ∘ \mathrm{ELU} ∘ \mathrm{Affine} ∘ \log ∘ \mathrm{Affine}$$
  where only the affine transformations are trainable. The flow acts on each dimension independently, that is $T_φ(\br) = [T_{φ_1}(r_1), \dots, T_{φ_n}(r_n)]^\T$.
  • ZIFFA includes a zero-inflated component to capture zero spikes.
  • Dependency of noise correlation on the stimulus can be indirectly captured during this transformation.

• Stimulus-conditioned variability:: The second-order variability is captured using a factor analysis model, which is a covariance matrix with rank equal to the number of latent variables $\bz$. The self-variance $Ψ_i$ is distinct for each neuron.
• Stimulus-driven activity: The means of the Gaussian transformed spikes $\E[\br\mid\bx] = \bf_θ(\bx)$ are modeled based on their previous paper (Lurz et al., 2020).

Rationale

Factor analysis assumes Gaussianity whereas spike rates are strictly positive and are distributed like a zero-inflated overdispersed Poisson distribution. Fixed transformations may be too limiting and cannot capture neuron-specific transformations.

Related Papers

(Keeley et al., 2020)

Data

6,000 images from ImageNet were shown in each scan.

• 1,000 images consist of 100 unique images each repeated 10 times to allow for an estimate of the neural response variability. We used the repeated images for testing.
• 4,500 training and 500 validation images.

~1,000 neurons from LM and V1 recovered.

Performance

Flow Transformation Performance

Comparisons

FA models outperform independent models. Although fixed transformations work better for conditional correlation.

Latent variables

The latent variables describe noise correlations (neuron-neuron correlations) from FA. These are inferred without repeats.

Correlations of the extracted latent variables ($k=3$) to various values. Position does seem to be a significant contributor to intrinsic variability, in addition to pupil size.

References