Efficient sensory coding of multidimensional stimuli
Fig 3
Example 2-D stimulus probability distributions and the resulting optimal encoding populations.
A) Each row represents a different example probability distribution over two stimulus dimensions (s1 and s2). For each panel, the probabilities are defined over a lattice ranging from -1 to 1 (cropped to the central 65% to remove boundary artifacts). Top: uniform over s2 and Gaussian distributed over s1 (μ = 0, σ = 0.25). Upper middle: isotropic bivariate Gaussian (μ = 0, σ = 0.25 in both dimensions). Lower middle: bivariate generalized Gaussian (μ = 0, σs1 = 0.75, σs2 = 0.25, power = 1.1). Bottom: Gaussian distributed over s1, with a σ that varies non-linearly with s2 (this distribution is non-separable). B) For each probability distribution, we show a down-sampled and scaled visualization of the inverse density mapping function. The direction and length of the arrows illustrate how density will be mapped from sensory space into the stimulus space. C) For each probability distribution, we show an example neuronal population that has been warped to optimally encode the stimulus. For these visualizations, we chose a population of neurons with isotropic bivariate Gaussian tuning curves (σ = 0.05) tiling the space on a hexagonal lattice (spacing ≈ 0.2). Though these choices for the population are arbitrary, varying them does not change the qualitative properties of the warped populations. Circular domain boundaries were used for the bottom three examples. To account for the uniform probability in panel A (top row), the population illustrated in panel C (top row) was defined with a square rather than a circular domain boundary. D) On the right side of each population, a pair of 1-D samples are illustrated. For each sample, s2 is held constant and the tuning curves are visualized over s1. Neurons with a maximum normalized response of less than 0.2 within the sample are not visualized. The distribution of Fisher information in each 2-D population is shown in S2 Fig).