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Jointly efficient encoding and decoding in neural populations
Fig 4
Characterization of the optimal solutions as functions of the target rate.
In all simulations, N = 12, 
. Solid curves illustrate the mean across different initializations and shaded regions correspond to one standard deviation. (A) Solutions of the ELBO optimization problem as a function of target rate, 
(blue curve), and theoretical optimum, 
(black curve), in the rate-distortion plane. Values of 
where the solutions coincide with the theoretical optimum (grey region). Solutions depart from the optimal line when the rate is very low (poor generative model) or very high (saturated distortion). Inset: mutual information between stimuli and neural responses as a function of 
. (B) Kullback-Leibler divergence between the stimulus and the generative distributions, as a function of 
. (C) Optimal tuning curves for different values of 
. Each dot represents a neuron: the position on the y-axis corresponds to its preferred stimulus, the size of the dot is proportional to the tuning width, and the color refers to the amplitude (see legend). Tuning curve parameters are averaged across 16 initializations, ordering the neurons as a function of their preferred position. The curve on the right illustrates the data distribution, π(x). (D) Entropy of the prior distribution over neural activity, pψ(r), as a function of 
. Insets show two configurations of the coupling matrices, with rows ordered according to the neurons’ preferred stimuli, and coupling strengths colored according to the legend. (E) MSE in the stimulus estimate, obtained as the MAP (blue curve, scale on the left y-axis), or from samples (orange curve, scale on the right y-axis), as a function of 
. Inset: MSE (MAP) as a function of the average tuning width.
doi: https://doi.org/10.1371/journal.pcbi.1012240.g004