The Equivalence of Information-Theoretic and Likelihood-Based Methods for Neural Dimensionality Reduction
Fig 3
Effects of the number of histogram bins on empirical single-spike information and MID performance.
(A) Scatter plot of raw stimuli (black) and spike-triggered stimuli (gray) from a simulated experiment using two-dimensional stimuli to drive a linear-nonlinear-Bernoulli neuron with sigmoidal nonlinearity. Arrow indicates the direction of the true filter k. (B) Plug-In estimates of p(k⊤s|spike), the spike-triggered stimulus distribution along the true filter axis, from 1000 stimuli and 200 spikes, using 5 (blue), 20 (green) or 80 (red) histogram bins. Black traces show estimates of raw distribution p(k⊤s) along the same axis. (C) True nonlinearity (black) and ML estimates of the nonlinearity (derived from the ratio of the density estimates shown in B). Roughness of the 80-bin estimate (red) arises from undersampling, or (equivalently) overfitting of the nonlinearity. (D) Empirical single-spike information vs. direction, calculated using 5, 20 or 80 histogram bins. Note that the 80-bin model overestimates the true asymptotic single-spike information at the peak by a factor of more than 1.5. (E) Convergence of empirical single-spike information along the true filter axis as a function of sample size. With small amounts of data, all three models overfit, leading to upward bias in estimated information. For large amounts of data, the 5-bin model underfits and therefore under-estimates information, since it lacks the smoothness to adequately describe the shape of the sigmoidal nonlinearity. (F) Filter error as a function of the number of stimuli, showing that the optimal number of histogram bins depends on the amount of data.