The Equivalence of Information-Theoretic and Likelihood-Based Methods for Neural Dimensionality Reduction
Fig 7
Two examples illustrating sub-optimality of MID under discrete (non-Poisson) spiking.
In both cases, stimuli were uniformly distributed within the unit circle and the simulated neuron’s response depended on a 1D projection of the stimulus onto the horizontal axis (θ = 0). Each stimulus evoked 0, 1, or 2 spikes. (A) Deterministic neuron. Left: Scatter plot of stimuli labelled by number of spikes evoked, and the piece-wise constant nonlinearity governing the response (below). The nonlinearity sets the response count deterministically, thus dramatically violating Poisson expectations. Middle: information vs. axis of projection. The total information Icount reflects the information from 0-, 1-, and 2-spike responses (treated as distinct symbols), while the single-spike information Iss ignores silences and treats 2-spike responses as two samples from p(s|spike). Right: Average absolute error in and as a function of sample size; the latter achieves 18% lower error due to its sensitivity to the non-Poisson structure of the response. (B) Stochastic neuron with sigmoidal nonlinearity controlling the stochasticity of responses. The neuron transitions from almost always emitting 1 spike for large negative stimulus projections, to generating either 0 or 2 spikes with equal probability at large positive projections. Here, the nonlinearity does not modulate the mean spike rate, so Îss is approximately zero for all stimulus projections (middle) and the MID estimator does not converge (right). However, the estimate converges because the LNC model is sensitive to the change in conditional response distribution. Equation (37) details the relationship between Icount and ℒlnc, so that this figure can be interpreted from either an information-theoretic or likelihood-based perspective.