Two examples illustrating sub-optimality of MID under discrete (non-Poisson) spiking.

<p>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 (<i>θ</i> = 0). Each stimulus evoked 0, 1, or 2 spikes. <b>(A)</b> Deterministic neuron. <i>Left</i>: 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. <i>Middle</i>: information vs. axis of projection. The total information <i>I</i>_<i>count</i> reflects the information from 0-, 1-, and 2-spike responses (treated as distinct symbols), while the single-spike information <i>I</i>_<i>ss</i> ignores silences and treats 2-spike responses as two samples from <i>p</i>(<b>s</b>|<i>spike</i>). <i>Right</i>: Average absolute error in <math><mrow><msub>k<mo>̂</mo><mrow><mi>M</mi><mi>I</mi><mi>D</mi></mrow></msub></mrow></math> and <math><mrow><msub>k<mo>̂</mo><mrow><mi>c</mi><mi>o</mi><mi>u</mi><mi>n</mi><mi>t</mi></mrow></msub></mrow></math> 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>(B)</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 <i>Î</i>_<i>ss</i> is approximately zero for all stimulus projections (middle) and the MID estimator does not converge (right). However, the <math><mrow><msub>k<mo>̂</mo><mrow><mi>c</mi><mi>o</mi><mi>u</mi><mi>n</mi><mi>t</mi></mrow></msub></mrow></math> estimate converges because the LNC model is sensitive to the change in conditional response distribution. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004141#pcbi.1004141.e068" target="_blank">Equation (37)</a> details the relationship between <i>I</i>_<i>count</i> and ℒ_<i>lnc</i>, so that this figure can be interpreted from either an information-theoretic or likelihood-based perspective.