Illustration of MID failure mode due to non-Poisson spiking.

<p><b>(A)</b> Stimuli were drawn uniformly on the unit half-circle, <i>θ</i> ∼ Unif(−<i>π</i>/2,<i>π</i>/2). The simulated neuron had Bernoulli (i.e., binary) spiking, where the probability of a spike increased linearly from 0 to 1 as <i>θ</i> varied from -<i>π</i>/2 to <i>π</i>/2, that is: <i>p</i>(<i>spike</i>|<i>θ</i>) = <i>θ</i>/<i>π</i>+1/2. Stimuli eliciting “spike” and “no-spike” are indicated by gray and black circles, respectively. For this neuron, the most informative one-dimensional linear projection corresponds to the vertical axis (<math><mrow><msub>k<mo>̂</mo><mrow><mi>B</mi><mi>e</mi><mi>r</mi></mrow></msub></mrow></math>), but the MID estimator (<math><mrow><msub>k<mo>̂</mo><mrow><mi>M</mi><mi>I</mi><mi>D</mi></mrow></msub></mrow></math>) exhibits a 16^∘ clockwise bias. <b>(B)</b> Information from spikes (black), silences (gray), and both (red), as a function of projection angle. The peak of the Bernoulli information (which defines <math><mrow><msub>k<mo>̂</mo><mrow><mi>B</mi><mi>e</mi><mi>r</mi></mrow></msub></mrow></math>) lies close to <i>π</i>/2, while the peak of single-spike information (which defines <math><mrow><msub>k<mo>̂</mo><mrow><mi>M</mi><mi>I</mi><mi>D</mi></mrow></msub></mrow></math>) exhibits the clockwise bias shown in A. Note that <math><mrow><msub>k<mo>̂</mo><mrow><mi>M</mi><mi>I</mi><mi>D</mi></mrow></msub></mrow></math> does not converge to the optimal direction even in the limit of infinite data, due to its lack of sensitivity to information from silences. Although this figure is framed in an information-theoretic sense, equations (19) and (20) detail the equivalence between <i>I</i>_<i>Ber</i> and ℒ_<i>lnb</i>, so that this figure can be viewed from either an information-theoretic or likelihood-based perspective.