Optimal output modality in cascades with unimodal input. (

*A*) Plots of optimal input distribution

*p**(

*n*_{1}) (black) and optimal output distributions

*p**(

*n*_{L}) (colors; see legend) for different cascade lengths

*L* (optimal input distributions are qualitatively the same; only that for

*L* = 2 is shown), corresponding to regulation functions

*q*_{n} (identical for each step) plotted underneath (ρ = 1, and solutions used

= 10 and

= 〈

*g*_{n}〉). Mutual information

*I* is optimized as a function of the mean

*g* of the Poisson input distribution. Magenta numbers on plots correspond to magenta points in

*B*.

*Insets* show plots of average output 〈

*n*_{L}〉 vs. cascade length

*L*. In the first column of

*A*, the output is always unimodal; in the second column, the output is bimodal for cascade lengths

*L* ≥

*L** for some

*L** (“persistent” bimodality); in the third column, the output is bimodal for a range of

*L* values, then unimodal once more for large

*L* (“localized” bimodality). The first row shows “DC” cascades, in which each step is up-regulating, and the second row shows “AC” cascades, in which each step is down-regulating. (

*B*) Phase diagram of optimal output modality as a function of

*q*_{−} and

*q*_{+} (

*n*_{0} = 8). White is unimodal, and color is bimodal, with color corresponding to

*L**. Distinction between persistent (no ‘x’) and localized (‘x’) bimodality is shown up to

*L* = 10. Dashed line separates DC cascades from AC cascades. (

*Inset*) Capacity

*I** in bits as a function of

*q*_{−} and

*q*_{+} for the same data, with

*L* = 2.

## PubMed Commons