Functional diversity among sensory neurons from efficient coding principles
Fig 3
Optimal linear decoding of stimuli.
A. Framework schematic. A stimulus s from a probability distribution p(s) is encoded by the spiking responses of a population of ON (red) and OFF (blue) cells. We optimize the cells’ nonlinearities by minimizing the mean squared error (MSE) between the original stimulus s and the linearly reconstructed stimulus y from the spiking response. B. Minimizing the MSE between a stimulus s (black) and its linear estimate y (blue) by a population of (6) ON and OFF cells, in the absence of noise. We show the optimal weight w1 and the center of mass 〈s〉1 of the first threshold interval (red dashes). C,D. Any ON-OFF population can achieve the same error with the same set of optimal thresholds and weights but a different constant, w0. C. 6 ON cells (w0 < 0). D. 3 OFF and 3 ON cells (w0 = 0). E. The optimal thresholds equalize not the area under the stimulus density (as in the case of the mutual information), but the area under its one-third power (Eq 8). The optimal thresholds are shown for the Laplace distribution. F. The information maximizing thresholds partition the Laplace distribution into intervals that code for stimuli with higher likelihood of occurrence (bottom), while minimizing the MSE pushes thresholds to favor rarer stimuli near the tails of the distribution (top). Threshold distributions are the same as in E. G. The cumulative optimal thresholds (compare to E).