### Optimal neural tuning curves for arbitrary stimulus distributions: Discrimax, infomax and minimum Lp loss

Zhuo Wang and

**Alan A Stocker** and

Dan D Lee
NIPS Advances in Neural Information Processing Systems 25, Lake Tahoe CA,
December 2012, MIT Press. p. 2177–2185

In this work we study how the stimulus distribution influences the optimal coding of an individual neuron. Closed-form solutions to
the optimal sigmoidal tuning curve are provided for a neuron obeying Poisson statistics under a given stimulus distribution. We
consider a variety of optimality criteria, including maximizing discriminability, maximizing mutual information and minimizing
estimation error under a general Lp norm. We generalize the Cramer-Rao lower bound and show the Lp loss can be written as a functional
of the Fisher Information in the asymptotic limit, by proving the moment convergence of certain functions of Poisson random
variables. In this manner, we show how the optimal tuning curve depends upon the loss function, and the equivalence of maximizing
mutual information with minimizing Lp loss in the limit as p goes to zero.

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