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Optimality principles have been useful in explaining many aspects ofbiological systems. In the context of neural encoding in sensory areas,optimality is naturally formulated in a Bayesian setting, as neural tuningwhich minimizes mean decoding error. Many works optimize Fisher information,which approximates the Minimum Mean Square Error (MMSE) of the optimal decoderfor long encoding time, but may be misleading for short encoding times. Westudy MMSE-optimal neural encoding of a multivariate stimulus by uniformpopulations of spiking neurons, under firing rate constraints for each neuronas well as for the entire population. We show that the population-levelconstraint is essential for the formulation of a well-posed problem havingfinite optimal tuning widths, and optimal tuning aligns with the principalcomponents of the prior distribution. Numerical evaluation of thetwo-dimensional case shows that encoding only the dimension with highervariance is optimal for short encoding times. We also compare direct MMSEoptimization to optimization of several proxies to MMSE, namely Fisherinformation, Maximum Likelihood estimation error, and the Bayesian Cram\'er-Raobound. We find that optimization of these measures yield qualitativelymisleading results regarding MMSE-optimal tuning and its dependence on encodingtime and energy constraints.

Optimal Multivariate Tuning with Neuron-Level and Population-Level Energy Constraints