Optimality principles have been useful in explaining many aspects of biological systems. In the context of neural encoding in sensory areas, optimality is naturally formulated in a Bayesian setting as neural tuning which minimizes mean decoding error. Many works optimize Fisher information, which approximates the minimum mean square error (MMSE) of the optimal decoder for long encoding time but may be misleading for short encoding times. We study MMSE-optimal neural encoding of a multivariate stimulus by uniform populations of spiking neurons, under firing rate constraints for each neuron as well as for the entire population. We show that the population-level constraint is essential for the formulation of a well-posed problem having finite optimal tuning widths and optimal tuning aligns with the principal components of the prior distribution. Numerical evaluation of the two-dimensional case shows that encoding only the dimension with higher variance is optimal for short encoding times. We also compare direct MMSE optimization to optimization of several proxies to MMSE: Fisher information, maximum likelihood estimation error, and the Bayesian Cramér-Rao bound. We find that optimization of these measures yields qualitatively misleading results regarding MMSE-optimal tuning and its dependence on encoding time and energy constraints.

中文翻译：

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具有神经元级和种群级能量约束的最优多元调节

最优性原理在解释生物系统的许多方面很有用。在感觉区域的神经编码的背景下，最优性在贝叶斯设置中自然地被表述为神经调整，从而最小化平均解码错误。许多工作优化了 Fisher 信息，该信息近似于长编码时间的最佳解码器的最小均方误差 (MMSE)，但可能会误导短编码时间。我们通过均匀的尖峰神经元群体研究多元刺激的 MMSE 最优神经编码，在每个神经元以及整个群体的放电率约束下。我们表明，总体水平约束对于具有有限最优调谐宽度和最优调谐与先验分布的主成分对齐的适定问题的公式化是必不可少的。二维情况的数值评估表明，对于较短的编码时间，仅对方差较大的维度进行编码是最佳的。我们还将直接 MMSE 优化与多个 MMSE 代理的优化进行了比较：Fisher 信息、最​​大似然估计误差和贝叶斯 Cramér-Rao 界限。我们发现，这些措施的优化会产生关于 MMSE 优化调整及其对编码时间和能量约束的依赖性的定性误导结果。