A simple measure of the coding efficiency of a neuronal population

We derive a simple measure for quantifying the average accuracy with which a neuronal population can represent a stimulus. This quantity, the basis set error, has three key properties: (1) it makes no assumptions about the form of the neuronal responses; (2) it depends only on their second order statistics, so although it is easy to compute, it does take noise correlations into account; (3) its magnitude has an intuitive interpretation in terms of the accuracy with which information can be extracted from the population using a simple method-"simple" meaning linear. We use the basis set error to characterize the efficacy of several types of population codes generated synthetically in a computer. In general, the basis set error typically ranks different encoding schemes in a way that is qualitatively similar to Shannon's mutual information, except when nonlinear readout methods are necessary. Because this measure is concerned with signals that can be read out easily (i.e., through linear operations), it provides a lower bound on coding accuracy relative to the computational capabilities that are accessible to a neuronal population.