Efficient population coding of sensory stimuli

The efficient coding theory postulates that single cells in a neuronal population should be optimally configured to efficiently encode information about a stimulus subject to biophysical constraints. This poses the question of how multiple neurons that together represent a common stimulus should optimize their activation functions to provide the optimal stimulus encoding. Previous theoretical approaches have solved this problem with binary neurons that have a step activation function, and have assumed that spike generation is noisy and follows a Poisson process. Here we derive a general theory of optimal population coding with neuronal activation functions of any shape, different types of noise and heterogeneous firing rates of the neurons by maximizing the Shannon mutual information between a stimulus and the neuronal spiking output subject to a constraint on the maximal firing rate. We find that the optimal activation functions are discrete in the biological case of non-negligible noise and demonstrate that the information does not depend on how the population is divided into ON and OFF cells described by monotonically increasing vs decreasing activation functions, respectively. However, the population with an equal number of ON and OFF cells has the lowest mean firing rate, and hence encodes the highest information per spike. These results are independent of the shape of the activation functions and the nature of the spiking noise. Finally, we derive a relationship for how these activation functions should be distributed in stimulus space as a function of the neurons' firing rates.