  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 constrain 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.
