Design of efficient channels with given input statistics

The classic problem for a channel with inputs X, outputs Y, and conditional probability p_{Y |X}(y|x) is to find the distribution p_X(x) that maximizes Shannon´s I(X; Y) subject perhaps to constraints imposed on p_X(x). Here, we seek instead, for a specified p_X(x), the channel p_Y|X(y|x) that maximizes I(X; Y) subject to constraints on p_{Y |X}(y|x). That is, we investigate the part of joint source-channel coding that matches channels to sources. We assume that p_X(x) and p_{Y |X}(y|x) are pdfs, eventually relaxing this assumption somewhat. We consider only time-discrete memoryless channels. Our motivation for studying this problem stems from neuroscience. Energy costs therefore must be analyzed and addressed in detail if one hopes to understand how Nature has built neuron “channels” that are so astoundingly energy efficient. However, our general theory is not limited to neuroscience nor limited solely to constraints on energy expenditure.