Capacity of a single spiking neuron

It is widely believed the neurons transmit information in the form of spikes. Since the spike patterns are known to be noisy, the neuron information channel is noisy. We have investigated the channel capacity of this "Spiking neuron channel" for both of the "temporal coding" and the "rate coding," which are two main coding considered in the neuroscience [1, 2]. As the result, we've proved that the distribution of inputs, which achieves the channel capacity, is a discrete distribution with finite mass points for temporal and rate coding under a reasonable assumption. In this draft, we show the details of the proof.