Temporal code versus rate code for binary Information Sources

Neuroscientists formulate very different hypotheses about the nature of neural code. At one extreme, it has been argued that neurons encode information in relatively slow changes of individual spikes arriving "rates codes" and the irregularity in the spike trains reflects noise in the system, while in the other extreme this irregularity is the temporal codes thus the precise timing of every spike carries additional information about the input. It is known that in the estimation of Shannon information the patterns and temporal structures are taken into account, while the rate code is determined by firing rate. We compare these types of codes for binary Information Sources which model encoded spike-trains. Assuming that the information transmitted by a neuron is governed by uncorrelated stochastic process or by process with a memory we compare the information transmission rates carried by such spike-trains with their firing rates. We showed that the crucial role in the relation between information and firing rates is played by a quantity which we call "jumping" parameter. It corresponds to the probabilities of transitions from no-spike-state to the spike-state and vice versa. For low values of jumping parameter the quotient of information and firing rates is monotonically decreasing function of firing rate, thus there is straightforward, one-to-one, relation between temporal and rate codes. On the contrary, it turns out that for large enough jumping parameter this quotient is non-monotonic function of firing rate and it exhibits a global maximum, in this case optimal firing rate exists. Moreover, there is no one-to-one relation between information and firing rates, so the temporal and rate codes differ qualitatively. This leads to the observation that the behavior of the quotient of information and firing rates for large jumping parameter is important in the context of bursting phenomena.