Normalized entropy as a measure of randomness is explored. It is employed to characterize those properties of neuronal firing that cannot be described by the first two statistical moments. We analyze randomness of firing of the Ornstein-Uhlenbeck (OU) neuronal model with respect either to the variability of interspike intervals (coefficient of variation) or the model parameters. A new form of the Siegert's equation for first-passage time of the OU process is given. The parametric space of the model is divided into two parts (sub-and supra-threshold) depending upon the neuron activity in the absence of noise. In the supra-threshold regime there are many similarities of the model with the Wiener process model. The sub-threshold behavior differs qualitatively both from the Wiener model and from the supra-threshold regime. For very low input the firing regularity increases (due to increase of noise) cannot be observed by employing the entropy, while it is clearly observable by employing the coefficient of variation. Finally, we introduce and quantify the converse effect of firing regularity decrease by employing the normalized entropy.