  Branching with immigration is one of the most common models for the
stochastic processes observed in neuronal circuits. However, it is not observed
directly and, in order to create branching-like processes, the observed spike
time series is processed by attaching time bins to spikes. It has been shown
that results such as criticality and size distributions depend on the chosen
time bin. A different methodology whose results do not depend on the choice of
time bin might therefore be useful and is proposed in this article.
  The new methodology circumvents using time bins altogether by replacing the
previously used discrete-time models by continuous-time models. First, the
article introduces and characterises a continuous-time version of the branching
process with immigration, which will be called pumped branching process, and
second, it presents an analytical derivation of the corresponding spike
statistics, which can be directly compared to observed spike time series. The
presented approach allows determining the degree of criticality, the average
number of overlapping avalanches, and other observables without using a time
bin. Furthermore, the effects caused by using time bins are analyzed and the
influence of temporal and spatial subsampling discussed, all of which is
compared to experimental data and supported by Monte Carlo simulations.
