  During early development, waves of activity propagate across the retina and
play a key role in the proper wiring of the early visual system. During the
stage II these waves are triggered by a transient network of neurons, called
Starburst Amacrine Cells (SACs), showing a bursting activity which disappears
upon further maturation. While several models have attempted to reproduce
retinal waves, none of them is able to mimic the rhythmic autonomous bursting
of individual SACs and reveal how these cells change their intrinsic properties
during development. Here, we introduce a mathematical model, grounded on
biophysics, which enables us to reproduce the bursting activity of SACs and to
propose a plausible, generic and robust, mechanism that generates it. The core
parameters controlling repetitive firing are fast depolarizing $V$-gated
calcium channels and hyperpolarizing $V$-gated potassium channels. The
quiescent phase of bursting is controlled by a slow after hyperpolarization
(sAHP), mediated by calcium-dependent potassium channels. Based on a
bifurcation analysis we show how biophysical parameters, regulating calcium and
potassium activity, control the spontaneously occurring fast oscillatory
activity followed by long refractory periods in individual SACs. We make a
testable experimental prediction on the role of voltage-dependent potassium
channels on the excitability properties of SACs and on the evolution of this
excitability along development. We also propose an explanation on how SACs can
exhibit a large variability in their bursting periods, as observed
experimentally within a SACs network as well as across different species, yet
based on a simple, unique, mechanism. As we discuss, these observations at the
cellular level have a deep impact on the retinal waves description.
