  In an extended superconductor-topological insulator-superconductor (S-TI-S)
Josephson junction in a magnetic field, localized Majorana bound states (MBS)
are predicted to exist at the cores of Josephson vortices where the local phase
difference across the junction is an odd-multiple of $\pi$. These states
contribute a supercurrent with a $4\pi$-periodic current-phase relation (CPR)
that adds to the conventional $2\pi$-periodic sinusoidal CPR. In this work, we
present a comprehensive experimental study of the critical current vs. applied
magnetic field diffraction patterns of lateral Nb-Bi$_2$Se$_3$-Nb Josephson
junctions. We compare our observations to a model of the Josephson dynamics in
the S-TI-S junction system to explore what feature of MBS are, or are not,
exhibited in these junctions. Consistent with the model, we find several
distinct deviations from a Fraunhofer diffraction pattern that is expected for
a uniform sin$({\phi})$ CPR. In particular, we observe abrupt changes in the
diffraction pattern at applied magnetic fields in which the current-carrying
localized MBS are expected to enter the junction, and a lifting of the
odd-numbered nodes consistent with a $4\pi$-periodic sin$(\phi/2)$-component in
the CPR. We also see that although the even-numbered nodes often remain
fully-formed, we sometimes see deviations that are consistent with
quasiparticle-induced fluctuations in the parity of the MBS pairs that encodes
quantum information.
