  The interplay of spin-polarized electronic edge states with the dynamics of
the host nuclei in quantum Hall systems presents rich and non-trivial transport
physics. Here, we develop a Landauer-B\"uttiker approach to understand various
experimental features observed in the integer quantum Hall set ups featuring
quantum point contacts. The approach developed here entails a phenomenological
description of spin resolved inter-edge scattering induced via hyperfine
assisted electron-nuclear spin flip-flop processes. A self-consistent
simulation framework between the nuclear spin dynamics and edge state
electronic transport is presented in order to gain crucial insights into the
dynamic nuclear polarization effects on electronic transport and in turn the
electron-spin polarization effects on the nuclear spin dynamics. In particular,
we show that the hysteresis noted experimentally in the conductance-voltage
trace as well as in the resistively detected NMR lineshape results from a lack
of quasi-equilibrium between electronic transport and nuclear polarization
evolution. In addition, we present circuit models to emulate such hyperfine
mediated transport effects to further facilitate a clear understanding of the
electronic transport processes occurring around the quantum point contact.
Finally, we extend our model to account for the effects of quadrupolar
splitting of nuclear levels and also depict the electronic transport signatures
that arise from single and multi-photon processes.
