We studied the temporal dynamics of perceptual switches that are known to occur during prolonged presentations of a multi-stable visual motion stimulus using a neural fields, population-level representation of cortical activity in the middle temporal (MT) visual area. The so-called 'barber pole'' stimulus is considered with an aperture configuration that supports horizontal (H), diagonal (D) or vertical (V) perceived directions for the same input. It was established in previous modelling work and psychophysical experiments that, for short presentations, there is a shift in perceptual dominance from D to either H or V with increasing duration. The experiments described in the companion experimental abstract (I) show that for longer term presentations on the order of tens of seconds, the perceived direction remains multi-stable, with perception switching every few seconds between H and V. Here, we present an extension our previous motion integration model with the addition of a firing rate (spike frequency) adaptation dynamic that enables switching between the mutually inhibitory representations of H and V. A parameter study is performed using the tools of bifurcation analysis and numerical continuation that shows the model supports two different switching characteristics. In the bump oscillation case, with more weight given to the local input, switches between H and V occur at regular intervals with a very fast transition between the two states. In the travelling wave case, with more weight given to lateral connections, switches also occur at regular intervals but the transition is slower and a wave propagates from H to V (or vice-versa) via the diagonal percept D. Interestingly, the experimental results (I) also show a range of subject responses with respect to the switching characteristic involving D as an intermediate percept; this suggests variable dynamical interplay between intracortical connections in MT and the input received from lower cortical areas.