  Topographic maps are a brain structure connecting pre-synpatic and
post-synaptic brain regions. Topographic development is dependent on
Hebbian-based plasticity mechanisms working in conjunction with spontaneous
patterns of neural activity generated in the pre-synaptic regions. Studies
performed in mouse have shown that these spontaneous patterns can exhibit
complex spatial-temporal structures which existing models cannot incorporate.
Neural field theories are appropriate modelling paradigms for topographic
systems due to the dense nature of the connections between regions and can be
augmented with a plasticity rule general enough to capture complex time-varying
structures.
  We propose a theoretical framework for studying the development of topography
in the context of complex spatial-temporal activity fed-forward from the
pre-synaptic to post-synaptic regions. Analysis of the model leads to an
analytic solution corroborating the conclusion that activity can drive the
refinement of topographic projections. The analysis also suggests that
biological noise is used in the development of topography to stabilise the
dynamics. MCMC simulations are used to analyse and understand the differences
in topographic refinement between wild-type and the $\beta2$ knock-out mutant
in mice. The time scale of the synaptic plasticity window is estimated as
$0.56$ seconds in this context with a model fit of $R^2 = 0.81$.
