Dimension Reduction in Pattern Forming Systems with Heterogeneous Connection Topologies

Spatiotemporal pattern formation in physical and chemical systems is typically based on a dynamics with a homogeneous, i.e. translationally invariant, connection topology. However, biological systems like the human cortex show homogeneous connectivity with additional strongly heterogeneous projections from one area to another. Here we use nonlinear retarded integral equations to describe the spatiotemporal dynamics of systems with heterogeneous connections. Systematic changes in the connection topology may be used as a control parameter to guide the system through a series of phase transitions. Here we show how the dynamics of the entire system may be reduced to a few degrees of freedom at these bifurcation points.