Low Dimensional Dynamics of Globally Coupled Complex Riccati Equations: Exact Firing-rate Equations for Spiking Neurons with Clustered Substructure

We report on an exact theory for ensembles of globally coupled, heterogeneous complex Riccati equations. A drastic dimensionality reduction to a few ordinary differential equations is achieved for Lorentzian heterogeneity. By applying this technique, we obtain low-dimensional firing-rate equations for populations of spiking neurons with a clustered substructure.