Data-driven identification of macroscopic dynamics with implicit equation-free sampling and Gaussian process regression: for the example of an integrate-and-fire neural network

We investigate a data-driven approach to derive low-dimensional macroscopic models of complex systems with only high-dimensional microscopic descriptions available. This is achieved by sampling of the macroscopic behaviour at selected points using an implicit equation-free approach with appropriate initialisation of the microscopic system. This enables subsequent data-driven identification of the macroscopic dynamics with Gaussian process regression. We demonstrate the technique on a high-dimensional neural network of integrate-and-fire neurons. A numerical bifurcation analysis of the obtained macroscopic model is performed, showing both stable and unstable branches. The appropriate sampling using the implicit equation-free approach avoids grid distortion and prevents spurious states as well as other artefacts.