Nilpotent singularities and bifurcations structure of the Landau–Ginzburg theory of cortex dynamics

Abstract

The dynamics research of cerebral cortex activity is an important part of neuroscience. A previous work (Proc. Natl. Acad. Sci. USA 115 (2018) E1356-E1365) put forward a cortex dynamics model based on Landau–Ginzburg theory and it generated an oscillation and bistability. However, the full dynamics of cerebral cortex is still ill-defined. In this paper, we rigorously prove that this model can undergo Bogdanov–Takens bifurcation of codimension 2, Bogdanov–Takens bifurcations of codimension 3 with cusp, focus and elliptic types, and doubly degenerate Bogdanov–Takens bifurcation of codimension 4. Our work complements and enriches the results in the field of cerebral cortex activity.