Heaviside world: Excitation and self-organization of neural fields

Mathematical treatments of the dynamics of neural fields become much simpler when the Heaviside function Heaviside function is used as an activation function. This is because the dynamics of an excited or active region reduce to the dynamics of the boundary. We call this regime the Heaviside world. Here, we visit the Heaviside world and briefly review bump Bumps dynamics in the 1D, 1D two-layer, and 2D cases. We further review the dynamics of forming topological maps by self-organization. The Heaviside world is useful for studying the learning or self-organization equation of receptive fields. The stability analysis Stability analysis shows the formation of a continuous map or the emergence of a block structure responsible for columnar microstructures. The stability of the Kohonen map is also discussed.