Neural Field Theory — Amari Bump Attractor

Localized persistent activity as a neural model of working memory

Amari Neural Field (1977)

A continuum model of cortex: neural field u(x,t) evolves under lateral inhibitory interactions. Stable bumps of activity emerge and persist — a model of spatial working memory.

τ ∂u/∂t = -u + ∫w(x-y)f(u(y))dy + I(x)

The Mexican-hat connectivity kernel w(x) (local excitation, lateral inhibition) supports bumps:

w(x) = A_e·exp(-x²/2σ_e²)
- A_i·exp(-x²/2σ_i²)

The bump's existence and stability depend on the kernel parameters. The bump position encodes a memorized location — a continuous attractor.

Multiple bumps can coexist if inhibition is sufficiently local.

Bump width: —
Peak activity: —