Amari Neural Field

Amari Neural Field

Audience

Difficulty	Intermediate

Notation on this page

The Amari neural-field equation (Shun-ichi Amari, 1977) is the spatial-continuum generalisation of Wilson-Cowan. Instead of two lumped populations, it treats neural activity as a continuous scalar field u(x, t) defined over cortical position x.

This is the mathematically natural setting for studying spatial patterns of cortical activity: travelling waves, localised "bumps" (working memory), pinwheels (orientation columns), and pattern formation.

Statement

$ \tau \,{\frac {\partial u(\mathbf {x} ,t)}{\partial t}}=-u(\mathbf {x} ,t)+\!\int _{\Omega }\!w(\mathbf {x} -\mathbf {x} ')\,f{\bigl (}u(\mathbf {x} ',t){\bigr )}\,d^{n}x'+h(\mathbf {x} ,t) $

Components

Symbol	Meaning	Units
u(x, t)	Synaptic-input field	mV (or arbitrary)
Ω	Cortical domain	m2 (or m3)
x, x'	Cortical positions	m
w(x − x')	Synaptic-connectivity kernel	m−2
f(u)	Firing-rate function (sigmoid)	Hz
h(x, t)	External input	mV
τ	Membrane time constant	~ 10 ms

Derivation sketch

Start from Wilson-Cowan with the position index x. In the large-population limit:

1. Replace the discrete-population weights w_EE, w_EI, ... by a spatial kernel w(x − x'). For a single population (effective net coupling), this kernel commonly has the Mexican-hat shape: positive at short range, negative at intermediate range.
2. Replace the population activity by a continuous field u(x, t).
3. Replace the sum over connections by a spatial integral.

The result is the Amari equation.

Mexican-hat connectivity

A canonical choice is:

$ w(\mathbf {x} -\mathbf {x} ')=A_{+}\,\exp \!\left(-{\frac {|\mathbf {x} -\mathbf {x} '|^{2}}{\sigma _{+}^{2}}}\right)-A_{-}\,\exp \!\left(-{\frac {|\mathbf {x} -\mathbf {x} '|^{2}}{\sigma _{-}^{2}}}\right) $

with $ A_{+}>A_{-}>0 $ and $ \sigma _{-}>\sigma _{+} $ — short-range excitation, longer-range inhibition. This is the empirical pattern of intracortical connectivity for many cortical regions.

Localised solutions: "bumps"

A central result: with Mexican-hat connectivity, the Amari field admits stable localised solutions — bumps — that persist in the absence of input. These bumps:

• Carry localised activity at position x₀ indefinitely.
• Are the mathematical basis for working memory (the bump stores "the cup is to the left").
• Are the basis for attentional spotlights.
• Can drift, split, merge under perturbation.

Travelling waves

The Amari field also supports travelling-wave solutions:

 u(x, t) = U(x − v · t)

These propagate at a velocity v determined by the kernel and gain. They are the mathematical model for:

• Cortical waves observed in electrophysiology (e.g. during slow-wave sleep).
• Spreading depression and migraine aura.
• Wave-propagation models of epileptic seizures.

Dynamics

Stability of bumps depends on the kernel and the gain of f(u). Amari's original 1977 paper showed:

• For Heaviside firing functions f(u) = Θ(u − θ), bumps exist for a window of input strengths.
• Wider bumps are unstable; narrower bumps stable (counter-intuitive but rigorous).
• For sigmoidal f, the analysis becomes more involved but the bump phenomenology persists.

Sanity-check limits

• No spatial coupling (w → δ-function at zero): reduces to point Wilson-Cowan (u_x ≡ u_local). ✓
• Uniform input h(x,t) = h₀: solution u(x,t) ≡ u₀ uniform (no spatial structure). ✓
• Δ → 0 or w → 0: pure exponential decay; no patterns. ✓

Connection to ψ

In the ψ-coupled extension, the Amari equation gains a ψ-driven term:

$ \tau \,\partial _{t}u=-u+\!\int \!w(\mathbf {x} -\mathbf {x} ')\,f{\bigl (}u(\mathbf {x} ',t){\bigr )}\,d^{n}x'+h(\mathbf {x} ,t)+\beta \,\psi (\mathbf {x} ,t) $

and sources ψ via:

$ J_{\psi }(\mathbf {x} ,t)=\kappa _{J}\,f{\bigl (}u(\mathbf {x} ,t){\bigr )} $

ψ propagates from x to x' as a separate, relativistic field (□ψ − m²ψ − λψ³ = ...) — so ψ provides a non-local coupling channel that cortical-network connectivity alone does not capture.

Experimental status

Amari fields are standard in computational neuroscience:

• Working memory in prefrontal cortex — bump solutions match electrophysiology of delay-period activity (Wang 1999, 2001).
• Cortical-wave observations — match Amari-style propagation (Rubino et al. 2006; Townsend et al. 2015).
• Pinwheels and orientation maps — emerge from Amari fields with self-organising plasticity (Wolf-Geisel 1998 and successors).
• MEG / EEG spatial spectra — fit to Amari + delay extensions (the Robinson-Rennie family of models).

The Amari equation is mainstream, well-validated computational neuroscience.

See Also

• Wilson-Cowan_Model
• Neural_Field_Equations
• Wilson-Cowan_Coupled_to_Psi
• Jansen-Rit_Neural_Mass
• Effective_Field_Theory_of_Consciousness

References

• Amari, S. (1977). "Dynamics of pattern formation in lateral-inhibition type neural fields." Biological Cybernetics 27: 77–87.
• Coombes, S., Beim Graben, P., Potthast, R., Wright, J. (eds.) (2014). Neural Fields: Theory and Applications. Springer.
• Wang, X.-J. (1999). "Synaptic basis of cortical persistent activity: The importance of NMDA receptors to working memory." Journal of Neuroscience 19: 9587–9603.
• Bressloff, P. C. (2012). "Spatiotemporal dynamics of continuum neural fields." Journal of Physics A: Mathematical and Theoretical 45: 033001.