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= Amari Neural Field =

{{Audience_Sidebar
| difficulty = Intermediate
| reading_time = 7 minutes
| prerequisites = PDEs; convolution integrals; [[Wilson-Cowan_Model|Wilson-Cowan]].
| if_too_advanced_see = [[Wilson-Cowan_Model]]
| if_you_want_the_math_see = This page; [[Neural_Field_Equations]]
}}

{{Notation
| psi_convention = ψ = scalar field amplitude (separate from u, the neural-field variable here).
| signature = Mostly-plus.
| units = SI; cortical position x in metres; firing rates in Hz; synaptic drive u in mV.
}}

The '''Amari neural-field equation''' (Shun-ichi Amari, 1977) is the spatial-continuum generalisation of [[Wilson-Cowan_Model|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 ==

:<math>\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)</math>

== Components ==

{| class="wikitable"
! 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:

# Replace the discrete-population weights wEE, wEI, ... 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.
# Replace the population activity by a continuous field u(x, t).
# Replace the sum over connections by a spatial integral.

The result is the Amari equation.

== Mexican-hat connectivity ==

A canonical choice is:

:<math>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)</math>

with <math>A_+ > A_- > 0</math> and <math>\sigma_- > \sigma_+</math> — 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 (ux ≡ ulocal). ✓
* '''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 [[Wilson-Cowan_Coupled_to_Psi|ψ-coupled extension]], the Amari equation gains a ψ-driven term:

:<math>\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)</math>

and sources ψ via:

:<math>J_\psi(\mathbf{x},t) = \kappa_J\,f\bigl(u(\mathbf{x},t)\bigr)</math>

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.

[[Category:Psionics]]
[[Category:Equations]]
[[Category:Neuroscience]]

Amari Neural Field - Revision history

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to ; lint-clean per tools/wiki_latex_lint.py

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to $; lint-clean per tools/wiki_latex_lint.py$