The Wilson–Cowan and Amari neural-field formulations describe the same mathematical object from two perspectives

The two canonical 1970s neural-field frameworks — the Wilson–Cowan field equations and Amari's lateral-inhibition neural field (claim-amari-1977-neural-field-paper-rooted-in-reaction-diffusion-morphogenesis-literature) — differ in how they set up the state variable (activity-based versus voltage-based formulation), but a 2022 mathematical overview of neural field theory treats them as one system seen from two angles. Cook, Peterson, Woldman and Terry write: "Despite this difference in formulation, both equations give rise to similar dynamics. This is unsurprising since they describe the same physical system from two different perspectives."

The equivalence is load-bearing for the historiography of the field, not merely a tidy remark. It means that later analytical work built on the Wilson–Cowan equations — for example the bifurcation-and-group-theory treatment of cortical pattern formation (claim-ermentrout-cowan-1979-applied-neural-field-bifurcation-to-visual-hallucinations) — reaches the Amari-family formalism even when it does not cite Amari (1977) directly. Claims about where the neural-field lineage went therefore need not be partitioned by which of the two formulations a given paper happened to name.

The 2022 review is a recent, comprehensive survey tracing neural field theory from its 1970s origins to contemporary applications; this equivalence claim sits in its Section 3.2 model-derivation material. For the author whose 1977 paper anchors the Amari side of the equivalence, see entity-shunichi-amari.