We study coupled FitzHugh-Nagumo oscillators on a small-world network in which the activator and the inhibitor diffuse through independent channels, with an added three-body coupling whose structure is varied among an all-to-all form, a simplicial complex built on the network triangles, and a random hypergraph. At fixed coupling strength and a fixed number of three-body interactions, the structure decides whether multistability and coexisting oscillatory-quiescent states survive. With matched triple count on the same backbone, the simplicial complex retains most of its partial-quenching basin whereas the random hypergraph removes nearly all of it, a separation far larger than the variation across network realizations, while total death falls comparably under both. The difference is not set by the spectrum of the normalized higher-order Laplacian, whose leading eigenvalue the normalization forces to be the same for both structures, but by the alignment of that operator with the pairwise eigenbasis: simplicial triples join adjacent nodes and leave long-wavelength perturbations almost undamped, whereas random triples damp them as strongly as short ones. Randomizing this alignment at fixed spectrum moves the transverse stability of synchrony under the simplicial complex onto that under the hypergraph. Sampling the attraction basins shows that the total-death basin shrinks under both structures, but only under the simplicial complex does the partial-quenching basin grow, and the quiescent domains of these mixed states are long-wavelength patterns. An all-to-all three-body control acts approximately as a pairwise mean field whose strength is reduced by an amplitude gate, set by the cycle-averaged squared activator of the collective oscillation.

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• Dynamics of networks
• Neuronal dynamics
• Coupled oscillators