Liley, Robinson, Wong--Wang, and David--Friston: Four ``other'' neural mass models, viewed as one-step extensions of the rate/PSP lineage

Four widely-used whole-brain neural models aren't separate traditions — each is exactly one step away from a single unified lineage.

The companion paper "Rosetta Stone of Neural Mass Models" built a clean ladder connecting oscillator math to biophysical brain models: phase oscillator → Stuart-Landau → Wilson-Cowan → Jansen-Rit (NMM1) → Montbrió-Pazó-Roxin (NMM2). That ladder is tight — each rung follows from the previous by an explicit, named approximation. But four popular models used in whole-brain neuroimaging research were left off: Liley, Robinson, Wong-Wang/DMF, and David-Friston/DCM. This note explains why they were left off, and the answer is clarifying: none of them needs its own derivation chain. Each sits exactly one step outside the ladder, and the step is different in each case.

The Liley model looks like NMM1 — same second-order synaptic filters, same sigmoid — but adds one thing: a voltage-dependent "shunting" weight on synaptic inputs. As the membrane potential approaches a synapse's reversal potential, that synapse loses driving force and its effective contribution shrinks toward zero. This is conductance-like physics without explicit conductance variables. Near resting potential, Liley collapses back to NMM1. The shunting nonlinearity only matters when the membrane swings hard toward a reversal potential — relevant for divisive inhibition, NMDA voltage-dependence, and anaesthetic mechanisms that shift chloride reversal potentials. The Robinson model is also NMM1 on the synaptic side, but adds a wave equation describing how firing-rate activity propagates across cortex at finite axonal speed. That spatial propagation operator — with its Laplacian term — is what puts Robinson in neural-field territory. Clamp space or discretize onto a parcellation and you get NMM1 back. The full continuum model earns its keep only when travelling waves and centimeter-scale spatial coherence patterns matter.

The Reduced Wong-Wang / DMF node has a different flavor. It was derived from leaky-integrate-and-fire neurons with NMDA recurrence, so its transfer function reflects spiking biophysics rather than a phenomenological sigmoid — that's the spirit of NMM2. But the closure is heuristic (slow NMDA, Gaussian inputs, mean-field self-consistency), not exact the way the Ott-Antonsen reduction is for quadratic-integrate-and-fire networks. The result is a single state variable per region — the NMDA gating fraction — which is more compact than either Wilson-Cowan or NMM2. It's the right tool for fitting whole-brain fMRI attractor dynamics, where the empirical validation record is strong. Finally, David-Friston/DCM is simply NMM1 — Jansen-Rit dynamics, same equations — wrapped in a variational Bayesian inference engine. The dynamics contribute nothing new; the scientific payload is the ability to invert the model against M/EEG data and estimate effective connectivity and synaptic gains as latent variables.

The payoff of this framing is practical. Choosing between these models stops being a choice between unrelated traditions and becomes a question of which single extension your scientific question requires: shunting physics (Liley), spatial propagation (Robinson), heuristic LIF/NMDA reduction (DMF), or Bayesian inversion (DCM). Everything else — bifurcation structure, phase reductions, field perturbation responses — transfers directly from the lineage rung each model extends.