Rosetta Stone of
Neural Mass Models

phase only · no amplitude · synchronization, stripped to essentials

θ̇ᵢ = ωᵢ + (G/N)Σⱼ sin(θⱼ−θᵢ). The leanest possible skeleton for synchronization — every later model on this ladder adds detail, starting with amplitude, on top of this bare rotation.

linear · closed-form solution · a mass on a spring

Amplitude re-enters the picture: ż = (α + iω)z. One complex equation captures damping, resonance, and coupled networks; the model's undamped, amplitude-locked limit is the Kuramoto model, the rung just below.

adds nonlinearity · self-sustaining limit cycle

A cubic nonlinearity turns the neutral circle at HO's Hopf point into a genuine limit cycle — the orbit locks onto a fixed size no matter where it starts. The first real rhythm.

excitatory + inhibitory populations · sigmoid · Hopf bifurcation

Two neural populations replace the single oscillator. Turn up the background drive and the E–I loop spontaneously starts oscillating — a biologically grounded rhythm.

explicit synaptic filtering · realistic time constants

Synaptic dynamics get their own rise and decay time constants, refining the frequency and shape of the rhythm that emerges from the E–I loop.

exact QIF reduction · dynamic firing-rate/potential pair

Replaces the static sigmoid with an exact mean-field reduction of spiking QIF neurons (Montbrió–Pazó–Roxin) — same E/I push-pull skeleton, but the transfer function itself now has its own dynamics.