How the models connect

The Ladder

Six models, three ways: what their equations are made of, what a single node and a coupled brain network physically look like, and the ladder they build up along — starting from Kuramoto's bare phase oscillators, the leanest possible skeleton every later model adds detail to.

The ladder, read across

Every rung on one table — read each row left to right: the model and the one idea it adds, its single-node equation (coloured by role), what a single unit looks like, how units couple, and the coupled network. Diagrams are animated.

rotationdamping / relaxationamplitude brakeE–I couplingself-couplingsynaptic filtermean fieldexternal input
Model & what it adds
Single-node equation
One unit
Coupled equation
Coupled network
Phase-only
the starting point
θ̇ᵢ = ωᵢ
one node = a bare phase θᵢ rotating at ωᵢ — no amplitude at all
θ̇ᵢ = ωᵢ + G (1/N) Σⱼ sin(θⱼ − θᵢ) phase coupling · §2.1.4, Eq. 2.20, App. D
coupling “rope”start out of phase → the rope pulls the two into sync
not yet synchronised: dots scattered, order-parameter arrow (coral) short
▾ + amplitude (a mass on a spring)
HO
rung 2 · harmonic oscillator
ṙ = α rθ̇ = ω
one node = one oscillator (a coarse-grained population)
żᵢ = (α+iω)zᵢ + G Σⱼ Cᵢⱼ(zⱼ − zᵢ) diffusive coupling · §2, Eq. 2.30
coupled masses: start out of phase → the coupling spring syncs them
diffusive: each node is pulled toward its neighbours
▾ + cubic nonlinearity
Stuart–Landau
rung 3
ṙ = α r − γr³θ̇ = ω − βr²
one node orbits a fixed ring (limit cycle)
żᵢ = (α+iω)zᵢ − (γ+iβ)|zᵢ|²zᵢ + G Σⱼ Cᵢⱼ(zⱼ − zᵢ) diffusive coupling · §3, Eq. 3.9
coupling “rope”start desynced → diffusive coupling brings them into sync
diffusive: pulled toward neighbours, amplitude self-limited
▾ + E/I populations, sigmoid
WILCO
rung 4 · Wilson–Cowan
τₓẋ + x = σ(wₓₓx − wₓᵧy + Pₓ)τᵧẏ + y = σ(wᵧₓx − wᵧᵧy)
E I
one node = excitatory + inhibitory populations
τₓẋᵢ + xᵢ = σ(wₓₓxᵢ − wₓᵧyᵢ + Pₓ + Σⱼ Cᵢⱼ xⱼ)τᵧẏᵢ + yᵢ = σ(wᵧₓxᵢ − wᵧᵧyᵢ) additive coupling, inside σ · §5, Eq. 5.5
EIEIthe excitatory rate of one region drives the other (E→E)
additive, E→E: node j’s rate feeds node i’s input
▾ + synaptic time constants
NMM1
rung 5
τ²ẍ + 2τẋ + x = γ σ(−wₓᵧy + F̂ₑ)τ²ÿ + 2τẏ + y = γ σ(wᵧₓx)
EI
E and I populations (like WILCO) — but each synapse adds a rise/decay + delay
Lₓ[xᵢ] = σ(−wₓᵧyᵢ + F̂ₑ + Σⱼ Cᵢⱼ xⱼ(t−τᵢⱼ))Lᵧ[yᵢ] = σ(wᵧₓxᵢ) additive, delayed, E→E · §6, Eq. 6.6
τEIEIsame E→E drive, delayed by the synapse (t − τ)
the same, E→E, but arriving with a conduction delay τᵢⱼ
▾ + exact QIF mean-field derivation
NMM2
rung 6 · next-gen mean field
ṙ = Δ/π + 2rvv̇ = v² − π²r² + Js + η̄ + I(t)
r v
one node = a population of spiking neurons → (r, v)
ṙᵢ = Δ/π + 2rᵢvᵢv̇ᵢ = vᵢ² − π²rᵢ² + η̄ + Σⱼ Cᵢⱼ sᵢⱼ, sᵢⱼ = K̂[rⱼ] synaptic-current coupling · §7, Eq. 7.11
a synapse turns rⱼ into a current sᵢⱼ into the other
node j’s synaptic activation sᵢⱼ drives node i’s voltage
Explore the models →