Here's the payoff of the whole quest: you can predict a circuit's fate just by looking at the eigenvalues of its weight matrix — without simulating anything.

Start the activity exactly on an eigenvector (a₀ = v). Then each step just scales it by λ:

a₁ = W·v = λv
a₂ = W·(λv) = λ²v
aᵢ = λⁱ·v

The activity stays on the eigenvector's line, growing or shrinking by λⁱ.

aᵢ = λⁱ · v

Along an eigenvector, the whole matrix collapses to repeatedly multiplying by the single number λ.

The rules that fall out (memorize these):
• |λ| > 1 → λⁱ grows → activity explodes.
• |λ| < 1 → λⁱ shrinks → activity decays to zero.
• |λ| = 1 → activity is sustained (stays steady).

With several eigenvalues, the largest one wins in the long run — it dominates as the power grows.

Any starting point can be written as a mix of eigenvectors, so in general aᵢ = c₁λ₁ⁱv₁ + c₂λ₂ⁱv₂. Whichever eigenvalue is biggest eventually dominates, dragging the trajectory toward its eigenvector. Two numbers — the eigenvalues — tell you the destiny of the entire system.