🏔️ That's the Differential Equations summit. Two analogies carried the whole climb — let's tie them together.

What a differential equation is:
• A rule for change — it gives the rate (dy/dt), not the value. Like a speedometer; solving it rebuilds the trip.
• It's self-referential (change depends on the current state) — the Escher hands, the compound-interest loop.
• Solve it analytically (exact formula, rare) or numerically (step-by-step, usually — next quest).

The bank-account warm-up (population):
• dp/dt = αp → solution p(t) = P₀·e^(αt) (exponential, because it's an eigenfunction).
• α > 0 explodes, α < 0 decays, α = 0 is the equilibrium where nothing changes.

The leaky bucket (the LIF neuron):
• τ·dV/dt = −(V − E_L) + R_m·I — leak pulls to rest E_L, input I pushes up.
• No input → always drifts back to the resting equilibrium.
• Add a threshold + reset → it spikes. More input → faster firing.
• The F–I curve = the transfer function, now derived from a real model. Its slope is gain.

Where it goes next: the population and LIF equations were the lucky ones — they had exact solutions. Most differential equations don't. The next quest, Numerical Methods, shows how a computer solves any of them step by step using Euler's method — which is just the Riemann sum you already know, wearing a new hat. 🚀