🏔️ You've reached the top of Math Mountains. Numerical methods are how we solve the differential equations that have no tidy formula — which is most of them.

The one idea:
• A derivative is a slope between two points Δt apart.
• Replace dy/dt with that slope → Euler's method: Future = current + (step × dynamics).
• Loop it to trace the whole solution.

The catch:
• Each step has a small local error; they pile up into global error.
• Smaller Δt = more accurate but more computation. Always a trade-off.

What we solved with it:
• The population equation (checked against its exact answer).
• The LIF neuron with time-varying input — bursts of spikes.
• A system of two equations (Wilson–Cowan) → oscillations, phase planes, nullclines, and how one knob changes everything.

Where it goes next: fancier methods like RK4 (in the bonus) are far more accurate for the same step size, and nonlinear oscillators like Stuart–Landau let researchers reason about real brain rhythms. But every one of them is built on the humble idea you just learned: take a small step, re-check the slope, step again. 🚀