Let's strip the equation down. Turn off the input (I = 0) and just watch the leaky bucket drain. This is where the neuron's resting behavior lives.

With I = 0, divide through by τ_m and the equation becomes:

dV/dt = −(V − E_L) / τ_m

Notice the shape: it's just like the population equation dp/dt = αp, but pulling toward E_L instead of zero.

Read the sign of the change — this is the whole intuition. Look at where the voltage V is relative to rest E_L (= −75 mV):
• V above E_L → −(V − E_L) is negative → dV/dt < 0 → voltage leaks down toward rest.
• V below E_L → −(V − E_L) is positive → dV/dt > 0 → voltage climbs up toward rest.
• V exactly at E_L → dV/dt = 0 → the voltage sits still.

That third case is our friend the equilibrium point again. Just like the population at α = 0, the neuron has a resting voltage E_L where nothing changes. A leaky neuron with no input always drifts back to rest — like a ball rolling to the bottom of a bowl. The bucket always drains to its resting level.

The exact solution (same integration trick as the population equation — needs an initial voltage V_reset at t = 0):

V(t) = E_L + (V_reset − E_L) · e^(−t / τ_m)

'Current voltage = resting potential + (how far you started from rest) × an exponential decay.' The exponential shrinks over time, so the gap to rest melts away.

See it for yourself. Drag V_reset — the voltage where the neuron starts — and watch it always relax back to rest. Left: dV/dt vs V (the pull). Right: V(t) over time (the relaxation). The green dot is your starting point.

The green dot just acts out the three sign-cases above: start it above −75 mV and dV/dt is negative, so the curve slides down to rest; start it below and it climbs up; park it exactly at −75 mV (where the line crosses zero) and it sits flat — that crossing is the equilibrium.

The speed of the return is set by τ_m. A bigger time constant means a lazier, slower drift back to rest; a smaller one snaps back fast. This 'leak back to baseline' is why it's called leaky — and it's the same math as a capacitor discharging, or a hot coffee cooling to room temperature.