The Biophysics of Electrical Stimulation: What a Pulse Actually Does

An axon fires not where the outside voltage is highest, but where that voltage changes shape most sharply along the fiber. This formula measures exactly that curvature — the second difference — so bends, endings, and the spot right under the electrode light up first.

f_n

The drive to fire at node n — how strongly this spot is pushed toward an action potential.

V_e

The voltage the electrode creates outside the fiber, in the surrounding tissue.

\frac{\partial^2 V_e}{\partial x^2}

The curvature of that outside voltage along the fiber; big wherever it bends sharply.

\Delta x

The spacing between neighboring nodes used to approximate the curvature.

Along a straight stretch of fiber far from the electrode the voltage changes smoothly, the curvature is near zero, and nothing fires; right at a bend the curvature spikes and that node crosses threshold first.

Two rules of a safe stimulation pulse. First, the "dose" that actually excites tissue is charge — current multiplied by how long the pulse lasts, not current alone. Second, a biphasic pulse must push exactly as much charge one way as the other, so nothing accumulates and corrodes the electrode.

Q

Charge per phase — the true excitation "dose" of one pulse.

I

The current amplitude during the pulse.

t_{\mathrm{pw}}

Pulse width — how long the current flows.

\int I(t)\,dt = 0

The charge-balance condition: the net charge over the whole biphasic pulse is zero.

A 100 µA current held for 200 µs delivers a charge Q of 20 nanocoulombs; a matching reverse phase then returns that same amount so the net charge is zero.

A safety line drawn in log-log space: the higher the charge density D you use, the smaller the charge-per-phase Q you're allowed, and vice versa. The constant k marks how close to the danger line you sit — staying underneath is necessary for safety, but not a full guarantee.

D

Charge density — charge spread over the electrode's surface area (charge per unit area).

Q

Charge per phase — the same "dose" as in the previous formula.

k

The Shannon constant setting how aggressive the waveform is (about 1.5 conservative, up to ~2 near injury).

Small electrodes squeeze charge into a tiny area, driving D up, so the line forces Q (and thus each pulse) to stay small — one reason microelectrodes must use gentle pulses.

To make an axon fire, a briefer pulse must be stronger. This curve says the threshold current equals a "floor" current I_{\mathrm{rh}} (for a very long pulse), scaled up by a factor that blows up as the pulse gets short. Because large fibers have a shorter chronaxie, narrow pulses recruit them first — a selectivity knob.

I_{\mathrm{th}}

The threshold current needed to fire, for a given pulse width.

I_{\mathrm{rh}}

Rheobase — the minimum current that still works when the pulse is very long.

t_{\mathrm{chr}}

Chronaxie — the pulse width at which threshold is exactly twice rheobase; a fiber's characteristic time.

t_{\mathrm{pw}}

The pulse width you choose.

When t_{\mathrm{pw}} equals the chronaxie, the term in parentheses becomes two, so you need twice the rheobase current; make the pulse ten times shorter and the required current climbs about elevenfold.