Formulas

Computed in SI units; the build recomputes every worked example and fails if one disagrees. Formulas we could not source yet are left out rather than written from memory.

Membrane and neuron

Hodgkin–Huxley sodium currentHodgkin and Huxley 1952

I_{Na} = \bar g_{Na}\, m^{3} h\, (V - E_{Na})

C_m \frac{dV}{dt} = -\left(I_{Na} + I_K + I_L\right) + I_{stim}

I_K = \bar g_K\, n^4 (V - E_K), \qquad I_L = g_L (V - E_L)

\phi = Q_{10}^{(T - 6.3\,^{\circ}\mathrm{C})/10}

The sodium current through a patch of membrane is its maximum conductance, scaled by three activation gates (m) and one inactivation gate (h), times the driving force: how far the membrane potential is from the sodium reversal potential. Negative current flows into the cell. The potassium and leak currents have the same form; the capacitance equation adds them up; φ scales the gate rates with temperature.

Variables of Hodgkin–Huxley sodium current
Symbol	Meaning	Unit
$\bar g_{Na}$	maximum sodium conductance	mS/cm²
$m$	activation gate (fraction open, 0 to 1)	dimensionless
$h$	inactivation gate (fraction not inactivated, 0 to 1)	dimensionless
$V$	membrane potential	mV
$E_{Na}$	sodium reversal potential	mV
$I_{Na}$	sodium current density	µA/cm²

Valid when

Squid giant axon membrane, the preparation it was fitted to; other neurons have other channels and constants
Potentials here are shifted so that rest is −65 mV, the common modern convention (Hodgkin and Huxley measured from rest, with the opposite sign)
Gate values m and h come from their own rate equations; the example fixes them to show one instant

Worked exampleHodgkin and Huxley 1952

With the published maximum conductance and reversal potential (in the shifted convention), and gates half open and 60% available, at −40 mV the sodium current is strongly inward.

gNa = 120 mS/cm², m = 0.5 , h = 0.6 , V = −40 mV, ENa = 50 mV → INa = −810 µA/cm²

Timing and conduction

Conduction velocity of a myelinated fibre (Hursh)Hursh 1939

v \approx k\, D, \qquad k \approx 6\ \tfrac{\mathrm{m/s}}{\mu\mathrm{m}}

For myelinated nerve fibres, conduction velocity grows in proportion to the fibre’s outside diameter. Hursh measured a ratio of about 6 metres per second for every micrometre.

Variables of Conduction velocity of a myelinated fibre (Hursh)
Symbol	Meaning	Unit
$D$	fibre diameter (outside the myelin)	µm
$v$	conduction velocity	m/s

Valid when

Myelinated peripheral fibres of the cat, where it was measured
Using it for central fibres, or for human nerves, is an approximation
Not for unmyelinated fibres, where speed grows more slowly with diameter

Worked exampleHursh 1939

A 10 µm fibre conducts at about 60 m/s by this ratio.

D = 10 µm → v = 60 m/s

Conduction time along a fibreHursh 1939

t = \frac{L}{v} = \frac{L}{k\, D}

The time a signal needs to travel along a fibre is its length divided by its speed; with Hursh’s ratio, the speed comes from the fibre diameter. Synaptic delays add to this; the signal journey lab adds them up for a whole pathway.

Variables of Conduction time along a fibre
Symbol	Meaning	Unit
$L$	fibre length	m
$D$	fibre diameter	µm
$t$	conduction time	ms

Valid when

Assumes one constant diameter along the whole fibre
Ignores synaptic and neuromuscular delays
Inherits the limits of Hursh’s ratio (myelinated fibres, measured in cat)

Worked exampleHursh 1939

One metre of 10 µm myelinated fibre takes about 16.7 ms.

L = 1 m, D = 10 µm → t = 16.7 ms

Decoding and learning

Cosine tuning of a motor cortex neuronGeorgopoulos et al. 1982

f(\theta) = b_0 + b_1 \cos(\theta - \theta_{pref})

A motor cortex neuron fires most for movements in its preferred direction and less the further the movement turns away from it, following a cosine. Decoders for brain–computer interfaces build on this.

Variables of Cosine tuning of a motor cortex neuron
Symbol	Meaning	Unit
$b_0$	baseline rate	spikes/s
$b_1$	modulation depth	spikes/s
$\theta$	movement direction	°
$\theta_{pref}$	preferred direction	°
$f$	firing rate	spikes/s

Valid when

Two-dimensional arm movements of monkeys, where it was described
A fit to average rates over trials, not a description of single spikes
Rates cannot go below zero, so a large b₁ with a small b₀ is not physical

Worked exampleGeorgopoulos et al. 1982

A neuron with baseline 20 and depth 10 spikes/s, moving 60° away from its preferred direction: 20 + 10 × cos 60°.

b0 = 20 spikes/s, b1 = 10 spikes/s, theta = 60 °, thetaPref = 0 ° → f = 25 spikes/s

Information transfer rate (Wolpaw)Wolpaw et al. 1998; Wolpaw et al. 2002

B = \log_2 N + P \log_2 P + (1 - P) \log_2 \frac{1 - P}{N - 1}

How many bits each selection of a brain–computer interface carries, given the number of possible targets and the chance of choosing the right one. Multiply by selections per minute for bits per minute.

Variables of Information transfer rate (Wolpaw)
Symbol	Meaning	Unit
$N$	number of targets	dimensionless
$P$	accuracy	%
$B$	bits per selection	bit/selection

Valid when

Assumes every target is equally likely and errors spread evenly over the wrong targets
Accuracy below chance (1/N) gives misleading values
Says nothing about how long each selection takes; that comes in through selections per minute

Worked exampleWolpaw et al. 1998

Four targets chosen correctly 90% of the time carry about 1.37 bits per selection, against 2 bits if it were always right.

N = 4 , P = 90 % → B = 1.37 bit/selection

Mechanics and alignment

Pelvic incidence, pelvic tilt and sacral slopeLazennec et al. 2011

\mathrm{PI} = \mathrm{PT} + \mathrm{SS}

Pelvic incidence is a fixed property of a person’s pelvis. It is shared between two angles that change with posture: pelvic tilt, how far the pelvis is rotated back, and sacral slope, how steep the top of the sacrum is. When the pelvis tilts back, the sacral slope falls by the same amount.

Variables of Pelvic incidence, pelvic tilt and sacral slope
Symbol	Meaning	Unit
$\mathrm{PT}$	pelvic tilt	°
$\mathrm{SS}$	sacral slope	°
$\mathrm{PI}$	pelvic incidence	°

Valid when

Angles measured on a lateral radiograph that shows the femoral heads and the top of the sacrum
Pelvic incidence is constant for an adult; tilt and slope change with posture, sitting or standing

Worked exampleLazennec et al. 2011

A pelvic tilt of 15° and a sacral slope of 40° add up to an incidence of 55°.

pt = 15 °, ss = 40 ° → pi = 55 °

Cobb angle from the end-vertebra tiltsWang et al. 2018

\angle\mathrm{Cobb} = \alpha + \beta

The Cobb angle of a curve in the coronal plane equals the tilt of the upper end vertebra plus the tilt of the lower end vertebra, each measured from the horizontal. Wang and colleagues derived this from plane geometry and used it to measure the angle without drawing the classical perpendicular lines.

Variables of Cobb angle from the end-vertebra tilts
Symbol	Meaning	Unit
$\alpha$	upper end vertebra tilt	°
$\beta$	lower end vertebra tilt	°
$\angle\mathrm{Cobb}$	Cobb angle	°

Valid when

Tilts measured on the same coronal image, on opposite sides of the horizontal
Which vertebrae are the end vertebrae is a judgement; the angle depends on it

Worked exampleWang et al. 2018

The worked case in Wang et al. (their figure): tilts of 41° and 30° give a Cobb angle of 71°.

alpha = 41 °, beta = 30 ° → cobb = 71 °

Clinical scores

ISNCSCI motor scoreRupp et al. 2021; Kirshblum et al. 2011

\mathrm{MS} = \mathrm{UEMS} + \mathrm{LEMS},\quad \mathrm{UEMS} = \sum_{C5}^{T1} (R + L),\quad \mathrm{LEMS} = \sum_{L2}^{S1} (R + L)

Each of the ten key muscles is graded 0 to 5 on each side. The upper-limb motor score adds the five arm muscles (C5 to T1) on both sides, up to 50; the lower-limb score adds the five leg muscles (L2 to S1), up to 50.

Variables of ISNCSCI motor score
Symbol	Meaning	Unit
$\mathrm{UEMS}$	upper extremity motor score	dimensionless
$\mathrm{LEMS}$	lower extremity motor score	dimensionless
$\mathrm{MS}$	sum of the two motor scores	dimensionless

Valid when

A scoring convention for examination after spinal cord injury, done by trained examiners
For learning only: this site does not score or interpret anyone’s examination

Worked exampleRupp et al. 2021

Full strength in all ten key muscles on both sides gives 50 + 50.

uems = 50 , lems = 50 → ms = 100