Membrane Length Constant Equation, Derivation & Rearrangements

Core idea

Overview

The membrane length constant (λ) is a crucial parameter in neurophysiology, representing the distance over which a steady-state voltage change across the neuronal membrane decays to 1/e (approximately 37%) of its initial value. It is determined by the square root of the ratio of membrane resistance (Rm) to internal resistance (Ri). A larger length constant indicates that a signal can propagate further along the neuron with less attenuation, which is vital for efficient signal transmission in the nervous system.

When to use: Use this equation to understand the passive electrical properties of neurons and how far a local potential change will spread. It's applied when analyzing the efficiency of signal propagation in axons and dendrites, particularly in response to subthreshold stimuli. Knowing the length constant helps predict how effectively a synaptic input at one point will influence the membrane potential at a distant point.

Why it matters: The membrane length constant is fundamental to understanding neuronal communication. It dictates the spatial summation of synaptic potentials and the speed and reliability of action potential propagation. Variations in length constant due to changes in membrane or internal resistance can significantly impact neuronal function, affecting everything from sensory perception to motor control and learning.

Symbols

Variables

\lambda = Membrane Length Constant, R_m = Membrane Resistance, R_i = Internal Resistance

λ

Membrane Length Constant

c m

R_{m}

Membrane Resistance

Ω \cdot c m^{2}

R_{i}

Internal Resistance

Ω \cdot c m

Walkthrough

Derivation

Formula: Membrane Length Constant

The membrane length constant quantifies the distance over which a voltage change decays to 37% of its initial value along a passive cable.

The neuron acts as an infinitely long, uniform cylindrical cable (cable theory).
The membrane is passive, meaning no active voltage-gated channels are involved.
The external resistance is negligible compared to internal and membrane resistance.

1

Define the Cable Equation:

This is the steady-state cable equation, describing voltage (V) change along distance (x) in terms of internal (Ri) and membrane (Rm) resistance.

\frac{d ^{2} V}{d x ^{2}} = \frac{R _{i}}{R _{m}} V

2

Introduce Length Constant:

By definition, the length constant $\lambda$ is introduced such that $\lambda^2 = \frac{R_m}{R_i}$. This simplifies the cable equation.

\frac{d ^{2} V}{d x ^{2}} = \frac{1}{λ ^{2}} V

3

Solve for Lambda:

Taking the square root of both sides yields the formula for the membrane length constant. This constant determines the spatial decay of voltage.

λ = \frac{R _{m}}{R _{i}}

Result

λ = \frac{R _{m}}{R _{i}}

Source: Kandel, E.R., Schwartz, J.H., Jessell, T.M., Siegelbaum, S.A., Hudspeth, A.J. (2013). Principles of Neural Science (5th ed.). McGraw-Hill.

Free formulas

Rearrangements

Solve for $λ$

Make lambda the subject

λ = \frac{R _{m}}{R _{i}}

lambda is already the subject of the formula.

Difficulty: 1/5

Solve for $R_{m}$

Membrane Length Constant: Make R_m the subject

R_{m} = λ^{2} R_{i}

To make $R_{m}$ (Membrane Resistance) the subject of the Membrane Length Constant formula, square both sides and then multiply by $R_{i}$ (Internal Resistance).

Difficulty: 3/5

Solve for $R_{i}$

Membrane Length Constant: Make R_i the subject

R_{i} = \frac{R _{m}}{λ ^{2}}

To make $R_{i}$ (Internal Resistance) the subject of the Membrane Length Constant formula, square both sides, then multiply by $R_{i}$ and divide by $λ^{2}$ .

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows a square root curve starting at the origin and increasing at a decreasing rate as membrane resistance grows. For a biology student, this means that increasing membrane resistance allows the electrical signal to travel significantly further along the neuron, though with diminishing returns at higher resistance levels. The most important feature is the square root relationship, which shows that the length constant grows more slowly than the membrane resistance, meaning large increases in resistance

Graph type: power_law

Why it behaves this way

Intuition

Visualize an electrical signal as current flowing along a cylindrical, leaky cable, where the length constant defines how far the current travels before most of it dissipates through the cable's walls.

λ

The distance over which a steady-state voltage change across a neuronal membrane decays to 1/e (approximately 37%) of its initial value.

A larger length constant means electrical signals spread further along the neuron before significantly weakening, which is crucial for efficient long-distance communication.

R_{m}

Membrane resistance, which is the resistance to current flow across the neuronal membrane, reflecting how easily ions leak out.

Higher membrane resistance means less current leaks out of the neuron, allowing the electrical signal to propagate further along its length.

R_{i}

Internal (axial) resistance, which is the resistance to current flow along the inside of the neuron (cytoplasm), influenced by cytoplasm resistivity and diameter.

Lower internal resistance means current flows more easily along the neuron's interior, allowing the electrical signal to propagate further before leaking out.

Signs and relationships

\sqrt{...}: The square root indicates that the length constant scales non-linearly with the ratio of membrane to internal resistance. It arises from the balance of current flowing axially and leaking radially, as described by the

Free study cues

Insight

Canonical usage

This equation calculates a length, so the output unit must be a unit of length (e.g., cm or m) when the input resistance terms are consistently expressed as resistance per unit length.

Common confusion

Students often confuse specific membrane resistance (R_M, in Ω·cm2) or specific axoplasmic resistivity (ρ_i, in Ω·cm) with the resistance per unit length terms (R_m in Ω·cm and R_i in Ω/cm) required for this formula.

Unit systems

$λ$ cm | m · The membrane length constant, representing a distance.

$R_{m}$ Ω·cm | Ω·m · Membrane resistance per unit length of the neuron (e.g., axon or dendrite). This is distinct from specific membrane resistance (Ω·cm2).

$R_{i}$ Ω/cm | Ω/m · Internal (axial) resistance per unit length of the neuron. This is distinct from specific axoplasmic resistivity (Ω·cm).

One free problem

Practice Problem

A neuron has a membrane resistance (Rm) of 10000 Ω·cm² and an internal resistance (Ri) of 100 Ω·cm. Calculate its membrane length constant (λ).

Membrane Resistance10000 Ω·cm²

Internal Resistance100 Ω·cm

Solve for: lambda

Hint: Remember to take the square root of the ratio.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Predicting how far a synaptic input on a dendrite will influence the axon hillock.

Study smarter

Tips

Ensure Rm and Ri are in consistent units (e.g., both in Ω·cm² or Ω·cm).
A larger λ means better passive signal conduction.
λ is influenced by myelination (increases Rm) and axon diameter (decreases Ri).
The length constant is a measure of passive spread, not active propagation (like action potentials).

Avoid these traps

Common Mistakes

Confusing length constant with time constant.
Incorrectly interpreting the units of Rm and Ri.
Assuming active propagation when only passive spread is considered.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The membrane length constant quantifies the distance over which a voltage change decays to 37% of its initial value along a passive cable.

Use this equation to understand the passive electrical properties of neurons and how far a local potential change will spread. It's applied when analyzing the efficiency of signal propagation in axons and dendrites, particularly in response to subthreshold stimuli. Knowing the length constant helps predict how effectively a synaptic input at one point will influence the membrane potential at a distant point.

The membrane length constant is fundamental to understanding neuronal communication. It dictates the spatial summation of synaptic potentials and the speed and reliability of action potential propagation. Variations in length constant due to changes in membrane or internal resistance can significantly impact neuronal function, affecting everything from sensory perception to motor control and learning.

Confusing length constant with time constant. Incorrectly interpreting the units of Rm and Ri. Assuming active propagation when only passive spread is considered.

Predicting how far a synaptic input on a dendrite will influence the axon hillock.

Ensure Rm and Ri are in consistent units (e.g., both in Ω·cm² or Ω·cm). A larger λ means better passive signal conduction. λ is influenced by myelination (increases Rm) and axon diameter (decreases Ri). The length constant is a measure of passive spread, not active propagation (like action potentials).

References

Sources

Principles of Neural Science, Sixth Edition by Kandel, Schwartz, Jessell, Siegelbaum, Hudspeth
Neuroscience, Sixth Edition by Purves, Augustine, Fitzpatrick, Hall, LaMantia, Mooney, Platt, White
Wikipedia: Length constant
Principles of Neural Science, 6th Edition, Kandel, Schwartz, Jessell, Siegelbaum, Hudspeth
Neuroscience, 6th Edition, Purves, Augustine, Fitzpatrick, Hall, LaMantia, McNamara, White
Kandel, Schwartz, Jessell, Siegelbaum, Hudspeth, 'Principles of Neural Science', 6th ed., Chapter 8
Johnston and Wu, 'Foundations of Cellular Neurophysiology', 2nd ed., Chapter 3
Purves, Augustine, Fitzpatrick, Hall, LaMantia, Mooney, Platt, White, 'Neuroscience', 6th ed., Chapter 2

Membrane Length Constant

Overview

Variables

Derivation

Define the Cable Equation:

Introduce Length Constant:

Solve for Lambda:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Membrane Time Constant

Ohm's Law (Membrane)

Frequently Asked Questions

Sources