Cable Equation Axon Length Constant Calculator

Find a neuron's electrotonic space constant lambda, how far a passive voltage change decays along an axon or dendrite, from membrane resistivity, axial resistivity, and diameter.

📏 Cable Equation Axon Length Constant Calculator
Specific membrane resistance (Rm)
Ω·cm²
Specific axial resistivity (Ri)
Ω·cm
Axon diameter (d)
µm
Length constant (λ)
—
Length constant (μm)
—
Step-by-step working

📏 What is the Cable Equation Axon Length Constant Calculator?

The cable equation axon length constant calculator finds a neuron's electrotonic space constant, lambda, the distance along an axon or dendrite over which a passive voltage change decays to about 37% (1/e) of its original size. It uses the diameter-based form of the classic cable theory equation, lambda = sqrt(Rm x d / (4 x Ri)), where Rm is specific membrane resistance, Ri is specific axial resistivity, and d is the fiber diameter.

Neuroscience courses use the length constant to explain why signal conduction speed and distance depend so heavily on axon diameter and myelination. Computational neuroscientists use lambda, alongside the membrane time constant tau, as one of the two fundamental parameters in passive multi-compartment cable models of dendrites and axons. Comparative physiologists use it to explain why invertebrates like squid evolved unusually thick axons, since they lack myelin, the vertebrate strategy for boosting effective membrane resistance.

A common misconception is that a wider axon conducts proportionally faster, doubling diameter does not double lambda, since lambda scales with the square root of diameter, not diameter itself. Another point of confusion is treating lambda as a fixed anatomical property, it is actually a purely passive electrical quantity determined by Rm, Ri, and diameter together, and changes in any of the three (such as demyelination reducing Rm) directly change lambda.

This calculator removes the unit-conversion and square-root arithmetic from computing lambda by hand, letting students verify homework problems and quickly compare how much lambda changes across a realistic range of axon diameters.

📐 Formula

λ  =  √(Rm × d / (4 × Ri))
λ = electrotonic space (length) constant
Rm = specific membrane resistance (Ω·cm²)
Ri = specific axial (intracellular) resistivity (Ω·cm)
d = axon or dendrite diameter, converted to centimeters inside the calculation (1 µm = 10−4 cm)
The units under the square root work out to cm², so λ itself comes out in centimeters; this calculator also reports it in millimeters and micrometers.
Example: Rm=2000 Ω·cm², Ri=100 Ω·cm, d=20 µm → λ = 1.000 mm (1000 µm).

📖 How to Use This Calculator

Steps

1
Enter the membrane and axial resistivity. Type Rm, the specific membrane resistance, and Ri, the specific axial resistivity, in ohm-cm-squared and ohm-cm.
2
Enter the axon diameter. Type d, the axon diameter, in micrometers.
3
Read the length constant. See lambda in both millimeters and micrometers, with the lambda-versus-diameter curve plotted.

💡 Example Calculations

Example 1 — Typical Mammalian Myelinated Axon

Rm=2000 Ω·cm², Ri=100 Ω·cm, d=20 µm

1
d = 20 µm = 0.002 cm
2
λ = √(2000 × 0.002 / (4 × 100)) = √(4 / 400) = √0.01
3
λ = 0.1 cm = 1.000 mm = 1000.0 µm
λ = 1.000 mm (1000.0 µm)
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Example 2 — Squid Giant Axon

Classic unmyelinated squid giant axon: Rm=1000 Ω·cm², Ri=35.4 Ω·cm, d=500 µm

1
d = 500 µm = 0.05 cm
2
λ = √(1000 × 0.05 / (4 × 35.4)) = √(50 / 141.6) = √0.3531
3
λ = 0.5942 cm = 5.942 mm = 5942.3 µm, far larger than a thin mammalian axon, this large diameter is exactly why the unmyelinated squid axon can still conduct quickly
λ = 5.942 mm (5942.3 µm)
Try this example →

Example 3 — Thin Unmyelinated Fiber

A thin unmyelinated axon: Rm=5000 Ω·cm², Ri=150 Ω·cm, d=1 µm

1
d = 1 µm = 0.0001 cm
2
λ = √(5000 × 0.0001 / (4 × 150)) = √(0.5 / 600) = √0.000833
3
λ = 0.02887 cm = 0.289 mm = 288.7 µm, much shorter than the previous two examples, showing how strongly λ shrinks for very thin fibers
λ = 0.289 mm (288.7 µm)
Try this example →

❓ Frequently Asked Questions

What is the axon length constant (space constant) lambda?+
The length constant lambda, also called the space constant, is the distance along an axon or dendrite over which a passive voltage change decays to about 37% (1/e) of its original value. It is calculated as lambda = sqrt(Rm x d / (4 x Ri)), where Rm is specific membrane resistance, Ri is specific axial resistivity, and d is the diameter.
What is the formula for the cable equation length constant?+
lambda = sqrt(Rm x d / (4 x Ri)), where Rm is specific membrane resistance in ohm-cm-squared, Ri is specific axial (intracellular) resistivity in ohm-cm, and d is the axon or dendrite diameter, giving lambda in centimeters when consistent units are used.
Why does lambda depend on the square root of diameter?+
As diameter increases, both the axial resistance (which decreases, since a wider core conducts current more easily along its length) and the membrane surface area (which increases, adding more parallel paths for current to leak out radially) change together in a way that produces a square-root, not linear, scaling of lambda with diameter.
How does lambda relate to conduction velocity?+
A larger length constant generally supports faster passive signal spread and, in unmyelinated axons, correlates with faster action potential conduction velocity, which is why some invertebrate axons (like the squid giant axon) evolved very large diameters, since myelin was not available to them as an alternative strategy for increasing effective Rm.
What are typical values for Rm, Ri, and lambda in real neurons?+
Specific membrane resistance Rm typically ranges from about 1,000 to 10,000 ohm-cm-squared for unmyelinated membrane, specific axial resistivity Ri is typically 35 to 200 ohm-cm, and the resulting length constant lambda for real axons and dendrites commonly falls between about 0.1 mm and several millimeters, depending heavily on diameter.
Why is the squid giant axon so large in diameter?+
The squid giant axon can reach up to about 1 mm in diameter specifically because a wider unmyelinated axon is one of the only ways to substantially raise the length constant lambda and therefore conduction speed, since squid (and invertebrates generally) lack myelin, the vertebrate strategy of raising effective membrane resistance without added bulk.
How does myelin change the length constant?+
Myelin wraps the axon in many insulating lipid layers, which dramatically increases the effective specific membrane resistance Rm. Since lambda scales with the square root of Rm, myelination substantially increases the length constant without requiring the axon to grow physically thicker, a much more space- and energy-efficient way to speed up conduction.
What units does this calculator use?+
Specific membrane resistance Rm is entered in ohm-cm-squared (ohm x cm^2), specific axial resistivity Ri in ohm-cm, and axon diameter d in micrometers (µm), the typical scale for a single neuron. The calculator converts diameter to centimeters internally and reports lambda in both millimeters and micrometers.
Is the length constant the same as the membrane time constant?+
No, they describe different things. The length constant lambda = sqrt(RmD/4Ri) describes how far a signal decays in space along the axon, while the membrane time constant tau = Rm x Cm describes how quickly the membrane voltage responds in time at a single point. Both are key parameters of passive cable theory, describing spatial and temporal decay respectively.
Does a larger lambda always mean a 'better' axon?+
Not necessarily, a larger lambda means passive signals travel further with less decay, which is generally desirable for fast, reliable conduction, but achieving it costs either more physical space (larger diameter) or more myelin (more lipid material and wrapping time during development), both real biological tradeoffs.
Tags: Cable Equation Calculator Axon Length Constant Calculator Space Constant Lambda Calculator Electrotonic Length Constant Cable Theory Neuron Calculator Formula Free Online

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What is the axon length constant (space constant) lambda?

The length constant lambda, also called the space constant, is the distance along an axon or dendrite over which a passive voltage change decays to about 37% (1/e) of its original value. It is calculated as lambda = sqrt(Rm x d / (4 x Ri)), where Rm is specific membrane resistance, Ri is specific axial resistivity, and d is the diameter.

What is the formula for the cable equation length constant?

lambda = sqrt(Rm x d / (4 x Ri)), where Rm is specific membrane resistance in ohm-cm-squared, Ri is specific axial (intracellular) resistivity in ohm-cm, and d is the axon or dendrite diameter, giving lambda in centimeters when consistent units are used.

Why does lambda depend on the square root of diameter?

As diameter increases, both the axial resistance (which decreases, since a wider core conducts current more easily along its length) and the membrane surface area (which increases, adding more parallel paths for current to leak out radially) change together in a way that produces a square-root, not linear, scaling of lambda with diameter.

How does lambda relate to conduction velocity?

A larger length constant generally supports faster passive signal spread and, in unmyelinated axons, correlates with faster action potential conduction velocity, which is why some invertebrate axons (like the squid giant axon) evolved very large diameters, since myelin was not available to them as an alternative strategy for increasing effective Rm.

What are typical values for Rm, Ri, and lambda in real neurons?

Specific membrane resistance Rm typically ranges from about 1,000 to 10,000 ohm-cm-squared for unmyelinated membrane, specific axial resistivity Ri is typically 35 to 200 ohm-cm, and the resulting length constant lambda for real axons and dendrites commonly falls between about 0.1 mm and several millimeters, depending heavily on diameter.

Why is the squid giant axon so large in diameter?

The squid giant axon can reach up to about 1 mm in diameter specifically because a wider unmyelinated axon is one of the only ways to substantially raise the length constant lambda and therefore conduction speed, since squid (and invertebrates generally) lack myelin, the vertebrate strategy of raising effective membrane resistance without added bulk.

How does myelin change the length constant?

Myelin wraps the axon in many insulating lipid layers, which dramatically increases the effective specific membrane resistance Rm. Since lambda scales with the square root of Rm, myelination substantially increases the length constant without requiring the axon to grow physically thicker, a much more space- and energy-efficient way to speed up conduction.

What units does this calculator use?

Specific membrane resistance Rm is entered in ohm-cm-squared (ohm x cm^2), specific axial resistivity Ri in ohm-cm, and axon diameter d in micrometers (µm), the typical scale for a single neuron. The calculator converts diameter to centimeters internally and reports lambda in both millimeters and micrometers.

Is the length constant the same as the membrane time constant?

No, they describe different things. The length constant lambda = sqrt(RmD/4Ri) describes how far a signal decays in space along the axon, while the membrane time constant tau = Rm x Cm describes how quickly the membrane voltage responds in time at a single point. Both are key parameters of passive cable theory, describing spatial and temporal decay respectively.

Does a larger lambda always mean a 'better' axon?

Not necessarily, a larger lambda means passive signals travel further with less decay, which is generally desirable for fast, reliable conduction, but achieving it costs either more physical space (larger diameter) or more myelin (more lipid material and wrapping time during development), both real biological tradeoffs.

📌 Quick Tips

💡The space (length) constant lambda is the distance over which a passive (non-regenerating) voltage change decays to 1/e (about 37%) of its value at the injection site, it is the spatial counterpart to the membrane time constant tau, which describes decay over time instead of distance.
💡Lambda scales with the square root of diameter, not diameter itself, doubling an axon's diameter only increases lambda by a factor of about 1.41 (the square root of 2), not by a factor of 2.
💡This is exactly why myelination is such an effective strategy for fast conduction, myelin dramatically raises effective membrane resistance Rm without requiring the axon to simply grow thicker, which would waste far more space and metabolic energy for the same increase in lambda.
💡The squid giant axon, famous from Hodgkin and Huxley's original experiments, is enormous by neuron standards (up to 1 mm diameter) specifically because a wide unmyelinated axon is one of the few ways evolution found to raise lambda and achieve fast conduction without myelin.
💡A longer space constant means a synaptic input further out on a dendrite has more influence on the cell body (soma), because less of the voltage change decays away before reaching it, this directly affects how neurons integrate spatially distributed synaptic inputs.