Action Potential & Cable Equation

Neuronal Signaling

Neurons transmit electrical signals along axons. The cable equation describes how voltage spreads along a cylindrical axon, balancing membrane leakage against axial current flow.

Space Constant (λ)

The space constant λ determines how far a subthreshold voltage deflection spreads:

$\lambda = \sqrt{\frac{R_m \cdot a}{2 R_i}}$

Where:

• a — axon radius (m)
• R_m — specific membrane resistance (Ω·m²); typical: 0.1 Ω·m²
• R_i — intracellular resistivity (Ω·m); typical: 1.0 Ω·m

A larger λ means signals travel farther without decaying.

Time Constant (τ)

The time constant τ governs how quickly the membrane voltage responds:

$\tau = R_m \cdot C_m$

Where:

• C_m — specific membrane capacitance (F/m²); typical: 0.01 F/m²

Typical τ ≈ 1 ms for unmyelinated axons.

Propagation Velocity (rough estimate)

$v \approx \frac{\lambda}{\tau}$

This approximation shows that thicker axons (larger a → larger λ) conduct faster.

Giant Squid Axon

The squid giant axon (diameter ~1 mm, radius ~500 μm) is a classic model system:

• λ ≈ 5 mm
• Conduction velocity ~20 m/s

Myelination

Myelin sheaths increase effective R_m and decrease C_m dramatically, increasing both λ and decreasing τ, yielding saltatory conduction at speeds up to 120 m/s in humans.

Typical Parameters

Functions to Implement

• space_constant_m(a_m, R_m_ohm_m2=0.1, R_i_ohm_m=1.0) — λ in meters
• time_constant_s(R_m_ohm_m2=0.1, C_m_F_m2=0.01) — τ in seconds
• propagation_velocity_m_s(a_m, R_m_ohm_m2=0.1, R_i_ohm_m=1.0, C_m_F_m2=0.01) — v in m/s