Cable Theory — Space Constant λ & Time Constant τ

The Cable tab shows a 2D heatmap: x-axis = position along dendrite (mm), y-axis = time (ms), colour = voltage (warm=depolarised, cool=rest). When current is injected at x=0, voltage rises locally and spreads with exponential decay in space (e^{−x/λ}) and exponential decay in time (e^{−t/τ_m}). Unlike a simple RC (uniform decay everywhere), the cable shows a spatiotemporal wavefront. A synapse at x=2λ delivers e^{−2} ≈ 14% of its voltage to the soma. The Lambda tab shows V(x) at steady state as the exponential decay curve — drag the diameter slider and watch λ change.

λ = √(R_m d/(4R_a)) ∝ √d. Thicker dendrite: lower axial resistance per unit length r_a ∝ 1/d² (larger cross-section carries more current) but higher membrane resistance per unit length r_m ∝ 1/d (more perimeter = more leak). Net: r_m/r_a ∝ (1/d)/(1/d²) = d, so λ ∝ √d. Doubling d increases λ by √2 = 1.41×. The squid giant axon (d~1 mm) has λ~10 mm (signals travel far); a thin cortical dendrite (d~0.1 μm) has λ~0.1 mm (signals decay quickly). Myelination effectively increases d by adding insulating layers.

MS destroys myelin, which normally increases R_m×100 and decreases C_m×100, giving λ_myelinated ≈ 10× λ_unmyelinated and conduction velocity 50–70 m/s vs 0.5–2 m/s. When myelin is lost: λ decreases dramatically, C_m increases, action potential conduction fails (conduction block). MS symptoms — visual loss (optic nerve), weakness (corticospinal tract), sensory loss — correspond to demyelination in specific white matter tracts. The cable equation directly predicts the minimum diameter for successful AP conduction after demyelination. Treatment: 4-aminopyridine (K⁺ channel blocker) prolongs AP duration to compensate for reduced λ.

Passive cable theory predicts strong location dependence — a distal synapse at L=2 delivers only 14% voltage to soma. Yet Magee & Cook (2000) showed somatic EPSPs are similar regardless of synaptic location. The resolution: (1) Synaptic strength scaling — distal synapses are physically larger (more AMPA receptors) to compensate for attenuation (homeostatic normalisation); (2) Active dendritic conductances — NMDA receptors, Ca²⁺, h-current amplify distal EPSPs locally; (3) Dendritic spikes — strong distal inputs trigger local Na⁺/Ca²⁺ spikes that propagate to soma with less attenuation. Passive cable theory is incomplete — active dendrites change the computation fundamentally.

For explicit Euler on the cable equation (diffusion coefficient D = λ²/τ_m): dt < δx²/(2D) = δx² τ_m/(2λ²). For λ=0.5 mm, τ_m=20 ms, δx=0.1 mm: dt_max = 0.01×20/0.5 = 0.4 ms — safe with dt=0.01 ms. Violating stability: spatial oscillations appear and grow exponentially. Crank-Nicolson (semi-implicit) is unconditionally stable for any dt and is preferred for production cable simulations. Rule: always check if δx < λ (otherwise the spatial grid cannot resolve the decay profile).

Rall's rule: a branch point is impedance-matched when d_parent^(3/2) = Σ d_children^(3/2). Under this condition, no voltage reflection occurs at the branch point, and the entire dendritic tree can be 'unfolded' into a single equivalent cylinder — enormously simplifying analysis. Real dendritic trees often violate the 3/2 rule, creating impedance mismatches that reflect voltage waves back toward the soma. These mismatches allow individual dendritic branches to implement logical AND operations: two inputs must both be active to trigger a local dendritic spike (threshold nonlinearity). This means a single neuron with active dendrites has computational power equivalent to a multi-layer network.

L = ℓ/λ is the physical length normalised by the space constant. L=1: 37% voltage at distal end; L=2: 14%; L=3: 5%. Electrotonically compact (L < 0.3): voltage changes propagate nearly undistorted from one end to the other — any synapse effectively drives the soma. Electrotonically extended (L > 2): distal synapses are electrically isolated. Most cortical pyramidal neurons have L ≈ 1–3 (partial attenuation). Interneurons are typically compact (L < 1) — their small size means any synaptic input reaches the soma effectively. This is why interneurons integrate inputs globally while pyramidal neurons can perform compartmentalised dendritic computations.