The Membrane as an RC Circuit

Membrane

Resistance R 200 MΩ

Capacitance C 100 pF

Holding potential E_hold 0 mV

Stimulus

Mode

Amplitude I +100 pA

Positive = depolarizing, negative = hyperpolarizing.

Pulse onset (ms)

Pulse duration (ms)

Sweep length (ms)

Keep previous sweeps (up to 6) Show reference markers

Predicted from R and C

τ = RC

20.0 ms

ΔV_∞ = IR

+20.0 mV

Membrane potential V_m (mV) Parameters changed — press Run sweep.

Currents (pA) — outward positive I_inj I_C I_R

Equivalent circuit

Timeline —

Arrow size = current magnitude. Auto-syncs with sweep.

Measured time to 63.2%

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Peak ΔV reached

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Charge moved Q = C·ΔV

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Last sweep — every 10th sample. Copy gives every sample.

t (ms)	Vm (mV)	I_inj (pA)	I_C (pA)	I_R (pA)

Current clamp: Injected current I first charges C (I_C = C·dV/dt), then leaks through R, until V_m = E_hold + IR with τ = RC. Voltage clamp: The command voltage is stepped; a brief capacitive transient I_C = C·ΔV/τ_r is followed by steady resistive current I_R = ΔV/R. After 3τ the response is 95% settled.

Membrane properties

Specific capacitance C_m 1.00 µF/cm²

≈ 0.9–1 µF/cm² for most neurons.

Specific resistance R_m 20 kΩ·cm²

Set by channel density, not cell size.

Give cell B its own R_m

Cell A

Total membrane area 5,000 µm²

Cell B

Total membrane area 30,000 µm²

Stimulus

Current I +50 pA

Pulse duration (ms)

Sweep length (ms)

Normalize each trace to its ΔV_∞

V_m (mV) Cell ACell B Parameters changed. Press Run sweep.

A ΔV this large would make a real neuron fire. This model is purely passive.

Single-compartment model.

Cell	Area (µm²)	Ø (µm)	C (pF)	R_in (MΩ)	τ (ms)	ΔV_∞ (mV)	I for 10 mV (pA)

C, R_in and τ vs membrane area (log–log) C (pF)R_in (MΩ)τ (ms)

Capacitance grows in proportion to area (slope +1); input resistance falls in proportion (slope −1); their product τ = R_m·C_m does not depend on area at all. A small and a large cell with the same membrane charge with the same time course, but the small one reaches a much larger voltage for the same current.

Compartment

Geometry

Diameter 1.20 mm

Specific capacitance C_m 1.00 µF/cm²

Voltage change

ΔV 100 mV

Number of APs (no pumping) 1

Ion carrying the charge

Intracellular concentration (mM)

Flux relative to minimum 1.00×

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Membrane capacitance

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Charge moved ΔQ = C·ΔV

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Ions crossing

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Ions inside

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Concentration change

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Relative change

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APs to change [ion] by 10%

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without pumps

Ions crossing vs ions inside (log scale)

Relative Δ[ion] per event vs diameter (log–log) spherecylindercurrent

Current calculation.

Quantity	Value	Unit

The relative change in concentration scales with surface-to-volume ratio (6/d sphere, 4/d cylinder). In thin axons and dendritic spines the fraction per AP becomes significant; for Ca²⁺ (resting ~100 nM), even a small fraction of the charge produces a huge relative change — which is why Ca²⁺ is such an effective intracellular signal.