= Passive membrane response in current clamp
{title2=$\tau_m=R_mC_m$}
Model a spatially uniform <cell membrane> as a <capacitor> of <membrane capacitance> $C_m$ in parallel with membrane <electrical resistance> $R_m$ and a fixed leak <reversal potential> $E_L$. Exclude active <ion channels>, <seal leak current> and pipette <capacitance>. For <electric current> $I$ injected into the <biological cell>, <electric charge conservation> yields
$$
C_m\frac{dV}{dt}=I-\frac{V-E_L}{R_m}.
$$
Starting at $V(0)=E_L$ and applying constant $I$ gives the <ordinary differential equation> solution
$$
V(t)=E_L+IR_m\left(1-e^{-t/\tau_m}\right),\qquad \tau_m=R_mC_m.
$$
The <steady state> rise is $IR_m$; the <time constant> is $R_mC_m$. With $I=50\,\mathrm{pA}$, $R_m=200\,\mathrm{M\Omega}$, $C_m=50\,\mathrm{pF}$ and $E_L=-70\,\mathrm{mV}$, the rise is $10\,\mathrm{mV}$ and $\tau_m=10\,\mathrm{ms}$. At one <time constant> the rise is $10(1-e^{-1})\,\mathrm{mV}\approx6.321\,\mathrm{mV}$. The figure's uncompensated pipette voltage adds $IR_s=1\,\mathrm{mV}$ for $R_s=20\,\mathrm{M\Omega}$. The membrane voltage is continuous at step onset in this model, while the pipette voltage immediately acquires that access drop.
Checking the units is instructive: <electrical resistance> times <capacitance> has units $\Omega\,\mathrm F=\mathrm s$, and <electric current> times <electrical resistance> has units of voltage. Unlike the passive <voltage clamp> transient under <patch-clamp series resistance>, this ideal <current clamp> membrane <time constant> is set by $R_mC_m$, not $R_sC_m$. Strong current steps can recruit <voltage-gated ion channels>; the passive model then fails and cannot generate an <action potential>. Small pulses of both signs, stable access and repeated baseline measurements help test whether the passive approximation is appropriate.
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