= Patch-clamp series resistance
{title2=$R_s$}
= Patch-clamp access resistance
{synonym}
<Patch-clamp series resistance>, also called <patch-clamp access resistance>, is the effective <electrical resistance> between the pipette's sensed voltage and the cell interior. It includes tip access and any remaining obstruction after break-in. For injected <electric current> $I$ into the <biological cell>, an ideal resistive access path gives
$$
V_p-V=IR_s.
$$
Thus $1\,\mathrm{nA}$ through $20\,\mathrm{M\Omega}$ produces a $20\,\mathrm{mV}$ difference: a substantial possible <voltage clamp> error. The <electric current> in this equation is the access-path current, excluding any bypass <seal leak current>. In <current clamp>, the same access drop contaminates the pipette voltage during injection. Bridge compensation subtracts an estimated $IR_s$; an estimate that changes during recording leaves an error.
An independent passive <voltage clamp> calculation makes both speed and error visible. Write $v=V-E_L$ and $v_p=V_p-E_L$, with fixed leak <reversal potential> $E_L$, constant membrane <electrical resistance> $R_m$ and <membrane capacitance> $C_m$. Assume an ideal command at the pipette node and omit <seal leak current> and pipette <capacitance>. <Electric charge conservation> gives
$$
C_m\frac{dv}{dt}=\frac{v_p-v}{R_s}-\frac{v}{R_m}.
$$
For a step $v_p=\Delta V$ from $v(0)=0$, the <ordinary differential equation> gives
$$
v(t)=\Delta V\frac{R_m}{R_s+R_m}\left(1-e^{-t/\tau_{\mathrm{vc}}}\right),\qquad \tau_{\mathrm{vc}}=C_m\frac{R_sR_m}{R_s+R_m}.
$$
With the figure's parameters and $\Delta V=10\,\mathrm{mV}$, the <time constant> is $0.909\,\mathrm{ms}$, the final membrane rise is $9.091\,\mathrm{mV}$, and access-path <electric current> falls from $500$ to $45.45\,\mathrm{pA}$. Only when $R_m\gg R_s$ does the <time constant> approach $R_sC_m$ and the final voltage approach the command. Actual <voltage clamp> also depends on amplifier <feedback control> and the <space clamp> problem.
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