% TCI Wrap/Unwrap % % If you're separating the files in this message by hand, % % files followed by (macdir) go into your macros/styles directory, % % and the remaining files go into your TeX source file directory. % % Files included: % % NOISE.TEX (srcdir) 9374 bytes % %%%%%%%%%%%%%%%%%%%%%%%%% Start NOISE.TEX %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% This document created by Scientific Word (R) %% Version 1.1 \documentstyle[12pt,thmsa,sw20jart]{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{TCIstyle=Article/ART4.LAT,jart,sw20jart} \input tcilatex \begin{document} \author{{\bf Richard A. Levis} \\ %EndAName Department of Molecular Biophysics \& Physiology\\ Rush Medical College\\ 1750 W. Harrison Street\\ Chicago IL 60612 USA\\ (312)-942-6454\\ {\sc FAX:} (312)-942-8711 \and Slightly modified by \and Bob Eisenberg \\ %EndAName Department of Molecular Biophysics \& Physiology\\ Rush Medical College\\ 1750 W. Harrison Street\\ Chicago IL 60612 USA\\ (312)-942-6467\\ {\sc FAX:} (312)-942-8711\\ {\sc Email:} bob@aix550.phys.rpslmc.edu} \title{Current Noise in Recordings of Currents through Single Channels} \maketitle \date{March 30, 1993\\ } \newpage\ \section{Noise of Amplifier} The power spectral density {\sf PSD }of the current noise of the amplifier $% S_{hs}(f)$ used to record single channel currents is well described by \begin{equation} \label{1} S_{hs}^2\left( f\right) \simeq \stackrel{\text{{\it white\ noise}}}{% \overbrace{\ 9\times 10^{-32}}}+\stackrel{\text{{\it dielectric\ noise}}}{% \overbrace{7\times 10^{-35}f}}+\stackrel{\QATOP{\text{{\it differentiated}}}{% \text{{\it white\ noise}}}}{\overbrace{1.65\times 10^{-38}f^2}}\qquad {\rm % units:amp}^2{\rm /Hz} \end{equation} if the amplifier is the Axopatch 200A made by Axon Instruments of Foster City California. $f$ is the frequency in Hz. $S_{hs}^2(f)$ describes the noise measured from the headstage itself with no connections made to it of any kind; in particular, it is the noise without the electrode holder, pipette, bath, or preparation attached. The first term of the current noise ($9\times 10^{-32}$) describes the{\sf \ white noise} current arising (mostly) from the shot noise $i_{n\text{ }}$of the gate leakage current of the input JFET of the amplifier, with a small part coming from the leakage current of the FET used to reset the amplifier (which is in effect in parallel with the input JFET). The total leakage current of this term is about equal to 0.3 pA and produces noise equivalent to the thermal current noise of a 180 Gohm resistor. The {\sf PSD }of the second term of the current noise $\left( 7\times 10^{-35}f\right) $ is proportional to frequency and arises mostly from the {\sf dielectric noise} of the capacitance of physical capacitors, input capacitance of the JFET, and packaging of the amplifier. This term also includes the so called $1/f$ noise of the amplifier (i.e., that produced by the $1/f$ component of the amplifier input voltage noise $e_n$ once it is `differentiated' by the associated capacitance), although it is negligible compared to the dielectric noise in the present situation. The {\sf PSD} of the third term $\left( 1.65\times 10^{-38}f^2\right) $ describes the {\sf differentiated white noise} current that is produced mostly by the white component of amplifier input voltage noise $e_n$ once it is `differentiated' by the effective input capacitance at the headstage of the amplifier. This term varies as the square of the frequency and so dominates the total current noise at high frequencies. The effective input capacitance is around 12 pf and is made of the effectice input capacitance of the JFET input stage; the capacitors used for feedback and compensation; plus stray capacitance, including the capacitance with respect to infinity describing the charge necessary to create an electric field, even{\it \ in vacuo}. The noise of the signal applied to the compensation capacitance and the reset FET contribute a small component to this $f^2$ noise, as does the differentiator and subsequent stages of the amplifier. \section{Total Noise in Single Channel Measurements} In general, the total background noise in measurements does not all arise from the amplifier noise just described. It also has components coming \begin{enumerate} \item from the electrode (i.e., pipette) holder; \item from the part of the pipette in the air; \item from the part of the pipette bathed in salt solution; \item from the leakage of current in the gigaseal between the pipette and the biological membrane; and perhaps \item from some components associated with the lipid bilayer of the membrane and the access regions of the channel itself. \end{enumerate} \medskip\ The {\bf total current noise} {\sf PSD }can be described by an equation of the same form as eq.1 \begin{equation} S_\Sigma ^2\left( f\right) \simeq \ a_1+a_2f+a_3f^2\qquad {\rm units:amp}^2% {\rm /Hz} \label{2} \end{equation} The noise variance (units: amps$^2$) in a bandwidth from {\sc DC }to a frequency of $B$ (units: Hz) is then given by \begin{equation} I_\Sigma ^2\left( B\right) \simeq \ c_1a_1B+c_2\dfrac{a_2}2B^2+c_3\dfrac{a_3}% 3B^3\qquad {\rm units:amp}^2 \label{3} \end{equation} where the coefficients $c_j$ depend on the filter characteristic used to establish the bandwidth $B.$ The{\sc \ RMS} current noise (units: amps) in this bandwidth $B$ is simply the square root of the variance, namely \begin{equation} I_\Sigma \left( f\,\text{; {\sc RMS}}\right) \simeq \sqrt{\ c_1a_1B+c_2% \dfrac{a_2}2B^2+c_3\dfrac{a_3}3B^3}\qquad {\rm units:amps}\text{ }\left( \text{{\sc RMS}}\right) \label{4} \end{equation} If the filter establishing the bandwidth has a `brickwall' characteristic $% H\left( f\right) $, the coefficients $c_1=c_2=c_3=1$, that is to say,\medskip% \ \begin{equation} \stackrel{\text{{\sf Brickwall}}}{\overbrace{% \begin{array}{cccccc} H\left( f\right) & = & 1 & & \text{for }fB & \\ & & & & & \end{array} }} \label{5} \end{equation} For other types of low pass filters (with $-3$ dB bandwidths of $B$), these coefficients will exceed 1. For three commonly used filters with $B=5\times 10^4\ $Hz, these parameters $% c_{j\text{ }}$are\medskip\ \begin{eqnarray} && \begin{array}{llllll} \text{{\sf FILTER}{\bf \qquad }} & & \text{{\sf Characteristics}} & & \quad & \text{{\sf NOISE}} \\ & & & & & \end{array} \label{6} \\ &&{ \begin{array}{ccccccccc} & & \text{c}_1 & & \text{c}_2 & & \text{c}_3 & & \begin{array}{c} \text{{\sc RMS}} \\ \text{noise} \end{array} \\ & & & & & & & & \\ \text{Brickwall} & & 1 & & 1 & & 1 & & 45\text{ fA} \\ & & & & & & & & \\ \text{4 pole Butterworth} & & 1.02 & & 1.1 & & 1.2 & & 47 \\ & & & & & & & & \\ \text{8 pole Bessel} & & 1.04 & & 1.3 & & 1.9 & & 54 \\ & & & & & & & & \\ \text{Gaussian} & & 1.06 & & 1.4 & & 2.3 & & 57 \\ & & & & & & & & \end{array} } \end{eqnarray} The {\sc RMS }noise shown is the value of the amplifier alone (without anything else attached) calculated from equation \ref{1}. Because Gaussian filters are difficult to implement in analog circuitry, Bessel filters are usually used for typical time domain measurements. (Cauchy elliptic filters, which have much steeper cut-offs, are suitable for frequency domain measurements only). Typical parameters $a_j$ for good experiments (i.e. `good patches' of membrane and good gigaseals) are\medskip\ \begin{equation} \label{8} \begin{array}{l} \stackrel{ \begin{array}{c} \text{{\sf PARAMETERS of the NOISE}} \\ \text{amps}^2/\text{Hz} \\ \ \end{array} }{ \begin{array}[b]{cccccccc} & & \begin{array}{c} \text{amplifier } \\ \text{plus holder} \end{array} & & \begin{array}{c} 20 \text{ G}\Omega \\ \text{ seal} \end{array} & \begin{array}{c} 100 \text{ G}\Omega \\ \text{ seal} \end{array} & \begin{array}{c} \text{Quartz} \\ \text{Pipette} \end{array} & \begin{array}{c} \text{Glass} \\ \text{Pipette} \end{array} \\ & & & & & & & \\ a_1 & & 9\times 10^{-32}\text{ } & & 8\times 10^{-31} & 1.6\times 10^{-31} & & \\ & & & & & & & \\ a_2 & & 1.0\times 10^{-34} & & & & 1.5\times 10^{-35} & 3\times 10^{-34} \\ & & & & & & & \\ a_3 & & 2.2\times 10^{-38} & & & & 5\times 10^{-39} & 3\times 10^{-38} \\ & & & & & & & \end{array} } \\ \text{Empty locations represent zeros. The quartz pipette was coated } \\ \text{with a thick layer of Sylgard; the glass pipette was not.} \\ \\ \end{array} \end{equation} \section{Examples of Expected Noise} The tables and equations can be used to determine the total noise expected in a given experimental situation by adding together the appropriate terms.% \medskip\ (1) For example, if the experiment uses a headstage, electrode hold, quartz pipette, and has a 100 G$\Omega $ seal, the total noise $S_\Sigma ^2\left( f\right) $ is \[ \text{100 G}\Omega \text{ seal: }S_\Sigma ^2\left( f\right) \simeq \ 2.5\times 10^{-31}+1.15\times 10^{-35}f+2.7\times 10^{-38}f^2\text{;\qquad units: amp}^2\text{/Hz} \] {\sf Typical total values} for a quartz pipette with 100 G$\Omega $ seal is \fbox{87 fA {\sc RMS}}{\sc \ }in 5 kHz bgandwidth established by an 8 pole Bessel filter.\medskip\ (2) For example, if the experiment uses a a headstage, holder, glass pipette, with 20 G$\Omega $ seal gives \[ \text{20 G}\Omega \text{ seal: }S_\Sigma ^2\left( f\right) \simeq \ 8.9\times 10^{-31}+4\times 10^{-34}f+5.2\times 10^{-38}f^2\text{;\qquad units: amp}^2\text{/Hz} \] {\sf Typical total values }for a glass pipette with 20 G$\Omega $ seal is \fbox{173 fA {\sc RMS}}{\sc \ }in a 5 kHz bandwidth established by an 8 pole Bessel filter. \end{document}