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\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 }f<B &  \\ 
&  &  &  &  & \Longleftrightarrow c_1=c_2=c_3=1 \\ 
H\left( f\right)  & = & 0 &  & \text{for }f>B &  \\ 
&  &  &  &  & 
\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  }} & \qquad \quad  & \text{{\sf %
Characteristics}} &  & \quad \  & \text{{\sf NOISE}} \\ 
&  &  &  &  & 
\end{array}
\nonumber  \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}
\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. } \\ 
\text{The quartz pipette was coated with a thick layer of Sylgard;} \\ 
\text{ the glass pipette was  coated with a moderate layer.} \\ 
\\ 
\end{array}
\label{8}
\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}
