Current Noise in Recordings of Currents through Single Channels

Richard A. Levis

Department of Molecular Biophysics & Physiology Rush Medical College 1750 W. Harrison Street Chicago IL 60612 USA (312)-942-6454 FAX: (312)-942-8711

Slightly modified by Bob Eisenberg

Department of Molecular Biophysics & Physiology Rush Medical College 1750 W. Harrison Street Chicago IL 60612 USA (312)-942-6467 FAX: (312)-942-8711 Email: bob@aix550.phys.rpslmc.edu

February 27, 1996 1. Noise of Amplifier

The power spectral density PSD of the current noise of the amplifier S_{hs}(f) used to record single channel currents is well described by

S_{hs}^{2} (f) ≃\overbrace{9 × 10^{−32}}^{white noise} + \overbrace{7 × 10^{−35}f}^{dielectric noise} + \overbrace{1.65 × 10^{−38}f^{2}}^{differentiated white noise}  units : amp^{2}/Hz (1.1)

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 (9×10^{−32}) describes the white noise current arising (mostly) from the shot noise i_{n} 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 PSD of the second term of the current noise (7 × 10^{−35}f) is proportional to frequency and arises mostly from the 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 PSD of the third term (1.65 × 10^{−38}f^{2}) describes the 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 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.

2. 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

1. from the electrode (i.e., pipette) holder;

2. from the part of the pipette in the air;

3. from the part of the pipette bathed in salt solution;

4. from the leakage of current in the gigaseal between the pipette and the biological membrane; and perhaps

5. from some components associated with the lipid bilayer of the membrane and the access regions of the channel itself.

The total current noise PSD can be described by an equation of the same form as eq.1

S_{Σ}^{2} (f) ≃ a_{1} + a_{2}f + a_{3}f^{2} units : amp^{2}/Hz  (2.1)

The noise variance (units: amps^{2}) in a bandwidth from DC to a frequency of B (units: Hz) is then given by

I_{Σ}^{2} (B) ≃ c_{1}a_{1}B + c_{2} \frac{a_{2}}{2} B^{2} + c_{3} \frac{a_{3}}{3} B^{3} units : amp^{2}  (2.2)

where the coefficients c_{j} depend on the filter characteristic used to establish the bandwidth B.

The RMS current noise (units: amps) in this bandwidth B is simply the square root of the variance, namely

I_{Σ} (f ; RMS) ≃ \sqrt{c_{1}a_{1}B + c_{2} \frac{a_{2}}{2} B^{2} + c_{3} \frac{a_{3}}{3} B^{3}} units : amps (RMS) (2.3)

If the filter establishing the bandwidth has a ‘brickwall’ characteristic H (f), the coefficients c_{1} = c_{2} = c_{3} = 1, that is to say,

Brickwall

\overbrace{\begin{aligned}H (f) & = & 1 & for f < B \\ & & & & ⟺ c_{1} = c_{2} = c_{3} = 1 \\ H (f) & = & 0 & for f > B \end{aligned}} (2.4)

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 × 10^{4} Hz, these parameters c_{j} are

FILTER  Characteristics  NOISE

c_{1}  c_{2}  c_{3}  RMS noise
Brickwall  1  1  1  45 fA
4 pole Butterworth 1.02 1.1 1.2  47
8 pole Bessel  1.04 1.3 1.9  54
Gaussian  1.06 1.4 2.3  57

(2.5)

The RMS noise shown is the value of the amplifier alone (without anything else attached) calculated from equation 1.1. 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 PARAMETERS of the NOISE amps^{2}/Hz

amplifier plus holder  {20 GΩ \atop seal}  {100 GΩ \atop seal}  Quartz Pipette  Glass Pipette
a_{1}  9 × 10^{−32}  8 × 10^{−31} 1.6 × 10^{−31}
a_{2}  1.0 × 10^{−34}  1.5 × 10^{−35} 3 × 10^{−34}
a_{3}  2.2 × 10^{−38}  5 × 10^{−39} 3 × 10^{−38}

Empty locations represent zeros.

The quartz pipette was coated with a thick layer of Sylgard; the glass pipette was coated with a moderate layer.

(2.6)

3. 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.

(1) For example, if the experiment uses a headstage, electrode hold, quartz pipette, and has a 100 GΩ seal, the total noise S_{Σ}^{2} (f) is

100 GΩ seal: S_{Σ}^{2} (f) ≃ 2.5×10^{−31}+1.15×10^{−35}f+2.7×10^{−38}f^{2}; units: amp^{2}/Hz

Typical total values for a quartz pipette with 100 GΩ seal is 87 fA RMS in 5 kHz bgandwidth established by an 8 pole Bessel filter.

(2) For example, if the experiment uses a a headstage, holder, glass pipette, with 20 GΩ seal gives

20 GΩ seal: S_{Σ}^{2} (f) ≃ 8.9×10^{−31}+4×10^{−34}f +5.2×10^{−38}f^{2}; units: amp^{2}/Hz

Typical total values for a glass pipette with 20 GΩ seal is 173 fA RMS in a 5 kHz bandwidth established by an 8 pole Bessel filter.