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Impedance Measurement of the Electrical Structure of Skeletal Muscle

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Abstract

The sections in this article are:

1 Methods and Techniques
1.1 Microelectrode Techniques
1.2 Analysis of Sinusoidal Data
1.3 Impedance Analysis With Fourier Techniques
1.4 Instrumentation Noise
1.5 Manipulation of Impedance Data
1.6 Theory and Curve Fitting
1.7 Electrical Models of the T System
1.8 Necessity for Morphometry
2 Results of Impedance Measurements
2.1 Impedance Measurements of Normal Frog Fibers
2.2 Other Preparations of Skeletal Muscle
2.3 Impedance Measurements of Muscle Fibers in Various Conditions
2.4 Comparison With Other Results
3 Discussion
3.1 Impedance Measurements of Nonlinearities
3.2 Other Methods
Figure 1. Figure 1.

A: simple circuit with 2 different values of series resistance. Other panels show response of the circuit plotted in different ways. B: transient response to a step function of current (1 μA) applied at time 0. Dashed line, response with 100‐Ω series resistance, is a vertical displacement of response with no series resistance (solid line). Therefore these responses are hard to tell apart and are hard to use to measure the series resistance. C: magnitude of impedance of circuit measured with sinusoidal currents of the frequency shown on the abscissa. The effect of a series resistance is upward displacement of the curve without change in shape. D: plot of imaginary part of the impedance vs. real part of the impedance. Although frequency is not an explicit variable, variation of frequency and the subsequent variation in the real and imaginary parts of the impedance produce the curve. Again the effect of a series resistance is a simple translation of the curve without change in shape. E: phase angle between sinusoidally applied current and voltage. Effects of series resistance are substantial, making it easy to measure series resistance.

Figure 2. Figure 2.

A: lumped approximation to properties of 1 cm2 of surface membrane and associated T system of frog skeletal muscle with 2 different values of series resistance indicated. B: transient response of lumped circuit to a step function of applied current. Tiny difference between the 2 curves makes measurement of series resistance difficult from transient responses. C: magnitude of the response to sinusoidal currents of different frequencies. Effect of series resistance is small, making measurement difficult. D: a plot of imaginary vs. real part of the impedance. Effects of series resistance and of the 1‐μF capacitor are hard to see in this plot. E: phase angle between sinusoidally applied current and voltage. Effects of series resistance are substantial, making it easy to measure series resistance.

Figure 3. Figure 3.

Three sinusoids show effects of digital sampling. Samples taken at 8/s (filled circles) from a sinusoid of 4‐Hz frequency are the same as samples taken from a sinusoid of 0 Hz (i.e., DC). Thus samples taken at this rate cannot distinguish between the 2 sinusoids—the sinusoids are said to be aliases of one another. Similarly samples taken at 5/s (open circles) from a sinusoid of 1‐Hz frequency cannot be distinguished from samples from a sinusoid of 8 Hz; they also are aliases. Aliasing is a direct consequence of sampling. Once digital samples are taken, aliases cannot be distinguished. Precautions can be taken before sampling to minimize problems of aliasing.

Figure 4. Figure 4.

Idealized filter response shown in a log‐log plot. Ordinate is the gain in dB. Frequency at which the gain is −3 dB is shown as Bw; frequency at which the gain is down −A dB is shown as FA = kBw. Filters recommended for use in impedance measurements have more complicated responses, with ripple in the pass band (DC to Bw), nonlinear dependence on frequency, and a finite amount of gain in any range of frequencies. All these features of practical filters are introduced to allow a very steep dependence of gain on frequency, i.e., as small a value of k as possible.

Figure 5. Figure 5.

Optimal relation between sampling rate, bandwidth, and folding frequency. All signals above Bw contain aliased energy. Adjustment of the folding frequency to halfway between Bw and FA allows closest approach of FA to Bw without introduction of aliased energy below Bw. This adjustment maximizes bandwidth and fraction of usable frequency points. Linear plot is used for convenience. Filter is shown with a limiting gain at high frequencies to be more realistic.

Figure 6. Figure 6.

Instrumentation noise in the output signal. Transfer function H(f) might be impedance of a muscle fiber. Input x(t) is applied current. Noise‐free measurements of x(t) are assumed to be available. Pure output y(t) is voltage that would be measured in the absence of instrumentation noise. Noise n(t) is noise introduced by voltage‐recording amplifier and associated electronic devices. Noisy output ŷ(t) is the sum of pure output and noise; ŷ(t) is the best available estimate of membrane voltage. Transfer function should be estimated from estimates Ĝxy(f) and Ĝxx(f) of cross‐power and power spectra, respectively. Those estimates should be the mean of cross power and power recorded from k blocks of data, as indicated. Cross power and power of each block of data are determined from the Fourier transforms X(f) and Y(f) of signals x(t) and y(t), respectively. *Complex conjugate. Other estimates of the transfer function (e.g., made by dividing the power spectra and then averaging) lead to incorrect results, as discussed by Bendat and Piersol .



Figure 1.

A: simple circuit with 2 different values of series resistance. Other panels show response of the circuit plotted in different ways. B: transient response to a step function of current (1 μA) applied at time 0. Dashed line, response with 100‐Ω series resistance, is a vertical displacement of response with no series resistance (solid line). Therefore these responses are hard to tell apart and are hard to use to measure the series resistance. C: magnitude of impedance of circuit measured with sinusoidal currents of the frequency shown on the abscissa. The effect of a series resistance is upward displacement of the curve without change in shape. D: plot of imaginary part of the impedance vs. real part of the impedance. Although frequency is not an explicit variable, variation of frequency and the subsequent variation in the real and imaginary parts of the impedance produce the curve. Again the effect of a series resistance is a simple translation of the curve without change in shape. E: phase angle between sinusoidally applied current and voltage. Effects of series resistance are substantial, making it easy to measure series resistance.



Figure 2.

A: lumped approximation to properties of 1 cm2 of surface membrane and associated T system of frog skeletal muscle with 2 different values of series resistance indicated. B: transient response of lumped circuit to a step function of applied current. Tiny difference between the 2 curves makes measurement of series resistance difficult from transient responses. C: magnitude of the response to sinusoidal currents of different frequencies. Effect of series resistance is small, making measurement difficult. D: a plot of imaginary vs. real part of the impedance. Effects of series resistance and of the 1‐μF capacitor are hard to see in this plot. E: phase angle between sinusoidally applied current and voltage. Effects of series resistance are substantial, making it easy to measure series resistance.



Figure 3.

Three sinusoids show effects of digital sampling. Samples taken at 8/s (filled circles) from a sinusoid of 4‐Hz frequency are the same as samples taken from a sinusoid of 0 Hz (i.e., DC). Thus samples taken at this rate cannot distinguish between the 2 sinusoids—the sinusoids are said to be aliases of one another. Similarly samples taken at 5/s (open circles) from a sinusoid of 1‐Hz frequency cannot be distinguished from samples from a sinusoid of 8 Hz; they also are aliases. Aliasing is a direct consequence of sampling. Once digital samples are taken, aliases cannot be distinguished. Precautions can be taken before sampling to minimize problems of aliasing.



Figure 4.

Idealized filter response shown in a log‐log plot. Ordinate is the gain in dB. Frequency at which the gain is −3 dB is shown as Bw; frequency at which the gain is down −A dB is shown as FA = kBw. Filters recommended for use in impedance measurements have more complicated responses, with ripple in the pass band (DC to Bw), nonlinear dependence on frequency, and a finite amount of gain in any range of frequencies. All these features of practical filters are introduced to allow a very steep dependence of gain on frequency, i.e., as small a value of k as possible.



Figure 5.

Optimal relation between sampling rate, bandwidth, and folding frequency. All signals above Bw contain aliased energy. Adjustment of the folding frequency to halfway between Bw and FA allows closest approach of FA to Bw without introduction of aliased energy below Bw. This adjustment maximizes bandwidth and fraction of usable frequency points. Linear plot is used for convenience. Filter is shown with a limiting gain at high frequencies to be more realistic.



Figure 6.

Instrumentation noise in the output signal. Transfer function H(f) might be impedance of a muscle fiber. Input x(t) is applied current. Noise‐free measurements of x(t) are assumed to be available. Pure output y(t) is voltage that would be measured in the absence of instrumentation noise. Noise n(t) is noise introduced by voltage‐recording amplifier and associated electronic devices. Noisy output ŷ(t) is the sum of pure output and noise; ŷ(t) is the best available estimate of membrane voltage. Transfer function should be estimated from estimates Ĝxy(f) and Ĝxx(f) of cross‐power and power spectra, respectively. Those estimates should be the mean of cross power and power recorded from k blocks of data, as indicated. Cross power and power of each block of data are determined from the Fourier transforms X(f) and Y(f) of signals x(t) and y(t), respectively. *Complex conjugate. Other estimates of the transfer function (e.g., made by dividing the power spectra and then averaging) lead to incorrect results, as discussed by Bendat and Piersol .

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Robert S. Eisenberg. Impedance Measurement of the Electrical Structure of Skeletal Muscle. Compr Physiol 2011, Supplement 27: Handbook of Physiology, Skeletal Muscle: 301-323. First published in print 1983. doi: 10.1002/cphy.cp100111