Fluctuation Analysis in Neurobiology1

FLUCTUATION ANALYSIS IN NEUROBIOLOGY' By Louis J. DeFelice Department of Anatomy

Emory University Atlanta, Georgia

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Scope . ........................... B. Early Studies.. ............................................... 11. Methods ......... ....................... A. Conductance Fluctuations ....................... B. Voltage and Current Noise . . . . . . . . . . . . . . . . . . . C. General Relationships ................................................ 111. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Nerve A x o n . . . . . . . . . . ..................... B. Drug-Induced Noise ................................................. C. Sensory Systems ...... ........................ D. Other Preparations ... ..... IV. Summary ............................... ..... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 169 171 175 1176 76 I80 181

183 183 191 200 205 206 206

1. Introduction

Electrophysiology has progressed continuously from whole body surface recordings, through compound-action potentials, single-unit recordings, transmembrane-action potentials, and graded and postsynaptic transmembrane responses of ever decreasing size. The minute random fluctuations which occur spontaneously in the steady state of biological membranes reveal further information about the elementary events associated with electrical conduction. The measurement and interpretation of electrical noise from biological membranes may be thought of as the next level of awareness in electrophysiology. A. SCOPE

In this chapter I want to describe the basic concepts and tools required for understanding fluctuation analysis in membranes and to re-

' This work was supported in part by the National Library of Medicine (LM02505). 169

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view recent progress in this field. Several reviews are already available (Stevens, 1972; Verveen and DeFelice, 1974; Conti and Wanke, 1975; Neher and Stevens, 1977). There are two important areas of fluctuation analysis which will not be covered in this chapter. The mere characterization of membrane noise may be important as a data base for the description of the macroscopic random behavior of cells. For example, in very small cells, spontaneous firing or spontaneous transmitter release may be related to membrane noise. Membrane noise may also be implicated in the probabilistic response to a constant or regular stimulus, including selfoscillatory systems such as heart cells or pacemaker neurons. None of these questions will be dealt with here. The second area of fluctuation analysis to be omitted is the use of noise as an external stimulus to characterize a system. In linear systems, the response to any input is known once the response to a known source of noise has been measured. This is the transfer function of the system. The input noise must contain all frequencies of interest. It is convenient if each frequency component in the noise has the same average amplitude (white noise). The system to be characterized may range from a single membrane to a large group of cells. Measurement of the transfer function of a linear system implies an equivalent electrical network made of passive linear elements (the conventional R, L, and C of circuit theory). Noise is a convenient way to measure the impedance of any electrical equivalent circuit of a linear system. In nonlinear systems an analogous situation exists. There are two general ways to approach nonlinear systems. One assumes that for a particular state of the system, small perturbations from that state may be treated as linear. Other states are treated the same way, but the equivalent linear networks used in each state are not the same. This “piecewise” linear approximation is used to describe the impedance of nerve membranes. A second approach is the Wiener theory of nonlinear systems (Wiener, 1958). This method is not restricted to small perturbations. T h e full dynamic range of the nonlinear system may be probed with a noise stimulus. The output from the system to be described is expanded as a series of terms, each of which is defined by an operation between the input noise and the Wiener kernel for that term. T h e operational procedures are explained very clearly in Lee and Schetzen ( 1965) using crosscorrelation analysis. Once measured, the set of Wiener kernels completely defines the nonlinear system for an arbitrary input and is analogous to the transfer function for a linear system. Wiener kernels may also be measured using a fast Fourier transform algorithm (French and Butz, 1973). White noise analysis of nonlinear systems has become a useful tool in neurophysiology. This subject is discussed in detail by Marmarelis and Naka (1974) and Naka et al. (1974) and was the

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topic of a recent conference (McCann and Marmarelis, 1975). The Wiener theory has not been applied to excitable membranes, which are usually treated as quasilinear for small perturbations or by nonlinear differential equations for large perturbations. This chapter will be restricted to the study of indigenous membrane noise. Since the inherent spontaneous electrical fluctuations from biological membranes are necessarily the smallest which may occur, it has been generally assumed that linear system analysis may be applied. To what extent we may continue to ignore nonlinearities in spontaneous membrane noise remains to be seen. B. EARLYSTUDIES Fluctuation analysis in neurobiology has been applied to three general areas: excitable nerve membranes with voltage-controlled conductances, postsynaptic or extrajunctional membranes whose conductance is altered by chemical transmitters, and sensory receptors whose conductance is altered by light or mechanical stimulation. In many preparations, noise phenomena have been known for years. Some early observations are discussed here. In 1952, Brock et nl., in their study of electrical potentials from motoneurons, reported: “It has regularly been observed that, as the microelectrode entered into close contact with a neuron, the noise level increased several times . . , the peak voltages were as much as 0.5 mV.” This was apparently a chance observation, since the paper deals primarily with intracellular neuronal potentials. They regarded the extracellular noise as being actually produced by the surface membrane, and not by increased microelectrode resistance as the electrode approached the membrane. No further analysis of the effect was made. Similar observations are well known to present-day neurophysiologists, although there has been no systematic study of the phenomenon (Krnjevic, personal communication). Extracellular noise analysis is now regarded as a usefiil tool in analyzing transmitter-induced noise at cell junctions. This will be reviewed below. Brock et nl. (1952) concluded that the noise they observed might possibly represent normally occuring voltage fluctuations from a small area of neuronal membrane. Approximately ten years later, Verveen and Derksen began their pioneering noise studies on the node of Ranvier (see Verveen and Derksen, 1968, for an early review). These investigators measured the transmembrane voltage fluctuations directly, introduced the use of the power spectral density to describe membrane noise quantitatively, and started a new level of research in electrophysiology .

FIG. IA. An intracellular recording from the frog end-plate (left panel) and from a distance 2 mm away (right panel) in the same muscle fiber. The lower part of each panel shows the response to a nerve stimulus (the scales are 50 mV and 2 msec). The upper part of each panel shows the spontaneous activity (the scales are 3.6 mV and 47 msec). The spontaneous discharge is seen where the end-plate potential is large and where the spike originates. (From Fatt and Katz, 1952.)B. lntracellular recording from the frog end-plate. In each photograph the upper trace was recorded from a low-gain dc channel (10 mV

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In the same volume of the Journal of Physiology as the Brock et al. paper, another chance observation of a different type of noise was reported: Fatt and Katz (1952), in their microelectrode studies of the acetylcholine (ACh) receptors at the neuromuscular junction, state: “. . . that end-plates of resting muscle fibers are the seat of spontaneous electrical discharges which have the character of miniature end-plate potentials.” These observations were described initially as “biological noise” by Fatt and Katz (1950) in a brief note to Nature. It apparently took some time for these authors to become convinced that the now familiar miniature end-plate potentials were characteristic of normal nerve-muscle preparations. The miniature end-plate potentials are not caused by the insertion of the microelectrode since in some cases the spontaneous discharge could be detected at a distance of 1-1.5 mm from the end-plate. Fatt and Katz compared their internally recorded spontaneous discharges with those recorded externally. Figure 1A is reproduced from their paper. The potential drop recorded externally arises from a flow of current across the postsynaptic membrane and through the external fluid around the electrode tip. These local currents are more rapid than the miniature transmembrane potentials since the latter are recorded across the membrane capacitance. Local extracellular currents more nearly reflect membrane conductance changes than d o transmembrane voltages. This same distinction is made below when comparing voltage noise and current noise spectral densities. Fatt and Katz speculated that the miniature discharges might be attributed to molecular leakage of ACh from nerve endings. This was rejected, largely from negative results obtained by external application of ACh. Moderate concentrations of ACh depolarized the end-plates by a few millivolts with no appreciable effect on the miniature discharges. They conclude: “. . . if individual molecular collisions between ACh and the end-plate builds u p a steady depolarization then the molecular units of this depolarization must be much smaller than the recorded miniature end-plate potential.” scale). The lower trace was recorded simultaneously from a high-gain ac channel. The top row shows control experiments (no ACh). The bottom row shows the depolariTation and the increased membrane noise which are induced during ACh application. Two spontaneous miniatures are also seen. (From Katz and Miledi, 1972.) C. Current recorded through a patch of membrance approximately 1 +m’ from the extrasynaptic region of denervated frog muscle fibers. A downward deflection represents inward current. The membrane was exposed to SubCh and voltage clamped at -120 mV at 8°C. Histograms of the single events give a value of 3.4 pA for the elementary current pulses. Channel conductance was estimated to have a mean value of 22.4 pS. The mean open time of a single-channel current was 45 msec. (From Neher and Sakmann, 1976b.)

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Approximately 20 years after this discovery, Katz and Miledi (1970, 1972) reported the measurement of membrane noise which results from the molecular bombardment described above. The miniature end-plate potential is approximately 1/100 the size of the normal end-plate potential. T h e molecular shot effects are approximately 1/1000 the size of the miniatures. In fact, the shot effects are normally too small and too numerous to be recorded individually, and require statistical analysis of the spontaneous fluctuations resulting from a steady exposure of ACh molecules to the postsynaptic membrane. In a lecture to the Fifth International Congress of Biophysics (Copenhagen, August, 1975) B. Katz said that we have yet to see the elementary conductance change associated with a single agonistreceptor interaction. At that time, only the statistical analysis of a random sum of many events was available (Fig. 1B). Recently, Neher and Sakmann (1976b) have reported seeing individual elementary events which show many of the features derived from noise measurements (Fig. 1C). Observing both the noise and the elementary event from the same preparation enables one to compare the two methods and to decide between ambiguous models in the statistical method. These results will be reviewed in detail. The ACh-induced channel is probably the best characterized channel in biological membranes at the present time. AChinduced noise and the observation of individual elementary events have given indirect support to those cases where only noise data are available. In 1909, N. Campbell discussed a similar problem in relation to the measurement of the electrical charge carried by alpha-particles from spontaneously radioactive material. Initially, Campbell was concerned with a more accurate disclosure of von Schweidler’s theory of the discontinuous emission of rays by radioactive substances. The theory, Campbell (1909) said, lost much of its importance when Rutherford and Geiger measured the charge on an alpha-particle “by the more direct and probably essentially more accurate method of counting the number of particles one at a time.” Campbell felt, however, that a more thorough exposition of the theory of discontinuous processes could enable fluctuation experiments to “rival in accuracy” those obtained by individual measurement. He pointed out that radioactivity is not the only discontinuous process amenable to study by such a theory. His study of discontinuous phenomena resulted in Campbell’s theorem which relates the average effect of a random process to the average fluctuation from the mean effect. This phenomenon is often called shot noise. It is Campbell’s theorem which Katz and Miledi used so effectively to study the molecular bombardment of transmitter molecules on receptors at the neuromuscular junction.

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It is difficult to isolate the initial investigation of membrane noise in receptor physiology. An early application of fluctuation analysis was made by Dodge et al. (1968). In 1959, Hagins reported to the Biophysical Society (Hagins, 1959; Wagner and Hagins, 1959) that the predicted minimum primary receptor current should be the order of several thousand electronic charges per absorbed quantum. This estimate was based on the idea that the thermal fluctuations in the receptor membrane must set a lower limit to the size of the signal which results in response to light stimulation. In 1965, Hagins summarized his work in this field. The measurements were done on squid photoreceptors using extracellular field potential records. He reasoned that the photovoltage of the retina milst be made u p of the superposition of many absorbed photons and should show a shot noise effect. The light-induced fluctuations were found to be proportional to light intensity. By measuring the current flow in response to incident light, he was able to calculate the frequency content of the extracellularly recorded noise records. The spectrum of the noise was consistent with randomly spaced pulses whose mean duration was about 100 msec. He calculated that approximately 60,000 electronic charges flowed per incident photon, although values four times higher were also obtained. Hagins (1965) concluded that the ions carrying the current were Na and that the magnitude of the Na conductance produced by 1 quantum response (QR) to light is about 20 pmho. A previous result (Hagins et al., 1962) showed that the average depolarization produced by a single event is about 20 pV. Hagins (1965) concluded that “isolated molecular events in a cell membrane can, in principle, produce electrophysiologically significant currents” and that, as regards the quanta1 effects known to exist at that time at the frog neuromuscular junction, the “. . . miniature end-plate potential . . . yielded flows of 10:’ charges. Since the miniature responses are known to be caused by quanta of ACh containing thousands of molecules, however, it is possible that the single transmitter molecules acting at the postsynaptic end-plate membrane produce conductances of the order of that estimated here for the QRs. Thus, it may be that the QRs exemplify the unit of change in the membrane conductance on the molecular scale.” This hypothesis has been well documented in recent years, and is the main subject of this chapter. II. Methods

The basic goal of noise measurements is to be able to deduce something about the elementary events which give rise to the observable fluc-

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tuations. The three general areas to which fluctuation analysis has been applied are all concerned with conductance changes in a membrane, either in response to a voltage gradient, a transmitter molecule, or a sensory stimulus. Therefore, the elementary event will initially be considered as a change in membrane conductance. A. CONDUCTANCE FLUCTUATIONS

"++

Introduce the symbol " for the phrase "has the units of." As a first example, consider the conductance event g(t) corresponding to a rapidly opening membrane channel which then closes exponentially, thus: g(t) = ye-"e

where y -8- Sz-' (or mho) and 0 -0- sec. I will adopt the unit Siemans (S) for Sz-'. Consider identical changes g(t) occurring in parallel at uncorrelated times with an average frequency of occurrence of v -€+ sec-I. Since these events individually occur in one direction, they will sum to an average effect in the same direction. Since the events are occurring randomly, there will be instantaneous deviations from the average effect. However, these deviations also tend toward an average value. Let G(t) represent the instantaneous value at time t of the total conductance due to the random addition of all the elementary conductances g(t). Let ( ) represent a time average of the quantity in the brackets. Then Campbell's theorem states that

((G - ( G ) ) ' ) = v loffig2 (t) dt

Notice that (G) and ( ( G - (G))') are no longer functions of time. In words, these equations state that by simply knowing the size and shape of the elementary event and the mean frequency of their occurrence, their average effect, ( G ) , and the mean square deviation from the average, ( ( G - ( G ) ) ' ) may be calculated. The mean square deviation is called the variance. The ratio of the two quantities is independent of the mean frequency of the elementary events. The average effect and the mean square deviation from the average effect are measurable quantities from which properties of the elementary event g(t) may be deduced. There are two important considerations yet to be made. If the interval between elementary events is small (i.e., v large) compared to the shape of g(t), then the resulting amplitude fluctuations about the mean will be symmetric even though g(t) itself always occurs in one direction. Also, measuring a quantity proportional to the time integral over g(t)

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cannot, in general, reveal information about the temporal form of g(t), but only its size. Our present example illustrates this point. Evaluate the two integrals above for g(t) = then

,

v /om

ye-""dt =

v8y

y2e-2t/Odt = - v8y2

The ratio of the second to the first expression is simply b.Other shapes for g(t) would change the factor of $, but there is not a unique relationship between shape and this factor. The shape of the elementary event has been integrated out and replaced by a number. Conductance fluctuations may be analyzed in a way which does give temporal information. One method is called autocorrelation. The autocorrelation of the conductance fluctuations is a way of comparing the random noise signal in time with itself in an average way. The random signal is initially compared with itself by multiplication of every point of G(t) with itself and averaging the result. The autocorrelation operation shifts one copy of the signal with respect to the original by a fixed time and makes the same comparison by again multiplicating and averaging the two signals. The result is plotted as a function of the shift. This operation results in the autocorrelation function C ( T ) .Symbolically,

c(7)= (G(t)G(t + 7 ) ) Notice that C -€+ F.It is standard to use the symbol T for thecorrelation function; this prevents the use of T as the time constant of the elementary event, for which 8 is used above. When G(t) is the result of the random linear sum of the exponential g(t) described above, the predicted autocorrelation function is

This is the correlation function of the noise due to elementary events, each with the shape ye-"e. The correlation function decays with the same time course as the elementary event and is therefore a way to determine g(t). At T = 0: v8y~ C(0) = 2

+ (v8y)'

which is the sum of the mean square deviation of the conductance (the variance) and the square of the mean conductance caused by the random summation of the elementary events. Since the frequency of the events,

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v, is often a very large number, the first term, which varies directly as v, may be much smaller than the second term, which carries as v’. This fact has important practical significance. In order to amplify the measured noise signal to sufficient levels for analysis, the mean (dc) component usually has to be removed. If ac coupling is used, the shape of the entire correlation function will be distorted. The distortion must either be negligible or be taken into account. Examples may be found in DeFelice and Sokol (1976a,b). In the statistical literature (e.g., Bendat and Piersol, 19’71) it is common to use the symbol R ( T )to represent the total correlation function, including the dc term, and to reserve C ( T )for the total correlation function minus the dc term. This convenient notation is usually not followed by people working in membrane noise and has not been adopted here. I t is obvious that certain practical considerations have to be dealt with carefully when interpreting the shape of the measured correlation function. Assuming that all these considerations have been met, does the shape of the correlation function uniquely represent the shape of the elementary event? Unfortunately, it does not. As an example, consider the elementary event as a channel which fully opens or closes rapidly, but stays open or closed for random lengths of time. This will be called the random-switch model. In our previous example, each event g(t) was mathematically determined, but their time of occurrence was random, In the random switch, a n event is not defined by a deterministic formula. Nevertheless, each channel has average properties, such as the average duration of the open state, and the sum of N randomly opening and closing channels in parallel does define a noise signal with average properties. Let the open conductance of a single channel be y and the closed conductance be zero. Let the average open-chamel lifetime be 8 seconds. The mean frequency of closings is therefore v = 1/8. The autocorrelation function for noise due to the parallel sum of N random switches is given by

The meaning of is the same in both cases; y is the maximum openchannel conductance. However, 8 (and v) have a completely different interpretation. From the autocorrelation function alone’it is not possible to distinguish between the randomly occurring exponential and the random switch model of the elementary event. In Section 111 below, it will be seen that both models of the ACh-receptor interaction have been used to describe electrical noise at the neuromuscular junction. The measurement of isolated elementary events should help resolve conflicting views.

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The above examples illustrate some of the more important points about the use of noise analysis to study underlying events. In both of the examples, “random” was used to imply Poisson-distributed events and the results obtained depend on this assumption. Other models and assumptions are reviewed in Verveen and DeFelice (1974) and Holden and Rubio (1976). Membrane noise has been studied more often by using the spectral density analysis than by using autocorrelation analysis. Temporal information about the elementary events is given indirectly in the frequency domain. From an analysis point of view, there is little difference between methods. However, there may be some practical difference which favors one or the other in particular circumstances. To obtain a spectral density, the noise signal may be passed through a series of narrow band filters. The mean square output of each filter, divided by the filter bandwidth, is plotted against the center frequency of each filter. The result is an estimate of the spectral density. Theoretically, the power spectral density contains the same information as the autocorrelation function, and one may be transformed into the other. As an example, consider the two autocorrelation functions described above for the random sum of exponential events or the random-switch model. The general relationship between the autocorrelation function C ( r ) and the power spectral density S(f) is: S(j) = 4

or d r C(T)COS

For convenience, we have written w = 2 ~ f Notice . that) -@- H z (= sec-’) S2 . sec (= R-’/Hz). Substitution o f t h e expression for C ( r ) and S(,f) for the first model described above results in

+

The second term is an impulse at the origin ofS(f), corresponding to dc information in the noise signal. The usual way of measuring power spectra excludes dc components since each sharp filter has zero transmission at zero frequency. Assuming perfect removal of the dc component:

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This function is called a Lorentzian. The expression for random switch model is

S(f)

for the

The two spectral densities have the same shape but totally different interpretations. Let us restrict ourselves to the second (random-switch) model. The results interchange for W - + ud. The zero frequency limit of the power spectrum is

S(0) = Ny'd It is common to define d = 1/(2n-fc), wheref, is called the half-power or cutoff frequency. Therefore do =fFc. When w = l/d, i.e.,f =f r , then

S(fr) = Ny28 = BS(0) A convenient way to measure 8 is to find the frequency at which the power spectral density is down by half the zero frequency limit. B. VOLTAGE A ND CURRENT NOISE We have considered the elementary event to be a conductance change and the noise to be a fluctuation from the total mean membrane conductance: Conductance is a quantity which must be measured indirectly. It may actually represent a molecular conformational change in channels in the membrane. Such changes are expressed as an electrical event either as a voltage or a current. The two extremes of the measurement may be illustrated by a simple example. Consider a parallel RC circuit. The resistor is a source of noise due to the thermal agitation of the charge carriers in the resistor. The noise has a flat power spectrum. In this case, the voltage spectral density is given by

S ( f ) = 4kTR where K is Boltzmann's constant and T is the absolute temperature. Notice that S ( f ) VVHz. The capacitor is considered noiseless. T h e resistor noise is measured with a perfect voltage meter (noiseless, with infinite input resistance) across the capacitor of the RC circuit. Since high frequencies of the noise will be shorted to ground more easily than lower frequencies, the output spectrum will not be white, even though the only elementary noise source which is present, thermal noise in the resistor, is white. In fact, the output spectrum from the RC circuit is

+

s(f)

= 1

4kTR

+

(@o)Z

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where 8 = RC. Notice that white noise from an RC circuit has a voltage spectral density whose shape is equivalent to the conductance spectral density from the models considered above. In order to measure the current spectral density, the RC circuit is shorted with a perfect current meter (noiseless, with zero input resistance). T h e meter is always a better pathway for the noise current than the capacitor. T h e time constant 8 is effectively made zero. In this case, the spectral density of the current noise is: S(f) = 4 T / R

Notice that Scf, -0- A'/Hz and that the current spectral density has the same shape of the actual noise source (white) unspoiled by the capacitor. Current noise gives more direct information about the actual noise source than does voltage noise. If the impedance of the system is known, the two contain equivalent information. In excitable membranes, the equivalent impedance may be a rather complex LRC circuit which itself depends on the physiological stimulus. For this reason, it is usually desirable to measure current noise under voltage clamp. This was first done by Poussart (1969, 1971) in his studies of noise from nerve membranes. C. GENERAL RELATIONSHIPS I t is convenient to use the subscripts V, I , and G to denote voltage, current, and conductance. Capital letters will be reserved for a property of the entire membrane, i.e., the combined property of many ionic channels. Lower case letters will be used to represent individual channels. Voltage spectral densities and current spectral densities are related by the general expression:

whereZ is the membrane's impedance. In the example of the parallel RC circuit described above:

z=-1 + i w 0 and

IZI2 =

1

R'

+ (80)'

The relations given for the expected voltage and current noise spectral densities from this circuit satisfy the general expression. Current spectral densities are related to conductance spectral densities by the general expression:

s, = (V -E)'s(;

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where V is the potential across the membrane and E is the equilibrium potential associated with the particular current whose fluctuations are being measured. In our example of the random-switch model, the current spectral density due to the conductance fluctuations is given by:

s/ =

Ny'O(V - E )' 1 (60)'

+

The current fluctuations are measured about some mean current I . The mean current goes to zero when V = E . By fitting the above expression to measured current spectral densities under voltage clamp, the value of N y 2 may be estimated. If we assign the membrane property associated with an equilibrium potential to every channel in the membrane, then for each ionic pathway:

i = (V

-E ) y

where i is the current through the channel and y is the open-channel conductance. T h e average current through N channels in the membrane will be given by: I = iNl2

Therefore:

- E)I s/ = 2y6(V 1 + (we)? By fitting this expression to measured current spectral densities under voltage clamp, when the current I through the N channels is directly measured, the value of y may be estimated. Therefore, N and y are known independently. There are other equivalent expressions for current spectral densities which are based on these same ideas. For example, the expression for current spectral density for the random-switch model may also be written I'

s1

4e

= N 1 + (we)'

which leads directly to an estimate of N . If the area of the membrane is also well known, the channel density may be estimated. These examples cover the basic concepts of the noise measurement. Certain simplifying assumptions have been made which are not discussed in this chapter. Details may be found in the articles reviewed below. Some of the applications made to biological membranes have been quite direct. For example, at the neuromuscular junction the

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ACh-receptor interaction is well modeled by a random-switch conductance change. The induced channel is assumed to open and close rapidly, but to remain open for random times. There is an average open time 0; for the ACh-receptor model, one of the most important features of tlie average open time is that it is voltage dependent: 8 = e(V). This model and tlie results obtained from it will be discussed in detail below. In the squid giant axon, there are two populations of channels due to K and Na ionic pathways. The kinetics of-each pathway may be described by a model derived from the Hodgkin and Huxley (1952) picture of axon excitability (Hill and Chen, 1972; Stevens, 1972). For example, the K system is described by a sum of four Lorentians which correspond to then' kinetic scheme of the Hodgkin-Huxley model. However, the principles involved are similar to those outlined above, namely, obtaining information about the elementary conductance change associated with a particular process or ionic pathway from an analysis of membrane noise. 111. Results

In this section I will summarize experimental results and conclusions that have been derived from an analysis of membrane noise. I will concentrate on biological membranes, with occasional reference to model systems where it seems appropriate. A recent review of ionic channels in lipid bilayers is available (Ehrenstein and Lecar, 1977). The results will be organized around the type of preparation studied, rather than the class of observed noise or the nature of the conclusion. No attempt will be made at an historical presentation. Each section will build from a selected key article which has emphasized interpretation of the data in terms of a specific model. A. NERVEAXON 1. The Node of Ranvier

The most thorough study at the time of writing has been done on the node of the myelinated nerve fiber of the sciatic nerve of Rana esculenta (Conti et al., 1976a,b).T h e data represent current fluctuations measured under conditions of voltage clamp. An important feature of this study is the separation of the components of current fluctuations due to Na and K by using the poisons tetrodotoxin (TTX: to selectively inhibit the Na membrane conductance) and tetraethylammonium (TEA: selective for K). I t was shown, for example, that the normal current spectral density

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(no toxin added) minus the spectrum with TEA was equivalent to the spectrum with TTX minus the spectrum with both TEA and TTX. In principle, both of these conditions yield the current spectral density of the fluctuations in the K current alone. Na current fluctuations were measured using similar techniques. Conti et al. fitted their results from the Na system only to a theoretical model, although some K data are presented. The model they used is based on an interpretation of the quantitative description of membrane currents, similar to the Hodgkin-Huxley equations for the squid giant axon. It assumes independent Na channels with only two conducting states, open or closed. Each Na channel is controlled by three m gates and one h gate. Each gate may be likened to the random-switch model discussed in Section 11. The channel is multistate in that various conditions of the four gates may exist for the closed state. However, each channel has only one open state. The kinetics of the channels conform to the model of the macroscopic Na conductance. The macroscopic parameters necessary to fit the current spectral densities predicted by the model were measured from the same nodes used for the noise experiments. The steady-state Na current was measured simultaneously with the Na current fluctuations. These assumptions allow the calculation of the conductance of a single open Na channel. Referred to the resting potential, the value is 3/Nn

= 7.9

* 0.9 p s

The measurement of yNadid show slight voltage dependence, being higher at greater depolarizations. In the presence of 0.1 mM external nickel, the value of yNais about half that in normal nickel-free Ringer’s solution. The power spectra were fit by a theoretical sum of m-gate and h-gate fluctuations. The h-gate fluctuations contribute more to the low frequencies than the m-gate fluctuations. To some extent, these contributions to the total spectrum may be varied independently. Because of the greater confidence in the data at higher frequencies (due in part to the presence of a background noise, which is presumably not related to the conductance fluctuations being discussed here, and which varies inversely with frequency), the values of yNamay be regarded as having been calculated from the m-gate fluctuations. The value of rm obtained from macroscopic voltage clamp records was compared with the microscopic values of r, derived from current noise spectra. In both normal and in Ni-treated nodes, qualitative agreement was obtained. In principle, this comparison allows one to determine the exponent of the kinetic variable rn, i.e., the number of m gates which control each channel. NO

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firm conclusion was reached because of insufficient data, and m“ was used in the theoretical calculations. By selective interference with the inactivation process (h gates) using scorpion venom, anemonia toxin, or iodate, it was shown that the current noise spectra due to the m kinetics were little affected. The intensity of the spectra is enhanced because blocking the Na inactivation increases steady-state Na current. The suggestion is that h and m kinetics are separate, and that inactivation interacts with activation of Na channels in a simple on-off fashion. We note that the value of yNahas also been obtained from TTX binding experiments. From the number of TTX binding sites and the measurement of the gating currents, which yield the maximum charge movements involved in Na activation, one gets an upper bound on the possible number of channels. If the maximum Na conductance, GNa,is also known, an estimate of the conductance of a single channel is obtained. Almers and Levinson (1975) estimate yNaas 1 pS using what is probably a low value of GNa.This point is discussed by Conti et al. who derive from Almers and Levinson’s binding data a value of yNaof 8.6 pS The value of 8.6 pS by using more recent and higher values of agrees well with the value obtained from their current fluctuations experiments described above. Similar values have also been obtained for the squid giant axon Na channels (see below). The current values of yNa from nerve axon are much smaller than earlier estimates of 100 pS or greater (Hille, 1970). The new values of yNalead to the interesting conclusion that the higher value of GNain the node than in the squid giant axon is due to a greater density of Na channels in the node rather than a difference in the Na channel itself. There are about 10”Na channels per node, or about 2000 per square micrometer. I n the squid giant axon there are less than 500 Na channelslpm’. Nonner et al. (1975) have calculated that the maximum sodium conductance in the node of Ranvier is about 10 times that of the squid giant axon; thus, 15 nS/pm2 for the node and 1.2 nSlpm’ for the giant axon. They also report an insensitivity to external Na concentration of ions of the Na slope conductance for large depolarizations in giant axons. From their measurements of the displacement current in nodes they estimate about 5000 Na channels/ pm2. Using this value, the ratio of Na channel density in the node to the density in the giant axon agrees roughly with the ratio of 10 for the maximum conductance. Lastly, since the external solution used in squid axon experiments contains more external Na than the node, it is noteworthy that the two values of yNain the node and in the squid agree as well as they do. Conti et al. imply that the channels in the two cases are similar in nature. The channel is normally “filled” and may not be

cNa.

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further influenced by more external Na ions. Similar situations exist in simple model membrane systems. DeFelice and Michalides (1972) studied noise from collodian membranes. It was shown that the resistance of the membrane was well described by a model in which the internal ionic concentration of the membrane remained independent of the external bathing solution. On the other hand, fairly large channels induced in bimolecular lipid membranes by proteins added to the surrounding medium act to a certain extent like watery channels. That is, their conductance is influenced by external ionic concentrations (Lattore et al., 1972). A direct test for the nerve axon would be to measure current noise under conditions of varying external ionic concentrations. Electrical noise from the node of Ranvier of Rana temporaria was originally studied by Verveen and Derksen (see Verveen and Derksen, 1968, for an early review). One of the more recent studies is by van den Berg et al. (1975). By internal application of both Cs and TEA, these authors report the existence of a noise component due to the Na conduction system. Their data represent measured voltage fluctuations under current clamp. In order to block the K system, cut ends outside the central measurement node are exposed to Cs and TEA ions. The central node is bathed in normal or TTX-Ringer. In normal internal conditions with the K system intact, the variance of noise voltage increases monotonically with depolarizations from rest. In the Cs-TEA-treated nodes, the variance (between 20 and 1000 Hz) has a maximum at -38 mV. This bell-shaped voltage dependence of the variance disappears with TTX or in Na ion replacement experiments. The maximum noise in Cs-TEA is only about 1/10 the value of the normal voltage noise from the node at that same voltage. Voltage noise spectral densities were measured in the range 10-1000 Hz for membrane potentials between -70 and -20 mV. These spectra are fitted to a model which includes a simple on-off conductance component (random switch) due to the open-close kinetics of Na channels. The magnitude of this component also has a maximum value near -40 mV. When TTX is added, this spectral component disappears. Steady-state Na current is known from other studies to peak at about -40 mV. Van den Berg et al. interpret their results as representing the h process of the Na conduction system. They reasoned that the m process should have a much lower magnitude if any of the models they used were interpreted with the help of other voltage clamp data on a similar preparation (Hille, 1970). They also report that qt is within 30% of the measured time constant from the power spectra (at 20°C). Conti et al. ( 1976a,b) have interpreted their current noise data (at higher frequencies) primarily in terms of the m process. The major assumption in inter-

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preting voltage noise spectral densities in terms of conductance fluctuations is that the nodal impedance is not frequency dependent. The voltage clamp equations of Hille (1970) used by van den Berg et al. do predict a rather complicated frequency- and voltage-dependent smallsignal impedance (Clapham and DeFelice, 1976). At the present time, it is unclear whether or not the nodes used by van den Berg et al. (1975) displayed the resonance behavior expected from excitable axon membranes. Capacitance neutralization amplifiers were used to help eliminate the rather large capacitative component of the input impedance. Until recently, no voltage clamp data existed from the preparations used by van den Berg et al. Two reports are now available in which current noise has been measured (van den Berg, 1976: van den Berget al., 1977). Large differences between long-term and short-term (less than 600 msec) voltage clamp records are discussed which are pertinent to their earlier studies. Van den Berg et al. (1975) conclude from their experiments that the density of Na channels is about 20,00O/pm'. This is roughly 10 times the figure arrived at by Conti et al. (1976a). Each Na channel has a yNa'v 2 pS, or about f the value of Conti et nl. In an earlier paper, Siebenga et al. (1974) reported K noise from the node of Ranvier of Rana tempmaria from essentially the same preparation as that used by van den Berg et al. (1975). Voltage noise was measured between -70 and +30 mV under current clamp at room temperature. Na-replacement blocking agents and metabolic inhibitors had no measurable effect on the spectra. The observed noise was most affected by TEA, although in a rather complex manner. The spectra were interpreted in terms of a leakage conductance plus battery, which are in parallel with a similar branch for the K system. The magnitude of the observed noise was therefore dependent on the relative magnitude of the K and the leakage conductances. The random-switch channel model was used for the K system to interpret part of the observed spectra. N o voltage dependence of the time constant obtained from voltage spectral densities could be measured. The data were not fitted to the model quantitatively, so that no elementary properties of the K conductance system were deduced. However, in an earlier study by Siebenga et (11. (1973), values of yKbetween 10 and 40 pS are reported. Siebenga et al. investigated the impedance of their node preparation under current clamp up to 150 Hz and report that in this range the impedance is frequency independent. In any case, their voltage noise spectral data extend to beyond 1000 Hz. These impedance considerations are implicit in the interpretation made by van den Berg et al. (1975) in their interpretation of voltage noise from the Na system discussed above. Van den

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Berg et al. (1977) report a value of yK = 2.9 & 0.4 pS from their recent current noise experiments. N o voltage dependence of 3/k on voltage was found for V between -30 and +5 mV. Begenisich and Stevens (1975) have reported results of current noise measurements from the K system of the node of Rana. Their preliminary results use the variance of the noise to estimate the single channel conductance to be If the K channels have only two states, open and closed, then yk represents the conductance of an open state. In this case, the voltagedependent conductance of the membrane is not due to a dependence of yk on voltage. If the K channels have many conductance states, then the value of y k above represents some sort of average over these states; in this case, yK could be expected to be voltage dependent. In the Begenisich and Stevens study, the Na and K systems were selectively blocked by the use of TTX and TEA. Current noise was measured under voltage clamp between 4 and 2000 Hz. The steady-state K conductance was measured for each noise experiment. Fluctuations around the mean membrane current varied directly as GK and were effectively removed by TEA. Between V = -48 and +16 mV, at 15"C, the value of yKdid not significantly depend on voltage. One interpretation of their results is that K channels have only two conductance states which differ by about 4 pS. Begenisich and Stevens compare their results with an estimate of yKfrom the squid giant axon. yK in the squid is more than three times larger than in the node (see below). They make the interesting suggestion that this discrepancy might be expected since seawater has a higher conductance than frog Ringer's solution. This would be in contrast to the Na-channel values of yNa,which are very similar in the node and the giant axon in spite of large differences in external solutions. Yi-der Chen has emphasized the theoretical use of membrane noise to distinguish between various kinetic models of the elementary ionic conductance changes. Chen (1976, 1977) and Chen and Hill ( 1973) have written extensively on the use of the voltage dependence of current spectral densities to decide between alternative kinetic schemes for the elementary conductance change. Chen (1976) has stressed the use of the variance of the noise to test for the two conduction-state random-switch model as opposed to the multiconduction-state model. Begenisich and Stevens (1975) tentatively concluded that the relative insensitivity of yK to membrane voltage argues favorably for the random-switch model. Chen concludes that this is unlikely and that a careful examination of the voltage dependence of yK may indicate multiconducting states.

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It is worth stressing at this point that an ensemble of independently acting channels may well give rise to nonlinear electrical current-voltage relationships even though each open channel is a linear ohmic device. Thus, the macroscopic behavior of a membrane's conductance need not be reflected in every channel. An ensemble of two-state random switches, where the duration of one of the states is voltage dependent, can account for current-voltage relations similar to those seen in biological membranes. This has been shown rather directly in an electronic model by Mauro and Rossetto (1976). A similar phenomenon exists in lipid bilayers modified by a proteinaceous substance which forms channels in the membrane (Ehrenstein et al., 1970). 2. Squzd Giant Axon Membrane noise from squid p a n t axon has been studied by Conti, DeFelice, and Wanke (Wanke et al., 1974; DeFelice et al., 1975; Conti et al., 1975). Voltage noise, current noise, and impedance were measured from the same preparations. Large areas of the axon membrane were isolated by means of air gaps. Initially, internal air gaps were used to isolate areas of about 0.03 cm'. For technical reasons related to amplifier noise, small areas may have advantages. This is one advantage the node preparations have. Subsequent experiments in the giant axon showed that membrane areas up to 0.3 cm' could be used. These larger areas were isolated with external air gaps. This provided a fairly standard axial wire space-clamped preparation, not unlike those used for macroscopic kinetic measurements. Noise and impedance were studied in the frequency range 1-1000 Hz between temperatures of 2°C and 26°C and over a membrane voltage range of - 100 to -40 mV. Normal rest potential at 6°C is about -60 mV. Both TTX and TEA were used to separate K and Na components of conductance fluctuations. The initial observations emphasized the experimental relationship between S v, S,, and IZ '. Z is the small-signal impedance of the excitable nerve membrane. The passive RC component of the membrane is only part of the total impedance. The rest is due to the voltage-dependent, time-varying conductances in the membrane. I t was established that in well-isolated and well-controlled preparations, voltage noise spectra are dominated by the shape of the total membrane impedance. Similar behavior is expected for any excitable membrane, since the conductance kinetics responsible for excitation are reflected in the membrane smallsignal impedance as voltage-dependent components of the equivalent circuit. By comparing current noise spectra in normal seawater and in the presence of TEA and TTX, both K and Na components of conductance fluctuations were observed. The two ionic components were compared

I

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with a theoretical model which assumes that the channels conform to a Hodgkin-Huxley-like description for macroscopic behavior. The parameters for this description were obtained from unpublished data on the same preparations. This model was used to fit the shape of the current noise spectra. From the dependence of noise amplitude on voltage, both NK and NNa could be deduced. The elementary conductances of K and Na channels were obtained from the experimental values of the maximum K and Na conductances for these axons = 70 mS/cm’, GNa= 130 mS/cm2).The results are:

(c,

NK= 60Ipm‘ NNa = 330Ipm‘

YK

= 12 p s

yNa = 4 pS

There are fewer K channels than Na channels per unit area, but every K channel has a larger conductance. These data from the squid were used in the comparisons made above with the K and the Na system for the node. Electrical noise has also been studied from giant axons by Fishman, Poussart, and Moore (Fishman, 1973, 1975; Fishman et al., 1975a,b). In their experiments, small areas of membrane ( 10-4-10-5 cm’) were isolated on the external surface of squid giant axons using sucrose gap techniques. The “isolated” patch (10-100 m a ) is shunted by a relatively low resistance pathway (1 MR) through the sucrose solution in the Schwann cell interstitial space. Voltage and current noise spectral densities are similar because of the sucrose shunt pathways. The authors claim that the patch method improves channel noise resolution in comparison to large-area axial-wire measurements. Fishman et al. have not made a quantitative comparison between their data and a model. They report that the shapes of their K-noise spectra (Fishman et al., 1975a) are not compatible with either first-order reaction kinetics (single Lorentzian) or the probabilistic versions of conduction fluctuations based on the Hodgkin-Huxley model (Hill and Chen, 1972; Stevens, 1972). They conclude that the use of two-state conductance models in axons is questionable. New impedance data from this group (Fishman et al., 1977) purport to verify their interpretation of the observed spectral densities from the patch-clamped axon membrane. The noise analysis of kinetic systems and its application to channel kinetics has been thoroughly discussed by Chen (1977). The entire question rests on whether the measured spectra truly reflect conductance fluctuations. Since conductance fluctuations are only observed indirectly through current or voltage fluctuations, interpretation is often difficult. Fishman et al. (1976) have recently verified a noise component due to the Na system which had been suspected from some of their earlier data.

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B. DRUGINDUCED NOISE 1. The Neuromirscular Junction The most complete noise data, in terms of a correspondence between the macroscopic and the microscopic, exist for the neuromuscular junction. The interaction between the ACh molecule and the ACh receptor in the postsynaptic membrane has been studied from the natural release of' packaged ACh, through the effect of individual ACh molecules iontophoretically applied to the ultimate observation of single-receptor interactions. This study is summarized in Fig. 1 (see Section I , p. 172). The most quantitative study of ACh-induced noise has been done by Anderson and Stevens (1973). Their voltage clamp analysis of the conductance fluctuations caused by the interaction of the ACh molecule and its receptor has led to a fairly complete biophysical description of the elementary conductance change associated with that event. This analysis and its relationship to other work on the postjunctional conductance increase induced by ACh is nicely summarized in Stevens (1975). Anderson and Stevens used the sartorius nerve-muscle preparation of Rana pipiens. The excitation-contraction coupling was disrupted by either ethylene glycol or glycerol treatment. High Ca and Mg Ringer's solution was used to improve input resistance and resting potential. Two intracellular glass microelectrodes, about 50 P apart, were necessary for their voltage clamp circuit; one electrode provided the holding current needed to clamp the voltage at a desired level, the other electrode monitored the voltage. A third external electrode, positioned near the intracellular electrodes, was used for releasing ACh to the end-plate. Under voltage clamp, the spontaneous miniature end plate currents were used as criteria for recording. End-plates with low spontaneous rates of firing of the miniature end-plate currents (less than l/sec at 18°C) were used. The current noise which was induced between the spontaneous miniatures was studied. The relationship between current noise, voltage noise, and impedance o f the postsynaptic membrane was discussed. Voltage noise and impedance were also measured, but only current noise data under voltage clamp is shown and analyzed. The analysis made use of difference spectra derived by subtracting control data without applied ACh from spectra in the presence of a steady-state application of ACh. This was done at each level of membrane potential, even though in most cases extraneous noise was only a few percent of ACh-induced noise. The primary data are an increase in the mean membrane current, and the fluctuations about the mean (the variance) upon application of ACh to the end-plate (cf. Fig. 1B). Relative to the equilibrium potential

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for evoked end-plate currents, the variance and the mean are linearly related. That is, if one calculated conductance fluctuations from the observed current fluctuations, the variance and the mean are related by a constant. The current spectral density of the induced ACh noise were measured in the frequency range 1-300 Hz. The spectra were obtained over a voltage range of - 150 to 50mV. T h e shape of the spectra were well described by a single Lorentzian. Although the increase in membrane current and the fluctuations about the mean gradually diminish during constant ACh iontophoresis, the spectral shape (notably the half-power frequency) does not change. Spectra were taken from noise samples before appreciable desensitization during steady application of ACh had occurred. The ACh molecules are viewed as binding rapidly to the receptors in the postsynaptic membranes. This transmitter-receptor combination is fluctuating randomly between open and closed states. The elementary conductance change is considered to be a rectangular pulse with constant amplitude and random duration, i.e., equivalent to the random-switch model discussed earlier in Section 11. The rate-limiting step in the ACh-receptor interaction was considered to be the relaxation of an open channel. For low ACh concentrations, the model predicts the observed spectral density. The only arbitrary constant in the measured spectrum is 7, the single-channel conductance of the open ACh-receptor complex. The single-channel open conductance was found to be y = 32 pS

This value is sensibly independent of voltage. However, the rate-limiting relaxation is voltage dependent. On the average, a channel stays open longer at hyperpolarizations (the mean open time at - 150 mV is about 15 msec; at 50 mV it is about 4 msec). The mean open time (0) depends exponentially on membrane potential. The constants which fit this dependence are interpreted as the change in a dipole moment associated with the transition from an open to a closed configuration. Anderson and Stevens com pared their microscopic noise data with macroscopic voltage clamp data from the same preparation. The kinetics of ACh-receptor interaction may be determined from the spontaneous miniature end-plate currents. Miniature end-plate currents decay exponentially. The time constants depend exponentially on membrane potential in precisely the same way that the noise spectra time constants depend on membrane potential. This statement is also true of end-plate currents. It is important to realize that, while the macroscopic relaxation is truly exponential, the microscopic relaxation is viewed as a random switch with exponentially distributed durations of the open state.

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The estimates of y are also independent of temperature. However, increasing the temperature does decrease the mean open time of a channel. The rate constants depend exponentially on temperature. A Q l o of 2.77 near rest is reported, but this value may vary between 2.4 and 3.6 depending on membrane potential. The spectral density of the spontaneous conductance fluctuations may be derived from the decay of end-plate currents. Unfortunately, the formal relationship between the macroscopic and the microscopic is not unique. The two-state conductance model used by Anderson and Stevens could be replaced by a channel which opened instantaneously and then decreased its conductance exponentially. All of their data would be equally well described under this assumption, although the values of parameters used to fit the data would take on different meanings in the two models. Evidence for the random switch model was quite indirect when Anderson and Stevens first proposed it for the end-plate. The assumption of the two-state model was partly made on the basis of analogies with work on artificial membranes (e.g., Ehrenstein, et al., 1970), where individual conductance changes are seen as rectangular pulses. As we now know, individual conductance changes have been observed for the receptor interaction which are compatible with this model (Neher and Sakmann, 1976b; Fig. 1C). Assuming that there are l o xACh channels at the end-plate, the single channel conductance of 30 pS would imply a total possible conductance of 3 mS. The observed peak conductance of 4 pS implies that only 0.13% of the total number of channels are open at the peak conductance. This result is important for certain simplifying assumptions made in the statistical analysis. The biophysical model of the postjunctional conductance increase was discussed by Stevens (1975) in a Cold Spring Harbor Symposium. The main features of this model are given here. Either one or two ACh molecules bind the ACh-receptor. This induces a conformational change which allows Na, K, and C1 ions to flow through the postsynaptic membrane. There are two principal conformational states; the difference between the two corresponds to an elementary conductance change of about 25 pS (independent of whether one or two ACh molecules open the channel). The two conductance states are separated by an energy barrier and do not easily flip from one to the other. The binding of ACh reduces the energy barrier and allows the channel to open and close randomly. These two states correspond to a change in a dipole moment perpendicular to the membrane of about 50 D. The electric field-dipole interaction is the mechanism for altering the energy barrier between the two conducting states. Depending on the field strength, the channels

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normally stay open between 1 and 10 msec. The current which flows through an open channel corresponds to about 2 x lo7 iondsec, or i=3pA ACh binding is rapid compared to the conformation change and the channels behave independently. This fairly detailed picture of the ACh-receptor interaction does not rely entirely on noise analysis. Stevens first describes the postjunctional conductance increase (similar to Hagins’ “quanta1 response”) in response to brief ACh transients. However, the ACh transients are neither under experimental control nor are they directly measurable. Noise analysis, in the presence of a constant concentration of ACh, was considered advantageous although the two kinds of information converge. Also, information about the size of the elementary open-channel conductance y is only obtained from the noise analysis. The discovery of ACh noise and the first use of spectral analysis to help describe the underlying elementary events was made by Katz and Miledi (1972).The work is reminiscent of the suggestion made by Hagins et al. (1962) referred to in Section I, namely, that single transmitter molecules might act perceptibly at the postsynaptic junction. Katz and Miledi’s original photograph of the effect still remains the most illustrative, and is reproduced in Fig. 1B. Their analysis of ACh voltage noise is related to underlying conductance fluctuations, although to a lesser degree than transmembrane current noise. They were able to estimate the elementary conductance change to be the order of 100 pS. The current pulse associated with this change was approximately lo-’’ A lasting for about 1 msec. This produces minute voltage depolarizations the order of 0.3 pV which sum in random fashion to the effect shown in the baseline of Fig. 1B (lower panels). It was evident to Katz and Miledi that the time course of the voltage fluctuations were being filtered by the impedance of the postsynaptic membrane. By using focal extracellular recordings they, in effect, were able to measure current noise albeit indirectly. This allowed the more direct observations of conductance fluctuation kinetics. From these records they were able to show that carbachol produces a more brief current pulse than ACh and is therefore less effective. The comparative kinetics of drug-receptor interactions has been one of the major contributions of the technique of noise analysis. Katz and Miledi were led to a rather specific picture of the interaction. The macroscopic effects of prolonged and more stable depolarizations due to carbachol as compared with ACh are because carbachol is not hydrolysed by AChesterase. Noise analysis studies the effectiveness of the drug-receptor

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interaction and not local drug concentrations. Katz and Miledi also showed that at low temperature the duration of the elementary current pulse increases. This was also found to be true for chronically denervated preparations. Katz and Miledi (1974) have also studied the effect of procaine on the action of ACh at the neuromuscular junction. I t was suggested that the elementary potential change, calculated from the ratio of the variance of the noise to the mean depolarization, is substantially reduced by procaine. The spectral analysis of the ACh-induced voltage noise in the presence of procaine is more complex than a single relaxation spectrum. This is related to a similar complex shape observed in the end-plate current of procaine-treated muscle. Ruff (1976) concludes that anesthetic molecules reversibly block ion conduction while preventing gates from closing. One view of ionic channels which are induced in the end-plate region by drugs is that the channel, once open, has a conductance which is independent of the drug used to open it. For example, Neher and Sakmann (1976a) have studied the effect of carbachol and suberyldicholine (SubACh) at the end-plate (and in regions outside the end-plate in denervated preparations, see below) and conclude that the single-channel conductances for these two are not substantially different from the value of y obtained for ACh-induced channels. In contrast, Colquhoun et nl. (1975) report that four cholinomimetic agonists each have different values for y. One of the drugs studied by Colquhoun et nl. in the endplates of summer Rnna Pipiens cutaneous pectoris muscle was also SubACh. In Colquhoun et al., the mean single-channel conductances were calculated from the variance of current noise and the average membrane current under voltage clamp. N o spectral densities of the noise are shown. I t is stated, however, that ACh and SubACh have simple Lorentzian spectra while the other two agonists have more complicated spectra well fitted by the sum of two Lorentzians. The method used assumes that the current fluctuations represent random opening and closing of channels similar to the model of Anderson and Stevens (1973). ACh and SubACh were found to have single channel conductances which differ by about 20%. The values are 25.0 k 0.9 pS and 28.6 2 1 .O pS, respectively. SubACh has a longer mean open lifetime than ACh, a result also found by Neher and Sakman (1975) and Katz and Miledi (1973). However, two agonists known to cause contraction in frog muscle have substantially lower values of y than either ACh o r SubACh. These are 3-(m-hydroxphenyl) propyltrimethylammonium or HPTMA and 3-phenylpropyltrimethylammonium or PPTMA (see Colquhoun et nl., 1975, for details). The values of the single-channel conductances are 18.8 k 0.8 pS for HPTMA and 12.8 k 1.1 pS for PPTMA. The analysis assumes low-

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receptor occupancy. If this is not the case, the values given for y are lower limits for the single-channel conductances. The receptor of the normal cholinergic synapse has also been altered and changes in noise spectra observed. Landau and Ben-Haim (1974) studied voltage noise from frog sartorius muscle end-plates. T h e preparation was treated with dithiothreitol (DTT) which is known to decrease the postsynaptic sensitivity to ACh by altering the receptor molecule. Elementary voltage events were analyzed from the voltage noise data assuming a shot model similar to that used by Katz and Miledi (1972). The amplitude of the elementary pulses derived from the data were decreased by about half in DTT from normal values of 0.18 p V . They conclude that the elementary conductance event is probably reduced both in amplitude and duration. As in Katz and Miledi, extracellular noise was measured to help separate conductance fluctuations from membrane impedance. Their results show that changes in observed noise occur not only by modifying the agonist, but also by changing the structure of the receptor molecule. This problem has been studied in greater detail by Ben-Haim et al. (1975). Noise has also been studied during the decay phase of miniature end-plate currents prolonged by ethanol (Quastel and Linder, 1976). The amplitude of the noise corresponds to unit events similar to those produced by applying ACh directly to the end-plate.

2. Extrajunctional Receptors Chronically denervated frog muscle fibers have been studied recently under voltage clamp by Dreyer el al. (1976) and Neher and Sakmann (1976a). I t is well known that the entire extrasynaptic muscle membrane becomes sensitive to ACh following denervation. These extrajunctional receptors may have different macroscopic pharmacological properties than the normal junctional receptors. Neher and Sakman studied the effects of ACh, carbachol, and suberyldicholine (SubCh) on the denervated and normal cutaneous pectoris muscles from Rana esculenta and tempmaria. Experiments were done 40-70 days after denervation. T T X was used in the extracellular bath. The two-microelectrode clamping circuit was similar to that used by Anderson and Stevens. Autocorrelation functions were calculated from the current noise data and these were compared with simple exponential functions to extract one or more time constants from the process. The first point (7 = 0) of the autocorrelation function is the variance (see Section 11). By knowing the mean current (Z) associated with the current fluctuations, the elementary current L = C(O)/Z and the single-channel

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conductance y = d(V - E ) were calculated. T h e equilibrium potential ( E ) for the channels was determined separately to be near 0 mV. The conductance of a single extrajunctional ACh-induced channel is: y = 1 5 2 1.8 pS

This is somewhat lower than the junctional open-channel conductance measured both for these preparations (23 & 2 pS) and the preparation used by Anderson and Stevens (1973). The average open time ( 8 ) of an extrajunctional ACh channel is about 1 1 msec at -80 mV and 8°C. This is reported as being three to five times the average open time of a normal junctional channel. Under the same conditions, carhachol extrajunctional channels are open less on the average (4 msec) and SubACh are open longer (19 msec). Both carbachol- and SubACh-induced extrajunctional channels are open three to five times longer than junctional channels. The extrajunctional singlechannel conductances for carbachol and SubACh are similar to the ACh value given above. An interesting result obtained by Neher and Sakmann is that former end-plate regions of chronically denervated fibers often (1 1 out of 15 experiments) show two time constants in the autocorrelation functions of the drug-induced current noise. The faster time constant corresponds roughly to the normal fiber. The slow component was similar to that derived from the extrajunctional denervated experiments. They conclude that this could indicate two populations of channels at former end-plate regions of denervated fibers. Neher and Sakmann make the assumption in their analysis that only a small percentage of the channels are in the open state during their current noise measurements. This is fairly well documented in the druginduced noise in normal fibers but has not been unequivocally established for extrajunctional denervated preparations. If the assumption were not valid, and high-receptor occupancy occurs in the extrajunctional experiments, the lower values of y given for this region may have to be raised. It is more difficult to see how the larger values of 8 which were obtained for extrajunctional regions could be explained by highreceptor occupancy. A genuine difference in the kinetics of extrajunctional versus normal ionic channels appears well established. A new study of junctional and extrajunctional ACh receptors that compliments the noise analysis and deals with the question of cooperativity, has been made by Dreyer et al. (1977). ACh-induced current fluctuations from muscle cells in tissue culture has been studied by Sachs and Lecar (1973, 1977). Chick skeletal muscle

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cells were used. These cells possess ACh receptors spread over their entire surface. Current spectral densities were also obtained using carbachol as the agonist. A standard two-microelectrode voltage clamp was used on cells transformed to an approximately spherical shape by treatment with vinblastine. A useful analysis of noise sources from microelectrode voltage clamped cells is also given. Sachs and Lecar conclude that, in this preparation, y = 39 pS at 25°C

but increases with temperature with a Ql0of 1.7. The current spectral densities are well described by single Lorentzians. The relaxation time of the channel conductance is more markedly temperature dependent. The mean open duration is about 3 msec at 30°C. The Ql0is about 5 in the direction of increased 8 with decreased temperature. The relaxation time for carbachol is shorter than for ACh, in agreement with earlier work on the end-plate. 3. Glutamate Receptors Crawford and McBurney (1976) have studied voltage noise in the giant muscle fibers of the walking legs of the spider crab Maia squinado. These fibers may be cannulated and a large Ag-AgC1 wire inserted into the fiber. The fibers have a low membrane resistance (about 360 R * cm2). Relatively large areas (0.6 cm') gave preparations with an input resistance of about 600R. This is considerably lower than other preparations discussed so far. Extracellular noise was also recorded from these preparations using micropipettes. These experiments are similar to those of Katz and Miledi (1972) described above. The spontaneously occurring miniature excitatory junctional potentials (ejp's) have a mean amplitude of about 5 pV, which is about 1/100 the value of the frog end-plate spontaneous miniatures. This is due mainly to the difference in input resistance in the two fibers. O n applying L-glutamate to the external bath, transmembrane voltage recordings show a steady depolarization accompanied by an increase in voltage noise. The mean and the variance of this glutamate-induced noise were linearly related. The ratio of the noise variance to the mean depolarization was 2.2 X IO-'"V. A shot model was used which assumes numerous and random exponentially decaying elementary voltage events. The mean size of an event was therefore calculated to be about 5 x lo-'" V. Comparison of this value with the size of the ejp's led to the conclusion that each ejp contains 5000-10,000 elementary events. This is considerably larger than the number of events probably occurring during quanta1 release of ACh at the frog end-plate.

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The power spectrum of the intracellularly recorded voltage noise was fitted by a single Lorentzian with a time constant 8 o f 8 msec. This cutoff frequency (20 Hz) was thought to be mainly due to the membrane’s impedance. Extracellularly recorded noise, which more nearly reflects current fluctuations through the membrane, was fitted to a Lorentzian with a time constant of 1.4 msec. Similar voltage noise does not occur in the presence of y-aminobutyric acid (GABA) in place of the t-glutamate. The effect of GABA on crayfish muscle is known to be the induction of changes in chloride conductance. The driving force for chloride is small near the resting potential (about -50 mV) of the fibers. Crustacean muscle fibers are known to have excitatory junctions distributed over their entire surface. Focal electrode recording showed that only where depolarizing extracellular miniatures were recorded did L-glutamate produce extracellularly recorded noise. Thus “silent” regions of the muscle surface could be found. An estimate was made of the current associated with the glutamateinduced elementary conductance event. A lower limit of 0.5 pA was established. I t was concluded that the size and duration of the glutamate elementary event is very nearly equal to values obtained for ACh elementary events at the frog end-plate. Crawford and McBurney believe that the quanta1 current delays are probably limited by the closure of conductance channels and not by the relaxation of the concentration of transmitter molecules in the synaptic cleft, which they consider to be very fast. There is no known enzyme system at crustacean neuromuscular junctions for the rapid denaturing of the transmitter. Crawford and McBurney point out that the observation of transmitter noise is a useful criterion for the identification of a putative molecule as the transmitter substance. Dionne and R ~ i f f(1976) has made a similar point by using drug-induced noise to measure the equilibrium potential of a postsynaptic membrane current. A current noise-voltage clamp analysis of glutamate noise has been done by Anderson et ul. (1976). The metathoracic extensor tibiae of the adult locust Scistocercn greguria was used. This preparation is composed of fibers about 1500 pm long by 100 pm in diameter. In Cl-free medium, the input resistance is about 5-8 MQ. Chlorine-free bathing solutions were used to restrict the transmitter responses to the depolarizing type. Hyperpolarizing effects are due to the opening of CI channels and are avoided in the absence of the C1 ion. Spectral analysis of the current noise fluctuations plus a knowledge of the equilibrium potential for the glutamate-induced channels (0 mV) and the mean current through these channels were used to calculate the elementary open-channel conductance assuming a model similar to An-

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derson and Stevens (1973). y was found to be independent of membrane voltage and current, but dependent on temperature. The mean value of the open-state conductance of the glutamate induced channel was y = 231 pS

at 23°C

Recent measurements give a final mean value of 122 pS (S. G. CullCandy, personal communication). Measured values varied between 100 and 260 pS. y decreases as the temperature is lowered. For example: y = 39 pS

at 8.5"C

This corresponds to a Qloof 2.12. Recall that ACh-induced channels in the frog end-plate were found to have much smaller conductances and are insensitive to temperature. Temperature also prolongs the open state of the glutamate-induced channel from 1.59 msec at 23°C to 3.23 msec at 8.5"C. This corresponds to a Qlo of 1.45 which is lower than that observed for ACh channels. 8 also depends on membrane voltage. T h e duration of an open-single-channel decreases with hyperpolarization, from 2.12 msec at -50 mV to 1.56 msec at - 110 mV at 23°C. ACh channels have increased 8 with hyperpolarization. The glutamate analog, quisqualic acid, also produced a mean depolarization and increased current noise. In general, the time constants 8 were nearly twice as long in quisqualic noise compared with glutamate noise, but the single-channel conductances were almost identical. Macroscopically, quisqualic acid is known to be several times more effective on glutamate receptors than the natural agonist. Anderson et al. close with the interesting remark that, relative to the dipole model discussed above for the ACh-receptor complex in the frog end-plate (Stevens, 1975), the glutamate-receptor complex in invertebrate muscle has opposite polarity. C. SENSORY SYSTEMS The unifying concept in sensory mechanisms is that external forms of energy ultimately result in the conductance change in either the receptor cell membrane or in a related cell in the sensory pathway. One is usually concerned with average changes, or large transient changes such as the production of generator potentials, that result from the interaction of a physiological stimulus and the sensory system. Individual QRs have been reviewed by Fuortes and O'Bryan (1972). The steady-state fluctuations from some average change reflect the underlying elementary events in a way that is analogous to the internal forms of communication between cells already discussed.

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20 1

1 . Photoreceptors Simon et al. (1975) and Lamb and Simon (1976b) have investigated voltage noise from bipolar cells and cones and rods in the retina of the turtle Pseudemys scriptu eleguns. Cells in the isolated eyecup of the turtle were stimulated optically using either spots, annuli, or narrow slits. Transmembrane voltage was recorded with intracellular glass microelectrodes. The original observations showed that when the eyecup is exposed to a spot of light, the bipolar cells hyperpolarize and show a decrease in voltage noise. One value showed a change in noise variance V 2 for a corresponding hyperpolarization of from 1.53 to 0.045 X 6.5 mV. The ratio of the change in the variance between light and dark to the mean change in potential was about 0.24 mV. This was interpreted as the average size of the elementary voltage event in the bipolar cell membrane. Bipolar cells receive information from the cones in the form of a transmitter which is continuously released in the dark but is suppressed by light. Therefore, the hyperpolarization and decrease in noise in bipolar cells might be expected. The value 0.24 mV for the elementary event is approximately the size of a miniature end-plate potential at the frog neuromuscular junction. However, the current associated with this event in the bipolar cell is probably much smaller than at the neuromuscular junction since bipolar cells have much larger input resistances. Exposure of the turtle retina to an annulus of light showed either a slight depolarization or no effect in bipolar cells. The slight depolarization was also accompanied by a decrease in noise. Since horizontal cells are implicated in annulus stimulation, these data suggested to the authors that horizontal cells release a hyperpolarizing transmitter in the dark and that this release is reduced in the light. Cones and rods also show a similar decrease in noise when illuminated by bright steady light. However, the magnitude of the observed intrinsic dark noise varied widely between cells. The noisiest cones had a V 2while quiet cones could be as low as 0.0 1 X lo-’’ variance of 0.4 x V 2 .This difference is explained by Lamb and Simon as being due to the relative amounts of intercellular coupling between cones. The noisiest cones are those with narrow receptive fields. Quiet cones have relatively wide fields, indicating that they are well coupled to other cells. Their model assumes that similar random processes occur in all cones in the retina and that only the degree of intercellular coupling varies. Three different passive electrical models of coupling were developed. The essential problem to be solved is to calculate the sum of all the elementary noise sources in a coupled network of cells at some point in the network. It was required to measure intercellular coupling indepen-

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dently. This was done by measuring the spatial profile of the voltage response to light. For a slit stimulus, the peak flash response showed an exponential decay with distance. The length constant (A) of the exponential decay was used as a measure of intercellular coupling. The models investigated predicted a strong dependence of input resistance and voltage noise variance on A. For well-coupled cells, the mean square voltage noise varies inversely as A?, Through the model one is able to determine the magnitude of the voltage noise from a cell as though no coupling existed. These calculated values agreed with occasional noise records from truly isolated cells (small A) which are much noisier than well-coupled cells (large A). It was noted by the authors that noise from isolated cells might mistakenly be taken as an increase in microelectrode noise. Unlike bipolar cell noise, intrinsic cone and rod noise was an unexpected observation. Small spots of light 6 pm in diameter could cause a 10-fold decrease in voltage noise from an isolated cone. Lamb and Simon propose that photoreceptor noise results from the opening and closing of light-sensitive ionic channels. In darkness, isolated cones have a V' for an estimated 4 mV hyperchange in noise variance of 0.4 x polarization from the level with all channels open. This implies an elementary event of about 100 pV, or 40 simultaneous events to produce a 4 mV change. Taking the input resistance of the isolated cone to be 200 M a (from other work) they estimated the current associated with an elementary event to be:

i = 0.5 pA The driving force for this current was taken to be 40 mV, the same as the voltage required to reverse the macroscopic light response. Thus an estimate of the elementary conductance change would be about y = 10 ps

This is only approximate. Since the value is nearly the same as the values found for other single-channel conductances, the authors propose that photoreceptor noise results from the opening and closing of individual membrane ionic channels. The voltage spectral density of the noise from turtle cones was also investigated (Lamb and Simon, 1976a; Simon and Lamb, 1977). The spectra were fitted to the product of two Lorentzians with time constants O1 and 02. The shorter (0,) is associated with the receptor cell's capacitive time constant. The longer (0,) is associated with the temporal property of the elementary conductance events. Values were scattered. A typical

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example used 12 and 25 msec for 0, and 0 2 . 0, averaged 8 msec. 0, ranged from 17 to 60 msec with a mean value of 40 msec. Two interpretations were given for the temporal property of the elementary conductance events. 0, could correspond to the average time that molecules blocking the conductance channels are activated, or the average time a channel exists in the open and closed states. Simon and Lamb compare their noise data with specific kinetic models and photoisomerization data. They consider it likely that the measured elementary event (100 p V , 40 msec) is the same as that evoked by one photon. An event could either be the production of one blocking molecule or the closure of one ionic channel. Lastly they propose that the role of intercellular coupling in photoreceptors may be to improve the signal-to-noise ratio for diffuse stimuli. The induction of photoreceptor voltage noise in the dark in Drosophila has been studied by Minke et nl. (1975). A mutant was used which has a large initial photoresponse with increased stimulus but decays during stimulus to the level observed with dim stimuli. The decay of the receptor potential was investigated. By applying noise analysis, the authors show that at the saturated steady-state level the response to strong and weak stimuli is made up of elementary events of approximately the same size. Thus, the hypothesis (Dodge et nl., 1968) that light adaptation is due to reduced elementary events was considered to be improbable. 2. Mechanoreceptms Voltage noise has been studied by DeFelice and Alkon (1977) from the hair cells of the nudi-branch mollusc Hermimendn drassicornk in response to rotation of the statocyst. The statocyst is usually 70-100 p in diameter and is composed of 12-13 cells 40-50 pm in diameter and 5-10 pm thick. The cells form the periphery of the statocyst. A ciliated surface projects inward toward a cluster of 3-10 pm particles (statoconia) suspended in the cyst fluid. The receptor cells, unlike vertebrate hair cells, have axons. The axons leave directly from the cell bodies and join to form the static nerve (Alkon and Bak, 1973; Alkon, 1975). The statoconia are in constant motion. Rotation of tile statocyst may be used to promote an average movement of the statoconia toward some cells and away from others. Intercellular recordings may be made during rotation of the statocysts from cells either in front of or behind the vector of rotation. In the voltage noise studies (DeFelice and Alkon, 1977), spontaneous firing was suppressed by hyperpolarizing the soma under single-electrode current clamp. Cells with cut and uncut axons were

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used. Noise was measured under constant rotational acceleration u p to 1 g. Cells in front of the force vector depolarize and the variance of the voltage noise increases. Cells behind the force vector hyperpolarize and the noise decreases. A 1 mV change in mean membrane potential produces about a 100 p V change in the rms of the noise. The data are consistent with a shot model of the statoconia-hair cell interaction. At -80 mV, the elementary voltage event which results from this interaction is about 20 p V in a cell with an input resistance of 80 M a . This corresponds to a current of 2.5 x A. Although the exact driving force for the current is unknown, an estimate of the elementary conductance change associated with event was found: y=5ps The voltage spectral density shows that most of the power is located at low frequencies (below 50 Hz). This probably reflects the impedance of the membrane more than the temporal events associated with the conductance change. Correlation functions of the noise were fitted to a single exponential at low frequencies. A hyperpolarizing response of 6 mV increases 0 from 65 to 166 msec. This nearly 3-fold increase may be interpreted in part as an increase in average membrane resistance as the statoconia are drawn away from the ciliated surface. In 1974, Wiederhold (1974) suggested that observed fluctuations in mechanoreceptor membrane voltage result from the statoconia striking the cilia of the receptor cell, and directly measured the input resistance in response to physiological stimulation. Recently, a thorough study has been made of the response ofAplysia calafornica statocyst receptor cells by Gallin and Wiederhold (1977), who studied the intracellular electrical responses of the ciliated mechanoreceptor cells as a function of tilting. Tilting in the excitatory direction (statoconia move toward the cell) caused a depolarizing potential and large membrane potential fluctuations. The voltage noise was reduced or absent during opposite tilting. The voltage spectral density of the noise is reported as being primarily below 3 Hz. Removing the synaptic input of the receptor cells did not abolish the increase in noise due to tilting, and a decrease in input resistance is reported during the depolarizing response (see also Wiederhold, 1974). By Na replacement experiments it was argued that the basic mechanism underlying the receptor potential in this mechanoreceptor is an increase in Na permeability. Wiederhold (1977) has shown that the effect of tilting on membrane resistance is more complicated than previously thought. Slope resistance depends on the membrane potential and includes anomalous and delayed rectification. The results of his studies are critical to a complete interpretation of voltage noise from hair cells.

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3. Stretch Receptors Membrane noise from the slowly adapting stretch receptor neuron of the lobster has been studied by Sjolin and Grampp (1975). A singlemicroelectrode voltage clamp technique was used. Current noise was measured in the 2-400 Hz range. Depolarization from the resting potential (-60 mV) caused the current noise to increase. This increase was abolished with TTX or by replacement of the extracellular Na but was unaffected by TEA. For moderate depolarizations, the computed correlation functions of the current noise were approximated by single exponentials. Near the firing threshold, a sinusoidal component was observed indicating a subthreshold periodicity, perhaps from peripheral regions not under good voltage control. Time constants between 1.6 msec (near rest) and 7.9 msec (near threshold) were obtained from the exponential component of the noise. These time constants were considered to reflect Na inactivation and were compared to the nonadapting firing mode in the preparations. Sjolin and Grampp report that a distinct Na noise was not found in the rapidly adapting stretch receptor neuron of the lobster. In 1966, Firth (1966) investigated the role of membrane thermal and shot noise on the firing pattern of the crayfish stretch receptor. These noise sources are not large enough to explain the observed interval fluctuations. This simple system has an unusually small variation in the impulse train, much less than motoneurons where the fluctuation in the interval is due in large part to random synaptic input. The paper is of interest because of the analysis of the relation of membrane noise and its relation to firing patterns of the cell. A useful expression is given for the variance of the voltage noise expected from a long discrete cable composed of axoplasmic resistors and an RC equivalent for the membrane. D. OTHERPREPARATIONS Fluctuation analysis has been applied to many biological membranes besides those described above. ,For example, noise has been measured during ionic transport in the frog skin (Lindemann and Van Driessche, 1977; Van Driessche and Lindemann, 1976) and possibly in active transport systems (Segal, 1972, 1974). DeFelice and DeHaan (1975, 1977) have measured voltage noise from heart cells in tissue culture. Fluctuation analysis of membrane noise may be used to measure electrical coupling between cells without the injection of extraneous current. Periodicities were observed in the correlation functions derived from voltage noise from heart cell membranes whose spontaneous beating has been suppressed by TTX. This periodic component is related to the normal oscillatory properties of the heart cell membrane.

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IV. Summary

This chapter has been developed around the use of a relatively new technique in electrophysiology to study basic membrane phenomena in excitable nerve axons, muscle cells, and sensory receptors. The survey of literature has emphasized those studies which compare membrane noise with specific models of conductance change. Thus llf noise, which appeared frequently (and in some cases exclusively) in earlier studies, has not been discussed. The explanation of l/f noise remains an important but as yet unsolved problem of nerve physiology. The general impression one obtains at this stage of the literature is that certain parameters, such as single-channel conductance or the density of channels, may be reliably obtained from noise data. This can apparently be done even when the correct kinetic model of the elementary conductance change is not known. Where individual channel conductances have been observed directly, good agreement exists between these events and those derived from noise measurements. The single event remains high on the list of experimental objectives and may one day be observed in the excitable nerve membrane. The use of noise measurements to distinguish between kinetic models of conductance change appears less well determined at the present time, although this is potentially the most important application of fluctuation analysis. The information we have obtained from fluctuation analysis has been useful in discussing possible physical mechanisms of conductance change in membranes. This information must ultimately conform to the molecular models of ionic conduction in biological membranes that are now emerging. REFERENCES Alkon, D. L. (1975).J . Gen. Physiol. 66, 507-530. Alkon, D. L., and Bak, A. (1973).J. Gen. Physiol. 61, 619-637. Almers, W., and Levinson, S. (1975).J. Physiol. (London) 247,483-509. Anderson, C. R., and Stevens, C. F. (1973).J. Physiol. (London) 235, 655-692. Anderson, C. R., Cull-Candy, S. G., and Miledi, R. (1976).Nnture (London) 261, 151-153. Begenisich, T., and Stevens, C. F. (1975). Biophys. J. 15, 843-846. Ben-Haim, D., Dreyer, F., and Peper, K. (1975). Pjiregws Arch. 355, 19-26. Bendat, J. S., and Piersol, A. G. (1971). “Random Data: Analysis and Measurement Procedures.’’ Wiley (Interscience), New York. Brock, L. G., Coombs, J. S.,and Eccles, J. C. (1952).J. Physiol. (London) 117, 431-460. Campbell, N. (1909). Proc. Cambridge Phil. SOC. 15, 117-136. Chen, Y.-D. (1976). Bio,bhy~.J. 16, 965-971, Chen, Y.-D. (1977).J. Chtm. Phys. (in press). Chen, Y.-D., and Hill, T. L. (1973). Biophys. J . 13, 1276. Clapham, D. E., and DeFelice, L. J. (1976). Pjhregers Arch. 366, 273-276.

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