quantification of membrane properties of trigeminal root...

JOURNALOF NEUROPHYSIOLOGY Vol. 55, No. 5, May 1986. Printed in U.S.A.

Quantification of Membrane Properties of Trigeminal Root Ganglion Neurons in Guinea Pigs

E. PUIL, B. GIMBARZEVSKY, AND R. M. MIURA

Department of Pharmacology and Therapeutics, FucuIty of Medicine, The University of British Columbia, Vancouver, British Columbia V6T I WS, Cunudu

SUMMARY AND CONCLUSIONS

I. Passive and active (voltage- and time- dependent) membrane properties of trigeminal root ganglion neurons of decerebrate guinea pigs have been determined using frequency- domain analyses of small-amplitude perturbations of membrane voltage. The complex impedance functions of trigeminal ganglion neurons were computed from the ratios of the fast Fourier transforms of the intracellularly recorded voltage response from the neuron and of the input current, which had a defined oscillatory waveform. The impedance magnitude functions and corresponding impedance locus diagrams were fitted with various membrane models such that the passive and active properties were quantified.

2. The complex impedances of less than one-quarter of the 105 neurons which were investigated extensively could be described by the complex impedance function for a simple RC-electrical circuit. In such neurons, the voltage responses to constant-current pulses, using conventional bridge-balance techniques, could be fitted with single exponential curves, also suggesting passive membrane behavior.

3. A nonlinear least-squares fit of the complex impedance function for the simple model to the experimentally observed complex impedance yielded estimates of the resistance of the electrode, and of input capacitance (range, 56 to 490 pF) and input resistance (range, 0.8 to 30 MQ) of the neurons.

4. The majority of trigeminal ganglion neurons were characterized by a resonance in the 50- to 250-Hz bandwidth of their imped-

ante magnitude functions. Such neurons when injected with “large” hyperpolarizing current pulses using bridge-balance techniques showed membrane voltage responses that “sagged” (time-dependent rectification). Also, repetitive firing commonly occurred with depolarizing current pulses; this characteristic of neurons with resonance in their impedance magnitude functions was not observed in neurons with “purely” passive membrane behavior.

5. A nonlinear least-squares fit of a five- parameter impedance fitting function based on a membrane model to the impedance locus diagram of a neuron with resonance yielded estimates of its membrane properties: input capacitance, the time-invariant part of the conductance, the conductance activated by the small oscillatory input current, and the relaxation time constant for this conductance. The ranges of the estimates for input capacitance and input resistance were comparable to the ranges of corresponding properties derived for neurons exhibiting “purely” passive behavior. The membrane resistance which apparently was voltage- and time-dependent usually ranged between 1.6 and 300 MQ with a relaxation time constant of 0.7 to 8,O ms.

6. Membrane models given by 1) an equivalent electrical circuit consisting of a parallel RC circuit in parallel with a series resistance- inductor branch; 2) a linearized Hodgkin- Huxley model with one active conductance component; and 3) a linearized Goldman or permeability model with one active permeability component, were all shown to have complex impedance functions with the same frequency dependence.

0022-3077/86 $1.50 Copyright 0 1986 The American Physiological Society 995

996 PUIL, GIMBARZEVSKY, AFJD MXURA

INTRODUCTION

An understanding of signal transfer in vertebrate nervous systems requires study of the electrical behavior of the membrane in individual neurons. The membrane electrical properties and morphology influence the temporal and spatial distributions of the electrical response of a neuron that has been stimulated by a synaptic input or by injection of current into the soma. Ever since intracellular recording in spinal ganglion cells was initiated by Svaetichin (5 1 ), the cell bodies of sensory neurons of vertebrates have been popular targets for electrophysiological investigations, partly because of their relatively large size and approximately spherical shape (4 1). It is widely assumed that the unipolar geometry and the absence of synaptic impingements on their perikarya (41) permit an unambiguous interpretation of electrical behavior of these cells. Some revision of this belief may become nec- essary in view of the recent demonstration of synapses on dorsal root ganglion cells of cats (36) and because of the established presence of receptors for y-aminobutyric acid, a neu- rotransmitter (15, 20). On the other hand, neurons of the mesencephalic trigeminal nucleus that receive synaptic input and are primary afferents of the trigeminal nerve (41), appear to be insensitive to y-aminobutyrate (16), suggesting that trigeminal root ganglion cells may differ from dorsal root ganglion cells.

The combination of their large size, an approximately spherical soma, and a stem process that is small at its origin (41) make trigeminal ganglion cells ideal for certain electrophysiological studies because spatial variations and the effects of attached processes may be neglected in the modeling of neuronal responses to stimuli. However, membrane electrical properties of trigeminal ganglion cells have received little attention compared with those of other ganglionic neurons. The only results of intracellular recordings from trigeminal ganglion neurons are contained in brief accounts of passive membrane properties in tissue culture (22) and the in vitro dem- onstrations of an ATP sensitivity and a highly specific Ht sensitivity of the neuronal membrane (39,40). This is surprising, especially in view of evidence that trigeminal neuralgia in humans may be due to pathogenesis in the trigeminal ganglion or the sensory root (37, 38).

In the experimental studies reported here, we have used temporal variations in voltage responses to injected currents to infer membrane properties of trigeminal ganglion neurons. For small-amplitude injected currents, resulting in small perturbations of the membrane potential from its resting value, one might expect membrane behavior to be strictly passive, i.e., the voltage- and time-dependent (active) conductances do not contribute. However, neuronal membranes are not solely passive near rest ( IX), and the calculated time constant, assuming passive behavior, may be very different from the true membrane time constant. Thus the estimates of input resistance and capacitance of neurons need to account for these active properties.

In principle, the assumption of linear superposition permits time-domain responses of the membrane potential to step current injections to be decomposed into a sum of exponentials using the method of peeling. Unfor- tunately, the accuracy of peeling techniques is questionable when more than two exponentials are needed to approximate the response (53). A more accurate method for determining membrane properties is to use frequency-domain analysis of the data (34). Injection of sinusoidal currents at certain frequencies yields a relationship between the membrane properties of the cell body and the given input frequency. Conventional methods for accom- plishing this go back more than 40 years to Cole (cf. Ref. lo), but these are time consum- ing and were superceded largely by the analyses based on the Hodgkin-Huxley model of an excitable membrane (30). Sophisticated high-speed computers obviate a number of problems associated with the time constraint and make frequency-domain analyses more appropriate. One is no longer restricted to giving a sequence of single-frequency current inputs (lo), but more general current inputs can be applied with responses decomposed into their individual frequency components using Fourier transform techniques. Furthermore, off-line computer-intensive analysis of the data in conjunction with linearized membrane models permits determination of values of certain membrane properties and with an accuracy heretofore unobtainable.

We present here for the first time measurements of membrane potential responses of trigeminal ganglion neurons in vivo to oscillatory

MEMBRANE PROPERTIES OF TRIGEMINAL NEURONS 997

current inputs. Then, the membrane properties are obtained using our frequency-domain methodology. In addition to comparisons with estimates of membrane properties obtained by bridge-balance techniques, these analyses (25, 46) permit accurate quantification of activated voltage- and time-dependent conductances (30). Some aspects of this recently developed methodology have been described previously (24-26).

METHODS

Animal preparation All experiments were performed on decerebrate

guinea pigs (300-400 g). The animals were anes- thetized initially either by an intraperitoneal injection of urethan (ethyl carbamate, 1.2- 1 S g/kg) or, in a chamber and subsequently by mask, with di- ethyl ether or isoflurane. After tracheotomy, the general anesthetic state was supplemented if nec- essary or maintained by the administration of either inhalational agent. The animal then was placed in a stereotaxic head holder, and an extensive bilateral craniotomy was performed. The encephalon rostra1 to the superior colliculus was removed, with hem- orrhage being minimized by brief (- 3-min) ligation of both carotid arteries. Further anesthesia was not required. The rectal temperature of the animal was maintained with a heating pad at -37OC. In about one-third of the experiments, blood pressure was monitored from a cannulated carotid artery. Pan- curonium bromide was used in most of the experiments for muscle paralysis to provide improved conditions for stable intracellular recording. End- tidal CO2 was controlled at ~4-4.5% by artificial respiration.

Electruphysiological recording About 2 h after discontinuation of anesthesia,

entry of the microelectrode was facilitated by a small tear opened in the dura of the middle fossa overlying the maxillary or mandibular portions of one trigeminal ganglion. The indifferent electrode was in- serted into the dorsal muscles of the neck or the adjacent subcutaneous tissues. The microelectrode was fixed to a mechanical micromanipulator and connected to a conventional amplifier for sensing the intracellular voltage and providing the current injection (WP Instruments, model 701). Impale- ment of a neuron, presumably its soma, in the pos- terolateral region of the ganglion was accomplished using a glass microelectrode containing 3 M KC1 (tip resistances 10-80 MQ). Neurons were selected for study which initially had resting membrane potentials more negative than -50 mV and which were able to generate action potentials of at least 60 mV amplitude in response to injections of depolarizing

current pulses. The experimental arrangement for recording, current injection, and data acquisition is shown in Fig. 1.

Input waveform, frequency and data acquisition

spectrum,

For the frequency-domain studies presented here, an oscillatory input current [I(t)] was applied intracellularly to the cell in order to measure its complex impedance (2). This input current had the form

I(t) = a sin(bt’ + d) (1)

a “ZAP function” [ZAP = impedance (2) amplitude profile], where a is the peak amplitude of the current and 6, c, d are empirically assigned constants (Fig. 2A). The values of these constants were chosen such that the frequency spectrum of the waveform was relatively smooth in a restricted range of frequencies (Fig. 2C). The complex-valued impedance function is defined by

z = FFT of voltage response FFT of current input

= G-4 + jZim+w (2)

where FFT means a fast Fourier transform (3), and Ltl and Zimaginary are the real and imaginary components with j = fi. The ZAP function (or input current) was used to generate an “impedance amplitude profile” (impedance magnitude function) from the neuron (e.g., Fig. 4A). The ZAP function was comprised of 1,024 discrete time points in order to use an FFT algorithm in the computerized analysis of the data. In the majority of experiments, these discrete time points were spaced 200 ps apart.

The calculated ZAP function was stored in a PDP 1 l/23 processor (Digital Equipment), which was used as the front end processor for the data acquisition system (Fig. 1). Once this processor was loaded with the ZAP function, only the amplitude and the output clock rate were changed. The amplitude of the input current was adjusted to keep the perturbation of membrane potential between l-5 mV peak-to-peak (e.g., after accounting for the voltage response of the electrode, -2.4 mV in Fig. 2B). In some experiments, the ZAP function was inverted, or reversed in time, such that the validity of linear superposition of the cell’s voltage response could be examined.

The digitized input current waveform was delivered to a digital-to-analog (D/A) converter at the specified clock rate. The 12-bit output voltage of the D/A converter was low-pass filtered and fed into the WPI amplifier, which was connected to the microelectrode. The ZAP function had a theoretical range of 4.883 Hz to 2.5 kHz in the frequency spectrum. However, in practice slightly less than the lower half of this range was used, the range of frequencies in the ZAP function being limited to a

998 PUIL, GIMBARZEVSKY, AND MIURA

i D/A --:-,

; D/A -I

Acqulsltiurl ;

r : System i

. . . . . . . . . . . . . . * . . . . . . . . . . . *.. . . . . . . . 4 4 V out

Bd out

73 Isolation -

Unit l!Y a: ml+

(UK/

FIG. 1. Schema of electronic hardware for stimulation, intracellular recording, and data acquisition from trigeminal ganglion neuron, Oscillatory functions generated within PDP computer system were converted to current waveforms with a digital-to-analog converter (D/A) of the data acquisition system (DAS, outlined by broken lines), delivered to a WPI amplifier, and then to the cell. Neither bridge balance nor capacitance compensation was employed to offset the impedance properties of the microelectrode when used for frequency-domain analysis. The voltage response of the electrode-cell system was summed upon entry into the DAS with a calibrated offset voltage and then digitized by an analog-to-digital converter (A/D). Voltage responses were averaged by the PDP 1 l/23 processor and sent to the host PDP 1 l/34 computer for further processing. Data storage was in both analog and digital forms for off-line analysis by the PDP 1 l/34 and array processors. Measurements also were made for time-domain analyses by inclusion of three switches at the WPI amplifier stage for “enabling” a stimulus isolator, bridge balance, and capacitance compensation (not shown).

maximum of - 1 kHz (Fig. 2C). This limitation tenuation of the low-pass filter was chosen to be allowed the use of noncritical low-pass filters, since - 18 dB/decade at 2.5 kHz. with a 1.5kHz “guard band,” the first frequency Before the input current was injected into the that would alias into the frequency spectrum of the cell, resting membrane potential was measured at response signal was 4 kHz. Thus, the stop-band at- low gain. This value then was used to compute an

Frequency (Hz)

FIG, 2. Variable-frequency oscillatory waveforms representing the input current (A) recorded from the D/A converter of the DAS, and output voltage response (B) from a neuron are shown with their corresponding power spectra (C and D, respectively). The response in B to the input ZAP function ofA was the membrane voltage recorded from a neuron with an input resistance of - 6 MO in series with a 26MQ microelectrode; i.e., the voltage response of the electrode has not been subtracted from the record. The input power spectrum in C shows an approximately exponential distribution of amplitudes with increasing frequency, ensuring more power in the low-frequency region (<250 Hz).


offset voltage such that the resting membrane potential would be centered in the middle of the analog-to-digital (A/D) conversion range.

Data from experiments were recorded in analog form on a tape recorder (Hewlett-Packard model 3968A) or initially were digitized using a data acquisition system (DAS) and then stored on magnetic tape (Fig. 1) With oscillatory input current, the bridge-balance technique and capacitance compensation were not used; the voltage response of the electrode-cell system was sampled by a 12-bit A/D converter simultaneously with the generation of this current waveform. When current pulses were used for comparative studies in the time-domain, the unity gain output signal of the WPI amplifier was fed into a bridge-balance circuit where it was summed with the balance output of the WPI amplifier (Fig. I). The output from the WPI amplifier (in the case of oscillatory input current or from the bridge-balance circuit in the case of current pulses) monitored by the oscilloscope was fed through a low-pass filter and into the input amplification circuit of the DAS. This stage of the DAS consisted of a summing amplifier that added an offset voltage to this input signal. The signal then was led via a variable gain amplifier (range 1-100X), subject to sample and hold commands, into the A/D converter. The offset voltage range (& 128 mV) was used to shift the membrane potential values so that the neuronal voltage response to the ZAP function could be recorded in a voltage range that was sym- metric about 0 mV.

The DAS was controlled by the PDP 1 l/23 processor through a l&bit parallel interface. This processor translated high-level commands from the PDP 11/34 host computer (Digital Equipment), which specified measurements of voltage, changes in ZAP-function amplitude, etc., into the correct sequence of commands to the DAS required to im- plement the particular experimental operation.

Dais collech~ procedures Sixteen values of membrane potential were mea-

sured and averaged over a 3.2-ms period before the start of each current application (which was repeated up to 24 times in each cycle; a cycle lasted for 5 s). Averages of voltage responses and normalization of raw data were performed by the PDP 1 l/23 computer.

The voltage response, consisting of 1,024 digitized points transmitted to the PDP 1 l/34, was converted to the frequency-domain by use of FFTs. The impedance magnitude of the cell (cf. Fig. 4A) was calculated by dividing the output frequency spectrum by the input frequency spectrum (cf. Fig. 2, C-O), yielding the complex impedance, and taking its magnitude [[Z] = (I& + Zfm,g,,,)“]. The cal- culation time required by the PDP 1 l/34 program allowed the generation of one impedance function

with estimates of electrode and membrane properties at the end of every cycle. Within this cycle time, 24 ZAP functions were applied, although usually only 20 averaged responses were recorded in order to ensure that the sampling occurred during periods of voltage stability, i.e., recordings were free from interfering action potentials or artifacts.

On-line and of-line data analysis procedures Digitized records of each experiment contained

voltage responses, calibration values, values of resting membrane potential, and other relevant variables. During the experiments, impedance magnitude functions (displayed visually) and estimates of electrode resistance (R,), input resistance (Ri), and input capacitance (Ci) were computed on-line and stored on magnetic tape. These rough estimates were used as initial guesses for more accurate parameter estimations off-line. The algorithm used for these on-line approximations was to estimate the total resistance, RT = R, + &, from the impedance magnitude function at low frequencies. The equation for the impedance magnitude function then was rewritten such that it was linear in the square of the frequency; this allowed estimation of the remaining parameters using a linear least-squares procedure (the latter linearization procedure is not justified statistically). The above estimation algorithm was inappropriate for cells that exhibited resonance in their impedance magnitude functions, but permit- ted rough estimations of R, and Ri (and Ci) for ad- justment of the ZAP-function amplitude. The same algorithm was employed for estimation of electrode capacitance prior to penetration of a cell. Electrode capacitance usually was found to be negligible (~2 pF) and therefore not included in the fitting models.

Upon withdrawal of the electrode from a neuron, the impedance magnitude function for the DAS and electrode in presumed extracellular space became flat over the entire frequency range (5- 1,000 Hz), and usually equal in magnitude to the previously computed values for R,.

Voltage-response data were read from digital tapes off-line for recomputation of the complex- valued impedance function, Z. Each complex impedance function was displayed graphically as a plot of real (resistance) and imaginary (reactance) components, i.e., as an impedance locus diagram or Cole-Cole plot (12). Cells were classified according to whether their impedance data were best fitted by a model comprised of three or five parameters using a nonlinear least-squares procedure consisting of a grid search method (2). These computations were performed on an array processor (Computer Design Associates, MSP-3000).

Membrane models Three different model representations of the

neuronal membrane are given by 1) an equivalent


electrical circuit (Fig. 3B), 2) the Hodgkin-Huxley where I(t) is the applied current and Ic, IA, IL are (HH) equations (30), and 3) the Goldman perme- the currents flowing in the capacitive, resistive, (R,), ability model (27). All three of these are based on and inductor-series-resistance branches, respectively different assumptions and various levels of physi- (see Fig. 3B). (This is simply Kirchoff’s law.) For a ological conformity. In the APPENDIX we show that voltage Jr across each branch of the circuit (repre- a linearized analysis of each of the above models senting the membrane potential), the currents Ic, results in complex impedance functions [Z(o)], IR, and IL satisfy which have the same functional dependence on frequency. Thus, determination of the values of the parameters in the general complex impedance

I&t) = Cid V/dt

function come from fitting the data, and a distinc- tion between the models must come from a more LdIJdt + RLIL = V (6)

explicit interpretation of these models in a physiological context. The “linearizations” of the HH and Goldman equations are mathematical linearizations and & not correspond to assuming time- invariant membrane properties.

The standard representation of neuronal membrane is an electrical circuit. For a passive membrane, the electrical circuit is given by a parallel RC circuit (see Fig. 34. One of the simplest representations of an adve neuronal membrane with one active component is a generalization of the RC cir-

Note that for constant circuit elements, Eqs. 3-6 are linear constant coefficient equations in the currents and voltage. For a purely sinusoidal applied current I(t) = I exp(jot) at frequency w in radians per second (where w = 2rf) and a corresponding voltage response Y(t) = Y exp(jwt), the complex impedance fitting function is easily computed to be

Z(0) = voltage response applied current

t cuit in which a resistor and inductor in series are =-

added in parallel to the parallel RC circuit (Fig. i 39. Note that whereas the parallel resistance (R,) I and capacitance (Ci) branches of the electrical circuit = R, +

GL. (7)

have physiological interpretations, the inductor-se- G, + j&i + -

ties-resistance branch may be considered simply an 1 + jc37,

electrical circuit analog of the active component of where the series electrode resistance has been added, a membrane conductance, since a true inductor G, = l/R, and GL = 1 /R,. The relaxation time (7,) may not exist in the membrane. for the unknown membrane conductance is related

With constant circuit elements, the linear model to inductance (L) and inductor-series-resistance (RL) equation for the current is given by bY

L =- I(t) = Ic + IR + Ir. (3)

TU

RL (8)

A B

FIG, 3, Equivalent electrical circuit models for neurons. Both models include a series electrode resistance (R,) but not an electrode capacitance. The “active” model in B possesses a series resistance (Rt) and inductor (L) branch, which is in parallel with the “passive” RiCi model in A.


For a more general applied current, P and 17 are replaced by the Fourier transforms of the applied current and voltage response, respectively, but the frequency dependence remains unchanged.

For RL - 00, the circuit in Fig. 3B becomes the simple RC circuit in Fig, 3A, and the explicit frequency dependence of the fitting function is given by

Z(w) = R, + & . wRi2Ci

- 1 + 02R.2C.2 J 1 + a2Ra2C2 (9) 1 1 1 1

Neither the HH nor the Goldman permeability interpretations were used in the three-parameter case because they are identical to the case of the paradigm consisting of constant circuit elements.

In their famous paper, Hodgkin and Huxley (30) gave a linearized version of the HH equations, which has served as the basis for more recent studies of the active electrical properties of the membrane of the squid giant axon (cf. Refs. 21, 44). We note that in the HH model, time- and voltage-dependent conductances replace the constant inductance and series resistance branch in the simpler equivalent circuit model.

The use of linearized equations introduces two difficulties: 1) in the voltage-clamp paradigm, there are technical difficulties, and 2) the identity and properties of the major ionic channels in the membrane are not known a priori. The first difficulty is obviated by considering the membrane-potential equation, and the second requires a generalization of the linearized HH equations (see APPENDIX).

For a small injected current that passes through the membrane, the linearized HH equations become (see Eqs. 35 and 36 in Ref. 30)

WV) I(t) = c 7 + (&mo3ho + &no4 + gi)AV

+ 4&no3( V. - v,)An (10)

Wn) 1 dn An >AV--,

dl =T,dV 7,

ffn 1 n, = -

an + Pn ’ 7, - ~

an + Pn ’ (10

where rn have let

and dn,/dV are evaluated at VO, and we

V= V,+AV, m=mo+Am,

h = ho + Ah, n = no + An

with VO, mo, ho, ~2~ equal to the values of V, m, h, and y2 at the resting state of the membrane potential. The quantities AV, Am, Ab, and An are the perturbations from these resting values. Relaxation equations for Am and Ah are similar to Eq. Il. The complex impedance function for this linearized HH

model is identical to Eq. 7 but with different interpretations of the coefficients (see APPENDIX).

A more accurate description of ionic current flow through neuronal membrane, which accounts for extracellular and intracellular ionic concentrations, is given by a constant-field membrane model (27, 31). The applied membrane current is given by a generalization of Eq. 3 (see Eq. A. 1 in APPENDIX). The membrane currents now are given by

Ii = -IhlFPiV ci - p&iv

’ 1 - k-w (13)

with fl: = ziF/RT where zi, F, R, T are the valence of the ith ion, the Faraday, the gas constant, and the temperature, respectively; Pi is the membrane permeability to the ith ion, and C! and CF are the intracell ular and extracell ith ion, respectively.

ular concentrations of the

In the HH and Goldman interpretations of Eq. 7, all but one of the ionic conductances are assumed to be constant. Thus, G, is the time-invariant part of the membrane conductance, and CL is that part activated by perturbation of membrane potential with the ZAP current. In this case, the sum of the conductances (G, + GL) is the input conductance (GJ of the cell, and the input resistance is given by Ri = 1 /Gi l In the HH and Goldman interpretations of E’q. 7, 7, is the relaxation time constant for the conductance and permeability, respectively.

Initial estimates for the parameters of each fit were obtained automatically from the rough estimates calculated on-line during the experiments in the case of the three-parameter model (single time constant model), but were guessed in the case of the five-parameter model. A nonlinear least squares fitting procedure (2) was used to fit the above complex impedance functions to the data points in the impedance locus diagram. Such a fit of the full set of 200 data points would yield a single estimate of the three or five parameters. In order to obtain an indication of the variation in the values of these parameters as a result of experimental error, we used a modification of the bootstrap statistical method (19) for determinations of the distributions and mean values of the parameter estimates. Our modified bootstrap method utilized a random selection of 100 of the 200 data points (with replication allowed) in the impedance locus diagram, and then, a nonlinear least squares fit of the five-parameter model to these 100 points was made. This procedure was repeated for a minimum of 2,000 bootstrap samples or until no further increases in the range of estimated values were obtained.

In order to have some confidence in the modified bootstrap procedure used here, we have applied the true bootstrap procedure (i.e., randomly selected 200 points from the available 200 data points al- lowing replication) and a cross-validation procedure (cf. Ref. 19) to three of the cells listed in Tables 1


and 2. The confidence limits obtained with the

increased. The cross-validation procedure chooses

modified

100 of the 200 data points, and a fit of the three-

and true bootstrap procedures approached identical values as the number of bootstrap samples

within 5%. The precision of the estimates, i.e., the reproducibility of an

RESULTS

estimate upon surements, was withi n 1%.

repeated mea-

or five-parameter model to these 100 data points is made. This provides information as to how well the curve fit predicts the remaining 100 data points. The averaged error between the predicted values and the excluded data points is the prediction error (19). The distributions of the sum of squares error of the fit and the prediction error did not differ sig- nificantly in the three cases.

The accuracy and precision of the FFT methodology were checked by the application of the ZAP function to various electrical networks comprised of resistors and capacitors with values approximat- ing those of the resistances and capacitances of electrodes and neurons. (It was not practical to in- corporate inductors in such networks because of the large values of required inductance.) For the model with a single time constant, accuracy for the estimation of the three electrical components was

In the development of the methodology described herein, measurements of complex impedance were made on -250 neurons. The results presented here were selected from this pool and are based on data obtained from 105 trigeminal ganglion neurons with resting membrane potentials of -50 to -80 mV (typically -55 to -65 mV). All of the neurons initially displayed action potentials of 60-90 mV amplitude (typically 60-70 mV) in response to intracellular injections of depolarizing current pulses (using bridge techniques; cf. Figs. lo- 12). The physiological state of the recorded neuron was evaluated on the basis of resting membrane potential, spike height (and shape), repetitive firing, and changes in

200 600 1000

Frequency (Hz)

Resistance (Ma) 20 22 24

f/ I I I I no 0 I L ’

FIG. 4. Experimental and theoretical impedance functions for a neuron (cf. cell no. 5 in Table 1). A: impedance magnitude function vs. frequency. Smooth cuwe is a nonlinear least-squares fit of an impedance magnitude function computed for model A (Fig. 3). B: the same data (200 points), which represent responses to 20 ZAP functions, are shown as an impedance locus diagram. The complex impedance (smooth cuwe) is approximately a semicircle with center on the resistance axis.


TABLE 1. Summary of membrane properties for six neurons with simple electrical behavior

Cell No. ci, PF Ri, mQ 7 rn7 ms

1 74-t 5 8.1 ztr 0.4 0.6 * 0.1 2 142 I!I 29 7.3 t 1.0 1.0 k 0.3 3 153 * 10 7.6 t 0.4 1.2 k 0.2

4 183 k 13 5.0 t 0.3 0.9 Y!z 0.1 5 202 4 16 4.1 k 0.3 0.8 t 0.1 6 319 k 32 4.2 + 0.3 1.4 t 0.2

Electrodes used for these cells had mean resistance of 27.1 MS2 (range, 11.9-46.2 MQ). Results are expressed as a mean k SD for 95% confidence limits derived from probability density functions based on bootstrap estimates. Each impedance magnitude function from which >2,000 bootstrap samples were drawn was calculated from 6 to 140 average voltage responses of a neuron. Mean value of resting membrane potential was -60 mV with a SD of 11 mV. Ci 7 input capacitance; Ri , input resistance; rrn, membrane time constant.

input resistance. Many cells, when tested at - lo-20 min after their impalement, continued to generate spikes. However, some neu-

rons failed to generate action potentials when depolarized with injection of current pulses at this time (- 10 min or more after impalement), while showing no ancillary evidence of injury; indeed, such cells often displayed 5- 10 mV of hyperpolarization and a 5-20% increase in input resistance at the time of putative “injury”.

Neurons with simple (passive) electrical behavior

Regardless of continued spike generation, a minority (~25%) of the neurons (24/105) exhibited impedance magnitude functions (Fig. 4A), which could be fitted closely by a theoretical impedance magnitude function computed for a resistance in series with a parallel RC circuit model. The impedance locus diagram, yielding the impedance magnitude function, was approximately a semicircle with center on the resistance-axis (Fig. 4B). The semicircular fit was computed using the parameter values which were estimated from the impedance magnitude function as initial

TABLE 2. Summary of membrane electrical pruperties in 20 trigeminal ganglion neurons with resonance in their impedance magnitude functions

Cell NO.

1 205 (-45,95) 2 121 (-18,40) 3 185 -+ 48 4 115 k 16 5 160 k 21 6 167 k 30 7 138 k 31 8 259 -+ 36 9 186 122

10 146 k 22 11 104 * 16 12 71k 5 13 72k 5 14 102 k 12 15 76k 19 16 98k 13 17 119zk 9 18 66k 6 19 86+ 10

20 109k 7

4.5 (-1.9, 5.5) 8.0 (-1.6, 10.1) 9.3 (-3.5, 6.2)?

14.5 (-3.7, 12.4) 9.1 (-1.6, 6.1) 4.8 (-1.0, 2.1) 9.8 (-3.6, 12.0) 7.1 (-2.2, 10.0)

11.3 (-3.3, 14.4) 11.4 (-4.2, 12.8) 16.3 (-5.1, 14.5)*

25.8 (-5.4, 13-O)*

19.4 (- 1.9, 4.4) 6.4 I 0.5

13.0 k 2.6 9.8 f 2.7

11.2 k 1.6 17.4 k 2.9

7.8 t 1.3 9.9 f 1.5

1.6 k 1.1 7.8 k 2.1 2.7 (-0.9, 1.7) 5.7 (-1.0, 2.2) 8.0 (-1.8, 4.4) 7.7 zk 3.7 4.5 k 2.2 3.7 k 1.3 3.9 ~fi 0.8 3.3 k 0.9 6.2 k 1.2*

11.7 2 2.9 11.8 k 2.0

7.1 (-1.8, 3.7) 15.2 (-9.0, 34.O)t

8.8 (-2.2, 7.6) 12.1 -+ 4.5 12.5 -+ 2.0* 17.8 k 7.6 11.9 k 2.1

1.1 (-0.9, 0.5) 2.4 (- 1.4, 3.4)? 3.9 f 0.7 1.6 (-0.8, 1.4) 2.0 t- 0.5 0.4 (-0.2, 0.5) 4.1 k 0.6 1.2 k 0.6 4.2 k 0.7 2.2 k 1.0” 2.7 (-2.1, 0.8) 2.6 (-1.7, 3.7)” 3.0 k 1.0 1.6 (-0.9, 1.6) 2.4 k 0.4 1.2 k 0.5 2.9 k 0.5 1.7 k 0.7 2.6 k 0.6 1.3 * 0.7” 4.4 k 0.7 1.0 -t 0.5 8.1 1 1.4 1.5 k 0.5 7.4 (-1.1, 2.2) 3.2 k 1.4

3.4 z!I 0.7 4.1 (-1.5, 2.4) 6.4 k 1.97 9.3 (-7.0, lO.Oyf 4.6 2 0.8 1.3 k 0.8 5.8 (- 1.7, 0.9) 2.5 (-1-2, 2.8)

7.3 (- 1 .o, 2.0) 1.6 I!I 0.5 5.4 t- 0.4 0.9 f 0.5 5.4 4 0.3 1.4 * 0.4

0.9 (-0.5, 1.3) 1.0 (-0.4, 2.2) 1.8 (-1.2, 4.9) 1.8 (-0.7, 3.9) 1.5 (-0.4, 1.0)

0.8 (-0.3, 0.5) 1.4 (-0.7, 1.9) 1.9 (-0.8, 3.3) 2.2 (-0.9, 2.9) 1.7 (-0.9, 2.1) 1.8 (-0.9, 4.2) 1.9 (-0.7, 2.0) 1.4 r!z 0.2 0.7 4 0.1

1.0 k 0.4 1.0 k 0.4 1.3 k 0.3 1.1 * 0.3 0.7 k 0.2 1.1 & 0.2

Results (including A;, which was computed for the parallel RLC-circuit model of Fig. 3B) are expressed as a mean f SD for 95% confidence limits derived from probability density functions based on bootstrap estimates. In cases where distribution of values about the mean is not symmetrical, the deviations below and above the estimated values are given in parentheses. Each impedance locus

diagram from which >2,000 bootstrap samples were drawn, was calculated from 4 to 200 averaged voltage responses of a neuron. For each neuron, this procedure required -2.5 h of computations with an array processor. Electrodes used for the 20 cells had a mean resistance of 28.4 Ma (range, 17.9-4 1.1 MQ). Mean value of resting membrane potential was -60 mV with a SD of 9 mV. Ci 7 input capacitance; R,, resting resistance; RL, inductor-series-resistance; Ri, input resistance; rU, conductance (GL) time constant; TV, membrane time constant. * 90% confidence limits; t 80% confidence limits.

1004 PUIL, GIMBARZE VSKY, AND MIURA

2 - c

z 0

8

s

-2

I’u : -4

p1 -6

200 600

Frequency (Hz) 200 600

Frequency (Hz)

4 D r Resistance (Ma)

.

FIG. 5. Experimental and theoretical impedance functions representing 2 neurons (cf. Table 2 for membrane properties of cell no. 4 in A and B and of cell no. 13 in C and D). Impedance magnitude functions vs. frequency are shown in A and C. A: impedance magnitude was computed from the averaged voltage responses of a neuron (-73 mV resting potential) to 4 ZAP functions. C: impedance magnitude was computed from the averaged voltage responses of a neuron (-58 mV resting potential) to 200 ZAP functions. The data (200 points) in A and C are plotted in B and D as impedance locus diagrams. Clockwise movement along smooth czuws corresponds to increasing frequencies from 5 to 1,000 Hz. Positive (inductive) reactance (at low frequencies) is evident in both cells. Nonlinear least-squares fits of a linearized model (cf. METHODS) to the data in B and D are spiral curves. The impedance magnitude curves corresponding to the spiral curves are superimposed in A and C.

guesses. In this diagram, the frequency increases as one moves clockwise along the impedance locus curve. Assuming the membrane behavior has passive but no active properties, the projected right-angled intersec- tions of the impedance locus curve with the resistance-axis occur at - 19.9 and 24.1 MQ, corresponding to the values of R, and the total resistance for the electrode-neuron system, respectively. Table 1 summarizes the bootstrap estimates for the membrane properties of six representative neurons with simple electrical behavior. For these neurons, the mean membrane time constant is 1.0 $- 0.3 ms (SE).

Neurons with resonant impedance functions The majority (>75%) of trigeminal neurons

was characterized by resonant peaks in their impedance magnitude functions (Fig. 5, A and C). The impedance magnitude function attained its maximum value IZI,,, at an average frequency fmax of 159 Hz (range 66 to 3 12 Hz) for 20 neurons (see Tables 2, 3). The fmax usu-

ally occurred in the 8 l/ 105 cells between 100 and 180 Hz (175 Hz in Fig. 5A and 120 Hz in Fig. 5C). A further characterization of the impedance magnitude functions is obtained by defining the bandwidth between the two half-power frequencies where IZl = Ri +

(14 - Ri)/ fie This value, which is an ap- prozzation of the range of frequencies over which resonance occurred, yielded an average bandwidth of 124 Hz (range 86 to 194 Hz) for the resonance in 20 neurons (see Tables 2, 3). The impedance magnitude function decreased to a relatively constant level at frequencies higher than about 500 Hz. Figure 5A shows an impedance magnitude function computed from averaged voltage responses of a trigeminal neuron to four successive ZAP functions, which perturbed the membrane potential by 3.5 mV (peak-to-peak) from an initial resting potential of -7 1 mV. Figure 5C shows this resonant behavior in a different neuron; in this case, the impedance magnitude function was computed from 200 averaged voltage respon-


Resistance (Ma) 0

Resistance (Ma)

FIG. 6. Impedance locus diagrams for a neuron (cf. cell no. 1 in Table 2). A: data points (200) are from 5 averaged voltage responses. Note positive reactance points lying above the resistance axis at low frequencies. B: estimations of the complex impedance based on a modified bootstrap procedure using 2,000 bootstrap samples, each containing 100 data points (induding possible replications) from A. The white curve (shown in black in A) is the mean of the 2,000 complex impedance curves.

ses. Note that the pattern of data points for 200 averaged responses is much tighter about the theoretical impedance locus curve (Fig. 5D) than in the case of four averaged responses (Fig. 5B).

Examination of impedance locus diagrams of neurons that exhibited resonance in their impedance magnitude functions usually re- vealed a crossing over of the locus into the inductive (positive or upper) half-plane (Fig. 5, B and D). The impedance locus diagrams of all neurons could be approximated by the linearized models giving estimates corresponding to the four membrane properties (R,, Ci, GL, 7,) plus the electrode resistance.

Variability in frt to Jive-parameter model Each theoretical fit of the five-parameter

model to the 200 data points of the impedance

locus diagram (e.g., Fig. 64 was calculated from the mean values of the parameter estimates obtained by the bootstrap statistical technique (cf. METHODS; Ref. 19). Figure 6B shows a superposition of impedance locus curves obtained from 2,000 bootstrap fits. This representation, in its graphical form, provides a qualitative indication of the uncertainty associated with the particular fit (white curve, Fig. 6B) of the five-parameter model to the experimental data points derived from voltage responses of a cell (cf. Figs. 5, B and D, 64. Note that the majority of the 2,000 bootstrap fits, as indicated by the density of curves in Fig. 6B, lie very close to the mean curve fit (white curve).

To investigate the sensitivity of the impedance locus curves to variations in the parameter values, each of the four membrane pa-


L

I t 40 30

Resistance (MSZ)

FIG. 7. Clam-shell diagrams showing effects of parameter variations on the impedance loci. Doffed curve in each diagram corresponds to a nonlinear least-squares fit of experimental data (see Fig. 5B). Direction of increasing 1 parameter (while others are kept constant) is indicated by the number sequence, 1-4. A: Ci parameter was increased in 20-pF steps from 50 pF (curve I) to 250 pF. B: R, parameter was increased in 2.SMQ steps from 5 to 27-5 MR C: RL parameter was increased in 2.0-MQ steps from 2 to 24 MQ. D: 7, parameter was increased in 0.2-ms steps from 0.5 to 2.3 ms.

rameters (Ci , R,, RL , TJ was varied separately in uniform steps with a range bracketing the parameter values for cell 4 of Table 2 (Fig. 7). As expected, RL has the most pronounced effect of the four parameters on the amount of positive reactance in the impedance locus diagram, Note that in the “clam shell” diagram given by Fig, 7C, there is no positive reactance portion of the impedance locus curve for values of RL greater than ~8 MQ. This indicates that if the activated conductance is small enough, then the reactance is dominated by the capacitance. In this case, the bulge in the diagram disappears as RL is increased. For the other three parameters, an increase in their values is manifested in a shrinking (Ci) or ex- pansion (R,, TJ of the impedance locus curves. In addition, no positive reactance is evident with large values of Ci or small values of 7,.

Estimates of model parameters The estimates for each membrane param-

eter and their ranges of variability obtained by the fit of the impedance locus diagram using the five-parameter model were found quan- titatively for 20 neurons exhibiting resonance in their impedance magnitude functions, using the bootstrap procedure (Table 2). The large variation in the estimates of input capacitance (66-259 pF) is not unexpected in view of the variability in cell size and unknown sampling

3 a c - 506 1 I I I 1 1 I t

2 5 IO resistancy 40

Resting (lO”S2)

L- 3 00 B

eu =.,, t r0

‘.., ‘C ‘-... - 200

-+ ‘.. ‘.. ‘-..,

Q, 2 “A.. u ‘..

.= ‘A. “.

g Q 100

1 ..t

s “.., -L L

-

I

G- I

-‘. 3

--P ‘.A., ‘.. -... L

- sol 1 I 11 I I 1

1 2 Input5resisto2 20

(10%)

FIG. 8, Log-log plots relating input capacitance ( Ci) and resting resistance (R,) or input resistance (Ri) of 20 neurons with values (dots) and ranges (vertical and hori- zontal lines) listed in Table 2. A: dotted line corresponds to a linear least-squares fit of the curve CiR, = constant. Interpretation of this product as the membrane time constant results in a value of ~~ = 1.25 ms. B: dotted line (slope = - 1) corresponds to a linear least-squares fit of data.

MEMBRANE PROPERTIES OF TRTGEMINAL NEURONS 1007

bias, which was attributable partly to electrode tip size.

If one assumes that the larger values of input capacitance correspond to cells of larger diameter, then it is reasonable to expect a reciprocal correlation between input capacitance and resting resistance or input resistance (Fig. 8). Because the membrane time constant equals the product of Ci and R,, a straight line fit (slope = - 1) of the log-log plot of the data for the 20 cells of Table 2 results in a membrane time constant, 7, = 1.3 ms (Fig. 8A). (The mean value of 7, from Table 2 is l-4 ms with SE = 0.2 ms.) A closer correspondence occurs between estimates of Ci and Ri for the same 20 cells of Table 2 (Fig. 8B). In this case, since Ci is proportional to cell membrane area and Ri is inversely proportional to the area, their product equals the product of the specific input capacitance and specific input resistance (see Table 3); again, a log-log plot of the data can be approximated by a straight line with slope = -1.

The reciprocal of the inductor-series-resistance values ( l/RL = GL) listed in Table 2 can be interpreted as the membrane conductances activated by perturbations of the resting membrane potential. For many of the 20 cells in Table 2, the estimates of activated conduc-

TABLE 3. Relationship of membrane properties to the peak impedance magnitude,

tance are comparable to the values of the time- invariant part of the membrane conductance ( l/R,). Furthermore, the relaxation time constants for the activated conductance (with the exceptions of cells 13 to 15) are comparable to the membrane time constants.

The fact that Ci/Ri is proportional to the square of the membrane area of the neuron leads to some interesting observations on the relationships between membrane properties and peak impedance magnitude (Table 3). An increase in the area seems to correspond to a decrease in the peak value of the impedance magnitude function and in the inductor-series- resistance, RL. We point out that from the theoretical calculations leading to the clam- shell diagrams for variations in Ci and RL (cf. Fig. 7, A and C), decreases in Ci and increases in RL result in an increase in ]ZI,, , as observed experimentally. On the other hand, there appears to be no correlation between cell area and the frequency (fmax) at the peak of the impedance magnitude function. However, theoretical computation of the natural frequency of the system, fn = [4LCi - (L/R, - RLCi)2]“/4rLCi (based on the estimated parameters in Table 2, including damping) shows f max > fn for all neurons, except cell no. 3,

relative cell urea, and resonance characteristics

Cell No. IZI max7 MQ CiIRi 3 PF/MQ ci, PF RL, MQ fmax, I-Jz fn, l-Iz

12 18 13 15 19 20 17 16 11 4

14 2 5

10 6 9 3 8

13.8 8.7 11.5 9.0 14.7 9.7 12.1 11.9

5.9 15.9 6.9 20.2 8.4 20.5 6.7 21.3 7.7 23.6 7.3 28.0 5.8 30.0 5.9 31.0 6*4 38.1 6.0 46.0 5-7 56.2 4.1 61.9 6.0 64.1 2.6 92.5 3.6 107.9 3.4 186.3

71 66 72 76 86

109 119

98 104 115 102 121 160 138 146 167 186 185 259 205

11.7 12.5 11.8 15.2 17.8 11.9 12.1 8.8 6.2 5.7 7.1 7.8 8.0 4.5 3.3 7.7 3.9 2.7 3.7 1.6

155 142 165 142 114 91 66 51

154 132 137 116 103 79 176 149 201 187 192 178 122 41 152 121 112 93 178 159 214 198 116 54 149 137 312 322 156 147 202 171

Cj , input capacitance: Rj 9 input resistance; RJ + inductor-series-resistance,:J-,, frequency at jZ]ma,;J, natural frequency; lzlmax, maximum value of impedance magnitude function.

PUIL, GIMBARZEVSKY, AND MIURA

3mV

c--q/-- ,.,- 20 ms

FIG. 9. Membrane-potential responses of neuron (cf. cell no. 11 in Table 2) to hyperpolarizing current pulse. A: membrane potential response to a hyperpolarizing current pulse of 0.5 nA. B: theoretical reconstruction of membrane potential response of A using linearized membrane model equations. C residual membrane potential obtained by subtracting theoretical curve from experimental response,

Theoretical transformation time-dumain response

Gtf

A test for the precision of the theoretical estimates of the neuronal membrane properties obtained from frequency-domain analysis is to reconstruct the cell’s membrane potential response to step currents in the time domain. This reconstruction was carried out for several cells (see Fig. 9 for cell no. 11 of Table 2). There is a close correspondence between the time evolution of the membrane potential, calculated theoretically (Fig. 9B), with the experimental record obtained in the time domain (Fig. 9A).

Comparutive behavior in time domain Neurons that had no resonance in their

impedance magnitude functions exhibited the single exponential behavior in the time domain in their hyperpolarized voltage responses of lo-20 mV amplitudes to injected step currents (Fig. 1 OB) as expected from a passive model. Such neurons did not exhibit off-responses (e.g., a spike) following termination of a hyperpolarizing current pulse. The current-voltage relationship in the hyperpolarizing direction was approximately linear when measured over the lo- to 50-mV range (Fig. 1OC). These cells also showed no tendency to discharge repetitively when injected with depolarizing current pulses.

The majority of the neurons that had sat- isfactory spikes and resonance in their impedance magnitude functions also did not exhibit simple electrical behavior when injected with step currents, at least during the first 10 min after the initial impalement. Time-dependent rectification was observed in responses to depolarizing step currents that were subthreshold for the production of action potentials, as well as in responses to hyperpolarizing current pulses (Fig. 11). The initial depolarizing response was followed by a small sag in membrane potential (Fig. 1lA). In this regard, the rectification appeared to be similar to that observed in cultured dorsal root ganglion neurons (45). Some membrane potential recordings of trigeminal root ganglion neurons ad- ditionally showed a depolarizing trend that continued for the duration of the injected step current (Fig. 1lA). A possible explanation of this trend is that slow changes in ionic concentrations or conductances are occurring. However, there is the possibility that such trends are a consequence of time-dependent changes in electrode properties. When an action potential was elicited, a short-lasting ( 1.5- to 4-ms) afterhyperpolarization followed the spike (Fig. 1 lA, uppermost trace).

These cells also showed a time- and voltage- dependent sag in their membrane potentials when injected with a hyperpolarizing current (Fig. 11 C). This time-dependent rectification


50 mv (A)

25 mV (B)

20 ms

B

FIG. 10. Membrane-potential responses of neuron (cf. cell no. 4 in Table 1) in time domain. A: action potential evoked by a 7.6nA depolarizing current pulse B: voltage response as a function of time to 2.1-nA hyperpolarizing current pulse of 62 ms duration. Smooth curve is an exponential fit to voltage response. Membrane time constant was 1.2 ms, C: plot of current-voltage relationship shows linear slope resistance with large hyperpolarizing current pulses.

A 1

I 50 mV

20 ms

B 3.O-

nA O----

-3.o-

C

FIG. 11. Voltage responses from a neuron (cf. cell no. 4 in Table 2) showing effects of depolarizing (A) and hyperpolarizing (C) current pulses (cf. B). Rectification and overshooting of “off-responses” were evident with large depolarizing and hyperpolarizing current pulses. Increasing the magnitude of depolarizing current pulses evoked larger local responses of short latency (A) until threshold for an action potential was exceeded (1 l 1 times) in uppermost trace. Current pulse of maximum amplitude in the hyperpolarizing direction evoked time-dependent %ag” of membrane potential evident in C (cf. broken line for visual reference). Traces in B have been redrawn from original current monitor traces. Voltage calibration at top is for responses in A and C.

1010

A

PUIL, GIMBARZEVSKY, AND MIURA

1 1

20 ms

1

50 mv

2.6 nA I

C

II.

L FIG. 12. Voltage responses of a neuron (cf. cell no. 8

in Table 2) showing repetitive firing behavior elicited by 3 depolarizing current pulses. Single action potential was evoked using a current stimulus of 0.9 nA (not shown). Small increase in current stimulus resulted in appearance of second spike (A). Further increases in stimulus resulted in increasingly longer spike trams (B, C) until a maximum rate of firing of -300 Hz was attained (C). Traces below each voltage response have been redrawn from original current monitor traces.

was apparent from careful examination of voltage responses to step currents of 50- 100 ms duration, which &placed membrane potential by at least 10-20 mV, and was similar to that described for dorsal root ganglion neurons in vivo (13) and in vitro (45). Both the characteristic resonance and the sag of trigeminal cells disappeared with intracellular injections of strong constant hyperpolarizing current (not shown).

In most cases, the membrane potential did not return immediately to its resting value after termination of the hyperpolarizing step current but showed a rebound depolarizing overshoot similar to that observed in other mammalian neurons, notably cat spinal motoneurons (1, 33). Sometimes a spike occurred on top of the overshoot followed by an oscillation of membrane potential.

Repetitive firing characleristic In contrast to neurons that did not show

membrane rectification, those with resonance in their impedance magnitude functions, and showing depolarizing overshoots following their hyperpolarizing responses to current steps, displayed an ability to discharge action potentials in a repetitive manner. This was re- vealed by the intracellular application of suprathreshold pulses of depolarizing current. An example of the repetitive firing characteristic is shown in Fig. 12. A single action potential was elicited with a just suprathreshold pulse of depolarizing current (0.9 nA; not shown). Further increases in the depolarizing current resulted in a train of action potentials with amplitudes that decreased after the first spike (Fig. 12). The maximum rate of discharge that could be elicited in the cell was -300/s (Fig. 12C). This maximum rate, typ- ical for neurons with resonance, appeared to be partly a consequence of the presence of a slow background oscillation in membrane potential and, partly the result of the presence of postspike afterhyperpolarization (cf. Fig. 12C). The maximum discharge frequency was approximately the same as fmax in many cases, or was some integer multiple of the peak resonant frequency of the neuron.

DISCUSSION

The necessity of accurate measurement of membrane impedances for the interpretation of the electrical behavior of invertebrate nerve has led to extensive studies using a frequency- domain analysis of small perturbations of membrane current. Such v01tage-cJmp ba- vestigations in squid axon, in particti@mw

facilitated a description of the linear kinetics of its active ionic conductances (cf. Refs. 10, 2 1). Our results, which were obtained from a frequency-domain analysis of membrane potential perturbations using a computer-intensive methodology, demonstrate for the first


time that an accurate characterization of membrane impedance may be obtained from vertebrate neurons, in vivo. The confidence that one may ascribe to the accuracy and precision of these impedance measurements of trigeminal ganglion neurons partly results from the analysis of membrane potential responses to near sinusoidal currents over a wide range of frequencies (cf. Ref. 34) using high- speed computers (cf. Ref. 2 1).

Comparison with time- domain measurements

The electrical properties of various sensory ganglion neurons have been assessed by many investigators using intracellular application of hyperpolarizing current pulses and subsequent measurements of the perturbations of membrane potential in the time domain (5, 13, 18, 22, 23, 28, 32, 35, 45, 52). The responses to current pulses have been interpreted on the basis of an equivalent electrical circuit model for the passive membrane, i.e., simple RC circuit.

In the present investigation, only a small proportion of neurons studied could be mod- eled with this equivalent circuit. However, the estimates of input resistance for neurons with simple electrical behavior were within the same range as those for neurons with active behavior (cf. Tables 1 and 2). In general, the values of input resistance reported here are much lower than the estimates for Ri of various types of ganglion cells in in vitro preparations (23, 32, 35) or in tissue culture (5, 22, 29). The average of the estimates of input resistance in dorsal root ganglion cells obtained from in vivo investigations ( 13, 18) and in certain identified classes of dorsal root ganglion cells in the pigeon (28) and mouse (52), fall within the range of estimates for input resistance reported here.

Similarly, estimates for input capacitance of trigeminal ganglion neurons (Tables 1 and 2) a-e cumpambk tu tliose determined for other types of ganglion cells (5, 23, 32, 35) including trigeminal root ganglion cells in tissue culture (22). These previous estimates were deduced from input resistances and membrane time constants obtained by using time-domain analyses in step-current experiments. In most cases, the assumptions are made that the small perturbations of the membrane potential fol- low a simple exponential time course and that activation of voltage- and time-dependent

conductances is negligible. Brown et al. (5) found, for mouse dorsal root ganglion neurons in tissue culture, that injections of hyperpolarizing step currents resulted in membrane potential responses that could be fitted by a sum of two exponentials, thus leading to two time constants. These time constants were interpreted as the membrane time constant and the equalizing time constant, i.e., a measure of the time for the membrane to reach iso- potentiality upon local polarization of the membrane by the intracellular injection of current. In our step-current experiments on trigeminal ganglion neurons (e.g., those listed in Table l), we observed some membrane potential responses that could be fitted with a single exponential. It is not clear if this contrast with the results from cultured dorsal root ganglion cells may be accounted for by certain differences in morphology such as differences in the initial tract of the axon attached to the nearly spherical cell body. For example, very large stem processes with diameters up to 15 pm have been observed in adult cat and mon- key dorsal root ganglia (50; cf. Fig. 1 of Ref, 5). It may be significant that cells of well maintained cultures of dorsal root ganglia do not develop axonal glomeruli (6), which are known to exist in trigeminal root ganglia. [See Lieberman (41) for further descriptions of morphological variations in sensory ganglion cells.] Differences in the estimates of the membrane time constants between our in vivo and the tissue culture preparations (5) may be accounted for by consideration of the larger values (- 10 times) of input resistance for dorsal root ganglion cells compared with the values reported here for trigeminal ganglion neurons. Since the corresponding input capacitance values of the tissue culture cells are approximately half the values reported here, we conclude that the cell membrane areas of the dorsal root ganglion cells are approximately half of those for the trigeminal cells. Ths, the longer membrane time constant (- 5 ms) in cultured dorsal root ganglion cells may result from the larger input resistances.

Active membrane properties and signiJicance of resonance

“Active” behavior such as inward rectification (45) also may be identified by following the perturbation of the membrane potential produced by a current pulse, particularly when current pulses produce a large displacement

1012 PUIL, GIMBARZEVSKY, AND MIUR4

of membrane potential (lo-20 mV). In such cases, estimates of the resting conductance may be misleading because of activation of voltage- and time-dependent ionic conductances (30). Indeed, our results suggest that in the majority of cases, active properties of the membranes of trigeminal ganglion cell bodies can be measured even when the membrane potential is near its resting level. For this rea- son, and because of possible variations arising from differences in animal species, experimental conditions, microelectrode sampling bias, etc., the estimates of membrane properties obtained in the previous investigations cited above are not strictly comparable with those obtained from trigeminal ganglion neurons by means of the frequency-domain methodology.

In the frequency domain, the active component in membrane behavior in trigeminal ganglion neurons is expressed in their impedance magnitude functions as a resonant peak; this was apparent in the impedance locus diagrams as a bulge in the capacitive half-plane, or for most cells, as an inductive reactance. Over 20 years ago, Araki et al. ( 1, 33) postu- lated the existence of an inductive reactance, such as that demonstrated previously for squid axon (11; cf. Ref. 30) to explain the time-dependent rectification they observed in mammalian motoneurons injected with hyperpolarizing current pulses. The majority of trigeminal ganglion neurons investigated exhibited active behavior in the time domain similar to that of motoneurons. In particular, many of the trigeminal neurons also showed an inductive reactance in the impedance locus diagram corresponding to analogous behavior in squid axon ( 10, 11,43) and lamprey spinal neurons (46). These observations suggest that this electrical behavior may be a basic property of many types of neurons. The possibility may not be ruled out, however, that the absence (or presence) of resonance in the impedance magnitude functions of trigeminal ganglion neurons determined near their resting membrane potentials or when the cells are depolarized might be attributed indirectly to some damage to the membrane or be a consequence of an unusual positioning of the microelectrode within the cell. In such cases, the ability of the neurons to fire action potentials would be impaired. In contrast to other trigeminal ganglion cells with resonance in their impedance magnitude functions, those neurons

without resonance often failed to discharge in a repetitive manner in response to injections of suprathreshold depolarizing current pulses.

A consequence of membrane resonant behavior appears to be a restriction of the range over which a cell would discharge spikes repetitively in response to an applied constant current. For example, the application of a depolarizing current to a space-clamped squid axon will evoke a train of action potentials at a frequency which is close to the “subthreshold” natural frequency of - 100 Hz (44). This frequency corresponds approximately to the peak resonant frequency in lamprey spinal neurons at depolarized levels (< 100 Hz; 46) and in the impedance magnitude functions of a majority of trigeminal ganglion neurons at their resting membrane potentials (fmax - 1 OO- 180 Hz); cf. Table 3). The higher fmax in trigeminal neurons can be attributed to the differences in resting membrane potential and the higher temperature of mammals as compared to invertebrates (44).

The relationship of the resonant frequencies of the trigeminal neurons to membrane potentials that approach the firing thresholds was not investigated systematically, although it was apparent that the resonance was strongly voltage-dependent. The importance of the frequency-voltage relationship in the entrain- ment of spike discharge near the peak resonant frequency has been emphasized in the case of frog node of Ranvier (4, 8) and in the case of heart cells (9). Thus, at a given subthreshold membrane potential, the impedance magnitude function of a trigeminal ganglion neuron exhibiting resonance may be considered analogous to those of certain band-pass filters (cf, Refs. 14, 46) based on an RLC circuit and possessing a low quality factor. According to this analogy, the voltage responses generated by input currents with frequencies quite different from the resonant frequencies would be attenuated relative to the voltage responses to input currents with frequencies nearf,, . Such a resonant system, while experiencing a relatively low energy loss, would verge on damped oscillation. These natural frequencies of oscillation (cf. Table 3) have been determined mathematically from the linearization of the HH equations (30) for squid axon and in other equations for the node of Ranvier (3, 14, 17).

Our results, obtained with small oscillatory perturbations of membrane potential, suggest that the use of the ZAP function in a fre-


quency-domain analysis of the membrane electrical behavior of neurons may be a more sensitive indicator of active properties than step-current measurement techniques. One might predict that the active component in the membrane would influence the cablelike properties of mammalian neurons that have extensive dendritic arborization (cf. Ref. 49). Such contributions may not be always evident from a time-domain analysis of responses to applied current pulses. Complex impedances corresponding to a model comprised of a dis- tributed capacitance and resistance in parallel, as in the case of spinal motoneurons (42, 47), is no assurance that active properties, however small, do not exist. For example, some trigeminal ganglion neurons with relatively large values of the time-invariant conductance exhibited no inductive reactance in their impedance locus diagrams, but these diagrams were closely approximated by the equivalent circuit (Fig. 3B) or linearized models. The estimates of Gr in these neurons were associated with small values of GL, the membrane conductance activated by the oscillatory perturbations of membrane potential. In squid axon (7, 44, 48) and in lamprey spinal neurons (46), the inductive reactance has been attributed to time- and voltage-dependent activation and inactivation of certain ionic channels in the membrane. The demonstration that the inward rectification in cultured dorsal root ganglion cells of the mouse is mediated by Na+ and K’ currents (45) raises our expectations that the voltage responses of trigeminal ganglion neurons to current inputs, which we have observed in both the time- and frequency do- mains, may be a consequence of similar ionic mechanisms.

APPENDIX

To generalize the HH and constant-field equation, we note that the applied membrane current [1(t)] is equal to the sum of the capacitive current and the currents carried by the cations and anions

dV I(t) = c dr + z Ii + MgN Ii (A-0

i=l i=M+ 1

wherei= 1,. . . , M corresponds to M cations andi=M+ 1,. . . , M + N corresponds to N anions. This generalizes Eq. 3 for the electrical circuit model and Eq. 10 for the I-IH equations. A generalization of the current for the ith ion in the HH and constant-field models is given by

where qi generalizes the conductance in the HH model and the permeability in the constant-field model. The function I& depends only on the membrane potential I/ and is parameterized by Cp, C;, and zi, which correspond to the extracellular and intracellular concentrations and valence of the ith ion, respectively.

We assume that the qi depend on L - 1 auxiliary variables, say WI, where 2 = 2, 3, . . . , L. These auxiliary variables generalize pn, h, p2 in the HH equations. Thus

i= 1,2,. . l ,N (A-3)

where the functional dependence of qi on the WI values is determined by the physiological interpretation of the W( and corresponding experimental data. Each WI is assumed to be voltage- and time- dependent and to satisfy a relaxation equation given bY

dW, WY(V) - WI -= dt r1W)

, I= 2,3,. . . , L (A-4

where Wla(V) is the resting state value of WI at membrane potential V and there is a single voltage- dependent time constant T](V) for each W1. Note that each equation contains only WI and this occurs linearly. The linearized equations for both models are obtained by assuming that I(t) is small and let- ting

V = V. + v, Wi = Wlo + wI, I= 2, 3, . . . , N (AS)

where v is the perturbation of V from the resting membrane potential and WI is the perturbation of WI about its resting value WlO* At the resting state of the membrane, there are constraints on the parameters. From Eg. A.1 with no applied current, i.e., 1(t) = 0, and dV/dt = 0, we have the constraint

where subscript 0 means that the membrane potential is at its resting value VO, and WI is given by WI0 = W1”“(VO) with 7 given by 7zo = 7/(Vo).

To simplify the presentation here, we assume that there is only one active ionic component (qI) in the membrane and that it depends on a single auxiliary variable WZ. (The general case of more than one active component with several auxiliary variables is more complicated but can be treated similarly.) We thus have the linear approximations (the first 2 terms of a Taylor series)

W2YV N wo + Y2V (A- 7)

where y2 = dW2”(Vo)/dV and a2 = dT2(V0)/dV. Here (~4.2) q, has the linear approximation


SlW2) = 9lW20) + 2 W2ob2 2

= 910 + 412w2

Taking account of the constraint Eq. A-6, the linearized forms of Egs. A. 1 and A.4 become

dv c; + av + bw, = I(&) (A. 10)

dW2 dt + cw2 - dv = 0 (AX)

where c = l/rzo, d = y2/720 (A.12)

For the HH model, the coefficients a and b are given by

N a= cs iO? b = g12wo - w (A.13)

i=l

where V1 is the Nemst potential for the ionic species with active membrane component. I f the active component in the HH membrane were the potas- sium conductance, then Egs. A.10 and A. I I with w2 = An reduce to Eqs. 10 and 1 I with Am = 0 = Ah.

For the Goldman model with one active component in the membrane, the coefficients a and b are given by

M+N IAIFpio[ 1 -Jyo - e-Avo][C; - Cye-‘;‘vO] a=-C [ 1 - &ivD]2 (A. 14)

i= 1

b = -IflIm2~0

cl’ _ cloe-fi~o

1 _ ,-f v. . (A. 15) 1

If one accounts only for the three major ions, K, Na, Cl, and the active component is due to K permeability, then these expansions become (subscripts K, Na, Cl correspond to the ions)

a= fF(P+ + P-)

[ 1 - epo12 (A. 16)

b = -fFPK2Vo cg - cKiefvo

1 - eflo (A.1 7)

REFERENCES

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4. BROMM, B, Spike frequency of the nodal membrane generated by high frequency alternating current. Pfluegers Arch. 353: l-19, 1975.

where f = F/RT and

P+ = [l + (fvo - l)eflo][PKo(Cg - CKi eflo)

+ PNaO(C& - CNa’eJ”“)] (AM)

p-+1 +fv,- eflo]Pcl&/ - C&elvo) (A.19)

For the generalized model given by Eqs. A.10 and A. 11, we note that Eq- A. IO has the same form as Eq. 3 for the electrical circuit if we let

I= = C dvldt (A.20)

IR = av (A.21)

IL = bw2 (A. 22)

Rewriting Eq. A. 11 using Eq. A.22, we obtain

I dl, -- bd dt

CJ =2, +bd ’

(A-23)

which has the same form as Eq. 6. Thus, since Eqx A.10, A.20, A.21, and A.23 have the same form as Eqs. 3-6, the complex impedance for the generalized model has the same dependence on frequency as that for the electrical circuit, namely

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