physiology of peripheral neurons innervating semicircular ...mined by the mechanics of the cupula-...

Physiology of Peripheral Neurons Innervating

Semicircular Canals of the Squirrel Monkey.

II. Response to Sinusoidal Stimulation and

Dynamics of Peripheral Vestibular System

CESAR FERNANDEZ AND JAY M. GOLDBERG

Departments of Surgery (Otolaryngology), Physiology, and Theoretical Biology, University of Chicago, Chicago, Illinois 60637

A CENTRAL PROBLEM in vestibular physiology has been the elucidation of the dynamics of the peripheral vestibular apparatus. Following the development of the torsion-pendulum model by Steinhausen (24, 25), many efforts have been directed toward the determination of the time constants of the model. That the model, itself, might not adequately describe the transfer characteristics of the peripheral system has seldom been questioned. Further, the pro- cedures employed to estimate the time constants have had their drawbacks.

The dynamics of the system have some- times been deduced from hydrodynamic principles (5, 6, 15, 17, 22, 23). Such an analysis permits the inference of only some of the relevant variables. In the torsion- pendulum model, it will be recalled, the angular deflection of the cupula g(t) is related to the angular acceleration a(t) by the equation

d2W + n W) o- dt2

- + AE(t) = @a(t) dt (1)

0 being the moment of inertia of the cupula-endolymph, II the viscous-damping couple, and A the elastic-restoring couple. The response of the system is governed by two time constants, zl = II/A and z2 = O/ II. Some estimate can be made by hydrodynamic calculations of the time constant x2. However, the more important time constant z1 cannot be determined, since the

Received for publication December 31, 1970.

elastic-restoring couple has not been di- rectly measured. Another limitation of the hydrodynamic approach is that it ignores the question as to what extent the dynamics of the peripheral apparatus are determined by the mechanics of the cupula- endolymph system and to what extent by the physiology of the hair cells and of the nerve fibers innervating them.

An alternate approach has been to study the dynamics of the overall response of the system, usually in humans. Response measures have ,included subjective estimates of the sensation of bodily rotation (5, 10, 13), vestibular-induced nystagmus (10, 13, 19), and the oculogyral illusion (11). Such studies have resulted in estimates of both zl and z2. But it is not clear whether these estimates simply reflect peripheral sensory mechanisms, as is usually assumed, or are influenced by the complicated physiology. of the central vestibular pathways.

One aim of the present series of experiments was to describe the dynamics governing the response of peripheral vestibular afferents to angular accelerations. Some observations of relevance were presented in the previous paper (9). There the response to steps of constant angular acceleration was considered. The present paper contin- ues this analysis. The response to sinusoidal stimulation will be described. From these observations, a descriptive model of the peripheral vestibular system is devised; the model, even though it is the simplest linear system which will approximate the response dynamics, is more elaborate than

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662 C. FERNANDEZ AND J. M. GOLDBERG

a torsion pendulum. The theoretical response of the model to acceleration steps is then calculated and compared to the actual step response of the neurons.

METHODS

Observations were made in the same animals as were used in the previous paper (9), where a full description of techniques may be found. Sine-wave inputs to the velocity servomecha- nism were produced by a function generator (Wavetek, model 116), whose distortion was less than 0.5%. The frequency was varied over a range extending from 0.006 to 8.0 Hz (see Table 1). Over this range, the velocity of the rotating device faithfully followed the input signal except that, at 8.0 Hz, the device exhibited a phase lag of some 8O. A stimulus sequence consisted of an initial stationary period of 20 set, during which the resting discharge could be determined. There followed a period of sinusoidal oscillations, succeeded by a stationary or recovery period of 40 set duration. The number of oscillations was varied from 4 at the low end of the frequency spectrum to 256 at the high end. The response was assumed to reach a steady state in not less than 20 set after the start of stimulation.

Estimates of the amplitude and phase of the steady-state response were made from its fundamental component, extracted by a Fourier analysis of the average response to several successive sine-wave cycles. Gains were calculated by di- viding the response amplitude, obtained from the Fourier analysis, by the magnitude of the peak acceleration (or that of the peak velocity). The analysis was also used to measure the nonlinear distortion and to compute the 2nd through 10th harmonics. Such calculations are -meaningful only if the neuron’s discharge is not silenced during a portion of the stimulus cycle. When the discharge was silenced, ampli- t-tide, phase and distortion measurements were based on a nonlinear regression scheme which fitted the average response to a sine wave clipped at a value of zero.

RESULTS

The response to sinusoidal stimulation was studied in 57 units; 28 of the neurons innervated the horizontal canal, 19 the superior canal, and 10 the posterior canal. In all the units, sinusoidal stimulation was begun only after the response to a series of velocity trapezoids had been studied. We first investigated the response at several sine-wave frequencies, the peak accelera-

TABLE 1. Stimulus conditions

Frequency, Hz

.006 .0125 .025 .050 .lO 025 .50

1.00 2.00 4.00 8.00

Maximum Peak Acceleration,

deg/se&

10 20 40 80 80 80

80 or 160 160 or 320 1600r 320 160or320 160or320

No. of Units

6 30 52 36 50 54 56 51 27 22

8

tion for each being that listed in Table 1. The table also includes the number of units studied at each frequency. Then, if the unit was still isolated, we held the frequency constant and varied the peak acceleration in 6-dB steps from the maximum listed in the table down to 10” (and some- times S”)/sec 2. Intensity series for at least one frequency were completed in 12 units.

Pattern of response to sinusoidal stimulation

The typical course of the response to sine waves is illustrated by data obtained from unit 51-8 (Fig. 1). The stimulus frequency was 0.0504 Hz. The discharge, after a transient period lasting roughly .5 cycle, waxes and wanes around the resting level in an approximately sinusoidal manner. When the stimulus ends, the firing rate returns to the resting level with an expo- nential time course.

A more detailed steady-state analysis of the unit’s response is presented in Fig. 2. The stimuli were all 0.0504 Hz and the peak acceleration was varied from 5.0 to 80.6”/sec? Each curve represents the average of the response to 6 successive cycles, the. period of analysis beginning 1.5 cycles (29.76 set) after the start of stimulation. The resulting functions are roughly sinusoidal in shape, though it will be noted that they are not symmetrically disposed around the resting level. When the different curves are compared, the response is seen to be proportional to acceleration magnitude and the phase lag, amounting to some 58O, is more or less constant. Gain

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TIME (seconds)

FIG. 1. Response of unit 51-8 (superior canal) to 8 cycles of a sinusoidal stimulus, 0.0504 Hz, 80.6°/sec2. Stimulus ends at 185.72 sec. Each point, average discharge rate for IL0 th of sine-wave cycle (0.496 set). Vertical marks, instants of peak excitatory acceleration. Lower and upper horizontal lines, respectively, resting discharge before and after stimulation.

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FIG. 2. Steady-state responses of unit 51-8 (superior canal) to series of 0.0504-Hz sine waves. See key for accelerations. Zero degrees, peak excitatory acceleration. Inset plots gain and phase as functions of acceleration; included are points for one acceleration (10.1 ~/se&) not shown in main graph,

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C. FERNANDEZ AND J. M. GOLDBERG

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0 90 180 270 360 0 90 180 270 360

400.

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200

100

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PHASE ANGLE re MAXIMUM EXCITATORY ACCELERATION

FIG. 3. Steady-state response to sinusoidal stimulus. Points, average response to three successive sine-wave cycles; curve, best-fitting sine-wave function centered around resting discharge (horizontal continuous line). A: unit 25-8 (horizontal canal), 0.0243 Hz, 38.9” /se&; resting discharge, 114.0 spikes/set; average discharge during sine-wave cycle, 114.3 spikes/set. B: unit 49-3 (superior canal), 0.0241 Hz, 38.6O/sec2; resting discharge, 150.8 spikes/ set; average discharge (horizontal dashed line), 159.9 spikeslsec.

and phase data are summarized in the inset. The gain did not vary by more than 10% over the 16-fold range of stimulus magnitudes and the phase angle varied by no more than 3’. A similar constancy of gain and phase values was observed in the other 11 units studied in this way.

Distortion analysis

In the preceding paper (9) it was shown that the response of the peripheral vestibular system could, under certain circum- stances, be quite nonlinear. Such nonlin- earities are also seen in the response to sine waves. Unit 5.5-8 (Fig. 3A) was unusual in that the data points were quite closely fit by a sinusoidal function symmetrically placed around the resting level. The total nonlinear distortion was 3.5%. Unit 49-3 (Fig. 3B) was more typical. Here there are obvious discrepancies between the best-fitting sine wave and the actual response. The distortion is largely of an asymmetrical type, the peak excitatory response being larger than the peak inhibitory response. One consequence is that the average dis-

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FIG. 4. Distortion analysis based on sample of 19 units. Bars, relative amplitudes; dashed lines, relative amplitudes corrected for random variation in discharge rate. Left: U, unrestricted distortion; R, restricted distortion. Right: amplitude spectrum.

charge rate during stimulation is higher than the resting level. In such a situation, it is useful to quote two distortion figures. The first, the unrestricted distortion, reflects the deviation of the response from a sine wave whose mean value is adjusted to the average rate and, hence, involves only the harmonics of the fundamental frequency. The other, the restricted distortion, also includes the distortion introduced by the shift in d-c level. For unit 49-3, the unrestricted distortion was S.S%, the restricted distortion 13.9%.

Figure 4 summarizes the distortion data for 19 units. In all cases, the stimulus had a frequency of 0.025 Hz and a peak acceleration of 40” /sec2. The 19 units were chosen from a larger population of 52 units. To be included, a unit had to meet two criteria. First, the resting discharge, measured after stimulation, had to differ from that measured before stimulation by less than 5% of the amplitude of the response. And second, the response amplitude could not exceed the resting level, i.e., the neuron’s discharge could not be silenced during any portion of the sine- wave cycle. The response to the 0.025-Hz stimulus was reasonably large, the response amplitude in the 19 units averaging 65.0 spikeslsec. The mean unrestricted distor-

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tion was 7.3%, the mean restricted distortion 13.6% (Fig. 4, left).

The asymmetrical nature of the distortion is reflected in the amplitude spectrum (Fig. 4, right). The major distortion prod- ucts are the d-c component and the even harmonics, particularly the 2nd harmonic. That the d-c distortion was not simply due to random variations in the resting level is indicated by the fact that, in 16 of the 19 units, the average discharge was larger than the resting discharge, the probability of this occurring by chance being small (P < .005). Further, in calculating the average amplitude of the d-c component, the sign of the component was taken into consideration. The average d-c distortion was 11.6 & 1.6%, which was significantly different from zero (t test, P < .OOl). Finally, the variation between the resting levels measured before and after stimulation aver- aged, in this selected sample of units, only 2.9% of the response magnitude and could have had only a negligible influence on the d-c distortion figure of 11.6%.

A source of spurious distortion needs to be considered. This is the random distortion introduced by moment-to-moment variations in discharge rate. A study of such variations occurring during the resting activity provided an estimate of 3.2% for the average amplitude of this distortion, the value for any particular neuron being closely related to the regularity of spacing of its action potentials. When the contribu- tion of the random component was sub- tracted, the average unrestricted distortion was reduced to 6.6y0, the average restricted distortion to 13.27& Further, on the not unreasonable assumption that the power contained in the random distortion was uniformly distributed throughout the spectrum, the amplitudes of the individual harmonics could be corrected. The corrected values are indicated in Fig. 4 by dashed lines. As might be expected, the random distortion had an appreciable effect only on the amplitudes of the upper harmonics.

Gain and phase as functions of sine-wave frequency

Gains and phase lags were relatively un- affected when stimulus magnitude was var-

ied over wide limits (see Fig. 2). This justifies presenting the data in the normalized form of Bode plots. Such plots for two units-M-11 and 51-8-are included in Fig. 5. The points at the left are the gains re acceleration, those at the right the phase lags re acceleration.

Consider first unit 64-11 (Fig. 5, solid circles). The behavior of the unit deviates from that of a torsion pendulum (indicated by the solid line) at both low and high frequencies. At high frequencies, there is a gain enhancement; also the phase lag, rather than continuously increasing with frequency, reaches a maximum at 0.25 Hz and then begins to diminish. The discrepancy at low frequencies is mainly reflected in the phase lag being less than expected. The simplest linear transfer function which will approximate the data is of the form

H(s) ZaS (1 + TLS>

=-

(1 + w) (1 + w)(l + T#) (2)

The factor HTP = I/[(1 + w)(l + zzs)] is the transfer function of the torsion pendulum with z1 = II /A and z2 = O/II. The choice of the values z1 = 5.7 set and x2 = .003 set will be gone into in the next sec- tion. The term HA = z,s/( 1 + QS) results in a phase lead at low frequencies and is the frequency-domain representation of the Young-Oman adaptation operator (9, 27). The Constant TA is the adaptation time constant. The operator HA should also lead to a gain attenuation at very low frequencies, well below 0.0125 Hz. The term HL = (1 + zLs) is a lead component and re- produces the high-frequency deviations from the torsion-pendulum model, including the gain enhancement and the progressive Phase lead. If one assumes that the transier function bP provides an ade- quate depiction of the relation between angular acceleration and cupular displacement, then the meaning of the operator HL can be deduced. The presence of the high- frequency lead component HL would then imply that the system is sensitive both to cupular displacement and to the velocity of the displacement, the time constant zL reflecting the relative sensitivities to these two aspects of cupular motion.

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; TORSION 1 - :, PENDULUM (1+5.7s)(l t 003s)

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I I .0125 ’ .050 ’

I .250 ’ ,

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.0250 .I00 .500 2.00 8.00 .0250 .I00 .500 2.00 800

FREQUENCY (Hz) FIG. 5. Bode plots (re acceleration) for two units. Left: gain plots. Right: phase plots. Points, experi-

mental values; curves, values calculated from transfer functions. In Figs. 5, 7, and 12, log-gain at 0.25 Hz arbitrarily set to value expected from torsion-pendulum model. Acceleration magnitudes as in Table 1, Unit 64-11, posterior canal; unit 51-8, horizontal canal.

In unit 51-8 (Fig. 5, open circles), there is less discrepancy from the torsion-pendulum model. This is reflected in the corresponding transfer function. The adaptation operator is omitted and the value of zL is only 0.015 sec.

A comparison of units 64-11 and 51-8 suggests that neurons exhibiting a relatively high degree of adaptation are also characterized by a pronounced high-frequency lead component. This was gener- ally true. The phase lag at 1.0 Hz may be taken as a measure of the magnitude of the lead component, that at 0.0125 Hz as a measure of the degree of adaptation. Fig- ure 6 plots these two phase lags against one another for 29 units. A strong positive relation is indicated. The product-moment correlation is 0.59 (t test, P < .OOl). The figure also indicates a great variability in the observed phase angles for the 0.0125- and LO-Hz stimuli. The spread of phase angles at low frequencies reflects, in large part, differences among units in their adaptive proper ties; that at high frequencies differences in the magnitude of the high- frequency lead component. Figure 7 (right) shows the mean and standard deviation of phase angles at different frequencies, as determined from the entire population of

units. The standard deviation is least at 0.25 Hz-a frequency where the phase angle should be only imperceptibly affected by either adaptation or the high-frequency lead component- and systematically in- creases as one departs from this frequency in either direction.

Determination of time constants

The transfer function presented in equation 2 requires the specification of four time constants. We begin with a considera-

40 50 60 70 80 90

PkASE LAG AT 1.0 Hz

FIG. 6. Comparison for 29 units of phase lags (re acceleration) at 0.0125 and 1.0 Hz. Accelera- tion magnitudes as in Table 1. N, number of units; r, product-moment correlation; P, significance level of correlation (t test).

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- 3.0

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73 l ------a 80s (I + 049s)

t-s 1 +8Os (I +57s)(l+d03s)

& s 0 TORSION 1 --- T PENDULUM (I+ 5 7s) (I+ c)O3s)

Le. _- ~-- ---- --- .0125- T .250 1 l.bo 4.x -

.025 .I 00 ,500 2.00 800 025 .I 00 .500 200 8.00

FREQUENCY (Hz)

FIG. 7. Average Bode plot (f-e acceleration). Points, mean values; bars, standard deviation of values; curves, values calculated from transfer functions. See Table 1 for sample size at each frequency and for acceleration magnitudes.

tion of the time constants of the torsion pendulum. The constant z1 = ll/A can be estimated from the phase lag + at a suffi- ciently low frequency f by the formula z1 = (1/2nf)tan 4. The equation is valid provided that adaptation has a negligible effect. Otherwise the value of + and, hence, that of z1 will be reduced. A convenient frequency is 0.025 Hz. The only neurons used in the estimation were those-the LA units (see ref 9)-whose response to velocity trapezoids was characterized by a relatively small degree of adaptation. Twelve such units were available. Their average phase lag was 42.0 k 1.1 O, equivalent to a zl of 5.73 t- 0.23 sec. This value corresponds reasonably well with that determined from previous time-domain analyses (9). The estimated time constant, based on the response during constant accelerations, was 5.41 -+ 0.38 sec. That based on the recovery from such accelerations was 5.99 k 0.16 sec.

Our observations, since they extended only to 8.0 Hz, are insufficient to provide a direct experimental measure of r2 = O/II. Some estimate of this constant can, however, be made from hydrodynamic consid- erations. What seems to be required is a solution of the Navier-Stokes equation (ref

18, p. 642) for the complicated geometry represented by the canal, the associated ampulla, and the utriculus. No one to our knowledge has accomplished this, though Steer (23) has solved the equation for a straight tube. The approximate value of the time constant, so derived, is

where p1 is the first zero of the zeroth-order Bessel function of the first kind; P and q are, respectively, the density and viscosity of the endolymph; and r is the internal radius of the tube, in this case the radius of the membranous canal. This radius has been measured by Igarashi (14) and leads to a value of r2 equal to 0.005 set in man and 0.003 set in the squirrel monkey.

Once zl and r2 have been calculated, some estimate can be made of the other two time constants. Figure 7 presents Bode plots based on the entire population of units. The gain plot is to the left, the phase plot to the right; mean values and standard deviations are included. In deter- mining time constants, phase information is more revealing than is gain information. Note that at’ all frequencies the phase lag falls short of that predicted by the torsion-

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pendulum model. The discrepancy will be denoted by A+ and, by convention, will be taken as positive. At a very low frequency ft TA should be given by the expression (l/2 nf)tan(z/2+). Thirty units were studied at a frequency of 0.0125 Hz. The average phase angle was 15.1 O, giving a A+ of 9.1 O and a %A of 80 sec. The standard deviation of A+ at this frequency was appreciable, in large part presumably reflecting variations in zg. From the standard deviation, it may be surmised that ZA ranges among the population from 30 set to values which are so large as to be indistinguishable from infin- itv.

-The time constant zL may be estimated from the formula (1/2nf)tan(A+), where A+ is the phase discrepancy at a relatively high frequency f. The average value of zL, based on all data available between 1.0 and 8.0 Hz, was 0.049 sec. The standard deviations of A+ at these frequencies indicates that ~~ may vary within the population from 0.013 to 0.094 sec.

The average transfer function thus ar- rived at is

80s H(s) =

(1 + .049s)

(1 + 80s) (1 + 5.7s)(l + .003s) (4)

The agreement between the transfer function and the data is, as indicated by Fig. 7, reasonably good in the low- and high-frequency ranges. There is, however, a discrepancy of some 5” between the observed and predicted phase angles in the range of 0.1-0.5 Hz. The discrepancy cannot be significantly reduced by adjusting any of the four time constants in H(s), without at the same time upsetting the -fit in other parts of the frequency spectrum. One could of course add an appropriate lead-lag compensation factor to the transfer function. A factor-(1 + .64s)/(l i- .52s)-would eliminate the discrepancy. But such an ad hoc procedure has little to recommend it since the compensation factor does not have a ready physiological interpretation.

Comparison between time- frequency-do imain analyses

and

The transfer function of equation 2 can be used to obtain a theoretical response to

60 80

TIME (seconds)

FIG. 8. Comparison for unit 69-3 (superior canal) of experimental response (points) and theoretical response (curve) to a constant-acceleration stimulus, 40 set duration, 12.5°/sec2. Each point is based on five stimulus presentations and is the average of excitatory and inhibitory responses. Theoretical calculations based on equation 2 with

Zl = 5.7 set; z2 = .003 set; TA = 60 set; zL = -075 sec.

velocity trapezoids. A comparison between a theoretical and an actual response is presented in Fig. 8. The empirical points rep- resent the average of the excitatory and inhibitory responses to a 40-set velocity trapezoid. The parameters for the calcula- tion were, with the exception of the sensitivity factor, determined from sine-wave data. Thus, ZA was deduced from the phase lags at 0.006 and 0.0125 Hz, zL from the phase lags at 1.0 and 2.0 Hz. The value of the sensitivity factor used in the calcula- tion, 3.44 spikes-set-l/deg-set-2, is in reasonable agreement with the value of 3.00 derived from the response to a sine wave of 0.25 Hz.

As is exemplified in Fig. 8, there should be a close relation between the responses to sine waves and to velocity trapezoids. Each kind of stimulus may be used to measure the sensitivity of the neuron. The degree of adaptation, observed in the response to velocity trapezoids, should be reflected in the response to low-frequency sine waves. And the high-frequency lead component, best seen when sine waves are presented, should also affect the initial

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PERIPHERAL VESTIBULAR NEURONS IN MONKEY II

0

GAIN re VELOCITY

.6 . N=29

r =-0.66 P< .OOl

-.I

FIG. 9. Sensitivity of individual units determined by response to constant excitatory acceleration (ordinate) and by response to sinusoidal stimulus (abscissa in spikes-sec- l/deg-sec- 1). Constant acceleration: 10 set duration, 30°/secX Sine-wave stimulus: 0.25 Hz, 80” /sec2 (peak acceleration). Solid curve, regression line; dashed curve, theoretical relation based on equation 4. N, number of units; r, product-moment correlation; P, significance level of correlation (t test).

portion of the response to constant accelerations. These relations are explored in Figs. 9-11, where each point represents a comparison for an individual neuron of its response to sine waves and to velocity trapezoids.

Figure 9 indicates the precise relation between the sensitivity, as measured by the cumulative excitatory response to a constant acceleration of 10 set duration (see ref 9), with that determined from the steady-state response to a sine wave of 0.25 Hz. The solid line is based on a regression analysis. The dashed line is the relation expected from the average transfer function (equation 4). The slopes of the two lines differ by less than 10%.

The greater the adaptation of the unit, the smaller should be the phase lag at low frequencies. This expectation is verified in Fig. 10. The ordinate plots the adaptation index (AI) derived from the response to velocity trapezoids (see ref 9), the abscissa the phase lag to a 0.0125-Hz sine wave. A fairly high correlation exists between these two measures of adaptation and the theoretical relation calculated from the transfer function (equation 2) provides a fair ap- proximation to the data.

FIG. 10.

termined Adaptation of individual units as de-

by recovery from constant accelera ti .ons (ordinate) and by phase lag (re acceleration) of response to 0.0125-Hz, 20°/sec2 sinusoidal stimulus (abscissa). Curve, theoretical relation based on equation 2 with zl, r2, and zL as in equation 4. N, number of units; r, product-moment correlation; P, significance level of correlation (t test).

-10 .I0 20 30

PHASE LAG AT 0.0125 Hz

Perhaps the most unexpected finding of the present paper, given -our observations on velocity trapezoids, was the demonstra- tion of a high-frequency lead component. Can the effects of this factor be identified

the response to constant-acceleration in stimuli? Were no lead component present, the initial response to such a stimulus should increase linearly with time. Con- sider the transfer function (equation 4) with the lead component HL removed. The ratio of the average responses during the periods 0.5-1.0 and 0.0-0.5 set after the start of the constant acceleration should be slightly greater than 3.0. Reasonably wide variations in time constants, including large, but reasonable, decreases in z1 and TAP would only reduce the constant-acceleration ratio to on the order of 2.9. In contrast, the lead component can have a dra- matic effect. If, for example, zL is set to 0.05 set, the resulting ratio will be 2.66. In- creasing the time constant to 0.1 set will

the ratio to 2.41. . Measures of the decrease ratio were obtained in 50 neurons. The accelerations were 150°/sec2 and only excitatory responses were used. The average value of the ratio was 2.47 -t- 0.06, which would correspond to an average rL of 0.086

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N=50 f = 0.34 P< .Ol

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50

PHASE LAG AT 1.0 Hz

FIG. 11. Magnitude of high-frequency lead component as measured by early part of excitatory response to constant-acceleration stimulus (ordinate) and by phase lag (re acceleration) of response to l.O-Hz sinusoidal stimulus (abscissa). Constant acceleration, 150O/se&. Peak acceleration of sine wave as in Table 1. Dashed curved line, theoretical relation based on equation 2 with zl, rz, and %A as in equation 4. N, number of units; r, product- moment correlation; P, significance level of correlation (t test).

-+ 0.013 sec. Note that this estimate of zL is roughly double that obtained from the sine-wave analvsis.

Despite the discrepancy, the results may be taken as a verification in the time- domain of the existence of a high-frequency lead component. A further comparison is possible. It would be expected that the phase lag in the response to a l-Hz sinusoidal stimulus would be posi- tively related to the value of the constant- acceleration ratio. These two variables are plotted against one another in Fig. 11. There is a wide spread in the data, which is largely attributable to the fact that the acceleration ratios were calculated, with few exceptions, from the response to a single velocity trapezoid. Nevertheless, a statistically significant trend is observable. Again, the discrepancy is apparent. In 33 of the 50 units, the ratio is less than that predicted from the sine-wave response (P < .05).

DISCUSSION

Most treatments of the dynamics of the peripheral vestibular apparatus (5, 6, 10, 11, 13, 15, 17, 19, 22-25) have assumed that the system behaves like a heavily damped torsion pendulum. It is probable that the torsion-pendulum model adequately represents the motion of the cupula and endolymph (22, 23). But the model cannot account for the response dynamics of the first-order afferents innervating the semicircular canals. Here two additional factors come into play. They affect the response to both velocity trapezoids and sinusoidal stimulation. One factor is adaptation. The second is a high- frequency lead component, which suggests that the neurons, in addition to being sensitive to cupular displacement, are also sensitive to the velocity of the displacement. The adaptation need not be considered further, since it was dealt with in a previous paper (9). The high-frequency lead component does merit some comment.

One of the main theoretical deductions made on the basis of the torsion-pendulum model was that the peripheral vestibular system acts as a quite faithful velocity transducer for the brief angular accelerations encountered in everyday life. The conclusion must be modified. Figure 12 replots the data of Fig. 5 in terms of angular velocity, rather than angular acceleration. Consider first the behavior expected of the torsion pendulum (solid lines). The gain re velocity is flat in the physiological range between 0.25 and 8.0 Hz and the phase lag is less than -+ 10’. Introduction of the lead component, as is illustrated by the curves for unit 64-11, results in a high-frequency gain enhancement and a progressive phase lead. Both effects become substantial only for fre- quences above 1 Hz. At 0.5 Hz, for example, the phase lead is only some 15”, an observation in reasonable accord with the phase measurements made by Melvill Jones and Milsum (16) in second-order neurons; further,’ there is no appreciable gain enhancement. In contrast, the gain enhancement at 8.0 Hz amounts to 10 dB and the phase lead to some 60’.

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E 0 t s .25’ w r > t

g-25

-l

I I .0125 1 ,050 1

I .250 1

I I 1.00 1 4.00 1

.0250 .I 00 ,500 2.00 8.00 .0250 .I00 .500 2.00 8.00

FREQUENCY (Hz)

- ..-- ._---- --~ (I + .063s)

--?1+32.4s) (1+57s)(1+.003s) -100

'3i s(l+ 015s)

ii fi+ 5 7s)(l+.OO3s)

&

t

4 TORSION ~~ a -80

E \ PENDULUM (1+5.7sh +.003s) \

0 9

-60

Y ; t?! -40 t

FIG. 1% Bode plots (re velocity) for two units. Data replotted from Fig. 5.

That the presence of the lead component may profoundly affect the response to a physiological stimulus, such as a rapid rotation of the head, is depicted in Fig. 13A. Shown are a series of calculated responses to a rotation whose velocity profile is .5 cycle of a ~-HZ sine wave. The value of xL, the time constant of the lead component, is varied over the range encountered experimentally. As might be-expected from the steady-state analysis, an increase in zL produces an increase in gain and a phase lead re velocity. The phase lead, in turn, results in the response becoming progres- sively biphasic. These tendencies become more pronounced, the higher the frequency (Fig. 13B).

What might be the functional significance of the lead component? One way to view its function is in terms of dynamic load compensation. Certainly the lead component compensates, within the frequency range of interest, for the inertia of c the cupula- and endolymph. But, if this were ail that was involved, the value of zL need not be larger than 0.003-0.005 sec. Probably of greater consequence are the loads represented by the various vestibular reflex pathways. Such pathways may in- troduce delays, phase lags, and high-fre-

quency gain attenuations. The lead component could compensate for these effects. Optimum compensation would be accomplished were each vestibular afferent mainly routed into those pathways whose dynamics most closely matched the magnitude of the neuron’s lead component.

Unfortunately, it is not possible to specify how the lead component works in most vestibular reflexes. The simplest way to gain some insight into its role would be to compare the response dynamics when natural stimuli are employed with the dynamics characterizing the response to direct electrical stimulation of the vestibular nerve. There have been numerous studies of the reflex responses to vestibular nerve stimulation (14, 7, 26), but few of these have concentrated on dynamic properties. An exception is a paper by Partridge and Kim (20). Individual ampullary nerve bundles were stimulated with sinusoidally modulated pulse trains and the resulting variations in the isometric tension of the triceps surae muscle recorded. The dynamics resembled those of a first-order lag element with a time constant of 0.18 set, in series with a 15-msec conduction delay. Obviously the gain attenuations and phase shifts, observed by Partridge and Kim in

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0

TIME (seconds)

FIG. 13. Theoretical responses to series of head rotations, each consisting of .5 cycle of a sine wave. Calculations based on equa tioq 2 with -cl, z2, and ZA as in equdon 4. Peak velocity assumed same for all rotations. Ordinate in units of response expected from corresponding torsion-pendulum model. A: effect of variation of zL on response to 4.0-Hz stimulus. B: effect of variation of sine-wave frequency, rL = .045.

the frequency range above 1 Hz, could be most effectively compensated were the afferents involved in the reflex characterized by a lead compone nt with a tim e constant zL near the time constan t of the afore- mentioned lag element. The lead components observed in our experiments were never this prominent, the highest value of zL being on the order of 0.1 sec. But even such a lead component could serve to extend the bandwidih of the reflex and would bring the variations in isometric tension mori closely into phase with angular velocity of head movement.

the

An interesting question which may be raised is the extent to which the dynamics of vestibular reflexes are determined by peripheral mechanisms, rather than by the central pathways. The reflex dynamics observed by Partridge and Kim (20) were similar to the dynamics relating motor nerve stimulation to muscle contraction. Similarly, if natural stimulation had been used, it - may be assumed that the overall dvnamics of the reflex would also reflect the dynamics of the peripheral vestibular apparatus. Although the central nervous system may affect the dynamics in any of a number of ways, its only certain con- tribution is to the 15msec conduction delay. That the central nervous system might play only a minor role in deter- mining reflex dynamics is, at first glance,

not unreasonable. On both the afferent and efferent sides, the reflex components consist of mechanical systems characterized by time constants which are much larger than those thought to govern the discharge of central neurons and these relatively slow mechanical systems may dominate the dynamics of the overall pathway.

Can it be argued that the central nervous system plays a minor role in the dynamic properties of all vestibular reflexes? One need only consider the vestibulo- oculomotor system to be disabused of this notion. The fast phase of nystagmus is unquestionably of central origin. Even the dynamics of the slow phase cannot be ex- plained on the basis solely of peripheral mechanisms. This is most clearly indicated by the fact that the discharge of abducens motoneurons, during both phases of nys- tapus, parallels the movement of the eye and not the time course of afferent nerve discharge (12, 21). What is suggested then is that, in the study of any particular vestibular function, consideration must be given both to central and peripheral mechanisms. Unfortunately, there has been a tendency in much of vestibular research to explain all observations, including the results of human psychophysical experiments, on the basis of the motion of the cupula and endolymph. This may lead to erroneous conclusions concerning the physics of the semicircular canals. One example may be cited.

In their pioneering studies, van Egmond, Groen, and Jonkees (5) attempted to de- termine the time constants of the torsion- pendulum model from human judgements of the sensation of turning. The first-order time constant z1 = II /A was estimated from cupulometric studies to be about 10 sec. As was reviewed in a previous paper (9) this value is probably low, in part due to the intrusion of sensory adaptation. In any case, nystagmographic studies consis- tently provide somewhat larger values of zl, usually on the order of 15 set (10, 13). A more serious ‘discrepancy was encountered in the estimation of zz = @/II. Here van Egmond and his colleagues placed subjects on a torsion swing and determined the resonant frequency mo, i.e., the frequency for which the subjective sensation

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would just be in phase with the velocity of the swing. The resonant frequency was 1 rad/sec, which corresponds to a value of z2 = .05 sec. This differs from hydrodynamic calculations by one order of magnitude. Further, any relation between subjective estimates of u. and peripheral mechanisms is, at best, indirect. None of the neurons examined in the present study had a resonant point. This was largely a consequence of the high-frequency lead component. At all frequencies, the discharge led the velocity of the stimulus, though this phase lead reached a minimum at 0.25-0.50 Hz. In some units, the minimum phase lead amounted to no more than 5O. Perhaps coincidentally, the frequency of 0.25-0.50 Hz is quite close to the resonant point defined in the human experiments.

There is no reason to believe that the dynamics governing the discharge of human vestibular afferents differs in any fundamental way from those described in the squirrel monkey. One quantitative difference, though, is striking. The average value of z1 in the squirrel monkey is 5.7 set, roughly 2-3 times smaller than the values estimated from human studies. The difference can be partly related to the physical dimensions of the canals in the two species. Let R be the radius of cur- vature of the canal and r the internal radius of the membranous canal. According to the theory elaborated by van Egmond et al. (5) and by Melvill Jones and Spells (17), the time constant z1 should be proportional to R2 and inversely proportional to r? From the physical measurements of Igarashi (14), the expected time constant in man should be 1.68 times larger than that in the monkey. Steer’s (22, 23) somewhat more detailed analysis of the motion of the cupula-endolymph system leads to a not dissimilar conclusion. Note that the calculated ratio of 1.68 is less than that obtained experimentally. The discrepancy probably reflects the fact that we still do not possess a complete physical theory of cupular motion. What is particularly lack- ing is a knowledge as to how the elastic- restoring forces of the cupula are gener- ated.

SUMMARY

A study was made as to how peripheral vestibular afferents innervating the semicircular canals of the squirrel monkey (Saimiri sciureus) respond to sinusoidal stimulation. Frequency was varied from 0.006 to 8.0 Hz. In many ways the responses resembled those expected of a linear system. The nonlinear distortion, which mainly reflected asymmetries between excitatory and inhibitory responses, was reasonably low, averaging some 13y0. For a given sine-wave frequency, gains and phase lags were relatively independent of stimulus magnitude. This justified presenting the data in the normalized form of Bode plots. From such plots, linear transfer functions were constructed.

The response to sinusoidal stimulation deviates in two respects from that predicted by the torsion-pendulum model. At low frequencies, the phase lag re acceleration is less than expected. This is a consequence of sensory adaptation. At high frequencies, there is a gain enhancement; also the phase lag re acceleration, rather than continuously increasing with frequency, reaches a maximum and then begins to diminish. One interpretation of this high-frequency lead component is that the afferents are sensitive both to cupular displacement and to the velocity of that displacement. A possible function of the presumed velocity sensitivity is that it may permit the peripheral vestibular system to compensate for the dynamic loads represented by the various vestibular reflex pathways.

The prominence of both sensory adaptation and the high-frequency lead component varied from neuron to neuron and were related to one another. The greater the adaptation exhibited by a unit, the more pronounced tended to be the lead component.

There is, as expected, a close relation between the responses to sine waves and to velocity trapezoids. The transfer function, deduced from the response of a neuron to sinusoidal stimulation, may be used to reconstruct the major features of the response to velocity trapezoids. The two kinds of response provide equivalent measures of the unit’s sensitivity. The de-

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gee of adaptation observed in the response Domizi was of particular help in the design and

to velocity trapezoids is also reflected in implementation of computer systems. Mr. William

the response to low-frequency sine waves. Abend participated in some of the experiments.

Finally, the velocity sensitivity, best seen This study was supported by Grants NB-01330,

NB-05237, and FR-00013 from the National Insti- when high-frequency sine waves are pre- tutes of Health. Publication costs were partly de-

sented, also affects the initial portion of frayed by a Sloan Foundation Grant.

the response to constant accelerations. A preliminary account of this work was presented at the 1969 Spring meeting of the Acoustical

ACKNOWLEDGMENTS Society of America (8). -

Mr. Bernard Murphy was responsible for the Both authors are Research Career Development construction of electronic equipment. Dr. Dario Awardees, National Institutes of Health.

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physiology of peripheral neurons innervating semicircular ...mined by the mechanics of the cupula-...

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