Multivariate statistical formulas were used to infer the morphological type and longitudinal position of extracellularly recorded afferents. Efferent fibers were stimulated electrically in the nerve branch interconnecting the anterior and posterior VIIIth nerves. Responses of bouton (B) units depended on their inferred position: BP units (near the planum semilunatum) showed small excitatory responses; BT units (near the torus) were inhibited; BM units (in an intermediate position) had a mixed response, including an initial inhibition and a delayed excitation. Calyx-bearing (CD-high) units with an appreciable background discharge showed large per-train excitatory responses followed by smaller post-train responses that could outlast the shock train by 100 s. Excitatory responses were smaller in calyx-bearing (CD-low) units having little or no background activity than in CD-high units. Excitatory response-intensity functions, derived from the discharge during 2-s angular-velocity ramps varying in intensity, were fit by empirical functions that gave estimates of the maximal response (r MAX), a threshold velocity (v T), and the velocity producing a half-maximal response (v 1/2). Linear gain is equal to r MAX/v S,v S =v 1/2 − v T.v S provides a measure of the velocity range over which the response is nearly linear. For B units,r MAX declines by as much as twofold over the 2-s ramp, whereas for CD units, r MAXincreases by 15% during the same time period. At the end of the ramp,r MAX is on average twice as high in CD as in B units. Thresholds are negligible in most spontaneously active units, including almost all B and CD-high units. Silent CD-low units typically have thresholds of 10–100 deg/s. BT units have very high linear gains and v S < 10 deg/s. Linear gains are considerably lower in BP units and v S> 150 deg/s. CD-high units have intermediate gains and near 100 deg/sv S values. CD-low units have low gains andv S values ranging from 150 to more than 300 deg/s. The results suggest that BT units are designed to measure the small head movements involved in postural control, whereas BP and CD units are more appropriate for monitoring large volitional head movements. The former units are silenced by efferent activation, whereas the latter units are excited. This suggests that the efferent system switches the turtle posterior crista from a “postural” to a “volitional” mode.
In the preceding paper (Brichta and Goldberg 2000), we characterized the discharge properties of labeled bouton, calyx, and dimorphic afferents in the turtle posterior crista, including their discharge regularity and their responses to 0.3-Hz sinusoidal head rotations. The behavior of bouton (B) units varied with their longitudinal position in each hemicrista: units located near the planum were regularly discharging and had small gains and small phase leads re angular head velocity; other units, located near the torus, were irregular, and had large gains and large phase leads; and still other units, found in midportions of the hemicrista, had intermediate properties. Because calyx and dimorphic units had similar discharge properties, they were placed into a single calyx-bearing (CD) category. CD units have an irregular discharge resembling bouton units near the torus, but the gains and phases of the CD units are similar to those of bouton units located in midportions of the hemicrista. Almost all B units had an appreciable background discharge as did many CD units. Other CD units could be silent or nearly silent at rest and many of their other discharge properties were distinctive. From these data we developed multivariate statistical formulas that reliably distinguished between B and CD units and accurately predicted their locations within a hemicrista.
In the present paper, we used these formulas to classify extracellularly recorded units and then studied two other aspects of their discharge. One of these was the response to electrical stimulation of efferent fibers. In mammals (Goldberg and Fernández 1980; McCue and Guinan 1994) and in the toadfish (Boyle and Highstein 1990;Highstein and Baker 1985), efferent activation excites afferents. More heterogeneous responses are observed in the posterior crista (Bernard et al. 1985; Rossi et al. 1980; Sugai et al. 1991) and other vestibular organs of anurans (Sugai et al. 1991). In the present study, heterogeneous responses were found in afferents innervating the turtle posterior crista. As in anurans, some turtle afferents are excited by efferent activation, whereas others are inhibited. In an attempt to understand the functional significance of the heterogeneity, we compared efferent responses in the several afferent groups.
The second discharge property studied was the relation between excitatory response magnitude and stimulus intensity. We became interested in response-intensity relations for three reasons. First, there is an ≈200-fold variation in the linear (near-threshold) gains of individual afferents (Brichta and Goldberg 2000). The variation is 20 times larger than seen in mammals (Baird et al. 1988; Lysakowski et al. 1995). One notion is that the most sensitive fibers monitor small head movements and that progressively less sensitive fibers monitor progressively larger head movements. To evaluate such a recruitment scheme, we needed to determine how the sensitivity of afferent groups was related to their maximal response rates and the stimulus ranges over which their sensory coding was linear. Second, it has been suggested that one function of type I hair cells and calyx endings is to increase the linearity of vestibular transduction (Baird et al. 1988;Goldberg 1996) To test the suggestion, we compared response-intensity functions obtained from B and CD units. The third reason concerned our efferent results. A comparison of the linear stimulus ranges and efferent responses of individual afferents led to the hypothesis that efferent activation switches the turtle posterior crista from a “postural” to a “volitional” mode.
Methods were similar to those described in previous papers (Brichta and Goldberg 1998a, 2000). In brief, we used male and female red-eared turtles [Pseudemys (Trachemys) scripta elegans] weighing 200–400 g and having carapace lengths of 11–14 cm. After decapitation, the head was sectioned in the midsagittal plane, and one of the two half-brains was blocked at the levels of the trigeminal nerve rostrally and the glossopharyngeal nerve caudally. In most instances, the left half-head was chosen. Pivoting the brain stem exposed the dorsal surface of the posterior division of the VIIIth nerve, including the fibers innervating the posterior crista. The isolated half-head was placed on its lateral surface in a recording chamber that was bolted to the superstructure of a computer-controlled rotating device. Extracellular recordings were made with glass micropipettes filled with 3 mM NaCl and having impedances of 20–40 MΩ.
As described elsewhere (Brichta and Goldberg 2000), rotations were used to identify posterior-crista (PC) fibers. For all PC units, a 5-s sample of background discharge was recorded, as was the response to a 0.3-Hz sinusoidal head rotation. The coefficient of variation (cv*), normalized to a standard mean interval of 50 ms, provided a measure of discharge regularity (Brichta and Goldberg 2000; Goldberg et al. 1984). A Fourier analysis was used to extract the fundamental components of the response and the angular-velocity signal from the servo's tachometer. Gains were calculated as the ratio of the fundamental components of the response (in spikes/s) and the angular velocity (in deg/s). Phases were calculated as the difference (in deg) between the response phase and the angular-velocity phase; positive phases correspond to response phase leads. During testing, the posterior canal was tilted 45° from the horizontal plane of rotation. As a result, reported gains should be multiplied by 1/cos(45°) = to get their maximal values. Phases are unaffected (Brichta and Goldberg 1998a).
Classification of units
Extracellularly recorded units were assigned to the CD or B categories based on their quadratic discriminant scores, g(Brichta and Goldberg 2000). The coefficients of the discriminant formulas were set so a unit with a positive score was assigned to the B group, whereas one with a negative score was assigned to the CD group. A z score, z =g/SD, was obtained where SD is the pooled intragroup standard deviation obtained from labeled B and CD units. From thez score, a misclassification probability (p M) was calculated based on the assumption that B and CD units were equally probable (seeBrichta and Goldberg 2000, Fig. 7 D). In addition, a linear regression was used to infer the normalized location (ℓ) of each afferent within a hemicrista (see Brichta and Goldberg 2000, Fig. 9).
Based on their ℓ values, B units were assigned to one of three categories: BP (near the planum, ℓ > 0.8), BM (midportions of the hemicrista, 0.4 ≤ ℓ ≤ 0.8), or BT (ℓ < 0.4). CD units were assigned to CD-high and CD-low categories depending on their having background discharge rates >5 and <5 spike/s, respectively.
All efferent fibers destined for the posterior crista travel in the nerve bundle connecting the anterior and posterior branches of the VIIIth nerve (Fayyazuddin et al. 1991) (Fig.1 A). A Teflon-coated chlorided silver wire with a 0.5-mm exposed tip was placed on the nerve bundle. A second Ag-AgCl electrode was placed on the skull. Electrical stimuli consisted of trains of 100-μs constant-current shocks delivered from a World Precision Instrument 1850A stimulus isolator to the two electrodes. The first electrode was the cathode. To record afferent activity during repetitive shocks, the associated artifacts were canceled by a computer program written in Microsoft C5.0 for a PC computer. On a first pass, an average shock artifact was computed. On subsequent passes, whenever a shock was issued by the program, an inverted version of the average was produced on-line by a DAC and was summed in an operational-amplifier with the raw record. The procedure reduced the artifact by ≈40 dB (Fig. 1 B). The program also collected voltages and spike times and stored them on disk, provided on-line displays, and generated voltages to control devices.
These were obtained from the discharge during 2-s ramps of angular velocity that were the leading edges (up ramps) of velocity trapezoids. Only excitatory ramps were analyzed. For the left half-head, which was used most often, the excitatory direction is counterclockwise (as viewed from above). Maximum ramp amplitudes were 320 deg/s in early experiments and 640 deg/s in later experiments. To achieve the larger velocities, the chair first was rotated in the inhibitory direction to 320 deg/s and kept there until unit discharge returned to its background value. Maximal (640 deg/s) rotations reached 320 deg/s in the excitatory direction at the end of the ramp. Typically, responses were obtained as amplitude was increased in 6-dB steps, usually starting at 10 or 20 deg/s. If the maximum amplitude was reached while the unit remained isolated, a descending series was done, starting 3 dB below maximum and continuing in 6-db steps. At the lowest intensities, responses to several identical ramps sometimes were averaged.
Rates were calculated for each 0.5 s of the ramp and responses (actual rate minus background rate) were plotted against stimulus velocity. Responses were fit by the formula Equation 1and where r(v) is the response to head velocity v, r MAX is a maximum response,v T is a threshold head velocity, andv S is a stimulus (head-velocity) scale factor. At low intensities (v −v T ≪v S) andr(v) ≈r MAX(v −v T)/v S= gV (0)(v −v T) is linear with a gain re velocity of gV (0) =r MAX/v S. We refer to gV (0) as a linear, near-threshold gain. As intensity is raised, gain declines according to the formula, gV (v) =gV (0)/[1 + (v −v T)/v S]2. We take v S =v 1/2 −v T =r MAX/gV (0) as a measure of the stimulus range over which response in nearly linear; v 1/2 is the velocity at which response is 50% of r MAX and also 50% of its linearly extrapolated value, i.e.,r(v 1/2) =r MAX/2 =gV (0)(v 1/2− v T)/2.
Estimates of r MAX,v T, andv S were obtained by nonlinear regression (Levenberg-Marquardt method). If the estimated value ofv T was positive and significantly different from zero (P < 0.05), these values of the three parameters were used. Otherwise, a second regression withv T constrained to zero provided new values of r MAX andv S. Reliable estimates of the parameters required that the maximum velocity,v MAX, exceedv S. So as not to bias the samples toward low values of v S, we did a post hoc analysis to determine an adequatev MAX that would exceedv S for all units in each class. For the last 0.5 s of the 2-s ramp, values ofv MAX (in deg/s) for the various unit categories are followed (in parentheses) by the number of qualifying units over the total number: BP, 450 (8/17); BM, 160 (6/7); BT, 40 (17/17); CD-high, 320 (31/32); and CD-low, 640 (9/24). Linear gains and thresholds could be estimated even whenv MAX <v S.
In analyzing how response-intensity functions change with elapsed time during the ramp, it became important to predict how response dynamics would affect gV (0). To accomplish this, we used a simple linear model of canal dynamics Equation 2where H(s) is a transfer function relating response to angular head velocity,g MID is a midband gain at a frequency ωMID = 1.88 rad/s (≡ 0.3 Hz),TP(s) is a torsion-pendulum model with rate constants α = 0.37 rad/s (≡ 0.059 Hz) and β = 75 rad/s (≡ 12 Hz), and (s/ωMID)k is a fractional operator (Brichta and Goldberg 1998a). By varying the value of k in the transfer function, we generated a predicted relation between the ratio ofgV (0) at two different times during the velocity ramp and the phase of the response to a 0.3-Hz sinusoid (Fig.14 A). The phase was calculated directly from the transfer function, whereas the gain ratio required invertingH(s) (see Brichta and Goldberg 1998a).
Unless otherwise stated, means are presented ±SE.
Responses to electrical activation of efferent fibers
For every unit, we first obtained a sample of its background discharge and a response to 0.3-Hz sinusoidal head rotations. The discharge regularity and rotational responses were used to assign the unit to the B or CD categories and to infer its normalized location (ℓ) within a hemicrista (see Brichta and Goldberg 2000); ℓ = 0 corresponds to the torus and ℓ = 1 to the planum. The central zone of the hemicrista extends from ℓ = 0.4 to ℓ = 0.8. On the basis of their presumed locations, B units were called BT, BM, or BP (see methods for details). CD units were distinguished into those with background discharges >5 spikes/s (CD-high) or <5 spikes/s (CD-low). Efferent responses were collected for a standard train consisting of 20 shocks delivered at a rate of 200/s. Discernible responses were seen at shock strengths of 10–40 μA. Except where noted, responses were obtained with shock amplitudes two to four times threshold and were found to be nearly maximal. Only responses obtained in the absence of head rotations were studied in any detail.
Units differed in their responses to the standard efferent shock train. This was so even in units obtained in a single preparation. Responses from eight units recorded from one half-head are seen in Fig.2; the units are presented in the order in which they were encountered. Six of the units (Fig. 2, A, B, D, E, G, and H) are excited, whereas two show a mixed response including an inhibition during the shock train and a post-train excitation (Fig. 2, C and F). Excitatory responses can be small (Fig. 2, B, E, andH) or large (Fig. 2, A, D, and G). Although not seen in this particular preparation, in some units an inhibitory response during the shock train was not followed by a post-train excitation (see, for example, Fig. 7, A–C).
Relation between unit type and efferent response
The efferent responses shown by units are related to their head-rotation responses and, by inference, to their morphological types and longitudinal positions in the crista. Figure3 illustrates the relation between the cv*, rotational gain, and rotational phase for units distinguished by their efferent responses. Included in the figure are discriminant curves separating units into B and CD categories (see Brichta and Goldberg 2000, Fig. 7). Table1 presents the mean locations for B and CD units showing different kinds of efferent responses.
Both B and CD units showed excitatory efferent responses (Fig. 3,A and B). On the basis of their regular discharge, small rotational gains, and phases, the excited B units were categorized as BP (and less frequently as BM) units. Excitatory responses, measured during the first 100 ms after the standard efferent shock train, were considerably smaller in BP and BM units than in CD units. Mean excitatory responses were 9.9 ± 2.4 spikes/s for 22 B units as compared with 28.8 ± 1.8 spikes/s for 65 CD units. One of the B units was probably misclassified. It had a z score of 0.097 and a misclassification probability,p M > 0.4. In addition, the unit had a large excitatory efferent response of 57 spikes/s, which was unlikely to have been drawn from the same population as the responses seen in the other B units (P ≪ 0.001). Removing the atypical unit reduced the mean excitatory response of the B units to 7.6 ± 0.9 spikes/s.
There was a difference in excitatory-response size between CD-high units (30.1 ± 2.0 spikes/s, n = 58) and CD-low units (17.6 ± 2.1 spikes/s, n = 7). The difference between the two kinds of CD units may be more substantial than indicated by these statistics because they were compiled only for CD-low units with definite efferent responses. In our best preparations, almost all CD-high units showed large excitatory responses, whereas many CD-low units were unresponsive. The lack of responsiveness of CD-low units was not merely due to their being silent. Many CD-low units with background rates of 1–5 spikes/s were unresponsive, and many silent CD-low units remained unresponsive in the presence of activity produced by excitatory rotations.
Inhibited units, whether their responses were purely inhibitory or inhibitory-excitatory, had a background discharge and this was abolished during the efferent shock train (Fig. 2, C andF; see Fig. 7). Most (50/57) inhibited afferents were B units (Table 1). Of the 75 CD units studied, 7 (9.3%) were inhibited during the shock train; each of the 7 inhibited units had a misclassification probability, p M > 0.10, consistent with the possibility that they were actually B units. From their irregular discharge, large rotational gains, and phases, inhibited B units were classified most often as BT units and less often as BM units. Units showing purely inhibitory responses had mean longitudinal positions nearer the torus than did those showing inhibitory-excitatory responses and both groups were nearer the torus than were excited B units (Table 1). There was only a small degree of overlap in the presumed longitudinal positions of B units having inhibitory, inhibitory-excitatory, and excitatory efferent responses (Fig. 4).
Responses are shown for six excited afferents in Fig.5, including three B units with small responses (<15 spikes/s, Fig. 5, A–C) and three CD-high units with large responses (>30 spikes/s, Fig. 5, E–G). Responses begin shortly after the start of the shock train and persist for 500–1,000 ms after the train is terminated. In most excited units, the response shows a single peak, occurring early in the post-train period. In a few CD units, the excitatory response consists of two peaks, one during the shock train and the other in the post-train period (see, for example, Fig. 5 H). Units with two peaks were not placed in a separate category.
The presence of two peaks suggests that per-train and posttrain excitation might reflect separate mechanisms. Figure 5 Dillustrates a variant of the typical excitatory response that also suggests the presence of a distinct posttrain excitation. Here there is no significant response during the shock train, but an excitatory response is seen in the post-train period. Such “delayed excitation” was observed in two B and three CD units. The two B units had near-zero z scores suggesting that they might be CD units. Among the three CD units with delayed excitation, one was highly unlikely to have been misclassified (p M < 0.001).
Longer (10-s) shock trains were used in several CD units. In some of them, shock frequency was varied. Results for one such unit are presented in Fig. 6, left.Per-stimulus responses are seen down to 20 shocks/s. At higher rates (Fig. 6, A and B), there is a response decline during the shock train and a posttrain undershoot. This is followed by a slow response of 10–15 spikes/s that takes 30 s to reach its peak and almost another 100 s to return to the baseline. A slow response is seen down to 10 shocks/s, a rate at which fast, per-stimulus responses are no longer observed. Most (26/29) CD-high units tested with 10-s trains had slow responses, as did 6/9 CD-low units. As was the case for fast (per-stimulus) responses, slow (poststimulus) responses were generally larger in CD-high than in CD-low units. Both fast and slow excitatory responses commonly were seen at shock rates as low as 20/s and occasionally were observed down to 5/s. In all cases, slow responses outlasted the shock train by 50–100 s. The peak post-train response could occur immediately after the train or, as exemplified by Fig. 6 (left), be delayed.
Slow responses also were seen in two of three BP units tested with 10-s shock trains. In the two positive cases, responses peaked immediately after the train and had durations of 20–30 s, shorter than those of CD units.
Responses of three units with purely inhibitory responses are shown in Fig. 7, A–C. For the first two units, the inhibition outlasts the shock train by 2,000 ms. Post-train inhibition is shortened to 500 ms in the third unit and to 150–250 ms in three other units in which a per-train inhibition is followed by a post-train excitation (Fig. 7, D–F).
A simple interpretation of the post-train excitation is that it represents an adaptive rebound from the preceding inhibition. We now present evidence that the excitation cannot be explained in this way. The mixed inhibitory-excitatory response of a BT unit to our standard efferent shock train is seen in Fig. 7 F. Longer (10-s) shock trains varying in shock rate were studied in the same unit (Fig.8). For rates of 100/s and 50/s (Fig. 8,A and B), inhibition persists throughout the shock train and is immediately followed by a post-train excitatory response of 60 spikes/s, which declines to background rates over the next 20–30 s. At 20 shocks/s, inhibition lasts for 2.5 s and is replaced by a small (3 spikes/s) excitatory response (Fig.8 C). Even though inhibition has ceased 7.5 s previously, there is still a large post-train excitation of 40 spikes/s lasting 10 s. It would be even more difficult to ascribe the post-train excitation seen with 10 shocks/s to the consequences of inhibition. Here, the small per-train response consists of a gradually increasing excitation without any sign of a net inhibition (Fig.8 D). Despite this, there is a 15 spikes/s post-train excitation.
An alternative explanation for the post-train excitation would envision that each shock evokes both an excitatory and an inhibitory process (see discussion). The two processes could arise from the convergence of separate excitatory and inhibitory inputs or could represent separate processes in the hair cells and/or afferent fiber triggered by activity in individual efferent fibers. Were separate inputs involved, one might expect that the balance between excitation and inhibition would be altered as shock intensity was raised and as successive inputs were recruited. Such an effect was not seen in three units in which it was sought. The unit shown in Fig. 8 illustrates the point. Shocks were delivered at 20/s, while current strength was varied between a near-threshold 12 μA (Fig. 8 H) and 50 μA (Fig.8 E). Even though there was a large variation in response amplitude, the same response pattern was seen throughout the intensity range. During the shock train, there was a relatively brief inhibition, followed by a more prolonged, but weaker excitation. This was followed by a post-train excitation.
It might be expected that units showing purely inhibitory efferent responses would show relatively simple responses as shock frequency is lowered. Consider the unit (Fig. 7 B) whose response to the standard efferent shock train showed a prolonged inhibition with only a hint of a post-train excitation. The unit was tested with 10-s shock trains (Fig. 6, E–H). As shock rate was varied, only inhibition was seen during the train. Note that there is still a clear inhibitory response at a shock rate of 5/s. A post-train excitation is seen. It differs from that observed in units with mixed (inhibitory-excitatory) responses in being smaller, of shorter duration, and in reflecting the magnitude and duration of the preceding inhibition. These are properties that would be expected of a postinhibitory rebound.
As in our efferent studies, the background discharge and responses to 0.3-Hz head rotations were used to classify afferents into several classes. After this preliminary testing, response-intensity functions were determined for each of the four consecutive 0.5-s intervals (designatedt 1–t 4) in 2-s velocity ramps.
General features of the functions and their fit by Eq. 1 are illustrated in Fig. 9, A andB, based on t 4 data from a BP unit. The function is plotted in both semilogarithmic (Fig.9 A) and double logarithmic coordinates (Fig. 9 B).Equation 1 provides an excellent fit;r MAX is denoted by the thick horizontal lines and v 1/2 =v S by the thick vertical lines. The function in semilogarithmic coordinates is sigmoidal with an inflection point at v S. Departures from linearity are better illustrated in the double-logarithmic plot. Because the threshold is zero, the low-intensity points in Fig. 9 B are fit by a unity-slope (- - -) line. Data for a BM unit with a nonzero threshold is seen in Fig. 9, C and D. As a result of the nonzero threshold, the double-logarithmic plot (Fig.9 D) is concave downward even near threshold.
Figure 10 includes results for four units illustrative of the relations for the several unit classes. In Fig. 10, A–D, response rates are plotted as a function of time within the ramp and are compared with linear extrapolations, which for each unit were calculated from the expressions,gV (0)(v −v T), separately calculated for the four periods,t 1–t 4. Companion response-intensity functions are presented for the four periods (Fig. 10, A1–D1). Response-intensity functions for the t 4 period are plotted for units of each particular class in Fig. 11 and mean values of the t 4 and (when available) the t 1 parameters ofEq. 1 are included in Table 2. Unless otherwise stated, parameters cited in the text are for thet 4 period. The exception isv T, which was more accurately estimated from the lower velocities obtaining duringt 1.
These units have modest values ofr MAX, combined with small values of linear (near-threshold) gain, gV (0). As a result, v S is relatively large, approaching 200 deg/s (Table 2). Each of the 17 BP units in the sample had an appreciable background discharge and only one of them had a statistically significant t 1threshold. Based on their responses to sinusoidal head rotations (Brichta and Goldberg 2000, Fig. 14), BP units can be described as approximately velocity sensitive with k ≅ 0 in Eq. 2. Reflecting these response dynamics, linear gains re head velocity change only slightly with elapsed time during velocity ramps. The mean (±SE) ratio between thet 1 andt 4 linear gains was 0.99 ± 0.06 (•, Fig. 14 A).
The large values of v S imply that the responses of BP units remain approximately linear for large stimulus amplitudes. In addition, the units are nearly velocity sensitive. These two features can be seen in the ramp responses of an individual BP unit (Fig. 10 A). For the ramp to a final velocity of 40 deg/s, responses increase almost linearly with time, more or less paralleling angular velocity. The responses to the 80 and 160 deg/s ramps show a partial saturation that becomes prominent fort 3 andt 4. As the final velocity reaches 320 or 640 deg/s, the response is linear only duringt 1 and reaches a nearly constant value of 70–80 spikes/s duringt 2–t 4.
The t 4 response-intensity functions for BP units show large linear stimulus ranges, withv S values typically being between 120 and 240 deg/s (Fig. 11, C and F).
Linear (near-threshold) gains of BT units are much higher than those of BP units and are associated with small values ofr MAX and especially small values ofv S (Table 2). All 17 BT units had background rates >5 spikes/s; only 4 of them had a statistically significant t 1 threshold, the largest of which was 0.24 deg/s. From their responses to sinusoidal head rotations between 0.1 and 1 Hz, BT units can be approximated byEq. 2 with a fractional exponent, k ≈ 0.7 (Brichta and Goldberg 2000, Fig. 14), closer to acceleration (k = 1) than to velocity (k = 0). As a consequence of the partial acceleration sensitivity of BT units, responses to small-amplitude (1 deg/s) ramps reach nearly constant values from t 2onward (Fig. 10 B) even though head velocity continues to grow. As a result, there is a time-dependent decrease ingV (0). The mean ratio of the linear velocity gains for t 1 andt 4 is 2.9 ± 0.4 (n = 17) (▾, Fig. 14 A).
Because of their small v S values, BT units have very small linear stimulus ranges. This can be seen in the responses of the unit in Fig. 10 B. Responses are nearly linear for a ramp with a final velocity of 1 deg/s. For ramps to 5 deg/s, responses are close to linear for the first 0.5 s but then become saturated. At higher intensities (20–320 deg/s), even the early response falls short of linearity.r MAX declines with time, from 134 spikes/s for t 1 to 57 spikes/s fort 4 (Fig. 10 B1).
Response-intensity functions for several BT units are presented in Fig.11 A. Typically, responses become nonlinear for head velocities of 5 deg/s and saturate between 10 and 50 deg/s.
There were only a few BM units and a disproportionate number of them (4/7) had low background rates (<5 spikes/s). Two of the low-rate units had nonzero t 1 thresholds.r MAX andv S values for BM units were intermediate between those for BT and BP units (Table 2 and Fig.11 B). Misclassification probabilities (p M) are high, ranging from 0.10 to 0.49 and suggesting that some of the units may have been misclassified.
Linear (near-threshold) gains are intermediate between those of BT and BP units (Table 2). v S values overlap those of BM and BP units and are considerably higher in CD-low than in CD-high units. CD units have distinctively hight 4 r MAX values, 1.5 to 3 times those of B units. Correlated with the difference in their background rates, 21/24 CD-low units, but only 1/32 CD-high units had statistically significant nonzero t 1 thresholds. Thresholds of five CD-low units were so high that they did not respond or only responded at the very highest intensity duringt 1. For this reason, we could not compute a t 1 threshold or linear gain. These five “very insensitive” units were compared with other CD-low units during t 4. Thresholds ranged from 37 to 102°/s for the five units (Fig.12) and from 0 to 67°/s for the other 19 CD-low units. In addition to their high-thresholds, the very insensitive CD-low units had distinctively low linear gains, with a mean (in spikes · s−1/deg · s−1) of 0.37 ± 0.11 as compared with 1.21 ± 0.15 for the other CD-low units and 0.67 ± 0.09 for BP units (Fig. 12).
Responses of individual CD units are illustrated in Fig. 10,C and D. Time profiles of responses to low-velocity ramps fall between those of BP and BT units, consistent with CD response dynamics being between those of the other two groups (Brichta and Goldberg 2000, Fig. 15). Unlike the responses of BT units, those of CD units continue to grow during velocity ramps. At the same time, and unlike the situation in BP units, the increase falls short of the linear growth in stimulus velocity.
Response-intensity curves for t 4 are shown for several CD-high (Fig. 11, D and F) and CD-low units (Fig. 11, E and F). As can be seen in Table 2, CD-low units have v Svalues during t 4 that are higher than those of BP units and considerably higher than those of CD-high units. None of the aforementioned very insensitive units were tested with velocities that allowed us to estimatev S orr MAX, so they are not included in Fig.11, E and F, or Table 2. As can be seen in Fig.12, responses of the very insensitive units remain approximately linear to 280 deg/s, the highest velocity tested in them. The figure also includes the mean response-intensity functions for BP and CD-low groups, other groups with large values ofv S. Clearly, the very insensitive fibers have smaller gains and considerably larger linear stimulus ranges than either of the other groups.
Changes in response-intensity functions with elapsed time
Response-intensity functions are shown in Fig. 10,A1–D1, for the four time periods,t 1–t 4. The functions for the BP unit at different times are almost superimposable (Fig. 10 A1). In contrast, BT functions are shifted both horizontally and vertically with elapsed time (Fig.10 B1). CD units show horizontal shifts larger than those of BP units and smaller than those of BT units (Fig. 10, C1 andD1). Vertical shifts are not evident in CD units. In this section, we consider how time-related changes in response-intensity functions are affected by the choice of stimulus dimensions and by time variations in each unit's r MAXvalues.
There are three conditions required for response-intensity functions to be superimposed as time progresses. First, the stimulus dimension,x, has to be chosen so that the linear (near-threshold) gain, gx (0,t) =ḡx (0), remains constant with time (Fig. 13). The other two conditions are that the threshold,x T(t) =x̄ T, measured in the xstimulus dimension, and the maximal response,r MAX(t) =r̄ MAX, are both time invariant. The constancy of ḡx (0) andx̄ T assures the superimposition of the linear (near-threshold) part of the function. In a similar way, the constancy of r̄ MAX andḡx (0) assures the time invariance ofx̄ S =r̄ MAX/ḡ x(0). Finally, the constancy of r̄ MAXassures the time independence of the high-intensity portion of the function. The entire function, when stated in terms of x, isr(t,x) =r̄ MAX(x −x̄ T)/[x̄ S+ (x − x̄ T)] and is time invariant.
To examine whether these conditions held, we first determined that the ratio betweengV (t 1) andgV (t 4) reflected response dynamics, in particular, the transfer function,H V (Eq. 2 ). Figure14 A plots thegV ratio versus φ, the phase of the response to 0.3-Hz rotational sinusoids. A semilogarithmic regression provided a good fit (—, r = 0.83, n = 92) and was statistically indistinguishable from the relation predicted from Eq. 2 (- - -). These results show that the transfer function is sufficiently accurate to predict salient features of the ramp response (see also Brichta and Goldberg 1998a). The conclusion is important because calculation of the appropriate stimulus dimension is more easily done in the frequency domain rather than in the time domain.1
For the response-intensity functions to be superimposable also requires that r MAX be independent of time. The condition is met only approximately by CD units (Fig. 14 C). For these units, r MAX increases by 10–15% between t 1 andt 2 and then remains almost constant for the remainder of the ramp. An even larger discrepancy in the opposite direction is seen in BT units, where there is a twofold decrease in r MAX betweent 1 andt 4, with most of the decrease occurring between t 1 andt 2 (Fig. 14 B). Althoughr MAX fort 1 could not be calculated for most other B units, for several of them we could calculate values for t2–t 4. A decline in r MAX betweent 2 andt 4 was seen in BP and BM, as well as BT, units. The ratio between the r MAXvalues for t 4 andt 2 for eight BP and BM units had a mean of 0.76 ± 0.08, compared with 0.70 ± 0.02 for 17 BT units. Both ratios are significantly different from unity (2-sidedt-tests, P < 0.01) but not from each other.
Figure 15 depicts how response-intensity functions change with elapsed time for BP, BT, and CD-high groups. In the top row, the stimulus dimension is head velocity and the functions compare favorably with those for individual units (Fig. 10, A1–D1). On thebottom, the stimulus dimension appropriate to each kind of unit is used. Because the dimension was chosen so that the linear gains for the two times coincide, the low-intensity parts of the functions superimpose. The high-intensity portions diverge reflecting time-dependent changes in r MAX. So for BP and BT units, the high-intensity responses are higher fort 1 than fort 4. For CD-high units, there is a smaller effect in the opposite direction.
Relation between sensory-coding and efferent-response properties
In mammalian vestibular organs (Goldberg and Fernández 1980; McCue and Guinan 1994) and in the toadfish horizontal crista (Boyle and Highstein 1990), efferent activation predominantly excites afferents. More heterogeneous efferent responses are seen in the posterior crista (Bernard et al. 1985; Rossi and Martini 1991; Rossi et al. 1980; Sugai et al. 1991) and other anuran vestibular organs (Sugai et al. 1991). Reminiscent of the latter studies, some units in the turtle posterior crista are excited, while others are inhibited when efferent axons are stimulated. Furthermore units differing in the polarity and magnitude of their efferent responses have distinctive sensory coding properties. Using multivariate statistical formulas (Brichta and Goldberg 2000), we have been able to distinguish extracellularly recorded units into B and CD categories, to estimate the longitudinal position of either kind of unit in a hemicrista and to relate the kind and location of an afferent with its response to efferent activation.
In the turtle posterior crista, B units with small excitatory efferent responses are located near the planum and are regularly discharging, have small rotational gains, and near-zero rotational phases. B units with large inhibitory efferent responses are found near the torus and are irregularly discharging, have large rotational gains, and large phase leads. Still other B units have mixed efferent responses, consisting of an initial inhibition followed by an excitation. The sensory coding properties and inferred longitudinal positions of the units with mixed responses are intermediate between those of the excited and inhibited B units. Among CD units, those with an appreciable background discharge (CD-high units) have large excitatory efferent responses, whereas those with little or no resting activity (CD-low units) have smaller excitatory efferent responses or no efferent responses at all.
We now compare our findings concerning B units with those in anamniotes (fish and amphibians), which lack CD units. B units located near the planum (toadfish: Boyle et al. 1991; frog:Honrubia et al. 1989; Myers and Lewis 1990; turtle: Brichta and Goldberg 2000) have excitatory efferent responses (toadfish: Boyle and Highstein 1990; turtle: present study). In all species studied, such units have a regular discharge together with low rotational gains and phases. Efferent responses in these units are usually small; this may explain why most regular fibers in anurans were described as nonresponsive (Sugai et al. 1991). B afferents located nearer the center of the crista or isthmus have higher rotational gains and phases (toadfish: Boyle et al. 1991; anurans:Honrubia et al. 1989; Myers and Lewis 1990; turtles: Brichta and Goldberg 2000). In most species, these units have an irregular discharge, but in the toadfish some of the units are regular (Boyle et al. 1991). As compared with the fibers located near the planum, those found near the isthmus have large efferent responses the polarity of which depends on species. Almost all isthmus units are excited in the toadfish (Boyle and Highstein 1990), about equal numbers are excited or inhibited in anurans (Bertrand et al. 1985; Rossi and Martini 1991; Rossi et al. 1980; Sugai et al. 1991) and almost all are inhibited in the turtle (the present study). In the previous paper (Brichta and Goldberg 2000), we emphasized that in all nonmammalian species there is a similar longitudinal gradient in the sensory-coding properties of B units as one proceeds from the planum toward the isthmus There are also gradients in efferent response properties, but these vary across species.
Results in mammals resemble those in turtles in two ways. First, regularly discharging mammalian afferents resemble comparable turtle fibers in their sensory-coding properties (Baird et al. 1988; Lysakowski et al. 1995) and in their small excitatory efferent responses (Goldberg and Fernández 1980). In terms of their structure, the mammalian regular fibers differ from those in nonmammalian species in including dimorphic, as well as bouton, fibers and in being distributed throughout a peripheral zone that surrounds the central zone both near the planum and at the base of the neuroepithelium (Baird et al. 1988). Second, large excitatory efferent responses are obtained from irregularly discharging mammalian afferents, which include calyx and dimorphic fibers in the central zone. In their sensory-coding and efferent-response properties, the centrally located mammalian units resemble those of turtle CD-high units but not CD-low units. Mammals may be unique in one way. Unlike turtles and anamniotes, they do not have irregular B units with large rotational gains and phases and large efferent responses. Rather all mammalian B units are regularly discharging (Lysakowski et al. 1995) and, so, may be presumed to have small, excitatory efferent responses (Goldberg and Fernández 1980).
Mechanisms of efferent response heterogeneity
Some of the afferents innervating the turtle posterior crista are excited by efferent activation, others are inhibited, and still others show mixed responses. There are several mechanisms that could be responsible for this response heterogeneity. In anurans, both the efferent inhibition and the efferent excitation seen in irregular fibers are mediated by presynaptic efferent inputs that modulate the frequency of miniature excitatory postsynaptic potentials (mEPSPs) arising in the hair cell and recorded in the postsynaptic fiber (Bernard et al. 1985; Rossi et al. 1980,1994; Sugai et al. 1991). Rossi et al. (1980, 1994) have found that efferent inhibition is mediated by nicotinic receptors, whereas efferent excitation involves purinergic transmission. In contrast, Bernard et al. (1985)implicated nicotinic receptors for both kinds of efferent responses. Efferent inhibition involves the entry of Ca2+through nicotinic channels and the subsequent turning on of Ca2+-activated K+(K Ca) channels that hyperpolarize the hair cell (Fuchs and Murrow 1992a,b). Nicotinic channels not linked to K Ca channels should cause excitation both by depolarizing the hair cell and increasing intracellular Ca2+. There are also postsynaptic efferent terminals on calyx endings (Lysakowski 1996;Lysakowski and Goldberg 1997; Wersäll and Bagger-Sjöbäck 1974) and on afferent processes terminating as boutons (Lysakowski and Goldberg 1997;Sans and Highstein 1984). The large efferent excitation seen in turtle CD afferents and in CD afferents located in the mammalian central zone is likely the result of postsynaptic efferent inputs. It is unclear whether the smaller excitation seen in regularly discharging fibers involves presynaptic, postsynaptic, or both kinds of efferent synapses.
A mixed efferent response consisting of per-train inhibition followed by a post-train excitation is seen not only in the turtle posterior crista but also in anuran vestibular organs (Rossi and Martini 1991; Sugai et al. 1991) and in lateral-line organs (Russell 1971a; Sewell and Starr 1991). Two of our observations suggest that the post-train excitation is not merely a release from inhibition. First, by lowering shock frequency, we could replace the per-train inhibitory response by an excitatory response without abolishing the post-train excitation. Second, the profile of the response did not change as shock intensity and the number of efferent fibers recruited were altered. The results suggest that an impulse in each efferent fiber gives rise to a fixed sequence, f(t), of inhibition followed by excitation. To explore this idea we setf(t) = A Eexp(−αEt) − A Iexp(−αIt) and assumed that the responses to successive shocks summed linearly. To obtain a per-train inhibition followed by a post-train excitation required that of the two rate constants, αE < αI. KeepingA E and A I fixed resulted in similar response profiles as shock frequency was changed. For the particular values of A E andA I used in Fig.16 A, the time at which the response changed from inhibition to excitation increased only slightly as shock rate was lowered. This is in contrast to observed responses (Fig. 8, A–D), which consist almost entirely of inhibition at high shock rates and of a brief inhibition followed by a more prolonged excitation at low shock rates. As can be seen in Fig.16 B, these features can be reproduced by assuming thatA I increases with shock rate whileA E remains constant. The assumption is equivalent to assuming that the inhibitory component shows facilitation during repetitive activation of efferent synapses while the excitatory component does not.
Excitatory efferent responses in mammalian afferents consist of fast and slow response components (Goldberg and Fernández 1980). A slow excitatory response also has been seen in lateral lines after the fast inhibitory response is blocked pharmacologically (Russell 1971a; Sewell and Starr 1991). We looked for slow responses in turtle posterior-crista afferents in the period after 10-s shock trains. A slow excitatory response was commonly seen in CD-high units and could take 30 s to reach its peak and another 100 s to decline to zero. A smaller, shorter slow response was seen in BP units. In BT and BM units showing mixed inhibitory-excitatory responses, the posttrain excitatory response could last ≥20 s and might qualify as a slow response. Even in units showing purely inhibitory response to 100-ms shock trains, a small, brief excitation was seen after 10-s shock trains. Here a distinction between fast and slow responses could not be based on kinetics, which was comparable for the per-train inhibition and the post-train excitation. In general, distinctions based solely on kinetics are likely to become obsolete as more is learned about the neurochemical and cellular bases of the response components.
Excitatory response-intensity relations
The turtle vestibular system has to monitor a wide range of head velocities. Postural control during quiet standing requires the monitoring of small rotations, possibly ≤1 deg/s. In contrast, velocities of 200–600 deg/s occur during rapid forward extensions of the head and neck or during diving behavior; the movements themselves last 200–300 ms (J. Brannigan, A. M. Brichta, and R. A. McCrea, unpublished observations). Hence the velocities of natural head movements match those used in our experiments, but natural accelerations can be considerably higher than those we could produce. As a result our velocities are relatively small at brief times.
One way of interpreting our results is that very sensitive BT units are designed to monitor the small head rotations involved in postural control, whereas less sensitive BP and CD units are more appropriate for monitoring rapid, volitional movements. To illustrate the situation, we first consider variations inv S, the head velocity above threshold leading to a half-maximal response. As such,v S may be taken as a measure of the range of angular head velocities over which unit response is approximately linear. Even if we confine ourselves to B units,v S varies almost 30-fold duringt 4, from 5–10 deg/s in BT units to 150–200 deg/s in BP units. An even larger, possibly >100-fold variation may occur for earlier times becausev S for BT units is ≈2.5 deg/s duringt 1, whereas the corresponding value ofv S for BP units is expected to be >300°/s (Table 2).
There is an enormous difference in the linear stimulus ranges for BT and BP units. But if we assume that all units operate in the same part of their normalized response-intensity function, their differential sensitivity, expressed as a Weber fraction, will be similar. To see this, we can state stimulus velocities, v −v T =fv S, as a fraction ofv S and assume that all units operate around a stimulus level, f̄ v S, which is a fixed fraction of v S. The Weber fraction is defined as W = Δs/s, where Δs is the stimulus increment needed to produce a fixed response, Δr. Algebraic manipulation of Eq. 1 yields W = Δr (1 +f̄)2/f̄r MAX, inversely proportional to r MAX. The results in Table 2 indicate that for eithert 1 ort 4 there is less than a twofold variation in r MAX between BP and BT units. Many sensory systems obey Weber's law, so that the just-noticeable-difference (Δs) is proportional to stimulus magnitude (s) or, alternatively, the Weber fraction (Δs/s) is constant (see, for example,Stevens 1951). The presence of a variety of B units allows the turtle posterior crista to maintain an almost constant Weber fraction over a large dynamic range.
What about CD units? In previous papers (Baird et al. 1988; Goldberg 1996), we suggested that type I hair cells and calyx units are intended to extend the dynamic range of vestibular transduction. The electrophysiology of type I and type II hair cells is consistent with the suggestion. In particular, the input impedance of type I hair cells is as much as 20-fold smaller than that of many type II hair cells (Brichta et al. 1998;Rennie and Correia 1994; Rüsch and Eatock 1996). A smaller input impedance would serve to extend the dynamic range of type I cells by reducing their receptor potentials. This is the first study to test the suggestion at the level of afferents. Results are equivocal and depend on elapsed time. Table 2presents data for t 4 and data or extrapolations for t 1. For the later time, there is an overlap in the linear stimulus range for B and CD units. The v S values of CD-high units are intermediate between those of BT and BP units, whereas those of CD-low units are only slightly larger than those of BP units. For the earlier time, BP units probably have the largest values ofv 1/2 although this conclusion rests on the unproven assumption that the decline in r MAXwith time is similar in BP units to the decline observed in BT units.
There is one group of presumed CD units that have high-thresholds coupled with very low linear gains. These very insensitive CD units extend the linear stimulus range beyond that reached by BP units. We relied on a unit's background firing and its responses to modest head rotations to detect its presence. As a result, we may have underestimated the proportion of CD-low units, especially those that were very insensitive. At the same time, one has to be suspicious of such units when encountered in an in vitro preparation. Because of this concern, we looked for (and found) such units in an intact, barbiturate-anesthetized preparation (Brichta and Goldberg 2000), even though blood gases were maintained at normal concentrations by artificial respiration (Frankel et al. 1969). We calculate for the very insensitive units av S=r MAX/g V(0) +v T = 630 deg/s. In doing so, we have used the mean values of g V(0) = 0.37 spikes · s−1/deg · s−1and v T = 70 deg/s from the very insensitive units (Fig. 12) and the mean value ofr MAX = 230 spikes/s from other CD-low units (Table 2).
The situation in the turtle posterior crista can be compared with that existing in mammals. In the latter, there are no units resembling BT units in having large linear (near-threshold) gains and in becoming saturated at relatively low excitatory head velocities. Nor are there many silent units with low sensitivities. The highest discharge rates seen in mammalian vestibular afferents are 400–600 spikes/s (J. M. Goldberg and C. Fernández, unpublished observations). Subtracting a background discharge of 50–100 spikes/s leaves anr MAX of 300–550 spikes/s. Linear gains in mammals range from 0.2 to 2 spikes · s−1/deg · s−1 (Baird et al. 1988;Lysakowski et al. 1995), sov S is 150–275 deg/s in the most sensitive units and 1,500–2,750 deg/s in the least sensitive fibers. This may be compared with estimated v S values in the turtle of 150–300 deg/s in BP and in many CD-low units and 600–650 deg/s in very insensitive CD-low units. These figures suggest that canal units can have more than twice the dynamic range in mammals as compared with turtles. Voluntary head movements, at least in humans, do not reach 1,000 deg/s (Grossman et al. 1988). This implies that the least sensitive afferents in mammals will remain linear for even the largest naturally occurring head movements. At the same time, there are no afferents in mammals specialized to monitor the small head movements involved in postural adjustments or silent, very insensitive afferents seemingly specialized to extend the dynamic range. In this sense, the turtle crista is more highly differentiated than is that of mammals.
One difference between turtle B and CD units involves the time dependence of their r MAX values. The values for B units decline by as much a twofold fromt 1 to t 4, whereas those for CD units show a slight increase over the same time period. One result is that r MAX values of CD units are two to four times higher than those of B units duringt 4, but the difference is obscured duringt 1. As the time dependence is one of the few differences seen between B and CD units, its cellular basis is of interest. In the case of auditory afferents, comparisons between hair-cell receptor potentials and afferent discharge have suggested that the saturation of response-intensity functions occurs at later stages of transduction, possibly involving depletion of afferent neurotransmitter (Eatock et al. 1992; Smith 1985). It is unclear whether a similar conclusion holds for vestibular transduction. Assuming that it does, it is possible that the persistently high values of r MAX for CD units may reflect an unusual mode of synaptic transmission between them and type I hair cells. In particular, it has been suggested that the postsynaptic depolarization produced by the afferent neurotransmitter is supplemented by the accumulation of K+ ions that are released by the hair cell, accumulate in the synaptic cleft, and depolarize the postsynaptic membrane (Goldberg 1996). Accumulation is calculated to have slow kinetics (≈100 ms); this may explain the slight increase in r MAX betweent 1 and t 2. Because the concentration of K+ ions in the cleft depends almost entirely on the current entering the mechanoelectric transducer channel, depolarization from this source may be expected to persist even when neurotransmitter is depleted.
Results from the two parts of this paper suggest a specific function for efferent control in the turtle posterior crista. BT units are of particular interest. From their response-intensity functions, including their very large linear (near-threshold) gains and their limited r MAX values, these units have a limited dynamic range (<10 deg/s), which would make them useful in monitoring the small head movements involved in postural control but not in the guidance of larger, voluntary head movements. BT units are inhibited by activation of the efferent system. Most other afferents show efferent excitation (BP and CD-high units) or no efferent responses (many CD-low units).
To draw functional inferences, we need to consider the conditions leading to efferent discharge. There is some suggestion that efferents respond to voluntary head movements, either by an efference-copy mechanism (Boyle and Highstein 1990; Roberts and Russell 1972; Russell 1971b; Tricas and Highstein 1990, 1991) or by a reafference mechanism (Blanks and Precht 1976; Dickman and Correia 1993, Gleisner and Henriksson 1964; Myers et al. 1997; Precht et al. 1971; Schmidt 1963). Many of the aforementioned studies show that the efferent activation is not directionally specific in the sense that the efferents going to a specific organ respond in anticipation of or as a direct consequence of voluntary head movements that would activate any vestibular organ. Such an efferent activation would silence the BT units involved in postural control while at the same time exciting other afferents involved in the guidance of voluntary movements. Were the responses of BT units merely saturated by large-amplitude movements the signals need not disrupt central processing. But we have seen that the BT cells show time-dependent nonlinear distortions, which might interfere with central processing and make it advantageous to silence their discharge peripherally. The background discharge of excited units would be increased, and this would serve to reduce inhibitory silencing, especially in CD-high units with large efferent responses. At the same time, excitatory dynamic ranges would be curtailed. This may provide a rationale as to why the efferent responses in CD-low units are so weak because these units seem specialized to ensure the largest possible excitatory dynamic range.
Viewed more generally, the efferent system can be thought of as switching the turtle posterior crista from a postural to a voluntary mode. Three limitations of these ideas can be mentioned. First, we need more detailed information on the physiological conditions leading to efferent discharge. Second, the scenario depends on the efferent inhibition of BT units, which are presumably specialized for the monitoring of postural movements. But only in turtles are almost all such cells inhibited. In other species, many of the comparable cells are excited (Bernard et al. 1984; Boyle and Highstein 1990; Rossi et al. 1980; Sugai et al. 1991). One suggestion is that nonlinear distortion is less disruptive of central processing in these other species. Third, the scenario presumes that efferent inhibition in the turtle abolishes the responses of BT units to even intense head rotations. This last point has yet to be examined.
Drs. R. A. Eatock and A. Lysakowski made helpful comments on the manuscript.
This research was supported by National Institute on Deafness and Other Communication Disorders Grant DC-02508 (J. M. Goldberg, principal investigator).
Present address of A. M. Brichta: Discipline of Anatomy, Medicine and Health Sciences, University of Newcastle, Callaghan, NSW 2308 Australia.
Address for reprint requests: J. M. Goldberg, Dept. of Neurobiology, Pharmacology and Physiology, University of Chicago, 947 E. 58th St., Chicago, IL 60637.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
↵1 Let R(s) andV(s) be the Laplace transforms of the response and of the stimulus waveform expressed in terms of head velocity. Then R(s) =H V(s)V(s), where H V(s) is the transfer function re velocity. The appropriate stimulus dimension is defined byX(s) =H V(s)V(s), in which case the transfer function relatingR(s) andX(s) isH X(s) = 1. It follows that the gain is constant in both the frequency and time domains.
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