Figure 1.
Central vestibular neurons respond nonlinearly to sums of noise stimuli.
(A) Vestibular information is transmitted from the sensory end organs through two types of afferents (regular and irregular) that converge on first order central neurons within the vestibular nuclei. (B) During the experiment the monkey was comfortably seated in a chair placed on a motion platform. (C–E) The firing rate (red traces) of an example central vestibular neuron in response to noise stimuli (black traces) whose frequency content spanned 0–5 Hz (C), 15–20 Hz (D), and 0–5 Hz+15–20 Hz (E). The upper insets show the power spectrum of each stimulus, while the lower insets show the power spectrum of the firing rates (red). (F) Population-averaged normalized gains curves for central neurons. Note the attenuated response at low frequency (0–5 Hz, arrow). (G) Population-averaged normalized gains for central neurons. Here and in all subsequent figures, the bands (F) and error bars (G) show 1 SEM. The firing rate estimates were obtained by convolving the spike trains with a Kaiser filter (see Materials and Methods).
Figure 2.
Afferents respond linearly to sums of noise stimuli.
(A, B) Population-averaged normalized gain curves as a function of frequency for regular (A) and irregular (B) afferents. (C, D) Population-averaged normalized gains for regular (C) and irregular (D) afferents. (E) Population-averaged attenuation indices for central neurons, regular afferents, and irregular afferents.
Figure 3.
Central vestibular neurons but not afferents display a nonlinear relationship between output firing rate and input head velocity.
(A) Output firing rate as a function of head velocity. The inset shows the instantaneous firing rate and the head velocity stimulus as a function of time and the various symbols correspond to different values of the head velocity and the corresponding firing rates. If the firing rate is related linearly to the head velocity stimulus, then the curve relating the two should be well fit by a straight line. The slope of this line is then the response gain. (B) Population-averaged firing rate response as a function of head velocity for afferents when stimulated with 0–5 Hz noise alone (solid blue) and concurrently with 15–20 Hz noise (solid black). In both cases, the curves were well fit by straight lines (dashed lines) and largely overlapped (0–5 Hz alone: R2 = 0.99, slope = 0.70 (spk/s)/(deg/s), y-intercept = 98 spk/s; 0–5 Hz with 15–20 Hz: R2 = 0.99, slope = 0.72 (spk/s)/(deg/s), y-intercept = 98 spk/s). (C) Population-averaged firing rate response as a function of head velocity for afferents when stimulated with 15–20 Hz noise alone (solid red) and concurrently with 0–5 Hz noise (long dashed black). Both curves were again well fit by straight lines (short dashed lines) and largely overlapped (15–20 Hz alone: R2 = 0.99, slope = 1.97 (spk/s)/(deg/s), y-intercept = 102 spk/s; 15–20 Hz with 0–5 Hz: R2 = 0.99, slope = 2.06 (spk/s)/(deg/s), y-intercept = 102 spk/s). Note, however, the increased slope with respect to panel B. (D) Population-averaged firing rate response as a function of head velocity for central neurons when stimulated with 0–5 Hz noise alone (solid blue) and concurrently with 15–20 Hz noise (solid black). In both cases, the curves were well fit by straight lines (dashed lines) although the solid black curve had a lower slope (i.e., gain) than the solid blue curve (0–5 Hz: R2 = 0.98, slope = 1.56 (spk/s)/(deg/s), y-intercept = 67 spk/s; 0–5 Hz with 15–20 Hz: R2 = 0.87, slope = 0.83 (spk/s)/(deg/s), y-intercept = 81 spk/s). (E) Population-averaged firing rate response as a function of head velocity for central neurons when stimulated with 15–20 Hz noise alone (solid red) and concurrently with 0–5 Hz noise (long dashed black). While both curves were similar and largely overlapped, they were not well fit by straight lines (short dashed lines) that underestimated the firing rate for head velocities <−10 deg/s (15–20 Hz: R2 = 0.64, slope = 2.32 (spk/s)/(deg/s), y-intercept = 79 spk/s; 15–20 Hz with 0–5 Hz: R2 = 0.27, slope = 2.78 (spk/s)/(deg/s), y-intercept = 79 spk/s). We note that central neurons did not display rectification since the firing rate was always above zero.
Figure 4.
Central neurons display a static nonlinear relationship between their output firing rate and their afferent input.
(A) Low (top) and high (bottom) frequency head velocity stimuli (gray) cause smaller and larger changes in afferent firing rate (green), respectively. These differential changes in afferent firing rate in turn cause differential changes in central neuron firing rate (purple), respectively. Notably, the changes in afferent firing rate caused by high frequency head velocity stimuli are distributed over a greater range and thus elicit nonlinear responses from VO neurons, whereas this is not the case for those caused by low frequency head velocity stimuli. Note that the same scales were used for corresponding panels in the bottom and upper rows. (B) Population-averaged firing rates of central VO neurons as a function of afferent firing rate for low (blue) and high (red) frequency noise stimuli presented in isolation. Note that the curve obtained for the low frequency stimulus (blue) extends over a smaller range than that obtained for high frequency (red) stimuli. Further, both curves are linear over the range for which they overlap. Also shown are best linear fits to the portion of the curve below and above 90 Hz (dashed red lines). As such, the curve can be approximated by a piecewise linear function. Inset: population-averaged firing rates of afferents as a function of the head velocity stimulus for low (blue) and high (red) frequency noise stimuli presented alone. (C) Population-averaged firing rates of central VO neurons as a function of afferent input firing rates: (1) for the low frequency stimulus when presented alone (blue) and concurrently with the high frequency stimulus (solid black); (2) for the high frequency stimulus when presented alone (red) and concurrently with the low frequency stimulus (dashed black). Note that the curves obtained in response to the high frequency stimulus when presented alone (red) and when presented concurrently with the low frequency stimulus (dashed black) overlapped before (Figure 3E) and thus, not surprisingly, also overlap. Note also that only the curve obtained when the low frequency stimulus was presented concurrently with the high frequency stimulus (solid black) does not overlap with the others. This is because the central VO neuron firing rate is higher than that obtained for the low frequency stimulus when applied alone for values lesser than 110 Hz. Inset: population-averaged normalized slopes under all four conditions. The afferent activity was estimated by fitting a linear model to previous experimental recordings from a large population of afferents (see Materials and Methods).
Figure 5.
A simple model accurately predicts nonlinear central VO neuron responses to sums of low and high frequency stimuli.
(A) Model (solid) and data (dashed) relationships between afferent firing rate and central VO neuron firing rate when the low frequency stimulus was presented alone (blue) and concurrently with the high frequency stimulus (black). Note that the model accurately reproduces the decrease in slope seen experimentally as evidenced by the large overlap between the model and data curves (R2 = 0.92). (B) Model (solid) and data (dashed) relationships between afferent firing rate and VO neuronal firing rate when the high frequency stimulus was presented alone (red) and concurrently with the low frequency stimulus (black). Note that the model also accurately reproduces the lack of change seen experimentally as the model curves largely overlap with the experimental ones (R2 = 0.99). (C) % gain attenuation plotted as a function of signal and masker frequency. The stimulus for which the response is computed is referred to as the signal, while the other stimulus is referred to as the masker. Maskers with higher frequency content lead to greater gain attenuation. (D) % gain attenuation as a function of masker amplitude and frequency. Maskers of greater amplitude and frequency lead to greater gain attenuation.
Figure 6.
A linear-nonlinear (LN) cascade model reveals that central vestibular neurons respond nonlinearly to broadband noise stimulation.
(A) Schematic showing the LN model's assumptions. The stimulus (left) is convolved with a filter H(t) that is given by the inverse Fourier transform of the transfer function in order to generate the linear predicted firing rate (middle). This linear prediction is then passed through a static function f (which can be linear or nonlinear) to give rise to the predicted output firing rate (right). (B) Population-averaged function f for afferents. Also shown is the best-fit line (R2 = 0.998±0.001, n = 15) (red) whose slope did not significantly differ from unity (p = 0.99, n = 15, pairwise t test). Inset: population-averaged filter H(t) for afferents. (C) Population-averaged function f for central VO neurons. Also shown are the best-fit straight lines for the intervals (0–80 Hz) and (80–160 Hz) (red) whose slopes were significantly different from one another (p = 0.0014, n = 13, pairwise t test). Inset: population-averaged filter H(t) for central VO neurons.
Figure 7.
Schematic showing how a nonlinear static relationship between input and output can lead to attenuated sensitivity to sums of low and high frequency stimuli.
(A) Input-output relationship showing a vertex (i.e., a sudden change in slope) (black curve). If we assume that the input is normally distributed with low intensity (i.e., standard deviation) such that all the input values are to the right of the vertex (light green distribution on x-axis), then the corresponding output distribution will also be normally distributed (light purple distribution on y-axis). The mean output (light purple circle on y-axis) corresponds to the image of the mean input (dashed purple circle on y-axis; note that the light purple and dashed purple circles were offset for clarity) as both input and output are linearly related. In contrast, for a higher intensity input that extends significantly past the vertex (dark green distribution on x-axis), the corresponding output distribution (dark purple on y-axis) is skewed with respect to the linear prediction (dashed purple on y-axis). The mean output (dark purple circle on y-axis) is thus greater than the linear prediction (dashed purple circle on y-axis). (Note that here and below, we represented the distributions to have the same maximum value in order to emphasize the fact that we are changing the standard deviation.) (B) Increasing the input distribution intensity for a given mean (compare red, yellow, and blue distributions) causes a greater skew in the corresponding output distribution (unpublished data) and thus an increased bias in their means (red, yellow, and blue dots on the y-axis and inset) as compared to the linear prediction (dashed yellow and blue dots on the y-axis). (C) Shifting the mean of the high intensity input distribution to the left (compare points 1, 2, and 3 on the x-axis and the inset) makes it extend to the left of the vertex more and more (compare the green curves on the x-axis), causing greater skewness in the corresponding output distributions (purple curves on the y-axis), which creates a greater bias in the mean (dark purple points on y-axis) with respect to the linear prediction (light purple points on y-axis). As a result, the mean output in response to a given value of the low intensity input (points 1, 2, and 3 on the x-axis) when the high intensity signal is present (dark purple line) has a lower slope (i.e., gain) than when the high intensity signal is absent (light purple line). (D) Shifting the mean of the high intensity input distribution to the left (compare points 1, 2, and 3 on the x-axis and the inset) makes the corresponding distributions of the low intensity input extend to the left of the vertex more and more (green curves on the x-axis), causing greater skewness in the output distribution (purple curves on the y-axis), which creates a greater bias in the mean (dark purple points on y-axis) with respect to the linear prediction (light purple points on y-axis). Note, however, that the bias in the mean will be lower than in (C) since the input distributions now have a lower intensity as explained in (B). Thus, the input-output relationship when the low intensity signal is present (dark purple line) will have a lower slope (i.e., gain) than when the low intensity signal is absent (light purple line) but the effect will be weaker than in (C).