INTRODUCTION

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The vestibular system is essential for the detection of the position of the head in space and the general maintenance of balance and posture. Furthermore, during head movements, reflexes such as the vestibulo-ocular reflex (VOR) and the vestibulocollic reflex stabilize the eyes and head in space, respectively (Leigh and Zee 1999; Wilson et al. 1995). An understanding of the computational abilities of vestibular neurons is necessary to elucidate the neural substrate of all these reflexes.

Primary otolith afferent neurons encode head translational acceleration, including gravity (Angelaki and Dickman 2000; Fernandez and Goldberg 1976; Goldberg et al. 1990), but understanding of the signal processing at subsequent stages in the vestibular nucleus remains unclear. One outstanding question is how the vestibular system produces multiple integrations of the translational acceleration signal. These computations are necessary to provide the eye plant with the appropriate velocity and position signals, while maintaining the high-pass filtering characteristics of the translational VOR (Angelaki 1998; Telford et al. 1997). Although several models have been proposed (Angelaki 1992; Green and Galiana 1998; Musallam and Tomlinson 1999, Telford et al. 1997), experimental support is lacking. Circuits in the prepositus hypoglossi and the medial vestibular nucleus could function as a velocity to position integrator (Cannon and Robinson 1987; Cheron and Godaux 1987; Lopez-Barneo et al. 1982; Robinson 1989). How velocity is obtained from otolith afferent signals is debatable.

In this study, we have chosen to record the activity of VO cells in response to position transients. Traditionally, a sinusoidal stimulus has been used. Such a stimulus complicates the elucidation of calculus-like operations because the derivative and integral of sine waves are again sine waves. Any conclusion as to the calculus being performed by a system is determined from the phase difference between the input and output sine waves. In a linear system, a 90° phase lag in the output with respect to the input may imply that the input signal has been integrated. Delays and nonlinearities (Musallam and Tomlinson 2001), such as asymmetries in the response, may also produce a phase change, making the conclusion as to whether a system is differentiating or integrating a given signal unclear. However, the biphasic acceleration and monophasic velocity waveforms of position transients differ not just temporally but also spatially, allowing the neural correlates of velocity and acceleration to be easily discerned. In this paper, we show that the initial integration of acceleration to velocity does indeed take place in the vestibular nucleus but in a surprising manner. We propose that a traditional linear integrator is not needed, and our modeling efforts show that these signals can be obtained through synaptic dynamics and membrane properties.

	METHODS

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Surgical procedures

Extracellular recordings were obtained from the vestibular nuclei of two female rhesus monkeys. Binocular search coils (Robinson 1963) were used to measure eye position. The search coils were implanted subconjucativally according to the methods of Judge et al. (1980). The monkeys underwent two surgical procedures; during the first procedure, one eye coil was implanted and the head-holding bar anchored to the skull by four stainless steel screws and two inverted titanium T-bolts. The monkeys were allowed to recover for 6 weeks before the next surgery. During this recovery time, the monkeys were trained to fixate a visual target projected onto a tangent screen at a distance of 100 cm from the eyes in exchange for a juice reward. During the second procedure, the second eye coil was implanted along with the recording chamber, which was stereotaxically anchored to the skull with stainless steel screws. It was placed at an orientation 30° from the stereotaxic vertical axis so that an electrode going through the origin of an x-y grid centered on the chamber passed directly between the abducens nuclei. Once the location of the abducens nuclei had been mapped, the vestibular nuclei were found by moving laterally or posteriolaterally (Smith et al. 1972). All surgical procedures were carried out in sterile operating conditions at the University of Toronto under the supervision of a veterinarian.

Position transients were used as the primary translational stimulus. Transients were delivered while the monkeys faced 90° counterclockwise (CCW, Fig. 2A), 60°CCW (B), 30°CCW (C), 0° (D), 30° clockwise (CW, E), and 60°CW (F) relative to the sled [0° refers to naso-occipital (NO) and 90° refers to interaural (IA)]. About 100 cycles (1st row of Fig. 1 is 1 cycle) were recorded for each orientation. The translational acceleration was monitored using a three-dimensional linear accelerometer (Crossbow) placed on the head holding bar centered between the monkeys' ears on the interaural line. The sled position, eye positions, accelerometer output, and neural spike train were digitized at 1,000 Hz using Labview (National Instruments).

View larger version (35K): [in this window] [in a new window]   Fig. 1. A typical position transient cycle used in this study. Top trace, the sled position. Middle trace, the velocity of the transients, obtained by integrating the acceleration (3rd trace). Bottom trace, the unaveraged firing rate in response to 1 transient cycle. The numbers (1-4) are behavior in the firing rate described in the text. IF: inhibitory first. EF: excitatory first.

Recordings were obtained from tungsten electrodes (impedance = 1 M) coated with paralene "C" and fitted into a polyamide tube for additional insulation. Initially, the monkeys were translated sinusoidally in the NO direction until a neuron that responded to the oscillation was located. Once isolated, the neuron was then tested for eye-movement sensitivity by having the monkeys saccade between eccentric targets and use smooth pursuit to follow a target oscillating with a variable frequency ranging between 0.2 and 1.5 Hz. Only cells without eye-movement sensitivity (including no saccade responses) are reported in this paper. The firing rate of each neuron was calculated by convolving the spike train with a Gaussian curve with a width of 8 ms (Richmond et al.1990). Because of the firing variability of some of the neurons in our sample, the baseline firing rate was taken to be the average rate for the 60 ms before the onset of the stimulus. All data analysis was conducted using custom-written software in Matlab. Fits to the firing rate were also performed in Matlab with a nonlinear least-squares algorithm weighted by the inverse of the SD using the Levenberg-Marquardt method. The functions used to fit the firing rate were any combination of

(1)

where B is the bias and V_IA and V_NO are the integral of the accelerometer's output in the IA and NO directions, respectively. Note that n and m do not have to be integers but can take on any real number (n, m < 0 implies integration and n, m > 0 implies differentiation). Position transients in the IA or NO directions were fit with the term that corresponded to the respective direction of motion. All the variables (B, a, b, n, m) were optimized during the fitting process. The stimulus being used for fitting was temporally shifted relative to the firing rate until the optimal fit was obtained. Fractional integrals (or derivatives) were computed using the following formula: D¹f(t) = 1/(l)(t  x)^l1f(x)dx where (l) is the gamma function where 0 < l < 1 for integration (note the negative on the exponent of D). (This definition differs from Eq. 1, where n > 0 implies differentiation). The fractional derivative can also be calculated in the frequency domain by noting that multiplication by complex frequency is equivalent to differentiation in the time domain. Therefore Dⁿ(A) = ¹[(A)*(i)ⁿ[ where  and ¹ are the Fourier and inverse Fourier transforms respectively. Integration of the acceleration signal was also computed in the time domain by using the trapezoid method with a base equal to half the sampling period (0.5 ms) (Kahaner et al.1989). Both methods produced near identical results. All derivative fractional exponents are reported from Eq. 1 and are relative to velocity (an exponent of 0 is velocity and an exponent of 1 is acceleration).

Position transients of several amplitudes and durations were used resulting in several peak accelerations for each direction of motion. Peak accelerations used in this experiment were approximately 0.25, 0.5, and 0.75 g where g = 9.81 m/s² (see Table 2).

	RESULTS

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A total of 62 cells were recorded but not all of these cells were held long enough to complete the six possible orientations mentioned above. Of a possible 372 different orientations (62 cells × 6 orientation/cell), 241 orientations were completed. Only eight cells were recorded along all six orientations. In addition, 25 neurons were omitted from population analysis due to excessive ringing in the sled. This ringing was dependent on the center of mass of the monkeys relative to the center of mass of the pole that supported the sled. When the position of the monkeys was eccentric relative to the center of mass of the sled, an increased moment arm resulted in increased ringing. Of the remaining 37 cells, recordings were made from 145 orientations (Table 1).

Figure 1 depicts a typical single position transient cycle with the velocity and acceleration of that cycle as reported by the accelerometer. A cycle here is defined as a position transient in one direction followed by a transient in the opposite direction, back to the starting position (Fig. 1, 1st row) and takes approximately 1 s to complete. The position trace is the feedback signal from the sled while the velocity trace is the trapezoidal integral of the accelerometer's output. The firing rate that a single cycle elicited in a VO neuron is shown at the bottom. The peak acceleration of the stimulus shown in Fig. 1 varied between 0.72 and 0.74 g in the positive direction and was 0.68 and 0.70 g in the negative direction. The rising and falling phases of the transients elicited different responses. Specifically, the first acceleration pulse (t = 0.0 to t = 0.3 s) [we shall refer to this stimulus as inhibitory first (IF)] elicited a biphasic response in the firing rate, which initially drove the response of this cell toward zero (Fig. 1, 1) before it excited it to fire at approximately 200 spikes/s (Fig. 1, 2). After the stimulus was over, (ignore the ringing in the stimulus for now), the cell decayed back to baseline (taken here to be 66 spikes/s) which took approximately 130 ms (Fig. 1, 3), much longer than it took the stimulus to return to baseline. On the other hand, once the sled reversed direction (t = 0.5 to t = 0.8 s) and the acceleration reversed polarity [referred to as excitatory first (EF)], the firing rate (Fig. 1, 4) no longer represented the biphasic nature of the acceleration but instead adopted a response that was more monophasic than biphasic and therefore could approximate the velocity of the stimulus (compare the firing rate at t =0.5 to t = 0.8 s with the velocity trace in Fig. 1, 2nd row). Even though the acceleration had gone from 0.74 to 0.70 g in 44 ms (32.7 g/s) and maintained the minimal value for approximately 20 ms, the firing rate of the cell decreased at a much slower rate. This extension of the firing also occurred for the positive acceleration portion of the IF trial because, as can be seen, acceleration in the excitatory direction is a much more powerful stimulus than an equal acceleration in the inhibitory direction. Thus the responses evoked by oppositely directed acceleration pulses were clearly different.

Figure 2 depicts the response of another cell recorded during translation along all the orientations used in this study. (The responses shown are single cycles not averages). During these experiments, the monkey is always being translated along the thick arrow but is facing in the same direction as the thin arrows. The stimulus while the monkeys are in the naso-occipital orientation is depicted in the top and bottom right corners of Fig. 2. As can be seen, the asymmetry was present along all orientations tested. Note also that the difference in the behavior of the neuron to a stimulus in the IF and the EF directions was not generated gradually as the angle of the translation axis swept through different orientations. Instead, the cell's response approximated the velocity of the motion in all forward directions spanning 180°.

View larger version (24K): [in this window] [in a new window]   Fig. 2. The response of a cell to translations spanning 360°. NO, naso-occipital; IA, interaural. Shown are the responses at 30° increments. The translation is always in the forward direction (along the thick line). The black axes with the arrows are the direction the monkey is facing. For each axis, the stimulus is composed of a transient in the direction of the arrow, and a transient back in the opposite direction. The firing rate scale is shown on the left. The stimulus is shown in the top right and bottom right corners. Note that the responses only differ when the translation is in the EF or IF direction. Due to the excessive ringing in the stimulus, this cell was not included in the analysis presented in subsequent figures.

During translation along the other directions, the response of the neuron is a biphasic signal reminiscent of the acceleration of the stimulus (Fig. 2). This reversal always occurred at 180°. The only difference among the responses along different orientations in the EF direction is the modulation in amplitude and variation in the SD of the firing rate from one orientation to the next. We will only address the asymmetry of the response, and therefore we shall lump all responses for the EF and the IF directions into two separate groups. This cell, like all the cells recorded in this study, responded to acceleration in all tested directions. It also responded to the ringing in the sled, which does not contradict the claim of asymmetry but may bias the estimate of the time taken for the cell's activity to decay. Due to the excessive ringing, this cell was excluded from further analysis. In total, 25 of the 62 cells were excluded due to the ringing (Fig. 2 is the only cell with all orientations that was excluded). Note that the ringing in the sled extends the time of decay, and so including these cells would support and not contradict our conclusions.

Figure 3 depicts the response of four cells recorded while the monkey was translated along the NO direction (red) and the IA direction (black). The stimulus has a peak acceleration of 0.52 g. As can be seen, the time taken for the firing rate to return to baseline for the IF trials in the IA direction is smaller than that of the NO trials. In addition, the peak firing rate is reduced in the IA direction. Nevertheless, the hypothesis of asymmetry still holds for translation in both orientations. If this neuron encoded the linear integral of the acceleration, then the plots in the IF trials should be opposite to those in the EF trials because the stimulus has simply reversed. Instead, what is observed in the IF trials is a weak inhibitory effect at t = 0.1 s followed by a strong excitatory response. In contrast, the EF trials, which have acceleration profiles that initially go positive, strongly excite the neuron, eliciting a response that resembles the integral of the stimulus.

View larger version (28K): [in this window] [in a new window]   Fig. 3. The response of 4 cells during NO and IA translation. Depicted is the mean of 16-31 cycles depending on the trial. Black and red traces, responses to translations in the IA and NO directions, respectively. The time taken for the firing rate to return back to baseline for the IF trials in the IA direction is smaller than that of the NO trials. No such discrepancy is seen in the EF trials. Nevertheless, the response asymmetry is consistent across orientations. If this was an integrator cell then the plots in the IF trials should be opposite to those in the EF trials because the stimulus has simply been reversed.

The means of the responses of another cell elicited by three stimuli of different peak accelerations for all orientations is depicted in Fig. 4. As in Fig. 3, the responses along different orientations are similar, and so we have averaged across different orientations. As can be seen, the asymmetry holds for all accelerations used. By combining the response to a single stimulus along all orientations for each cell, the ringing in the sled was minimized. The similarity between the response and the stimulus acceleration shown in the bottom row of Fig. 4 (D-F) is no longer evident once the stimulus reversed direction (Fig. 4, A-C). In the top row, the cells were excited by the positive swing of the acceleration, and the excitation lasted much longer than the stimulus. The persistent firing prevented the cell from encoding the reversal of the acceleration as robustly as it could when it was initially inhibited, and as a result the firing rate resembles the velocity of the motion. In fact, the firing rate persists in both EF and IF for excitatory (positive) accelerations. In both cases, the firing rate decayed back to baseline slower than a linear response to the stimulus would suggest. Increasing the amplitude of the acceleration only leads to a modest increase in the firing rate (Musallam and Tomlinson 2001) and an increase in the time (T_f) that the firing rate is above baseline after reaching its maximum. Table 2 summarizes the mean value of T_f for the six figures depicted in Fig. 4 [null hypothesis is: T_f entries in Table 2 between rows x and y (different accelerations) are equal: IF trials: t_0.05(2),10. Rows 1 and 2, t = 1.71; not significant (accept null hypothesis); rows 1 and 3, t = 6.46, P < 0.001; rows 2 and 3, t = 3.95, P < 0.005. EF Trials: t_0.05(2),10. Rows 1 and 2, t = 7.02, P < 0.001; rows 1 and 3, t = 10.75, P < 0.001; rows 2 and 3, t = 3.40, P < 0.01]. The real value of the time to decay is not as important as the result that the time these cells are active is extended whenever the stimulus goes positive. T_f for the whole population (37 cells, 147 orientations) is given in Table 3. As the magnitude of the acceleration increases, so does the amount of time the firing rate is above baseline (Null hypothesis is same as above. IF trials: t_.05(2),290. Rows 1 and 2, t = 0.891, Not Significant, Rows 1 and 3 t = 2.46, P < 0.05; Rows 2 and 3, t = 2.08. P < .05; EF Trials. t_.05(2),290. Rows 1 and 2, t = 2.26, P < .05; Rows 1 and 3, t = 3.87, P < .001; Rows 2 and 3, t = 2.22, P < .05). The variability in the EF direction is less than the variability in the IF direction. This is partly due to the result that the response during the EF trials is below baseline when the ringing commences. The persistence of the firing rate beyond the time that the stimulus reaches its minimum indicates that the inhibiting drive is weaker than the excitatory drive. The extension of the time constant is the basis of our model that attempts to explain this asymmetric behavior.

View larger version (34K): [in this window] [in a new window]   Fig. 4. The mean response of another neuron (see Table 2) (mean ± SE) to accelerations of different amplitudes. The plots depict the response averaged across different orientations for the same stimulus (see Fig. 3 for the similarity in the response along perpendicular orientations). The orientations are in the 1st row of Table 1. The horizontal red line indicates the baseline firing, whereas in the 2nd row (the IF trials), the vertical blue lines indicated the peak and the 0 crossing of the stimulus. A and D: row 1 in Table 2; B and E: row 2 in Table 2; C and F: row 3 in Table 2.

The difference in the response to a stimulus composed of a transient forward and a transient backward is best presented by a phase plot (Fig. 5). The acceleration and its integral (velocity) are plotted against the firing rate for both directions of motion (labeled as EF and IF in Fig. 5). For example, Fig. 5, C and D, depicts the response during the IF stimulus, plotted against the velocity (D) and the acceleration (C) of the stimulus. Similarly, Fig. 5, A and B, shows the response of a neuron translated in the EF direction plotted against velocity (B) and acceleration (A). A collapsed plot (Fig. 5, B and C) indicates that the stimuli and the response are well correlated with features occurring together, while an inflated surface indicates that the biphasic stimulus is being plotted against a monophasic curve which is the description of its integral (Fig. 1, 2nd and 3rd row). The existence of both an inflated and collapsed curve for a single cycle (transient forward and then backward) indicates that stimulus velocity is a better fit to the data in one direction while acceleration is a better fit in the opposite direction.

View larger version (28K): [in this window] [in a new window]   Fig. 5. Phase plot of another neuron indicating the asymmetry present in the response of the neurons responding to transients. Shown are plots of the acceleration and velocity of the stimulus vs. the responses elicited during the EF or IF trials. A: acceleration vs. firing rate during the EF trial. B: velocity vs. firing rate during the EF trial. C: acceleration vs. firing rate during the IF trial. D: velocity vs. firing rate during the IF trial.

More accurately, using Eq. 1, the best fit to the response of this cell is the fractional derivative of velocity. A fractional exponent of 0 indicates a pure velocity fit, whereas a fractional exponent of 1 indicates an acceleration fit. These fractional dynamics (plus a bias) were better at fitting the responses (higher correlation index) than a bias plus any combination of velocity, acceleration, jerk or dFR/dt (low-pass filter fit). Because this is a neural system, there is no reason why exact velocity or acceleration should be encoded centrally. Figure 6 depicts the mean of four cells translated in the interaural direction but to two different stimuli with the fit that yielded the best correlation coefficient. For example, with a stimulus of a = 0.51 g the EF stimulus yielded a response that was fit with FR = 70 + 15.2(d^0.24v/dt^0.24) (r² = 0.90) while the IF stimulus yielded a fit of FR = 81 + 11.4(d^0.69v/dt^0.69) (r² = 0.77). As the amplitude of the stimulus increased, the fractional exponent appears to decrease, better able to approximate velocity for greater acceleration amplitudes. The fits for stimulus in the IF direction had lower correlation coefficients. A lower exponent would better fit the response to the positive acceleration (t > 0.15 s), while a higher exponent would better fit the response to the negative acceleration portion of the IF cycle (t < 0.15 s).

View larger version (39K): [in this window] [in a new window]   Fig. 6. Fractional derivative fits of the response of the mean of 4 neurons translated in the inter-aural direction. The responses have been smoothed to ensure convergence. The equations that best fit these responses along with the correlation coefficient are depicted in the figure. Note that an increase in acceleration results in a lower fractional exponent.

The best fractional exponent fit for 17 neurons that underwent the set of three different accelerations is plotted in Fig. 7A against the peak acceleration in the IF and EF directions along with a linear regression fit and a 99% confidence interval. The slope for the EF fit is m = 0.251, which is significantly different from 0 (t-test, P < 0.001). The slope for the regression fit in the IF direction is m = 0.104, which is significantly different from zero only at P > 0.1 (t-test). This may be caused by the limited accelerations used. If accelerations less than 0.25 and greater than 0.75 g, where utilized, then the slope may become steeper. Nevertheless, for the EF direction, increasing the acceleration resulted in the fractional exponent approaching velocity. The population asymmetry between the IF and EF directions is shown in Fig. 7B. Figure 7B shows the fractional exponent fit (in black) and mean fit ± SD (in red) for 32 neurons recorded while translating in the NO direction. There is a large distribution of fractional fits, with some overlap between the EF and the IF direction. Indeed, the distribution of fractional exponents during a single direction is diverse. For example, during the IF direction, the range of fractional exponents is 0.49-1.09 (1.09 being the 0.09th derivative of acceleration; 0.823 ± 0.147). In contrast, in the EF direction, the range of fractional exponents is 0.01-0.55 (0.24 ± 0.142). The difference between the fractional exponent in the IF and EF direction for 145 trials is plotted in Fig. 7C. The distribution of differences is skewed with most differences having values greater than 0.5 indicating that this behavior is consistent. The difference represents a measure of the degree of temporal integration of the primary afferent signal between the EF and the IF responses. The clustering of the data around a difference of 0.7 indicates that there is fractional integration in one direction of motion.

View larger version (30K): [in this window] [in a new window]   Fig. 7. A: fractional exponent fits for 17 neurons in response to accelerations of different amplitudes in the IF and EF stimulus. Lines are linear regression ±99% confidence intervals. The slope of the regression line for the IF response is 0.104 which is not significantly different from 0 at the  = 0.01 level. The slope for the IF stimulus is 0.251 which is significantly different from 0 (t-test, P < 0.001). B: distribution of fractional exponents for 32 neurons translated in the NO direction. Fractional exponent value of 1 is acceleration while 0 is velocity. As can be seen, the fractional exponents during IF are clustered close to 1 while those recorded during EF are clustered toward 0. C: difference between IF and EF fractional exponent fits for the whole population. As can be seen, more neurons are clustered at a difference greater than 0.5, indicating a tendency to integrate in the EF direction.

The responses reported here are asymmetric, with the integration of the afferent response being performed in one direction but not the other. If the stimulus was a sinusoid, one would expect an asymmetric slope between the rising and falling phase of the sinusoid. Figure 8 depicts the response of another neuron recorded while the monkey was oscillated in the naso-occipital direction at a frequency of 4 Hz. The vertical drop lines connecting the firing rate trace to the time axis indicate the difference between the rise time and the time to reach minimum. The asymmetry between the response and the stimulus is clear. Note that no asymmetry is present in the stimulus. The asymmetry in response to sinusoids was not discernible in all cells and only at high frequencies. This is in contrast to the recordings during position transients, where every cell recorded exhibited an asymmetry.

View larger version (25K): [in this window] [in a new window]   Fig. 8. Asymmetry in sinusoidal data. Top: sled position. Bottom: firing rate. Stimulus is 4 Hz. The vertical lines connecting the firing rate with the time axis indicate where on the time axis features of the response occur. As can be seen, there is an asymmetry between the rising and falling phase of the sinusoid.

	DISCUSSION

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It is often assumed that a single and a double integration of the otolith primary afferent signal is required to extract position and velocity information. Little is known about how this is accomplished. Here we present evidence that the first integration may be accomplished without requiring a dedicated neural integrator circuit. Accelerations of the head are generally of high frequency (Grossman et al. 1988), and therefore single neurons can emulate integrations if they have appropriate membrane time constants. In the following text, we shall present a model that utilizes the long membrane time constant as a possible means in which this could be accomplished. This organization suggests that single neurons in the vestibular neuron have sophisticated computational abilities, and that circuits are not always needed (Angelaki 1991). Although the neurons in this paper did not respond to eye movements, the results in this paper could possibly apply to the many reflexes driven by the otolith systems. The tVOR, for example, could benefit from the signal processing reported here because the velocity of translation is readily available and can be delivered to the various oculomotor nuclei. Thus a single neural integrator is all that is needed in the tVOR circuit, one that can integrate the velocity into position.

The novel stimulus used in this study not only elucidated the asymmetries present in the VO cells in response to translation but also illuminated the inadequacies of sinusoids. Given that the derivative and the integral of a sine wave is again a sine wave, any conclusion about the calculus performed by neurons that receive this stimulus is ambiguous. Position transients have a spectrum composed of many frequencies, which better approximates natural movements and makes no such linearity assumptions. Both sinusoidal stimuli and position transients need to be used to fully characterize the behavior of vestibular neurons.

Response asymmetry in other systems

In a linear system, except for a change in sign, it makes no difference to the system whether the stimulus is going from an amplitude of x₁ to x₂ or from x₂ to x₁ (Dorf and Bishop 1998). There is a clear violation of this rule presented in this report as oppositely directed stimuli did not elicit opposite responses. This has long been known to occur in the smooth pursuit system where a step in target velocity elicits eye velocity overshoot and subsequent oscillations while stopping the target elicits an exponential decay in eye velocity with a time constant of 90 ms (Robinson et al.1986). This pursuit behavior has been attributed to separate subsystems controlling eye movements and fixation (Huebner et al.1992; Leubke and Robinson 1988), but we show below that asymmetric behavior can arise within a single system by prolonging the time course of the response and by having the prolongation be a function of the stimulus.

Model of TCE

The model presented here is based on the result that T_f is extended beyond that predicted by a linear system. Time constant enhancement (TCE), as the name suggests, is a mechanism that elongates the time constant of the decay of a postsynaptic response. Although this is a theoretical algorithm used to simulate the behavior of neurons recorded here, it has a direct correlate with neuronal activity. It functions to perseverate the activity of the postsynaptic cell rewarding the postsynaptic neuron for being activated. TCE simply works by eliciting an excitatory postsynaptic potential (EPSP) in the postsynaptic cell with a time constant that is a function of the interspike interval of the presynaptic cell (the input). For example, a spike arriving at a synaptic terminal will elicit an EPSP. If a subsequent spike arrives within a time window, another EPSP is elicited with an increased time constant with the size of the increase of the time constant being proportional to the inverse of the interspike interval. This is a form of adaptive filtering as the EPSPs can be viewed as kernels of low-pass filters whose corner frequency changes because the time constant of decay of the EPSP changes. The sum of all EPSPs was then smoothed and taken to be the firing rate. As depicted in Fig. 2 and shown by others (Angelaki and Dickman 2000), otolith afferents have been shown to converge onto central vestibular neurons, behavior that is not accounted for in this model. Nevertheless, this is a highly conceptual model used to show that single neurons with certain synaptic dynamics can solve the integration problem.

Figure 9 depicts the output of a simulation using TCE. What is shown is several EPSPs with variable time constants convolved¹ with the spike train representation of the input. The input, which we took to be the output of the accelerometer, was deconvolved so that a spike train mimicking the output of afferents could be utilized. (The spike train of an afferent convolved with a Gaussian will reproduce the analog acceleration trace.) This representation ignores the dynamics of the afferents but is a generalized case. Equivalently, the analog acceleration signal was turned into a series of spikes by using the amplitude of the signal as an estimate of the interspike interval. This method was preferred to the deconvolution methods because the accurate representation of the signal by spikes is highly dependant on the accurate heuristic choice of Gaussian characteristics. Both methods were used, one being used as a test for the other. Once the spike train representation of the acceleration signal was obtained, it was convolved with several EPSPs that differed in time constant. An EPSP is simply defined by e^-t/ where  is the time constant of decay and a function of the interspike interval. The time constant of the EPSP (e^-t/) was defined as
where t_i is the time of spike occurrence. For example, if the firing rate at any particular instance is 250 spikes/s, then t_i - t_i-1 = 4 ms (the interspike interval), and the time constant becomes 3₀. The upper limit that we allowed the reconstructed firing rate to reach was 350 spikes/s. Therefore the maximum amount by which the time constant was extended was 3.55₀. These values are arbitrary and highly dependant on the average firing rate assigned to the acceleration signal. Any other representation of the acceleration as a series of spikes can easily be simulated by adjusting the definition for . The acceleration waveform (thin trace), before being converted to a spike train, along with the simulated firing rate (thick trace) is depicted in Fig. 9A. The acceleration (input) was transformed from a peak of 0.7 g to the firing rate shown in Fig. 9A (thin trace) while the simulation is shown as a thick trace. The desired behavior is easily replicated (thick line in Fig. 9A). When the acceleration rises first (EF direction; Fig. 9A at t = 0.2 s), the response of the model is to produce an approximate integral of the input. However, when the input reverses direction, so that it is inhibited first, then, the response is biphasic, decaying with a time constant greater than the time constant of the EF direction. Comparing the actual firing rate along with the simulated trace (Fig. 9, C and D), one can see that time-constant enhancement of the input signal produces a good approximation to the firing rate observed in this cell. For the simulation shown in Fig. 9, ₀ = 16 ms so that the maximum time constant was 56 ms. This amount of enhancement is sufficient in order for us to simulate our data. Figure 9B depicts the normalized frequency of various time constants used in the simulation for Fig. 9, A, C, and D. As can be seen from Fig. 9B, the time constant was low for most of the duration of the motion. Less than 1% (0.7%) of all the points were convolved with an EPSP having a time constant of 56 ms while 78% of the points were convolved with an EPSP with a time constant less than 25 ms. Figure 9, E and F is another simulation to an acceleration profile (peak 0.25 g) that is contaminated by oscillation. The same value for ₀ was used as in the preceding text. In transforming the acceleration trace into a firing rate to be used with the model, the gain of the transformation was increased so that we have assumed a nonlinear transformation between the measured acceleration and the input to the model. The nonlinearity comes about from a sigmoid function that softly saturates the modeled firing rate of the afferents as the acceleration increases.

View larger version (32K): [in this window] [in a new window]   Fig. 9. The output of simulations using time constant enhancement (TCE). A: output of model (thick black line) in response to the acceleration (thin line) after the acceleration signal was transformed into a spike train and convolved with a variable excitatory postsynaptic potential (see text for detail). The acceleration trace (thin black line) is an example of the transformation that the input (actual acceleration trace) underwent before being passed to the TCE model. The original acceleration trace had a peak of 0.7 g, which we transformed into the firing rate depicted in the plot, reaching a discharge rate of 400 spikes/s for an acceleration of 0.7 g. B: the relative amount of enhancement to the time constant necessary to produce the results shown in C and D. Note that 40% of all points received no enhancement (0 = 16 ms) while greater than 75% of all points receive less than 24 ms enhancement. C and D: simulation (thick black line) using TCE superimposed onto firing rates (black) for EF and IF input profiles with a peak acceleration of 0.70 g. E and F: simulation (thick black line) using TCE for a noisy input from 1 of our cells. Peak acceleration is 0.25 g

Increasing the time constant of the decay of the response can easily be accomplished by having the excitation of the postsynaptic neurons mediated by a balance of N-methyl-D-aspartate (NMDA) and AMPA receptors (Landfield and Deadwyler 1988). NMDA receptors are known to exist in the vestibular nucleus, are implicated in restoring the VOR after a lesion (Caria et al. 1996; Grassi et al. 1998; Grassi and Pettorossi 2000), and participate in gaze holding (Mettens et al. 1994). Extending the time constant of the membrane would lead to a more robust and stable velocity to position integrator (Seung et al. 2000; Shen 1989). Serafin et al.(1991) showed that there do exist NMDA receptors in the guinea pig's vestibular nucleus and that these receptors control the level of resting discharge of vestibular neurons. Indeed, it has been estimated that there is a large number of NMDA receptors in the vestibular nucleus of the rat (Fujita et al. 1994) and gerbil (Kinney et al. 1994). By balancing the activation of fast and slow receptors and by recruiting NMDA receptors for increased firing rate, TCE could be accomplished.

Conclusion

The data presented in this paper shows that the activity of VO neurons is asymmetric in response to oppositely directed stimuli. We showed that in one direction, the velocity of the translation better describes the cell's activity, whereas in the opposite direction, the response is better characterized by the acceleration of the translation. We propose a model that suggests that this integration in one direction does not need to occur in a neural integrator circuit, but that the estimate of the velocity of motion can be produced by using synapses with long time constants that have adaptive properties.