Biological Cybernetics

Biol. Cybern. 66, 231-240 (1992)

9 Springer-Verlag 1992

A model for the characterization of the spatial properties in vestibular neurons D. E. Angelaki l, G. A. Bush 2, and A. A. Peraehio 2, 3,4 Department of Physiology, University of Minnesota, Minneapolis, Minn, USA 2 Department of Otolaryngology 3 Department of Physiology and Biophysics, and 4 Department of Anatomy and Neuroscience University of Texas Medical Branch, Galveston, Texas, USA Received May 16, 1991/Accepted in revised form July 15, 1991

Abstract. Quantitative study of the static and dynamic response properties of some otolith-sensitive neurons has been difficult in the past partly because their responses to different linear acceleration vectors exhibited no "null" plane and a dependence of phase on stimulus orientation. The theoretical formulation of the response ellipse provides a quantitative way to estimate the spatio-temporal properties of such neurons. Its semi-major axis gives the direction of the polarization vector (i.e., direction of maximal sensitivity) and it estimates the neuronal response for stimulation along that direction. In addition, the semi-minor axis of the ellipse provides an estimate of the neuron's maximal sensitivity in the "null" plane. In this paper, extracellular recordings from otolith-sensitive vestibular nuclei neurons in decerebrate rats were used to demonstrate the practical application of the method. The experimentally observed gain and phase dependence on the orientation angle of the acceleration vector in a head-horizontal plane was described and satisfactorily fit by the response ellipse model. In addition, the model satisfactorily fits neuronal responses in three-dimensions and unequivocally demonstrates that the response ellipse formulation is the general approach to describe quantitatively the spatial properties of vestibular neurons.

1 Introduction

The information processing that takes place in the central vestibular system is of extreme importance in understanding its complex motor outputs for the control of eye, head and body in space. Signals arising from the semicircular canals which sense the angular acceleration of the head in space and from the otolith organs which encode linear accelerations and orientation relative to gravity are transformed to appropriate motor commands for each set of motor neurons and muscles. Understanding the neural processing that takes place in the vestibular nuclei complex is critical

since it constitutes the center for such computations that initiate the sensory-to-motor transformations appropriate for each of the vestibulo-ocular and vestibulospinal systems (Wilson and Peterson 1981). Based on the vectorial coding of hair cells (Shotwell et al. 1981), it may be expected that a similar behavior governs the response of vestibular neurons. It has long been shown that semicircular canal primary afferents (Blanks et al. 1975; Blanks and Precht 1976) and the static or low frequency responses from most otolith afferents (Fernandez and Goldberg 1976a, b; Loe et al. 1973) have indeed vectorial characteristics. Therefore, they are characterized by a functional polarization vector, namely, a vector in the three dimensional space such that stimulation along its direction produces maximum response, whereas stimulation at perpendicular directions produces no response ("null" plane). A vectorial description of the spatial properties of a neuron requires that the response gain be a rectified cosine function of stimulus orientation ("cosine rule", Fernandez et al. 1972; Blanks et al. 1975; Fernandez and Goldberg 1976a; Loe et al. 1973). During dynamic stimulation the "cosine rule" would also require the response phase to be constant along different directions of stimulation. Vectorial spatial characteristics and "cosine-like" behavior with dynamic stimulation have been observed in some vestibular nuclei neurons that are sensitive either to angular (Baker et al. 1984b; Fukushima et al. 1990) or to linear acceleration (Schor et al. 1984). However, more complicated spatial characteristics have also been described (Baker et al. 1984a; Perachio 1981; Bush et al. 1992; Perachio et al. 1992). These neurons are characterized by a response phase that depends on stimulus orientation and a response gain that is not described by a cosine function. At extreme cases, the response gain does not show a clear maximum and the response phase changes almost linearly with stimulus orientation (Baker et al. 1984a). The complicated "noncosine-like" spatial characteristics of such neurons require a different mathematical formulation to fit and quantitatively characterize their responses.

232 Recently, a theoretical model has been presented which describes and characterizes quantitatively the complicated spatial properties of such neurons (Angelaki 1991a). According to this model, a response ellipse is defined at each stimulus frequency from the neuron's responses during acceleration along three linear independent axes. The semi-major axis of the ellipse gives the direction of the polarization vector and estimates the response properties of the neuron during stimulation with linear acceleration along that direction (Angelaki 1991 a). In addition, the semi-minor axis provides the maximum sensitivity in the "null" plane. According to the model, a "cosine-like" behavior in the response gain and a constancy of the response phas e are equivalent to a zero semi-minor axis and an ellipse that has degenerated to a straight line. In this case, the neuron's spatial tuning follows the vectorial rules (for example, the direction of maximal sensitivity for stimulation in any plane is the vectorial projection of the three-dimensional polarization vector on that plane). In contrast, a neuron with a response gain which does not exhibit "cosine-like" behavior and with a response phase that depends on stimulus orientation is characterized by an ellipse which has a non-zero semi-minor axis. In other words, the neuron does not exhibit a "null" plane with zero sensitivity for stimulation along any direction in that plane and its spatial properties do not follow vectorial rules. In the present paper, the theoretical model is used to fit experimental data obtained during extracellular recordings from vestibular nuclei neurons in decerebrate rats. The major purpose of this work is to present a practical application of the response ellipse formulation to analysis of neuronal responses. It is shown that the dependance of neuronal gain and phase on stimulus orientation in two dimensions is precisely described and adequately fit by the model. A further purpose of this presentation is to extend the application of the model in three dimensions. Finally the functional and practical applications of the response ellipse formulation are discussed. The application of the response ellipse to responses to pure linear head acceleration suggests that some vestibular nuclei neurons do not behave like one-dimensional linear accelerometers. Instead, such neurons encode two-dimensional linear acceleration.

2 Experimental methods Long Evans rats which were decerebrated and paralyzed were used in acquiring the experimental data described here. The rats were anesthetized with methohexital sodium (Brevital; 65 mg/kg i.p.) prior to the decerebration which was accomplished through a combination of aspiration and transection at the rostral pole of the superior colliculus. To minimize bleeding during decerebration the carotid arteries were ligated bilaterally. The trachea was cannulated allowing the animal to be artificially respirated during the recording session. In addition, the femoral vein was cannulated for subsequent intravenous administration of a paralytic agent, pancuronium (Pavulon), during the recording sessions. All procedures were conducted under a protocol approved by the

institutional animal care and use committee. Extracellular recordings of neuronal discharges were made through glass micropipettes filled with 2M NaC1 and 2% Alcian Blue stereotaxically driven into the vestibular nuclei through a craniotomy over the cerebellar cortex. The recording sites were verified with histological reconstruction of dye spots ionophoretically ejected at the bottom of each electrode track. Sinusoidal linear head acceleration was achieved by mounting the animal on a cart driven by a 54.23 Nt m (40 ft lb) DC torque motor which was coupled to a set of parallel linear rails oriented parallel to the earth-horizontal plane and attached to a supporting frame. The neurons reported here were studied at one of three frequencies (0.2, 0.6 or 0.8 Hz) with peak accelerations of +_0.1 g, _+0.145 g and • g, respectively. In order to align the animal's horizontal semicircular canals and the utricular macula near the earth-horizontal plane, the animal was mounted in the cart with its head rotated around the interaural axis 26 ~ nose-down from the stereotaxic horizontal plane. The results are expressed relative to a head-fixed, right handed, orthogonal coordinate system defined in this standard animal position with the head tilted 26 ~ nose-down, such that the y axis was the interaural axis, whereas the x and z axes described the longitudinal and the vertical, respectively. The positive directions were dorsal, left and forward. In the following discussion, the horizontal plane refers to the x - y plane that was tilted 26 ~ up from the stereotaxic head plane and parallel to the earth-horizontal plane. The animal could be repositioned relative to the direction of travel on the linear track by manually rotating the animal around the z and/or y axes of the head coordinate system. For example, by statically repositioning around the z axis, the animal could be linearly translated along any direction in the head horizontal plane. Also, by statically pitching the animal 10-30 ~ nose-up or nose-down from the horizontal plane before it was statically repositioned around the z axis, the neuronal response could be measured during linear acceleration stimulation in various head planes. Responses during stimulation along axes that do not all belong to the same plane were often recorded in order to calculate the direction of the pMarization vector in three dimensions. The neuronal activity (average of 5-15 cycles) and the stimulus profile were fit with sinusoids using a linear least squares algorithm. The first and second harmonic of the stimulus frequency were sufficient to satisfactorily fit the neuronal responses. The response ellipse formulation has been based on the assumption that the neuron's response to sinusoidal stimulation along any axis is satisfactorily described by a single sine wave of the same frequency (i.e., there is no significant harmonic distortion). The neurons whose responses are reported here had a second harmonic sensitivity that was less than 10-15% of the fundamental. Therefore, only the first harmonic component of the fit to the neuronal responses was further processed and the second harmonic component was ignored. The terms gain and sensitivity are used interchangeably here and refer to the

233

first harmonic of the response expressed relative to the component of peak linear acceleration in the horizontal plane (in spikes/s/g). The phase of the response was measured relative to the cart displacement (i.e., head position along the track). The gain and phase of the response of a selective number of cells to stimulation with pure linear acceleration at a single frequency but at multiple orientations were used to demonstrate the application of the response ellipse formulation.

3 Theoretical description of the response ellipse The formulation of the response ellipse has been described in detail (Angelaki 1991a). Responses measured during stimulation along at least three linear independent axes (i.e., they do not all lie on the same plane) define the response ellipse that, generally, lies on a plane different from the coordinate planes. When only responses during stimulation in one plane are used, the resulting ellipse is coplanar and merely represents the projection of the response ellipse on that plane. For this reason, the response ellipse defined by the three-dimensional responses is called the "parent" ellipse in order to distinguish it from its projection ellipses on different planes. Therefore, when the response gain and phase of a neuron have only been measured during stimulation along axes lying on the horizontal plane, an ellipse in the horizontal plane can be constructed (referred to as the horizontal ellipse and being the projection of the "parent" ellipse on the horizontal plane). Let 0~be the polar angle of an axis in the horizontal x - y plane (i.e., angle formed with the positive x axis; positive angle corresponds to counterclockwise rotation as defined by the right hand rule). The gain of the neuron for stimulation along that direction, as predicted by the response ellipse formulation, is given by (Angelaki 1991a): S, (~) = (S2x cos 2 ~ + Syz sin 2

+ S~Sy sin 2~ cos(Ox - Oy)) 1/2

(1)

In addition, the response phase - 180 ~ < ~b1 < 180 ~ is such that: cos ~1(~) =

sin q~l(~) =

Sx cos 0~cos Ox + Sy sin ct cos Oy S, (00

(2)

S~ cos ~ sin ~gx + Sy sin ~ sin Oy S~ (~)

In the above equations S~ and Sy are the gain and ~9~ and Oy are the phase values that have been measured during stimulation with sinusoidal linear acceleration at a certain frequency along the x and y axes which, in this application, could correspond to the naso-occipital and interaural axes, respectively. As seen from the above equations, the neuronal gain, $1 (~), and phase, ~b,(~), for any direction in the horizontal plane are uniquely defined by the response gains Sx and S. and phases Ox and Oy. If the neural response has been ~neasured during stimulation along more than two directions in the plane, a more accurate calculation of the horizontal ellipse

could be obtained by fitting (1) and (2) to the data and estimating the response parameters Sx, Sy, ~gx and ~gy. Any parameter estimation software that minimizes a "least square error" of the gain and phase values simultaneously can be used for fitting the equations to the data. In this application, Matlab (Mathworks Inc.) was used. A two-column matrix was defined with one column containing the measured gain and the other consisting of the measured phase values. A second two-column matrix included the estimated gain and phase values given by (1) and (2) for the current values of S x, Sy, Ox and Oy. The elements of both matrixes were first suitably normalized (each value in the same column was divided by the square root of the sum of the squares of the elements of the column) so that the response gain and phase would be equally weighted in order to determine the estimated parameters. The number of rows in the matrix was determined by the number of different stimulus orientations used to test a neuron. The simplex algorithm (Nelder and Mead 1965) was used to estimate the values of the parameters Sx, Sy, O~ and ~gy that minimize the norm of the difference of the two matrixes (Strang 1980). After the values of Sx, Sy, ~9~ and Oy are available either through direct measurement (if the neuron has been tested along only two orthogonal directions in the plane) or through the fitting procedure outlined above (if the neuron has been tested in more directions), the direction of maximal sensitivity in the horizontal plane (i.e., direction of the semi-major axis of the ellipse) is specified by its polar angle ~o (angle formed with the positive x axis; Angelaki 1991 a): ~0 = 1/2( 180 - z), where r is an angle (0 ~ < z < 360 ~ defined by: -

COS "r = -

-

2#

'

sin z = SxSy cos(Ox - Oy) # and 2 2 + 4Sx2 S y2 # = 1/2(S 4 +S 4-2S~Sy

c0s2( O x

O y ) ) 1/2

The gain and phase of the neuron's response to stimulation along the direction of maximal sensitivity (semi-major axis of the ellipse) in the horizontal plane is predicted by (1) and (2) with ~ = cto. Similarly, the gain and phase of the neuron's response to stimulation along the direction of minimal sensitivity (semi-minor axis of the ellipse) in the horizontal plane is predicted by (1) and (2) with ~ = u0 - 90. Both values of ~ = ~o --- 90 give equal sensitivity values (I) and phase values (2) that differ from that of the maximum sensitivity vector by 90 ~ (one leads and the other lags by 90~ In the examples demonstrated here, the neural phase has been expressed in the interval [ - 9 0 ~ 90~ Therefore, if the direction of the polarization vector (~o) calculated from the above equations gives a phase value (2) that is outside this interval, the direction of maximum sensitivity is corrected to 0t0 + 180 that will give a phase value within the interval [ - 9 0 ~ 90~

234

When the response of the neuron has been measured during stimulation in the horizontal plane, it is only the projection of the three-dimensional ellipse that can be determined and, therefore, the neuron's maximum and minimum sensitivity vectors can be estimated only in the same plane. If the response of a neuron has been tested in more than one plane, the "parent" ellipse can be constructed and the direction of the polarization vector in three dimensions can be determined from its semi-major axis. For neurons that have a pure linear response with no significant harmonic distortion along all stimulus directions, the "parent" response surface is an ellipse and not an ellipsoid. Thus, there will always be a single direction in space (and not a whole plane), which is defined by the normal to the plane of the "parent" ellipse, for which the neuron exhibits no response (null direction). Therefore, the semi-minor axis of the "parent" ellipse does not represent the minimum sensitivity direction in the three-dimensional space. Instead, it represents the minimum sensitivity in the plane of the "parent" ellipse. Also, the semi-minor axis of the "parent" ellipse corresponds to the maximum sensitivity in the "null" plane (i.e., the plane that is formed by the semi-minor axis and the null direction and is perpendicular to the semi-major axis). In order to calculate the gain and phase of a neuron for any direction in space defined by two angles (rotation around the z axis) and/3 (rotation around the y axis) under the assumption that the gain (Sx, Sy and S.) and phase (Ox, Oy and Oz) for stimulation along the axes x, y and z are known, the order of the two consecutive rotations must be defined (since three-dimensional rotations do not commute). When rotation by angle ~ is performed first, the two angles ~ and 90-fl are the spherical coordinates that define that direction. The appropriate equations for this case have been derived (Angelaki 1991a). In the present application, the order of rotations was reversed since the experimental protocol was such that multiple orientations (a minimum of four) were tested on each plane tilted from the horizontal by an angle ft. Therefore, it would be easier to visualize the experimental data in three dimensions if they were plotted at different "pitch planes" (i.e., fl = constant). Similar calculations to those presented by Angelaki (1991a) give the neuronal gain during stimulation along a direction in space that lies on a plane that has been rotated by an angle fl around the y axis and forms an angle ~ with the new x axis (xfl): S ( a , fl) = (SI (fl) 2 c o s 2 (x -J- S y2 sin 2 Ct

+ S,(fl)Sy sin 2a cos(q51(fl) - Oy)) '/2

(3)

The neuronal phase for stimulation along that direction is: cos O(~,/~) =

In the above equations, S1 (fl) and 4h (fl) are the gain and phase of the neuron on the x - z plane and can be calculated from equations similar to (1) and (2). That is: sl (/~) = (S2x cos 2 fl + S 2 sin 2 fl + S~S= sin 2fl cos(Ox - 0~)) 112 cos ~bl(fl) = Sx cos fl cos Ox + Sz sin fl cos Oz sl (/~) Sx cos fl sin Ox + ~ sin fl sin O: sin (~)l (]~) =

s,(fl)

In the above equations positive values of ct and fl are defined to be counterclockwise. The direction of the polarization vector (defined by the angles ~ = a p and fl = tip) could be calculated by maximizing the value of the function S(a, fl). For a = ~p and fl = tip, (3) and (4) define the phase angle and the gain of the polarization vector. Here, the values of % and tip were calculated as follows: A four-column matrix was created with the first two columns consisting of the values for (0 ~ < ~ < 360 ~ and fl ( - 9 0 ~ < fl < 90~ The third and fourth columns had elements that were the calculated gain and phase values as predicted by (3) and (4) for the values of ~ and fl of the same row. The maximum value of the third column gave the maximal sensitivity of the neuron and the elements of the same row gave the other measurements of the polarization vector (i.e., the first and second column were the ~p and tip values and the fourth column provided the response phase). The accuracy in the calculation of ~p and tip is limited only by the accessible computer memory since accuracy of many decimal points would require large matrixes. In the present application and since the purpose was merely to demonstrate the use of the model and its ability to qualitatively and quantitatively fit the experimental data, the angles that define the three dimensional polarization vector have been calculated with an accuracy of + 2 deg (the matrix had 181 x 91 = 16,471 rows).

4 Comparison with experimental results

4.1 Two-dimensional case Figure 1 shows the measured gain and phase of two different neurons in response to a sinusoidal linear acceleration stimulus as a function of the stimulus orientation angle ~ in the horizontal plane. The superimposed solid lines are the fits of (1) and (2). The neuron on the left (v174d) is closely "cosine-tuned". That is, the response of the neuron to stimulation in the horizontal plane exhibits maximum sensitivity (So = 36.7 sp/s/g) when the head is accelerated along one axis (ct0 = 249 ~ and minimum sensitivity that is close to

Sx cos Ox cos fl cos ~ + Sy cos Oy sin ct + S~ cos Oz sin fl cos S

and sin O(~z, fl) =

(4) Sx sin Ox cos fl cos 0~+ Sy sin Oy sin cz+ ~ sin Oz sin fl cos S

235

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Fig. I h - l ) . Examples of responses from vestibular nuclei neurons to sinusoidal linear acceleration in the animal's horizontal plane. A, B Response gain (top panel) and phase (bottom panel) are plotted as

functions of the stimulus orientation angle a. The solid lines are fits of (1) and (2). A Neuron v174d was stimulated at a frequency of 0.2 Hz. The maximum sensitivity in the horizontal plane (ao = 249~ was calculated to be So = 36.7 sp/s/g with response phase of ~bo = -22% whereas the minimum sensitivity was estimated as So= 1.1 sp/s/g (tuning ratio = 0.03). B Neuron h177d was stimulated at a frequency of 0.8 Hz. The estimated parameters were: So = 50.7 sp/s/g, ao = 305~ ~b0 = 66~ and So= 14.9 sp/s/g (ratio = 0.29). C and D The corresponding horizontal ellipses for neurons v174d and h177d, respectively.The polar angles of the directions of maximum sensitivity for each neuron are indicated by arrows. The x and y-components of gain are plotted in sp/s/g

zero during acceleration along an axis perpendicular to the first. At intermediate orientations the neuronal gain is equal to the product of S O and the cosine of the angle that it forms with the direction of maximal sensitivity as predicted by (1) if x is the direction of maximal sensitivity and y the direction of zero sensitivity (it is always true that the main axes of an ellipse have O x -- O y = "~-90~ In order to characterize the vectorial specificity of neurons, we defined the tuning ratio as the ratio of the semi-minor over the semi-major axis. Therefore, neuron vl74d (Fig. 1A) had a tuning ratio of 0.03. This behavior is expected f r o m neurons which behave as ideal one-dimensional linear accelerometers measuring linear acceleration with maximal sensitivity along one direction and showing no response when stimulated along directions lying on the perpendicular plane. Neurons that behave like this are characterized by a constant phase versus stimulus orientation shifting

abruptly by 180 ~ as the stimulus direction approaches the anti-parallel orientation (Fig. 1A, b o t t o m panel). In addition, the horizontal ellipse degenerates to almost a straight line (Fig. 1C). The direction of the m a x i m u m sensitivity (with phase in the interval [ - 9 0 , 90]) specified by the polar angle ~o is also demonstrated in Fig. 1C. The neuron on the right (Fig. IB) is an example of a different neuron (h177d) that responds during head acceleration at any stimulus direction in the horizontal plane. A m a x i m u m sensitivity of S O= 50.7 sp/s/g is observed at an angle ct0 = 305 ~ but the minimum sensitivity at the perpendicular direction is nonzero (So = 14.9 sp/s/g), giving the neuron a tuning ratio of 0.29. The neuronal sensitivity at intermediate directions is not a simple cosine function of stimulus orientation (top panel). In addition, phase changes gradually with angle ~, having a value of ~b0 = 66 ~ at the direction of maximal sensitivity ( b o t t o m panel). The response phase of any neuron for acceleration in the minimal response direction always differs 90 ~ from that for acceleration in the maximal response direction (Angelaki 1991a). The orientation of the direction of m a x i m u m sensitivity and the non-zero length of the semi-minor axis are also seen in the plot of the horizontal ellipse (Fig. 1D). It is important to appreciate the exact meaning of the response ellipse representation. Even though the semi-major and semi-minor axes (in terms of direction and actual length) directly correspond to the m a x i m u m and minimum sensitivity of the neuron, one should be careful in generalizing this to any direction. The picture of the response ellipse does not represent the tuning curve of the neuron. These statements will be more obvious by examining Fig. 2. Each trace in the Figure is plotted in the x - y plane (i.e., any direction is defined by the angle ~ with the positive x axis). The top traces of Fig. 2 A - C show two different curves. The response ellipse in the x - y plane is plotted with dotted lines, whereas the gain of the neuron as given by (1) is plotted with solid lines. It is the solid lines that represent the "tuning curves" of the gain of the neurons in the x - y plane. In other words, the gain of a neuron for stimulation along any direction defined by the polar angle ~ is given by the length between the coordinate center and the intersection of the direction with the solid curve. It can be seen from the plots that it is only in the directions of the major and minor axes (specified by the arrows in Fig. 2B) that the two curves coincide. The curves on the b o t t o m traces of Fig. 2 show the representation of the response phase in the x - y plane as given by (2). T h a t is, the neuronal phase along any axis is given by the length of the vector that connects the coordinate center with the intersection of the axis with the curve. The traces on the left (A) and the traces in the middle (B) represent the gain and phase "tuning curves" of neurons v174d (Fig. 1A) and h177d (Fig. 1B), respectively. A hypothetical neuron that follows precisely the cosine rule and whose ellipse has been degenerated to a straight line is plotted on the right (C) for comparison. The two circles that comprise the gain "tuning curve" are tangent (Fig. 2C, top panel) and a

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geometric interpretation of the cosine rule is apparent (any cord of the circle is given by the product of the diameter and the cosine of the angle that it forms with it). In addition, for this ideal case, the phase is represented by two concentric half circles whose radii differ by 180 ~. As the neuron's spatial properties diverge from the simple "cosine-like" behavior, the two circles that describe the tuning of the gain bisect each other and the two half-circles that describe the tuning of the response phase become distorted (Fig. 2B). As the tuning ratio increases the neuronal gain deviates more and more from the rectified cosine function and the phase has an increasing dependance on stimulus orientation. Figure 3 shows two examples of the most "broadly tuned" neurons encountered thus far. They had ratios of 0.57 (h78p; left) and 0.50 (h76p;

100

0

Fig. 2 A - C . Relation between the response ellipse and the gain and phase "tuning curves". The top panels show the response ellipses (dotted lines) and the "tuning curves" of the neuronal gain as given by (1) (solid lines). The bottom panels show the phase "tuning curves" in the x y plane as given by (2). The x and y-components are plotted in sp/s/g (top traces) and degrees (bottom traces). A Polar plots for neuron v174d (Fig. 1A). B Polar plots for neuron h177d (Fig. IB). The directions of maximum (Smax) and minimum (Stain) sensitivity are specified as vectors from the coordinate center to the intersection of the solid and dotted lines. C Polar plots of a hypothetical neuron whose ellipse is degenerated to a straight line (S O=46.1 sp/s/g, ~ o = 4 1 ~ ~b0=80 ~ and so=O ) . Unlike Fig. 1, the response phase has been calculated in the interval (0, 360), in order to be plotted in polar coordinates

so. B

right), indicating that the minimum sensitivity in the horizontal plane is larger than half of the maximum. 4.2 Three-dimensional case

The neurons whose responses are shown in Figs 1 and 3 were not tested at different planes of stimulation. In such cases, no predictions (not even its vectorial projection in the horizontal plane) can be made about the magnitude and direction of the polarization vector in three dimensions. If it were known that the "parent" ellipse collapsed to a straight line (i.e., if the neuron had a "null" plane with zero sensitivity along any direction in the plane), it would always be true that the direction of maximal sensitivity in any plane would be the vectorial projection of the three-dimensional polar-

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Fig. 3A, B. Examples of neurons which have spatial response characteristics that deviate significantly from the simple "cosine-like" behavior. Neuronal gain (top traces) and neuronal phase (bottom traces) are plotted as a function of the orientation angle ct of the linear acceleration in the animal's horizontal plane ( f = 0 . 2 Hz). A Neuron h78p: S o = 4 7 . 5 s p / s / g , % = 9 5 ~, 4,0 = 59 ~ and So = 27 (ratio of 0.57). B Neuron h76p: S O = 13.4 sp/s/g, cto = 347 ~ 4'0 = - 2 2 ~ and s o = 6 sp/s/g (ratio of 0.50). Both neurons show a high sensitivity for stimulation along the axis perpendicular to the direction of maximal sensitivity in the horizontal plane

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237

ization vector. However, when the "parent" ellipse does not degenerate to a straight line, what is projected on the plane is not the three-dimensional polarization vector but the whole "parent" ellipse. It is easy to intuitively understand this by considering the shadow of an ellipse or a circle on a screen. Depending on the orientation of the figure relative to the light beam, its shadow would change9 Not only would its size vary but also the direction of the major and minor axes would not be constant. It is important to clarify that even if a neuron is characterized by a horizontal ellipse that is degenerated to a straight line (i.e., it seems "cosine-tuned" in the horizontal plane), it can not be concluded that the "parent" ellipse would also be a straight line. This is because the projection (or similarly, the shadow) of the "parent" ellipse on a plane could change from a straight line to a circle depending on the orientation of the projection plane relative to the plane of the "parent" ellipse9 For example, the projection on any plane that is perpendicular to the "parent" ellipse (one of those is the "null plane") would always be a straight line. Therefore, the neuron shown on the left of Fig. 1 might in reality have a "parent" response ellipse with a large tuning ratio if the plane of the "parent" ellipse was nearly perpendicular to the horizontal plane, or equivalently, if the direction of zero sensitivity laid the horizontal plane. Examples of neurons whose responses were obtained at different planes of stimulation are shown in Fig. 4. When the animal is pitched backwards (/3 = - 3 0 ~ or forwards (/3 = + 30 ~ so that it is stimulated in a plane that forms an angle of 30 ~ with the horizontal, the neurons exhibit maximum sensitivity at different angles ~, depending on the plane of stimulation. The neuron on the left (h183a) exhibits maximal sensitivity at 5_30=38 ~ ( / 3 = - - 3 0 ~ and 0~3o=69 ~ (/3= +30 ~ compared to 0~o=53 ~ in the horizontal plane (/3 = 0~ In addition, its maximum gain increases when the animal is pitched nose-up (fl < 0). The actual coordinates of the direction of the three-dimensional polarization vector have been calculated to be /3p = --42 ~ and ~p = 42 ~ For the neuron on the right (h196b), ~-30 = 67~ and % = 55 ~ with a decrease in the sensitivity when the animal is pitched nose-down. The three-dimensional polarization vector for this neuron has been calculated at /~p=34 ~ and % = 5 0 ~. It is important to clarify that the direction of the maximal sensitivity in the different planes, 5_3o, 50 and 530, and the coordinates of the three-dimensional polarization vector have been calculated simultaneously by fitting both gain and phase at all planes by the equations described with the "parent" response ellipse model. For example, the three data points in Fig. 4A (squares; /3 = - 3 0 ~ were not fit independently but in combination with the responses at the other orientations (circles and diamonds). Theoretically, three data points which do not all lie on the same plane would be sufficient to estimate the "parent" ellipse and its projection onto different planes. The projections of the "parent" ellipse on the different "pitch planes" of stimulation have been

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Fig. 4 A - l ) . Examples of response from vestibular nuclei neurons to sinusoidal linear acceleration in different planes o f stimulation (0.6 Hz). A, B Response gain (top panel) a n d phase (bottom panel) are plotted as functions o f the stimulus orientation angle ct. The lines are fits of (3) and (4). Note that both gain and phase and all data points (at all planes of stimulation) have been fit simultaneously. A The polarization vector of neuron hi 83a (left) was calculated to be at /~p = - 4 2 ~ and ~p = 42 ~ with a gain o f Sp = 28.8 sp/s/g and a phase of Op = - 2 9 ~ The minor axis of the "parent" ellipse was estimated to be sp = 6 s p / s / g . B For neuron h196b (right), tip = 3 4 ~ a n d ~p = 50 ~ Sp = 50.2 sp/s/g, Op = - 3 1 ~ and sp = 16.8 sp/s/g. The phase of the m i n i m u m sensitivity response always differs by 90 ~ from that of the m a x i m u m sensitivity response. C a n d D Projections of the "parent" ellipse on each-of the planes defined by the angle/~ for neurons h183a and h196b, respectively, xp is the new x-axis after rotation around the y axis by an a n g l e / L Dashed lines are used for /~ = 0 ~ solid lines for /~ = - 3 0 ~ and dashed lines for /~ = 30 ~ The x and y-components are plotted in sp/s/g

plotted in the bottom traces of Fig. 4 (Fig. 4C and D). It can be seen that the ellipses of the same neuron at each plane of stimulation have different orientation and tuning ratios. The fact that maximum sensitivity at different "pitch planes" is observed at different directions (i.e., 0~_30~ 530 ~ 0 ~ 0 ) is a direct indication that the spatial properties of these neurons can not be accurately described by only one vector (the polarization vector). For example, simply knowledge of its direction (i.e., tip and ~p) is not enough to give the direction of maximal sensitivity at different planes9 Equivalently, the polar angle of the direction of maximal sensitivity at one pitch plane (for example, 50 in the horizontal plane) does not coincide with the 5p of the three-dimensional

238

polarization vector. Instead, complete description of the spatial properties of a neuron requires knowledge of the "parent" ellipse (i.e., its plane and both principal axes). Only then can the neuron's response be estimated for any direction in any plane. It would be helpful to visualize the predictions of (3) and (4) for the gain and phase of a neuron in three-dimensions. Figure 5 shows the gain (top panel) and the phase (bottom panel) of neuron h196b (shown in Fig. 4B) as a function of the two angles ~ and ft. The solid and dashed curves of Fig. 4B are simply the intersections of the three-dimensional gain and phase surfaces with the planes at fl = - 3 0 ~ and fl = 0 ~ Gain and phase tuning surfaces (extension of the tuning curves shown in Fig. 2 in three-dimensions) are plotted in Fig. 6 for the same neuron (hl96b). For better visualization of the three-dimensional surface, three perspectives of each of the gain and phase tuning surfaces are shown. The intersection of any axis in the x - y - z coordinate system with the surface that describes the gain tuning surface (top traces; Eq. 3) and the phase tuning surface (bottom traces; Eq. 4) gives the response gain and phase, respectively, for stimula-

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239 tion along that direction. There is only one direction in space which lies in the "null" plane that is characterized by zero sensitivity (i.e., along that direction, the gain surface of Fig. 6A, B and C has zero thickness; see arrow). The "parent" ellipse (not shown) touches the gain surface of Fig. 6A, B and C at two points (the principal axes) and its plane is perpendicular to the direction of zero sensitivity. The tuning surfaces for the gain are always symmetrical around the principal axes and any asymmetry seen in Fig. 6 is due to different perspectives. 5 Discussion

It has been shown in this paper that the response ellipse model (Angelaki 1991a) qualitatively and quantitatively describes and satisfactorily fits the response gain and phase recorded from vestibular nuclei neurons in response to linear acceleration. The model could, therefore, be used in quantifying the spatial properties of neurons, irrespective of whether they exhibit "cosinelike" or "non-cosine-like" spatial characteristics. The only assumption required for the model to hold is the lack of any significant harmonic distortion in the neuronal responses. The response ellipse is defined and calculated at each frequency of stimulation. It is likely that the orientation, the principal axes and the tuning ratio (i.e., the ratio of the minimal over the maximal sensitivity) of the ellipse could be different at different frequencies of stimulation (Baker et al. 1984a, 1985; Kasper et al. 1988). The response ellipse model is equivalent to the definition of two response vectors in spatial and temporal quadrature (since the semi-major and semi-minor axes are perpendicular to each other and have a phase difference of 90~ It has been shown that when .the "parent" ellipse has not degenerated to a straight line, determining a direction in space that corresponds to the polarization vector is not sufficient to describe the spatial properties of a neuron. Additional knowledge of the second response vector (minor axis of the ellipse) is required in order to define the tuning surfaces of the response gain and phase. In addition, it has been demonstrated that unless the neuron has been stimulated in three-dimensions (i.e., along axes that lie in more than one plane), the information that is obtained is limited and can not be used to estimate any aspect of the three-dimensional vector, not even its projection onto the plane of stimulation. Spatial tuning characteristics that are not "cosinelike" have been reported in vestibular nuclei neurons of alert cats (Baker et al. 1984a; Kasper et al. 1988) and in vestibular reflexes (Baker et al. 1985). In all of these reports, the stimuli consisted of both angular and linear accelerations. These observations of complex spatial properties have been attributed to misalignment in the spatial-temporal convergence between the canal and otolith input (Baker et al. 1984a). However, the neurons showing the "non-cosine-like" spatial characteristics in this report have been stimulated with pure linear

acceleration. It is, therefore, possible that spatial and temporal misalignment in converging otolith inputs is present. How could these complex spatial tuning characteristics arise, assuming that hair cells are characterized by a true null plane (i.e., stimulation along any direction in the plane would produce no response) (Shotwell et al. 1981)? Under the assumption that converging inputs to a neuron add linearly, it has been proposed (Baker et al. 1984a) and theoretically demonstrated (Angelaki 1991b) that "non-cosine-like" tuning characteristics can arise from inputs which exhibit "cosine-like" spatial properties only if the inputs differ in both their temporal and spatial properties. An interesting question is whether the otolith afferents show "non-cosine-like" spatial properties. Several observations and reports would suggest that this might be the case for some fibers. Fernandez and Goldberg (1976b) have reported a nonzero sensitivity in "null" directions that was significant in some primary otolith afferents recorded from anesthetized squirrel monkeys. In addition, evidence for "non-cosine-like" spatial tuning characteristics in primary otolith afferents comes from the different response phase values reported during sinusoidal pitch and roll rotation in the frog (Blanks and Precht 1976). Finally, the present model applied to responses during small angle sinusoidal pitch and roll tilts showed that approximately 24% of otolith primary afferents in gerbils have minimum sensitivity that is more than 10% of the maximum sensitivity in the horizontal plane (Dickman et al. 1991). It has been shown that some otolith afferents innervating the peripheral extrastriola may branch and innervate several hair cells within a terminal field diameter of approximately 100 ~t (Fernandez et al. 1990). The non-zero sensitivity of primary otolith afterents in "null" orientations, whenever present, could be created by different temporal characteristics of inputs from hair cells which have somewhat different polarization vectors and are innervated by a single afferent fiber (Angelaki 1991b). Even though there is evidence that "non-cosinelike" spatial properties might be present at the level of otolith primary afferents, this observation by itself is not sufficient to explain the spatial characteristics of vestibular nuclei neurons. The largest tuning ratios observed in the otolith afferents of the gerbil almost never exceed the value of 0.25 (Dickman et al. 1991). In contrast, ratios of 0.25-0.65 is not a rare occurrence in the vestibular nucleus (Bush et al. 1992). In addition, the percentage of neurons having tuning ratios larger than 0.10 is significantly higher for neurons recorded in the vestibular nucleus compared to the afferent population. Furthermore, the direction of maximal sensitivity in three-dimensions does not necessarily lie on the plane of either the utricle or the saccule (Fig. 4). All of these observations suggest that the spatial properties of vestibular nuclei neurons could only be explained by a significant convergence of otolith afferents from the same and/or different end-organs. Whether the temporal and spatial misalignment exists between converging saccular and utricular afferents, between afferents from

240 t h e s a m e o r o p p o s i t e l a b y r i n t h s , o r b e t w e e n afferents f r o m t h e s a m e e n d - o r g a n is still t o be d e t e r m i n e d . N e u r o n s t h a t b e h a v e like ideal o n e - d i m e n s i o n a l line a r a c c e l e r o m e t e r s h a v e s p a t i a l c h a r a c t e r i s t i c s t h a t enc o d e o n l y o n e s p a t i a l d i m e n s i o n . It is i n t e r e s t i n g to n o t i c e t h a t n e u r o n s w i t h ellipses h a v i n g a n o n - z e r o m i n o r axis a r e c a p a b l e o f c o n v e y i n g a d d i t i o n a l i n f o r m a t i o n as a c o n s e q u e n c e o f t h e i r s p a t i a l p r o p e r t i e s . T h e y e n c o d e t w o - d i m e n s i o n a l s p a t i a l i n f o r m a t i o n , repr e s e n t e d b y t h e i r s e m i - m a j o r a n d s e m i - m i n o r axes. T h e f u n c t i o n a l s i g n i f i c a n c e o f n e u r o n s e x h i b i t i n g s u c h spatial p r o p e r t i e s is j u s t b e g i n n i n g to be i n v e s t i g a t e d . F o r e x a m p l e , w e h a v e r e c e n t l y d e m o n s t r a t e d t h a t t h e semim i n o r axis c o u l d c a r r y i n f o r m a t i o n r e g a r d i n g t h e t i m e d e r i v a t i v e o f t h e s e m i - m a j o r axis in h o r i z o n t a l c a n a l sensitive v e s t i b u l a r n u c l e i n e u r o n s ( A n g e l a k i et al. 1991; B u s h et al. 1992). S u c h n e u r o n s w h i c h e n c o d e f o r b o t h a l i n e a r a c c e l e r a t i o n v e c t o r a n d its d e r i v a t i v e m i g h t p a r t i c i p a t e in t h e g e n e r a t i o n o f t h e m a i n t a i n e d eye v e l o c i t y d u r i n g o f f - v e r t i c a l axis r o t a t i o n ( A n g e l a k i et al. 1991).

Acknowledgement. This work was supported by NASA grants NGT50165 and NAG2-26 and NIH grant DC00385 to AAP and by the Shevlin and Doctoral Dissertation Fellowship from the Graduate school of the Univ. of Minnesota to DEA.

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sponse to constant angular acceleration. J Neurophysiol 38:1250-1268 Bush GA, Perachio AA, Angelaki DE (1992) Quantification of different classes of canal-related vestibular nuclei neuron responses to linear acceleration. NY Acad Sci (in press) Dickman JD, Angelaki DE, Correia MJ (1991) Response properties of gerbil otolith afferents to small angle pitch and roll tilts. Brain Res 556:303-310 Fernandez C, Goldberg JM (1976a) Physiology of peripheral neurons innervating otolith organs of the squirrel monkey. I. Response to static tilts and to long duration centrifugal force. J Neurophysiol 39:970-984 Fernandez C, Goldberg JM (1976b) Physiology of peripheral neurons innervating otolith organs of the squirrel monkey. II. Directional selectivity and force response relations. J Neurophysiol 39:985995 Fernandez C, Goldberg JM, Abend WK (1972) Response to static tilts of peripheral neurons innervating otolith organs of the squirrel monkey. J Neurophysiol 35:978-997 Fernandez C, Goldberg JM, Baird RA (1990) The vestibular nerve of the chinchilla. III. Peripheral innervation patterns in the utricular macula. J Neurophysiol 63:767-780 Fukushima K, Perlmutter SI, Baker JF, Peterson BW (1990) Spatial properties of second-order vestibulo-ocular relay neurons in the alert cat. Exp Brain Res 81:462-478 Kasper J, Schor RH, Wilson VJ (1988) Response of vestibular neurons to head rotations in vertical planes. I. Response to vestibular stimulation. J Neurophysiol 60:1753-1764 Loe PR, Tomko DL, Werner G (1973) The neural signal of angular head position in primary afferent vestibular nerve axons. J Physiol 219:29-50 Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7:308-313 Perachio AA (1981) Responses of neurons in the vestibular nuclei of awake squirrel monkeys during linear acceleration. In: Gualtierotti T (ed) The vestibular system: function and morphology. Springer, Berlin Heidelberg New York, pp 443-451 Perachio AA, Bush GA, Angelaki DE (1992) A model of responses of horizontal canal-related vestibular nuclei neurons that respond to linear acceleration (in press) Schor RH, Miller AD, Tomko DL (1984) Responses to head tilt in cat central vestibular neurons. I. Direction of maximum sensitivity. J Neurophysiol 51:136-146 Shotwell SL, Jacobs R, Hudspeth AJ (1981) Directional sensitivity of individual vertebrate hair cells to controlled deflection of their hair bundles. Ann NY Acad Sci 374:1-10 Strang G (1980) Linear algebra and its applications. Academic Press, New York Wilson VJ, Peterson BW (1981) Vestibulospinal and reticulospinal systems. In: Handbook of physiology. The nervous system. Bethesda, MD. Am Physiol Soc Sect 1, vol II, part I, chap 14, pp 667-702 Dora Angelaki ENT Research OJSH 7-318 University of Texas Medical Branch Galveston, TX 77550 USA