Biological Cy metics

Biol. Cybern. 67, 523-533 (1992)

9 Springer-Verlag1992

Detection of rotating gravity signals Dora E. Angelaki Department of Neurology, University of Ziirich, CH-8091 Zfirich, Switzerland Received December 3, 1991/Accepted April 10, 1992

Abstract. It is shown in the preceding paper that neurons with two-dimensional spatio-temporal properties to linear acceleration behave like one-dimensional rate sensors: they encode the component of angular velocity (associated with a rotating linear acceleration vector) that is normal to their response plane. During off-vertical axis rotation (OVAR) otolith-sensitive neurons are activated by the gravity vector as it rotates relative to the head. Unlike "one-dimensional" linear accelerometer neurons which exhibit equal response magnitudes for both directions of rotation, "two-dimensional" neurons can be shown to respond with unequal magnitudes to clockwise and counterclockwise off-vertical axis rotations. The magnitudes of the sinusoidal responses of these neurons is not only directionally selective but also proportional to rotational velocity. Thus, responses from such "two-dimensional" neurons may represent the first step in the computations necessary to generate the steady-state eye velocity during OVAR. An additional step involving a nonlinear operation is necessary to transform the sinusoidally modulated output of these neurons into a signal proportional to sustained eye velocity. Similarly to models of motion detection in the visual system, this transformation is proposed to be achieved through neuronal operations involving mathematical multiplication followed by a leaky integration by the velocity storage mechanism. The proposed model for the generation of maintained eye velocity during OVAR is based on anatomical and physiological properties of vestibular nuclei neurons and capable of predicting the experimentally observed steady-state characteristics of the eye velocity. I Introduction

Traditionally, the otolith system has been considered to function like a set of one-dimensional linear accelerometers that generate the necessary signals for producing

ocular responses during linear head accelerations and translations (Niven et al. 1966; Hess et al. 1984; Baloh et al. 1988; Hess and Dieringer 1991; Paige and Tomko 1991), the static deviation of the eyes during head tilt (cf., Miller 1962; Diamond and Markham 1981; Collewijn et al. 1985), and provide the velocity storage with information regarding the orientation of the head relative to gravity (cf., Raphan and Cohen 1988; Angelaki and Anderson 1991; Angelaki et al. 1991; Merfeld et al. 1991). In addition to this role as a three-dimensional linear accelerometer, the otolith system has been shown to be capable of sensing head angular velocity during off-vertical axis rotations. Head rotations about axes tilted from the earth-vertical generate a compensatory nystagmus that is maintained throughout the constant velocity rotation (Guedry 1965; Benson and Bodin 1966; Correia and Guedry 1966; Correia and Money 1970; Young and Henn 1975; Raphan et al. 1981; Harris 1987; Darlot and Denise 1988; Darlot et al. 1988; Hess and Dieringer 1990; Hess and Haslwanter 1991; Haslwanter and Hess 1991). This ocular response during off-vertical axis rotation is in contrast to the transient nature of the nystagmus produced by constant velocity rotations about earth-vertical axes (cf., Wilson and Melvill Jones 1979). The steady-state eye nystagmus during constant velocity off-vertical axis rotation (OVAR) has been attributed to the otolith system (Correia and Money 1970; Janecke et al. 1970; Cohen et al. 1983). Given that naturally occurring head movements are often about axes not perfectly aligned with the earth-vertical, this low frequency improvement of the compensatory characteristics of the vestibulo-ocular reflex (VOR) by the otolith system is of special interest. Two experimental observations have been crucial for the present understanding regarding the generation of the steady-state ("bias") eye velocity. First, primary otolith afferents do not exhibit any bias component that could be responsible for the sustained nystagmus observed during OVAR (Goldberg and Fernandez 1982;

524 Raphan et al. 1983). Thus, the "bias" eye velocity is believed to be generated centrally (Goldberg and Fernandez 1981; Raphan et al. 1981; Hain 1986; Raphan and Schnabolk 1988; Hess 1992; Schnabolk and Raphan 1992). Second, the generation of the steady-state eye velocity is entirely dependent on a functional velocity storage mechanism (Cohen et al. 1983). Beyond that, little is known about how the steady-state component of the compensatory nystagmus during OVAR is generated. In the preceding paper (Angelaki 1992b) we presented evidence that neurons with two-dimensional spatio-temporal properties to linear acceleration, which have been recently demonstrated in the vestibular nuclei of rats (Angelaki et al. 1992; Bush et al. 1991), could function as one-dimensional rate sensors. Their firing rates during off-vertical axis rotations are related to the component of angular velocity that is normal to the plane defined by the two response vectors. It was then postulated that "two-dimensional" neurons provide an intermediate step in the process of computation of the steady-state eye velocity during OVAR (Angelaki 1992b). These neurons may, therefore, function as rate sensors provided that the rotational velocity information can be extracted and transformed into the appropriate oculomotor commands. This report deals with the additional steps in signal processing that are necessary to convert the sinusoidal output of "two-dimensional" neurons into an appropriate " D C " signal that can generate the steady-state ocular responses during OVAR. The main aspects of this work have been presented in abstract form (Angelaki 1992c).

2 Two-dimensional spatio-temporal linear accelerometers The properties of neurons with two-dimensional spatiotemporal properties to linear acceleration have been described in the preceding paper (Angelaki 1992b). Briefly, "two-dimensional" neurons are characterized by two response vectors with sensitivities $1 (maximum) and $2 (minimum). These two vectors are spatially and temporally orthogonal. That is, their spatial directions are perpendicular and their temporal dynamics are such that the minimum sensitivity vector $2 is proportional to the time derivative of the maximum sensitivity vector $1. Due to the temporal orthogonality, there is a phase difference of 90 ~ and a linear relationship between the sensitivities Sl and $2 (i.e. $2 = ko~Sl, where ~o is the rotational velocity and k is a proportionality constant; Angelaki 1992b). Neurons with two-dimensional sensitivity to linear acceleration can be created by spatio-temporal convergence (STC) of regular and irregular primary otolith afferents (Angelaki 1992a,c). It has been shown that simple convergence between regular and irregular otolith afferents with different polarization vectors result in "two-dimensional" neurons with one vector encoding linear acceleration and the other responding to the derivative of linear acceleration (jerk; Angelaki 1992c). It should be added that the response vectors $1 and $2 do not necessarily represent the converging

otolith afferent vectors (Angelaki 1992a). Generally, STC between any number of otolith afferents having polarization vectors with different orientations (not necesarily orthogonal) can create "two-dimensional" neurons. Thus, the S~ and $2 vectors do not reflect the polarization vectors of the converging otolith afferents but rather define the phasor-vector space that describes the spatio-temporal response properties of "two-dimensional" neurons. In the preceding paper (Angelaki 1992b), it was shown that "two-dimensional" neurons would exhibit different response magnitudes for CW and CCW rotations during OVAR. Neurons with larger responses during CW OVAR have been defined as CW, whereas neurons with larger response magnitudes during CCW OVAR are defined as CCW. The response of a "two-dimensional" neuron during OVAR can be described as (equations 10a and 11 in Angelaki 1992b): CW OVAR (CW neuron) or CCW OVAR (CCW neuron): R + = SI cos(~ot + A) + $2 sin(ogt + A + 90) = S~(I + k~o)cos(~ot + A)

(la)

CCW OVAR (CW neuron) or CW OVAR (CCW neuron): R - = $1 cos(~ot + A) + 5:2 sin(~ot + A - 90) = $1(1 - kog)cos(~ot + A)

(lb)

The angles A = - A c c w = --ogt0 + 01 (CCW OVAR) or A = Acw = ~ot0+ 01 (CW OVAR) depend on the orientation of the maximum sensitivity vector relative to the gravity vector at the beginning of rotation (~ot0) and the phase of the maximum sensitivity vector of the neuron (01) (Angelaki 1992b). According to (la) and (lb), the response amplitudes of "two-dimensional" neurons are not only different during the two OVAR directions but also proportional to the rotational velocity co. However, the rotational velocity information is contained in the sinusoidal response amplitude, whereas their firing rate modulates around a mean value (equal to their spontaneous discharge, which is represented as zero in la and lb) that is not dependent on the direction of rotation or the rotational velocity. To demodulate this signal, another stage of central processing is necessary. This additional step has to be nonlinear since the average output of a linear system is independent of the input modulation. As we will see, the final requirement can be fulfilled using a second-order nonlinearity, as suggested for motion detection in the visual system. The following presentation will concentrate on the generation of the horizontal steady-state eye velocity. In line with the current postulates regarding the organization of eye velocity in canal coordinates (Hain 1986; Schnabolk and Raphan 1992; see also Angelaki 1992b), these computations would take place within the horizontal VOR pathway. Thus, the generation of the horizontal steady-state eye velocity will be explained based on a model of interactions with otolith/horizontal canal convergent vestibular nuclei neurons.

525

B

A IN = f (x, t)

IN = t (x, t)

I

I

the head. A similar problem, that of detection of bidirectional motion in the visual system, has been a matter of investigation for some time. Most of the hypothesized motion detection mechanisms involve comparison of the outputs of two spatio-temporal filters. As originally proposed by Reichardt (1961) and later modified by van Santen and Sperling (1985), this concept has been used as a first-order approximation of motion detection in insects (Reichardt 1961), in the mammalian retina (Barlow and Levick 1965; Koch et al. 1982; Grzywacz and Poggio 1990) and in motion perception in humans (van Santen and Sperling 1984). The following sections will first present the principle of a basic motion detector as suggested by Reichardt (1961) and van Santen and Sperling (1985). Subsequently, a similar model will be proposed and developed for detection of the rotating gravity vector by the central vestibular system.

7 D

C

IN = I (x, t)

I I

I

I

I

I I, D

IN = f (x, t)

I

I

Z7

I

I

ZV

Zv

I

Z7

iI /,

ly Fig. 1A-C. Algorithmic models of motion detection. S F 1 and S F 2 represent spatial filters, TF, T F t and T F 2 are temporal filters and the symbol S x is used to represent cross-correlation (multiplication followed by integration). A: The basic Reichardt detector, as modified by van Santen and Sperling (1985), proposed to explain directional selectivity in the visual system. B: The basic linear subunit proposed in the visual system by Watson and Ahumada (1983). It corresponds to the output of "two-dimensional" neurons (la and lb). C: Algorithmic model of a complete motion detector based on the subunit of (B). D: An alternative model of the one shown in (C), in which the spatial filters S F , and S F 2 have been interchanged

3. Detection of the rotating gravity vector

The computation of the rotational velocity during OVAR can be regarded as a problem of motion detection of the constant gravity vector as it moves around

Algorithmic description Fig. 1A shows the simplified diagram describing the principle of a motion detector (van Santen and Sperling 1985). In the basic Reichardt detector two subunits (one with output YL and the other with output YR; Fig. 1A) share input channels, represented by the spatial filters SF~ and SF2, that sample the spatio-temporal input field IN =f(x, t) at two points in space. The Reichardt detectors operate on the basis of delaying one of the two input channels (i.e., SF~ or SF2) and comparing that delayed pattern with the non-delayed pattern of the other channel. The comparison operation is accomplished by multiplication followed by integration (i.e., by a cross-correlation of the two time functions). Algebraic subtraction of the two subunit outputs, Yr and YR, represents the detector's total output and gives a signal that is directionally selective (van Santen and Sperling 1985). Note that the temporal filters TF1 and TF2 are not necessarily pure time delays (as for example proposed by Raphan and Schnabolk 1988). They could be any linear, time-invariant, temporal filter (van Santen and Sperling 1985). An analogous model can easily be constructed to explain the directional selectivity of neurons encoding rotation of the gravity vector in the vestibular system. In this case, the outputs of the spatial filters could represent different vectors in space (or in a plane, since we limit this discussion to the horizontal plane). For example, the output of a ~z/2 spatial filter would be equivalent to a vector that is perpendicular to the input vector. Similarly, a re/2 temporal filter would create a phase difference of 90 ~ between two vectors. Using a combination of such filters, the responses of "two-dimensional" neurons during OVAR (la and lb) can be represented in algorithmic form as the subunit shown in Fig. lB. The output of this subunit (first proposed for the visual system by Watson and Ahumada 1983) is a linear combination of the two inputs and the linear filter TF introduces a phase shift of 90~ at all frequencies. Thus, the output of one of the spatial filters (SFO corresponds to the first term of (1), i.e., that due to the maximum sensitivity $1. The second filter, S F 2 , intro-

526 duces a spatial phase shift of 90 ~ and gives the minimum sensitivity $2. The output of SF2 goes through a temporal filter representing a differentiator (TF, Fig. 1B or TF(• Fig. 1C or 1D) characterized by either a phase advance or phase lag of 90 ~ depending on the direction of rotation. Summation of the outputs of SF1 and the temporal filter produces signals that are equivalent to the responses of "two-dimensional" neurons to CW and CCW OVAR ( l a and lb). The additional step required to specify how a time-varying signal is mapped into a single real number and transform the subunit shown in Fig. 1B into a full motion detector can be accomplished by a cross-correlation (Reichardt 1961; Hain 1986). A complete motion detector incorporating the "twodimensional" linear accelerometer subunits (Fig. 1B) and implementing a cross-correlation function is presented in Fig. 1C. The output of the linear summator (Z) is convolved with a direct output of the SF1 filter. The output of the subunit, YL, encodes both directions of movement (i.e., CW and CCW) with different amplitudes. However, the output for motion in one direction is not opposite to the output for motion in the opposite direction. Therefore, an identical subunit (on the other side of the brain) whose temporal filter responds opposite to that of the first subunit (i.e., when one is characterized by a phase lead the other is characterized by a phase lag) is required. Algebraic subtraction of the outputs of the two subunits gives a signal that is proportional to the velocity of rotation and ant• ric. In other words, the output is positive during rotation in one direction and (equal but) negative during rotation in the opposite direction. This is only one of many possible ways that a second-order nonlinearity could be implemented after the basic linear part of the subunit (i.e., output of the "two-dimensional" neuron). For example, an equivalent diagram would be created if the two spatial filters, SF~ and SF2 in Fig. 1C were interchanged. The resulting algorithmic model is shown in Fig. 1D. In this case, the vector that is temporally filtered through TFis the output of SFI, that is, the maximum sensitivity vector, $1. The equivalence of this type of subunit with the responses of "two-dimensional" neurons during OVAR and a detailed analysis of its response are presented in the Appendix. Even though these two examples are by no means exclusive, they will be further elaborated, with the main goal being to illustrate that the output of "two-dimensional" vestibular nuclei neurons is capable of generating the steady-state eye velocity during OVAR. A detailed mathematical description of these algorithmic computations is presented in the following section.

R ~ = S, cos(~ot + Acw)

As explained above, the output of the linear summator (E) corresponds to the response of a "two-dimensional" neuron during CW OVAR and can be expressed as (from la and lb):

To develop the mathematical description for the algorithmic model presented in Fig. IC, let's first consider a clockwise rotation. The output of the SF~ filter, which could correspond to the response of a "one-dimensional" neuron during CW OVAR can be expressed as (equations 7 in Angelaki 1992b):

(3)

R + = Sl(l • k~o)cos(~ot + Acw)

Equation (3) with " + " represents the response of a CW "two-dimensional" neuron, whereas " - " corresponds to the response of a CCW "two-dimensional" neuron. To simplify the calculations, the "one-dimensional" neuron whose response is described by R ~ and the "two-dimensional" neuron whose response is described by R + have similar sensitivity vectors S~ (in terms of direction ~Oto, phase 01 and gain S~). A cross-correlation of (2) and (3) gives: [ST cos2(ogt + Acw) • kogSTcos2(ogt + Acw)] dt = ST(1 ___kco) ~ COS2(~0t§ Acw )dt

(4)

The integral in (4) is equal to: S c~

t 1 ~ + Acw) dt = ~ + ~ sin[2(~ot + Acw)] + C,

(5) where C represents a constant. Equation (5) shows that the integral calculated in an interval [0, T] would consist of a constant, T/2, and a sinusoidal term [sin(2~oT + 2Acw ) - sin(2Acw)]/4~. The constant term, multiplied by ST( 1 _ keo), provides a change in the neuronal firing rate that is proportional to the rotational velocity during CW OVAR. However, the sinusoidal term in (5) would be inappropriate for the response of the neuron since it has double the frequency of rotation and, even though it would be minimal at high rotational velocities, its contribution to the neuronal response would be quite significant at low rotational velocities. This problem can be easily remedied by considering that many "two-dimensional" neuron subunits (Fig. 1B) interact (summate) before the cross-correlation step. It has been shown (Angelaki 1992b) that Acw is the sum of the polar angle of the maximum sensitivity vector of the neuron (~t0) and its neuronal phase (01). It is assumed here that the angles Acw of the "two-dimensional" neuron subunits are distributed throughout the horizontal plane. Under this condition, instead of (4), the output of the cross-correlation step would give: T 2n

I ~ [$2(1 _ kcolcosZ(~ot + Acw)] dAcw dt o

Mathematical description

(2)

(6/

0

Assuming that $1 and kco are independent of Acw, (6) becomes: T

2;~

ST(1 _ keo) ~ cosZ(~ot + Acw) dAcw dt o

0 T

= ~ [~zS2(1 ___k~o)] d t = nST(1 • kog)T 0

(7)

527 Thus, cross-correlation of multiple pairs of "one-dimensional" and "two-dimensional" neurons solves the problem that the sinusoidal term in (5) would impose. Equation (7), including only the constant term with no sinusoidal modulation at double the frequency, describes the outputs YL and YR of Fig. 1C during CW OVAR. During CCW OVAR, the " _ " signs are reversed for YL and YR. Thus, the output YL is opposite to that of YR during both CW and CCW OVAR. Because of the required antisymmetry and in accordance with the algorithmic model of Fig. 1C, the final output of the motion detector is given by the algebraic subtraction of YL and YR. Thus, during the two directions of OVAR, the eye velocity command to the eye muscles will be proportional to:

+_2k~or~S2 T

(8)

In the derivation of the equations it was assumed that CW and CCW "two-dimensional" neurons encode the same proportion k of the rotational velocity (see 1 and 3). This is not a necessary assumption for the presented model: CW and CCW could encode for a different portion of 09. It can be easily shown that, if kcw and kccw represent the two proportionality constants, the antisymmetric response during the two directions of OVAR would be (instead of 8): ___(kcw + kccw)~ozcS2 T

(9)

Neuronal implementation A simplified neuronal scheme that implements the mathematical functions described in the previous section is presented in Fig. 2. Neurons OR and OL represent "one dimensional" purely otolith-sensitive neurons (i.e., they do not exhibit any sensitivity to semicircular canal stimulation) and their responses are described by (2). IR and IL represent "two-dimensional" otolith/type I horizontal canal convergent cells (Type I and type H

I_

Iq

Fig. 2. A simple neuronal scheme, symmetrical on the left (L) and right (R) sides of the brain proposed to implement the functions of the algorithmic models presented in Fig. 1C and 1D. The vertical dashed line represents the midline. O: "one-dimensional" purely otolith-sensitive vestibular neuron; I: "two-dimensional" otolith/type I horizontal canal convergent vestibular nuclei neuron; Iv: type I horizontal canal vestibular nuclei neuron with mean firing rate that is proportional to the rotational velocityduring OVAR

are defined based on their canal sensitivity according to Duensing and Schaefer 1959: Type IR neurons are excited during angular head acceleration to the right, whereas type I L are excited during angular head acceleration to the left). During OVAR, the steady-state response of the IR and IL neurons is given by (3). The third neuron on either side of the brain (defined as IvR and IvL) receives convergent inputs from both OR and IR or OL and IL neurons, respectively. The interaction is assigned to be nonlinear in order to provide for the mathematical multiplication (see Discussion). Since neurons OR/OL do not receive semicircular canal inputs, Iv neurons respond linearly to canal stimulation. Thus, the canal response of IvR and IvL neurons will be identical to that of 1R and 1L (type I), yet their response to otolith stimulation will be considerably different from the responses of I R and IL cells (compare 3 and 7). For convenience, all horizontal canal-sensitive neurons in Fig. 2 have been considered to exhibit type I horizontal canal responses. It is likely, however, that these computations involve both type I and type H neurons. In the simplified neuronal diagram of Fig. 2, the third neuron (IvR or IvL) receives only one pair of inputs. However, as explained above, IvR and lvL neurons are assumed to receive inputs from many pairs of neurons with different polarization vectors and different phases. Further, the integration process shown in each subunit of Fig. I C has been assigned to the same neuron that performs the multiplicative interaction (i.e., IvR and IvL). However, the integration, which is performed by the velocity storage mechanism (cf., Raphan et al. 1979; Robinson 1981) is probably achieved through a combination of short or long term potentiation interactions (Shen 1989) and/or lateral interconnections between the two sides of the brain (cf., Anastasio 1991), possibly involving several neurons, rather than the single Ivn or IvL cell as indicated in the simple scheme of Fig. 2. Thus, it is important to point out that, despite the simplified neuronal scheme drawn in Fig. 2, integration over many "two-dimensional" neurons in (6) and (7) represents an example of distributing processing. A distributing processing has also been suggested for the canal driven vestibulo-ocular reflex (Anastasio and Robinson 1990). An ideal integration was used to derive (4) through (9). In reality, the velocity storage integrator is leaky with a time constant of approximately 15-20 s in monkeys (cf., Raphan et al. 1979). Therefore, the integration is similar to a low pass filter. Further, the velocity storage integrator must perform a continuous time integration, where the integration interval T is much smaller than its time constant. In order for the rising phase of the eye velocity to be smooth and for the steady-state response (after the integrator has been fully charged) to be independent of the duration of the angular velocity step during vertical or off-vertical axis rotation, a continuous time integration is required. Thus, T in ( 4 ) - ( 9 ) corresponds to a small, constant time interval which is an inherent characteristic of the velocity storage mechanism, equivalent to producing a continuous integration over time.

528 The described simplified neuronal scheme of Fig. 2 would also provide the functions described by the two subunits of Fig. 1D. The only difference would be that neurons OR/Oc would be characterized by a polarization vector that is similar to the minimum sensitivity vector of neurons IR/Ic. Using equivalent calculations (presented in the Appendix) the response of Iv neurons during CW and C C W O V A R can be described as: [rcS~kco( 1 + kco)T]

(10)

The antisymmetric response is: + 2(kco) 21tS2 T

(11)

Simulated responses o f the Iv neurons Figure 3A and 3B shows simulations of expected responses from Iv horizontal canal vestibular nuclei neurons (based on 7) after an instantaneous tilt (at time zero) of the axis of rotation away from the earth-vertical during constant velocity rotation. It is assumed that the tilt is delivered when the firing rate of the neurons has returned to the spontaneous discharge level after the transient canal response has subsided. This experimental condition was chosen to isolate the dynamic (governed by the properties of the velocity storage) and steady-state otolith-induced response from that due to canal stimulation (cf., R a p h a n et al. 1981). The firing rate of the neurons (Fig. 3A and 3B) increase gradually to a steady-state level that is different from the spontaneous firing rate (represented as zero in the figures). The proportionality constant in ( 7 ) - ( 9 ) was set to nS 2 T = 50 and the velocity storage mechanism had a

100-

A

--

/3co

--

ipsilateral rotation contralateral

B

time constant of 20 s (cf., Raphson et al. 1979). Two possibilities for the value of kco were considered. For Fig. 3A, kco = 0.2 approximated the value experimentally observed in rats for the C C W "two-dimensional" neuronal population at 0.6 Hz (Bush et al. 1991). Such values would produce an increase in the steady-state mean firing rate of 60 spikes/s and 40 spikes/s above the spontaneous level of the neuron during ipsilateral and contralateral rotation (relative to the cell body of type I cells), respectively. Similar responses are predicted for the type IIv horizontal canal neurons, however, the largest change in the steady-state firing rate would be observed during contralateral rotation. For Fig. 3B, kco = 0.6 corresponded to the experimentally observed value at 0.6 Hz for the CW population of vestibular nuclei neurons (Bush et al. 1991). In this case, the firing rate of the type Iv neuron increased by 80 spikes/s and 20 spikes/s above the spontaneous level during ipsilateral and contralateral rotation, respectively. The difference in the steady-state firing rates of vestibular nuclei neurons during the two directions of O V A R can be large (when kco is large; Fig. 3B) or small (when kco is small; Fig. 3A). Further, the larger the value of kco, the smaller the change in the mean firing rate during one Of the directions of O V A R compared to the resting discharge (e.g., contralateral rotation in Fig. 3B). Thus, the model predicts that Iv neurons would behave in one of the following ways: Their mean firing rate would either increase above the spontaneous level during both O V A R directions (although by different amounts to each direction of rotation), or it would increase during one direction of OVAR, while there will

--

rotation

ipsilateral rotation contralateral

rotation

80

~o q) Cl u3

6O 40

oo c 0 O_

20

0

100-

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D

6o g)

50-

9 09 ~D O_

-50-

E -100-

25

50 tTme ( s e c )

75

100

2'5

5'0 time (see)

7'5

1;0

Fig. 3A-D. Simulated responses from vestibular neurons (Iv) which change their mean firing rate during OVAR. For these simulations, a sudden instantaneous tilt (from the earth-vertical) of the rotational axis at time zero occurred while the constant velocity rotation was maintained. The velocity storage time constant was 20 s and the proportionality constant in (4)-(11) was set to nS~T = 50. A, B: Responses from vestibular nuclei neurons predicted from (7) with kco = 0.2 (A) and kco = 0.6 (13). Solid lines are used for ipsilateral rotation (CCW rotation for Ivz or CW rotation for IvR) and dashed lines are used for contralateral rotation (CW rotation for IV L o r CCW rotation for lvR). C, D: Simulated antisymmetric response during both directions of rotation predicted by (8) (C, with kco = 0.2) and (9) (D, with kccwCO= 0.2 and kcwcO = 0.6). Zero response amplitude represents baseline firing rate before instantaneous tilt

529 be almost no change in the mean firing rate during the opposite direction of rotation. Simulations of the antisymmetric response (predicted by 8 and 9) that is responsible for the "bias" component of eye velocity under the same experimental conditions as those in Fig. 3A,B are shown on the bottom traces (Fig. 3C,D). The antisymmetric responses shown in Fig. 3C are produced by subtraction of the responses shown in Fig. 3A and are based on the assumption that both IVl. and Ivn neurons encode the rotational velocity symmetrically. In this case, the antisymmetric response is given by (8). The antisymmetric responses shown in Fig. 3D are created from the outputs of a n IVL neuron with kccwO9 = 0.2 (Fig. 3A) and an Ivi~ neuron with kcwo~ = 0.6 (Fig. 3B), as predicted by (9). These simulations represent the output of the model. Even though the command from the oculomotor neurons to the eye muscles necessary to produce the "bias" eye velocity during OVAR is partially governed by these antisymmetric responses, the simulations of Fig. 3C,D do not actually represent firing rates of oculomotor neurons. Oculomotor neurons in alert animals have firing rates that are not directly correlated with the velocity of the slow phases of the nystagrnus, since their responses are further complicated by eye position sensitivity and fast phases which have not been considered here. 4 Discussion

In this report a model has been presented that would transform the sinusoidal responses of "two-dimensional" neurons during OVAR into a steady-state change in the firing rate that is proportional to the rotational velocity. Based on models of motion detection in the visual system, we have shown that an additional step involving second-order nonlinear processing is sufficient to transform the sinusoidal output of "two-dimensional" neurons into a steady-state change in the firing rate of "higher order" neurons that is proportional to rotational velocity. The cross-correlation principle used here to implement the required nonlinearity for "rectification" of the sinusoidal responses has been previously proposed by Hain (1986). The necessary calculations have been presented for the simplified case of generation of the horizontal "bias" eye velocity during head rotation about an axis normal to the plane of the horizontal semicircular canals. The model presented here is based on identified otolith/canal convergent central vestibular nuclei neurons exhibiting two-dimensional spatio-temporal response properties to linear acceleration (Angelaki et al. 1992; Bush et al. 1991). Based on a canal coordinate frame for the central otolith system (Hain 1986; Schnabolk and Raphan 1992; see also Angelaki 1992b) and the two-dimensional response properties of canal-sensitive cells to linear acceleration (Bush et al. 1991; Angelaki et al. 1992), we have assumed that the signal processing necessary to generate the "bias" eye velocity takes place within the horizontal VOR system. It is

understood that similar circuits would exist in the vertical VOR pathways. Assuming that canal-sensitive vestibular nuclei neurons encode two-dimensional linear acceleration in canal coordinates (i.e., their response planes to linear and angular accelerations are aligned), the three-dimensional angular velocity can be precisely reconstructed by the three populations of canal-sensitive cells (Angelaki 1992b). The model proposed here is based on the two-dimensional spatio-temporal linear accelerometer neurons and, as explained in the preceding paper (Angelaki 1992b), can be considered as an extension of previous models. Raphan and colleagues (Raphan and Schnabolk 1988; Fanelli et al. 1990; Schnabolk and Raphan 1992) have presented a "delayed pattern travelling wave model" for the generation of the "bias" eye velocity during OVAR. Their model assumes the existence of vestibular nuclei neurons that differ from tonic otolith afferents by a pure time delay. So far, there has been no experimental evidence for such neurons (Schor et al. 1985). As explained in the preceding paper (Angelaki 1992b), the travelling wave model implemented in a three layer neural network by Fanelli et al. (1990) utilizes two types of internal units that essentially behave like "two-dimensional" neurons. Neurons with two-dimensional spatio-temporal sensitivity to linear acceleration can simply be created by linear convergence of primary otolith afferents that differ in both spatial and temporal properties (Angelaki 1992a,c). Thus, the existence of two-dimensional spatio-temporal linear accelerometer neurons that could be created by spatio-temporal convergence and their capability to generate the "bias" eye velocity after some kind of response "rectification" makes the assumption underlying the "delayed pattern travelling wave model" proposed by Raphan and colleagues unnecessary. It should be mentioned that, in addition to the sustained compensatory nystagmus (or "bias" component), the eye movements elicited during OVAR are also characterized by an eye position modulation that oscillates at the same frequency as the rotating gravity vector (Guedry 1965; Benson and Bodin 1966; Young and Henn 1975; Hess and Dieringer 1990). The generation of the modulation component in the ocular responses during OVAR is not addressed in this paper since it is presumably organized differently from the steady-state eye velocity (Young and Henn 1975; Raphan et al. 1981; Cohen et al. 1983). These modulations superimposed on the sustained "bias" component are probably identical to the eye movements produced during linear translation (Paige and Tomko 1991; Hess and Haslwanter 1991).

Basic assumptions underlying the proposed model To formulate the model we assumed that there exist neurons with two-dimensional spatio-temporal response properties to linear acceleration. Neurons with such properties have been observed in the vestibular nuclei of rats (Angelaki et al. 1992; Bush et al. 1991). Two additional assumptions have been used for the deriva-

530 tion of (6)-(9). First, the presynaptic neurons that provide the input signals to the Iv cells are assumed to have a wide variety of angular orientations that are distributed throughout the horizontal plane. This assumption is satisfied by the maximum sensitivity vector distributions of "two-dimensional" neurons: Otolith/ horizontal canal convergent neurons with two-dimensional properties to linear acceleration have been shown to exhibit maximum sensitivity vectors that are distributed throughout the horizontal head plane (Bush et al. 1991). Second, $1 and k~o have been assumed independent of the Acw and Accw angles (7). This assumption is also supported by the properties of otolith/horizontal canal convergent vestibular nuclei neurons: The maximum sensitivity vectors and tuning ratios kco = $2/$1 did not depend on the direction of the maximum sensitivity vector and the neuronal phase (Bush et al. 1991).

Mathematical multiplication might be performed by single cells One of the basic aspects of the present model is the mathematical multiplication between the outputs of OR/OL and IR/IL (Fig. 2). A multiplicative interaction between gravity-related signals has been previously suggested to account for the dependance of the velocity storage mechanism on the static orientation of the head relative to gravity (Hain 1986; Angelaki et al. 1991). What could be the neuronal mechanism responsible for mathematical multiplication? It has been suggested that the interaction between excitatory and inhibitory (shunting inhibition) synaptic inputs to a cell can be strongly nonlinear and, under certain conditions, can be described as a mathematical multiplication (Torre and Poggio 1978; Poggio and Torre 1978). The nonlinear interaction between excitatory and inhibitory inputs is stronger in spines and distal dendrites that are characterized by high input and transfer resistances (Koch et al. 1982; 1983). Cells with relatively large dendritic trees, potentially composed of spines and regions with large input impedances, have been described in the vestibular nuclei (Mitsacos et al. 1983; Ohgaki et al. 1988; Highstein and McCrea 1988). In addition, the presence of GABA and glycine receptors that are associated with C1- channels and are possible candidates for shunting inhibition has been documented (cf., Precht 1979; Walberg et al. 1990). Therefore, it is reasonable to posulate that single vestibular nuclei neurons might be responsible for the required nonlinear information processing.

Model predictions for known characteristics of steady-state eye velocity during OVAR The model for the generation of the steady-state eye velocity during OVAR outlined above can provide for the eye velocity characteristics observed experimentally. The dynamics of the slow phase eye velocity build up in response to a sudden tilt while rotating (cf., Raphan et al. 1981), can be predicted by including the velocity

storage mechanism which provides the necessary integration in the present model. There is evidence in the monkey that the steadystate eye velocity during OVAR increases as a function of head velocity up to approximately 90~ and declines thereafter towards zero as head velocity increases further (cf., Raphan et al. 1981). This dependence of steadystate eye velocity on stimulus velocity is not consistent in all reports and might depend on individual animals. Young and Henn (1975) have described a linear dependance of the steady-state eye velocity on rotational speeds up to 130~ in monkeys. In cats (Darlot and Denise 1988) and humans (Darlot et al. 1988) saturation and sometimes a decrease in the steady-state eye velocity have been reported for high rotational velocities. Saturation at high rotational velocities might be expected, since the signal for the "bias" component is fed through the velocity storage mechanism which has been shown to exhibit nonlinear input-output characteristics at high input levels (cf., Raphan et al. 1979). A recent neural network model has shown that the magnitude-dependent nonlinear behavior of the velocity storage may be due to rectification of vestibular nuclei neurons (Anastasio 1991). Saturation of the velocity storage at high rotational speeds is not sufficient to explain the decrease in the steady-state eye velocity. The decrease of the steadystate eye velocity at high rotational speeds reflect a property of the specific circuits responsible for its computation. Based on the model presented here, the decrease in the steady-state eye velocity could be produced by the dependance of the steady-state firing rate of neurons on the square of rotational velocity. As mentioned earlier (see also Appendix), the output of the motion detector would be proportional either to [+nS~(kco)ZT] (11; Fig. 1D) or [+_nS2kogT] (8; Fig. 1C). An interplay of both algorithms could cause the steady-state eye velocity to be a parabolic function of the rotational velocityf(co) = aco2 + bog, co ~ [0, -b/a]. The function f(~o) would always be zero at co = 0 and co = - b / a and symmetrical around co = -b/2a. Different subjects might have different second-order dependence of the steady-state eye velocity on rotational head velocity and, depending on the relative values of the parameters a and b, the peak of the parabola would be at different values of head velocity. Contrary to the modulation component whose magnitude is proportional to the component of gravity in the plane normal to the axis of rotation, the steady-state eye velocity is independent of tilt angle above approximately 30~ (Young and Henn 1975; Darlot and Denise 1988; Harris 1987). The independence of the steady-state eye velocity on the angle of tilt of the axis of rotation could be accounted for by nonlinearities as proposed previously (Fanelli et al. 1990).

Vestibular nuclei neurons: experimental results and model predictions Sinusoidal responses of vestibular nuclei neurons during stimulation with a rotating linear acceleration vector without coactivation of the semicircular canals have

531

been observed in cats (Melvill Jones and Milsum 1969; Schor et al. 1984; 1985; Chan et al. 1987). Vestibular nuclei neurons that change their mean firing rate during constant velocity rotation of a linear acceleration vector have been reported by Benson et al. (1970) and recently by Reisine and Raphan (1992). Chan et al. (1987) observed periodic modulations in the firing rates of vestibular nuclei neurons during OVAR at 1.75~ (0.005 Hz), however, no changes in the mean firing rates were reported. The recordings were mostly concentrated in the lateral and descending vestibular nuclei with the study being limited to neurons which were sensitive to static head pitch or roll tilts of + 10~ from the horizontal (Chan et al. 1987). In contrast to the results reported by Chan et al. (1987), Benson et al. (1970) have presented evidence that neurons located in the medial (mostly), inferior and lateral vestibular nuclei exhibited different mean firing rates during constant velocity CW and CCW rotation of a linear acceleration vector at 187~ (0.52Hz) around the animal's horizontal plane compared to the resting discharge. There are two fundamental differences in the two studies. First, the cells tested came from distinctly different populations of vestibular nuclei neurons. All neurons tested by Chan et al. were tilt-sensitive and not identified in terms of their canal sensitivity. In contrast, the study be Benson et al. concentrated on neurons that responded solely to horizontal angular acceleration, did not receive any vertical canal input and did not respond to small amplitude oscillatory linear acceleration in the animal's horizontal plane. Second, Benson et al. (1970) employed stimuli at a frequency which was 100 times higher than that used by Chan et al. (1987). As shown in the simulations of Fig. 3, the change in the mean firing rate of Iv neurons during OVAR is proportional to the tuning ratio $2/S~ = kco. Since experimental evidence suggests that the tuning ratios of "two-dimensional" neurons are greater at higher stimulus frequencies (Bush et al. 1991), it is also expected that Iv neurons would exhibit the largest changes in their mean firing rates at higher frequencies (i.e., greater rotational velocities. Benson et al. (1970) reported that type I and type H vestibular nuclei neurons discharged more vigorously when the direction of rotation of the linear acceleration vector was opposite to the direction of angular motion that excited the neurons. Keeping in mind that rotation of a linear acceleration vector causes movement of the otolith membrane in the opposite direction than a similar rotation of the gravity vector, this behavior demonstrates a "synergistic" interaction between the semicircular canal and otolith inputs. Thus, Iv neurons on the right brainstem would have a larger increase in their firing rate during ipsilateral (CW) OVAR. Similarly, Iv neurons on the left brainstem would have a larger increase in their firing rate during CCW OVAR. As explained above, the final output of the model, i.e. the oculomotor command to the eye muscles, must be antisymmetric to CW and CCW OVAR, i.e., it should increase during OVAR in one direction and

decrease (by an equal amount) during OVAR in the opposite direction. Such property has not been observed in responses of vestibular nuclei neurons (Benson et al. 1970). Even though the change in the mean firing rate during the two OVAR directions was significant in most cells studied, there was no systematic difference from the spontaneous firing rate. This response variability is in agreement with the predicted change in the mean firing rates of the lv and IIv neurons during the two OVAR directions (7); depending on the values of kco and nS 2 T, several possibilities could occur (see model simulations in Fig. 3). Thus, the proposed model predicts the experimental observations of Benson et al. (1970) showing that the mean firing rates of Iv/IIv neurons may increase either during both directions of rotation or only during one direction while the other direction produces almost no detectable change from the resting discharge rate. The finding that the mean firing rate of most vestibular nuclei neurons does not usually decrease below the spontaneous level of the cell in response to either OVAR direction (Benson et al. 1970) could indicate that the last step in the proposed model, i.e., the algebraic subtraction of the outputs of Iv neurons might take place at the level of the oculomotor nuclei. The existing excitatory and inhibitory projections from vestibular nuclei cells onto oculomotor neurons innervating each of the lateral and medial rectus muscles (cf., Baker et al. 1969; Baker and Highstein 1978) could account for the postulated algebraic subtraction. In conclusion, the model proposed here implements a second-order nonlinearity and transforms the sinusoidal response of "two-dimensional" neurons into a signal appropriate to generate the "bias" eye velocity observed during OVAR. It should be added that it is possible for the stages following the linear computations (i.e., output of the "two-dimensional" subunit in Fig. I B) to be implemented differently. The multiplicative scheme used in the present model to provide the second-order nonlinearity could potentially be replaced by a squaring operation. For example, squaring of neural signals has been reported as part of retinal processing (Naka and Sakai 1991). It is theoretically possible that both multiplicative interactions of the form shown in Fig. 1C and squaring of the outputs from single "two-dimensional" neurons could generate the "bias" eye velocity and account for the parabolic dependence of the steady-state eye velocity on stimulus velocity. The constraints imposed by the limited experimental data available thus far do not allow one to choose between alternative schemes of implementing the second-order nonlinearity. The purpose of this work has been to propose one of these alternative schemes and demonstrate that two-dimensional spatio-temporal linear accelerometer vestibular nuclei neurons may represent an intermediate step in the generation of the steady-state eye velocity during OVAR. Acknowledgements. This work was supported by NASA NGT grant 50581.

532 Appendix

Implementation of the algorithmic model of Fig. 1D Equation (1) expresses the response of a "two-dimensional" neuron during CW and C C W O V A R relative to the phase of the m a x i m u m sensitivity vector (i.e., Acw = into + 01 and Acc w = COto- 01). The same equations can be expressed relative to the phase of the minimum sensitivity vector as: CW O V A R ( C W neuron) or C C W O V A R ( C C W neuron): R + = (Sj + S2)sin(cot + A')

(A1)

C C W O V A R ( C W neuron) or CW O V A R ( C C W neuron) : R - = (SI -- S2)sin(cot + A')

(A2)

where the angles A ' are defined as follows (02= 01 + 90~ CW OVAR: A ' = A ~w = Acw + 90 = cot0 + 02, C C W OVAR: A ' = - A ~cw = -- Accw + 90 =

-- (~to -

02)

The expressions described by (A1) and (A2) are completely equivalent to those presented by (1). During CW OVAR, the output of the linear summator (E) in Fig. 1D is ( f r o m A1 and A2): R -+ = $1(1 ___kco)sin(ogt + A~w) The output of the SF2 filter during CW O V A R is:

R ~ = kcoSl sin(e)t + A~w ) Finally, the output of the cross-correlation step in Fig. 1D during CW O V A R is calculated as: T

2n

S S2k~176 1 +-kco) ~ sin2(cot + A~w) dA~w dt 0

0 T

= ~ [rcS2km( 1 + kco)] dt = nS2kco( 1 + koo)T

(A3)

0

Equivalent calculations result in (A3) with reversed " + " signs during C C W OVAR.

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