a “beat-to-beat” interval generator for optokinetic nystagmus

Biol. Cybern. 66, 203-216 (1992) i~176 metic.s

�9 Spdnger-Verlag 1992

A "beat-to-beat" interval generator for optokinetic nystagmus Carey D. Balaban ~ and Michael Ariel ~

i Departments of Otolaryngology, Neurobiology, Anatomy and Cell Science and The Eye & Ear Institute, 2 Departments of Behavioral Neuroscience and Psychiatry, University of Pittsburgh, Pittsburgh, USA

Received March 11, 1991/Accepted in revised form August 26, 1991

Abstract. An analysis of optokinetic responses was used to derive an iterative model that reproduces the duration of nystagmus slow phases and eye position control during optokinetic nystagmus. Optokinetic nystagmus was recorded with magnetic search coils from red-eared turtles (Pseudernys scripta elegans) during monocular, random dot pattern stimulation at constant velocities ranging from 0.25-63~ The beat-to-beat behavior of slow phase durations was consistent with the existence of an underlying neural clock, termed the basic interval generator, that is based on an integrate-to-fire neuron model. This hypothetical basic interval generator produces an interval that is the product of the duration of the previous interval and a mean 1 truncated normal variate with variance tr 2. Data analyses indicated that the initial value of the interval generator during a period of nystamus, termed Zo, is proportional to the inverse square root of slow phase eye velocity. Further, if the eye was deviated in the slow phase direction (re mean eye position) when the slow phase began, the slow phase duration was consistent with a single cycle of the basic interval generator. However, if the eye was deviated in the fast phase direction, the distribution of the durations of the ensuing slow phases indicated that a proportion of the slow phases were produced by more than one cycle of the basic interval generator. This phenomenon is termed "skipping a beat" and occurs with probability Ps- Finally, the amplitude of fast phases behaved as a linear function of eye position at the fast phase onset and the product of Zo and slow phase eye velocity. A computer simulation reproduced the observed distribution of slow phase durations, the proportion of fast phases in the fast phase and slow phase directions and the distribution of eye positions at the onset and end of fast phases. This novel model suggests that both timing and eye position information contribute to the alternation of nystagrnus fast and slow phases.

1 Introduction

Nystagmus is an eye movement that consists of two alternating components, slow phases and fast phases (Purkinje 1819; Baloh and Honrubia 1990; Leigh and Zee 1983). Slow phase eye movements compensate for the velocity of movement of the head or visual sur- round to maintain gaze on a point in visual space. By contrast, the fast (or quick) phases (FPs) typically move the eye in the direction opposite the slow phases (SPs) and are thought to maintain eye position within appropriate limits. The alternation between these components produces a controlled, repetitive pattern of movement of the eye in the orbit.

An understanding of the neural control of eye position during nystagrnus requires a characterization of (1) the control of SP eye velocity, (2) the control of SP durations and (3) the control of FP amplitudes (FP velocity appears to be determined by the FP amplitude (Leigh and Zee 1983)). The neural control of nystagmus SP velocity has been characterized extensively in both experimental animals and humans (e.g., Raphan et al. 1977; Robinson 1977; Furman et al. 1989). How- ever, with the exception of data reported by Cheng and Outerbridge (1974a, b); Honrubia et al. (1971a-c); and Bosone et al. (1990), the issues of the control of the duration of SPs and FP amplitudes have received little attention. The major objective of this study is to derive a quantitative description of the timing of the alternation between FPs and SPs. The secondary objective is to describe quantitatively the control of FP amplitudes. The final objective is to incorporate these data in a computer simulation of constant velocity optokinetic nystagmus (OKN).

In his pioneering work on OKN, Ter Braak (1936) presented three possible explanations for the regular alternation of SPs and FPs. First, he proposed that FPs are elicited by an extraocular proprioceptive signal which indicates that eye position exceeds a position threshold. Since this proprioceptive signal is an eye

204

position signal, a general statement of this hypothesis is that FPs are triggered by eye position (eye position hypothesis). The second hypothesis, attributed to Ohm (1929), stated that the alternation between FPs and SPs is determined by a central pattern generator (internal clock hypothesis). Finally, by analogy with respiratory control, Ter Braak suggested that the alternating SPs and FPs are produced by coordinated actions of both a position control system and a central pattern generator (hybrid position-clock hypothesis).

Ter Braak's hypotheses provide a framework for analyzing the control of alternating SPs and FPs during nystagmus. The eye position hypothesis was extended by Chun and Robinson (1978), who assumed that two internal eye position signals specify the beginning and the end of vestibular nystagmus FPs. Lau et al. (1978), though, presented evidence that the threshold for FP generation is a function of both eye position and instantaneous SP eye velocity in humans. Furthermore, Cannon and Robinson (1987) questioned the adequacy of the eye position hypothesis because FPs were pre- served in monkeys after lesions that eliminated normal SPs. Thus, a strict eye position mechanism was not sufficient to explain the triggering of FPs.

This paper extends Ohm's (1928) timing approach, which is based upon the premise that SP durations are the basic temporal units of nystagmus. The control of SP durations was first incorporated in a model by Sugie and Jones (1971), who made the simplifying assumption that they were constant. However, Cheng and Outerbridge (1974a, b) subsequently reported that histograms of SP durations ranged in shape from unimodal to multimodal forms, with the higher order modes centered at approximately integral multiples of the initial mode. The present study extends these concepts and characterizes the beat-to-beat variations in SP durations in terms of intervals of an underlying neural clock.

2 Materials and methods

The apparatus and methods for using scleral search coils to measure optokinetic responses of turtles are described in detail elsewhere (Ariel 1990). Briefly, red- eared turtles, Pseudemys seripta elegans (carapace length 20-25 cm, n = 3), were placed in a head restrain- ing apparatus and search coil contact lenses were ap- plied to the eyes. One eye was occluded by a small matte black hemisphere. Each turtle exposed to nine consecutive, 26 min blocks of monocular, temporal-to- nasal optokinetic stimulation on a single day. Optoki- netic stimuli were generated by projecting the image of a high contrast random check pattern (1 ~ visual angle) at constant velocities ranging from 0.5-63~ A single stimulus velocity was presented during each stimulus block and the velocities were presented in a different order to each turtle. The eye position signals were sampled at a rate of either I0 Hz (2 turtles) or 20 Hz ( 1 turtle), digitized and recorded on a personal computer. Only the data for the stimulated (unoccluded) eye are presented in this paper. Local peaks of the first order

derivative of the eye position were calculated and displayed with the raw data record so that FP eye movements could be observed at the full resolution of the recording ( 10 or 20 Hz). Cursors were positioned man- ually to measure both the eye position and time at the beginning and end of each FP.

Since turtles were not trained to fixate, the absolute eye position in the orbit could not be determined from the records. However, the eye movement recording system defined a relative coordinate system for eye position throughout a stimulus condition because it provided an accurate determination of the orientation of a contact lens search coil in space. Thus, eye position was normalized for each block of trials by defining zero eye position (0 ~ as the mean eye position during each stimulus block.

Contact lens slippage occurred in only three (of 26) stimulus blocks. Slippage of the lens was observed directly during data collection and the times were noted; on the eye movement records, lens slippage appeared as a large amplitude, rapid change in eye position (due to catastrophic failure of methyl-cellulose/gelatin adhesive) and a subsequent shift in the mean eye position. Al- though the absolute eye position data were regarded as unreliable from the time of the slippage, the times of occurrence of the FPs were easily identifiable. The statistical properties of the timing data from these stimulus blocks were identical to data from other stimulus blocks when the lens did not slip on the cornea.

3 Experimental results

3.1 Introduction: OKN and control of SP eye velocity

3.1.1 Identification of FPs which then determine the SP durations. Typical O K N consisted of an alternating pattern of SP movements in the direction of visual simulation and FP movements in the opposite direction (Fig. 1A, B). However, some rapid eye movements were observed in the same direction as the SPs (Fig. 1C). These misdirected FPs comprised about 20% of the rapid eye movements at stimulus velocities below 1 ~ for each turtle, approximately 10% of the rapid eye movements at 1 ~ stimulation and constituted less than 1% of the rapid eye movements for stimuli between 2 and 42~ These rapid eye movements can be considered from two perspectives: (I) as saccadic intrusions into the basic nystagmus pattern and (2) as normal FPs that are programmed in the SP direction to facilitate eye position control during nystagmus. Since there was no a priori basis for differentiating between these views, the SP durations were calculated from both perspectives: (I) as intervals between FPs directed opposite the SPs (saccadic intru- sion perspective) and (2) as intervals between any two rapid eye movements (misdirected FP perspective). As described below, one implication of this study is that these eye movements represent misdirected FPs.

3.1.2 SP eye velocity during constant velocity stimuli. Individual SPs tended to be of constant velocity

205

RIGHT EYE RESPONSES TO LEFTWARD STIMULATION

I L MISDIRECTED l t ~ F A S T PHASE I~ *

0

TI

Fig. IA-C. Horizontal eye movement recordings from the fight eye during leftward optokinetic stimulation (turtle TCK). The stimulus velocities were 21~ in A, 0.5~ in B and 10~ in C. The mean eye position (0 ~ is indicated by a horizontal dotted line A Leftward SPs and rightward FPs of OKN are illustrated; the values of ~ (the ratio of durations of successive SPs) are graphed for each SP. Note. that two long duration SPs in A that have values of s > 2. B Example of a misdirected FP. C The decay of SP velocity during long duration SPs. Eye position records are shown in the upper traces and eye velocity records (differentiated eye position data) are illustrated in the lower traces. The center traces show a constant velocity SP; they are flanked by traces illustrating the roughly exponential decay of SP eye velocity when eye position exceeds an eye position threshold in the SP direction

(Fig. 1A-C). However , nonlinear SPs (Fig. 1C) were often observed at stimulus velocities greater than 10~ they were extremely rare at lower velocities�9 The velocity o f these SPs declined with a single exponential time course. This velocity decrement began when the eye was deviated 5 - 1 5 ~ (mean 8 ~ in the SP direction and was independent o f the dura t ion o f the SP. The eye velocity then decreased until the onset o f the FP, even if it resulted in a SP velocity near 0~ This posit ion-dependent SP eye velocity decay is incompatible with a simple posit ion hypothesis for triggering FPs: if a FP is generated whenever the eye reaches a posit ion threshold, a decrement in SP velocity is unnecessary. However , within the f ramework o f an internal clock hypothesis for SPs, the SP velocity decrement simply limits eye

A_

i 10

5 10 15 20 25 CUMULATIVE STIMULATION TIME (mln)

O8 / TCJ 21 deg/s

0 r ~ _

S u )

~>25 I I _ [7 TCJ 0.5 deg/s

1 jj .~,sHIHL I'oHHIk ~ 0111 t i l l R �89 m m Z 0 50 100 0 5 10 15

SLOW PHASE DURATIONS (s)

Fig. 2.A Weak stationarity of SP durations. The SP durations recorded from the right eye are plotted as a function of the time during a session of leftward optokinetic stimulation at 21~ The mean interval is indicated by a dashed line. There is no trend in the data throughout the stimulation period. B Histograms of SP durations. The left histogram shows the distribution of SP durations recorded from the right eye during leftward optokinetic stimulation at 0.5~ The fight histogram plots data from the right eye during leftward stimulation at 21~ These skewed data deviated significantly from normality (K-S and R tests, p < 0.05)

deviation in the SP direction until a timing circuit triggers a FP.

3.2 Control of SP durations

3.1.2 SP durations show broad-sense stationarity across an experimental session�9 Dur ing a stimulus block, the interval between successive FPs was considered as a time series, with the first th rough n th intervals designated as T1, T2 . . . . . T,. The stat ionari ty o f this time series was examined by analyzing the SP durat ions as a funct ion o f the elapsed time of stimulus presentat ion (Fig. 2A), The intervals did no t show any consistent trend (increase or decrease) over the period o f stimula- t ion at each velocity in each animal; rather, they appeared to vary r andomly about a mean interval. Thus, the mechanism producing these intervals displayed broad-sense (or weak) stationarity (Godf rey 1980) during a period o f 26 min.

3.2.2 SP durations as a lognormal variate. Several ap- proaches indicated that the SP dura t ion data were no t normal ly distributed. His tograms o f the distr ibution o f SP durat ions were skewed for each turtle at each stimulus velocity (Fig. 2B). Both the Ko lmogorov -Smi rn o v (K-S) test and the R test (Pearson and Hart ley 1976) demonst ra ted that SP durat ions were no t distributed normally. These tests were per formed separately on data f rom each stimulus block in each turtle; thus, they were performed on a total o f 26 SP dura t ion distributions. The K-S test (two-tailed, Lilliefors p correction) rejected the hypothesis of normal i ty for 24/26 stimulus blocks (p < 0.05), while the R test rejected 25/26 blocks

206

at the p < 0.05 level. Thus, the distributions of SP durations are inconsistent with a normal distribution.

The two parameter lognormal distribution is a skewed distribution function such that the natural loga- rithm of the observed variate is normally distributed with a mean value of #' and variance a '2 (Aitchison and Brown 1957; Johnson and Kotz 1970). It has been useful in problems in economics, biology and lifetime testing because it describes the skewed distribution of data generated by some classes of iterative processes (Aitchison and Brown 1957). Lognormal probability plots of SP durations were highly linear for data from each of the 26 stimulus blocks; the correlation coefficients of these plots of the ordered variates as a function of the expected value of each ordered variate (David 1970; King 1971) ranged from 0.93-0.996. K-S tests also revealed that the data from only 5/26 stimulus blocks deviated significantly from a lognormal distribution (p < .05). Thus, the SP durations are consistent with observations from a lognormal distribution.

3.2.3 Integrate-to-fire models as neural clocks. Log- normal distributions can be generated by an iterative process that follows the law of proportional effect (Aitchison and Brown 1957). This section develops an equivalent statement of this process as a special case of an integrate-to-fire model neuron (e.g., Pavlidis 1965; Bayly 1968), which is commonly regarded as a physiologically plausible model for neuronal function. In this particular class of models, a neuron or a group of neurons is viewed as a resettable integrator which integrates a series of inputs until a threshold value is reached. The neuron then discharges or bursts, resets to zero, and repeats these operations. Let us consider one special case of this model: the responses of a "burst neuron" or circuit that integrates a constant input equal to the current instantaneous frequency of nystagrnus (i.e., the reciprocal of the previous SP duration); when a random, normally distributed threshold value a~ is reached, the model responds with a burst discharge. The interburst interval of this model is described math- ematically as:

/ + r i + l

ai+, = ~ z,-' dt (1) t

where ai is a random observation from a normal distribution and r~ represents the duration of the i th SP. The mean of ~ is the threshold which sets the clock cycle length in units the initial cycle length, %; if cr is assumed to be a random variate (r.v.) with a mean 1 and variance a 2 (N(1, a2)), then the process will generate cycles of approximately Vo duration. An equivalent formulation of this model, then, is

~ ' i + I = (~ i+ l~ ' i (2)

where ai is a N(1, a 2) r.v. and z0 and ~'i are defined as above. Equation (2) has been examined previously in statistics and economics as the description of an independent variable that follows the law of proportionate effect, which specifies that the change in a variate in an

iterative series is a random proportion of its previous value (Aitchison and Brown 1957). Since large numbers of observations generated by (2) show a lognormal distribution (review: Aitchison and Brown 1957), this iterative process has the potential to reproduce the observed distribution of SP durations.

Biological events cannot be of negative duration. Thus, (2) must be constrained such that ~; > 0; there- fore, ~i will be a r.v. with a singly truncated normal distribution (Johnson and Kotz 1970) with the left truncation point fixed at zero. The degree of truncation of the left tail of the distribution depends upon the value of a 2 of the underlying N( 1, a 2) distribution; for example, if the lower 10% of the distribution is truncated, the expected value shift to 1 + .195a and the standard deviation decreases to .844a (Johnson and Kotz 1970). Thus, the ~ in (2) would increase because the expected value of at would be greater than 1. Alternatively, if ~i is a truncated normal variate from an underlying N(p, a 2) distribution a value of # < 1 can be calculated (see Appendix, eq. Ia) such that the expected value of ai is 1. This truncated normal distribution with an expected value of I, designated N*( 1, a2), has been adopted to preserve the assumption of a mean 1 variate in (2).

3.2.4 Stability o f the internal clock. The SP durations show broad-sense stationarity, oscillating about a mean value without degenerating to either zero or infinite length (Fig. 2A). However, the zi's in equation (2) will decrease toward 0 for a~ < 1, and increase for ~ > 1. Computer simulations showed that a series of r~'s generated by (2) can either be stable, increase monotoni- cally, degenerate to zero or show quasistable behavior; these non-stationarities result arise from ai's that are extreme outliers. As a result, it is necessary to modify equation (2) to prevent the emergence of extreme values of zi. This was accomplished by adding an interval outlier detector to reset the value of ~i in the basic interval generator to the initial value (r0) if the previously generated value is too large or too small. In terms of (1), this is equivalent to switching inputs to the integrator from zF ~ to Zo ~ if z~ is not contained in the interval [(1-2a)%,(1 + 2a)ro]. Thus, the i + 1 st SP duration can be defined as:

(a) z i + l = a i + , z i i f ( 1 - 2 a ) t o < t i < ( l + 2 a ) t o , (3)

(b) z~+ i = ~i+ l z0 otherwise.

The quantitative assessment of (3) as a model for SP durations requires estimation of the distribution of ~ ' s from the data. Let the observed duration of the i th SP be designated as Ti. From (3), it is clear that the ratio of successive SP durations provides a direct estimate of the distribution of the ~/s. However, this ratio will reflect a mixture of observations from two component distributions, ~,(Eq (3), case (a)) and O~i+lZO/'Ci (Eq. (3), case (b)), where

i

~i = I ] ~sr0 (4) j=k

for k < i - 2 iterations of the generator. For large values of i-k-l, Aitchison and Brown have shown that if ~ 's are N(1, a 2) independent r.v.s, then the distribution of z~ converges to the lognormal distribution. Be- cause the distribution of a N(1, tr 2) r.v. approximates the behavior of a N*(I, a 2) for small values of a 2 (see above), the distribution of (4) is assumed to be approximately lognormal. Since the ratio of the durations of successive SPs (T~ and T~_~) does not yield a pure distribution of ~, we have designated the parameter:

,(.= T, (5), T~_t

as an estimator of the ctt from the data. For the model in (3), then, each calculated value of ~ represents a sample from an underlying mixed distribution function 2, containing 95% of its observations from a N*(1, a 2) r.v. and 5% of its observations from a distribution approximated as the ratio of a N*( 1, tr 2) distributed r.v. to a lognormally distributed r.v. The behavior of 2 ~ did not show a significant linear trend over time in any data set (linear regression analysis, p >0.05). Since ~" is weakly stationary, the data can be used to explore the beat-to-beat behavior of SP durations within the framework of (3).

3.2.5 Eye position dependence of the distribution of s The distribution of ~" for all SP durations was positively skewed at each stimulus velocity in each turtle. Thus, (3) does not describe of the behavior of the duration of SPs initiated at all eye positions. An analysis of the distribution of ~ as a function of initial eye position (e.g., Fig. 3A) confirmed that large values of ~/ appeared preferentially when the eye was deviated in the FP direction at the beginning of the SP (Fig. 1A). Thus, when the eye was deviated in the direction of the SP movements at the onset of the i th SP, )~ showed a unimodal distribution (Fig. 3B) that was consistent with a truncated normal distribution. Data from 12/26 conditions did not deviate significantly from a truncated normal distribution (K-S test, p < .05); an additional 8 did not deviate significantly with a single outlier excluded (test N14; Barnett and Lewis 1978). By contrast, when the eye was deviated in the FP direction, the distribution of ~',. was broader and positively skewed (Fig. 3B). K-S tests rejected the hypothesis that these values reflect either a normal or a truncated normal distribution for all 26 data sets (p < .05). These analyses demonstrated that the distribution of 2"~ is consistent with (3) when the eye is deviated initially in the SP direction, but that the data deviated from the predictions of this simple model when the eye was deviated initially in the FP direction.

The dependence of s upon initial eye position was reproduced in the model by incorporating an eye position gate in (3), (Fig. 4). This mechanism produces a "skip-a-beat" phenomenon: when the initial eye position for a SP is deviated in the FP direction, the interval generator (3) is permitted to run a second time without generating a FP. For generality, it is assumed that

207

8

0 / -15

SLOW PHASE DIRECTION FAST PHASE DIRECTION

++ + +

+ * § + + + § + +~ + + + ++§ § §

+ +§ . - �82247 ~

-10 -5 5 10 15 EYE POSITION AT SLOW PHASE ONSET (deg)

u)

i l 0 r , ,

/ SLOW PHASE DIRECTION

,l IIlllJk

Z 0 1 2

20

FAST pHASE DIRECTION

2 4 6

Fig. 3.A Position dependence of the proportional effects parameter, ~'. These scatter plots illustrate the relationship between normalized eye position at the end of the previous FP (median eye position = 0) and -(i. The data were obtained from the right eye of a turtle (TCJ) receiving leftward optokinetic stimulation at a velocity of 21~ A vertical line at the median eye position demarcates the border between eye deviation in the SP and FP directions. B Histograms of the distribution of ~'i values from A for SPs generated when the eye was deviated in the SP versus FP direction. The symmetric distribution of ~ for eye deviations in the SP direction is consistent with successive single beats of the basic interval generator, mixed with a proportion of beats after skipped beats. The skewed distribution of ~i for intervals generated with eye deviation in the FP direction is consistent with the inclusion of skipped beats (see Fig. 5)

SLOW PHASE OURATION CONTROL

ESTIMATE OF SLOW PHASE

Interval ~ ~ _ _ D U R A . . T I O N OF.._ ~E IE~LOC lTY

- ] x l+l = I - s . ; c

Po'~ ~ "Yes

"skip a be~t" J RECYCLE GENERATOR ] PSj WITH PROBABILITY ps 1-Ps

PSL FROM OW PHASE

' VELOCITY CONTROL

TO FAST PHASE AMPLITUDE CONTROL

Fig. 4. Proportional effects model for the SP duration system with an eye position gate. When the eye is deviated initially in the FP direction, this model allows multiple operations of the basic interval generator without triggering a FP (skipped beats)

208

the "skip-a-bear' phenomenon is not obligatory; it occurs with probability p~ on a beat-to-beat basis when the eye is deviated initially in the FP direction. A hypothetical substrate for the "skip-a-bear' phenomenon would be an initial eye position-dependent synaptic inhibition of the output of the basic interval generator to prevent FP generation.

3.3 Testing the model for the distribution of ~ : estimation of parameters from data

The model in Fig. 4 generates SP durations on the basis of three parameters: o.z (the variance of the N*(1, o.2) distribution of the ~), Zo (the basic interval duration), and p~ (the probability of a skipped beat). These model parameters must be estimated from nystagrnus data that consist of SP durations T~, T2, T 3 , . . . , T,. This section discusses the distribution of the ratio of successive SP durations (termed 2) from the model (Fig. 4) as a function of the variance of ~ and the proportion of skipped beats in a data sample, termed p (Fig. 5 and Appendix). The ratio of successive SP durations from the data (termed ~) are then used to estimate s ~ (variance o.2 of the N*(1, o.2)-distributed ~) and p; these parameter values are then used for testing the correspondence between the predicted distribution of 2 and the observed distribution of ~..

3.3.1 The distribution of ~. Figure 4 predicts that the distribution function of ,~. is a hybrid of the distribution functions of a series of different operational cases. If we restrict consideration to instances where SPs reflect a maximum of two cycles of the basic interval generator, then the distribution function of /~ can be approximated by considering four major cases (Fig. 5):

(1) Single beats. The durations of two successive SPs (7,. and T~_~) reflect single cycles of the basic interval generator. If p designates the proportion of skipped beats of nystagmus in a data set, then single beats will comprise a proportion (1 _p)2 of a sample of ~.

(2) Skipped beats. The duration of T~ represents a skipped beat of the basic interval generator and T~_ represents a single generator cycle in a proportion p( 1 --p) of a )~. sample.

(3) Beats after a skipped beat. The duration of T~ represents a single cycle of the basic interval generator and T,_ ~ represents a skipped beat; this will be observed in a proportion p(1 - p ) of a sample of ~,..

(4) Two skipped beats. The durations of two successive SPs (7,. and T~_~) reflect skipped beats of the basic interval generator; this will occur in a proportion p: of a sample of ,~.

If L~, L2, L 3 and Z 4 a r e r.v.s that are distributed a s 2 i from cases (1)-(4) above, respectively, and N is the number of values of )~i in a period of nystagmus, then, the pooled 2 values reflect ( 1 -p )2 N observations from L1, p ( 1 - p ) N observations each from L2 and L 3 , and p ZN observations from L4. The derivation of each distribution component is summarized in the Appendix.

SINGLE B E A T ' ~

el-1 is an outlier? NO el.1 e. ~el.~

Yes e,.~ Oheo

~=TI/Ti.I i l

. O . E , P.OeESS ~ I {Intl11111 vlrllblel)

,~ ~; ~': a " ~ N'(x.o') ! I (x" ~ Eq.(I,c) ~ I

SKIPPED B E A T - " ]

T1-1 Ti

t,e, d a~ e~. d e, 4 is an outiied

No e,.~ ((x,+~.~ , Yes e,., (m+ ~ 7c ~ J

BEAT AFTER SKIPPED BEAT]. u

Ti -2 Ti -1 ~ T i _ -

_ _ .

I

a. e~la;,e, ! el .I is an outlier? NO r (a,.,+ a,.,) e. ~ a'.r

Yes e,~ (a,.,+ a~.,) e,, ~x, %

SECOND SKIPPED BEAT

]'i -2 Ti -1

r is an out/er No e,-2 ( a , + ~ , ) e~

Yes eJ.2 (au~+ a~) e~

a+~"

Ti

(~l+al)~Hr

~ a,~a*+ ~o

I d (ct** + a**~+a, "

Fig. 5, Diagram illustrating the possible behavior of the proportional effects parameter, 2~ = TiTs_ 1, for SPs of nystagmus generated by the model in Fig. 4. The data parameters T~ (duration of the i m SP) and ,~ are shown above a schematic nystagmus record. The model parameters that are reflected in these data parameters are shown below the schematic record: ct, et*, ~** and ~*** are N*( l, a 2) r.v.s, ct" is a r.v. with the density function in (Ic), z~ is the duration of the i th cycle of the interval generator and r o is the initial duration o f the basic interval generator

3.3.2 The basic interval generator: estimation of o.2. The value of o.2 can be estimated directly from the calculated variable ,~ (5). Since the model predicts that skipped beats do not occur when the eye is deviated initially in the SP direction, the distribution of ,~ reflects primarily two consecutive single beats, with a contribution from an unknown number of beats after skipped beats. A graphical analysis strategy was adopted for calculating s 2, an estimator of o -2. First, the ~,,. for movements initiated with the eye deviated in the SP direction were examined in normal probability plots of

7

6

5

_ §

i" Skipped

Beats

§ +

§ # § Two skipped beats

3 v g + + . r §

e z § § § +

. §

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 ~ - ~ ,,~ + * Beats after skipped beats

0 ~ , * *. , -§ . . . . . 0 1 2 3 4 5 6 7 8

F i g . 6. Representative scatter plot of ~'~ versus 3.~_ ] . Data were obtained from the right eye of a turtle (TCK) receiving leftward optokinetic stimulation at a velocity of 21~ The regions containing data from each model case in Fig. 5 are indicated. A circle of radius 2a demarcates the region occupied primarily by values representing two consecutive single beats of the interval generator. Vertical and horizontal lines demarcate additional regions occupied by values of ~'~ reflecting skipped beats after single beats, two successive skipped beats and single beats after skipped beats of the interval generator

each stimulus block (David 1970; King 1971). The right and left tails were trimmed to produce a single linear plot, which represents the central region of the data that is consistent with a normal distribution. Regression analysis of the normal probability plots and the standard deviation under the assumption of both normal and truncated normal distributions (Cohen 1961) were all used to calculate s 2 from the trimmed data. The calculated values of s 2 ranged from 0.046-0.318 and did not vary significantly across turtles or stimltlus velocities (Scheff6 test for homogeneity of variance, p > .05, Winer 1972).

The estimation, of 0 .2 from the data set permits visualization of the four model component distributions in scatter plots of successive values of :7 (Fig. 6). A circle of radius 2s has been drawn to indicate a region occupied by 95% of the values produced by consecutive single beats the basic interval generator in Fig. 6. Vertical and horizontal lines have been drawn at ~.- 1 = 1 + 2s and at the ~'i value representing the .97725 quantile of the distribution of beats after skipped beats, respectively, to demarcate regions occupied primarily

209

Fig. 6, the model predicts that the region labeled skipped beats will contain only skipped beats with ~. values greater than 1 + 2s; approximately 28% of the skipped beat cases will lie in the region containing consecutive single beats.

3.3.3 Estimation of the proportion of skipped beats (p) and the probability of skipping a beat (Ps). Since there is a substantial overlap in the distribution of )~ for consecutive single beats and for single skipped beats, it is impossible to objectively identify each skipped beat in eye movement records. However, it is possible to derive a solution for p as a function of s 2 and the expected value of 2, E(2). If the mean ~'; is used as an estimator of E(2), s z is used to estimate a 2 and L], L2, L3 and L4 are r.v.s distributed according to the values of 2 for single beats, skipped beats, beats after skipped beats and two skipped beats, respectively, then p can be calculated by the quadratic formula as the solution of:

E(2) ~ ( 1 - p)2E(L1) + p( 1 - p)(E(L2)

+ E(L3) ) + p2E(L4). (6)

The use of (6) hinges upon the accurate estimation of both E(2) and a 2 from each data set. The expected value of a sample from a skewed distribution can be estimated robustly by a winsorized mean or a trimmed mean (David 1970). We adopted 5%, 10% and 15% winsorized means (David 1970) of )7as a robust estimator of E(2); this procedure replaces the lower and upper 5%, 10% or 15% of the data points with the lowest and highest remaining values respectively, prior to calculating a mean. Equation (6) was then used to estimate a range of p values consistent with the data.

The most direct approach for estimating Ps is to apply (6) to ~'i from only SPs that were generated when the eye was deviated initially in the FP direction (designated ~'~). Since skipped beats are generated only when the eye is deviated initially in the FP direction (Fig. 4), it is predicted that all occurrences of (1) skipped beats after single beats and (2) consecutive skipped beats will be included in ~'~; their relative proportions are expected to be p s ( 1 - p ~ ) and p~, respectively. However, single beats and beats after skipped beats can be generated at any eye position. If we assume that consecutive single beats and beats after skipped beats are distributed uniformly across eye positions, then their relative contributions to the ~fi's will reflect the proportion of SPs represented in the ~ ' s , designated q. The quadratic equation then estimates Ps as a solution of:

( 1 - p,)2E(L,) + (p, -- p~)E(L2) + q(ps -- p~)E(L3) + pZ, E(L4) E( ,~:)

1 + (1 - - q)(ps --p~) (7)

by each component distribution. It is important to remember, though, that the distributions of these cases are partially overlapping. For the data set illustrated in

where E(2fi) denotes the expected value of the 2~'s. For each data set, a winsorized mean (David 1970) with a lower bound of 5% and an upper bound of 5-35% was

210

used to obtain a range of real solutions to equation (7). These values were regarded as boundaries for physiologically plausible values ofp~ for computer simulations of the model.

3.3.4 Model fitting for ,(,�9 The Appendix and Fig. 5 provide an expression for the distribution of 2 as a function of a 2 and p. The estimates of s 2 and p from the data were used as boundary conditions for permissible values of tr 2 and p; the parameter values were then adjusted within these bounds to optimize the goodness- of-fit (K-S test) of the model to the observed ;~ values for each turtle at each stimulus velocity (26 different data sets). Nineteen of these predicted distributions did not differ significantly f rom the data (K-S test, p > .05); four additional data sets did not differ significantly when 1 extreme outlier was excluded from the data. Thus, (6) and the procedures for estimating two parameters (with simplifying assumptions for the resetting of the interval generator) are adequate to describe the distribution of the ~ for nystagmus SP durations.

3.3.5 Estimation and stimulus dependence of %. Two factors preclude the estimation of z0 by iterative application of (2), for all i, as :

T i

C O - - i

l-Iaj j=l

First, during any given period of nystagmus the outlier detection mechanism can reset the value of zj(i < j ) to

c0 an unknown number of times, which will depend upon the FP amplitude, SP eye velocity and the values of the previous ~tj. Second, the values of each ~i of the basic interval generator cannot be determined directly from each ~ because there is considerable overlap in the distributions of the latter parameter for single beats, skipped beats, beats after skipped beats, and two successive skipped beats. Thus, c0 was estimated by two methods which gave comparable values: (1) the winsorized mean of the SP durations (no lower limit, upper limit: 1-p th quantile) and (2) the median SP duration. Log-log plots of the z0-stimulus velocity relationship departed from linearity when SP eye velocity saturated at stimulus velocities above 10~ By contrast, the log c 0 - log SP eye velocity relationship was linear throughout its range (r = - . 8 9 , p < .05); it is equivalent to a reciprocal square root relation (Fig. 7A), where Zo = 10/)-112, where/) represents the mean SP eye velocity. Thus, Co is apparently scaled with an internal motor command that determines the SP velocity.

3.3.6 Minimum SP durations: refractor&ess of FP generation. The minimum observed slow phase durations varied across stimulus conditions as a linear function of % (Fig. 7B). Regression analysis indicated that the SP durations were described by 0.2z0- .104 ( r = . 7 5 , p < .05), which does not differ significantly from 0.2%. This refractoriness was incorporated in the model (Fig. 8) by assuming that a FP could not be generated within 0.2% s of the previous FP; thus, the interval generator would produce another cycle.

A

3~ i 25

20

lO

B

r r

== a . ( n :S :S

+ + §

2 § +

�9 ++ + § + § ~ . ~ +

; 0 ~ lo 20 SP EYE VELOCITY (deg/s) ~o (s)

C

,,, 0

_. -5 Q .

X -10 .++ ,< I L l

-15 - r

~ -20

,,~ -25

TCJ 0.75 deg/s STIMULUS

�9 - § +++ § §

~ t+ + + + + + § § §

,k

+ § §

INITIAL EYE POSITION (deg)

5

,,, 0

-10

-15

mp -20

~ -25

TCJ 21 deg/s STIMULUS

+ § § § + 4

, , § 2 4 7 �9 § §

�9 ~ . + § + § §

'~ + 4 §

§

; ;o INITIAL EYE POSITION (deg)

Fig. 7.A Relationship between z o and mean SP eye velocity for three turtles. Each data point represents one animal tested at one stimulus velocity, z o was proportional to the reciprocal of the square root of SP eye velocity; the curve lO/x/v is also plotted (r = .87). B Relationship between z o and the minimum SP duration across data sets. Each point represents a different stimulus condition across turtles. C Scatter plots of the amplitude of FPs as a function of eye position. By convention, a negative position or amplitude represents the FP direction. The amplitude FPs was a linear function of eye position. The lines generated by (9) are also plotted

SLOW PHASE DURATION CONTROL

EYE VELOCITY

Interval �9 DURATION OF PREVIOUS INTERVAL ~ . , N o ~('r f |

J BaSic Interval

Genera~r

TIONATEE~N~ING

~No RECYCLE GENERATOR I WYI1H PROBABILITY p

h I ' I l - p ,

i~ r I CURRENT EYE VELOCITY

FAST PHASE AMPLITUDE CONTROL

SLOW PHASE VELOCITY CONTROL

EYE,g~ g,~I ~"~l

m 9

~ Extraocu,a~" Muse]es I

211

Fig. 8. Complete model for generating constant velocity OKN. This model contains three components, a SP duration control system, a FP amplitude control system and a SP velocity control system. The basic interval generator is engaged by a central signal representing the SP eye velocity, which determines the initial value for %. The z~ designates the duration of the i th iteration of the basic interval generator, p.~ represents the probability of "skipping a beat", ~ is a N*(1, a e) r.v. and E denotes eye position. The fast phase amplitude is determined by (9)

3.4 Control o f FP amplitudes

The control of FP amplitudes can be analyzed from two different perspectives, which make different predictions about the relationship between eye position and FP amplitude. The first approach, proposed by Chun and Robinson (1978) for nystagmus elicited by sinusoidal rotation, assumes that FP is generated to end at an internal target that can be represented as a random observation f r o m a continuous statistical distribution, whose parameters are independent of the eye position at the end of the SPs. The application of this approach to constant velocity O K N predicts an apparent eye position dependence for the amplitude of the i th FP (At):

A~ = K - - X~ + 5 (8),

where 5 is a zero mean r.v., K designates the FP target and X~ represents the eye position at the beginning of t h e i th FP; note that the slope is - 1 . The second approach to the control of FPs assumes that its amplitude is computed as a function of the eye position at the end of the previous SP. Thus, it predicts that the slope of (8) differs f rom - 1 .

The negative linear relationship between Ai and X~ is illustrated in Fig. 8. The relationships were significant for all 26 experimental data sets (p < 0.05, r = -0 . 245 to - 0.81). The slopes were all significantly greater than - 1 and less than zero (t-tests, p < 0.05), ranging from - 0 . 2 5 t o - 0 . 6 0 with a mean (and median) of -0 .40 ; they did not differ among turtles or stimulus conditions (least significant differences test, p > 0.05). This finding implies that the FP targets are calculated on the basis of the eye position at the end of the SP. Further analyses indicated that the y-intercept values of the Ai versus Xi relation were a function of Zo and v. The y-intercepts were significantly correlated with the amplitude of a SP of duration z 0 (i.e., ZoV); they could be expressed equiv-

alently as 0.85 ZoV (linear regression, r = 0.93). Thus, for computer simulations, (8) was expressed as:

Ai = -0.4)(,. - 0.85 T0 v + 5, (9)

where the positive direction of eye position is defined arbitrarily as the SP direction and 5 represents an error term. Since (a) the residuals from the regression analyses of the FP amplitude-eye position relationship were uncorrelated with either initial eye position or cumulative time of stimulation and (b) the estimated variances SyZx (the mean squared deviation about the regression line, Huntsberger and Billingsley (1977)) were homogeneous across data sets, the error (5) in (9) is apparently invariant across turtles and stimulus conditions. Finally, the apparent refractoriness of FP generation can be incorporated by setting A o = 0 for 0.2z0s after a FP.

The distribution of 5 has implications for potential mechanisms FP generation. Although least squares regression produces residuals with a mean value of zero, the residuals deviated f rom normality for 18/26 data sets (K-S test, p < 0.05). Histograms of the residuals were significantly skewed, resembling either a log-gamma or an extreme value distribution. A simple, biologically plausible method of generation of skewed residuals is to assume that they arise f rom mechanisms which select the largest of n parallel calculations, each possessing normal or exponential error (Gumbel 1958). Several alternative models were tested; the simplest adequate description was provided by a zero mean r.v. distributed as the maximum of two independent N(0, Sy2x) observations.

3.5 Computer simulation o f SP durations and eye position

3.5.1 Simulation parameters. Figure 9 shows the final form of a model that simulated constant velocity OKN.

212

A 2O

15

10

~ o

i., ul ~ - 1 0

-15

-2O

COMPUTER SIMULATION OF NYSTAGMUS

-25 10 15' 20

TIME (s)

C DATA FROM SIMULATED NYSTAGMUS

12 s low PHASE . FAST PHASE ~RECT~OH DIRECTION

10

8

< ~ 6

+ §

§

. . . : * , :

lo 2o ~o ,;o EYE POSITION ((:leg)

Fig. 9A-D. Computer stimulation of the model for the experimental condition used in Fig. 6. The parameters of the model were p~ = 0.18, r o = 3.0 s, a 2 = 0.15 and v = 7.5~ FP amplitudes were determined as the maximum of two parallel calculations from (9) with e denoting a N(0,25) r.v. Further details of the simulation are discussed in the text.

B STATIONARITY OF SIMULATED NYSTAGMUS 14

12

A

0 25 0 5 10 15 20 25

TIME (mini

D DISTRIBUTION OF SLOW PHASE DURATIONS: SIMULATION VERSUS DATA 1, . . . ,

0.9

0.8

1

~ I 0.1

O 5 10 1'5 2'0 2'5 30

SLOW PHASE DURATION (s)

A - C illustrates the simulated data in the format of Figs. IA, 2A and 3A respectively. D compares the cumulative distribution function of simulated SP durations (solid curve) with cumulative distribution of the actual SP durations

The SP duration generator is characterized by three parameters, to, a2(s 2) and p~, which were estimated directly from the data. The FP amplitude generator was (9); e was calculated as the maximum of two random observations from a zero mean normal distribution with a variance sy.x2 = 25 (estimated from the data), minus 0.56 to generate a zero mean variate (see Gum- bel 1958). The FP amplitude generator also had a refractory period of 0.2~ 0, during which its output was 0. During each SP, the eye velocity was fixed at the observed mean value for the appropriate experimental session plus a random Gaussian error of 15% (the average coefficient of variation in the data was 0.15). Finally, the SP eye velocity decelerated with an experi- mentally estimated time constant of t v ~ Zo when the eye position exceeded a threshold of 8 ~ in the SP direction.

The model makes two simplifying assumptions about the behavior of SP velocity. First, the internal SP eye velocity signal was assumed to be stationary; in fact, the SP velocity was weakly stationary in 23/26 data sets. Second, variations in SP velocity were assumed to be uncorrelated for successive nystagmus beats. Violations of this simplifying assumption are a potential source of error in the model because there was

a small but significant positive correlation between the mean eye velocity of successive SPs in 19/26 data sets. However, they were not incorporated to maintain sim- plicity of the model.

3.5.2 Simulation of SP durations. The computer simulations of the model provided an adequate characterization of the SP durations across turtles and stimulus velocities. Figure 9 shows the results from a simulated SP duration sample with the same number of beats of nystagmus from experimental data during 21~ optokinetic stimulation. The nystagmus generated by the model grossly resembles the data (compare Figs. 9A and 1A), showing the same weak stationarity as the data (compare Figs. 9B and 2A). Like the observed data, the calculated values of 2i from the stimulated data showed a clear eye position dependence (Fig. 9C). Finally, the distribution of the SP durations from the computer simulation was virtually identical to that of the experimental data. Figure 9D shows an example the close correspondence between the stimulated and experimental cumulative distribution functions of slow phase durations. The simulation failed to fit the distribution of SP durations for only 2/26 data sets (K-S test p < .05). In one condition that failed to fit, the model

assumption of stationary SP eye velocity was violated by the data.

3.5.3 Model simulations and eye position control. The overall ability of the model to predict eye position at the end of fast and SPs was assessed by comparing the results of one run of the simulation and the data from each stimulus condition in each turtle. Data from one stimulus block where the contact lens slipped and three stimulus blocks with nonstationary SP velocity were excluded. Average eye positions were reproduced accu- rately; simulations of only 8/22 data sets displayed average eye positions at the end of FPs or SPs that deviated significantly from observed values (p < 0.05, Wilcoxon two-sample test). There was no consistent tendency to either overestimate or underestimate mean eye position, since the average regression constants from quantile-quantile (Q-Q) plots (King 1971) of the observed versus stimulated eye positions at the end of FPs (0.25~ (S.D.)) and SPs ( -0 .42~ (S.D.)) did not differ singificantly from zero (two-tailed t-test p > 0.05). The common shapes of the stimulated and observed eye position distributions were also supported by two findings. First, only 7/22 or 10/22 simulation runs produced eye position distributions that differed significantly from the observed data (p < 0.05, chi-square tests) at the end of either SPs or FPs, respectively. Second, the correlation coefficients of Q- Q plots of eye position at the end of either FPs or SPs were high (0.92-0.998). The average slopes of these Q-Q plots of simulated as a function of observed eye position were 0.83 + 0.23 (S.D.) for the end of FPs and 0.80 + 0.25 (S.D.) for the end of SPs, indicating that the model underestimated eye position variability by about 20%.

Equation (9) permits the generation of both typical and misdirected FPs. Neglecting the contribution of e, a misdirected FP can be generated when the eye is deviated more than 2.125 deg in the FP direction. However, since the standard deviation of e(Syx) is approximately 5 ~ a misdirected FP can even be generated when the eye is deviated to a small degree in the SP direction. The simulation was a reasonably accurate predictor of the proportion of misdirected FPs for the 23 data sets with a stationary SP velocity. A regression analysis of the proportion from one run of the simulation as a function of the observed proportion of these FPs at each velocity had a slope of 1.34, which did not differ significantly from 1 (p > .05, 2 tailed t-test), and a constant coefficient that did not differ significantly from zero (p > .05, t-test). This finding supports the hypothesis that misdirected FPs are a normal output of OKN circuitry.

4 Discussion

The fundamental concept underlying this analysis is that "beats" constitute the basic temporal units of constant velocity OKN. Two types of analyses are consistent with this premise. First, the eye position

213

recordings (Fig. 1) showed that the SP eye velocity was constant for SPs near the median SP duration, but that it decelerated (and even stopped) during longer durations SPs. This deceleration suggested a mechanism to prevent excessive deviation of the eye while waiting for an internal timer to trigger a FP. Second, our analyses of nystagmus have demonstrated that the timing of the beats of nystagrnus is consistent with the operations of an internal clock, which determines its cycle duration as a proportion of its previous cycle duration. Thus, the behavior of a proportional effects clock, stabilized by an outlier detector, is sufficient to account for the durations of OKN SPs.

One important implication of these findings is that the restoration of eye position in the SP and FP direction is controlled by two distinct, parallel mechanisms. Eye movements in the SP direction occur in multiples of beats dictated by the basic interval generator; thus, at constant velocity, they are constrained to travel in multiples of the distance traversed during a single cycle. It is important to recognize the profound limitations that this clock imposes on the use of SPs to control eye position: the eye can only be recentered by a long duration SP that reflects the "skip-a-beat" phenomenon. By contrast, FP amplitudes are calculated as a linear function of the eye position at the end of the SP and the average SP amplitude. The computer simulation demonstrates that the net effect of these two mechanisms is a regulation of eye position within a symmetric region about an arbitrarily defined center of gaze (zero eye position).

The hypothetical "skip-a-beat" phenomenon has important teleologic implications. Since a single beat of the interval generator is the 'quantal' timing unit for generation of SPs, eye position can only be restored in the SP direction by allowing the SP to continue for a second timing unit. The effect of "skipping-a-beat" would be to double the SP duration. As a result, position control in the SP direction is programmed in units of approximately the product of the SP eye velocity and the average output of the basic interval generator.

Since our model limits the duration of a SP to cycles of an internal clock, a mechanism must be avail- able to prevent excessive deviations of the eye during long duration SPs. The observed exponential deceleration of the eye during long duration SPs provides this type of mechanism; it can regulate the maximum amplitude of nystagmus by limiting the maximum SP excursion. Since the exponential decay of the SP eye velocity (with time constant equal to %) begins when the eye exceeds a fixed position threshold, the maximum amplitude of a SP will be determined as the sum of (I) the position threshold and (2) the product of z0 and the SP eye velocity. Thus, the approximately inverse square root relationship between Zo and the SP eye velocity implies that the maximum eye deviation in the SP direction will increase with SP eye velocity, scaling approximately with the square root of the SP eye velocity for turtles exposed to these experimental conditions.

2 1 4

The control of eye position during OKN in turtles is not precise on a beat-to-beat basis; rather, the SPs and FPs interactively maintain eye position within a given window during OKN. It is obvious that the basic interval generator precludes precise control of eye position by SPs. Less obvious is the lack of precision in targeting the FPs: the unbiased standard deviation of the error about the regression relationship for the FP targets was 5 deg. The relative imprecision of beat-to- beat control of eye position is also evident in the process of restoration of eye position to a region near mid-gaze after large deviations in the slow or FP directions. For example, if error is neglected, (9) implies that eye position is not restored in the SP direction by a single FP for eye position deviations greater than 1.42 ZoV deg in the SP direction; rather, it is accomplished iteratively by a sequence of FPs and SPs representing single beats of the interval generator. Similarly, if there is a large deviation in eye position in the FP direction, it is recentered by a sequence of SPs representing skipped beats and single beats, misdirected FPs and typical FPs. In particular, (9) indicates that misdirected FPs are normally elicited when the eye is deviated greater than 2.125 %v deg in the FP direction at the conclusion of a SP, which is greater than the amplitude of an average long duration SP reflecting a skipped beat of the interval generator. This implies that misdirected FPs are a mechanism specifically for restoring eye position after large deviations in the FP direction, particu- larly if a SP does not produce an eye position within a 'skipped beat' distance of the intended center of gaze.

4.1 Possible neural bases for the nystagmus timing model

The simplest, physiologically plausible realization of this clocking process is an integrate-to-fire model (Pavlidis 1965; Bayly 1968), such that a pacemaker depolarization is proportional to the instantaneous frequency of previous beat of nystagmus. The signal to trigger the next FP is then generated when a normally distributed threshold value is attained. The model suggests four basic criteria for neurons contributing to a cellular substrate for a basic interval generator for SP durations. First, the cells should receive inputs that encode the previous SP duration. Second, their pattern of activity is initialized at each nystagrnus FP. Third, these cells should change their firing rates prior to the onset of FP motor discharges to the extraocular mus- cles. Fourth, and most importantly, the firing patterns should also change near the middle of very long duration SPs which originate in the FP direction; this change should be indistinguishable from the change that occurs at the onset of a FP. In other words, a burst or pause which normally occurs at a FP eye movement should also occur at a point which corresponds to a skipped beat. These criteria provide a basis for testing burst and pause neurons in the pontine reticular forma- tion, vestibular nuclei and perihypoglossal nuclei (Leigh and Zee 1983; Baloh and Honrubia 1990) for spike discharge behavior that is consistent with components of the model.

Appendix: Approximation of the distribution of 2

1.1 Calculation of parameters of a left truncated normal distribution with expected value of 1N*( 1, a 2)

The estimation of the parameters of truncated normal distributions and their relationships to the sample mean and variance have been discussed by Cohen (1961). If the expected value of a truncated normal variate is fixed at 1, then the mean p of a singly (left) truncated N(/~, 0-2), heretofore designated as N*( 1, 0-2), is a function of 0- (Johnson and Kotz 1971):

l _

where f (x ) and F(x) represent the probability density function and the cumulative distribution function of the standard normal distribution, respectively, and A is the lower truncation point. For our case (A = 0 and a 2 ~< 0.25), the mean of the distribution of e~ shifts at most 0.076 from 1, indicating that it approximates the behavior of a N(1, 0-2) distributed variate.

1.2 Estimation of distribution of 2

This section considers the distribution function of 2 as a hybrid of four cases of operation of the interval generator summarized in Fig. 5: two consecutive single beats, a skipped beat after a single beat, a single beat after a skipped beat and two consecutive skipped beats.

Case 1: The distribution of 2 for consecutive single beats. For this case, the behavior corresponds to (3). If the previous value of ~i-J does not exceed the range [To = ( 1 -20-)r0, To = (1 +20-)To], then 2i is distributed as ~i, a N*(1, 0-2) r.v. However, if zi-1 exceeds [To, T+], then 2i is distributed as

i - - 1 (Ib)

1-Iaj j - k

for k < i - 2 and aj . . . . . a; are N*(1, a 2) distributed r.v.s. Since a N*( 1, 0 -2) distributed r.v. can be approximated by a N(1, 0-2) r.v. for small values of a 2, the distribution of the denominator of (Ib) can be approximated by a lognormal distribution for large numbers of iterations (theorem 3.1 from Aitchison and Brown, 1957). Thus, the denominator was approximated by the distribution of observations from a lognormal r.v. that exceed the interval [ 1 -~20-, 1 + 2a], whose probability density is given by:

(0 .05) -~ ' (y) , i f y < 1 -20-

~"(y) = 0, if 1 - 20- < y < 1 + 20- (Ic)

(0 .05)-~ ' (y) , i f y > 1 + 20-,

where ~'(y) is the density function of a lognormally distributed r.v. The parameters of ~' can be estimated by using the property from Theorem 2.9 of Aitchison

and Brown (1957) that the dispersion parameter ( 'variance') of a' is given by n Var(log ~i), where n represents the number of cycles of the basic interval generator since the previous resetting to Zo ( i - k - 1 in (Ib)). This yields an expression for the dispersion parameter in terms of a 2 and n. The average value of n = 3 was estimated by calculating (a) the dispersion parameter of 2 under the assumption that the data are distributed lognormally (i.e., Var(log 2)) for each data set and (b) the predicted value of the dispersion parameter f rom the estimated a 2 (s2); the ratio of these values estimates n. Thus, ct' was approximated as a lognormally distributed r.v. with location parameter 0 and dispersion parameter 3 V a r ( ~ ) (designated A(0,3Var(c0) ) for each data set. The relative proport ion of 2~ that represent the case in (Ia) was estimated by Monte Carlo simulation of (2), the basic proport ional effects process. The pooled proport ion of outliers for 1 - 5 cycles of the model was approximately 0.05; this value was used for testing model predictions.

Case 2: A skipped beat after a single beat. The distribution function of 2~ for skipped beats that follow single operations of the basic interval generator can be re- duced to an examination of 2 conditions: (I) the condition when the previous SP duration (T~_ ~) is contained in the interval [To, Tff] and (2) the condition when T~_~ exceeds that interval. In the first condition, T i = ( a * - - ~ o ~ i ) ' C i _ l , with ct~ and ~* as independent nonzero r.v.s with a N*( 1, a 2) distribution; thus, 2 will be the sum of two r.v. with N*(1,0. 2) distributions. In the second condition, the SP duration T~ = (a* + ~i)Zo, while the previous SP duration

T,-_ ~ = ClzI ' c9) % (Id) X4=k

for j < i -- 1. Hence, following the approximation in the single beat case, 2 is a r.v. distributed as the ratio of the sum of two N*(1,0.2)-distributed r.v.s to cr

Case 3: A single beat after a skipped beat. The same two conditions must be considered in examination of the distribution of 2~ for a SP duration representing a single beat of the generator that follows a skipped beat. When the previous SP duration (T~_,) is contained in the interval [Td-, To], Ti = ~ ~_l*z i -2 ; thus, 2 is distributed as the ratio of the product of two N*(1, 0.2)_ distributed r.v.s, ~ ~*, to a r.v. distributed as the sum of two N*(1, a2)-distributed r.v.s, ~**. By contrast, when T i is exceeds [Td-, Tff], T i = (XiZ 0 and

Zi_l~-(i[-I 2 ~j)(ai_, + ct*_,)% (Ie) k/=k

for j _< i -- 2. I f we approximate the distribution of Ti_ as the product o f Zo and a A(0,3 Var(~t))-distributed r.v., the distribution of 2 f rom this condition is approximated by the ratio of a N*( 1, 0 -2) distributed r.v. to ~".

Case 4: Two consecutive skipped beats. When T~_ 1 is confined to the interval [Tff, Td-], 2 is distributed as the

215

ratio between two independent r.v.s that are each distributed as the sum of two N*(1, 0.2) r.v.s. I f Ti_ ~ is an outlier, though, Ti = (~i + ~*)~;_~*% and Ti_~ is given by (Id), where ~j's and ~ _ 1" are N*(1, 0.2)-distributed r.v.s. Employing the same approximation as in the previous case, the approximate distribution of 2 is the ratio between (1) the product of (a) a r.v. distributed as the sum of two N*(1, 0 "2) r . v . s and a N*(1, 0 -2) r.v. and (2) a A(0,3 Var(ct))-distributed r.v.

Computer simulation of the distribution of 2: a mixture of cases I-4. The predicted distributions of 2 were generated by Monte Carlo methods, using 10,000 random observations for each component distribution function. The random observations from truncated normal and lognormal distributions were obtained by an inverse cumulative distribution function mapping of random observations f rom a uniform distribution. Independent samples were generated for each component (Fig. 5) for each data. For example, the distribution of consecutive single beats was calculated as a mixed distribution of a proport ion 0.95 of observations f rom the quotient of two independent samples f rom N*(1, s 2) r.v.s and a proport ion of 0.05 of observations f rom an independent sample of a N*(1, s 2) distributed r.v. divided by an independent sample of observations f rom a A(0,3.s 2) r.v. that are also outliers f rom the N*(1, s 2) variate (i.e., exceeded the interval 1 +_ 2s). The predicted distribution of 2 was estimated as follows. Let p designate the proport ion of skipped beats in a session of N beats of OKN, and let L~, L2, L3 and L 4 be r.v.s distributed as 2 i f rom cases ( 1 ) - ( 4 ) above, respectively. I f we consider only cases where SP durations represent a maximum of one skipped beat of the basic interval generator, then the distribution of 2i was approximated by (1 - p ) 2 N observations f rom Ll, p( 1 - -p ) N observations each from L2 and L3, and p2N observations from L 4 . Thus, only two estimated parameters, s 2 and p, were needed to test the predicted distribution of 2.

Acknowledgements. We thank Mr. Keith Newell for data collection and Drs. Joseph M. R. Furman, Robert H. Schor, Charles Scudder and Mark Redfern for their constructive suggestions and stimulating discussion during preparation of this manuscript. This work was supported by NS 00891 (CB), EY05978 (MA), K02 MH00815 (MA), and BRSG 2S07RR0784-21 (University of Pittsburgh).

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Dr. Carey D. Balaban Department of Otolaryngology Eye & Ear Institute University of Pittsburgh 203 Lothrop Street Pittsburgh, PA 15213 USA

a “beat-to-beat” interval generator for optokinetic nystagmus

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