Chapter 12: The Pupil as a Paradigm Example of a Neurological Control System: Mathematical Approaches in Biology, by Lawrence Stark, M.D.

PART TWO SPECIAL FIELDS IN PUPILLARY PHYSIOLOGY CHAPTER 12 The Pupil as a Paradigm Example of a Neurological Control System: Mathematical Approaches in Biology by Lawrence Stark, M.D. CONTENTS 630 632 634 636 E. Experimental Evidence for Heuristic Nonlinear Models . . . . . . . . . . . . . . . . . .. 638 F. Chronology of Readings . . . . . . . . . . . . . . . 645 Mathematical models of the pupillary system were introduced in connection with a control or transfer function analy is of the pupil reflex to light (Stark and Sherman, 1957; Stark, 1959). Indeed, the pupil was the paradigm for the application of control theory to a biological system. A. B. C. D. The Pupil as a Control System for Light ... High-Gain Oscillations . . . . . . . . . . . . . . . . Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kernel Identification of Nonlinearities ..... A. The Pupil as a Control System for Light The classical analysis of the pupil as a control system indicates the difficulties in embedding a biological system in engineering terms. A feedback control system model is developed to encompass the light-regulator features of the pupil (Figure 12-la). Linearization of this system enables a dimensionless open loop gain to be calculated (Figure 12-lb): G(s) = AA IAvf M AAv = M!AAvlM/IAv This formulation is important since without a dimensionless gain, stability of the pupil system can be misinterpreted (Stegemann, 1957). Opening the loop can be achieved with a Maxwellian view (Figure 12-2), by electronic methods (Stark, 1962), or by drugs. Then classical sinusoidal experiments (Figures 12-3 and 12-4) yield the open loop transfer function: G(s) = 0.16e-0·2,/(1 + 0.ls) 3 This simple equation comprising a gain, a time delay, and a third-order lag approximates the Bode diagram data (Figure 12-4; confirmed by Sun et al., 1979). It accounts for the neuromuscular plant dynamics that are frequency-limiting but omits subtler features such as low-frequency retinal adaptation and various noise processes. Also nonlinearities present in great variety in the pupil reflex to light are eliminated in this linearized approach. Lawrence W. Stark, M.D., is Professor of Physiological Optics and of Engineering Science at the University of California, Berkeley, and of Neurology (Neuro-Ophthalmology) at the University of California, San Francisco. 630 A SIMPLE SERVO SYSTEM ( OPEN LOOP CONDITIONS \ \ RESPONSE ·>------ CONTROL \ DISTURBANCE) { CONTROLLED LIGHT SYSTEM FEEDBACK PATH FOR Le Figure 12-1a. A simple servo system. Note forward and feedback paths in the servo loop and different components therein. The error-actuated system is driven by the difference (Le - LREF) between the desired or reference light level (LREF) and the actual or controlled light level (Le)- Dashed lines indicate where the loop might be opened, a disturbance might be introduced, and the response around the loop might be measured. A LINEARIZED APPROXIMATION TO THE PUP1LLARY SERVO SYSTEM OPEN LOOP DIST~ANCE RESPONSE I I Figure 12-lb. Linearized approximation to pupil servo system. The actual pupil system is more complicated than this figure indicates. However, we linearized by using small variations about a fixed operating point. Thus, the necessary calculations are simplified and linear servoanalytic methods may be applied:ARff is reference area;A e is controlled area;A is an output change in area generated by control system whose transfer function is H(s) and whose input is error (L ); ! Av is the average intensity value used to multiply controlled area to yield controlled light flux. 12. The Pupil as a Neurological Control System: Mathematical Approaches / 631 Figure 12-2. Two techniques used for stimulation. Left, the normal closed-loop condition in which moveme nt of iris changes intensity at retina; Right, the open-loop or Maxwellian view technique used for obtaining amplitude phase frequency data. Light ente ring the center of the pupil is unaffected by movement of the iris. OPEN LOOP 0.6 OPEN-LOOP FREQUENCY RESPONSE Cps C-A.11-J = 0.16 AMPLITUDE IGAINI LIGHT FLUX O,l-+------+----1------+----"-'--",~-.1------, (millilumens) 0.4- o- 0.025-t----+-------f----+----+-----\----I PHASE SHIFT PUPIL AREA Sq.mm DEG-. O-t-- - - - - + - - - - + - - - - - - + - - - - - 1 - - - - - - - i 10.0- 90 15.0- 180-t--~::-c:---t----t-----<lb---"- ----l-----l 270--r--- - -r - - - - - t- - -- + - ---+-----4 2 5 . 0- i60-t-- --t----t----+----P'r.7-- -~ 30 .0I I I 0 5 10 15 I I 20 25 450--r-- --+-------!- - --+----+--'l~--4 SECONDS Figure 12-3. Pupil response to steady-state light flux changes. The stimulu trace shows undistorted sinusoid; the response trace shows a dominant fundamental response with harmonic distortion, high-frequency noi e, nonstationarity of response, and lowfrequency drifting. Open-loop experiment at 0.6 hz. 0.25 0.5 1.0 2JJ 4.0 FREQUENCY, CYCLES/SEC. Figure 12-4. Bode diagram: open-loop frequ ency response. Amplitu~e is plotted on a log-log scale while phase shift (lag) is on a log-linear scale. Points are experimental and continuous lines are fitted . Dashed lines are asymptotes. 632 / II. Special Fields in Pupillary Physiology B. High-Gain Oscillations The ability of the pupil transfer function to predict an entirely different behavior of the pupil- high-gain oscil.l ation-wa of timely ignificance. The theoretical and experimental succe s helped to convince biologists that application of the engineering principle of control and communication named "cybernetics" by Norbert Wiener (1948) was possible in the biomedical world. A different display (Figures 12-5a and b) of amplitude pha e data, the Nyquist di agram, has gain, modulu length a a function of phase angle; note that a polar plot ha frequency only as an implicit function (Stark and Sherman, 1957; Stark, 1959; Stark, 1962). The Nyqui t criterion for stability indicates how the highgain operating condition (Figure 12-6) transforms the pupil regulator for light into an unstable oscillating ystem (Figure 12-7) (Stark and Cornsweet, 1958; Stark and Baker, 1959; Stark, 1959). The frequency of this oscillation (Figure 12-8, solid vertical line) is predicted by the 180-degree phase cro sover frequency of the normal pupil (Figure 12-8, solid line fitted to experimental phase data; Stark and Cornsweet, 1958; Stark and Baker, 1959; Stark, 1959). An even more stringent test was carried out by using drugs to change the pupil transfer function (Figure 12-8, dashed line fitted to experimental phase data). When the drugged pupil was made to oscillate using the high-gain operating condition (Figure 12-6) the lower frequency of oscillation (Figure 12-8, dashed vertical line) was predicted by the lower 180-degree phase crossover frequency of the drugged pupil (Stark and Baker, 1959). The high-gain operating condition was generalized by electron means (Stark, 1962) and used for a number of other biological and neurological control systems (Stark, 1968). 0 ci1lations are almost always caused by abnorma l function of a feedback control system, and an explanation in quantitative terms is essential. Thus the quantitative prediction of the pupil high-gain oscillation represented a timely contribution to mathematical biology (Stark and Cornsweet, 1958; Stark, 1959). -270° I I _,e,'( I NCRE4s 1 'V<;;,.'"' _ _ _ _::..._ ,v / «,a; '<~ I I I CRITICAL POINT -180°--- I I I GAIN- -----r-,.o ------- ---o 0 I I I I I I I I Figure 12-5a. Nyquist diagram. A vector plot of gain and phase shift; the scale of modulus and a few frequencies are indicated. The curve is derived from fitted lines from gain and phase frequency-response graphs, while points are experimental. Figure I2-5b. Nyqu ist diagram. Compare the plot of the unstable (large outer curve) to the stable (small inner curve) servo system. The curve of the stable system lies between the Nyquist critical point and origin, indicating stability. Conversely, the curve of the unstable system encloses the critical point indicating instability. The critical point is that point in the graphical plan of openloop transfer function that represents gain of 1 at phase lag of 180°. Since no frequency-dependent parameters have changed, the 180° phase crossover frequency of high-gain and low-gain systems are identical. \ I 12. The Pupil as a Neurological Control System: Mathematical Approache / 633 ( PUPIL AREA \ LIGHT INTENSITY Tl ME DRUG EXPERIMENT .... ~ - ..... 90° :;~ (/) Lu SECONDS Figure 12-7. High-gain osci llations. Pupil area oscillation frequencies in cycles/sec of several such runs were determined and averaged to demonstrate the quantitative agreement between theoretical prediction and experiment. Figure 12-6. Technique used for high gain simulation. Light is here focused on the border of iri and pupil. Small movements of the iris result in large changes in light intensity at the retina. .,.............. IN . iFrequencyofHighGainOscillations l ,, . .' ,, :,~'~JI~' '".... "" "', I 80° 1 - - - - - - - - - - - - - - - - x > I -••---------- 1"-, o ~ I I ~ J I ~ ' X~ ~, A ~270° Drugged ', Pupil-,, ~ ~,,. J: a.. ', "\ 360° ,___ __ 450° 0.2 0 .4 I 3 FREQUENCY CYCLES/SECONDS Figure 12-8. Drug experiment. Data from experiment (FB220) with drugs u ed to shift system parameters. The solid curve represents phase-frequency data from a normal pupil in low-gain operating condition. The solid vertical line marks the frequency of spontaneous oscillation in high-gain operating condition that corresponds to 180° phase cro sover frequency. The dashed curve represents phase-frequency data from the drugged pupil in lowgain operating condition. The dashed vertical line marks the frequency of spontaneous oscillations of the drugged pupil in highgain operating conditions, again showing quantitative correspondence with the new 180° phase crossover frequency (Stark and Baker, 1959). 634 / II. Special Fields in Pupillary Physiology C. Noise The pupil ha al o been the paradigm example for tudy of biological and neurological noise (Stark, Campbell, and Atwood, 1958; Stark, 1962; Stanten and Stark, 1966). Noi e i defined a random output of a y tern uncorrelated with input. Clearly proce e within the pupil y tern either form or 'gate-in' the noise; the ' 1gnature" feature of the noi e may tell us about internal pupillary mechanism . Some noise characteri tic include amplitude or rm (root mean quare) value; di tribution function uch a Gau sian, and thence variance, ci1, and other moments; additive ver u nonlinear, i.e., multiplicative-dependence of the noi e on y tern tate ; pectral feature of the noise; and evidence of coupling within the y tern generally demontrated by cro -correlation function . The pupil evidence a great deal of noise (Figure 12-9), e pecially when the pupil area i made mall by high light level (lower trace Figure 12-9) (Stanten and Stark 1966). Accommodation level i equally effective in altering the amount of noi e. Thi nonlinear effect of a y tern tate n noi e amplitude is included as multiplier in a implified model (Figure 12-11). Other experiment demonstrated a Gaussian distribution for pupil noise (Stanten and Stark, 1966); more recent experiments indicate skewing at high and low pupil areas and attribute the multiplier gain dependence on the expanive range nonlinearity, a y terns representation of the length-tension relationship of iris muscles (Usui and Stark, 1978, 1982). The spectra (Figure 12-10) if normalized at 1.5 Hz (right-hand side, Figure 12-10) are easily interpretable. Pupil neuromuscular dynamics shapes the high-frequency cutoff for noise pulses and large and small sinusoids. Retinal adaptation accounts for the lowfrequency asymptotes; pulse responses are free of adaptation, inu oidal respon es are significantly reduced at low frequencies by adaptation, and the noise spectrum is intermediate (Stanten and Stark, 1966). By ensemb le-averaging many responses with the system in a stationary state, an average time function A(t), can be obtained (Figure 12-12, lower left). Furthermore, the instantaneous second moment, a(t), can al o be determined in the e computer experiments. The noi e plane (Figure 12-12, lower right) demonstrates a lag in noise with respect to the determinate component; in the simple model (Figure 12-11) this would be LOW LIGHT LEVEL 5fLL AREA 32768 DERIVATIVE MEDIUM LIGHT LEVEL 12fLL 2048 26 AREA "' !:: z ::::, ...~... iii DERIVATIVE ~ HIGH LIGHT LEVEL 120fLL "' 32 8 AREA DERIVATIVE .1 .4 6.4 1.6 (a) Normalized at 0.1 cps 0 10 20 TIME (sec) Figure 12-9. Pupil noise area and derivative. Light levels in microlumens; eyelid blinks can be seen. .1 .4 FREQUENCY ( cps ) ( b) Normalized at 1.5 cps Figure 12-10. Spectra. Comparison of response to large pulse (P), pupil noise (N), large sinusoidal stimulus (SL), and small sinusoidal stimulus (SS) are shown on a log-log scale. Normalization at 0.1 cycle/sec (a) and at 1.5 cycles/sec (b) are hown. Three smoothings arc used for all spectra. 12. The Pupil as a Neurological Control System: Mathematical Approaches / represented as a lag element in C2 (Stanten and Stark, 1966). Noise in the two eyes is highly correlated (Figure 12-13). This provides strong evidence that the peripheral neuromuscular iris does not contribute significantly to the noise. Indeed the noise must be generated by or introduced into the pupillary system at or before the Edinger-Westphal nucleus. That accommodation and light both control pupil noise indicates that the noise must be generated by or introduced into the pupillary 635 system at or after the Edinger-Westphal nucleus, after these two inputs have merged. Recent studies of " near" Edinger-Westphal neurons (Smith, Ichinose, Masek, Watanabe, and Stark, 1968, Smith, Masek, Ichinose, Watanabe, and Stark, 1970) indicated that brainstem signals may cross-talk or are gated into the pupillary system at this level. Many clinical observations relate the level of pupil noise or ' hippus" to arousal and other states. MAXIMUM PUPIL AREA LIGHT LEVEL + ---- b(t) L h(T) A - - - LINEAR ~ - SYSTEM PUPILAREtl MULTIPLIER n(t) Figure 12-11. Simplified model for pupil noise. Note the multiplicative gating of the noise source as well as a linear sy tern to account for the spectral characteristics of noi e. INDEPENDENT NOISE SOURCE SMALL STEP RESPONSE, LOW DC LIGHT ( 72 Re$p<>11sM) PUPIL AREA LIGHT STIMULUS TIME (sec) L(tl 16 (JL LI 4 NOISE PLANE '·:ttH Id 32.6 - --~ A(tl (mm 2 ) - I I 25.0 I 1- 17.4 0 ~ I 1- - ·- 1-2 CT(t) - - . (mm2 ) I 34:l 4 A(t)(mm2 ) 22.7 5 Time (sec) Figure 12-12. Noise plane display. Small step response, with low constant (DC) light level (Stanten and Stark, 1966). 636 / II. Special Fields in Pupillary Physiology D. Kernel Identification of Nonlinearities The pupil transfer function, oscillations and noise have all been approached by linear methods (although nonlinearities were considered to play a role in shaping ome noise features). What of nonlinearities? Two methods are available. In the most common, the differential equation method, a nonlinear model is developed, the tructure of which is fixed by con ideration of anatomical, phy iological and physical information about the pupil sy tern· then various parameter identification methods can be used to refine the nonlinear model (Stark 1959 1968; Lehman and Stark, 1979). Le u ual i the econd, the integral equation method, in which is developed an a umption-free, black box, input-output de cription. This de cription is then adequate for predicting experimental responses in normal and clinical situations. AJso the shapes of the series of generalized weighting functions that form the descrip- tion may lead to interpretations as to the underlying structure of the system. Such an approach was suggested to me by Norbert Wiener. "Stark, you will go a long way applying Wiener G-functionals to biological systems. The functional approach is ideal for complex systems and biological systems are complex! " He had developed a method to orthogonalize the Volterra kernels using a white noise stimulus (Figure 12-14), the Gram-Schmidt procedure, and the Cameron-Martin result (Wiener, 1958). The advantages of orthogonality is that as new higher-order terms are added to the series, the .lowerorder terms remain constant-they remain the best estimators of their respective series terms. While a full description of this method is beyond the scope of the present chapter, it may be helpful to point out that the Wiener G-functionals (or the related Volterra kernels) are generalizations of the weighting function (which is the kernel of the convolution integral of a linear system). The Wiener G-functionals may also be thought of as an extension of a power series with t: :'=· ;1,, 11~ ,:: ,~; ~!=;;~ \::· ~L -' time-dependent functions used instead of simple no' 1 ; 1•;,;::r::p ·, !"'." J~ E:· ~i i,;i memory nonlinearities. RIGHT EYE 1 &JL1,mJU11,ttl!i1ru . 1~ /:=1' 'Ti' ... , w. A 2 The experimental, data handling, computational, and (mm l grap h.1ca 1 teeh mques · l q. . • can b e b ur densome, b ut rea dy i 1:t1J11ti1it11u1ttf. . availability of microprocessors and attached specialized array processors should alleviate this situation. Averaging of responses to identical pseudorandom stimuli is an LEFT EYE interesting technique (Sandberg and Stark, 1968). This is also compatible with "blocking" of the experiment, so as to give the human subj ect time to rest between blocks and enables the system studied to remain " stationary" (Figure 12-15). 10 20 30 0 40 Many different methods can be used; the approxiTIME (sec) mate multiple-pulse stimulus (Figures 16a and b) has the advantage of ready interpretation with utilization of run I the experimenter's physical intuition about the pupillary CROSS CORRELATION cj,RL(T) system (Sandberg and Stark, 1968). Earlier quantitative FUNCTIONS researches represent a partial, heuristic approach to 0 1---~.--c-.------5 these multipulse phenomena (Baker, 1963; Varju, 1964a, b, 1965, a, b; van der Twee! and van der Gon, 1959; Webster, 1967). The 'base function set' method was 0.95 originally suggested by Wiener (1958) because of the 0.98 ~ possibility of constructing analog computational machinery in those days. The first third-degree kernel identification was carried out with this method implemented on a digital computer with modified discrete, finite-time LaGuerre functions as the basis function set (Watanabe and Stark, 1975) (Figures 12-17a and b ). The first kernel of a biological system (or perhaps any physical system) was identified at the Massachusetts Institute of Technology using white noise as input to the pupil (Figure 12-18) (Stark, 1963; Katzenelson, 1963). One of the most Figure 12-13. Correlation between pupil noise in right and left precise kernels of the pupillary system was obtained with eyes. Both raw data and three cross-correlation functions indicate the Lee-Schetzen correlation method (Lee and Schetzen, a very high degree of correlation (the electronic amplifier gain for 1965) by Sandberg and Stark (1968; Figure 12-19). It left-eye raw data record is slightly attenuated). j ® 12. The Pupil as a Neurological Control System: Mathematical Approaches / hows similar dynamics of the fir t-degree kernel and the main diagonal lice of the second-degree kernel, suggesting a Hammer tein-type cascade model, with a nomemory nonlinearity preceding the frequency-dependent linear element (as opposed to the Wiener-type ca cade model) . Off-diagonal terms, though sparse, are present and indeed demonstrate negative values (inset, Figure 12-19). Another set of pupil kernels (Figures 12-20a and b) were computed with a pectral method that has a beneficial property of providing confidence limits. Hung and Stark with Brillinger and Eykhoff (Hung, Stark, and Eykhoff, 1977; Hung, Brillinger, and Stark, 1979; Hung and Stark, 1979) utilized these kernels to argue that the pupil escape dynamical element complicates a basis asymmetrical nonlinearity of the pupil. An important question i whether termination of the kernel series can be determined, especially since the computational and experimental burden increases with degree (note that multiple symmetries make this increase less steep). If the harmonic distortion of a system is limited, for example, to third-order harmonics (Stark, 1968, pp. 152-56; Figure 21), then kernels up to only third-degree are sufficient for full description (Korenberg, 1972). The answer to this question can be obtained by considering the magnitude of error reduction, a benefit, against the number of parameters of a kernel model and the degree (order) of the model , a cost (Watanabe and Stark, 1975; Figure 22). Another means 637 of using kernels is to study the linear and nonlinear interactions between two inputs by measuring the crosskernels (Watanabe, 1975; Marmarelis and Naka, 1974; Yasui and Fender, 1975). An approximate use of the kernel method is to use a limited set of kernels, i.e., two, and to estimate a number of such pairs of kernels for different small segments of the entire range of inputs and domain of output (the L.A. heresy). The most challenging aspect of the kernel identification method is the development of standard interpretations of kernel shapes; one may perhaps even arrive at structural formulations (Stark and Dickson, 1966). X :, u: .E 0) ::::J ·a. :, 0.. X :, u: .E :3' x ( t\ -vWrrV-~l#~~JN~~/1NM1ttr Figure 12-15. Pupil response for two identical pseudo-random stimuli functions. y{t) • • • G c1[g1. x(t)J n a f gl(T) x (t -T) dr Gz[gz , x{t)] =f f g z (Tl , TZ)x(t-Tl)x(t-T2) dTldT2 -4fei_{T,T) dT c3[g3, x(t)] =ff! g3(T l , T2, T3) x (t -Tl ) x (t -Tz) x(t-T3) dTl dT2 dT3 -31[fa3(T2' Tz, Tl) dT2] x(t-Tl) dTl Figure 12-14. Orthogonal Wiener G-functionals. 638 / Il. Special Fields in Pupillary Physiology E. Experimental Evidence for Heuristic Nonlinear Models Thi final ection deals with important heuristic nonlinearities and the experimental evidence upon which they are based. That linear gain, !:J.A /A /M/1, works over a large range of input levels is due to Weber's law. This state that light timulus is considered to be a fractional one, i.e., t:,,.J/1 (equivalent to the logarithmic scale compres ion usually attributed to retinal operators). Only much later did the A multiplier experiment demon trate that respon e could al o be fractionalized (Stark, 1963; Figure 12-23). By this we mean that amplitude of re pon e, M , appears to be multiplied by average area A ; thu M IA remains constant over a considerable range of pupil area level. These two effects enable the linearized gain to be appropriate over a wide range of input and outputs. The expansive range nonlinearity developed by Semmlow, Hansmann , and Usui (Semmlow, Iida, Tamura and Stark, 1970; Semmlow and Stark, 1971) (and sometimes called SHUERNL) generalizes and extends the A-multiplier nonlinearity and connects it to the length-tension relationship of muscle (Hansmann, Semmlow, and Stark, 1974). Recall that the pupil changes nine-fold in linear dimensions. Experimen tal results using accommodation Figures 12-24 and 1225,A) and disparity vergence (Figure 12-25,B) as inputs demonstrated this length-tension expansive range nonlinearity (Semmlow and Stark, 1973; Semmlow, Hansmann, and Stark, 1974). Two similar results were obtained with noise; the more recent one (Figure 12-26) shows noise plane responses distorted by this effect (Usui and Stark, 1978, 1982). An older noise result (Figure 12-27) demonstrates the SHUERNL at the large area region for both light and accommodation inputs (Stanten and Stark 1966). Incidentally, the incorporation by Usui of the multiplicative ( equivalent gain control) noise effect into the biomechanical model nonlinearity is an aesthetic and parsimonious result. Usui has further shown that the sigmoidal shape of the classical static pupil input-output wave is a result of both the logarithmic scale compression and the expansive range nonlinearity (Usui and Stark, 1978, 1982; Bouma, 1962). Phase plane plots demonstrate clearly the asymmetrical nature of both pupil responses to light (Figure 12-28) f\-___ ~ ~ I 7MC ~ 11 I I 1- I I 7MC f\__ ~ 11 11 Figure 12-16a. Double-pulse light stimulus of the human pupillary system (Sandberg and Stark, 1968). Figure 12-16b. Associated second-degree kernels as derived from double-pulse experiments. 12. The Pupil as a Neurological Control System: Mathematical Approaches / 639 ~ / :.::::~. 50 \i/iY' -75 20 IO 30 h,(T,,;,'3) 40 Ji=0.4 TIME T Figure 12-17a. Pupil first- and second-degree kern els obtained by Watanabe and Stark (1975) using orthonormalized basis-function analysis of random stimulus data. Figure l2-17b. Third-degree pupil kernel of third-degree model: cross-section at hyperplane T, = 0.4 sec (Watanabe and Stark, 1975). h,(T) 15 10 0 0 .2 .4 .6 .8 1.0 12 1.4 1.6 1.8 TIME (SECONOS)(o) DELAY TIME (SECON DS) (bl Figure 12-18. First- and second-degree kernels of open-loop pupil light servo mechanism (Stark, 1963). Figure 12-19. Pupil first- and second-degree kernels obtained by means of cross-correlation analysis of random stimulus data (Sandberg and Stark, 1968). Also shown is a cross section of h 2 • 640 / II. Special Fields in Pupillary Physiology • 1500 •• • • N •• •• •• -' ...... I -' ...... I 1. 000 t: t: 0 .... v, ~ ~ 0 .... "'z - 1. 000 0 a. I VO .... 0 -2. 000 w "' .... "'-' :::, 1000 •., • E -------- -1.000 ~ 0 -2.000 -' :::, a. -3 . 000 t: 000 .... ..,a .... t: - L a: '-' -3. 000 0 - 1 .0 00 0 0 200 ... .... 0 •• •• •• •• • VO a. .... .... • • 500 I • a a. "'.... c., 1. 0 0 0 t: t: • • 1 00 600 800 100 0 0 X 10 VO TIME, 500 i-1 MSEC 0 1000 • 1500 msec z N Figure 12-20a. First-degree kernels ( olid) and second-degree main-diagonal kernels ( dashed) of the pupillary system, obtained from spectral program analysis of random-stimulus pupil experimental data (Hung, Brillinger, and Stark, 1979). Figure 12-20b. Contour print of pupil second-degree kernels (see Semmlow and Stark, 1973). The dashed line indicates the main diagonal. The number symbol for the most common level (7) is not plotted. Symbols are as follows (values in 10- 6 mm/(ft L) 2 ): Symbol 9 8 7 6 5 4 3 2 1 Value 1.5 0.8 0.1 -0.6 -1.1 -1.9 -2.6 -3.2 -3.9 The corresponding estimated standard error is -0.12 x 10- 6• 10 Figure 12-21. Power spectrum of input sinusoidal responses of 0.02, 0.2, and 2.0 cycles/ second taken from three different data runs: (variation in pupil area)2 in (mm) 4 vs. frequency (cycles/second). Note second- and third-order harmonics (Stark and Kuhl, 1961) . .06 a: 0 a: 05 ~ FIRST ORDER MODEL a: 58 w w04 SECOND ORDER MODEL a: <t 503 5 6 ~ .. ;:: ii: . . > .., a: • • • oci 0.02 Cp> INPUT SINUSOID 0 .2 ti>< INPUT SlNUSOIO 2 .0 cps INPUT SINUSOID (/) z 02 <t w ~ 01 6 00 0 10 10 20 30 40 50 60 70 80 90 NUMBER OF PARAMETERS oo'·L~~__._......_...u.L----'----'--'-~........,_ __._........_....,_~.,_...._ _.__,_.....___.~"': ) 001 001 01 I (cps) 10 ID Figure 12-22. Decrease in rms error between kernel model and pupillary responses as degree of kernel model used in computation is increased (Watanabe and Stark, 1975). eurological Control System: Mathematical Approaches I 12. The Pupil a a and to accommodation (Figure 12-29) (Stark, 1962· Stark van der Tweel, and Redhead 1962; Lorber Zuber, and Stark, 1965; emmlow and Stark 1973; SemmJow Han mann, and tark, 1974; Sun, 1981; Clyne 1961). In both, the velocity of dilation i less than that of con triction (we hall ee below that thi is not ~ue to a biomechanical effect). The pupil re ponse to light show an additional asymmetry, the pupil escape phenomenon (Levatin, 1959; himizu a nd Stark, 1977; Semmlow and Chen 1977). (Thi ha now bee n demonstrated to be in large part dependent upon the pupil ize effect: the amount of and occurrence of pupil escape i a function of the initial level of pupil ize; Meyers, Barez, Krenz, and Stark, 1990; un and tark, 19 3.) Of course retinal light adaptation i a fundamental part of the pupil escape phenomenon and al o plays a la rge role in pupilJary operational characteri tic . The asymmetry in velocity can be demonstrated, using a " double-step" experiment (Figure 12-30), to be caused not by asymmetry of biomechanical time constants but by preceding excitation ignal rate asymmetries (Semmlow and Stark, 1973). Static nonlinearities in excitation signals have been adduced (Figure 12-31) to explain the neurophy iological experimen ts stimulating isolated parasympathetic and ympathetic nerve in the retroorbital space (Terdiman, mith, and Stark, 1969, 1971; Han mann, emmlow, and Stark, 1974). Another important re ult of thi timulation experiment is that CONSTRICTION OF PUPIL DURING VOLUNTARY ACCOMMODATION PUPIL AREA LIGHT FLUX f with ymmetrical artificial electrical stimulation e~ual bandwidth gain and phase Bode diagrams are obtained for sphincter and dilator (Figure 12-32); these are also similar in bandwidth to light responses. The biomechanical model (Figure 12-33) summarizes the findings: the nonlinear static expansive range nonlinearity, with symmetrical time constraints that incorporates the multiplicative nonlinearities for gain, the A-multiplier, and for noise. The asymmetry of velocity, of pupillary escape, and the pupil ize effect lie in neural excitation pathways and operators (Semmlow and Stark, 1969; Stark, 1969; Hansmann, Semmlow, and Stark, 1974; Dennison, 1967). This review of mathematical approaches to the pupil has of necessity been brief. I have omitted detailed models that would have illustrate recent changing concepts of pupillary mechanisms and that also would serve to depict the modelling approach of bioengineers (Stark and Takahashi, 1965; Lehman and Stark, 1979). The interaction of the pupil with its triadic partners accommodation and vergence have only been touched upon (O'Neill and Stark, 1968); the pupil is even responsive to consequences of extraocular eye movements, as in saccadic suppression (Lorber, Zuber, and Stark, 1965; Zuber Stark, and Lorber, 1966). The physiology and clinical behavior of the pupil has been well reviewed in the main part of this handbook; my assignment was limited to presenting some of the flavor of the bioengineering approach. REDILATATION OF PUPIL WITH RELAXATION OF ACCOMMODATION I 641 \ f I I I TIME IN S E C O N D S _ _ ; - - - - - - - - - - - - - - - - - - - Figure 12-23. Light reflex movement in re pon e to a con tant amplitude p I h · red uced by teady accommodative effect ( tark, 1963b). u se owmg 3 reduction in amplitude as mean pupil size is 642 I II. Special Fields in Pupillary Physiology -.,,.... ·" > -_-c .3 zw 0. 0 6 ~ ... ., V, V, w ... z ., A , \ V, Q) E Q. ...... 0iCI... - E a.: V, w ~ _. E ., a.. :::, a.. LU rLU \ \ .2 I I .I E 1.0 ~ < _..J Cl V, V, Q.. C C w zw .,... :::> Q.. ~ ~ E 2.0 3.0 4' .0 .20 r- .... Q) 5 .-\ .15 u ~ 0 u" <- 1 z .,... 0a.. Q. .1 ~ V, w \ l 0 _. 2 3 .,...... ~ \ .05 a.. E :::, a..-= . 5 ..§. 6 7.0 B V, ~';; 6.0 5.0 PUPIL SIZE (millimeters) .;- 0 en /-, 1.0 2.0 3.0 4'.0 6.0 5.0 7.0 PUPIL SIZE (millimeters) TIME (seconds) Figure 12-24. Averaged pupil responses to a 5-diopter accommodative stim ulus for several values of m ean pupil size. The depende nce of responsiveness of pupil size is clearly demonstrated (Han ma nn , Semmlow, and Stark, 1974). Figure 12-25. Variation of pupillometer responsiveness, or gain, as a function of mean pupil size. A, range behavior averaged for two subjects as determined by using accommodative stimulation; B, similar behavior produced by dynamic disparity vergence stimulation for a single subject (Hansmann, Semmlow, and Stark, 1974). 1.4 07 ] z 0 ~ > w 06 05 ' x, 1.0 04 a: ~ 03 <{ 0 z <{ t;; 02 rms area 01 (mm 2 ) @ 0 X 0 0 0 fixation as variab le 3 4 5 6 .a Ught 7 PUPIL DIAMETER (mm) Figure 12-26. Random movements of the pupil displayed against instantaneous pupil size shows variation in activity produced by the " expansive range nonlinearity" (Usui and Stark, 1978, 1982) . .4 a, vadableA\ ~ 0 0 10 20 30 A average area (mm2 ) 40 F igure 12-27. Noise as a function of average area, both with accommodation (fixation) and with light level as area controlling variables, showing its multiplicative nature (Stanten and Stark, 1966) . . 12. The Pupil as a Neurological Control System: Mathematical Approaches / 643 e..• 6.0 . § ·e 5.0 a:: ....IIJ ~ II.I : 4.0 0 .~ 4.0 0 2.0 g C 0 l! -t t :-2. . r 6 0 DIAMETER (mil l imeters) ~-2.0 E a. 2.0 ....► u 0 ..J II.I > 0 1.0 2.0 3.0 4.0 5.0 TIME ( seconds) Figure 12-28. Averaged pupil light reflex movements to a 2-second-on, 4-second-off step change in light intensity from 3.5 to 4.5 log td. presented as time plots (left) and pha e plane trajectory 6.0 (right). Note the marked overshoot in the on response and the slower dilation movements (Semmlow and Stark, 1973). Figure 12-29. Averaged response of the pupil to a 2-second-on, 4-second-off, 6-diopter ( I to 7 diopters) step change in accommodative stimulation. Note the marked direction-dependent behavior (Semmlow and Stark, 1973) . 7.0 .......... -~ 1 a:: II.I I- "2 4.0 II.I ~ 0 ~ ._ 2.0 = 0 !. 3.0 ~ 0 ~ "'... !. 2.0 0 e - -4. ....► g -6.0 ..J II.I > '0 1.0 2.0 3.0 4.0 TI ME (seconds) 5.0 6.0 644 I II. Special Fields in Pupillary Physiology b 6.0 a: w ti a 2 0 2.0 .., .. ok----,--1------======== 8~ C1 .r -20 E 2.0 !! -~ >- ~ 5.0 ~ .., 8 4.0 ~! 2.0 .:, . g; 0 w .. !-----,-------~~ >j-2.0 i·4.0 oh--......--l-----__:::::::=== 0 ..., _J > -2.0 -4.0!:----,-'-::--~---,'---...J 0 1.0 2.0 TIME 30 4.0 Figure 12-30. Double-step input. a, model responses to teps applied while initial velocity was nonzero. Top: mechanical asymmetry; bottom: unequal excitation time constants. b, experimental 2.0 0 (seconds) 4.0 TIME (seconds) responses of pupil velocity characteristics show smooth transition .. cross zero velocity indicating the absence of an asymmetrical mechanical process (Semmlow and Stark, 1973). 08 _J 25 0: w tw ~ ~ ..,.,, 05 c0 ·.;; 0 C ...,r _J 0:::: ::) 0 I 0.. s C Q2 o.o 05 1.0 EXCITATION: E Figure 12-31. Model simulation ofTerdiman experiment. E, = 0.7 and Ed= 0.2 are the static antagonistic model excitation levels which may have an experimental counterpart in that these were varied in order to get usable levels (Hansmann, Semmlow, and Stark, 1974). Figure 12-32. Dynamic pupillary and pupillomotor response characteristics. Amplitude characteristics for light and electrical stimulation consist of the ratio of amplitude of the first harmonic component of pupil diameter variation to the stimulus modulation amplitude as a function of frequency, plotted on full logarithmic coordinates. Phase characteristics for light and electrical stimu lation consist of the phase shift of the first harmonic component of pupil diameter variation relative to the phase of stimulus modulation wave form, plotted on semilogarithmic coordinates (Tcrdiman, Smith, and Stark, l969, 1971). ...,., - 100 c,, .!! - 200 "' .,.,, C 0 0 €_ - 300 400 ' - - - - - - - - - - - - ' - - - - - - - - - - - - ' - - - - = - - . L 10 01 00 1 Modulation frequency, hertz 12. The Pupil as a Neurological Control System: Mathematical Approaches / 645 L L- X X p Ko 0 (4-ls +-so~ ~ •I 14 _t,d Figure 12-33. Biomechanical model: a simple push-pull system. The diagram represents two contractile components linked in a push-pull arrangement. Each component consists of a pure tension source, P, in parallel with a nonlinear spring, K. This spring is not to be confused with muscle passive parallel elasticity, which is not considered here, but rather is a mechanical equivalent to the second term in the Carlson equation for active contractile force. L equals the maximum system dimension (e.g., overall iris diameter) ; X equals the normalized characteristic length ( e.g., pupil diameter); and lo, and lod are lengths for the maximum active tension of the sphincter and dilator, respectively. Pis sum of muscle forces at "O" and is defined such that positive force tends to increase X (Hansmann, Semmlow, and Stark, 1974). F. Chronology of Readings Lowenstein, Otto, and I. Givner. 1943. Pupillary Reflex to Darkness. Arch. Ophthalmol. 30:603-9. Wiener, Norbert. 1948. Cybernetics, or Control and Communication in the Animal and the Machine. Cambridge: Massachusetts Institute of Technology Press. Stark, Lawrence, and Philip M. Sherman. 1957. A Servoanalytic Study of Consensual Pupil Reflex to Light.I. Neurophysio/. 20:17-26. Stegemann, J. 1957. Ueber den Einfluss sinusforminger Leuchtdichteanderunden auf die Pupillenweite. Pftuger's Arch. Gesamt. Physiol. Menschen Tiere 264: 113-22. Schweitzer, N.M., and M.A. Bouman. 1958. Differential Threshold Measurements on the Light Reflex of the Human Pupil. Am. Med. Assoc. Arch. Ophthalmol. 59:541-50. Stark, Lawrence, and Tom N. Cornsweet. 1958. Testing a Servoanalytic Hypothesis for Pupil Oscillations. Science 127:588. Stark, Lawrence, Fergus W. Campbell, and John Atwood. 1958. Pupil Unrest: An Example of Noise in a Biological Servomechanism. Nature 182:857-58. Wiener, Norbert. 1958. Nonlinear Problems in Random Theory. Cambridge: Massachusetts Institute of Technology Press. AJpern, M., S. Kitai, and J.D. Isaacson. 1959. The Dark-Adaptation Process of the Pupillomotor Photoreceptors. Am. J. Optom. 48. Levatin, Paul. 1959. Pupillary Escape in Diseases of the Retina or Optic Nerve. Arch. Ophthalmol. 62:768-79. Lowenstein, Otto, and lrene Loewenfeld. 1959. Influ- ence of Retinal Adaptation upon the Pupillary Reflex to Light in Normal Man. I. Effect of Adaptation to Bright Light on Pupillary Threshold. Am. J. Ophthalmol. 42:536. Stark, Lawrence. 1959. Stability, Oscillations, and Noise in a Neurological Servomechanism. J. Neurophysio/. 22:156-64. - - - . 1959. Stability, Oscillations, and Noise in the Human Pupil Servomechanism. Proc. Inst. Radio Eng. IRE 47:1925-1939. Stark, Lawrence, and Frank Baker. Stability and Oscillation in a Neurological Servomechanism. J. Neurophysiol. 22:156-64. Van der Twee), L.H., and J.S. Denior Van der Gon. 1959. The Light Reflex of the Normal Pupil of Man. Acta Physiol. Pharmacol. Neer/. 8:52-88. Stark, Lawrence. 1960. Vision: Servoanalysis of Pupil Reflex to Light.Med. Phys. 3:701-19. Clynes, M. 1961. Unidirectional Rate Sensitivity: A Biocybernetic Law of Reflex and Humoral Systems as Physiological Channels of Control and Communication. Ann. N. Y Acad. Sci. 92:946-69. Lowenstein, Otto, and Irene Loewenfeld. 1961. Influence of Retinal Adaptation upon the Pupillary Reflex to Light in Normal Man. II. Effect of Adaptation to Dim Illumination upon Pupillary Reflex Elicited by Bright Light. Am. J. Ophthalmol. 51 :644-54. Stark, Lawrence, and Frank Kuhl. 1961. Analysis of Pupil Response and Noise. Q. Prag. Rep. Res. Lab. Electron. MIT 62:255-62. 646 I U. Special Fields in Pupillary Physiology Bouma, H. 1962. Size of the Static Pupil as a Function of Wavelength and Luminosity of the Light Incident on the Human Eye. Nature 193:690-91. Campbell, F.W., and M. Alpern. 1962. Pupillometer Spectral Sensitivity Curve and Color of the Fund us. J. Opt. Soc. Am. 52:1084. Stark, Lawrence. 1962. Biological Rhythms, Noise, and Asymmetry in the Pupil-Retinal Control System. Ann. N. Y. Acad. Sci. 98:1096-1108. - - - . 1962. Environmental Clamping of Biological Systems: Pupil Servomechanism. J. Opt. Soc. Am. 52:925-30. Stark, Lawrence, Henk van der Twee!, and Julia Redhead. 1962. Pulse Response of the Pupil.Acta Physiol. Pharmacol. Neer/. 11:235-39. Baker, Frank H. 1963. Pupillary Response to DoublePul e Stimulation: A Study of Nonlinearity in the Human Pupil System. J. Optic. Soc. Am. 53:1430-36. Katzenel on, J. 1963. The Design of Nonlinear Filters. Sc.D. diss., Massachusetts Institute of Technology. Stark, Lawrence. 1963. Black-Box Description and Physical Element Identification in the Pupil System. Q. Prog. Rep. Res. Lab. Electron. MIT 69:247-50. - - - . 1963. Nonlinear Operator in the Pupil System. Q. Prog. Rep. Res. Lab. Electron. MIT 72:258-60. - - - . 1963. Stability, Oscillation, and Noise in the Human Pupil Servomechanism. Bol. Inst. Estud. Med. Biol. Mex.. 21 :201-22. Varju, Dezsoe. 1964. Der Einfluss sinusfoermiger Leuchtdichteanderungen auf die mittlere Pupillenweite und auf die subjektive Helligkeit. Kybemetik 2:33-43. - - - . 1964. Pupillenreaktionen auf sinusfoermige Leuchtdichteanderungen. Kybemetik 2:124-27. Bishop, Lewis G., and Lawrence Stark. 1965. Pupillary Response of the Screech Owl, Otus Asio. Science 148:1750-52. Lee, Y.W. , and M. Schetzen. 1965. Measurement of Wiener Kernels of a Nonlinear System by Crosscorrelation. Int. J. Control 2:237-54. Lorber, Martin, Bert L. Zuber, and Lawrence Stark. 1965. Suppression of the Pupillary Light Reflex in Binocular Rivalry and Saccadic Suppression. Nature 208:558-60. Stark, Lawrence. 1965a. Classical and Statistical Mathematical Models for a Neurological Feedback System. Neurosci. Res. Program Bull. 3:55-60. Stark, Lawrence, and James F. Dickson. 1965b. Mathematical Concepts of Central Nervous System Function. Neurosci. Res. Program Bull. 3:1-72. Stark, Lawrence, and Yoshizo Takahashi. 1965c. Absence of an Odd-Error Signal Mechanism in Human Accommodation. IEEE Trans. Biomed. Eng. BME-12: 138-46. ten Doesschate, J., and M . Alpern. 1965. Response of the Pupil to Steady-State Retinal Illumination Contribution of Cones. Science 149:989-91. Young, Laurence, and Lawrence Stark. 1965. Biological Control Systems-A Critical Review and Evaluation: Developments in Manual Control. NASA Contractor Report CR-190, pp. 1-221. Stanten, Saul F., and Lawrence Stark. 1966. A Statistical Analysis of Pupil Noise. IEEE Trans. Biomed. Eng. BME-13:140-52. Zuber, Bert L., Lawrence Stark, and Martin Lorber. 1966. Saccadic Suppression of the Pupillary Light Reflex. Exp. Neurol. 14:351-70. Dennison, B.L. 1967. A Mathematical Model for the Motor Activity of the Cat Iris. Ph.D. diss., Worcester Polytechnic Institute. Varju, Dezsoe. 1967. Nervose Wechselwirkung in der pupillomotorischen Bahn des Menschen. I. Unterschiede in den Pupillenreaktionen auf monoculare und binoculare Lichtreize. Kybemetik 3:203-214. - - - . 1967. Nervose Wechselwirkung in der pupillomotorischen Bahn des Menschen. [I. Ein mathematisches Modell zur quantitativen Beschreibung der Beziehungen zwischen den Reaktionen auf monoculare und binoculare Lichtreize. Kybemetik 3:214-26. Webster, J.G. 1967. Critical Duration for the Pupillary Light Reflex. J. Opt. Soc. Am. 59:1473-78. O'Neill, William D., and Lawrence Stark. 1968. Triple Function Ocular Monitor. J. Opt. Soc. Am. 58:570-73. Sandberg, Allen, and Lawrence Stark. 1968. Wiener G-Function Analysis as an Approach to Nonlinear Characteristics of Human Pupil Light Reflex. Brain Res. 11:194-211. Smith, James D., Lester Y. Ichinose, Gerald A. Masek, Takeshi Watanabe, and Lawrence Stark. 1968. Midbrain Single Units Correlating with Pupil Response to Light. Science 162:1302-3. Stark, Lawrence. 1968. Neurological Control Systems: Studies in Bioengineering. New York: Plenum. Stark, Lawrence, Robert Arzbaecher, Gyan Agarwal, Jerald Brodkey, Derek Hendry, and William O'Neill. 1968. Status of Research in Biomedical Engineering. IEEE Trans. Biomed. Eng. BME-15:210-231. Troelstra, A. 1968. D etection of Time-varying Light Signals as Measured by the Pupillary Response. J. Optic. Soc. Am. 58:685-90. Lee, R.E ., G.H. Cohen, and RM. Boynton. 1969. Latency Variation in Human Pupil Contraction due to Stimulus Luminance and/or Adaptation Level. J. Opt. Soc. Am. 59. Semmlow, John, and Lawrence Stark.1969. A Nonlinear Biomechanical Model of the Human Iris. Proceedings of the Conference on Applications of Continuous System Simulation Languages, San Francisco, pp. 149-57. Stark, Lawrence. 1969. Pupillary Control System: Its Nonlinear Adaptive and Stochastic Engineering Design Characteristics. Fed. Proc. 28:52-64. Terdiman, Joseph, James D. Smith, and Lawrence Stark. 1969. Pupil Response to Light and Electrical Stimulation: Static and Dynamic Characteristics. Brain Res. 16:288-92. Varju, Dezsoe. 1969. Human Pupil Dynamics. Proc. Int. Schoo/Phys. 43:442-57. Semmlow, John, Mitsuo Iida, Hiroshi Tamura, and Lawrence Stark. 1970. Biomechanical Model for the Pupil. Proceedings of the Fourth Annual Princeton Conference on Information Sciences and Systems, Princeton, N.J., pp. 40-45. Smith, James D ., Gerald A. Masek, Lester Y. Ichinose, Takeshi Watanabe, and Lawrence Stark. 1970. Single Neuron Activity in the Pupillary System. Brain Res. 24:219-34. 12. The Pupil as a Neurological Control System: Mathematical Approaches Semmlow, John, and Lawrence Stark. 1971. Simulation of a Biomechanical Model of the Human Pupil. Math. Biosci. 11:109-28. Terdiman, Joseph, James D . Smith, and Lawrence Stark. 1971. Dynamic Analysis of the Pupil with Light and Electrical Stimulation. IEEE Trans. Syst. Man Cybem. SMC-1:239-51. Korenberg, M. 1972. Aspects of Time Varying and Nonlinear Systems Theory with Biological Applications. Ph.D. diss., McGill University. Ohba, N., and M. Alpern. 1972. Adaptation of the Pupil Light Reflex. Vision Res. 12:953-67. Semmlow, John, and Lawrence Stark. 1973. Pupil Movements to Light and Accommodative Stimulation: A Comparative Study. Vision Res. 13:1087-1100. Hansmann, Douglas, John Semmlow, and Lawrence Stark. 1974. A Physiological Basis of Pupillary Dynamics and Behavior. In Pupillary Dynamics and Behavior, ed. Michel Pierre Janisse. New York: Plenum. Marmarelis, P., and K. Naka. 1974. Identification of Multi-input Biological Systems. IEEE Trans. Biomed. Eng. 21:88-101. Semrnlow, John, Douglas Hansmann, and Lawrence Stark. 1974. Va riation in Pupillomotor Responsiveness with Mean Pupil Size. Vision Res. 14:1-6. Watanabe, A. 1975. The Volterra Series Expansion of Nonlinear Functionals. Summer Computer Simulation Conference. Watanabe, Atushi, and Lawrence Stark. 1975. Kernel Method for Nonlinear Analysis: Identification of A Biological Control System. Math. Biosci. 27:99-108. Yasui, S., and D. Fender. 1975. Methodology for Measurement of Spatial-Temporal Volterra and Wiener Kernels for Visual Systems. Proceedings of 1st Symposium on Testing and Identification of Nonlinear Systems, pp. 366-83. Hung, George, a nd Lawrence Stark. 1977. The Kernel Identification Method (1910-1977): Review of Theory, Calculation, Application and Interpretation. Math. Biosci. 37:135-90. Hung, George, Lawrence Stark, and Pieter Eykhoff. 1977. On the interpretation of Quadratic Kernels: Computer Simulation of Responses to Impulse Pairs. / 647 Ann. Biomed. Eng. 5:130-43. Sernmlow, John, and D. Chen. 1977. A Simulation Model of the Human Pupil Light Reflex. Math. Biosci. 33:5-24. Stark, Lawrence. 1977. Models of Biocontrol Systems. Clin. All-Round [Jpn.] 26:9-26. Usui, Shiro, and Lawrence Stark. 1978. Sensory a nd Motor Mechanisms Interact to Control Amplitude of Pupil Noise. Vision Res. 18:505-7. Hung, George, and Lawrence Stark. 1979. Interpretation of Kernels. III. Positive Off-diagonal Kerne ls as Correlates of the Dynamic Process of Pupillary Escape. Math. Biosci. 46:189-203. Hung, George, David R. BriJlinger, and Lawrence Stark. 1979. Interpretation of Kernels. II. Same-signed Firstand Second-Degree (Main-diagonal) Kernels of the Human Pupillary System. Math. Biosci. 46:189-203. Lehman, Steven, and Lawrence Stark. 1979. Simulation of Linear and Nonlinear Eye Movement Mode ls: Sensitivity Analyses and Enumeration Studies of Time Optimal Control.]. Cybern Inf Sci. 2:21-43. Sun, Fu-chuan, and Xin-zhen Zhao. 1979. Frequency Analysis of the Pupil Control System Using the Fa t Fourier Transform. Proceedings of the Annual Conference, BM£ of Shanghai, p. 304. Sun, Fu-chuan, Xin-zhen Zhao, Shu-ping Dai, Hao-kun Liu, and Run-cai Yang. 1979. Dynamic Characteristics of the Pupillary Control System: Measurement and Mathematical Model.ActaAutom. Sin. 5:130-35. Sun, Fu-chuan, Hao-lam Liu, and Yu-min Liu. 1981. Dynamic Pupillary Response to Positive Differential of Light Stimulus. Sci. Sin. 24:872-84. Usui, S., and Lawrence Stark. 1982. A Mode l for Nonlinear Stochastic Behavior of the Pupil. Biol. Cybem. 45 :13-21. Sun, Fuchuan, and Lawrence Stark. 1983. Pupillary Escape Intensified by Large Pupillary Size. Vision Res. 23:611-15. Meyers, Glenn, Shirin Barez, William Krenz, and Lawrence Stark. 1990. Light and Target Distance Interact to Control Pupil Size. Am. J. Physiol. 258:81319.