A simulation model of the human pupil light reflex

A Simulation JOHN

Model

of the Human

Pupil light

Reflex

L. SEMMLOW

AND

DAWG C. CHEN Bioengineering Program, Vnioersi@ of Illinois at Chicago Circle-College

of Engineering,

Box 4348, Chicago, Illinois 60680

Communicated

by Lawrence

Stark

ABSTRACT A system-type model of the human pupil light reflex was developed which relates sensory and motor physiological processes to experimentally determined behavioral features such as the Weber-Fechner law, Bloch’s law, critical duration, and the expansive range nonlinearity. A major objective of the model was that it be capable of representing ten major light-reflex behavior characteristics with physiologically reasonable elements (i.e., integrators and lead and lag elements) and a structure compatible with the known light-reflex pathway. computer to match simulations produced was developed

Model parameters were adjusted through simulation on a digital human data for a wide variety of stimulus conditions. Further results which matched not only those behaviors for which the model

and adjusted,

but overall,

large-signal

responses

as well.

INTRODUCTION The human pupillomotor response to light (the light reflex) has proven to be a paradigm biological control system for application of servo-control theory with examples in transfer-function analysis and stability criterion [21,25], the treatment of pupillary unrest as a manifestation of signal noise [20], and the analysis of nonlinear physiological processes [ 14,231. Further, a quantitative analysis of light-reflex dynamical characteristics has become increasingly useful in diagnosing certain neurological disorders [ 11,9]. Recently, the use of two alternative stimulus modalities-accommodation (evoked by retinal blur of a target image) and vergence demand (produced by a disparity between the retinal target images on the two eyes)-has MATHEMATICAL

BIOSCIENCES

0 Elsevier North-Holland,

Inc.. 1977

33, 5-24 (1911)

5

JOHN L. SEMMLOW

6

AND

DAWG

C. CHEN

permitted Semmlow and Stark [19] to isolate and quantify features of the peripheral motor processes and led to the development of a “biomechanical model” for the iris neuromuscular apparatus. Here we return to the more complex light-reflex movements in an effort to isolate behavior attributable to sensory processes. Through the development of a computer-simulated model we expect to organize the dominant behavioral characteristics of the pupil light reflex and to relate them to possible sensory or motor mechanisms. The model should be a comprehensive representation of major processes in the pupil light reflex, capable of adequately predicting pupillomotor responses to a variety of stimulus patterns: impulse, step, and sinusoidal stimulus waveforms. Experiments on the model through the use of simulation methods allow us to test out alternative hypotheses, and should suggest new experimental procedures making the model a valuable aid to future pupillomotor studies. PUPIL-LIGHT-REFLEX

BEHAVIORAL

CHARACTERISTICS

If the pupil. light reflex in man were a linear biological system, its response to all possible stimuli could be completely described by a single differential equation, the transfer function. Indeed, in one of the earliest approaches to light-reflex models, Stark [21] approximated the system by a third-order low-pass transfer characteristic with transport delay. However, the relationship between pupil movement and light stimulation is highly nonlinear, so in addition to the dynamical properties approximated by the Stark model many nonlinear behavioral characteristics have been documented through quantitative experimentation. These nonlinear characteristics, while increasing the problems of the biological modeler, provide us with additional information concerning signal processing in the light-reflex pathway, since each nonlinear behavior is an external manifestation of an internal nonlinear process. The isolation and quantitative definition of these behaviors should lead to mathematical descriptions of the pupil light reflex which more closely overlie known physiology, linking gross pupillomotor phenomenon to processing features demonstrated by psychophysical and neurological experimentation. In a rough attempt to relate the nonlinear and other behavioral features of the light reflex to underlying physiological structure, we categorize these features into two major subdivisions: sensory behaviors (features attributable to sensory elements) and motor behaviors (features produced by elements in the final pathway). As any comprehensive model of the light reflex should simulate all major sensory and motor behaviors by the action of appropriately located model elements, we summarize below the major behavioral characteristics seen in static and dynamic pupillomotor activity.

SIMULATION SENSORY

MODEL OF HUMAN PUPIL LIGHT REFLEX

7

BEHA VIORS

Many features visual perception. (1) Static

seen in the light reflex parallel phenomena These sensory characteristics are described

characteristics:

steady-state

gain.

Above

threshold

long noted below.

in

the percep-

tion of brightness may be related as either a power function (Stevens power law), or logarithmic function (Weber-Fechner law) to stimulus amplitude. Though element,

retinal processing such a mechanism

state behavior [3,29]. After demonstrating amount of unbleached a relationship

probably includes a power-or logarithmic-type alone cannot accurately describe pupil steady-

a linear relationship between pupil diameter and the rhodopsin in the retina, Alpern and Ohba [l] derived

between

steady-state

pupil

diameter

and

equilibrium

light

intensity (under ganzfeld, or total retinal, illumination) which involved, in part, a logarithmic relationship, and which accounted rather well for actual pupil steady-state variations under the assumed conditions. The overall steady-state behavior of the pupil light reflex will also be influenced by the static characteristics of the pupil motor system later described. (2) Threshold characteristics: the Weber fraction. In visual perception as determined tectability

through psychophysical experimentation, the threshold for deof a small change in illumination, AZ, against a background level

I (the so-called brightness discrimination) is not stimulus amplitude AI, but is more closely related Weber fraction [6]) over a fairly wide range of demonstrated that a similar relationship (AZ/l= K)

linearly related to the to the ratio Al/Z (the I. Webster et al. [32] exists between light-re-

flex thresholds and increasing light stimulation, and Ohba and Alpern [ 131 have shown that, as in visual perception, the background intensity I can be either a real background or a bleaching background. characteristic: Bloch’s law. For brief (3) Amplitude- time-dependent flashes (less than 0.1 set) a given visually perceived the product of flash intensity and duration (also

magnitude depends on known as the Bunsen-

Roscoe Law [6]). Sandberg and Stark [ 141 showed an increase in pupillomotor response to light flashes with increased pulse duration, and Webster [31] demonstrated

light-reflex

behavioral

features

which

closely

parallel

the

psychophysical phenomenon of Bloch’s law for short pulse stimulation. (4) Time-dependent characteristics: critical duration. In visual perception, flashes longer than about 0.1 set produce effects related to the intensity only. Webster [32,3 I] has documented the fact that this same value applies to pupillomotor light-pulse responses, at least for stimulus amplitudes near threshold. (5) Time-dependent characteristics: frequency response. An intermittent light source flickering above the fusion frequency (no perceived flicker) will

8

JOHN

L. SEMMLOW

AND

DAWG

C. CHEN

produce a visually perceived magnitude proportional to the time-averaged intensity of the source (Talbot’s law). Troelstra [26] and Varju [28] using sinusoidally modulated light and a specific pupil-response criterion, demonstrated

an integrative

10 Hz (Fig.

feature

similar

to Talbot’s

law for frequencies

above

1).

01

I

I

I

I

I

I

I

0.2

05

10

2.0

5.0

10 0

20.0

FREQUENCY

(hr)

FIG. 1. Change in mean value of pupil diameter (in percent change) produced by sinusoidally varying light stimulation as a function of the frequency of modulation. Curve A, (dashed line and squares), from model simulations; curve B (solid line and triangles), replotted

from data on human

subjects

by Varju at one level of average

brightness.

While the remaining three prominent light-reflex characteristics do not correspond to any major visual perception phenomena, we still classify them as sensory behaviors, since they are not found to anywhere near the same extent in pupillomotor activity produced by other stimulus modalities. It is likely that these behaviors are attributable to sensory elements not shared by the visual system. (6) Time-dependent characteristics: light escape mechanism. An averaged pupillary response to an on-step of light [Fig. 2(a), curve A] shows a rapid construction followed by a slower redilatation. This redilatation movement, termed pupillary escape [lo], occurs despite the continuing stimulus and may be due to a short-term adaptation process.

SIMULATION

MODEL

OF HUMAN

0

PUPIL

LIGHT

10

9

REFLEX

2c

JO

TlME(sec)

FIG. 2 Human and simulated light-step responses. Upper traces show averaged human pupillomotor responses to small-step changes in light intensity for two mean values of pupil diameter (curves A and C, obtained by varying steady accommodative stimulus level) along with the response to a large-step input (curve B) [16]. Lower traces show responses from model simulations of the above experiment. (Plotted with latency added for comparative purposes.)

(7) Amplitude-dependent characteristics: escape nonlinearity. That pupillary escape phenomena are independent of steady background has been shown by Semmlow, Cahn and Stark [ 161; see Fig. 2(a), curves A and C. However, as was previously noted by Loewenfeld [9], it is dependent on input amplitude, with large amplitudes showing less escape; see Fig. 2(a), curve B. (8) Direction-dependent characteristics: sensory asymmetry. The response to a negative pulse or step change in light intensity is not the inverse of the response to a positive pulse or step, as would be the case in a linear system [22]. This aspect of pupillary light behavior has been described using asymmetrical nonlinear differential equations by Stark [21] and later termed

IO

JOHN

L. SEMMLOW

AND

DAWG

C. CHEN

by Clynes [5] “unidirectional rate sensitivity”. Essentially, the response to increasing light has considerably larger transient gain; thus, the dominate pupillomotor dynamic response to both positive (on) and negative (off) light pulses is one of constriction (see Fig. 3). Since this behavioral feature is not the result of either unidirectionality or rate sensitivity, we use the term “sensory asymmetry” in referring to it.

fJ\-:----

;r’--: =;w

“;w

2

FIG. 3.

Human

and simulated

(solid curves) to equal strong direction-dependent original

data.)

Dashed

light-pulse

positive (increasing) behavior (adapted curves

2 TIME(sec)

TIME(sec)

responses.

Human

pupillomotor

responses

and negative light-pulse stimulation show from Clynes [4]). (Note error in latencies of

are the responses

from

model

simulations

of this experi-

ment.

MOTOR

BEHAVIORS

Though the neural pathway involved in other pupil inputs-such as the accommodation response (the near response) and the response to vergence demand-are not as well defined, anatomically, as those in the light reflex, experimental evidence indicates that functionally these other input systems are relatively free of nonlinear processes [18]. Through the use of these alternative inputs a number of phenomena attributable to muscular or other peripheral motor processes have been isolated. (1) Static characteristics: steady-state gain. A variation in motor-system responsiveness as a function of the current motor output level has been isolated and quantitatively defined by Semmlow, Hansmann, and Stark [ 171. This nonlinear feature, termed the “expansive range nonlinearity”, results from steadily decreasing pupillomotor gain as the pupil size deviated from midrange levels (Fig. 4). (2) Direction-dependent characteristics: response asymmetry. Pupillomotor activity produced by either accommodation or disparity stimulation shows some direction dependence, as shown in Fig. 5(a) [12]. Though this is

SIMULATION

MODEL

OF HUMAN

PUPIL

LIGHT

PUPIL SIZE FIG. 4. a = subject

Variation

in pupillomotor

J.S.; 0 = subject

0

gain

FIG. 5.

(millimeters)

as a function

of mean

pupil

size [17].

D.H.

1

2

3

TIME

Average near and included modative

11

REFLEX

4

5

6

(XC)

Human and simulated responses to accommodative step stimulation. Upper: pupillomotor response to a step change in target viewing distance from far to near to far, a response associated with accommodative effort [ 181. (Latency not in original data.) Lower: Results from model simulations of the pupil accomresponse.

JOHN

12

L. SEMMLOW

AND

DAWG

C. CHEN

presumably a motor characteristic (since it is common to all responses), it is similar to the direction dependence found in the light reflex (sensory asymmetry). and may account for some portion of this behavior. However, the motor asymmetry is not sufficiently effective to produce the very marked asymmetry seen in light-induced movements. The above characteristics by no means form a completely comprehensive list of all light-reflex phenomena. Effects of stimulus size (retina1 area), color, subject fatigue and emotional states, binocular interaction, and adaptation characteristics are not dealt with here. Nor have we included noise phenomena [20], though they are a prominent feature of single, unaveraged pupillomotor responses, and have been incorporated into a previous light-reflex model [23]. The above features do, however, include the major behaviors associated with dynamic pupillary movement (after ensemble averaging tQ eliminate noise) induced by light stimulation under the controlled conditions normally found in laboratory experimentation. METHODS The pupil light reflex is mediated by highly nonlinear biological processes, and any model of this system would necessarily contain several nonlinear functions. For this reason, model simulations rely on digital-computer simulation techniques, in particular the digital-analog simulator language, 1130 CSMP [8]. By running 1130 CSMP on a dedicated IBM 1800 computer, model performance may be immediately evaluated without sacrificing the sophistication necessary to represent complex nonlinear biological processes. Digital-analog simulation languages are block-oriented and require the modeler to first reduce the mode1 to a network of processes, or function blocks, similar to those used on an analog computer (integrators, adders, multipliers, display units, etc.). At this stage techniques similar to those used in traditional analog-computer simulation are used. After suitable editing, the CSMP program translates the network topology into an appropriate set of mathematical equations which are then calculated numerically, using parameters and initial conditions specified by the modeler. Output is provided graphically or in tabulated form. Though digital-analog simulators require model description by traditional analog elements, the tedious chores of time and magnitude scaling are not necessary, and nonlinear elements may be easily simulated. MODEL

DEVELOPMENT

AND SIMULATION

RESULTS

Modeling the dynamical characteristics of the pupil light reflex has been a particularly active area since the initial efforts of Stark [21]. Here we refer to previous modeling efforts only as they relate to the construction of our

SIMULATION

model;

MODEL

a good

review

OF HUMAN

detailing

PUPIL

LIGHT

13

REFLEX

the configuration

and

simulation

behavior

of many earlier models can be found in Webster [29]. Our model, shown in Fig. 6, is an outgrowth of models suggested by previous workers [23,29] and the performance, under simulation, of several developmental models. As in most previous models the pupil light reflex system is represented as open-loop, the obvious feedback influences of pupil diameter on retinal light are ignored due to the low value of the system loop gain [21] and corresponding small effects (in terms of system behavior) from such feedback. Furthermore, many experimental situations inhibit the feedback pathway completely by using Maxwelliam view stimulation. Each element relates to one or more of the behaviors previously described. As in any complex nonlinear model the configuration cannot be proven unique; however, correlation of the model with both known physiology havior is good. The model will first be discussed and its simulation mance sensory

evaluated processes,

in terms of the two major after which total performance

SENSORY

MOTOR

subsections, the plant will be demonstrated.

and

Steadystate Pathway

PROCESSES I

and beperfor-

-

.35

I

PROCESSES

_DlA

FIG. 6. Block diagram presentation of the light-reflex submodels: the sensory and plant (motor) submodels.

PLANT

model

containing

two major

MODEL

The plant model represents the properties of the iris musculature and other elements of the final common pathway. It is essentially a system representation of the “biomechanical model” of Semmlow and Stark [19] derived from considerations of iris mechanical and muscle properties along with pupillomotor activity seen in the near response. Implicit here is the

14

JOHN

L. SEMMLOW

AND

DAWG

C. CHEN

assumption that iris motor mechanisms involved in the light reflex are the same as, or have similar behavior to those in the near response. Though there

is evidence

to suggest

modative and light-induced contain similar processes.

separate

final

common

pathways

[ 121. it is reasonable

signals

for accom-

to assume

they both

The first element in the plant submodel represents the direction-dependent behavior associated with motor mechanisms. This plant asymmetry, or direction

dependence,

can

be most

conveniently

evaluated

by comparing

the human pupil near response with model simulations, as in Fig. 5. Most of this dynamic asymmetry is represented by a rate-sensitive low-pass filter with time constants which vary with the sign of the input derivative. Physiologically, this process viscous properties of muscle,

could be or unequal

associated with muscle activation

the asymmetrical and deactivation

time constants [19]. Additionally, a second low-pass filter is present to produce the required second-order plant dynamics [ 15,181. The other plant submodel element represents the expansive range nonlinearity determined experimentally and presented in Fig. 4. Semmlow and Stark [19] showed that this behavior could be generated by the nonlinear length-tension characteristics of the pupillomotor muscles. This behavior is simulated

though

approximate any range nonlinearities postulated

the use of a nonlinear

function

element

which

can closely

general nonlinear function. Though slightly different for accommodative and light stimulation have been

[18], our

simulation

results

indicate

these

differences

do not

significantly affect behavior. The performance of this and the other plant elements have previously been evaluated with regard to data on the human near response

[19] and will be further

tions which, necessarily, pupillomotor processes. SENSORY

depend

supported on

here by light-reflex

appropriate

simula-

representation

of

MODEL

Unlike the plant section of the model, which was based, in part, on iris physiology, the sensory model is derived largely from behavioral characteristics. Each element was introduced to represent one or more of the sensory behaviors earlier described. In the model of Fig. 6, the sensory input divides after the first two elements, with the lower pathway processing transient signals only. Thus, the first two elements, along with the constant-gain term in the upper pathway and the “expansive range nonlinearity” in the plant, determine the steady-state light response. The first two elements are similar to, and serve equivalent functions to, initial elements found in other prominent models of the pupil light reflex [29,23]. The first element is a low-pass filter (second order) which produces a time

integration

for short

pulses

simulating

the amplitude-time

relation-

SIMULATION

ship

MODEL

described

OF HUMAN

by Bloch’s

PUPIL

LIGHT

law. The model

15

REFLEX

output

in response

to variable-

width pulses of fixed amplitude (Fig. 7) shows increasing response amplitudes with increasing pulse widths up to a critical duration, in agreement with experimental results. This low-pass element, in conjunction with a high-pass element and the rectifying effects of a direction-dependent element later described, accounts for the responses found by Troelstra [26] and Varju [28] to sinusoidally modulated light stimulation. demonstrates that model simulations under similar provide a reasonable match to experimental frequency The second

element

is a logarithmic

operator

Figure 1 (curve A) stimulus conditions behavior.

representing

the Weber-

Fechner law. While there is evidence that steady-state retinal processes are more complex [ 11, our later simulations of the overall light reflex show a simple logarithmic operator to be a good approximation. As stated previously, the signal pathway divides after the first two elements, with the upper pathway directly to motor processes while

carrying a small fraction of the signal the lower pathway performs additional

processing on transient signals. As the upper pathway contains no signalprocessing elements and has a comparatively low gain, it might be the result of a “leak”, or signal crosstalk, across the processes in the lower pathway. Since the lower pathway contains significantly higher gain (effectively 100 times that of the upper pathway for small fast transients) it determines threshold behavior, and its transient nature, along with a small threshold element and preceding logarithmic element, accounts for the threshold behavior of the pupil light reflex, i.e., the Weber fraction (al/ I = K, for threshold). The

elements

in

processes: a long-term tion. The first process

the

lower

pathway

adaptation, in the lower

represent

three

major

sensory

a short-term adaptation, and a saturapathway is a linear high-pass filter with

long time constant simulating a long-term adaptation also includes the threshold element. While the current

process. This process modeling effort is not

directed at a detailed representation of adaptation processes, such an element is required to simulate the amplitude dependence of the pupillary escape nonlinearity, to be discussed shortly. Visual perception experiments suggest a time constant of several minutes for this process [6]. The remaining processes of the lower pathway are principally responsible for short-term transient sensory characteristics. The first of these, a nonlinear short-term adaptation process, accounts for the related pupillomotor behavior of sensory asymmetry, critical duration, and pupillary escape. This process is modeled by a high-pass filter with a short time constant and a nonlinear gain having zero value for negative-going inputs. A relatively low-gain, direct signal pathway paralleling this process is required for proper operation of the following saturation element. Again. this pathway may be attributable to signal crosstalk, or leak, across the process itself.

16

JOHN

L. SEMMLOW

AND

DAWG

C. CHEN

Critical duration occurs when the stimulus pulse width is sufficiently long that it no longer appears as a pure impulse with respect to the integrative features of the system’s high-frequency dynamics. Thus, the critical duration is determined by the shorter time constants in the model, in particular those of this short-term adaptation process and of ‘the low-pass filter mentioned previously. The model has a critical duration of 0.1 set (Fig. 7) in good agreement with human data. In the model, pupillary escape is simulated through the falloff in highpass filter output with a sustained step input. The dynamics of pupillary escape are primarily determined by the longer time constants in the plant submodel and are seen to match quite closely those of actual human light-reflex step responses (Fig. 2). Sensory asymmetry is achieved by the nonlinear gain in the short-term adaptation process, which is zero for negative-going (light to darkness) inputs. The relatively high gain of this process is effective only for increasing transient signals; thus, pupillomotor threshold is defined only with respect to positive-going transient stimulation. Webster [29] used a similar element to simulate this behavior.

FIG. 7. Model responses to constant amplitude-duration product relationship

amplitude pulses of varying duration showing up to a critical duration of around 0.1 sec.

an

SIMULATION

MODEL

OF HUMAN

PUPIL

LIGHT

17

REFLEX

The last sensory process is a saturation nonlinearity responsible for the reduction of pupillary escape behavior with large step inputs. Due to a small direct signal pathway bypassing the short-term adaptation process, saturation input remains high even after short-term adaptation output has decayed to zero, provided the input signal is sufficiently large. However, saturation will not continue after the long-term adaptation process has decayed, as is required by experimental findings that pupillary escape is present even with high steady background levels once long-term adaptation has occurred [16]. Figure 2 demonstrates the model effectiveness in reproducing this behavior. LIGHT-REFLEX

by

SIMULATION

The mode1 development and parameter specific pupil light-reflex behaviors,

adjustment were largely directed ensuring relatively good perfor-

mance with respect to these features. As a test of the model’s generality, comparison of model and human light-reflex responses covering a range

a of

stimulus conditions will be presented. Figure 8 compares model and human responses to an on-and-off step of light showing the marked direction-dependent behavior in this pupil-light reflex.

In the off (or darkness)

response

the model

output

is not influenced

TIME Isecl

FIG. 8. Human and simulated responses to on and off steps of light stnnulation. Upper: Averaged pupil light reflex movement to a 2-set-on, 4-set-off step change in light intensity [ 181. Lower: Results of step-response simulations of the light-reflex model (plotted with latency added for comparative purposes).

18

JOHN

L. SEMMLOW

AND

DAWG

C. CHEN

by the lower sensory pathway; thus sensory involvement is minimal, and the dynamics are largely due to the plant submodel. Good agreement is seen between model and human responses for both movements. Human and model pulse and step responses are compared over a wide range of similar stimulus amplitudes in Figs. 9 and 10. The pulse responses show good agreement except for the last portion of the human responses, where the movement fails to return to the prestimulus level. This is probably due to some adaptation process (perhaps retinal bleaching) not covered by the model. The step responses of Fig. 10 clearly demonstrate pupillary escape and its associated amplitude-dependent nonlinearity in both human and model behavior. The final level reached by model-step responses is influenced by the expansive-range nonlinearity of the plant submodel and follows the general trend seen in human responses. An additional test of the response of the model to stimulus conditions for which it was not specifically developed is shown in simulations of the “comb” experiment [24]. In this experiment a series of light pulses are presented in a rapid pulse train, and the resultant pupillary constriction has the general appearance of a step response, but without the escape behavior normally seen in step responses of equivalent amplitude [27]; see Fig. 1 l(a).

0

2

1 TIME

3

4

lsecl

FIG. 9. Human and simulated light-reflex pulse responses. Lower: Averaged light reflex pulse response to 7 increasing (as a power series) stimulus amplitudes [7]. Upper: Simulation results of the light-reflex model to a similar sequence of pulse-stimulus amplitudes (plotted with latencies added for comparative purposes).

SIMULATION

MODEL

OF HUMAN

PUPIL

LIGHT

REFLEX

19

FIG. 10. Human and simulated light-reflex step response. Lower: Averaged light-reflex responses to the same stimulus amplitudes in Fig. 9 [7]. Upper: Simulation results of the light-reflex latencies added

model to a similar sequence for comparative purposes).

I

of step stimulus

I

I

I

I

amplitudes

(plotted

with

,

123456 TIME

(set)

FIG. 11. Human and simulated light-reflex responses to “comb” stimulation-a 4-set series of light impulses delivered at a rate of 10 per second. Upper: Single human light-reflex response [27]. Lower: Model simulation of an equivalent stimulus input.

20

JOHN L. SEMMLOW

AND DAWG C. CHEN

In the model, a repeating pulse input results in continued output from the high-pass filter, which after rectification by the direction-dependent element and shaping by the dynamically limiting elements in the plant produces an escape-free output, [Fig. 1 l(b)]. Thus, experimental results from the “comb” experiment support a model configuration wherein the high-pass filter responsible for pupillary escape precedes the direction-dependent element-which, in turn, precedes the dynamically limiting elements. DISCUSSION Each process in the overall light reflex model of Fig. 6 was justified the basis of either known physiology or behavior features, and an effort

on has

been made to relate these processes, at least grossly, to underlying physiological mechanisms. Thus, model elements have been identified as belonging to either sensory or motor structures. With regard to model performance the specific arrangements

configuration are

of each submodel

physiologically

more

is not unique; reasonable.

Plant

however, processes

certain have

been previously associated with muscular components, and there is evidence that nonlinear rate-dependent process is due to muscle activation-deactivation characteristics [ 191. Therefore, it is reasonable that this element precede other plant elements. As noted previously, many of the light-reflex behavioral features are analogous to well-known visual sensory phenomena implying a similar signal pathway for some of the major sensory processors. On the basis of critical-duration experiments. Webster [3 1] suggests that this shared pathway extends beyond the receptor level to deeper retinal structures, and from adaptation experiments Ohba and Alpern [13] find evidence for a common neural pathway at least through the stages mediating retinal sensitivity adjustment. In our model the features associated with visual perception (the Weber-Fechner law, the Weber fraction, Bloch’s law, Talbot’s law, and long-term

adaptation)

are represented

by initial

sensory

processes.

Those

unique to pupillary behavior (pupillary escape, escape nonlinearity, and sensory asymmetry) are represented by the final two sensory processes. The nonlinear short-term adaptation process accounts for both sensory asymmetry and critical duration, the former being unique to pupillomotor activity while the latter is in common with visual perception. Thus, we divide this process into a linear short-term adaptation element. representing critical duration (and pupillary escape), and a nonlinear gain, simulating sensory asymmetry. The light reflex signal pathway is presumed common to the visual pathway up to point A in the model (Fig. 6). Further correlation between model configurations and the light-reflex neuroanatomical pathway may be developed from the binocular interaction experiments of Baker [2] and Varju [28]. In these seemingly contradictory experiments Baker concluded that the “important” nonlinearities followed

SIMULATION

MODEL OF HUMAN

21

PUPIL LIGHT REFLEX

the merger of the light-reflex signals from the two eyes, while concluded the “essential” nonlinearity must precede the summation For Baker, using amplitude-saturation

pulse stimulation, element, while

the “important” for Varju, using

Varju point.

nonlinearity was an sinusoidal modulated

light stimulation, the “essential” nonlinearity was the demodulation or rectifying operation performed in our model by nonlinear gain simulating sensory asymmetry. Both these conclusions can be rationalized in our model by assuming binocular summation occurs at point B (Fig. 6) since demodulation of sinusoids occurs before this point and amplitude saturation afterwards. Alternatively, summation could occur at point X, since the expansive-range nonlinearity in the plant also provides a saturation-like operation. Further experimentation comparing monocular and binocular step responses, particularly variation in the pupillary escape nonlinearity, over carefully controlled response ranges would clarify and confirm this hypothesis. Though and

it is tempting

postretinal

mechanisms,

to further

divide

extensive

the sensory

neuroanatomical

process

into retinal

attribution

absence of adequate supporting evidence serves no purpose. at further physiological correlation is made at this time. Further analysis of the sensory submodel configuration

in the

Thus, no effort (Fig.

6) shows

that the lower pathway dominates the light-on response while only the upper pathway is active in the light-off response. This separation of the two sensory pathways is necessary for representation of the dissimilar behavior produced by the two stimulus conditions, a feature which has led previous investigators to imply separate mechanisms. Thus, the term “light reflex” is often applied only to the movement produced by increasing light stimulation, while the movement to decreasing light stimulation is termed the the low gain of the upper pathway results “darkness reflex”. Furthermore. in a slower and smaller response to off-light, and many previous investigators and modelers have been inclined to ignore this less dramatic, light-induced movement. While our model represents both movements as a result of a single system, the difference in sensory handling of positiveand negative-going signals lends support to a dualistic approach to pupillary light-reflex studies. In our earlier discussion. and subsequent representation of the pupillary escape phenonenon and its associate nonlinear behavior, we have implicity assumed that the small-signal response is a piecewise linear response with a saturation nonlinearity active for large-signal amplitudes. Since the largesignal response approximates a simple. nearly exponential function, it may be argued that this is the linear response with some nonlinear process contributing to the small-signal behavior. In fact, the overall behavior is nonlinear, and the choice of the linear response region is somewhat arbitrary in the light of currently available data. Accordmgly, our representa-

22

JOHN L. SEMMLOW

AND

DAWG

C. CHEN

tion of this behavioral phenomenon was chosen in keeping with the intuitively appealing concept of small-signal linearity, and for its structural simplicity (a single saturation element). Model parameter values have been adjusted to provide a reasonable match to the pupillomotor data presented. Each parameter is associated with a particular model element, and its value is presented with the element in Fig. 6. No formal parameter-sensitivity analysis is included, since we believe little additional information can be gained from such a study. Variations in a given element’s parameters will influence the behavioral features identified with that element; but without additional pupillomotor data, particularly intersubject variability, parameter variation cannot be linked to behavior changes. Similarly, no formal criterion for the “goodness of fit” between experimental and model results has been used here, since the accuracy of currently available pupillomotor response data does not justify such a rigorous approach. However, future pupil-model and experimental efforts will be directed towards quantitative refinement of model parameters. Though our model lacks the elegant simplicity of some previous models, each element is required to simulate one of the many behavior features which make up the complex pupillomotor light reflex. Parsimony is observed in the multiple behaviors represented by some of the elements. Thus, the nonlinear short-term adaptation accounts for critical duration, pupillary escape, and the low-frequency portion of the sensory-frequency characteristic, while the sensory low-pass filter accounts for Bloch’s law and the high-frequency portion of the sensory frequency characteristic. Finally, all model components describe physiologically realizable processes based on the known properties of receptors, nerve, and muscle tissue. CONCLUSION We have presented the development, structure, and simulation performance of a systems-type representation of the human pupil light reflex. A major objective of this effort was to provide an effective research tool which could be used for organizing previously identified behavioral features and for establishing possible links between these features and underlying physiological processes. To that end, a systems-type model was chosen, emphasizing the behavioral characteristics of individual processes, their influences, and their interactions, without regard to specific neurophysiological or neuromuscular mechanisms. Also, we have organized the major pupillomotor phenomena into two broad classes: sensory- and motor-produced behavioral features. Sensory behaviors were further divided into those having analogous features in visual perception and those unique to pupil activity. Specific characteristics were then described and classified into the appropriate categories.

SIMULATION

MODEL

OF HUMAN

PUPIL

LIGHT

23

REFLEX

Model components were generated largely by representing certain of the but physiologically reasonable, isolated features with independent, processes. Implicit in this approach is the assumption that those isolated behaviors chosen were a reflection of processes which can be uniquely identified and independently represented. This assumption can be tested, however, through experimentation designed to produce modifications in a particular process-experiments such as intersubject comparisons, drug studies, and multi-input experiments, particularly those utilizing the two retinas. Model performance in simulating pupillary activity produced by a wide variety of stimulus conditions was shown to match human data reasonably well, not only for those behaviors around which the model was developed, but for overall, large-signal light-reflex responses as well. Future simulation work will be done not only to expand the model’s range of application to include, for example, long-term adaptation phenomena, but, more importantly, to direct future experimental efforts. As pupillomotor experimentation becomes increasingly more sophisticated, involving multiple stimuli, null experiments, complex stimulus patterns, and drug effects, the model should prove particularly useful in relating the resultant motor activity to specific motor or sensory processes. The authors wish to thank Professor L. Stark for his valuable advice, and the staff of the IBM 1800 Computer, University of Illinois, Computer Center for their help. REFERENCES

1 M. Alpern 2 3

and N. Ohba,

The effect

of bleaching

and

backgrounds

on pupil

size,

&s/on Res. 12, 943-951 (1972). F. Baker, Pupillary response to double-pulse stimulation; a study of nonlinearity in the human pupil system, J. Opt. Sot. Amer. 53, 1430-1436, (1963). H. Bouma, Receptive systems mediating certain light reactions at the pup11 of the human eye, Philips Res. Rept. Suppl. No. 5, Eindhoven. Holland, 1965.

4

M. Clynes, Toward a view of man, in Biomedical Engineering Systems (M. Clynes and J. Milsum, Ed.), McGraw-Hill, New York 1970.

5

M. Clynes, Unidirectional rate sensitivity: a biocybernetic law systems as physiologic channels of control and communication. 92, 946-969 (1961). C. H. Graham. Vision and Perception, Wiley. New York, 1956. D. Hansmann, Human pupillary mechanics: physiology and Calif., Berkeley, 1972. R. N. Linebarger and R. D. Brennan, Digital analog simulator

6 7 8

of reflex and humoral

Ann. N. Y. Acad. Sci.

control,

Thesis,

programs,

Univ.

Simulation

3, 27-36 (1964). 9

Irene Loewenfeld, Pupillary movements associated with light and near vision: an experimental review of the literature, in Recent Decelopments in Vision Research, (M. Whitcomb, Ed.), NAS-NRC Pubi. No. 1272, 1966.

24 10 11 12 13 14

JOHN

L. SEMMLOW

AND

DAWG

C. CHEN

0. Lowenstein and Irene Loewenfeid, The pupil, in The Eye , Vo/. III (H. Davson, Ed.), 2nd. ed. Academic, New York, 1969. 0. Lowenstein and Irene E. Loewenfeld, Electronic pupillography. A new instrument and some clinical applications, Arch. Ophthalmol. 59, 352-363 (1958). P. Nathan,

and J. Turner,

(1942) Efferent

65, 343-35 1 (1942). N. Ohba and M. Alpern Adaptation (1972). A. Sandberg and L. Stark, Wiener characteristics

of the human

pathway

for pupillary

of the pupil light reflex, G-function

analysis

contraction,

Bruin

Vision Res. 12, 953-967

as an approach

to nonlinear

pupil light reflex, Brain Res. 11, 19&211 (1968).

18

J. Semmlow, Nonlinear pupil mechanisms, Thesis, Univ. Ill. Med. Center, Chicago, 1970. J. Semmlow, D. Cahn, and L. Stark, Pupillatory escape: bioengineering analysis permits identification of the mechanism behind a clinical phenomenon, Internal Rep., Univ. of Calif., at Berkeley, 1973. J. Semmlow, D. Hansmann, and L. Stark, Variation in pupillomotor responsiveness with mean pupil size, Vision Res. 15, 85-90 (1975). J. Semmlow, and L. Stark, Pupil movements to light and accommodative stimulation:

19

a comparative study, Vision Res. 13, 1087-l 100 (1973). J. Semmlow and L. Stark, Simulation of a biomechancial

20

Math. Biol. 11, 109-128 (1971). S. F. Stanten and L. Stark, A_ statistical

21

BME-13, L. Stark,

22

IRE IRE-47, 1925-1939 (1959). L. Stark, Biological rhythms, noise, and asymmetry

15 16

17

23 24 25 26 27 28 29 31 32

14%152 (1966). Stability, oscillations

analysis

model of the human

of pupil noise, IEEE

and noise in the human

Trans. Eioeng.

pupil servomechanism,

in the pupilretinal

pupil,

control

Proc. system,

Ann. N. I’. Acud. Sci. 98, 1096-1108 (1962). L. Stark, The pupillary control system: its nonlinear adaptive and stochastic engineering design characteristics, Automatica 5, 655-676 (1969). L. Stark, personal communication, 1970. L. Stark and P. M. Sherman, A servoanalytic study of consensual pupil reflex to light, J. Neurophysiol. 20, 17-25 (1957). A. Troelstra, Detection of time-varying light signals as measured by the pupillary response, J. Opt. Sot. Am. 58, 685-690 (1968). S. Usui, Functional organization of the human pupillary light reflex system, Thesis, Univ. of Calif., Berkeley, 1974. D. Varju, Human pupil dynamics, in Proc. Int. School of Phys. (W. Reichardt, Ed.), Academic, New York, 1969, pp. 442464. J. Webster, Pupillary light reflex: development Biomed. Eng. BME-18, 187-194 (1971). J. Webster, Critical duration of the pupillary 1473-1478 (1969).

of teaching light

reflex,

models, J.

IEEE

Opt. Sot.

Truns Am.

59,

J. Webster, G. H. Cohen, and R. M. Boynton, Optimizing the use of the criterion response for the pupil light reflex, J. Opt. Sm. Am. 58, 419424 (1968).