Equivalence between temporal frequency and modulation depth for flicker response suppression: analysis of a three-process model of visual adaptation

Eisner et al. Vol. 15, No. 8 /August 1998 /J. Opt. Soc. Am. A 1987

Equivalence between temporal frequency andmodulation depth for flicker response

suppression: analysis of a three-processmodel of visual adaptation

Alvin Eisner

Neurological Sciences Institute, Oregon Health Sciences University, 1120 N.W. 20 Avenue, Portland, Oregon 97209

Arthur G. Shapiro

Department of Psychology, Bucknell University, Lewisburg, Pennsylvania 17837

Joel A. Middleton

Department of Psychology, Brown University, Providence, Rhode Island 02912

Received November 13, 1997; revised manuscript received March 19, 1998; accepted March 23, 1998

We analyze adaptation processes responsible for eliciting and alleviating flicker response suppression, which isa class of phenomena characterized by the selective reduction of visual response to the ac component of a flick-ering light. Stimulus conditions were chosen that would allow characteristic features of flicker response sup-pression to be defined and manipulated systematically. Data are presented to show that reducing the sinu-soidal modulation depth of an 11-Hz stimulus can correspond precisely to raising the temporal frequency of afully modulated stimulus. In each case there is a nonmonotonic relation between flicker response and dc testilluminance. The nonmonotonic relation cannot be explained by adaptation models that postulate multipli-cative and subtractive adaptation processes followed by a single static saturating nonlinearity, even when tem-poral frequency filters are incorporated into such models. A satisfactory explanation requires an additionalcontrast gain-control process. This process enhances flicker response at progressively lower temporal re-sponse contrasts as the illuminance of a surrounding adaptation field increases. © 1998 Optical Society ofAmerica [S0740-3232(98)00308-1]

OCIS codes: 330.5510, 330.7320, 330.6790, 330.4060, 330.1800.

1. INTRODUCTIONThe visual system’s response to flicker often is highly lin-ear; the summed response to two flickering stimuli equalsthe sum of the responses to those two stimuli separately.1

Flicker response linearity provides the cornerstone ofphotometry2 and is essential for the widespread applica-tion of linear systems theory to the temporal aspects of vi-sual function.3 Thus any failures of flicker response lin-earity need to be specified and explained as well aspossible. Addressing this need is particularly importantwhen such failures are not due merely to compressivelynonlinear response behavior, which typically would be in-distinguishable from linear response behavior nearthreshold. Any nonlinear response behavior that is evi-dent near threshold may be especially useful for dissect-ing the visual system’s component response and adapta-tion properties.4 The present paper concerns such near-threshold nonlinear response behavior, manifested as aform of flicker response suppression.

Flicker response suppression is the name given to aclass of phenomena characterized by the selective reduc-tion of visual response to the ac (i.e., alternating) compo-nent of a flickering stimulus.5 Flicker response suppres-sion usually is assessed psychophysically by raising the

0740-3232/98/081987-16$15.00 ©

dc (i.e., time-averaged) illuminance of a flickering teststimulus at a constant temporal contrast, typically near100% modulation depth, until flicker is perceived.6–9 Inthis paradigm, flicker threshold is defined to be the dc testilluminance at which flicker first becomes perceptible.Other psychophysical paradigms have been used also.9–12

Flicker response suppression has been studied mostlyat low ambient illumination, both physiologically13–18 andpsychophysically,6–9,19,20 and can be identified psycho-physically by elevated flicker thresholds. Eisner5,21 ex-tended the usual psychophysical paradigm by continuingto increase the dc test illuminance beyond flicker thresh-old at higher ambient light levels. He showed that whenthe illuminance of a surrounding adaptation field washigh enough to reduce flicker threshold by more than alog unit, it still was possible to make flicker vanish byslightly increasing the dc illuminance of a circumscribedtest stimulus. In other words, not only could flicker re-sponse suppression exist at ambient illumination levelshigher than anticipated, but the relation of the flicker re-sponse to the dc light level of a constant-contrast teststimulus could be so nonlinear as to be nonmonotonic,even near threshold. The nonmonotonic relation be-tween flicker response and dc test illuminance needs to be

1998 Optical Society of America

1988 J. Opt. Soc. Am. A/Vol. 15, No. 8 /August 1998 Eisner et al.

explained. It may provide an especially useful means fordissecting the visual system’s component response andadaptation properties.

This paper examines the extent to which conventionalmodels of visual adaptation can explain flicker responsenonmonotonicities. In general, any adaptation processthat altered the response of the visual system at anygiven instant would affect response to flicker. Thus theentire class of response and adaptation models that con-tain instantaneous static nonlinearities would potentiallybe germane to the study of flicker response. The mostwidely used set of such models, known by their acronymas MUSNOL models,22–24 postulates two kinds of adapta-tion processes—referred to as ‘‘multiplicative’’ and‘‘subtractive’’ processes respectively. These processes arehypothesized to occur in successive stages that precede asingle instantaneous static nonlinearity.22–24 MUSNOLmodels were developed to account for the regulation of re-sponse to nonflickering stimuli.

An actively investigated question is whether MUSNOLmodels can account for the regulation of flicker responseas successfully as they can account for the regulation ofresponse for flashed stimuli. Since MUSNOL modelswere developed to account for the response to flashedstimuli, they would not include any adaptation processesthat selectively affect temporally modulated response.Nor would initial formulations of such models concern theability of temporal modulation to alter visual adaptation.MUSNOL models might not contain enough elements toaccount for the regulation of response to flickeringstimuli, even after modification to incorporate temporalfilters.

The first attempts to apply MUSNOL models to theregulation of flicker response naturally addressed the as-pects of flicker response that may be regarded as the mostconventional.22,25 Subsequent investigations addressedthe ways in which flickering adaptation stimuli affect de-tection of nonflickering stimuli.26,27 None of these stud-ies has addressed flicker response suppression. Con-versely, studies of flicker response suppression, withexceptions,11,16,28 have tended to evolve independently ofother types of adaptation studies. The analysis in thepresent paper was developed to identify some of themechanisms responsible for flicker response suppressionand to reconcile these mechanisms with conventionalmodels of visual adaptation and flicker response to the ex-tent possible. The emphasis is on MUSNOL models.

In principle, flicker response could become suppressedyet remain linear. For instance, flicker response sup-pression could result from the imposition of a temporal fil-ter. However, under the stimulus conditions used byEisner,5,21 it does not. For these conditions, flicker re-sponse suppression has some unique and well-definedproperties. For example, flicker threshold to a smalllong-wavelength test stimulus decreases abruptly at asufficiently high surround illuminance, which can bespecified to within 0.1 log unit.5,21 At test illuminancessufficiently above this new reduced threshold, flickerdisappears.5,21 The work described in the present paperwas designed to exploit these and other singular proper-ties to reveal adaptation processes that cause or allowflicker response to become measurably nonlinear with

psychophysical techniques. This strategy presumes thatthe component adaptation processes can be dissected attheir functional limits, where small stimulus changes canbe leveraged into large psychophysical effects.

This paper quantifies, extends, and discusses relationsreported in a previous paper.5 In that paper Eisner5

showed that the adaptation field illuminances at whichflicker response nonmonotonicities could be expressedpsychophysically depended critically on the modulationdepth and temporal frequency of the test stimulus. Aseither temporal frequency increased or modulation depthdecreased, progressively higher surround illuminanceswere required for flicker threshold to decrease abruptly.In the present paper, new data are presented regardingthese progressions as well as the accompanying flickerthreshold changes. We show that fully modulatedstimuli at relatively high temporal frequencies can be re-garded as operationally equivalent in a precisely quanti-fiable way to partially modulated stimuli at a fixed lowertemporal frequency. This operational equivalence pro-vides a necessary condition for applying MUSNOL modelsto flicker response suppression data. However, we showthat because this equivalence requires the action of an ad-aptation mechanism that acts as a temporal contrast gaincontrol, the data cannot be explained by a MUSNOLmodel to which only temporal filters are added.

2. METHODSSubjects. Two subjects were tested. Both were healthymale emmetropes with normal color vision. Subject TQN(age 34 years) was highly experienced at the flickerthreshold tasks conducted for this paper. However, hewas unaware of the purpose of the experiments. SubjectJAM (age 21 years) was inexperienced as a psychophysi-cal subject; he had about a dozen practice sessions prior todata collection for this paper. Subject JAM was aware ofthe purpose of the experiments; however, neither he norTQN was informed of the temporal parameters for anythreshold excursion.

Apparatus. The apparatus used for testing was aMaxwellian view with attached bite bar assembly. Theapparatus has been described in detail previously.5,29

Two channels were used, one to provide a flickering teststimulus and the other to provide an adaptation stimulusin the form of an annular surround. The light source forthe surround stimulus was a 450-W xenon arc (OSRAM).The light source for the flickering test stimulus was along-wavelength LED (Marktech MT 5000-UR, peakwavelength 662 nm, half-height bandwidth 22 nm). Thepsychophysical flicker stimulus waveform was controlledby pulse-density modulation of a train of constant-amplitude 1.85-ms pulses. The exit pupil of the appara-tus was 1.68 mm. Calibration was as described previ-ously for narrow-band stimuli.29 Relative illuminancecalibrations in the two channels were verified by bipartitecolor matching.

Stimuli. All stimuli were centered on the fovea andimaged in focus through the pupil’s optical axis. Thestimulus dimensions and temporal presentations wereidentical to those used previously.5,21 The test stimuluswas a 138-diameter disk. The annular surround had an


inner diameter of 188 and an outer diameter of 1°. Thetest stimulus was presented for 750-ms ON intervalsseparated by 1500-ms ‘‘off ’’ intervals. The surroundstimulus was always present. The nominal wavelengthof each of the two stimuli was 660 nm; the wavelengthswere created by placing separate 660-nm interference fil-ters (three cavities, half-height bandwidth of 10 nm,Ditric Optics) in a collimated portion of each of the twolight paths. The test stimulus was flickered sinusoidallyat temporal frequencies that ranged from 11 Hz to as highas 25 Hz (but usually to ;20–22 Hz). At 11 Hz, modu-lation depth ranged from 51% to as low as 13%. Athigher temporal frequencies, the modulation depth was99.5%.

Procedure. As in previous studies,5,21 subjects darkadapted monocularly for 1 h before any testing session.Similarly, after any change of surround illuminance, sub-jects viewed the new surround for 3 min before introduc-tion of the test stimulus. Whenever surround illumi-nance was changed, it was incremented by 0.10 log unit(with a single exception regarding the lowest surround il-luminance in Fig. 1 below).

Thresholds were determined with the same methodol-ogy used previously.5,21 At any given dc test illuminance,the flickering test was presented for four successive750-ms intervals. Then the dc test illuminance was in-cremented by 0.10 log unit to the next test illuminance.Threshold for detection of flicker was defined as the testilluminance for which the subject first reported seeingflicker for at least three of the four presentations. Afterthe threshold for flicker detection was reached, test illu-minance was incremented further until flicker became in-visible for at least three of the four stimulus presenta-tions. The test illuminance at which this happened wasdefined to be the vanishing threshold, or the threshold forthe disappearance of flicker.

After either the vanishing threshold was attained orthe expected flicker detection threshold had been ex-ceeded by at least 0.5 log unit with flicker still undetec-ted, the next stimulus condition was introduced. That is,the temporal parameters of the test were changed and/orthe surround illuminance was incremented. Flickerthreshold expectations were based (1) on the knowledge(Ref. 5 plus preliminary results) that flicker thresholdwould decrease profoundly and abruptly as a function ofsurround illuminance for any of the temporal frequency/modulation depth combinations used and (2) on prelimi-nary results that showed that the flicker threshold versussurround illuminance slope would not exceed unity on alog–log plot when either temporal frequency was raised ormodulation depth was reduced, as described in the nextparagraph.

Thresholds for the detection and disappearance offlicker were obtained for each of two independent se-quences: a temporal frequency sequence and a modula-tion depth sequence. For the temporal-frequency se-quence, modulation depth was fixed at 99.5%, buttemporal frequency could be raised by 1 Hz when sur-round illuminance changed. For the modulation depthsequence, temporal frequency was fixed at 11 Hz, butmodulation depth could be reduced by approximately 0.1log unit when surround illuminance changed. Modula-

tion depths tested were 51%, 40%, 32%, 25%, 20%, 16%,and 13%. Temporal frequency (T) and modulation depth(M) presentations were alternated in an ...MTTM... pro-gression. Within each sequence, parameters were al-tered only after elicitation of a flicker threshold. Thus,for any combination of temporal frequency and modula-tion depth (a Hz, b%), only enough threshold-versus-illuminance (tvi) data were collected to determine the sur-round and test illuminances for which flicker thresholdfirst decreased abruptly and for which flicker subse-quently disappeared. Except for one illustrative ex-ample, full flicker tvi curves were not obtained.

3. RESULTSDefinitions regarding characteristic features of flicker tvicurves. The analyses in this paper depend on propertiesof flicker response suppression that are reflected in char-acteristic features of flicker tvi curves, such as the one il-lustrated in Fig. 1. Figure 1 portrays a full flicker tvicurve for an 11-Hz, 40%-modulation-depth flickering teststimulus. This stimulus combination belongs to themodulation depth sequence used for this paper but is out-side the range of temporal frequencies and modulationdepths that were used for a previous paper5 regarding the

Fig. 1. Flicker tvi curve for 11-Hz, 40% modulation depth sinu-soidally modulated flicker for subject TQN. The lower branch(open squares) is comprised of thresholds for the initial appear-ance of flicker. The upper branch (crosses) is comprised ofthresholds either for the initial appearance of flicker at low sur-round illuminances or else for the reappearance of flicker athigher surround illuminances. [The absence of a cross at a sur-round illuminance of 1.0 log td is due to light level limitations inthe test channel.] The middle branch (filled squares) is com-prised of thresholds for the disappearance of flicker. The regionof flicker tvi stimulus space for which flicker response is sup-pressed and for which flicker consequently is invisible is repre-sented by the shaded area. Appearance corner and disappear-ance corner terminology is defined in the text.


Fig. 2. Continues on facing page.

effect of temporal frequency and modulation depth onflicker response suppression.

The flicker tvi curve in Fig. 1 may be described as hav-ing three branches. The lowest branch (open squares) oc-curs at surround illuminances that are high enough to al-leviate flicker response suppression to a degree thatallows flicker response to reach threshold at relativelylow test illuminance levels. The middle branch (solidsquares) denotes the disappearance of flicker. It occursat surround illuminances that are high enough to elicit alower branch but not high enough to preclude flicker fromvanishing as test illuminance is raised. The upperbranch (crosses) corresponds either to the initial appear-ance of flicker at relatively low surround illuminances orelse to the reappearance of flicker at higher surround il-luminances. This three-branch pattern is characteristicof the results that are obtained for a substantial range oftemporal frequencies and modulation depths.5

Flicker thresholds on the upper branch are mediatedvia middle-wavelength-sensitive (MWS) cones, at leastacross the range of surround illuminances for which theuppermost flicker thresholds remain approximatelyconstant21 before decreasing.30 The upper branch will bediscussed further only in the context of providing con-straints regarding possible sites of flicker response sup-pression for signals originating in long-wavelength-sensitive (LWS) cones.

Flicker thresholds on the middle and lower branchesare for LWS cones21 and constitute the subject matter of

the rest of this paper. Flicker thresholds are likely to bemediated via the same afferent pathways on these twobranches, particularly when the middle branch has notyet diverged much from the lower branch.

For the flicker tvi curve in Fig. 1, thresholds on thelower branch increased slowly as surround illuminanceincreased. On the basis of the data obtained for Refs. 5and 21 and on preliminary data obtained for the presentpaper, it appears that flicker thresholds on the lowerbranch never systematically decrease (as long as testwavelength is restricted to 646 nm or greater21). Simi-larly, for the stimulus conditions used for this paper thelarge decrease of flicker threshold from the upper to thelower branch always is abrupt.

Because the large decrease of flicker threshold alwaysis abrupt, any combination of temporal frequency andmodulation depth can be assigned the coordinate pair ofsurround and test illuminances for which flicker thresh-old abruptly becomes reduced. For the flicker tvi curvein Fig. 1, the combination of 11 Hz and 40% modulationdepth can be assigned the coordinate pair (x, y) log td5 (1.0, 1.0) log td. Since the abrupt decrease is so com-plete, this coordinate pair represents the corner of aflicker tvi curve. For this reason, any collection of suchcoordinate pairs will be referred to as corner data. Thecollection of coordinate pairs for an ordered series ofstimulus combinations traces out a curve, which will bereferred to as a corner data curve. The corner data curveportrays characteristic information from an entire series


Fig. 2. Corner data curves (see text for definition) for the initial appearance (open symbols) and disappearance (filled symbols) of flicker.Data are presented from four consecutive testing sessions, numbered sequentially. Squares, data from the modulation depth sequence;circles, data from the temporal frequency sequence. The fixed modulation depth for the temporal frequency sequence was 99.5%. (a)Results for subject TQN. The temporal frequency at which this sequence began was 14 Hz, and the incremental units were 1 Hz. Thefixed temporal frequency for the modulation depth sequence was 11 Hz, the modulation depth at which this sequence began was 51%,and the decremental units were approximately 0.1 log unit. (b) Same as (a) but for subject JAM. The temporal frequency sequencebegan at 16 Hz, the modulation depth sequence began at 40%, and the maximum dc test illuminance possible was 3.0 log td.

of flicker tvi curves. It can be plotted as a graph of log dctest illuminance (ordinate) versus log surround illumi-nance (abscissa) in the same way that any individualflicker tvi curve can be plotted.

The same corner data terminology will be used to referto coordinate pairs of test and surround illuminances forwhich flicker first disappears (i.e., at the beginning of themiddle branch). For the flicker tvi curve in Fig. 1, thecorner data point for the disappearance of flicker is givenby the coordinate pair (1.0, 1.3) log td.

Corner data curves can be generated for each of the twotypes of stimulus sequences that are used for this paper.One of these sequences—the modulation depthsequence—consists of a series of modulation depths at afixed temporal frequency. The other—the temporal fre-quency sequence—consists of a series of temporal fre-quencies at a fixed modulation depth. Each sequence canbe used to generate corner data curves for both the ap-pearance and the disappearance of flicker. The cornerdata curves from the two sequences can be compared (asin Fig. 2), and a relation between temporal frequency andmodulation depth can be extracted (as in Fig. 3).

Corner data graphs. Figures 2(a) and 2(b) plot cornerdata curves for the two subjects tested. For each subject,data are presented from four consecutive testing sessions.Corresponding to Fig. 1, open symbols represent cornerdata for the appearance of flicker, and filled symbols rep-resent corner data for the disappearance of flicker.Squares represent data derived from the modulationdepth sequence; circles represent data derived from thetemporal frequency sequence. The fixed modulationdepth for the temporal frequency sequence was 99.5%,31

and the fixed temporal frequency for the modulationdepth sequence was 11 Hz.31

Subject TQN’s corner data are plotted in Fig. 2(a). Thecorner data curves for the modulation depth sequence (11Hz, x% modulation depth) always coincided with the cor-ner data curves for the temporal frequency sequence ( yHz, 99.5% modulation depth) obtained at the same ses-sion. Any differences were within small error. The co-incidence was maintained for both types of corner data,i.e., for the appearance and for the subsequent disappear-ance of flicker, respectively. Subject TQN’s corner datacurves all had two features in common. First, test illu-


minance was fairly constant at relatively dim surround il-luminances, particularly for the initial detection of flicker.Second, the ratio of test to surround illuminance was ap-proximately constant at intermediate surround illumi-nances. There may have been an additional tendency fortest illuminance to begin to level off again at relativelyhigh surround illuminances.

Subject JAM’s corner data are plotted in Fig. 2(b). Thedata for JAM were less consistent than the data for TQN,both within- and between-session. At JAM’s first testingsession (panel 1 in Fig. 2(b)], the thresholds for the tem-poral frequency and modulation depth sequences ap-peared to follow separate patterns. The thresholds forthe modulation depth sequence changed relatively little

Fig. 3. Corresponding log modulation depths and log temporalfrequencies, derived from the four sets of corner data in Fig. 2(a)(see text for definition of ‘‘correspondence’’). For each temporalfrequency, error bars are standard deviations about the mean ofthe corresponding log modulation depths. (a) Results for subjectTQN. The solid line represents the least-squares linear regres-sion line through all the data, from 14–20 Hz; the solid circle onthe upper axis represents 11-Hz, 99.5% modulation depth. (b)Same as (a), except that correspondences are derived from thedata in Fig. 2(b) and are for subject JAM, and data are from16–21 Hz.

with surround illuminance, whereas the thresholds forthe temporal frequency sequence increased appreciablyas surround illuminance increased. At the next threetesting sessions this difference was not evident. Withineach of these sessions most of the between-sequence vari-ability appeared to be random. However, at the last twosessions [panels 3 and 4 in Fig. 2(b)] the vanishingthresholds for the highest temporal frequencies (22 Hz)were higher than they were for the lowest modulationdepths.

An operational equivalence between temporal frequencyand modulation depth. The extent to which TQN’s cor-ner data curves coincided suggested that each temporalfrequency could be assigned an operationally equivalentmodulation depth, one for which corner data would be in-distinguishable. However, a correspondence that wouldbe valid for a single session would not apply a priori toany other testing session. The temporal frequency dataand modulation depth data might have progressed at non-corresponding rates with increasing surround illumi-nance, particularly since the corresponding corner datacurves from different sessions did not coincide with oneanother.

The correspondence for within-session data was madefor each of two cases. When the corner data points fortemporal frequency and modulation depth shared thesame surround illuminance [e.g., as they do for the left-most data points in Fig. 2(a), panel 3], that temporal fre-quency and that modulation depth were considered to cor-respond. Otherwise, a temporal frequency was assigneda corresponding modulation depth by linearly interpolat-ing log modulation depth along the log-surround-illuminance axis.

When the mean correspondences derived from all foursessions are plotted along with their standard deviations,the resulting relation between log modulation depth andlog temporal frequency is seen to be highly linear. Thelinear regression line, displayed along with the summarydata for TQN in Fig. 3(a), accounts for 96% of all the vari-ance, between-session and within-session combined (r5 0.98). The regression line nearly intersects the singlepoint (11 Hz, 99.5% modulation depth, represented by thefilled circle on the upper axis) that would belong to boththe modulation depth and the temporal frequency se-quences. This inherently self-equivalent point was nottestable because an 11-Hz, 99.5% test stimulus would nothave elicited an abrupt decrease of flicker threshold forthe test stimulus that we used. Nevertheless, the linearregression line can be used to describe an across-sessionoperational equivalence between modulation depth andtemporal frequency from 11–20 Hz. Only the actualdata, from 14–20 Hz, were used to compute the regressionline, which has a noninteger slope of 23.35 6 0.14.

A similar straight-line relation was obtained for JAM,despite the substantial between-subject and within-subject differences in the raw data. The relation, whichfor JAM was derived from data ranging from 16 to 21 Hz,is graphed in Fig. 3(b). The linear regression line ac-counts for 89% of all the variance (r 5 0.94). The slopeof the regression line is 22.98 6 0.22. As it did for TQN,the linear regression line again passes near the singlepoint that would belong to both the modulation depth and


the temporal frequency sequences. The distance of theregression line from the self-equivalent point is withinstatistical error. In contrast to the raw data, there wasno hint of any systematic difference between the temporalfrequency–modulation depth correspondence derivedfrom JAM’s first testing session [panel 1 in Fig. 2(b)] andfrom subsequent testing sessions.

4. DISCUSSIONThe results in this paper extend and quantify previousresults5 regarding flicker response suppression, which re-fers generally to the selective reduction of visual responseto the ac component of a flickering stimulus. In this pa-per, as in two previous papers,5,21 flicker response sup-pression is manifested by the excessive loss of flicker sen-sitivity that occurs as surround illuminance decreasesand/or as test illuminance increases.

The precise equivalence between temporal frequencyand modulation depth suggests that flicker response sup-pression might depend on the physiologic modulationdepth of a response to the test stimulus. More specifi-cally, flicker response suppression might depend on therelative response magnitudes (i.e., amplitudes or firingrates) about some time-averaged response to the flicker-ing stimulus, with response excursions at progressivelyhigher temporal frequencies corresponding to responseexcursions at progressively shallower modulation depths.If flicker response suppression depends on the magnitudeof response excursions about some dc response, then wemight expect to evaluate flicker response nonmonotonici-ties successfully by using MUSNOL models, which evalu-ate peak (or trough) response magnitudes without neces-sarily considering the exact timing of the visual responsewithin any cycle.

Use of a MUSNOL model would be appropriate only ifthe model were compatible with flicker response non-monotonicities, which are manifested most plainly by thedisappearance of flicker with increasing test illuminance.Initially, a MUSNOL model does appear compatible, for astraightforward and plausible formulation of such amodel can account qualitatively for flicker response non-

monotonicities. In this formulation, reduction of the dcresponse component of the flickering stimulus becomesinadequate for preventing response compression of the accomponent. If there are only two kinds of adaptationprocesses, multiplicative and subtractive,23,24,32 thenflicker response suppression would have to be regarded asresulting from inadequacies of subtractive rather than ofmultiplicative adaptation. This is because flicker re-sponse suppression is alleviated by adaptation processesthat are spatially nonlocal, are slow, and are least effec-tive for those test stimuli that produce the least spectralantagonism.19,21

5. MODIFICATION AND AMENDMENT OFA MUSNOL MODELModification of models that work for flashes. In place ofthe peak increment response criterion that MUSNOLmodels use to account for thresholds to flashes, we as-sume that flicker is detected when the peak-to-trough re-sponse difference reaches a criterion magnitude. Thechoice of a peak-to-trough difference is based on the Kre-mers et al.33 analysis of primate ganglion cell response toflickering waveforms. However, the peak-to-troughchoice is not crucial. Choosing a mean-to-peak or anyother well-defined difference would also work, as long asthe presence of any nonlinear harmonic distortion prod-ucts is sufficiently small or otherwise negligible. Empiri-cally, we already know that responses to harmonic fre-quencies higher than the fundamental frequency do notalter the flicker tvi curve obtained for the fundamentalfrequency itself, at least not near the surround illumi-nances for which flicker threshold abruptly decreases.5

The MUSNOL model for flicker is diagrammed in Fig.4. Following Graham and Hood’s22 nomenclature, we as-sume that the ‘‘front end’’ of the model consists of a‘‘frequency-dependent gain-controlling process.’’ In ourgeneral formulation, this process is composed of a distaltemporal frequency response filter in tandem with a mul-tiplicative adaptation process, represented by g (for gain).The filter transforms the physical modulation depth of asinusoidally flickering test stimulus from m( f ) to a

Fig. 4. Schematic diagram of the MUSNOL model for flicker. The front end of this model is given by a distal temporal frequencyresponse filter m in tandem with a multiplicative adaptation process g (for gain). A subtractive feed-forward adaptation process s isinterspersed between the front end and a static saturating nonlinearity, assumed to be of the Michaelis–Menton or Naka–Rushton typewith half-saturation constant s and exponent n. There is a final temporal frequency filter T. The variable I represents the illumi-nance of the dc component of the test stimulus. The peak-to-trough response difference, R(I), that determines the flicker signal is givenby R(I) 5 @ gI(1 1 m) 2 s#n/$@ gI(1 1 m) 2 s#n 1 s n% 2 @ gI(1 2 m) 2 s#n/$@( gI(1 2 m) 2 s#n 1 s n%.


physiological modulation depth m( f ), where f signifiestemporal frequency. The single filter in our model couldbe composed of an assembly of various types of filters, forinstance as detailed by Wiegand et al.25 The multiplica-tive adaptation process may be able to follow the flicker tosome degree.22,25

The rest of the model is conventional.22,24 A subtrac-tive feed-forward adaptation process, represented by s, isinterspersed between the frequency-dependent gain-controlling process and a static saturating nonlinearity.We assume that the subtractive adaptation process is un-able to follow the flicker. The nonlinearity is of theMichaelis–Menton or the Naka–Rushton type. Its half-saturation constant is designated by s. For our pur-poses, it is immaterial how the model handles negative(i.e., subbaseline) responses, which could occur at re-sponse troughs if there were enough subtractive adapta-tion and the physiologic modulation were deep enough.For generality, the nonlinearity’s exponent is n ratherthan 1, and there is a final temporal frequency filter, T.The variable I represents the illuminance of the dc com-ponent of the test stimulus. Properly, the subtractive ad-aptation process is represented by s(I), a function of I.The dc signal input to the saturating nonlinearity is givenby F(I) 5 g(I)I 2 s(I), which itself is likely to be a posi-tive compressive function of I (i.e., F(I) . 0, dF/d I. 0, and d2F/d I2 , 0). If F is a compressive function,then so is f((F(I)), where f is any other compressive func-tion, such as the saturating nonlinearity in the MUSNOLpathway.

Figure 5 shows graphically how the MUSNOL modelcan yield a peak-to-trough flicker response that is non-monotonically related to dc test illuminance. Figure 5portrays the peak-to-trough response excursion (along theordinate) that results from incomplete modulation at con-stant temporal contrast about three separate dc test illu-minances (on the abscissa). The solid curve represents ahypothetical graph of the dc response output from thesaturating nonlinearity, plotted against I (the dc test il-luminance) in semi-log-x coordinates. The peak-to-trough response excursion, given by Ri , can be nonmono-tonically related to I because the flicker signal can berelatively small at sufficiently low and high test illumi-nances. At low test illuminances the peak-to-trough re-sponse excursion is small simply because I is small. Atintermediate test illuminances the peak-to-trough re-sponse excursion is larger because I is larger and there isrelatively little response compression. However, at suffi-ciently high test illuminances the peak-to-trough re-sponse excursion can decrease as the response compres-sion at trough catches up with the response compressionat peak.

If the multiplicative and/or subtractive adaptation pro-cesses reduced the dc signal sufficiently, then there wouldnot be enough response compression to cause the peak-to-trough response difference ever to decrease. Similarly, ifthe distal physiologic modulation depth were sufficientlyhigh, then there would never be enough response com-pression (in the positive response direction) at the troughto cause the peak-to-trough response to decrease. For ex-ample, if the trough response were constantly zero, thenthe peak-to-trough response would continue to increase as

long as the peak response continued to increase. The re-sponse at trough could become compressed in the nega-tive response direction (not shown on the graph) if therewere enough subtractive adaptation. In that case, how-ever, the peak-to-trough excursion would itself continueto increase, albeit progressively slowly, since the responsecompression at peak and trough would be in oppositedirections.34

With several simplifying assumptions, the dc test illu-minance I0 at which the peak-to-trough response differ-ence becomes maximal can be derived analytically andsubstituted into the equation that defines the model (seeAppendix A). If multiplicative adaptation were able tofollow the flicker, the net effect would resemble the com-bined effect of an additional front-end filter plus an addi-tional higher harmonic response component. However,Eisner5 has shown that adding higher harmonic stimuluscomponents does not materially alter the flicker tvi curveobtained for the fundamental frequency.

Assessment of the modified model. Although a plau-sible and straightforward MUSNOL model can yield a re-lation between flicker response and test illuminance thatis nonmonotonic, on close analysis such a model by itselfappears to be quantitatively incompatible with the data.

The nonmonotonicities generated for n < 1 are tooshallow to be consistent with the nonmonotonicities that

Fig. 5. The solid curve represents a hypothetical graph of the dcresponse output from a saturating nonlinearity as function of dcilluminance I. The intersections of each pair of vertical lineswith the abscissa define the ac excursion at a fixed modulationdepth about each of three test illuminances: low (I1), interme-diate (I2), and high (I3). The intersections of each pair of hori-zontal lines with the ordinate define the corresponding peak-to-trough response excursions, R1 , R2 , and R3 , output by thesaturating nonlinearity. The peak-to-trough response excursionis greatest for the intermediate dc test illuminance. For sim-plicity, the abscissa in this graph represents dc test illuminanceand the ac excursion is for incomplete physical contrast, i.e., for atemporal modulation depth ,100%. Equivalently, the abscissacould have represented the response output by the model’s frontend, in which case the ac excursion would represent the physi-ologic modulation of the front end’s response, which would be in-complete even if the modulation depth of the physical stimuluswere 100%.


are observed. Examples of nonmonotonicities generatedby the model are graphed in Fig. 6(a) for the extreme casewhen there is no subtractive adaptation, i.e., whens 5 0. This implausibly low value for s makes the non-monotonicity maximally steep when all other parametersor functions remain unchanged.35 Large values ofs, which represent nearly complete subtractive adapta-tion, eliminate the nonmonotonicity. When subtractiveadaptation is half-complete, i.e., when s 5 0.5gI, non-monotonicities exist but are extremely shallow, as in Fig.6(b). Even for the steepest function graphed [solid curveline in Fig. 6(a)], the dc test illuminances that correspondto flicker responses only 0.05 log unit below the peak of

Fig. 6. Peak-to-trough flicker responses generated by theMUSNOL model. The different curves represent the peak-to-trough flicker response function for different values of m. To de-rive the curve for m 5 70% in (b), we needed to extend the non-linearity in the MUSNOL model so that it was defined fornegative (i.e., subbaseline) stimuli. We did this by creating anodd-symmetric mirror image across the origin of the same in-stantaneous nonlinearity that we used for positive stimuli. Form 5 80%, the peak-to-trough response function would have be-come undefined at high enough I. The parameters or functionswere (a) g(I) 5 I/(I 1 100 td), s 5 0, n 5 1, s 5 25; (b) g(I)5 I/(I 1 100 td), s 5 0.5@ g(I)#I, n 5 1, s 5 25.

the flicker response function are separated by 0.8 log unit.This value is far greater than the distances actually foundbetween the thresholds for the initial appearance andsubsequent disappearance of flicker36 at the surround il-luminances for which a 0.1-log-unit decrease of modula-tion depth causes flicker to become subthreshold. More-over, any response noise would have huge consequencesfor the model, whereas the actual variation of flicker tvidata within a session often is quite small (see Fig. 1 ofRef. 21).

To generate the graphs in Fig. 6, we obtained or de-rived all parametric values or functions except s from theranges of values reported in the literature.23,37 We chosevalues that were realistic yet would make the flicker re-sponse slopes as steep as possible near the maximum ofthe flicker response function. With these criteria inmind, we chose s 5 25 td, n 5 1, and g 5 100/(1001 I). As long as m is not too large, its precise value isunimportant. The size of the flicker response is virtuallyproportional to m. In other words, the flicker responsecurves that are generated are virtually parallel to eachother. This can be seen by comparing the solid curves,which represent the flicker response functions derived form 5 10%, with the dotted and the dashed curves, whichrepresent the corresponding functions derived for m5 20% and 40%, respectively. The unimportance of theprecise choice of m for small-to-moderate m can be speci-fied and derived analytically from the same initial as-sumptions and equations that were used for Appendix A.

The choice of a small-to-moderate m is dictated by theuse of stimuli that are operationally equivalent to 11-Hzstimuli that have physical modulation depths of 51% atmost. The physiologic modulation depths that exit thefront end presumably are less. They can be greater onlywhen there is relatively little distal temporal filtering andwhen multiplicative adaptation follows the flicker incounterphase with the more distal flicker response.Since there is no discernible effect of temporal frequencyon the superimposition of corner data, at least for TQN,any phase differences probably are not important. Inany case, when m becomes too large, nonmonotonicityfails along with proportionality, as captured by thedotted–dashed curve derived for m 5 70%. To derive thedotted–dashed curve in Fig. 6(b), we extended the instan-taneous nonlinearity to account for negative stimuli, cre-ating an odd-symmetric mirror image across the origin ofthe same instantaneous nonlinearity that we used forpositive stimuli.

Based on unpublished analyses derived from the equa-tions in Appendix A, there are four ways to steepen theflicker response slopes that the model generates near themaximum of the flicker response function (on a log–logplot) for a low-modulation-depth stimulus: (1) the expo-nent n can be raised, (2) the half-saturation constant scan be reduced, (3) s can be reduced (i.e., subtractive ad-aptation can be weakened38), or (4) the gain factor g canbe raised (i.e., multiplicative adaptation can beweakened38). However, even if we steepen the curves byreducing s to 4 (which is the lowest half-saturation con-stant that we could find in the literature24), arbitrarilysetting n 5 1.5, and setting g and s to their extreme val-ues of 1 and 0, respectively, then the dc illuminances that


correspond to flicker responses only 0.05 log unit belowthe peak would be separated by 0.28 log unit at m5 1%. The 0.28-log-unit value probably is too large tobe compatible with the data, because the range of su-prathreshold test illuminances often is as small as 0.2 logunit at the surround illuminance for which flicker thresh-old becomes abruptly reduced. Moreover, the 0.28 valuedepends on assumptions that collectively are entirely im-plausible.

An amendment to the model. To account for the data,a satisfactory model would include at least one additionaltype of adaptation process. Such a process can be iden-tified by reconciling two descriptions of the flicker sensi-tivity changes (for the initial detection of flicker) that oc-cur with increasing surround illuminance. On the onehand, flicker sensitivity increases with increasing sur-round illuminance in the sense that progressively shal-lower modulation depths exceed threshold as surround il-luminance increases. On the other hand, flickersensitivity fails to increase with increasing surround illu-minance in the sense that flicker threshold to a stimulusof any given modulation depth never declines as surroundilluminance increases. The two uses of the term flickersensitivity (i.e., inverse modulation depth and inverse dclevel) are not interchangeable because the near-thresholdflicker response is not linear.

Reconciliation of the two flicker sensitivity descriptionsrequires that regulation of flicker response depend on

modulation depth.39 We postulate that as surround illu-minance increases, the flicker response to progressivelyshallower modulation depths are preferentially enhanced.In other words, a contrast gain function40–42 becomes pro-gressively steeper at low modulation depths as surroundilluminance increases. Of course, the visual system canhave access only to a physiologic correlate of modulationdepth rather than to the actual modulation depth of thephysical stimulus.

The addition of this contrast gain-control process to themodel is parsimonious, for it not only reconciles two sepa-rate effects of adaptation field illuminance, but it can si-multaneously account for the existence of flicker responsenonmonotonicities. Specifically, if the enhancement de-pends on the afferent modulated response output by aMUSNOL pathway, then the enhancement will decreaseas test illuminance increases because the ratio of the ac tothe dc response output by this pathway—i.e., the physi-ologic modulation depth at a proximal site—will decreaseas test illuminance increases. More generally, as provenin Note 43, the ac:dc ratio will decrease for any pathwaythat acts as an instantaneous compressive nonlinearitymonotonically ascended by sustained stimuli. Henceflicker will vanish with increasing test illuminance. Inaddition, the vanishing threshold will increase as sur-round illuminance increases. Both of these consequencescan be read from the graphs in Fig. 7, which portray hy-pothetical flicker response functions that are consistent

Fig. 7. The top graphs collectively represent a family of hypothetical flicker response versus dc test illuminance functions for a teststimulus of fixed modulation depth at a series of progressively higher surround illuminances, given by A, B, and C. The horizontaldashed lines represent the flicker response at threshold. The bottom graph portrays an idealized flicker tvi curve with the three sur-round illuminances, A, B, and C, noted on the abscissa. The region of flicker tvi stimulus space for which flicker response is suppressedand for which flicker consequently is invisible is represented by the shaded area.


with the properties of the postulated contrast gain-controlprocess and with the most frequently encountered pat-terns of actual data.

Figure 7 (top) portrays a family of flicker response ver-sus dc test illuminance functions for a test stimulus offixed modulation depth at a series of progressively highersurround illuminances, designated A, B, and C. Thehorizontal dashed lines represent the flicker response atthreshold. The shape and positioning of the curves arenot entirely arbitrary. If the curves were superimposed,they would nearly overlap at test illuminances that aresubthreshold for all three surrounds and for which flickerresponse is still increasing. This overlap is derived fromthe observation that once the surround illuminance be-comes high enough to reduce flicker threshold abruptly,that threshold often remains approximately constant.5

The steep falloff at relatively high test illuminances is de-rived from frequency-of-seeing data regarding the visibil-ity of flicker at a test–surround-illuminance combinationfor which 99.5% flicker was appreciably suprathresholdbut for which 80% flicker remained subthreshold.44 Thefamily of flicker response versus dc test illuminance func-tions at a fixed surround illuminance for a series of pro-gressively higher modulation depths would look similar toFig. 7 (top), except that there would be additional verticaldisplacement.

By comparison with Fig. 7 (bottom), which portrays anidealized flicker tvi curve, it is clear that the existence ofa nonmonotonicity can itself account for the exceedinglyabrupt reduction of flicker threshold that occurs with in-creasing surround illuminance. The abruptness does notrequire flicker response to increase much as surround il-luminance increases slightly. Whatever the underlyingreasons are for the nonmonotonic response behavior, thereduction will be abrupt as long as flicker response is non-monotonically related to the dc test illuminance and themaximal flicker response is elevated by increasing sur-round illuminance. In this way, small physiologic re-sponse changes can be amplified into large psychophysi-cal sensitivity changes.

Role of spatial factors for the model. The observed ef-fects of surround illuminance on flicker visibility implythat the hypothetical contrast gain-control mechanismmust be spatially extensive rather than strictly local. Onthe basis of previous work21 that used the same light-adaptation testing procedure and 138-diameter test size,there appears to be an adaptation pool with a diameter of;308 or more at the fovea. The use of a small flashedtest stimulus minimizes (but does not eliminate21) theability of the test stimulus to influence the adaptationpool. Had we used a test size larger than 138, the teststimulus itself would be expected to exert a greater effecton the adaptation pool, thereby facilitating its own tem-poral resolution and altering flicker response curves.This expectation follows from a set of corresponding dark-adaptation procedures in which the effect of increasingtest size is progressive19 and is interpretable in just sucha manner. Other investigators have reported effects oftest size and adaptation-field dimensions that are consis-tent with this general view, for cone–cone8 and forrod–cone9,13,16,17,45 interactions.

6. INTERPRETATION OF THE CORNERDATA CURVESAlthough an amended MUSNOL model—one that in-cludes a contrast gain control—can account successfullyfor various attributes of flicker response nonmonotonici-ties, it cannot account for all features of the flicker tvi/suppression data. For instance, the model has no obvi-ous bearing on the shape of the corner data curves. Nordoes the model explain why the corner data for the modu-lation depth sequence can coincide with the corner datafor the temporal frequency sequence, nor why the relationbetween log modulation depth and log temporal frequencyshould be so linear.

Shape of the corner data curves. It is possible to drawsome inferences from several features that appear consis-tently in TQN’s corner data curves.

At relatively dim surround illuminances, flicker thresh-old remained approximately constant as modulationdepth increased. The absence of a flicker threshold el-evation with progressively lower modulation depths itselfconnotes an enhancement of flicker response. This en-hancement is likely to be regulated by a temporal con-trast signal because flicker threshold was held approxi-mately constant. However, the enhancement mechanismthat we identified in the ‘‘amendment-to-the-model’’ por-tion of Section 5 was regulated by surround illuminance.Therefore, we must allow the possibility that a commonflicker response enhancement process is regulated partlyby local temporal contrast and partly by surround illumi-nance. Alternatively, there could be more than one en-hancement mechanism.

At intermediate surround illuminances, the ratio oftest to surround illuminance was approximately constant.This result suggests that the flicker response enhance-ment mechanism might also be regulated partly by spa-tial contrast, or more generally by a spatially weightedcomparison function.

Coincidence of the corner data obtained for the modula-tion depth sequence with the corner data obtained for thetemporal frequency sequence. Since the correspondencebetween modulation depth and temporal frequency wascalculated by use of surround illuminance levels only, ir-respective of any dc test illuminance levels, there is nological necessity for the independent corner data curves tocoincide, or even to parallel one another. Then why dothe corresponding corner data curves coincide so closelyfor TQN?

There may be a straightforward explanation. First,the 11-Hz stimuli must have been detected by the samemechanism that detected the higher-frequency stimuli.Second, the characteristics of any threshold-limiting tem-poral filters must have matched the correspondence be-tween modulation depth and temporal frequency. Thiswould happen if there were a single threshold-limiting fil-ter. This filter presumably would be unaffected by localand nonlocal adaptation changes because the correspon-dence between temporal frequency and modulation depthwas obtained across a substantial range of test and sur-round illuminances. These constraints suggest that thehypothetical filter would be situated distally in the visualpathway, perhaps in the cone outer segments, where tem-


poral response properties would be least affected byadaptation.46,47 Alternatively, there could be several rel-evant temporal filters, in which case the flicker signalprobably would be involved in its own self-regulation, assuggested above, based on the behavior of TQN’s cornerdata at dim illuminances.

Linearity of the relation between log modulation depthand log temporal frequency. The linear relation betweenlog temporal frequency and log modulation depth was im-mune to substantial between-subject and within-subjectdifferences in the raw data sets from which it was ex-tracted. The robustness of the relation plus its simplicitysuggests that it is a manifestation of some aspect of visualprocessing that either is fundamentally important for vi-sion or else is constrained by fundamental biochemical orneural principles. If the linear relation does indeed re-flect the temporal frequency properties of cone outer seg-ments, then it would be regarded as the high-frequency-falloff portion of the modulation transfer function of acone outer segment.

Although the linear relation between log modulationdepth and log temporal frequency had a nearly integerslope for JAM, it did not for TQN, whose data were lessvariable. Thus the linear relation did not result from acascade of first-order low-pass temporal filters all havingthe same time constant.25

An additional consideration. Since flicker responsesuppression can occur selectively for LWS cone-mediatedresponse in our paradigm,21 it is likely that relief of thissuppression occurs no more proximally than the outerplexiform layer, where signals from LWS and MWS coneshave not yet converged. However, we have presentedevidence that enhancement involves the action of a con-trast gain-controlling mechanism, one that is proximal toa saturating nonlinearity in a MUSNOL pathway. Onthe basis of physiological evidence, contrast gain-controlling mechanisms are likely to be located proxi-mally in the retina, for contrast gain strongly affects re-sponses of magnocellular-signaling ganglion cells but notparvocellular-signaling ganglion cells.40–42 Therefore,our results suggest that enhancement depends on regula-tion of response at the outer plexiform layer by signalsfrom the inner retina. This suggestion is consistent withdata presented by Eisner in Ref. 21, where adaptation pe-riods on the order of minutes could be required to demon-strably alleviate flicker response suppression.

7. RELATION TO OTHER STUDIESFlicker response suppression or enhancement. The sup-pression of flicker response at progressively dimmer am-bient light levels and, conversely, the enhancement offlicker response at progressively brighter light levels havebeen studied extensively with psychophysical and physi-ological techniques.5–9,12–21 Many studies have empha-sized the ability of rod–cone interactions to affect flickerresponse suppression.6,7,9,12–14,17,20,45 Because the stimu-lus conditions for our study involved the exclusive use offoveal long-wavelength stimuli and hour-long dark-adaptation periods, the results cannot be ascribed to dif-ferential rod responses at different light levels or times.

The extent to which the various suppression or enhance-ment phenomena involve shared rod and cone pathwaysis unknown. However, all physiologic studies agree thatthese phenomena occur quite distally, at the level of theouter plexiform layer.13–18 It appears that LWS cone-mediated response can be affected either preferentially orselectively,6,8,13,16,17,21,29 although not necessarily for allparadigms or procedures.7,18,20

The physiologic study most comparable to the presentstudy is the Cadenas et al.15 investigation of ‘‘cone-mediated surround enhancement (CMSE)’’ in turtle hori-zontal cells. As in the present study, Cadenas et al. usedconditions for which rod response was effectively heldconstant, and they measured the effect of annular-surround adaptation on flicker response to a centered teststimulus.

Our results and those of Cadenas et al. correspond inseveral important respects. In each case, enhancementof flicker response became less as the ratio of test to sur-round illuminance exceeded a criterion amount, which forCadenas et al. was 0% spatial contrast, i.e., equal bright-ness. In other words, in each study, flicker response canbe regarded as increasing nonmonotonically with dc testilluminance at a given surround illuminance. On largelyseparate grounds, both Cadenas et al. and we suggestthat the inner retina is involved in flicker response en-hancement that occurs at the outer plexiform layer.Cadenas et al. had a variety of reasons for suggesting in-ner retinal involvement. They showed, for instance, thatflicker response can be enhanced by light that is imagedoutside a horizontal cell’s receptive field.

However, our interpretations differ from those of Cade-nas et al. in some important respects, most notably re-garding the trade-off of temporal frequency for modula-tion depth. We found that a low-modulation-depth, low-temporal-frequency stimulus could be equivalent to ahigh-modulation-depth, high-temporal-frequency stimu-lus, whereas Cadenas et al. ascribed their results to ef-fects of temporal frequency only. Cadenas et al. did notvary modulation depth. They instead assumed that thehorizontal cell voltage response would be approximatelylinear with contrast for small perturbations of lightaround a mean level. Thus they would not have identi-fied an effect of modulation depth had one existed.

MUSNOL models. As stated in Section 1, whetherMUSNOL models can account for the regulation of flickerresponse as successfully as they can account for the regu-lation of response to flashed stimuli is a question underactive investigation. The present study shows thatMUSNOL models cannot generally account for the regu-lation of flicker response, essentially because MUSNOLmodels are two-process models that do not capture the ef-fects of a third, fundamentally different type of adapta-tion process. This adaptation process involves temporalcontrast and appears to exist proximal to MUSNOL activ-ity. Because this additional process involves temporalcontrast, it would not be expected to play any role regard-ing the regulation of sensitivity to flashed stimuli, whichMUSNOL models were developed originally to explain.

Wiegand et al.25 also identified a third adaptation pro-cess, which they incorporated into what they called a con-trol Low-Pass Filter (cLPF). The effects of the cLPF


were assigned to the front end of their model, where dis-tal filter characteristics and a multiplicative gain processwere adjusted. By specifying a cLPF, Wiegand et al.were able to reconcile MUSNOL models with the quasi-linear aspects of flicker response that occur for a widerange of stimulus conditions. However, the cLPF did notrepresent a fundamentally different type of adaptationprocess but rather a control mechanism for processes thatalready are incorporated into MUSNOL models. Sincethe cLPF depended mainly on integrated light levels, itwould be unable to deal with flicker response suppression,which is a grossly nonlinear phenomenon that depends ontemporal contrast. More generally, the strong depen-dence of flicker response suppression on temporal con-trast means that flicker response suppression cannot becharacterized by temporal filters, even near threshold.

Subsequently, Hood et al.26 extended the work of Wie-gand et al. by using a probed-sinewave paradigm, inwhich brief test probes were presented as a function ofthe phase and temporal frequency of sinusoidally flicker-ing background fields. Hood et al. found that the flicker-ing background field caused an elevation of the flickerthreshold, called the dc effect, which was separate fromphase-dependent flicker threshold excursions. Hoodet al. concluded that the Wiegand et al. extension of theMUSNOL model could not capture the dc effect unless atleast one more component were introduced to the model.They specified three possibilities: (1) introduction of ad-ditional multiplicative or subtractive adaptation pro-cesses, (2) introduction of an asymmetry between the ‘‘on’’and ‘‘off ’’ responses of a single channel, and (3) introduc-tion of a contrast gain control. Of these three possibili-ties, only the third would help explain our results.

In an even more recent study, Wu et al.27 used a para-digm similar to that of Hood et al. but also systematicallyvaried the modulation depth of the background, whichwas presented in Gaussian envelopes at constant time-averaged illuminances. Like Hood et al., Wu et al. foundphase-dependent threshold changes plus a dc shift thatoccurred at all phases of the background. However, Wuet al. were able to rule out temporal contrast gain-controlexplanations for their data on the basis of the rapiditywith which the dc shifts became evident. Wu et al. alsoproved that MUSNOL models could not account for the ef-fect of modulation depth on phase-dependent thresholdelevations since the range of threshold elevations inducedby shallow-modulation-depth flicker was outside therange of threshold elevations induced for fully modulatedflicker. On the basis of all the evidence, it appears that ahost of adaptation processes may affect the visual sys-tem’s response to flicker. Only some of these processescan be captured in a MUSNOL model.

What makes our study able to identify limitations of aMUSNOL model is that we treated modulation depth asan independent rather than dependent variable, we ma-nipulated test and surround illuminances in small incre-mental steps across substantial stimulus ranges, and wechose stimulus conditions (e.g., small test size and longtest wavelength) that would exacerbate flicker responsenonlinearities. In fact, we do not categorically rejectMUSNOL models. Instead, we show that additionaltypes of adaptation processes must be appended to them

if they are to account for the regulation of flicker re-sponse.

APPENDIX A: DERIVATION OF THE DCTEST ILLUMINANCE AT WHICHTHE MUSNOL MODEL’S PEAK-TO-TROUGHRESPONSE DIFFERENCE BECOMESMAXIMALWe use the same terminology as in Section 5 and Fig. 4.The hypothetical peak-to-trough response difference is re-ferred to as R(I). We assume that m( f ) and T( f ) do notvary with the dc test illuminance I, that g depends onlyon I, and that s is proportional to gI. [The assumptionsthat m( f ) and T( f ) are invariant follow from TQN’s datain Fig. 2(a), which suggest that any such variation is neg-ligible for changes of test illuminance on the order of 0.2–0.3 log unit. The reason for assuming that s is propor-tional to gI is given in the next-to-last-paragraph of thisappendix, with the ideas and terminology that are devel-oped for Eqs. (A9)–(A14)].

We begin with the special case, n 5 1 and s 5 0.Without loss of generality, we assume that R(I) equals

the peak-to-trough response difference output from thesaturating nonlinearity. In equation form,

R~I ! 5 $ gI~1 1 m!/@ gI~1 1 m! 1 s#%

2 $ gI~1 2 m!/@~ gI~1 2 m! 1 s#%. (A1)

Combining terms yields

R~I ! 5 2sgIm/@ g2I2~1 2 m2! 1 2sgI 1 s 2#. (A2)

Taking the derivative of Eq. (A2) and setting it equal tozero yields

0 5 @d~ gI !/dI#@~s 2 2 g2I2~1 2 m2!#/

@ g2I2~1 2 m2! 1 2sgI 1 s 2#2. (A3)

The extreme value (i.e., maximum or minimum) of R(I)occurs at the positive solution to Eq. (A3). This solution,if it exists, is given by

I0 5 s/@ g~1 2 m2!1/2#, (A4)

which is derived from the middle bracketed term in Eq.(A3). If gI is a monotonically increasing function of I,then d( gI)/d I Þ 0, and consequently the solution givenby Eq. (A4) is unique.

A solution to Eq. (A3) will exist at physiologic light lev-els if the modulation is not too full (m is not too near100%) and multiplicative adaptation is not too complete(g is not too small or decreasing too rapidly as a functionof I).

Note that substitution of Eq. (A4) into Eq. (A1) or Eq.(A2) yields a value of R that is independent of g. Inother words, the extreme value R(I0) of the peak-to-trough response difference function is independent of themultiplicative adaptation function.

It is straightforward to show that the d2R/d I2 is nega-tive at I0 when m , 100% and gI is a monotonic functionof I. The negative second derivative implies that R(I0)is a maximum.


For the case n Þ 1 and s 5 0, Eq. (1) is replaced by Eq.(A5):

R~I ! 5 @ gI~1 1 m!#n/$@ gI~1 1 m!#n 1 s n%

2 @ gI~1 2 m!#n/$@ gI~1 2 m!#n 1 s n%. (A5)

If we let z 5 (1 1 m)n 2 (1 2 m)n and y 5 (1 1 m)n

1 (1 2 m)n, then Eq. (A5) can be rearranged to take theform of Eq. (A2). We get

R~I ! 5 s n~ gI !nz/@~ gI !2n~1 2 m2!n 1 s n~ gI !ny 1 s 2n#.

(A6)The derivative of R(I) can be computed to be

d R/d I 5 ~s nz !@d~ gI !n/d I#@s 2n 2 ~ gI !2n~1 2 m2!n#

/@~ gI !2n~1 2 m2!n 1 s n~ gI !ny 1 s 2n#2.

(A7)

When d R/d I 5 0, the solution to Eq. (A7) is the value ofI that satisfies the following equation:

0 5 @d~ gI !n/d I#@s 2n 2 ~ gI !2n~1 2 m2!n#, (A8)

which has the same solution as Eq. (A3) since d( gI)n/d I5 n( gI)n21d( gI)/d I and gI . 0. Thus the extremevalue (i.e., maximum or minimum) of R(I) occurs at thesame test illuminance I0 , defined by Eq. (A4) for the casen 5 1. In other words, I0 is independent of the exponentn. As for the case n 5 1, R(I0) is independent of g.

It is straightforward to show that the d2R/d I2 is nega-tive at I0 when m , 100% and gI is a monotonic functionof I. The negative second derivative implies that R(I0)is a maximum.

We now address the case s 5 kgI, with k a positiveconstant ,1, by reducing it to the case s 5 0. Whenn 5 1, the peak-to-trough response output by the saturat-ing nonlinearity is given by

R~I ! 5 @ gI~1 1 m! 2 s#/@ gI~1 1 m! 2 s 1 s#

2 @ gI~1 2 m! 2 s#/@~ gI~1 2 m! 2 s 1 s#.

(A9)

We can define a new function of h(I) that satisfies

h~I !I 5 g~I !I 2 s~I ! 5 g~I !I 2 kg~I !I. (A10)

The function h, where h 5 g(1 2 k), can itself be for-mally regarded as a multiplicative adaptation function ifthe modulation depth is regarded as increasing from m to

ms 5 m/~1 2 k !. (A11)

Substitution of Eqs. (A10) and (A11) into Eq. (A9) yields

R~I ! 5 hI~1 1 ms!/@hI~1 1 ms! 1 s#

2 hI~1 2 ms!/@hI~1 2 ms! 1 s#, (A12)

which has no explicit subtractive adaptation term but hasa modulation depth ms that depends on the underlyingsubtractive adaptation function. Equation (A12) has pre-cisely the same form as Eq. (A1). Thus derivation of thedc test illuminance at which the peak-to-trough responsedifference becomes maximal proceeds exactly as it did forthe case n 5 1, s 5 0 as long as gI 2 s is a monotonicfunction of I. If ms > 1, then R(I) cannot have a maxi-mal value.

The analysis for n Þ 1 is addressed in a correspondingmanner. Since all adaptation precedes the saturatingnonlinearity, h(I) and ms are defined exactly as they arewhen n 5 1. The equation that describes the peak-to-trough response output by the saturating nonlinearity,which is given by

R~I ! 5 @ gI~1 1 m! 2 s#n/$@ gI~1 1 m! 2 s#n 1 s n%

2 @ gI~1 2 m! 2 s#n/$@ gI~1 2 m! 2 s#n 1 s n%,

(A13)

then can be reduced to

R~I ! 5 @hI~1 1 ms!#n/$@hI~1 1 ms!#

n 1 s n%

2 @hI~1 2 ms!#n/$@hI~1 2 ms!#

n 1 s n%,

(A14)

which has precisely the same form as Eq. (A5).The reason we cannot easily address the most general

case, i.e., when s 5 k(I)( gI) and k(I) is a positive func-tion ,1, is that all derivative equations would containdms /d I terms. Nevertheless, it can sometimes be usefulto think of subtractive adaptation as a form of multiplica-tive adaptation combined with amplification of temporalresponse contrast.

In principle, it is possible to estimate all the param-eters used in Eq. (A13), which represents the most gen-eral equation for describing the model. The parametersg, s, and n all can be estimated using the same probe-on-flash methodologies and curve-fitting analyses that arestandard for use with MUSNOL paradigms.23,24 The ex-istence of two types of flicker thresholds, one for the ap-pearance of flicker and one for the disappearance, pro-vides an equation that allows m to be computed once theother parameters have been estimated at each of the twothresholds.

ACKNOWLEDGMENTSThis research was supported by National Institutes ofHealth grant EY05047, by a grant from Alcon Laborato-ries, and by the Medical Research Foundation of Oregon.When it was conducted, the Neurological Sciences Insti-tute was affiliated with Legacy Good Samaritan Hospitaland Medical Center. Portions of the work were con-ducted while A. G. Shapiro and J. A. Middleton were atLewis and Clark College, Portland, Oregon.

REFERENCES AND NOTES1. D. H. Kelly, ‘‘Flicker’’ in Handbook of Sensory Physiology,

Vol. 7, D. Jameson and L. M. Hurvich, eds. (Springer-Verlag, New York, 1972), pp. 273–302.

2. P. Lennie, J. Pokorny, and V. C. Smith, ‘‘Luminance,’’ J.Opt. Soc. Am. A 10, 1283–1293 (1993).

3. A. B. Watson, ‘‘Temporal sensitivity,’’ in Sensory Processesand Perception, Vol. I of Handbook of Perception and Hu-man Performance, K. R. Boff, L. Kaufman, and J. P. Tho-mas, eds. (Wiley, New York, 1986), pp. 6-1–6-43.

4. S. A. Burns, A. E. Elsner, and M. R. Kreitz, ‘‘Analysis ofnonlinearities in the flicker ERG,’’ Optom. Vis. Sci. 69, 95–105 (1992).

5. A. Eisner, ‘‘Suppression of flicker response with increasingtest illuminance: roles of temporal waveform, modulation


depth, and frequency,’’ J. Opt. Soc. Am. A 12, 214–224(1995).

6. S. H. Goldberg, T. E. Frumkes, and R. W. Nygaard, ‘‘Inhibi-tory influence of unstimulated rods in the human retina:evidence provided by examining cone flicker,’’ Science 221,180–182 (1983).

7. K. R. Alexander and G. A. Fishman, ‘‘Rod influence on coneflicker detection: variation with retinal eccentricity,’’ Vi-sion Res. 26, 827–834 (1986).

8. N. J. Coletta and A. J. Adams, ‘‘Spatial extent of rod–coneand cone–cone interactions for flicker detection,’’ VisionRes. 26, 917–925 (1986).

9. G. Lange, N. Denny, and T. F. Frumkes, ‘‘Suppressive rod–cone interactions: evidence for separate retinal (temporal)and extraretinal (spatial) mechanisms in achromatic vi-sion,’’ J. Opt. Soc. Am. A 14, 2487–2498 (1997).

10. A. Eisner and D. I. A. MacLeod, ‘‘Flicker photometric studyof chromatic adaptation: selective suppression of cone in-puts by colored backgrounds,’’ J. Opt. Soc. Am. 71, 705–718(1981).

11. C. F. Stromeyer, A. Chaparo, A. S. Tolias, and R. E. Kro-nauer, ‘‘Colour adaptation modifies the long-wave versusmiddle-wave cone weights and temporal phases in humanluminance (but not red–green) mechanism,’’ J. Physiol.(London) 499, 227–254 (1997).

12. G. B. Arden and C. R. Hogg, ‘‘Absence of rod–cone interac-tion and analysis of retinal disease,’’ Br. J. Ophthalmol. 69,404–415 (1985).

13. M. Horiguchi, T. Eysteinsson, and G. B. Arden, ‘‘Temporaland spatial properties of suppressive rod–cone interaction,’’Invest. Ophthalmol. Visual Sci. 32, 575–581 (1991).

14. G. B. Arden and T. E. Frumkes, ‘‘Stimulation of rods canincrease cone flicker ERGs in man,’’ Vision Res. 26, 711–721 (1986).

15. I. D. Cadenas, E. S. Reifsnider, and D. Tranchina, ‘‘Modu-lation of synaptic transfer between retinal cones and hori-zontal cells by spatial contrast,’’ J. Gen. Physiol. 104, 567–591 (1994).

16. R. Nelson, R. Pflug, and S. M. Baer, ‘‘Background-inducedflicker enhancement in cat retinal horizontal cells. II.Spatial properties,’’ J. Neurophysiol. 64, 326–340 (1990).

17. T. E. Frumkes and T. Eysteinsson, ‘‘The cellular basis forsuppressive rod-cone interaction,’’ Visual Neurosci. 1 263–273 (1988).

18. N. S. Peachey, K. R. Alexander, D. J. Derlacki, and G. A.Fishman, ‘‘Light adaptation, rods and the human flickerERG,’’ Visual Neurosci. 8 145–150 (1992).

19. A. Eisner, ‘‘Losses of flicker sensitivity during dark adapta-tion: effects of test size and wavelength,’’ Vision Res. 32,1975–1986 (1992).

20. N. S. Peachey, K. R. Alexander, and D. J. Derlacki, ‘‘Spatialproperties of rod–cone interactions in flicker and hue detec-tion,’’ Vision Res. 30, 1205–1210 (1990).

21. A. Eisner, ‘‘Nonmonotonic effects of test illuminance onflicker detection: a study of foveal light adaptation withannular surrounds,’’ J. Opt. Soc. Am. A 11, 33–47 (1994).

22. N. Graham and D. C. Hood, ‘‘Modeling the dynamics of lightadaptation: the merging of two traditions,’’ Vision Res. 32,1373–1393 (1992).

23. M. M. Hayhoe, M. E. Levin, and R. J. Koshel, ‘‘Subtractiveprocesses in light adaptation,’’ Vision Res. 32, 323–333(1992).

24. M. M. Hayhoe, N. I. Benimoff, and D. C. Hood, ‘‘The time-course of multiplicative and subtractive adaptation pro-cess,’’ Vision Res. 27, 1981–1996 (1987).

25. T. E. von Wiegand, D. C. Hood, and N. Graham, ‘‘Testing acomputational model of light-adaptation dynamics,’’ VisionRes. 35, 3037–3051 (1995).

26. D. C. Hood, N. Graham, T. E. von Wiegand, and V. M.Chase, ‘‘Probed-sinewave paradigm: a test of models oflight-adaptation dynamics,’’ Vision Res. 37, 1177–1191(1997).

27. S. Wu, S. A. Burns, A. E. Elsner, R. T. Eskew, and J. He,‘‘Rapid sensitivity changes on flickering backgrounds:tests of models of light adaptation,’’ J. Opt. Soc. Am. A 14,2367–2378 (1997).

28. T. E. Frumkes and S. M. Wu, ‘‘Independent influences oncone-mediated responses to light onset and offset in distalretinal neurons,’’ J. Neurophysiol. 64, 1043–1054 (1990).

29. A. Eisner, ‘‘Losses of foveal flicker sensitivity during darkadaptation following extended bleaches,’’ Vision Res. 29,1401–1423 (1989).

30. We do not know whether the uppermost flicker thresholdsremain mediated by MWS cones at surround illuminancesfor which the uppermost flicker thresholds have systemati-cally decreased. It is possible that at those relatively highsurround illuminances, subthreshold MWS and LWS re-sponses combined to produce a suprathreshold flicker re-sponse.

31. The fixed modulation depth was set at 99.5% rather than100% to avoid potential artifacts resulting from the use ofpulse-density modulation, as discussed previously.5 Thechoices of temporal frequency were constrained by the needto obtain flicker tvi curves with abrupt decreases and by theintent to induce abrupt decreases with changes of surroundilluminance that were on the order of several tenths of a logunit for 0.1-log-unit decrements of modulation depth. Theupper limits of the variable temporal frequency sequencewere constrained mainly by the long duration of individualtesting sessions, particularly for TQN. For JAM we soughtto collect data over temporal frequency and modulationdepth ranges that were as comparable to TQN’s as feasible.

32. M. M. Hayhoe, ‘‘Spatial interactions and models of adapta-tion,’’ Vision Res. 30, 957–965 (1990).

33. J. Kremers, B. B. Lee, J. Pokorny, and V. C. Smith, ‘‘Re-sponses of macaque ganglion cells and human observers tocompound periodic waveforms,’’ Vision Res. 33, 1997–2011(1993).

34. There is a bound on how negative (i.e., how much belowbaseline) the trough response can become at the input tothe saturating nonlinearity. As this trough response ap-proaches 2s n, the saturating nonlinearity approaches asingularity. If the subbaseline response at the input to thesaturating nonlinearity cannot reach 2s n, then there willbe no singularity.

35. The nonmonotonicity can be steeper yet if s(I) Þ kg(I)Ibut instead s(I) 5 k(I)g(I)I, with k(I) being a decreasingfunction of I rather than a constant. However, the degreeof steepening is severely constrained if F(I) 5 g(I)I2 s(I) is constrained to be a positive compressive functionof I.

36. In fact, the illuminance level at the threshold for the disap-pearance of flicker would exceed the illuminance level of astimulus for which flicker visibility would equal that at thethreshold for the initial appearance of flicker. This is be-cause the threshold for the initial appearance of flicker isbased on a three-or-more-of-four detection criterionwhereas the threshold for the disappearance of flicker isequivalent to a one-or-fewer-of-four detection criterion.Therefore the distance between the thresholds for the ini-tial appearance and subsequent disappearance of flickerwould exceed the distance between two equally visiblethreshold-level flickering stimuli.

37. R. W. Massof, S. Marcus, G. Dagnelie, D. Choy, J. S. Sun-ness, and A. Albert, ‘‘Theoretical interpretation and deriva-tion of flash-on-flash threshold parameters in visual systemdiseases,’’ Appl. Opt. 27, 1014–1024 (1988).

38. We assume that gI and gI 2 s are smooth compressivelynonlinear positive functions of I. We define g8 to be astronger multiplicative adaptation function than g if g8I, gI and d( g8I)/d I , d( gI)/d I. Similarly, we define s8to be a stronger subtractive adaptation function than s ifgI 2 s8 , gI 2 s and d( gI 2 s8)/d I , d( gI 2 s)/d I or,equivalently, if s8 . s and d s8/d I . d s/d I.

39. An alternative solution, one in which flicker response wouldbe enhanced at progressively higher test illuminances assurround illuminance increased, is ruled out by the failureof subject TQN’s corner data to shift to higher test illumi-nances across relatively dim surround illuminances.

40. E. A. Benardete, E. Kaplan, and B. W. Knight, ‘‘Contrast


gain control in the primate retina: P cells are not X-like,some M cells are,’’ Visual Neurosci 8, 483–486 (1992).

41. B. B. Lee, J. Pokorny, V. C. Smith, and J. Kremers, ‘‘Re-sponses to pulses and sinusoids in macaque ganglion cells,’’Vision Res. 34, 3081–3096 (1994).

42. B. B. Lee, ‘‘Receptive field structure in the primate retina,’’Vision Res. 36, 631–644 (1996).

43. Proof that the ratio of ac:dc response decreases with in-creasing dc test illuminance I for any pathway that re-sponds instantaneously and compressively to dc stimuli:We denote the dc response output by the pathway as r(I).The ac:dc ratio is given by @r(I 1 mI) 2 r(I 2 mI)#/r(I),where m signifies modulation depth. Since r is compres-sive, r(I 1 mI)/r(I) decreases with I. Similarly, r(I)/r(I2 mI) decreases with I, which implies that 2r(I2 mI)/r(I) also decreases with I. Therefore @r(I 1 mI)2 r(I 2 mI)#/r(I), decreases with I.

44. Specifically, at a dc test illuminance 0.4 log unit above thethreshold for a 99.5% modulation depth test and at a sur-round illuminance 0.1 log unit above that which elicited anabrupt decrease of the flicker threshold to that 99.5%

modulation depth stimulus, 80% modulation depth flickerremained invisible at every flash for a period of at least 2min, whereas the 99.5% modulation depth flicker remainedvisible for at least 30 s before becoming invisible for even asingle flash. This experiment was conducted with 18-Hzstimuli for two subjects (TQN plus one other subject; JAMwas not tested). In contrast, at surround illuminances 0.1log unit below that which elicited an abrupt decrease of theflicker threshold, flicker often was visible for one or two outof four flashes at test illuminances that corresponded toflicker threshold at the higher surround illuminances.This observation was made for TQN and JAM.

45. N. Denny, T. E. Frumkes, and S. H. Goldberg, ‘‘Comparisonof summatory and suppressive rod–cone interaction,’’ Clin.Vision Sci. 5, 27–36 (1990).

46. J. L. Schnapf, B. J. Nunn, M. Meister, and D. A. Baylor,‘‘Visual transduction in cones of the monkey Macaca fasicu-laris,’’ J. Physiol. (London) 427, 681–713 (1990).

47. D. C. Hood and D. G. Birch, ‘‘Phototransduction in humancones measured using the a-wave of the ERG,’’ Vision Res.35, 2801–2810 (1995).

Documents

Equivalence between temporal frequency and modulation depth for flicker response suppression: analysis of a three-process model of visual adaptation