Perception & Psychophysics1973. Vol. 14. No. 3, 421-432

Bloch's law and a temporal integration modelfor simple reaction time to ligh t*

JAMES D. HILDRETH1877 Grand A venue, St. Paul, Minnesota 55105

In a series of experiments designed to determine whether Bloch's law holds for simple RT to low-energy visualstimuli, mean RTs were found to agree with Bloch's law to a close approximation only when a narrow range of stimulusintensities is used. However, they could be accounted for more generally by (1) assuming that detection depends on a"visual response function" (VRF) initiated and maintained by the light stimulus (when the time integral of the VRFreaches a criterion, S detects the light and initiates a response): and (2) the fact that VRF generated by a square-waveflash rises quickly to its maximum, remains at this value for the duration of the flash. and then decays exponentially tozero after flash offset. S continues to integrate the VRF throughout its lifetime, and consequently for a brief stimulus,detection will occur during the exponentially decaying portion of the response-the portion corresponding to "visualpersistence." Finally. when luminances used vary by more than a factor of four, Bloch's law fails to hold. while themodel succeeds. implying that the temporal integration model more generally accounts for RTs. .

Although luminance-duration reciprocity (constantbehavioral effect to stimuli with the same luminance xduration product) has been observed in many thresholdexperiments, e.g., Sperling & Jolliffe (1965), sinceBloch's classical work (Bloch, 1885), the use of RT asthe dependent variable in studies of both luminance andduration dates from an experiment by Raab and Fehrer(1962) in which intensities well above threshold wereused. While some less extensive reports exist, e.g., Lewis(1964) and Sticht (1969), only two other majorinvestigations of this general type have appeared: a set ofexperiments at threshold intensities by Grossberg (1968)and a set by Kietzman and Gillam (1972). In the Raaband Fehrer and the Grossberg studies, luminance wasfixed within any single block of trials and stimulusduration randomly varied over trials within a block. Andin both studies, the conclusion was essentially the same:Bloch's law (that luminance-duration reciprocity existsfor durations less than a critical time period) does nothold in general for RTs. Grossberg reported, however.that Bloch's law (BL) held approximately if RT wasmeasured from the end (offset) of the flash. Grossberg'sdiscussion 0 f a limi ted test by Lewis (1964) in anunpublished PhD dissertation suggested the hypothesisthat the asymmetrical treatment of the two independentvariables (luminance and duration) was responsible forthe negative Raab and Fehrer result. Indeed, this wasLewis's conclusion from his own experiment, in whichtwo flashes of equal energy, interspersed randomlyamong trials of the experiment, led to similar RTdistributions. When only duration is randomized,adaptation level, temporal expectancy, and response set

*Work on this project was supported by the National Instituteof Mental Health (Predoctoral Fellowships I-FOI-MH45396-01and 4-FOI-MH45396-02). I thank John Yellott. Jr. forsuggesting the terminology "visual response function" (VRF)and "temporal integration model," for his editorial advice in thewriting of this paper, and for his encouragement and assistancethroughout this research. Thanks are also due David LaBerge forthe use of his laboratory at the University of Minnesota. wherethis research was conducted.

may vary systematically between groups of trials offi xed Ium inance, confounding any attempt todemonstrate reciprocity. This suggests that Grossberg'sfailure to obtain reciprocity over the range of stimulusenergy used may also have been due to the sameasymmetry in experimental design.

This interpretation is supported both by our results(reported below) and a recent report by Kietzman andGillam (1972). The latter authors designed their firstexperiment to counter exactly this difficulty ofasymmetry. They compared effects on RT of equivalentstimulus intensity and duration combinations, avoidingmanipulation of only stimulus duration. Their resultssupported the conclusion that the BL holds for foveallight flashes of durations up to 11 msec. This rather lowlimiting duration may have been a result of the short(5 min) dark adaptation period used. Our results (below)indicate (1) that BL holds at least up to 32 msec, whenthe set of luminances used spans a factor of 4, (2) thatBL fails when the set of luminances used spans a factorof 10. The failure of BL in the latter case might be dueto loss of the state of dark adaptation caused by thehighest intensities, as the experiment progresses fromtrial to trial. Certainly, one would expect trial-to-trialeffects if intensity varied over a very wide range.

RT, Brightness, and VER Latency

Credit for discovering the relationship between acomponent of response latency and brightness isapparently due to H. Pieron, who, from his experimentson perceptual latency (Liang & Pieron, 1954) and RTand perceived brightness (Pieron, 1920, 1952), suggestedthat response speed, defined as the reciprocal of thereducible segment of response latency (T - To)' shouldbe directly proportional to brightness. (By To is meant apositive constant time with the restriction To < T. By 1.we mean the total response latency.) Brightness (S) hasbeen shown in many magnitude estimation experiments,

421

422 HILDRETH

e.g., Hanes (1949), Hopkinson (1956), Onley (1961),Raab (1962), and Stevens and Galanter (1957), to beclosely fit by a power function of physical intensity (S =QIl). The exponent ((3) of the power function in the caseof a visual stimulus ranges from about .3 to .5,depending on angular size (large to small, respectively)and retinal position (peripheral to foveal, respectively),and has been recognized to be about one-third, as ageneral rule of thumb, cf. Stevens and Galanter (I957).Pieron concluded from his results that speed of visualresponse also varies in approximate proportion to thecube root of stimulus intensity.

Vaughn (1964) provided the first explicitneurophysiological demonstration of this relationship bymeasuring the latency of the visual evoked response(VER) as a function of stimulus intensity. Vaughnobserved that the logarithm of the reduced VER latency,T - To, as a function of the logarithm of stimulusenergy was a linear function with a slope of aboutone-third. Taken together with the results of magnitudeestimation experiments, this result supports Pieron'scontention that "response speed" (the reciprocal of thereduced latency) is directly proportional to brightness;both are described by power functions of physicalintensity with exponents of about one-third.

The power law approach was used by Vaughn et al(1966) in a later experiment, in which the latencies ofthe VER and RT were compared concurrently as afunction of stimulus intensity (Q). Both were well fit bypower functions of intensity with exponents varyingfrom -.29 to -.44. It was concluded that the source ofthis variation was mainly retinal on the basis of studiesof human ERG amplitude (Armington et al, 1960), ofthe latency of discharge of the Limulus ommatidium(Hartline, 1934), and optic nerve' of the conger eel(Adrian & Matthews, 1927), data of the kind typicallyfitted by others with the function [a + b log(Q)] -1.

These data are well fit by power functions withexponents near one-third; cf. Vaughn et al (1966). Ittherefore appears that power functions with exponentsof about one-third are capable of fitting the same kindof data well fit by reciprocal linear functions of log (Q).Further, if brightness is proportional to (T - To )- l(reciprocal reduced latency), these two apparentlyconflicting representations parallel the historicalpsychophysical functions for scaling sensation:Fechner's logarithmic scale implies that visual brightnessis a linear function of log (Q), while Stevens's power lawstates that brightness is proportional to Q{3, where (3 =0.33. Evidently these functions are similar over narrowranges of Q when the exponent 13 is near one-third.

Although the exact mathematical relation ofreciprocal reduced latency (T - T0)-1 and brightness tophysical stimulus intensity is not specified in thetemporal integration model to be described below, wewill show below that this model is compatible withBloch's law for RTs and that the power law relationbetween rQ and Q must be approximately proportional

both to log (Q) and to Q{3 - 1, where the exponent (3must be near one-third, as was found in our experiments.

In summary, there is considerable evidence to supportPieron's hypothesis that the reciprocal reduced latencyof visual detection as measured by RT and VER latencyis inversely proportional to brightness as measured bythe method of magnitude estimation (see also Teichner& Krebs, 1972). It is tempting to infer from this thattemporal summation of transduced luminance (i.e.,"brightness") up to a threshold value determines the(reduced) latency of detection. The discussion below ofa class of temporal integration models is a formaldevelopment of this idea. These models assume that thelight stimulus gives rise to a "visual response function"(VRF), the amplitude of which is an increasing functionof stimulus luminance (e.g., a power function). Theexponential decay model, a member of this class, makesthe additional assumption that the VRF decaysexponentially following stimulus offset, a sort of visualpersistence. It is assumed that a detector mechanismintegrates the VRF, including the decay trace, until,with (reduced) latency T - To, a threshold value isreached.

Of course, if this threshold is never reached, detectioncannot occur. Specification of a VRF and a randomlyvarying threshold therefore allows theoreticalpredictions about threshold data, although such arebeyond the scope of this paper.

METHOD

Procedure

The purpose of the series of three experiments reported herewas to investigate simple RT as a function of stimulus luminanceand duration. A question of particular interest was whether BLholds for simple RT when complete symmetry is maintainedbetween the independent variables of luminance and duration,i.e., when both are presented randomly. Essentially, eachexperimental session consisted of a sequence of simple RT trials,25% of which were catch (no stimulus) trials. Over the remaining75% of the trials, stimulus duration and luminance variedrandomly, as described below. Altogether, there were threeexperiments, corresponding to three different ranges ofluminance. Except for differences in luminance values, theseexperiments were identical in procedure, and the same two Ssparticipated in all three.

The order in which the experiments were run was: for S 1-2,3, 1; for S 2-1,2, 3. Ss were extensively practiced before eachexperiment with new luminance parameters. About six l-hsessions were typically required to reach asymptoticperformance for a single experiment, after which six I-h datacollection sessions were conducted for each.

Stimuli

Stimuli were produced with two fluorescent lamps (GE FBTSDaylight) in each of three fields of a Scientific Prototypetachistoscope. Each field provided one of the luminance levels.The circuits produced nearly rectangular stimuli, which wasconfirmed by photographs of the electrical output of aphotocell. Stimulus duration was controlled by a small computer(CDC 160) and was tested to an accuracy of within .1 rnsec atthe electronic gate of the tachistoscope. Durations used in all

BLOCH'S LAW AND A TEMPORAL INTEGRATION MODEL 423

experiments were 1. 2,4, 8, 16,32,64, and 128 msec, Stimulusluminance was controlled by neutral density filters calibratedwith a Photovolt photometer (520'JE). The stimulus fields werecongruent, circular, subtended 1.6 deg. and appeared white whentransilluminated by the lamp. A dim red dol, 1 deg below thestim.ulus, served as a fixation point. The S. dark-adapted for '12 h,sat In a dark room and viewed binocularly. Head position wascontrolled by a rubber viewport. The response button was asensitive rnicroswitch mounted vertically below a handrest andand closed by pulling action.

Instructions

Table ILuminances by Experiment Number

Experi-ment Luminances C sed (fL)

1 .600 .300 .1502 .150 .075 .0383 .038 .019 .009

Luminances in Terms of L = .6 fL

1 L Ll2 L!42 L/4 L/8 L/163 L/16 L/32 L/64

Instructions to the 5 were to respond as quickly as possible tothe light flashes while maintaining a false alarm (F A) rate of lessthan 4'7<. Experimental Design

Event Sequence

The content and duration of fixed foreperiods and stimulusperiods, stimulus randomization and selection. latencymeasurement and recording, and conditional feedback wer-econtrolled by the computer. The trial event sequence, contingentfeedback, and sequencing control are shown in Fig. 1. Thissequence applied on every trial. Because of the criticalimportance of maintaining equal stimulus characteristics amongfields. and because onset reliability for short-duration stimulimade it necessary. each trial contained a special bulb warm-upperiod of 300 msec in all three fields. The resulting flash wasmasked from S's view by a shutter. The warm-up had the effectof minimizing sequential dependencies in latencies due to lamptemperature, and of increasing onset reliability at I-msecdurations to 100S'(.

Subjects

The author served as 5 1. A paid male college senior served asS 2. Both 5s served in all three experiments.

Dur.U.... (uee)

300 700

lampwn....

Three experiments were conducted, which were identical inprocedure except for luminance values used; these are shown inTable 1 by experiment number. Note that the lowest of thethree intensity values for Experiment j becomes the highest ofthe three intensities of Experiment j + 1 U = 1,2). This affords atest of replicability for these luminances and allows investigationof the effect of the context within which a stimulus appears (asthe brightest or dimmest of the set).

Each experiment consisted of 6 experimental sessions.Essentially, each experimental session consisted of a sequence ofsimple RT trials about 3.5 sec apart. with rest periods betweeneach block (32). Within each block, 25'7< (8) were catch trials (nostimulus). Over the remaining 759'c (24) trials, stimulus durationand luminance varied randomly; permutations of the 32 stimuli(3 x 8 luminance-duration combinations plus 8 catch trials) wereselected randomly and presented to the S. Altogether, withineach session, there were 360 stimulus trials and 120 catch trialsor 15 observed RTs for each of the 24 data points (means) persession. Each experiment of 6 sessions therefore consisted of2,160 observed RTs from stimulus trials and 720 catch trials,with each mean RT data point an average of 90 RTs. Todetermine stimulus intensity, the three stimulus fields were firstequated for intensity to 15 fL and the neutral density filtersinserted into fields of the tachistoscope. Within each session,one-third (6) of the blocks of 32 trials each were run with agiven filter-field assignment. To control for a possible effect ofdifferences in stimulus characteristics of the three fields, thefilter-field assignment (i.e., order) was shifted according to aLatin square design described below. Each session then consistedof 3 such orders of 6-block sets of trials, with the lust block ofeach order discarded as warm-up. This left each sessionconsisting of 3 orders by 5 blocks by 24 stimulus trials. for atotal of 360 stimulus trials per session. .

Analysis of Variance Design

CoIl~1aI_t '."baek .nei Sequenc.1D& Control:;

Fig. 1. The trial event sequence included fixed fore periodswith tones on each trial. Each stimulus or catch trial wasfollowed by feedback contingent on the S's response.

continue MISSto Dext (white: nolletTlal feedback) ,

cene , to next trial

FALS! ALARH ceee reue(apeec.h eeaee to nextf.adback) , trial (afterCOoC.. Co Dut 1 aecoll4)~1.al

In order to control for possible differences in stimuluscharacteristics due to field differences, the filter setscorresponding to, for example, L/4, L/2, and L, as referred tobelow, were rotated between fields each third part of a session.This procedure also made possible a test for order effects as wellas field effects. Data points of each three successive Latinsquares (three sessions) were arranged to comprise 3 orders by 3fields by 3 luminances. The resultant so-called "complete block"was then replicated twice, yielding a total of six sessions perexperiment, allowing an analysis of variance. The analysis,technically termed a "split plot" (Kempthorne, 1952), wasdesigned to test for the effects of field (F), order (0), luminance(L), and duration (0). Recall that of the six sets of 32 trials .collected in each one-third session, the first was discarded aswarm-up, the remaining five kept for analysis. Each set of trialsconsisted of 24 RTs, one to each of the 3 x 8luminance-duration stimulus combinations. Catch trial latencieswere not included in the analysis. Finally. the analysis of

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4~4 HILDRETH

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(L- Q6 FOOTLAMBERTS)

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SUBJECT IEXPERIMENTS 1,2,63(L- 0.6 fOOTLAMBERTS)

180

320

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360 .......-----------------,

8 16DURATION (msec:)

Figs. 2 and 3. The lines, solid, dotted, and solid, represent mean RTs of both Ss for Experiments I, 2, and 3, respectively.Luminance parameters on the right, which decrease successively by a factor of ~ may be converted to footlamberts via Table 1.Open triangles for S 2 are L/16 points displaced upward by 33 msec to show constant delay due to experiment context.

Table 2Estimated False Alarm and Miss Rates

. FAs-was at a cost of considerably increased latencies.Standard deviations (SOS) of RTs are shown in Figs. 4and 5 for Ss 1 and 2, respectively.

Bloch's law (BL) appears to hold approximately formean RTs to the range of luminances examined here.This is indicated in Figs. 2 and 3 by the approximatelyhorizontal arrowed lines between mean RTs to stimuli ofequal Lx D produce, except for the shortest durationsand lowest intensities. It should be noted in connectionwith this, however, that a small constant error orvariance in stimulus duration or luminance would bemagnified in this region because of the logarithmic scalesused for these variables. To show in another way thegoodness of fit of BL, mean RTs have been replotted asa function of Lx D product for Experiments 1-3,respectively, in Figs. 6-8.

variance model assumption of no session-to-session trends wasassured by providing enough practice days (about 6 perexperiment) until mean RT curves had clearly reachedasymptotic levels.

,EXPERIMENTAL RESULTS

Mean RTs as a function of stimulus duration (D) andintensity (L) are shown in Figs. 2 and 3. Each data pointrepresents an average of 90 RTs. Arrows are drawnbetween mean RTs for stimuli of equal energy (L x Dproduct); BL holds to the degree that they arehorizontal.

Note that for both Ss the same luminances, L/4 andL/16 gave rise to different latencies, depending onwhether they were the highest or lowest intensities ofthe experiment. This seems to be a constant additionwith respect to duration (0). This is shown most clearlyin the case of L/16 for S 2 (see caption). Here it appearsthat a constant of 33 rnsec has been added to mean RTsas a result of the context of Experiment (E) 3. Wesuggest below that this may be due to a greater degree ofdark adaptation with lower luminances.

It should be noted that this constant upwarddisplacement appears in the same experiment (S 2, E 3)in which the probability of FAshas been reduced throughpractice over that of other experiments. For thisexperiment in particular, the required accuracy-4%

S

2

E P (FA) P (Miss)

1 .0167 002 .0431 03 .0319 .2 @ (12,d) = (L/64,1)

1 .025 02 .0347 03 .0194 .2 @ (12,d) = (L/64,1)

BLOCH'S LAW AND A TEMPORAL INTEGRATION MODEL 425

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Figs. 4 and 5. Within each experiment, RT 50s to highest, medium, and lowest luminances (e.g., L, L/2, and L/4) are representedby filled circles, open circles, and open triangles, respectively. To facilitate comparison of asymptotic 80s across experiments, threeshort horizontal lines have been drawn near d =32 msec at 10, 12.5, and 15 msec. There appears to be relatively little change inobserved asymptotic 50s even across experiments and between Ss,

The interesting question of whether BL holds for RTstandard deviations is more difficult to answer; itappears, from examination of Figs. 4 and 5 that such aconclusion is not supported, i.e .. RT SDs to stimuli ofthe same L x D product are not in general similar enoughto justify this conclusion. They are similar enough,however, to suggest that a more carefully conductedexperiment with more precise stimulus generatingequipment might indeed show that BL holds for meansand variances, and perhaps even for RT distributions.

The analysis of variance for the six experimentsshowed that, in general. the only significant source ofvariance was due to luminance, duration, and theirinteraction. All were significant at better than the .001level. Some higher order interaction appeared inExperiment 1 data, however. as well as an effect of fieldand order significant at the .05 level. The importance ofthis analysis is that it shows that effects due to field andorder were negligible in general. and that the most

significant feature of the data is the strong interactionbetween stimulus intensity and duration. The latter is, ofcourse, evident in Figs. 2 and 3 and 6-8, where the goodfit of BL itself indicates strong interaction between thesevariables.

THEORY: TEMPORAL INTEGRATION MODELSFOR VISUAL REACTION TIME

Our goal is to develop a model that can account forthe relationships between RT and stimulus durationshown in Figs. 2-5. We begin with the well-knownassumption that simple RT (the total delay betweenstimulus onset and S's response) is the sum of twocomponents: Td. the time between stimulus onset anddetection; and Tr- the time between detection andresponse:

(1)

4~6 HILDRETH

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The temporal integration (TI) model, we now considerassumes that detection time, Td , is the time required forthe time integral of a nonnegative function vet) to reachsome fixed criterion, c. The function, vet), is called the"visual response function" (VRF); its specific formdepends on the visual stimulus. We assume vet) = 0before stimulus onset, i.e., for t = 0, let

yet) = fat v(t)dt. (2)

For the purposes of this paper, it is assumed that for agiven S and a given set of experimental conditions, all ofthe model parameters are fixed, i.e., c and T, areconstants, and vet) is the same function on every trialinvolving the same stimulus. Note, however, that themodel parameters may vary from experiment toexperiment, and from S to S. We acknowledge that thisis an idealized model, and it may be desirable later on toconsider some of these parameters as random variableswhich assume different values from trial to trial.

Then detection occurs when yet) = c, and Td is theinstant when this occurs, i.e.,

THE EXPONENTIAL DECAY MODEL

Thus

RT = y-l(c) + Tr .

To obtain quantitative predictions from the TI model,(3) we need to assume an explicit form for the YRF. Inthe

present application of the model, the visual stimulus isalways assumed to be a square wave with intensity (i.e.,luminance), Q, presented at time zero and having

(3b) duration d. To indicate explicitly the dependence of the

BLOCH'S LAW AND A 1EMPORAL INTEGRATION MODEL 427

TIME-

TIME ---

ASYMPTODIC CASE:

LATENCY = 71 + T"

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LATENCY =T d + 1;.

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intensity, Q(via AQ and fQ), and duration, d. The visual systemsimply integrates as much of it as is needed for detection. Whend > t, the latency is no longer a function of d, but of AQ alone,

(4)for EQ < t ~ d

a

v(t;d,£) = AQ

VRF on d and Q, we write v(t;d,Q), The exponentialdecay model we now consider assumes that

In other words, the VRF corresponding to a square-waveflash with intensity Q and duration d begins as a squarewave with amplitude AQ at t = E, is maintained until timed, and then decays exponentially following offset of theflash (see Fig. 9). Of course, there may be a lag similar toE before decay begins, but as long as E is small we willomit consideration of it for simplicity. The parameter rQdetermines the rate of decay: at time d + rQ- 1loge(2),

the VRF will have decayed to AQ/2. The threeparameters, EQ, AQ, and rQ are all assumed to depend onthe flash intensity, Q. When the criterion parameter. c,and the detection-to-motor-response parameter, Tt, areincluded, there are five free parameters altogether in thisexponential decay version of the Tl model. However, weshow below that for any fixed value of Q, the number ofparameters that must be estimated in order to generatethe predicted relationship between flash duration andRT is actually three, Put another way, the exponentialdecay model really involves only three "identifiable"parameters (Greeno & Steiner, 1964),

Detection occurs when the integral of vet) reaches thecriterion, c. For the model given by Eq. 4, we have

a

EQ < t ~ d It is convenient to introduce a parameter TQ

corresponding to the shortest duration for whichV(d;d,£) > c. In this case, V(TQ;TQ,£) = c, i.e. AQ(TQ - EQ)=c; hence

(5)

The quantity V(oo;d,Q) can be thought of as the "total(integrated) response" generated by the flash. IfV( co ;d,Q) < c, the flash will not be detected, It is naturalto introduce a parameter 0Q corresponding to the"threshold duration" for a flash of intensity Q: oQ is theshortest duration for which such a flash is abovethreshold; i.e., v(ooh,Q) = c. Thus, from Eq. 5,

(6)

(8)

TQ can be thought of as the "critical duration" forflashes with luminance Q; all flashes with d > TQ will havethe same latency (i.e" increasing the duration beyond TQ

does not lead to any further decrease in RT).Now, if oQ < d < TQ, the flash will be detected at

some point during the exponential decay portion of itsVRF. To determine Td in this case we solve the thirdline of Eq. 5 for VeTd :d,Q)=c and obtain

At the other extreme, we note that if d is sufficientlylong, V(d;d,Q) > c and detection occurs during the flashitself. For all such durations, the detection latency willbe the same, i.e., if V(d;d,Q) > c, then

(7)

Finally, then, combining Eqs. 7 and 9, and using Eq. 8to substitute for c/AQ + EQ. we have the following generalexpression of the detection latency Td of a flash withluminance Q and duration d:

428 HILDRETH

Table 3Parameter Estimates from Method 1 (L = .6 fL, All Other

Values are in Milliseconds)

duration for a stimulus with luminance Q. Using Eq. 11.the ED model prediction for this function is

TQ ~d (lIb)

THEORETICAL INTERPRETATlONOF PARAMETER ESTIMATES

RT(d;Q) =

The fact that l/rQ decreases with Qin Table 3 meansthat the half-life of the effect of a stimulus increases asits luminance decreases. This is rather surprising; onemight have expected that a more intense stimulus would

We denote by RT( 00 ;Q) the asymptotic latency (i.e., for dlarge). Then

RT(d;Q) - RT(oo;Q)

Empirically, Eq. 12 corresponds to the latency vsduration functions of Figs. 2 and 3 displaced down so asto be asymptotically zero. Estimates of the parametersoQ and r:Ql were obtained for each luminance in eachexperiment by means of a least-squares computerprogram which calculated the best fitting form of Eq. 12for each case separately. The resulting estimates areshown in Table 3. The goodness of fit was excellent, butnot surprising in view of the number of freeparameters-seven. The table also shows estimates of Tr

for each case; these were obtained by subtracting8Q+ ifl from the observed asymptotic mean latency[i.e., the empirical counterpart of RT(oo;Q)]. Table 3also includes an estimate of the critical duration 7Q foreach case; these estimates are simply the sums 8Q + ii 1 .

It will be noted that the estimates 8Qin Table 3 areapproximately constant within each experiment (exceptfor S 1 Eland S 2 E 1). This is also true of theestimates r. (except for S 2 E 3). However, it is quiteclear that t r changed considerably from experiment toexperiment-as much as 40 msec for S 2.

Because t rand 8Q seemed to be virtually constantwithin each experiment, a second method of parameterestimation was also applied to the data. In this method,all three curves from each experiment were fi}simultaneously by a least squares program in which 0and t r were required to be the same for all three, whileIQ was permitted to assume different values for differentluminances. The resulting estimates are given in Table 4;goodness of fit is shown in Figs. 10 and 11. It can beseen that even with these restrictions, the fit is quitegood-except perhaps for S 2 E 3, where thediscrepancies between Tr estimates in Table 3 now havethe effect of producing notable discrepancies betweenobserved and predicted values at asymptote.

c Tr /iQ ri2 1 rQ =re +6Q

L 165.8 .30 10.42 10.72L/2 167.3 .51 11.55 12.06L/4 167.6 .81 14.04 14.85

L/4 171.6 .86 13.25 14.11L/8 169.6 .87 19.16 20.03L!16 169.4 .85 26.16 27.01

L/16 180.3 .62 25.16 25.78L/32 180.3 .85 32.00 32.85L/64 182.7 .84 37.62 38.46

L 179.7 0 13.79 13.79L/2 183.6 .49 13.82 14.31L/4 183.3 .50 17.88 18.38

L/4 187.0 .46 17.77 18.23LIB 187.8 .47 22.06 22.53L/16 189.0 .38 27.88 28.26

L/16 219.8 .28 29.81 30.09L/32 226.4 .56 32.38 32.94L/64 233.6 .56 36.32 36.88

Expert-S ment

PARAMETER ESTIMAnONAND GOODNESS OF FIT

Let RT(d,Q) denote mean RT as a function of

1Td = d - -loge[I - rQ(7Q - d)] if oQ < d < 7Q(l0)

rQ

3

2

2

(II)

00 (no detection) if d ~ oQ

:2 3

2 2

In either case, three parameters must be estimated fromthe data. It is worth noting that Eq. 10 provides a checkon the assumption that Tr is independent of the stimulusparameters: By subtracting the observed critical duration(i.e., the observed duration value at which the RT vsduration function becomes asymptotic) from theasymptotic latency, we obtain an estimate of(7Q + Tr) - 7Q = Tr. These estimates should be roughlyequal for all luminances used in any given experiment.

Since RT = Td + Tr, the three parameters required to fitmean RT as a function of flash duration for a fixed valueof Q are Tr, 7Q, and rQ. Alternatively, we could expressTd in terms of rQ and the threshold duration oQ, usingEqs. 6 and 8:

BLOCH'S LAW AND A TEMPORAL INTEGRATION MODEL 429

12864

SUBJECT 2(L= 0.6 FOOTLAMBER"TS)

8 16 32DURATION (msec)

42

380 ..--.0------------------------,

340

320

360

~ I• Elljl

l.Ii6 3

SUBJECT I(L= 0.6 FOOTLAMBERTS)

8~TION

180

>-

~ 260r':5 \\Z ,

~ 240 '.::I: '.

.\ \

" "\ ", ,.... '" " .....

" '~" ~" ,..... .............!... LAG"""~... '~ ... - • ---- - ---'--Li8~~

--..! -------:----;-.-r -: 2P

: ~

• I

FiS". 10 and 11. Goodness of fit of the TI-ED model to the observed mean RTs.

have the effect of greater persistence, i.e., vet) requires a experiments. Since 82 is the threshold duration, welonger time to decay for any given fixed level. would expect it to increase as luminance decreases-the

A second surprising aspect of Table 3 is the fact that less intense the stimulus, the longer it must.last to reach82 seems to be roughly constant over Q in most of the threshold. Perhaps it is sufficient that the 82 values are

Table 4Parameter Estimates from Method 2·

Experi-1'r • -1 T2 =iii' +.5 2 1/0. ~S ment 2 112 r2

L 168.2 .74 7.93 8.67 6.4 .43L/2 10.20 10.94L/4 14.30 15.04

2 L/4 167.8 .83 14.65 15.48 6.6 .44L/8 20.20 21.03L/16 26.84 27.67

3 L/16 182.2 .82 22.66 23.48 7.0 .38L/32 31.97 32.79L/64 38.12 38.94

2 1 L 183.2 .44 ILl 1 11.55 9.5 .36L/2 14.12 14.56L/4 18.28 18.72

2 2 L/4 189.0 .44 17.23 17.67 9.3 .34L/8 21.89 22.33L/16 27.58 28.02

2 3 L/16 229.5 .53 25.35 25.88 10.1 .29L/32 31.88 32.41L/64 37.91 38.44

*L = .6 fl., P is a dimensionless constant. and all other values are in mil/iseconds. '" and p are optimal constants for powerfunction r2 =",flP.

430 HILDRETH

PILOT STUDY

A pilot study had been conducted using almostexactly the same procedure as in the experimentsreported above. The only important difference was thatthe luminances used were .15, .0475, and .015 fL,expanding the luminance range from 4 to 10. The S wasthe author. Observed and predicted mean RTs for thepilot experiment are shown in Fig. 13. To see that BLdoes not hold in this case, displace the observed datapoints of the figure right or left by ~ log, 0 units, or1.67 log2 units. The reduction in intensity by a factor of3.1 when exactly matched by a corresponding increasein stimulus duration of 3.1 = 21.67 leads to RTpredictions which consistently fall below observed meanRTs by 10·15 msec,or about the difference in estimatedcritical durations.

stochastic character of RTs. Although an extensivedevelopment of a stochastic version of the model isbeyond the scope of this paper, there are preliminaryindications that assuming c to be a random variableprovides a good quantitative fit to the observed variancesand explains the approximate invariance of the 8withineach experiment. At this time, however, we limitourselves to the qualitative argument that if c, All, or T2

is a random variable, the variances will be inflated forsmall d: then the observed variances are at leastqualitatively compatible with the model.

Although our finding that the half-life [the timerequired for vet) to decay to half its original value]increases with decreasing luminance seems paradoxical, arecent electrophysiological study by Levick and Zacks(1970) suggests that such "decay" may be a naturalconsequence of the way retinal ganglion cells respond tostimulus intensity changes for short light flashes. Thisstudy (see Fig. 2, p. 682) suggests that as stimulusintensity increases (for fixed duration flashes), theganglion cell response actually becomes more sharplylocalized in time rather than becoming more prolonged.

Finally, the apparent changes in Tr from experimentto experiment may be the result of changes in darkadaptation level. Since different stimulus intensities wereused in each of the three experiments, the effective darkadaptation level was greatest in E 3, next greatest in E 2,and least in E 1. There is considerable evidence, bothphysiological (Levick & Zacks, 1970) andpsychophysical (Standing, Dodwell, & Lang, 1968)showing that visual latency increases with the amount ofdark adaptation, so that on this basis we would expectthe longest "baseline" latency in E 3 and the shortest inE 1, which was found. If dark-adaptation level couldsomehow be held constant across all conditions, changesin T r from one condition to another could then beattributed to changes in the criterion, c. The lengthyprocedure required, however, might be impractical in

. view of the large number of observations necessary forparameter estimation.

(13)

Table 4 shows that a here was' consistently in theneighborhood of .35, roughly the same value found forbrightness as a function of luminance (cf. Stevens,1966). Since TQ =c/AQ + €lI =oQ + I/r2, and 62 here wassmall, it follows that c/A2 + €2 and l/ill wereapproximately equal here, and consequently (c/A2 +€lI)-l was also a power function of £-to a goodapproximation-with the same exponent as f2. If €lI == 0,this means that A2, the amplitude of VRF, is a powerfunction of luminance, with exponent in the range .3 to.5. This seems a reasonable result, since we shouldexpect brightness to be proportional to the maintainedlevel of the VRF. Or, put more realistically, if we wereto extend the model to account for brightnessphenomena, this would be a natural first assumption,and so it is gratifying to see that it is not disconfirrnedby our parameter estimates.

-1.6r----------------=-__:_

small, i.e., less than 1 msec; it may be unrealistic toexpect finer resolution from RT data of the sortconsidered here.

A natural question in the context of the exponentialdecay model is how the parameters rll and All are relatedto flash luminance. Figure 12 shows a plot of 10g(fQ) vslog (£). The relationship is approximately linear for bothSs in all experiments; best-fitting slope parameters aregiven in Table 4. A linear relationship between log(rll)and log(£) means that rll is a power function of £, i.e.,

SOME CONCLUDING REMARKS.ON THETI·ED MODEL

-ic

-l4

-1.5

Fig. 12. Logarithms of parameter estimates i Q as a function oflog(Q). The points fall in a nearly linear pattern with estimatedslopes near 1/3, except for the pilot experiments (large dots)with a slope of nearly 1/2.

We have indicated above an important limitation atthe TJ·ED model: As it stands, it is a deterministicmodel and does not provide any account of the

BLOCH'S LAW AND A TEMPORAL INTEGRATION MODEL 431

360r--

REFERENCES

128

•

L•

LItO

L/31 •

•

64

:

PILOT EXPERIMENT

(La 0.15 FOOTLAMBERTSI

2

320

340

300

8 16DURATION (msec)

Fig. 13. Goodness of fit of a pilot experiment usingluminances L, L/3.2, and LIIO, where L = 0.15 fL.

amplitude AQ with the onset of the flash and remains atthat level until flash offset. The VRF then decaysexponentially with a time constant, r£1. Both r£1and AQ depend on the stimulus intensity, Q.

Adrian, E. D., & Matthews, R. The action of light on the eye.Part 1. The discharge of impulses in the optic nerve and itsrelation to the electric changes in the retina. Journal ofPhysiology, 1927. 63. 378-414.

Armington, J. c.. Tepas, K. 1.. Kropfl, W. J., & Hengst. \Y. H.Summated retinal potentials. In E. P. Home and M. A.Whitcomb (Eds.), Vision research papers. Washington. D.C:National Academy of Sciences, National Research Council.1960. P. 112.

Bloch, A. M. Experiences sur la vision. Soc. BioI. Mern. (Paris).1885.27,493-495.

Greeno. J. G .. & Steiner. T. E. Markovian processes withidentifiable states: General considerations and application toall-ot-none learning. Psvchometrika, 1964.29,309-333.

Grossberg, M. The latency of response in relation to Bloch's lawat threshold. Perception & Psychophysics, 1968.4. 2:l9-232.

Hanes. R. M. The construction of subjective brightness scalesfrom fractionation data: A validation. Journal ofExperimental Psychology, 1949, 39. 719-728.

Hartline. H. K. Intensity and duration in the excitation of asingle photoreceptor unit. Journal of Cellular ComparativePhysiology, 1934,5.229-247.

Hopkinson, R. G. Light energy and the brightness sensation.Nature, 1956. 178. 1065-1066.

Kernpthorne, O. Design and analysis of experiments. New York:Wiley. 1952. Pp. 370-384,

Nevertheless, the ED model does an excellent job ofaccounting for the mean RTs. The parameter estimatesTil, however, depart from those obtained in the mainexperiments in that, for the pilot study, Til = 12. 5 VS TQ =£.33 for the major experiments (cf. Fig. 12-large dots).We see therefore that the conclusion that Bloch's lawholds must be conditioned by the observation that itfails when the range of luminances used is as large as 10(i.e., 120 to 101( 0 ) , Nevertheless, the ED model accountswell for the data (cf. Fig. 13).

One possible cause of the failure to obtain BL in thepilot experiment could be artifactual trial-to-trial lightadaptation. Trials were 3¥.1 sec apart, and sequentialeffects would be most severe in wide ranges of intensity.

We saw above that within each main experimentinvolving a fixed set of luminances and durations, BLheld for mean RTs to a close approximation (cf.Figs. 6-8). Now, a natural question at this point iswhether the ED model. with parameters rQ and AQconstrained as in i Q = 12 13 , actually implies BL will holdexactly. It is easily shown, even with [) = E. so that rQ =AI//c, that it does not (proof omitted for brevity).However, it can be shown that when iJ == 1/3, then theED model with rQ and AQ both proportional to 1213 doesimply a close approximation to BL over a limited rangeof 12 values-in particular. Q values such that 12. 3 3 - I andlog 10 (12) are approximately equal. The ranges of 12 valuesused in each of the main experiments were all of theform 12 0 < 12 < 412 0 , so the necessary approximationswere valid there. Consequently, even though the EDmodel does not strictly imply BL. it is compatible withthe approximate validity of that law found in thepresent experiments.

THE RELATIONSHIP BETWEEN BLOCH'S LAW,THE POWER LAW FOR SIMPLE RT,

AND THE ED MODEL

SUMMARY

In the foregoing, we have attempted to answer thequestion of whether Bloch's law (BL) holds for RTs tosuprathreshold light stimuli. Earlier research on thissubject had been limited to three studies, of which twohad concluded that BL did not, in general, hold for RTs.The present research, on the other hand, indicated theopposite. For the limited range of luminances-allsuprathreshold-that we investigated, BL did hold to avery good approximation. The results of the experimentswere orderly enough to be fit by a simple mathematicalmodel. A special case of the model-called the temporalintegration-exponential decay (II-ED) modele-provides avery good fit for the mean RTs of these experiments.

According to the model, RTs to a square-wave flashare determined by the time required to integrate a"visual response function" (VRF) generated by thatflash. For the II-ED model, the VRF rises instantly to

432 HILDRETH

Kietzman, M.. & Gillam, B. Visual temporal integration andsimple reaction time. Perception & Psychophysics. 1972, 11,333-340.

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Sticht, T. Effects of intensity and duration on the latency ofresponse to brief light and dark stimuli. Journal ofExperimental Psychology, 1969.80,419-422.

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(Received for publication November 6, 1972;revision received May 13, 1973.)