University of Groningen Asymmetric dynamics of adaptation ...Snippe, Poot, & van Hateren, 2000; Wu, Burns, Elsner, Eskew, & He, 1997). In these experiments, detection thresholds for

University of Groningen

Asymmetric dynamics of adaptation after onset and offset of flickerSnippe, H.P.; Poot, L.; van Hateren, J.H.

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https://doi.org/10.1167/4.1.1https://research.rug.nl/en/publications/asymmetric-dynamics-of-adaptation-after-onset-and-offset-of-flicker(94479c50-9354-47aa-9e80-0de868c00534).htmlhttps://doi.org/10.1167/4.1.1

Journal of Vision (2004) 4, 1-12 http://journalofvision.org/4/1/1/ 1

Asymmetric dynamics of adaptation after onset and offset of flicker

H. P. Snippe Department of Neurobiophysics, University of Groningen,

Groningen, The Netherlands

L. Poot Department of Neurobiophysics, University of Groningen,


J. H. van Hateren Department of Neurobiophysics, University of Groningen,


We measured human psychophysical detection thresholds for test pulses which are superimposed on spatially homoge-neous backgrounds that have abrupt onsets and offsets of high-contrast 25 Hz flicker. After the onset of the background flicker, test thresholds reach their steady-state levels within 20-60 ms. After the offset of the background flicker, test thresholds remain elevated above their steady-state level for much longer durations. Adaptation after onsets and offsets of background flicker is modeled with a divisive gain control that is activated by temporal contrast. We show that a feed-back structure for the gain control can explain the asymmetric dynamics observed after onsets and offsets of the back-ground contrast. Finally, we measure detection thresholds for tests presented on steadily flickering backgrounds as a function of the contrast of the background flicker. We show that the divisive feedback model for contrast gain control can describe these results as well.

Keywords: contrast adaptation, divisive gain control, feedback model, ideal observer

Introduction The natural input to the visual system consists of sig-

nals that contain large variations not only in local lumi-nance, but also in local contrast (Buiatti & van Vreeswijk, 2003; van Hateren, 1997; Ruderman, 1994). This poses a problem to the visual system because its neurons have a limited dynamic range which may well be smaller than the large variations in the input. The solution to this problem is well known: the visual system has gain controls both for luminance (Crawford, 1947; Fain, Matthews, Cornwall, & Koutalos, 2001; Geisler, 1978; Hayhoe, Benimoff, & Hood, 1987; Perlman & Normann, 1998) and for contrast (Baccus & Meister, 2002; Chander & Chichilnisky, 2001; Rieke, 2001; Shapley & Victor, 1978).

In the present paper our focus is on contrast gain con-trol. When the visual system encounters inputs with low contrast (locally in space and/or time), it will pass these small contrasts with high gain, thus protecting the resulting signals against noise further on in the visual system. How-ever, in an environment with high contrast, such a high contrast gain could well be inappropriate because it would lead to a saturation of the dynamic range that is available to the visual system, and hence to a loss of information. Thus, for these high contrasts it would be beneficial for the visual system to reduce its contrast gain in order to escape such a saturation. A reduction of the contrast gain can be tested in psychophysical experiments by presenting a test signal that is superimposed on the contrast background. A reduction

of the contrast gain that is induced by the background yields a reduced visibility of the test signal, compared to the visibility of the same test superimposed on a background with low (or zero) contrast. This reduced visibility for tests presented on backgrounds of high contrast yields detection thresholds for the test signal that are elevated compared to the detection threshold for the test presented on a back-ground with zero contrast. These elevations of test thresh-olds have been measured using the so-called probe-sinewave paradigm (Hood, Graham, von Wiegand, & Chase, 1997; Snippe, Poot, & van Hateren, 2000; Wu, Burns, Elsner, Eskew, & He, 1997). In these experiments, detection thresholds for brief test pulses superimposed on flickering (spatially homogeneous) backgrounds are measured for various moments of presentation of the test pulse relative to the flicker cycle of the background. Detection thresholds depend on the precise timing of the test pulse relative to the modulation of the background luminance. This de-pendence can be understood as a consequence of dynamic processes of light adaptation in the visual system (Snippe et al., 2000). When the test thresholds are averaged over a full cycle of the background flicker (to average out these effects of light adaptation), the averaged threshold is systematically elevated above the detection threshold measured for a test pulse superimposed on a steady (nonflickering) background with a luminance equal to the time-averaged luminance of the flickering backgrounds. Snippe et al. (2000) show how this threshold elevation can be explained by a process of contrast gain control in the visual system.

doi:10.1167/4.1.1 Received March 20, 2003; published January 16, 2004 ISSN 1534-7362 © 2004 ARVO

mailto:[email protected]?subject=http://journalofvision.org/4/1/1/http://hlab.phys.rug.nl/mailto:[email protected]?subject=http://journalofvision.org/4/1/1/http://journalofvision.org/4/1/1/

Journal of Vision (2004) 4, 1-12 Snippe, Poot, & van Hateren 2

In the present paper, we study the dynamics of such a contrast gain control, by measuring detection thresholds for tests that are presented shortly after an onset of the background flicker (Wolfson & Graham, 2000; Snowden, 2001), and also after the offset of the background flicker (Foley & Boynton, 1993). The comparison of threshold dynamics after the onset and the offset of the background contrast is especially interesting in view of the ideal-observer calculations of DeWeese & Zador (1998). These authors show that a statistically optimal observer can adapt faster after increments of the contrast than after decre-ments of the contrast. The present measurements are con-sistent with their prediction for an ideal observer: we find substantially faster adaptation after the onset of the back-ground contrast than after the offset of the background contrast. This match between the psychophysical results and the predictions for the ideal observer strengthens the belief that contrast gain control in the human visual system is functionally appropriate.

Methods

Apparatus and psychophysical procedure The experimental set-up was identical to that described

in Snippe, Poot & van Hateren (2000). Stimuli were pre-sented monocularly through a Maxwellian view system, us-ing two green (563 nm) Toshiba TLGD 190P light-emitting diodes (LEDs) as light sources. One LED provided a spa-tially homogeneous circular adaptation field of diameter 17°. The other LED was used for foveal presentation of a concentric, sharp-edged test stimulus with a diameter of 46 arcmin and a duration of 7.5 ms. A PC controlled the LED intensities at a rate of 400 Hz, through a 12-bit digital-to-analog converter. The LED outputs were linearized on-line using the photodiode-feedback design of Watanabe, Mori & Nakamura (1992). We collected psychophysical thresh-olds using a modified yes-no method, as described in Poot, Snippe & van Hateren (1997).

Observers Eight observers took part in this study, 5 male and 3

female. Ages ranged between 21 and 42 years. Observers used their normal optics to obtain good acuity. All observ-ers were aware of the purpose of the experiments.

Temporal dynamics of the stimuli In the main experiment, the background stimulus con-

sisted of alternating periods (of duration 1.28 s) of a 25 Hz sinusoidal flicker of 80% contrast, and a constant lumi-nance of 7500 Td equal to the average luminance level of the sinusoid (Figure 1). Brief (7.5 ms) incremental test pulses were superimposed on this background at times τ (positive or negative) relative to the beginning or the end-

ing of the sinusoidal flicker. Threshold strength for detec-tion of the test pulse was measured as a function of τ.

Test thresholds were measured for four different ver-

sions of the background stimulus (Figure 2). In the differ-ent versions the flicker started at a different phase of the sinusoid (0, 90, 180 and 270 deg). This procedure resulted in four detection thresholds for the test at each value of τ. Because the luminance on which the test pulse is superim-posed differs for the four different backgrounds (Figure 2), these four thresholds will be different (typically varying by a factor 2-3 during continuous 25 Hz flicker of 80% contrast; see Figure 1 in Snippe et al., 2000). These differences in thresholds can be understood as effects due to the dynam-ics of light adaptation (Snippe et al., 2000). However, in the present paper we are interested in dynamic effects of contrast gain control, rather than light adaptation. To re-

0

7500

15000

//// 264025602480136012801200800

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1280 ms constant1280 ms flicker

Lum

inan

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Td)

t (ms)

Figure 1. Temporal sequence of the background. A 25 Hz sinu-soidal flicker with a contrast of 0.8 is alternated continually with aconstant luminance of 7500 Td. Both alternating periods have aduration of 1280 ms.

-40 0 40 80

Lum

inan

ce

t (ms)

Figure 2. The four phase conditions used for the background,here shown with test pulses superimposed at a delay τ = 40 msafter the onset of the flicker of the background. The lowest curve(flicker onset at phase 0 deg) shows the luminance as used inthe experiments; the other three curves have been vertically dis-placed for clarity.


duce any effects of light adaptation we average the four thresholds obtained at each value of τ. This allows us to obtain an uncluttered picture of the effects of the onset of the background contrast, independent of the precise lumi-nance conditions of the 25 Hz carrier. The same averaging procedure was performed at contrast offset, since phase effects may still influence the thresholds after flicker has stopped.

We used a modulation frequency of 25 Hz for the background stimulus, because for continuous flicker the elevation of test thresholds is close to maximal at this fre-quency (Snippe, Poot & van Hateren, 2000). A second rea-son for choosing a high frequency of the background flicker is that the influence of the discontinuity in luminance which occurs when flicker begins or ends at a phase of 90 or 270 degrees is smaller at high frequencies, because of low-pass filtering early in the visual system (Levinson, 1968). A third reason is that the threshold modulations induced by light adaptation are smaller at 25 Hz than at lower frequencies (Snippe et al., 2000), and thus have less influence on the present results.

Results

Asymmetric dynamics after onset and offset of flicker

In Figure 3, phase-averaged detection thresholds for tests presented around the onset (A) and the offset (B) of the background flicker are shown for two individual ob-servers, and for the results averaged over the 8 observers who performed this experiment. A large asymmetry in the speed of adaptation after the onset and the offset of flicker is evident. After the onset of flicker, thresholds reach their steady state (the phase-averaged threshold level during con-tinuous flicker) within approximately 40 ms. Although not shown in Figure 3, this is the case not only for the phase-averaged thresholds, but also for each of the four phase conditions separately. After the offset of the background flicker, however, thresholds at τ = 40 ms are still much elevated above the steady state level for tests presented on a steady background. Denoting the measured threshold for observer i at time τ after contrast offset as Mi(τ), and the

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τ (ms)0 160 320 480 640

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τ (ms) τ (ms)

A

B

Figure 3. Detection thresholds for test pulses presented near the onset (A) and the offset (B) of the background flicker for two individualobservers (LP and JK), and for the results averaged over all the 8 observers who performed the experiment. The step functions in-cluded in each graph indicate the steady-state thresholds: the low level of each step function is the steady-state threshold for a testsuperimposed on a constant background, and the high level of each step function is the steady-state threshold for a test superimposedon a background with a continuous flicker of contrast 0.8. To reduce scatter in the data for the average observer due to differences inthe steady state thresholds of the individual observers, thresholds M(τj) for the average observer are obtained as follows. First the rawdata Mi(τj) for each observer i are scaled relative to that observer’s steady state thresholds, resulting in scaled data mi(τj) : mi(τj) = (Mi(τj)- Mi-) / (Mi C- Mi-), with Mi- the steady state threshold for that observer obtained with a constant background, and MiC the steady statethreshold obtained with a flickering background. Next, at each delay τj the scaled data mi(τj) are averaged over observers. Finally, tofacilitate comparison of these averaged data to the data obtained for the individual observers, the averaged data m(τj) are rescaled toM(τj) = M -+ (MC - M-) m(τj), with M- and MC the observer-averaged steady state thresholds. Note the logarithmic scale of the verticalaxis, and the different time scales of the horizontal axes.


steady-state threshold of this observer for a test presented on a steady background as Mi-, the normalised threshold elevation Ei(τ) above the steady state level can be quantified as

Ei τ( ) =Mi τ( )− M i−

M i− . (1)

At τ = 40 ms after contrast offset, threshold elevations Ei(40) for our observers range from 1.3 to 3.5; the arithme-tic mean of their threshold elevations is 2.2 ± 0.3(SE; n = 8).

A further asymmetry between adaptation after contrast onset and offset is that steady state is reached only during the “on” period of the contrast stimulus and not during the “off” period. After the onset of contrast, thresholds reach steady state within approximately 40 ms. But after the offset of contrast, a slow adaptation component appears to take over after about 80 ms, and the threshold curves flatten. At τ = 640 ms after contrast offset, threshold elevations Ei(640) above the steady-state level are still quite high (range Ei(640) = 0.1-0.7 ; mean± s.e.m. = 0.38±0.08). Thresholds do not further decline much below the level obtained at τ = 640 ms within the remaining 640 ms of steady background before flicker is switched on again.

Threshold recovery after contrast offset does not be-have as a simple exponential decay (Figure 4, dashed lines). Instead (Figure 4, full lines), a power law recovery produces an excellent fit to the data at τ ≥ 20 ms:

E τ( ) =τ1τ

γ

. (2) 10

10 100 10000.1

1

10

HS

Thr

esho

ld E

leva

tion

10 100 10000.1

1

10

RV

10 100 10000.1

1

Average Observer

τ (ms)

Figure 4. Threshold elevation (Equation 1) as a function of thetime τ after the offset of the background flicker for two individ-ual observers, and for the results averaged over all observers.In accordance with Equation 1, the threshold elevation E(τ) forthe average observer is E(τ) = (M(τ) - M-)/M-, with M(τ) and M-

defined in the legend of Figure 3. The straight lines are thebest power-law fits (Equation 2) to the data. Exponential func-tions (dashed lines) yield unacceptable fits to the data.

The value of the power exponent γ ranges approximately between 0.5 and 1 for our observers (Table 1).

Observer γ RV 0.67±0HS 1.06±0LP 0.62±0JB 0.57±0JK 0.63±0JH 0.63±0LT 1.06±0SW 0.54±0Average Observer 0.65±0

Table 1. Estimates of the parameues shown are means and SDs otime-parameter τ1 in Equation 2servers, and for the average obsFigure 3.

τ1 (ms) .06 90±5 .07 116±12 .12 104±18 .07 273±61 .08 92±10 .14 153±42 .06 63±8 .08 84±12 .05 122±9

ters γ and τ1 in Equation 2. Val-f the power exponent γ and the for each of the individual ob-erver defined in the Caption of


Control experiments Tapered contrasts

Some of our observers show overshoots of their thresh-olds for test pulses presented at times near the onset and offset of the contrast (e.g., Figure3A, LP). This could be caused by luminance artifacts due to the abrupt (instanta-neous) onset and offset of the contrast (Levinson, 1968). To test this possibility, we repeated the experiment for ob-server HS, but now using a more gradual (tapered) onset and offset of the flicker contrast:

Con (t) =12

C 1+ sin 2πt

T

, − T

4≤ t ≤ +

T4

(3)

Coff (t) =12

C 1− sin 2πt

T

,

44TtT +≤≤− . (4)

Between t = -T/4 and t = +T/4, for Con(t) the flicker contrast gradually increases from 0 to its full value C = 0.8. Likewise, at flicker offset Coff (t) gradually decreases from C = 0.8 at t = -T/4 to C = 0 at t = +T/4. We used T =120 ms, which should be sufficient to remove any luminance arti-facts due to the onset and offset of the contrast. Neverthe-less, detection thresholds for the test pulse show overshoots at the contrast onset and offset also for these tapered con-trasts (Figure 5). Also the asymmetric dynamics of thresh-olds after contrast onsets and offsets is equally strong for tapered and non-tapered (instantaneous) contrast switches. Thus we conclude that the asymmetric dynamics reported in Figure 3 is not caused by potential artifacts due to the instantaneous onsets and offsets of flicker contrast.

Noise carriers The long tail of threshold elevation at contrast offset in

our experiments could be a peculiarity of using a harmonic signal (25 Hz flicker) as the contrast carrier. To check the robustness of the result, one of the observers (RV) meas-ured test thresholds after the offset of a background con-trast of which the luminance values were samples of Gaus-sian noise, rather than harmonic modulations. Sample time (10 ms) and r.m.s. contrast (0.57) of the luminance noise were chosen such that during the noise, test thresholds were on average close to the phase-averaged threshold ob-tained with continuous 25 Hz flicker. As shown in Table 2, at 80 ms and 320 ms after the offset of the noise, threshold elevations above the steady-state for zero contrast are

somewhat larger than after the offset of 25 Hz flicker (a difference that is not accounted for in the models that we present later in this paper). However, from the experiment we can conclude that the long tail of threshold elevations after the offset of stimulus contrast is not limited to back-grounds with harmonic flicker. Further, since the offset of noise is free from luminance artifacts (Koenderink & van Doorn, 1978), the present results confirm that the long-term elevation of thresholds after the offset of harmonic flicker is not due to a luminance artifact for these stimuli.

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A

B

Figure 5. Detection thresholds for test pulses presented near theonset (A) and the offset (B) of the background flicker. The filledsquares are data obtained with an abrupt onset and offset of theflicker. The open circles are data obtained with a tapered onset(Equation 3) and offset (Equation 4) of the flicker. The step func-tions represent the steady-state thresholds for tests presentedon a constant background and for test presented during continu-ous flicker of the background. The dashed curves show the ta-pered onset (A) and offset (B) of the background contrast.

Contrast increments and decrements In the first part of the Results we reported asymmetric

dynamics of adaptation after contrast onsets and offsets, i.e. contrast steps in which the low contrast equals zero. To test whether this result generalises to contrast steps in which the low contrast is unequal to zero, two of our observers measured test thresholds after contrast increments (from C

aff

τ flicker noise

80 ms 1.09±0.12 1.45±0.14 320 ms 0.49±0.08 0.89±0.11

Table 2. Threshold elevations E(τ). Values shown are means nd SDs of the threshold elevations E(τ), defined in Equation 1, or observer RV at two delays τ after the offset of background licker, respectively background noise.


= 0.2 to C = 0.8) and contrast decrements (from C = 0.8 to C = 0.2). Results for observer SW are shown in Figure 6 (open symbols and broken lines refer to the results with steps between C = 0.2 and C = 0.8; as a reference, the filled squares are the results with steps between C = 0 and C = 0.8, as in Figure 3). Similar results were obtained for ob-server JB. Adaptation is fast (20-40 ms) after both onsets and increment steps of the contrast backgrounds. After contrast offset a prolonged (> 640 ms) threshold elevation occurs, similar to the results for other observers in Figure 3. Compared to the results at contrast offset, threshold eleva-tion after a contrast decrement is less prolonged: thresholds are close to the steady state level at C = 0.2 for times τ ≥ 160 ms after the decrement step. Nevertheless, adaptation is still substantially slower after the decrement step of con-trast (Figure 6B) than after the increment step of contrast (Figure 6A). At τ = 40-80 ms after an increment step the

test thresholds have reached the steady state, but at these times τ the test thresholds are still substantially elevated above the steady state level after a decrement step of the background contrast. Thus asymmetric adaptation occurs not only after contrast onsets and offsets, but also more generally after contrast increments and decrements.

A

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Divisive gain control In this Section we present models for contrast gain

control that can explain the psychophysical results. Our aim here is not to obtain a detailed quantitative correspon-dence with the psychophysical data. Rather, we concentrate on explaining the main trends of the present data with simple divisive models for contrast gain control. In divisive gain control, the output O(t) of the gain control equals the input I(t) divided by the control (adaptation) signal A(t):

( ) ( )( )tAtItO = . (5)

Divisive gain control has been observed both in the ret-ina (Baccus & Meister, 2002; Berry, Brivanlou, Jordan, & Meister, 1999; Kim & Rieke, 2001; Wilke, Thiel, Eurich, Greschner, Bongard, Ammermüller, & Schwegler, 2001) and in the visual cortex (Albrecht, Geisler, Frazor, & Crane, 2002; Carandini, Heeger, & Movshon, 1997). A further reason to study divisive models for contrast gain control is that the divisive control signal A(t) has a particu-larly simple relation to the thresholds p(t) measured for the test pulse in our psychophysical experiments. When a test pulse p is superimposed on a background I, the resulting output of the divisive gain control equals (I + p)/A. Hence the extra output ∆O generated by the test pulse p equals p/A. Using the traditional assumption (Fechner, 1860; Meier & Carandini, 2002) that the test pulse attains its detection threshold M when it generates a fixed extra out-put ∆O, at detection threshold

Figure 6. Detection thresholds for tests presented at times τ rela-tive to increments (A) and decrements (B) of the contrast of thebackground flicker. The filled squares are data obtained withtransitions of the background contrast between C = 0 (i.e. a con-stant background) and C = 0.8. The open circles are data ob-tained with contrast transitions between C = 0.2 and C = 0.8. Thestep functions included in both panels indicate the steady-statethresholds measured for test pulses on a constant backgroundand on a background with flicker contrast C = 0.8. The dashedhorizontal line is the steady-state threshold for test pulses pre-sented on a background with flicker contrast C = 0.2.

==∆AMO constant. (6)

Thus the detection threshold M of the test pulse will be proportional to A, the value of the divisive gain control. Hence we can relate the dynamics M(t) for the detection thresholds measured in our experiments to the dynamics of the adaptation signal A(t) in a model of divisive gain con-trol.

Does the adaptation signal A(t) in a divisive gain con-trol structure have the asymmetric dynamics that we see in our experiments: faster response at contrast onset than at contrast offset? Actually, it does not for a simple feedfor-ward gain control (Figure 7a) in which the dynamics of A(t) is governed by the contrast CI(t) of the input signal I(t):

)()(0 tAtCdtdA

I −=τ . (7)


The form of Equation 7 is such that A(t) is a low-pass fil-tered version of CI(t). This can be readily seen by Fourier transforming the equation, yielding 1/(1 + iωτ0) as the transfer function between the frequency domain versions of CI and A. Equation 7 yields identical dynamics for A(t) (an exponential response with time constant τ0) at the onset and the offset of the contrast CI(t). This would still be the case if a small positive constant ε would be added to the right-hand-side of Equation 7 (such that A attains a finite steady-state value ε, rather than zero, for a constant back-ground CI = 0). Although the resulting dynamics of adapta-tion is symmetric for these simple feedforward models, more complicated models for feedforward gain control can certainly respond with asymmetric dynamics to contrast onsets and offsets. However, rather than exploring such feedforward models, we will concentrate on a feedback structure (Victor, 1987) for contrast gain control (Figure 7b). In such a feedback structure the dynamics of the adap-tation signal A(t) is governed by the contrast CO(t) of the output O(t) of the gain control loop, rather than by the contrast CI(t) of the input I(t). The reason for exploring feedback structures is that they show asymmetric behavior already for a simple first order dynamics of A(t):

)()(0 tAtCdtdA

O −=τ . (8)

Using the divisive nature of the gain control: CO(t) = CI(t)/ A(t), Equation 8 can be rewritten by multiplying both sides of the Equation with A(t), and using the identity A dA/ dt = 1/2 dA2/dt:

)()(21 22

0 tAtCdtdA

I −=τ . (9)

Equation 9 shows that for the first-order feedback dynamics of Equation 8, the square A2(t) of the adaptation signal A(t) is a low-pass filtered version (with time constant τ0 / 2) of the input contrast CI(t), hence

[ ]( ) 21)()( passlowI tCtA −= . (10) The solid lines in Figure 8 show that the dynamics of

the resulting adaptation A(t) is asymmetric for onset versus offset steps in the input contrast CI(t). The asymmetric dy-namics for A(t) seen in Figure 8 can be simply explained. At contrast onset the adaptation A(t) is initially low, hence the output contrast CO(t) = CI(t)/A(t) is large (representing an overshoot in O(t)). This provides an especially strong drive to the dynamics in Equation 8. At contrast offset there is no similarly strong undershoot in the output con-trast CO(t),which explains the observed asymmetry in A(t) seen in Figure 8.

Although the solution (Equation 10) of Equation 8 re-acts fast at contrast onset, there is still a discrepancy of its response at contrast offset when compared with our psy-chophysical results. After a contrast offset (CI(t) = CO(t) = 0 for t > 0), the solution A(t) of Equation 8 is an exponen-tial: A(t) = A(0)exp(–t/τ0), which does not provide a good fit to our data (see Figure 4). However, by only slightly modifying the structure of Equation 8 we can obtain a low-pass filtering which yields the observed power-law behavior:

)()(0 tAtCdtdA mn

O −=⋅τ . (11)

Note that Equation 8 represents a special case of Equation 11, with n = m = 1. However, contrary to Equation 8, for m > 1 Equation 11 has a power-law solution for A(t) after

F00stiT

Figure modelstion’ petively Othe resnal thatof the g

LP

LP

Demodulation

Demodulation

O(t)

O(t)

I(t)

A(t)

A(t)

C (t)I

C (t)O

(a)

(b)I(t)

7. Schematic feedforward (a) and feedback (b) divisive for contrast gain control. The boxes labeled ‘Demodula-rform a demodulation of the flicker present in I(t) respec-(t). The boxes labeled ‘LP’ perform a low-pass filtering ofulting contrast signal. This yields A(t), the adaptation sig- divides the input I(t) to produce the output O(t) = I(t)/A(t)ain control loop.

0 200 400 600 800 10000.0

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0.8

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A(t

)

t (ms)

igure 8. Dynamics A(t) of the gain control after onsets (from C = to C = 1; increasing curves A(t)) and offsets (from C = 1 to C =; decreasing curves A(t)) of the contrast. The solid lines are theolutions of Equation 8. The dashed (dotted) lines are the solu-ons of Equation 11 with respectively n = m = 2 and n = m = 3.he parameter τ0 in Equations 8 and 11 has been set to 100 ms.


contrast offset:

( ) ( )( ( )) 1101)0(11)0()( −− ⋅⋅−+

= mmoff tAm

AtAτ

. (12)

For large enough t, the recovery of Aoff(t) in Equation 12 behaves as a power law (1/t)1/(m-1), thus (by using Equation 6) a power-law recovery with power exponent 1/(m – 1) is predicted for the test thresholds. Therefore, 1/(m – 1) cor-responds to the steepness parameter γ defined in Equation 2 (strictly speaking, Equation 1 is not defined for the pre-sent model because Mi- = 0, but this can be easily resolved by adding a small constant to the adaptation signal A(t) in Figure 7b). The typical range 0.5-1 obtained for γ in the psychophysical experiments thus corresponds to values m = 2 – 3 in Equation 11. The dashed and dotted lines in Figure 8 show the solution of Equation 11 to steps of the input contrast CI(t), for the choice n = m = 2, respectively n = m = 3. The response dynamics is much faster after a con-trast onset than after a contrast offset, and the response to the contrast offset shows a prolonged elevation as was seen in the psychophysical data. To further understand the be-havior of Equation 11, note that for n = m = 2 its right-hand-side can be rewritten as [CO(t) + A(t)][CO(t) – A(t)]. Hence, dividing both sides of Equation 11 by CO(t) + A(t), it can (for n = m = 2) be rewritten as

)()()()(

0 tAtCdtdA

tAtC OO−=

+τ . (13) 6000

From the similarity with Equation 8, Equation 13 can be understood as a low-pass filter with an effective time con-stant τeff = τ0 /[CO(t) + A(t)], which is small (fast adapta-tion) when CO and/or A are large, and which is large (slow adaptation) when CO and A are both small (as in the long-term behavior after contrast offset). This interpretation of Equation 11 further explains the increased asymmetry of the dynamics in Figure 8 for n = m = 2 (dashed lines) com-pared to the dynamics for n = m = 1 (solid lines). A similar decomposition CO3– A3 = (CO2 + A2 + COA)(CO – A) of the right-hand-side of Equation 11 can explain the strong asymmetry in Figure 8 for n = m = 3 (dotted lines).

Detection thresholds for pulses on backgrounds with steady contrast

As was shown above, threshold recovery after the offset of contrast constrains the power exponent m of A(t) in Equation 11 to values m = 2 – 3. Estimates of the power exponent n of CO(t) in Equation 11 can be obtained from the steady-state behavior (dA/dt = 0) of Equation 11 that is attained for backgrounds with a steady contrast CI (i.e. with steadily flickering backgrounds). Using the relation CO = CI /A, the steady-state solution of Equation 11 is

( )mnnICA

+= . (14)

Hence for n = m the adaptation signal A increases in pro-portion to the square root of the flicker contrast CI. For n < m the relation between A and CI would be more compres-sive than a square root, and for n > m it would be less com-pressive (i.e. more linear).

In order to test these predictions, we obtained detec-tion thresholds for test pulses presented on backgrounds with steady flicker as a function of flicker contrast CI for four observers. Thus in this experiment, contrary to Figure 1, the flicker now was “on” throughout the duration of the experiment. As in the previous experiments, we determined detection thresholds for the test pulse presented at four moments in the flicker cycle of the background (at phase 0, 90, 180 and 270 degrees). Reported (phase-averaged) thresholds are the mean of these four thresholds.

Figure 9 shows the phase-averaged threshold as a func-tion of the flicker contrast CI for two observers. Results for the other two observers who performed this experiment were similar, as were results for observer SW obtained for two different values of the flicker frequency (f = 6.25 Hz and f = 12.5 Hz). As expected from Equation 14, thresh-olds are a compressive function of the flicker contrast CI over most of the contrast range. An exception are the re-

0.0 0.2 0.4 0.6 0.80

2000

4000

8000

SW

Test

Thr

esho

ld (

Td)

0.0 0.2 0.4 0.6 0.80

2000

4000

6000

8000

JB

Background Contrast

Figure 9. Detection thresholds for two observers for tests pre-sented on continuously flickering backgrounds, as a function ofthe flicker contrast of the background. Solid lines are the fits ofthe model of Figure 10 (Equation 15). Parameters of the fits: SW,σ = 0.24, n/m = 0.94; JB, σ = 0.12, n/m = 0.78.


sults at low CI (CI ≤ 0.05), where the threshold function is accelerating, rather than compressive. This behavior can be accommodated in a divisive feedback structure for contrast gain control by assuming that the output O(t) of the divi-sive gain control consists of a deterministic part I(t)/A plus an additive noise N(t) with standard deviation σ (see Figure 10). Then, assuming that the noise N(t) is uncorrelated with I(t)/A,

22

σ+

=

ACC IO . (15)

The model of Figure 10 yields a non-zero value σ for the adaptation signal A when the contrast CI at the input equals zero (i.e. for a non-flickering background). An alter-native (and closely related) way to obtain a non-zero A at zero input contrast would be to include an additive con-stant in the feedback path of Figure 10. Each of these pos-sible elaborations of the model of Figure 7b would prevent a division by zero when the input contrast CI becomes zero for long durations. The lines in Figure 9 show that the model of Figure 10 can provide a good fit to the data over the complete range of flicker contrast CI. Using Equation 15), we determined the values for n/m in Equation 11 that yield optimal (minimalχ2) fits to the steady-state data. Re-sults as shown in Table 3 indicate that on average n/m ≈ 1 ,hence n ≈ m. Altogether we conclude that a feedback struc-ture for divisive gain control, as indicated by Equation 11 with n ≈ m ≈ 2–3, can explain both dynamic and steady-state results of contrast adaptation.

Discussion

Adaptation after contrast onsets Fast adaptation after the onset of contrast was shown

in previous psychophysical experiments. Wu et al. (1997) show that detection thresholds for brief test pulses can fol-low a Gaussian contrast envelope of a 30 Hz background flicker, with no evidence of delay in onset or in the thresh-old peak relative to the peak of the envelope. Using abrupt onsets of the background flicker, Wolfson and Graham

(2000) show that the detection thresholds for a brief test pulse reach steady state approximately 10-30 ms after the onset of a 9.4 Hz flicker of the background. Snowden (2001) shows that the detection thresholds for a 10 ms test pulse reach a steady plateau within 100 ms of the onset of a 16 Hz flicker of the background. From a theoretical analysis of their data, Foley and Boynton (1993) conclude that ad-aptation (desensitization) after the onset of background contrast occurs on a time-scale of 10-50 ms. These time scales of adaptation after the onset of contrast are similar to our results in Figure 3a. From their results on contrast dis-crimination under dynamic contrast conditions, Dannemil-ler and Stephens (1998, 2000) conclude that contrast gain control is much slower, with an integration time of at least 125 ms. However, in their model Dannemiller and Stephens take into account only the effects of the spatial contrast of their adapting stimuli, and ignore the temporal contrast that is induced by the instantaneous switch be-tween adapting stimuli with different spatial contrast. It can be shown that when this temporal contrast is taken into account, the results of Dannemiller and Stephens are in fact fully compatible with a fast contrast gain control. Fast gain control after contrast onsets has also been observed in physiological studies, both in the retina (Baccus & Meister, 2002) and in the visual cortex (Albrecht et al., 2002).

+

N(t)

LP Demodulation

O(t)

A(t)

C (t)O

I(t)

Figure 10. A dynamic model for contrast gain control that in-cludes an internal noise N(t) after the divisive gain control.

Adaptation after contrast offsets In comparison to adaptation after the onset of contrast,

adaptation is slower after the offset of contrast. At τ = 640 ms after the offset of a flicker that had a duration of 1280 ms, we still find threshold elevations E = 0.1 – 0.7. Similar threshold elevations are reported in Figure 4 of Foley and Boynton (1993) after the offset of flicker of durations of 200 ms and 2000 ms. That the dynamics of adaptation is slower after decrements of contrast than after increments of contrast was predicted by DeWeese and Zador (1998) for ideal observers. In fact, in Snippe and van Hateren (2003) we show that the dynamics of adaptation after the offset of contrast seen in our experiments is close to what would be expected for an ideal (statistically efficient) estimate of the

Observer n/m SW 0.94±0.15 JB 0.75±0.08 HS 1.37±0.18 RV 0.65±0.13

Table 3. Estimates of the power11. Values shown are means an power exponents n and m in Eq The estimates for n/m are determto the psychophysical data obtaibackground. The estimates for mmates of γ in Table 1, using the r + (1/γ). Finally, n is estimated from(n/m) × m.

m n 2.85±0.28 2.68±0.50 2.75±0.22 2.07±0.28 1.94±0.06 2.66±0.36 2.49±0.13 1.62±0.39

exponents n and m in Equation d SDs for four observers of the

uation 11, and of their ratio n/m.ined from the fit of Equation 15

ned with steady contrasts of the are determined from the esti-

elation γ = 1/(m - 1), hence m = 1 the values of n/m and m as n =


background contrast. However, it is known that for human observers the speed of recovery from adaptation decreases with increasing duration of the adapting contrast (Green-lee, Georgeson, Magnussen, & Harris, 1991; Rose & Lowe, 1982). It is at present unclear if this aspect of recovery from contrast adaptation can also be understood from the ideal-observer calculations of DeWeese & Zador (1998).

Demodulation of flicker Contrast C describes the size of the fluctuations of a

signal S(t) around its mean. We assume that the mean of S(t) equals zero, which is reasonable if S(t) is a signal in the visual system at a stage when processes of subtractive light adaptation and/or temporal high-pass filtering have oc-curred. Then a dynamic contrast C(t) for S(t) can be de-fined by writing S(t) = C(t)s(t), a product of a carrier signal s(t) with zero mean and unit variance and a contrast enve-lope C(t). Extracting the contrast C(t) of a signal S(t) hence amounts to demodulating the effects of the carrier s(t). Such a demodulation can be approximately attained in various ways. Perhaps the simplest way is through a full-wave rectification of S(t) (Victor, 1987). A more precise demodulation can be obtained by combining S(t) with its first and second order temporal derivatives (Snippe et al., 2000).

Yet another method of demodulation combines S(t) with its Hilbert transform (quadrature partner) SH(t), by using the relation C(t) = (S2(t) + SH2(t))1/2 (Adelson & Ber-gen, 1985; Klein & Levi, 1985; Morrone & Owens, 1987). Because the Hilbert transform is not a causal operation, an exact Hilbert transform cannot be implemented in real time. However, a causal Hilbert transform can be approxi-mated using band-pass filters. In the present paper, we have suppressed the exact form of the demodulation operation, since the qualitative dynamics of Equation 11 (including the asymmetry at contrast onsets and offsets) does not de-pend on the method used for demodulation. A fully quan-titative model for contrast gain control, however, would have to specify the operation used for demodulation.

Effects of contrast on the temporal response function

In the present paper we model contrast adaptation with a divisive gain control in which the output O of the gain control equals the input I divided by the gain control signal A, i.e. O = I/A. In fact, however, it is known that contrast gain control acts not only through a divisive operation, but also through a change of the temporal response of the vis-ual system. Contrast speeds up the visual system (Baccus & Meister, 2002; Benardete, Kaplan, & Knight, 1992; Chander & Chichilnisky, 2001; Kim & Rieke, 2001; Shapley & Victor, 1978; Stromeyer & Martini, 2003). To keep the modeling as simple as possible, we have ignored this dependence. Ignoring the effect of contrast on the shape of the temporal response function has the advantage that it yields a simple relation (Equation 6) between the

psychophysical detection thresholds M and the adaptation signal A in the model. Including the effects of contrast on the shape of the pulse response would necessitate a descrip-tion of which aspect of the response to the test pulse (e.g. the peak, variance, area, etc.) is most important for its de-tection. When such a more quantitative model would be desired, however, the effects of contrast on the temporal response can be incorporated by assuming that the relation between the output O and the input I of the contrast gain control is dynamic, rather than simply divisive. For in-stance, O could be an adaptively low-pass filtered version of I (Carandini & Heeger, 1994; Fuortes & Hodgkin, 1964; Sperling & Sondhi, 1968):

AOIdtdO

L −=τ . (16)

Note that a gain control that is purely divisive, O = I/A, corresponds to a value τL=0 in Equation 16. Alternatively, O could be an adaptively high-pass filtered version of I (Vic-tor, 1987):

AOdtdI

dtdO

HH −= ττ . (17)

Assuming that the adaptation signal A is related to O through Equation 11, both Equation 16 and Equation 17 can yield an adaptation after contrast onsets and offsets that is asymmetric, similar to the results shown in Figure 8 for a purely divisive contrast gain control. A quantitative analysis of the effects of Equations 16 and/or 17 on the detectability of brief test pulses, however, is beyond the aim of the present paper.

Conclusions We have shown that adaptation after onsets of flicker is

much faster than adaptation after offsets of flicker. This behavior of human observers is in good accord with theo-retical predictions for statistically optimal observers (DeWeese & Zador, 1998; Snippe & van Hateren, 2003). The asymmetric dynamics of adaptation after onsets and offsets of contrast can be implemented in a natural way using a feedback structure for contrast gain control.

Acknowledgements We wish to thank our observers Johan Kruseman,

René van der Veen, Lila Thymiati, Sri Wahyuni and Joep Bontemps for their input during the experiments. Com-mercial relationships: none.

Corresponding author: H. P. Snippe; email: [email protected].


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IntroductionMethodsApparatus and psychophysical procedureObserversTemporal dynamics of the stimuli

ResultsAsymmetric dynamics after onset and offset of flickerControl experimentsTapered contrastsNoise carriersContrast increments and decrements

Divisive gain controlDetection thresholds for pulses on backgrounds with steady contrast

DiscussionAdaptation after contrast onsetsAdaptation after contrast offsetsDemodulation of flickerEffects of contrast on the temporal response function

ConclusionsAcknowledgementsReferences

Documents

University of Groningen Asymmetric dynamics of adaptation ...Snippe, Poot, & van Hateren, 2000; Wu, Burns, Elsner, Eskew, & He, 1997). In these experiments, detection thresholds for