We measured human psychophysical detection thresholds for test pulses which are superimposed on spatially homogeneous 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.

*τ*(positive or negative) relative to the beginning or the ending of the sinusoidal flicker. Threshold strength for detection of the test pulse was measured as a function of

*τ*.

*τ*. Because the luminance on which the test pulse is superimposed 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 dynamics 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 reduce 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 luminance 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.

*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

*M*(

_{i}*τ*), and the steady-state threshold of this observer for a test presented on a steady background as

*M*

^{−}

_{j}, the normalised threshold elevation

*E*(

_{i}*τ*) above the steady state level can be quantified as At

*τ*= 40 ms after contrast offset, threshold elevations

*E*(40) for our observers range from 1.3 to 3.5; the arithmetic mean of their threshold elevations is 2.2±0.3(SE;

_{i}*n*= 8).

*τ*= 640 ms after contrast offset, threshold elevations

*E*(640) above the steady-state level are still quite high (range

_{i}*E*(640) = 0.1–0.7 ; mean±s.e.m. = 0.38±0.08). Thresholds do not further decline much below the level obtained at

_{i}*τ*= 640 ms within the remaining 640 ms of steady background before flicker is switched on again.

*τ*≥ 20 ms: The value of the power exponent

*γ*ranges approximately between 0.5 and 1 for our observers (Table 1).

Observer | γ | τ_{1}(ms) |
---|---|---|

RV | 0.67±0.06 | 90±5 |

HS | 1.06±0.07 | 116±12 |

LP | 0.62±0.12 | 104±18 |

JB | 0.57±0.07 | 273±61 |

JK | 0.63±0.08 | 92±10 |

JH | 0.63±0.14 | 153±42 |

LT | 1.06±0.06 | 63±8 |

SW | 0.54±0.08 | 84±12 |

Average Observer | 0.65±0.05 | 122±9 |

*t*= −

*T/4*and

*t*= +

*T/4*, for

*C*(

_{on}*t*) the flicker contrast gradually increases from 0 to its full value

*C*= 0.8. Likewise, at flicker offset

*C*(

_{off}*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 artifacts due to the onset and offset of the contrast. Nevertheless, detection thresholds for the test pulse show overshoots at the contrast onset and offset also for these tapered contrasts (Figure 5). Also the asymmetric dynamics of thresholds 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.

τ | flicker | noise |
---|---|---|

80 ms | 1.09±0.12 | 1.45±0.14 |

320 ms | 0.49±0.08 | 0.89±0.11 |

*C*= 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 observer 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 elevation 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 contrast (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.

*O*(

*t*) of the gain control equals the input

*I*(

*t*) divided by the control (adaptation) signal

*A*(

*t*):

*A*(

*t*) has a particularly 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

*Δ*

*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 output

*Δ*

*O*, at detection threshold = constant. 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 control.

*A*(

*t*) in a divisive gain control 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 feedforward gain control (Figure 7a) in which the dynamics of

*A*(

*t*) is governed by the contrast

*C*(

_{I}*t*) of the input signal

*I*(

*t*): The form of Equation 7 is such that

*A*(

*t*) is a low-pass filtered version of

*C*(

_{I}*t*). This can be readily seen by Fourier transforming the equation, yielding 1/(1 +

*iωτ*

_{0}) as the transfer function between the frequency domain versions of

*C*and

_{I}*A*. Equation 7 yields identical dynamics for

*A*(

*t*) (an exponential response with time constant

*τ*

_{0}) at the onset and the offset of the contrast

*C*(

_{I}*t*). This would still be the case if a small positive constant

*ɛ*would be added to the right-hand-side of Equation 7 (such that

*A*attains a finite steady-state value

*ɛ*, rather than zero, for a constant background

*C*= 0). Although the resulting dynamics of adaptation 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 adaptation signal

_{I}*A*(

*t*) is governed by the contrast

*C*(

_{O}*t*) of the output

*O*(

*t*) of the gain control loop, rather than by the contrast

*C*(

_{I}*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*): Using the divisive nature of the gain control:

*C*(

_{O}*t*) =

*C*(

_{I}*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

*dA*

^{2}/

*dt*: Equation 9 shows that for the first-order feedback dynamics of Equation 8, the square

*A*

^{2}(

*t*) of the adaptation signal

*A*(

*t*) is a low-pass filtered version (with time constant

*τ*

_{0}/ 2) of the input contrast hence

*C*(

_{I}*t*),

*A*(

*t*) is asymmetric for onset versus offset steps in the input contrast

*C*(

_{I}*t*). The asymmetric dynamics 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

*C*(

_{O}*t*) =

*C*(

_{I}*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 contrast

*C*(

_{O}*t*), which explains the observed asymmetry in

*A*(

*t*) seen in Figure 8.

*C*(

_{I}*t*) =

*C*(

_{O}*t*) = 0 for

*t*> 0), the solution

*A*(

*t*) of Equation 8 is an exponential:

*A*(

*t*) =

*A*(0)exp(−

*t*/

*τ*

_{0}), 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: 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 contrast offset: For large enough

*t*, the recovery of

*A*(

_{off}*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) corresponds to the steepness parameter

*γ*defined in Equation 2 (strictly speaking, Equation 1 is not defined for the present model because

*M*

^{−}

_{i}= 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

*C*(

_{I}*t*), for the choice

*n*=

*m*= 2, respectively

*n*=

*m*= 3. The response dynamics is much faster after a contrast 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 behavior of Equation 11, note that for

*n*=

*m*= 2 its right-hand-side can be rewritten as [

*C*(

_{O}*t*) +

*A*(

*t*)][

*C*(

_{O}*t*) −

*A*(

*t*)]. Hence, dividing both sides of Equation 11 by

*C*(

_{O}*t*) +

*A*(

*t*), it can (for

*n*=

*m*= 2) be rewritten as From the similarity with Equation 8, Equation 13 can be understood as a low-pass filter with an effective time constant

*τ*=

_{eff}*τ*

_{0}/[

*C*(

_{O}*t*) +

*A*(

*t*)], which is small (fast adaptation) when

*C*and/or

_{O}*A*are large, and which is large (slow adaptation) when

*C*and

_{O}*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) compared to the dynamics for

*n*=

*m*= 1 (solid lines). A similar decomposition

*C*

_{O}^{3}−

*A*

^{3}= (

*C*

_{O}^{2}+

*A*

^{2}+

*C*)(

_{O}A*C*−

_{O}*A*) of the right-hand-side of Equation 11 can explain the strong asymmetry in Figure 8 for

*n*=

*m*= 3 (dotted lines).

*m*of

*A*(

*t*) in Equation 11 to values

*m*= 2 − 3. Estimates of the power exponent

*n*of

*C*(

_{O}*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

*C*(i.e. with steadily flickering backgrounds). Using the relation

_{I}*C*=

_{O}*C*/

_{I}*A*, the steady-state solution of Equation 11 is Hence for

*n*=

*m*the adaptation signal

*A*increases in proportion to the square root of the flicker contrast

*C*. For

_{I}*n*<

*m*the relation between

*A*and

*C*would be more compressive than a square root, and for

_{I}*n*>

*m*it would be less compressive (i.e. more linear).

*C*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.

_{I}*C*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 (

_{I}*f*= 6.25 Hz and

*f*= 12.5 Hz). As expected from Equation 14, thresholds are a compressive function of the flicker contrast

*C*over most of the contrast range. An exception are the results at low

_{I}*C*(

_{I}*C*≤ 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

_{I}*O*(

*t*) of the divisive 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*,

*σ*for the adaptation signal

*A*when the contrast

*C*at the input equals zero (i.e. for a non-flickering background). An alternative (and closely related) way to obtain a non-zero

_{I}*A*at zero input contrast would be to include an additive constant in the feedback path of Figure 10. Each of these possible elaborations of the model of Figure 7b would prevent a division by zero when the input contrast

*C*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

_{I}*C*. Using Equation 15, we determined the values for

_{I}*n/m*in Equation 11 that yield optimal (minimal χ

^{2}) fits to the steady-state data. Results as shown in Table 3 indicate that on average

*n/m*≈ 1, hence

*n*≈

*m*. Altogether we conclude that a feedback structure 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.

Observer | n/m | m | n |
---|---|---|---|

SW | 0.94±0.15 | 2.85±0.28 | 2.68±0.50 |

JB | 0.75±0.08 | 2.75±0.22 | 2.07±0.28 |

HS | 1.37±0.18 | 1.94±0.06 | 2.66±0.36 |

RV | 0.65±0.13 | 2.49±0.13 | 1.62±0.39 |

*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 background contrast. However, it is known that for human observers the speed of recovery from adaptation decreases with increasing duration of the adapting contrast (Greenlee, 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).

*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 occurred. Then a dynamic contrast

*C*(

*t*) for

*S*(

*t*) can be defined 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 envelope

*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).

*S*(

*t*) with its Hilbert transform (quadrature partner)

*S*(

_{H}*t*), by using the relation

*C*(

*t*) = (

*S*

^{2}(

*t*) +

*S*

_{H}^{2}(

*t*))

^{1/2}(Adelson & Bergen, 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 approximated 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 depend on the method used for demodulation. A fully quantitative model for contrast gain control, however, would have to specify the operation used for demodulation.

*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 visual 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 description of which aspect of the response to the test pulse (e.g. the peak, variance, area, etc.) is most important for its detection. 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 instance,

*O*could be an adaptively low-pass filtered version of

*I*(Carandini & Heeger, 1994; Fuortes & Hodgkin, 1964; Sperling & Sondhi,1968):

*O*=

*I/A*, corresponds to a value

*τ*=0 in Equation 16. Alternatively,

_{L}*O*could be an adaptively high-pass filtered version of

*I*(Victor, 1987):

*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.

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