Light adaptation controls visual sensitivity by adjusting the speed and gain of the response to light

The range of c. 1012 ambient light levels to which we can be exposed massively exceeds the <103 response range of neurons in the visual system, but we can see well in dim starlight and bright sunlight. This remarkable ability is achieved largely by a speeding up of the visual response as light levels increase, causing characteristic changes in our sensitivity to different rates of flicker. Here, we account for over 65 years of flicker-sensitivity measurements with an elegantly-simple, physiologically-relevant model built from first-order low-pass filters and subtractive inhibition. There are only two intensity-dependent model parameters: one adjusts the speed of the visual response by shortening the time constants of some of the filters in the direct cascade as well as those in the inhibitory stages; the other parameter adjusts the overall gain at higher light levels. After reviewing the physiological literature, we associate the variable gain and three of the variable-speed filters with biochemical processes in cone photoreceptors, and a further variable-speed filter with processes in ganglion cells. The variable-speed but fixed-strength subtractive inhibition is most likely associated with lateral connections in the retina. Additional fixed-speed filters may be more central. The model can explain the important characteristics of human flicker-sensitivity including the approximate dependences of low-frequency sensitivity on contrast (Weber’s law) and of high-frequency sensitivity on amplitude (“high-frequency linearity”), the exponential loss of high-frequency sensitivity with increasing frequency, and the logarithmic increase in temporal acuity with light level (Ferry-Porter law). In the time-domain, the model can account for several characteristics of flash sensitivity including changes in contrast sensitivity with light level (de Vries-Rose and Weber’s laws) and changes in temporal summation (Bloch’s law). The new model provides fundamental insights into the workings of the visual system and gives a simple account of many visual phenomena.


Time-domain representation of the model
Going from the frequency domain to the time domain strictly requires knowledge of the phase response as well as amplitude response (Equation (2)) of the system. However, we note that the phase responses of LP-stages are determined by their temporal response (which is an exponential decay over time), and our model can be represented in the complex Fourier domain as, and, 1 1 Equation (B) has been plotted in the lower right orange panel of Fig 8 with k=0.80, fc=15 and fcL=30; g has been chosen to normalise the peak to 1.

The amplitude response of a cascade of leaky integrators is asymptotically a power law, but exponential for a range of visually significant frequencies
The amplitude response, A(f), of a single stage of leaky integration (low-pass filtering), or an "LP-stage" for short, is given by, where f is frequency in Hz, when f is large compared to fc, is given by, For values of f close to fc the expansion will be dominated by the constant and linear terms, i.e., been noted many times before [15,16] and has occasionally been used as justification for rejecting a leaky integrator model altogether [17] because the measurements at high frequencies don't assume this form. However, between about 0.36fc and 1.92fc a cascade of n identical LP-stages has an approximately exponential frequency response with an exponent of ( )

For a cascade of leaky integrators all the corner frequencies (or equivalently, all the time constants) should take similar values.
In addition to producing the largest range of frequencies over with the TCSF is approximately exponential, it is also the case that having identical corner frequencies produces maximal temporal contrast sensitivity at all frequencies. Consider the simplest case with two stages with corner frequencies C1 and C2. The amplitude response, A(f), is then given by: where f is temporal frequency in Hz. Note that the numerator is the product of the corner frequencies in order to set the low frequency asymptote to 1. The high-frequency asymptote on a log-log plot is then a straight line with slope -2 which passes through the point (C1C2, 1). The way we have defined our amplitude response means the low-frequency limb is fixed at A=1 for all C1 and C2 and the location of the high-frequency limb depends solely on the product of C1 and C2, so the family of curves that have the same low-and high-frequency limbs can be parametrised as 1 C C X = and 2 C CX = , for some C (which is the geometric mean of C1 and C2). C and X are both greater than 0. In the case when the corner frequencies are identical, i.e., X=1, Equation (G) reduces to: where the subscripts s and d denote corner frequencies that are the same or different. The question is which of these two amplitude responses is larger for any given frequency. We note that C, X and f are all strictly positive and so both As and Ad are positive, which therefore means that s . We have: A more intuitive explanation of why having identical corner frequencies is optimal is to consider a serial cascade of two filters (leaky integrators), one of which is slower (has a lower corner frequency) than the other. If the slower filter comes first, then the second filter will be capable of passing higher frequencies than it receives as input and its sensitivity would be wasted. Similarly, if the slower filter comes second then the first filter passes high frequencies which the second attenuates, which again would be wasteful. If the filters have the same corner frequency, then they are optimally tuned to pass (or attenuate) the same set of frequencies. While it is highly unlikely that all the significant LP-stages of light adaptation in the visual system have identical time constants, any process which seeks to optimise the efficiency of the system might be expected to make them as near to identical as possible.

Subtractive inhibition maintains the exponential fall-off but reduces its slope and shifts it to higher frequencies
While a cascade of LP-stages can produce an exponential decline in sensitivity over the visible range of flicker frequencies, it does not explain the loss of sensitivity at low frequencies. We model the loss of low-frequency sensitivity as the result of inhibition. A standard high-pass filter can be constructed by passing a signal through a leaky integrator with unity DC gain and subtracting the result from the original signal. If the DC gain is less than unity, the filter will only partially cancel signals at low frequencies. These kinds of filter can be thought of as partial high-pass filters and are often used in electric engineering (where they are referred to as "lead-compensators") in order to increase stability in control circuits and sharpen the temporal response. The amplitude response of such a filter is given by: where k is the gain of the filter whose response is to be subtracted and can be thought of as the strength of inhibition. Note that when k=1 there is complete inhibition at low frequencies; i.e., a standard high-pass filter, while for k=0 there is no inhibition and the numerator and denominator in

High-frequency linearity and low frequency Weber's law
At high frequencies (when f >> fc & fcL) the equation for our model (J) simplifies to: which, in the traditional double logarithmic coordinates of Bode [19], is a straight-line with slope Tds. However, as noted above, the region where the power law approximation holds lies largely above the temporal acuity limit above which sensitivity cannot be measured and below which TCSFs are approximately exponential functions. In semi-logarithmic coordinates the exponential losses in sensitivity appear as straight-lines of different slopes, and so cannot strictly conform to "highfrequency linearity", i.e., they could only coincide at a single intersection point, not over an extended range. Plotted on a logarithmic frequency scale, however, these lines accelerate downwards and only appear to coincide. We suggest that the notion of high-frequency linearity is an inappropriate inference based on the way in which amplitude sensitivity has been plotted in the past; it is not a feature of visual sensitivity.
where W is a constant that is proportional to the Weber fraction. Below about 3.4 log10 Td, g is constant (Fig 9[B]) so Weber's law will hold if fc increases in proportion to the 4 th root of I. However, the best fitting power law function relating fc to I (see Equation (3), plotted as the blue curve in Fig   9[A]) has an exponent of 0.181, which corresponds to the 5.5 th root of I. Above 3.4 log10 Td, fc is constant while g decreases in proportion to I (see Equation (4) and Fig 9[B]), so here W will be constant. Thus, according to our model, Weber's law is not strictly maintained at low frequencies at low-to mid-light levels. Note that low-frequency flicker thresholds are distinct from flash thresholds, which do indeed show good adherence to Weber's law over a large range of light levels [20], but this will likely depend on the shape and size of the impulse response rather than flicker sensitivity at any specific frequency.  Best-fitting variable corner frequencies (fc), logarithmic gains (log10(g)) and feedforward gain (k). Two LP-stages, which were fixed across observers, had corner frequencies of 30.92 ±2.23 Hz. Adjusted R 2 for simultaneous fit = 99.6% Levels are from low to high as listed in the keys of Figure 2-6. For details see text.