Figure 5.

The effect of temporal filtering. (a–d) The outputs to pulses separated by (a) 0, (b) 50 and (c) 100 ms. The filter is a one-lobe, low-pass causal filter given by

$$f(\tau )=\tau \text{exp}(-\tau /\mu ),$$ (2)

where μ is the time constant of the filter. Examples of the impulse response function (the response of the filter to a single brief pulse) are shown in (f). (a–c) The responses of a filter with μ=100 ms to double-pulse stimuli with a separation of 1, 50 and 100 ms. The blue-shaded area bounded represents the region of concavity in the curve, with the dashed line bounding this region and the tangent connecting the curve over this concavity. The concavity represents a qualitative difference in the output, clearly more different between separations of 50 and 100 ms than separations 0 and 50 ms. There are many ways to measure this qualitative difference (such as measuring the area of the concavity), but after some experimentation we chose a very simple nonlinearity, the squared response to double stimuli. (d) This measure, expressed as a difference from the response to a fixed separation Δτ and a separation of zero between the pulses. From this internal representation of duration, we calculated for each base duration τ the minimum increase in duration necessary for the response to increase by a threshold value (that we set at 10% of maximum). (e) The results of these simulations (varying μ to give the best fit of the data) against base duration for auditory, visual and bimodal conditions, together with the averaged data. (f) The impulse response functions of the filters that best fit the data. They have time constants μ of 9, 30 and 100 ms, respectively, for the auditory, visual and auditory–visual conditions (red, auditory; blue, visual; green, bimodal).