Probability Summation over Time

The sensory rcsponse lo ligltl persists in lime. This fact obliges us to ui'n.ierst:ind how the temporally distrihuted effects of visual slimukition collectively determine thrcshokl, A cl;tssieiiI solution to this problem has been to supposc lh;il the eye integrates the light sis,n;il ovrr somc inlcrviil of timc, the criliciil durcition. The light is visible whenever the intcgritl exceeds some fixed value. The lincling which nlotivated this iden was the apparent reciprocity between intensity inti lime for stimuli at threshold (Bloch, 1885; Grahum and Margiiriii, 1935). but this result is equally well i ~ c c o m n ~ < ~ l a l c d by a more general inodci. This mot-Id supposes the eye lo opcriilc in the vicinity of thrcsholil ;is a linc;ir temporal filter followed by ii thrcshold mcchtinism, which responds only when the excursions of the filtcrcd sirnal exceed some fixed magnilu~lc (Dc Lunge, 1952; Kelly, 1901; Spcrling and Sonilhi. 1968: Roilfs. 1972. I n this cwk-iir-trc~uilh di,lrclor, n,mporal sitmificilinm occurs as a consequence of the integrating action of the filter. In yet another model the line;ir filter is followed by ;I squaring device, and the lillercd, squiircd signiil is integritccl over some finite inlerviil. In the version of this p o w micip-itlor pi-oposrd by Rashhiiss (1970) the intcgmtion intcrviil is ;ihoul 200n1scc: in that of Kocnderink :mil v;in Doom (19781 it is about 500msec. A fcalnrc common to ;ill of thc^ie proposals is that they arc dclcrrninistic: thcy (lo not ziticmpt to represent the prohiihilistic milurc of detectioi~. We may imtiginc the variiihilily of clcleclion to be due to 'noise" in the visual process. I f thc lime scale of thc noise fluct~iiitions is long rcli~tive to that of the model, then it can properly he neglected. I f llic two scales are compiirdblr, however, then the intrusion of noise may render thc iihovc modcls inciipiihle of correctly predicting thc visibility of long ;is well as short signtil waveform, thc duration of each instililt mu\t be less than half the period of the highesi frequency present in the signal. The presence of noise ensures that within each instant there is some probability that thr~shold will be exceeded, ;ind so the overall probability that the signal is cleleclei.1 must t;ike inlo account ;ill of these momentary probabilities. L.ct P; be the probability that threshold is exceeded during the instant beginning ;it timc t ,Â¥ I f the fluctuations of the noise are suflicienlly rapicl (if the iiut~eorrcltition function of the noise process is sulficiently narrow), then these probabilities will he independent from instant to instant. 'Then supposing th;it the signal is detected if and only if threshold is exceeded n I s t one instant, the probribility of cletcctio~i, 1'. will be