Summation of Quantal Noise in Space and Time

Noise in visual neurons, or variability in psychophysical experiments, may be quantified in terms of photon fluctuations from an ‘equivalent’ steady illumination. The conversion requires assumptions on how photon signals are pooled in space and time, ie how to pass from the light flux to the numbers of photon events relevant to the Poisson statistics describing signal/noise. Real weighting profiles for the integration of photon events in space and time [the sensitivity distribution of the receptive field (RF) and the waveform of the impulse response (IR)] are commonly approximated by sharp-bordered apertures of ‘complete’, equal-weight summation of events. Such apertures based on signal equivalence cannot provide noise equivalence, however, because greater numbers of events summed with lower weights (as in reality) have lower variances than smaller numbers summed with full weight. Thus sharp-bordered apertures are necessarily smaller if defined for noise equivalence rather than for signal equivalence. We have calculated the difference for some commonly encountered RF and IR profiles. Summation areas, expressed as numbers of photoreceptors (cones or rods) contributing with equal weight, are denoted N_S for signal and N_N for noise, and sharply delimited summation times are correspondingly denoted t_S and t_N. We show that the relation in time is t _N=0.6t_S to 0.7t_S for realistic quantal response waveforms of photoreceptors. In space, the relation is N_N=0.5N_S for the Gaussian distribution (eg for the RF centre mechanism of retinal ganglion cells). For a photoreceptor in an electrically coupled network the difference is still greater, eg for rods in the toad retina N _N=0.2N_S (N_S=13.7 rods and N_N=2.8 rods). We introduce a third possible definition of sharp-bordered summation apertures: one that provides the same signal-to-noise ratio (SNR) for large-long stimuli as the real integration profiles. The SNR-equivalent summation area is N*=N_S²/N_N and the summation time is t*=t_S²/t_N.