Noise-equivalent and signal-equivalent visual summation of quantal events in space and time

Noise recorded in visual neurons, or variability in psychophysical experiments, may be quantified in terms of quantal fluctuations from an "equivalent" steady illumination. The conversion requires assumptions concerning how photon signals are pooled in space and time, i.e. how to pass from light fluxes to numbers of photon events relevant to the Poisson statistics describing signal/noise. It is usual to approximate 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]) by sharp-bordered apertures of "complete," equal-weight summation of events. Apertures based on signal-equivalence cannot provide noise-equivalence, however, because greater numbers of events summed with smaller 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- than for signal-equivalence. We here consider 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 NS for signal and NN for noise; sharply delimited summation times are correspondingly denoted tS and tN. We show that the relation in space is NN = 0.5 NS for the Gaussian distribution (e.g. for the RF center mechanism of retinal ganglion cells). For a photoreceptor in an electrically coupled network the difference is even larger, e.g., for rods in the toad retina NN = 0.2 NS (NS = 13.7 rods and NN = 2.8 rods). In time, the relation is tN approximately 0.7 tS for realistic quantal response waveforms of photoreceptors. The surround input in a difference-of-Gaussians RF may either decrease or increase total noise, depending on the degree of correlation of center and surround noise. We introduce a third useful 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* = N2S/NN and summation time t* = t2S/tN.