The transducer model for contrast detection and discrimination: formal relations, implications, and an empirical test.

The transducer function mu for contrast perception describes the nonlinear mapping of stimulus contrast onto an internal response. Under a signal detection theory approach, the transducer model of contrast perception states that the internal response elicited by a stimulus of contrast c is a random variable with mean mu(c). Using this approach, we derive the formal relations between the transducer function, the threshold-versus-contrast (TvC) function, and the psychometric functions for contrast detection and discrimination in 2AFC tasks. We show that the mathematical form of the TvC function is determined only by mu, and that the psychometric functions for detection and discrimination have a common mathematical form with common parameters emanating from, and only from, the transducer function mu and the form of the distribution of the internal responses. We discuss the theoretical and practical implications of these relations, which have bearings on the tenability of certain mathematical forms for the psychometric function and on the suitability of empirical approaches to model validation. We also present the results of a comprehensive test of these relations using two alternative forms of the transducer model: a three-parameter version that renders logistic psychometric functions and a five-parameter version using Foley's variant of the Naka-Rushton equation as transducer function. Our results support the validity of the formal relations implied by the general transducer model, and the two versions that were contrasted account for our data equally well.

Are long gamma-ray bursts biased tracers of star formation? Clues from the host galaxies of the Swift/BAT6 complete sample of bright LGRBs II. Star formation rates and metallicities at z < 1

Aims. Long gamma-ray bursts (LGRBs) are associated with the deaths of massive stars and might therefore be a potentially powerful tool for tracing cosmic star formation. However, especially at low redshifts (z< 1.5) LGRBs seem to prefer particular types of environment. Our aim is to study the host galaxies of a complete sample of bright LGRBs to investigate the effect of the environment on GRB formation. Methods. We studied host galaxy spectra of the Swift/BAT6 complete sample of 14 z< 1 bright LGRBs. We used the detected nebular emission lines to measure the dust extinction, star formation rate (SFR), and nebular metallicity (Z) of the hosts and supplemented the data set with previously measured stellar masses M_*. The distributions of the obtained properties and their interrelations (e.g. mass-metallicity and SFR-M_* relations) are compared to samples of field star-forming galaxies. Results. We find that LGRB hosts at z< 1 have on average lower SFRs than if they were direct star formation tracers. By directly comparing metallicity distributions of LGRB hosts and star-forming galaxies, we find a good match between the two populations up to 12 +log (O/H)~8.4−8.5, after which the paucity of metal-rich LGRB hosts becomes apparent. The LGRB host galaxies of our complete sample are consistent with the mass-metallicity relation at similar mean redshift and stellar masses. The cutoff against high metallicities (and high masses) can explain the low SFR values of LGRB hosts. We find a hint of an increased incidence of starburst galaxies in the Swift/BAT6 z< 1 sample with respect to that of a field star-forming population. Given that the SFRs are low on average, the latter is ascribed to low stellar masses. Nevertheless, the limits on the completeness and metallicity availability of current surveys, coupled with the limited number of LGRB host galaxies, prevents us from investigating more quantitatively whether the starburst incidence is such as expected after taking into account the high-metallicity aversion of LGRB host galaxies.