The Poisson shot noise model of visual short-term memory and choice response time: Normalized coding by neural population size
Section snippets
Introduction: the Poisson shot noise model of choice response time
One of the theoretical goals of cognitive neuroscience is to develop mathematical models that link behavioral and neural levels of description. In the area of speeded decision-making, progress towards this goal has received impetus from the discovery that models of choice response time (RT) developed in mathematical psychology to account for the speed and accuracy of behavioral decisions can also characterize the time course of the underlying neural processes. Smith and Ratcliff (2004) reviewed
The variance scaling problem
The assumption that discriminative information is carried by the difference between excitatory and inhibitory shot noise processes circumvents a problem that would otherwise arise when attempting to derive a choice RT model from assumptions of Poisson stimulus encoding, namely, the problem of variance scaling. The natural prediction of Poisson models is that the mean and variance of the encoded sensory representations will be equal to each other, because the number of Poisson distributed events
The information capacity of visual short-term memory
Fig. 3 summarizes the results of a VSTM experiment reported by Sewell, Lilburn, and Smith (2014) that was designed to investigate the memory representations that support perceptual decision-making. Their study was motivated in part by earlier work by Ratcliff and Rouder (2000) and Smith, Ratcliff, and Wolfgang (2004), who showed that speeded decisions about brief, backwardly-masked stimuli are well described by a diffusion model with constant drift, in which there is negligible decay of the
Normalization of Poisson shot noise stimulus variance by population size
The main result in this section makes use of a variant of the construction used to obtain a limiting OU velocity process from a high-intensity shot noise pair in Smith (2010, Section 5.3). That construction involves sequences of Poisson processes with rates , , , and a scaled sequence of random variables, , such that, as the Poisson rates become large, the size of the shocks they induce becomes small. It is convenient here to think of as indexing the size of the
Realization of normalization via nonlinear dynamics
The normalization of Poisson shock amplitudes discussed in the previous section can be realized dynamically by systems of shunting differential equations. In shunting equations, unlike the more commonly used additive equations of linear systems theory, the input to the system or the system’s own internal state multiplicatively gates the change in its response. The Hodgkin–Huxley equations of neural conduction are equations of this type (Tuckwell, 1988). The properties of shunting equations have
Link to random walk/diffusion measures of sensitivity
In Section 4, I showed how normalized coding, combined with the shot noise model of stimulus representation, leads to a signal detection model that realizes the predictions of the sample-size model. My aim in proposing the normalized coding condition was to show how the shot noise model leads to a diffusion process representation in which the drift rate scales in inverse proportion to the square root of the number of items in the display while the diffusion coefficient remains constant.
Discussion
The results in the previous sections follow straightforwardly from the properties of the shot noise decision model described by Smith (2010). Despite their simplicity, they are important theoretically because they allow us to show that models, which are not, a priori, theoretically and mathematically compatible with each other, can be made so by an appropriate scaling assumption. The sample-size model provides a parameter-free account of the VSTM representations that support two-choice
Acknowledgments
The research in this article was supported by Australian Research Council Discovery Grant DP140102970. I would like to express my appreciation for the hospitality of the Department of Psychology at Vanderbilt University, where preparation of this article was carried out and, in particular, to Gordon Logan, for research support. I thank E-J. Wagenmakers, Rani Moran, and Chris Donkin for helpful comments on an earlier version of the article.
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