The Poisson shot noise model of visual short-term memory and choice response time: Normalized coding by neural population size

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Highlights

  • Presents a normalized coding model of the statistics of stimulus representations.

  • Normalized coding predicts a constant diffusion coefficient decision model.

  • Normalized coding predicts the information capacity of visual short-term memory.

  • The model reconciles previously incompatible bodies of theory and experimentation.

Abstract

A normalized coding condition is proposed that provides a theoretical link between the Poisson shot noise model of choice response time and a Poisson neuron model of the information capacity of visual short-term memory (VSTM). In both models, noise in the cognitive representations of stimuli is attributed to Poisson variability in the neural processes that encode them. In VSTM, a Poisson coding model predicts the invariance of i(di)2 across displays of different sizes, as is found experimentally, but it incorrectly predicts that the diffusion coefficient of the approximating diffusion process will decrease with its drift rate. Normalized coding assumes that the squared magnitudes of the random perturbations to a stimulus representation are inversely proportional to the number of Poisson neurons that represent it. The normalized coding model correctly predicts both a constant diffusion coefficient and the invariance of i(di)2, as required by the experimental data. Normalized coding reconciles the theoretical and empirical properties of diffusion models of decision-making and the sample-size model of VSTM.

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 λj(n)=nλj, j{1,2}, n=1,2,, and a scaled sequence of random variables, Z(n)=Z/n, such that, as the Poisson rates become large, the size of the shocks they induce becomes small. It is convenient here to think of n 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.

References (86)

  • P.L. Smith et al.

    Psychology and neurobiology of simple decisions

    Trends in Neurosciences

    (2004)
  • P.L. Smith et al.

    Modeling perceptual discrimination in dynamic noise: Time-changed diffusion and release from inhibition

    Journal of Mathematical Psychology

    (2014)
  • P.L. Smith et al.

    Attention orienting and the time course of perceptual decisions: Response time distributions with masked and unmasked displays

    Vision Research

    (2004)
  • M.M. Taylor et al.

    Quantification of shared capacity processing in auditory and visual discrimination

    Acta Psychologica

    (1967)
  • X.-J. Wang

    Probabilistic decision making by slow reverberation in cortical circuits

    Neuron

    (2002)
  • A. Agresti

    Categorical data analysis

    (1990)
  • D.E. Anderson et al.

    Precision in visual working memory reaches a stable plateau when individual item limits are exceeded

    Journal of Neuroscience

    (2011)
  • P.M. Bays

    Noise in neural populations accounts for errors in working memory

    The Journal of Neuroscience

    (2014)
  • R.N. Bhattacharya et al.

    Stochastic processes with applications

    (1990)
  • A.M. Bonnel et al.

    Divided attention between simultaneous auditory and visual signals

    Perception & Psychophysics

    (1998)
  • A.M. Bonnel et al.

    Attentional effects on concurrent psychophysical discriminations: Investigations of a sample-size model

    Perception & Psychophysics

    (1994)
  • J. Busemeyer et al.

    Decision field theory: A dynamic-cognitive approach to decision making in an uncertain environment

    Psychological Review

    (1993)
  • N.R. Campbell

    The study of discontinuous phenomena

    Proceedings of the Cambridge Philosophical Society

    (1909)
  • M. Carandini et al.

    Normalization as a canonical neural computation

    Nature Reviews Neuroscience

    (2012)
  • M. Coltheart

    Iconic memory and visible persistence

    Perception & Psychophysics

    (1980)
  • N. Cowan

    The magical number 4 in short-term memory: A reconsideration of mental storage capacity

    Behavioral & Brain Sciences

    (2001)
  • D.R. Cox et al.

    The theory of stochastic processes

    (1965)
  • C.D. Creelman et al.

    Detection theory: a user’s guide

    (1991)
  • J. Ditterich

    Evidence for time-variant decision making

    European Journal of Neuroscience

    (2006)
  • C. Donkin et al.

    The overconstraint of response time models: Rethinking the scaling problem

    Psychonomic Bulletin & Review

    (2009)
  • C. Donkin et al.

    Discrete-slots models of visual working-memory response times

    Psychological Review

    (2013)
  • W. Gerstner et al.

    Spiking neuron models

    (2002)
  • I.C. Gould et al.

    Spatial uncertainty explains exogenous and endogenous attentional cuing effects in visual signal detection

    Journal of Vision

    (2007)
  • D.P. Hanes et al.

    Neural control of voluntary movement initiation

    Science

    (1996)
  • R.A. Heath

    A note on channel uncertainty in multidimensional signal detection tasks

    Psychonomic Science

    (1972)
  • C.H. Hesse

    The one-sided barrier problems for an integrated Ornstein–Uhlenbeck model

    Communications in Statistics. Stochastic Models

    (1991)
  • J.D. Ingleby

    The separation of bias and sensitivity in multiple-choice tasks

    Perception

    (1973)
  • G. Kallianpur

    On the diffusion approximation to a discontinuous model for a single neuron

  • G. Kallianpur et al.

    Infinite dimensional stochastic differential equation models for spatially distributed neurons

    Applied Mathematics and Optimization

    (1984)
  • S. Karlin et al.

    A second course in stochastic processes

    (1981)
  • D.R.J. Laming

    Information theory of choice reaction time

    (1968)
  • Cited by (0)

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