First Passage Time Properties for Time-varying Diffusion Models: A Martingale Approach

In this work, we use Martingale theory to derive formulas for the expected decision time, error rates, and first passage times associated with a multistage drift diffusion model, or a Wiener diffusion model with piecewise constant time-varying drift rates and decision boundaries. The model we study is a generalization of that considered in Ratcliff (1980). The derivation relies on using the optional stopping theorem for properly chosen Martingales, thus obtaining formulae which may be used to compute performance metrics for a particular stage of the stochastic decision process. We also explicitly solve the case of a two stage diffusion model, and provide numerical demonstrations of the computations suggested by our analysis. Finally we present calculations that allow our techniques to approximate time-varying Ornstein-Uhlenbeck processes. By presenting these explicit formulae, we aim to foster the development of refined numerical methods and analytical techniques for studying diffusion decision processes with time-varying drift rates and thresholds.

[1]  Xiao-Jing Wang,et al.  A Recurrent Network Mechanism of Time Integration in Perceptual Decisions , 2006, The Journal of Neuroscience.

[2]  W. Newsome,et al.  Neural basis of a perceptual decision in the parietal cortex (area LIP) of the rhesus monkey. , 2001, Journal of neurophysiology.

[3]  X. Lin,et al.  Double barrier hitting time distributions with applications to exotic options , 1998 .

[4]  Peter I. Frazier,et al.  Sequential Hypothesis Testing under Stochastic Deadlines , 2007, NIPS.

[5]  Philip Holmes,et al.  Dynamical Analysis of Bayesian Inference Models for the Eriksen Task , 2009, Neural Computation.

[6]  M. Steinhauser,et al.  A dual-stage two-phase model of selective attention. , 2010, Psychological review.

[7]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[8]  Miriam Kesselmeier,et al.  Fast and accurate calculations for cumulative first-passage time distributions in Wiener diffusion models , 2012 .

[9]  Jonathan D. Cohen,et al.  Reward rate optimization in two-alternative decision making: empirical tests of theoretical predictions. , 2009, Journal of experimental psychology. Human perception and performance.

[10]  A. Pouget,et al.  The Cost of Accumulating Evidence in Perceptual Decision Making , 2012, The Journal of Neuroscience.

[11]  D. Navarro,et al.  Fast and accurate calculations for first-passage times in Wiener diffusion models , 2009 .

[12]  Andreas Voss,et al.  A fast numerical algorithm for the estimation of diffusion model parameters , 2008 .

[13]  J. Gold,et al.  The neural basis of decision making. , 2007, Annual review of neuroscience.

[14]  A. Rangel,et al.  Visual fixations and the computation and comparison of value in simple choice. , 2010, Nature neuroscience.

[15]  Steven P Blurton,et al.  Even faster and even more accurate first-passage time densities and distributions for the Wiener diffusion model , 2014 .

[16]  Eduardo Calixto,et al.  Lifetime Data Analysis , 2016 .

[17]  Philip Holmes,et al.  Can Monkeys Choose Optimally When Faced with Noisy Stimuli and Unequal Rewards? , 2009, PLoS Comput. Biol..

[18]  James L. McClelland,et al.  The time course of perceptual choice: the leaky, competing accumulator model. , 2001, Psychological review.

[19]  Borís Burle,et al.  Using Covert Response Activation to Test Latent Assumptions of Formal Decision-Making Models in Humans , 2015, The Journal of Neuroscience.

[20]  Adele Diederich,et al.  Simple matrix methods for analyzing diffusion models of choice probability, choice response time, and simple response time , 2003 .

[21]  C. Gardiner.  Stochastic Methods: A Handbook for the Natural and Social Sciences , 2009 .

[22]  Christof Koch,et al.  The Drift Diffusion Model Can Account for the Accuracy and Reaction Time of Value-Based Choices Under High and Low Time Pressure , 2010, Judgment and Decision Making.

[23]  Roger Ratcliff,et al.  A note on modeling accumulation of information when the rate of accumulation changes over time , 1980 .

[24]  J. Gold,et al.  Neural computations that underlie decisions about sensory stimuli , 2001, Trends in Cognitive Sciences.

[25]  Julie Horrocks,et al.  Modeling Event Times with Multiple Outcomes Using the Wiener Process with Drift , 2004, Lifetime data analysis.

[26]  Roger Ratcliff,et al.  The Diffusion Decision Model: Theory and Data for Two-Choice Decision Tasks , 2008, Neural Computation.

[27]  R. Douady.  CLOSED FORM FORMULAS FOR EXOTIC OPTIONS AND THEIR LIFETIME DISTRIBUTION , 1999 .

[28]  Francis Tuerlinckx,et al.  Efficient simulation of diffusion-based choice RT models on CPU and GPU , 2016, Behavior research methods.

[29]  Jeffrey N. Rouder,et al.  Modeling Response Times for Two-Choice Decisions , 1998 .

[30]  J D Cohen,et al.  A network model of catecholamine effects: gain, signal-to-noise ratio, and behavior. , 1990, Science.

[31]  Bingni W. Brunton,et al.  Rats and Humans Can Optimally Accumulate Evidence for Decision-Making , 2013, Science.

[32]  Adele Diederich,et al.  Sequential sampling model for multiattribute choice alternatives with random attention time and processing order , 2014, Front. Hum. Neurosci..

[33]  J. Wolfowitz,et al.  Optimum Character of the Sequential Probability Ratio Test , 1948 .

[34]  Jonathan D. Cohen,et al.  The physics of optimal decision making: a formal analysis of models of performance in two-alternative forced-choice tasks. , 2006, Psychological review.

[35]  Xiao-Jing Wang,et al.  Probabilistic Decision Making by Slow Reverberation in Cortical Circuits , 2002, Neuron.

[36]  Philip L. Smith,et al.  Stochastic Dynamic Models of Response Time and Accuracy: A Foundational Primer. , 2000, Journal of mathematical psychology.

[37]  Pierre Hansen,et al.  Global optimization of univariate Lipschitz functions: I. Survey and properties , 1989, Math. Program..

[38]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae , 1996 .

[39]  J. Griffiths.  The Theory of Stochastic Processes , 1967 .

[40]  Philip L. Smith,et al.  A comparison of sequential sampling models for two-choice reaction time. , 2004, Psychological review.

[41]  R. Bogacz.  Optimal decision-making theories: linking neurobiology with behaviour , 2007, Trends in Cognitive Sciences.

[42]  Zeno Farkas,et al.  One-dimensional drift-diffusion between two absorbing boundaries: application to granular segregation , 2001 .

[43]  R. Ratcliff,et al.  Diffusion models of the flanker task: Discrete versus gradual attentional selection , 2011, Cognitive Psychology.