SDDM modified to encode leaky integration in the drift rate. Also known as an Ornstein-Uhlenbeck model, its drift rate is \(v(x,t) = \mu - L*x\) where \(L\) is the leakage rate. All other parameters are unchanged from the SDDM. Leakage describes the rate at which old information is lost from the accumulator, occurring on a time scale of approximately \(1/L\). The LIM is used to model decay of excitatory currents in decision neurons (Usher & McClelland, 2001; Wong & Wang, 2006) and has been proposed as a mechanism for preference reversals under time pressure (Busemeyer & Townsend, 1993). Due to its neural plausibility and simple functional form, recent work has proposed it as an alternative psychometric tool to the SDDM (Wang & Donkin, 2024).
dLIM(rt, resp, phi, x_res = "default", t_res = "default")pLIM(rt, resp, phi, x_res = "default", t_res = "default")
rLIM(n, phi, dt = 1e-05)
For the density a list of PDF values, log-PDF values, and the sum of the log-PDFs, for the distribution function a list of of CDF values, log-CDF values, and the sum of the log-CDFs, and for the random sampler a list of response times (rt) and response thresholds (resp).
vector of response times
vector of responses ("upper" and "lower")
parameter vector in the following order:
Non-decision time (\(t_{nd}\)). Time for non-decision processes such as stimulus encoding and response execution. Total decision time t is the sum of the decision and non-decision times.
Relative start (\(w\)). Sets the start point of accumulation as a ratio of the two decision thresholds. Related to the absolute start z point via equation \(z = b_l + w*(b_u - b_l)\).
Stimulus strength (\(\mu\)). Strength of the stimulus.
leakage (\(L\)). Rate of leaky integration.
Noise scale (\(\sigma\)). Model scaling parameter.
Decision thresholds (\(b\)). Sets the location of each decision threshold. The upper threshold \(b_u\) is above 0 and the lower threshold \(b_l\) is below 0 such that \(b_u = -b_l = b\). The threshold separation \(a = 2b\).
Contamination (\(g\)). Sets the strength of the contamination process. Contamination process is a uniform distribution \(f_c(t)\) where \(f_c(t) = 1/(g_u-g_l)\) if \(g_l <= t <= g_u\) and \(f_c(t) = 0\) if \(t < g_l\) or \(t > g_u\). It is combined with PDF \(f_i(t)\) to give the final combined distribution \(f_{i,c}(t) = g*f_c(t) + (1-g)*f_i(t)\), which is then output by the program. If \(g = 0\), it just outputs \(f_i(t)\).
Lower bound of contamination distribution (\(g_l\)). See parameter \(g\).
Upper bound of contamination distribution (\(g_u\)). See parameter \(g\).
spatial/evidence resolution
time resolution
number of samples
step size of time. We recommend 0.00001 (1e-5)
Raphael Hartmann & Matthew Murrow
Busemeyer, J. R., & Townsend, J. T. (1993). Decision field theory: A dynamic-cognitive approach to decision making in an uncertain environment. Psychological Review, 100(3), 432-459.
Usher, M., & McClelland, J. L. (2001). The time course of perceptual choice: The leaky, competing accumulator model. Psychological Review, 108(3), 550-592.
Wang, J.-S., & Donkin, C. (2024). The neural implausibility of the diffusion decision model doesn’t matter for cognitive psychometrics, but the Ornstein-Uhlenbeck model is better. Psychonomic Bulletin & Review.
Wong, K.-F., & Wang, X.-J. (2006). A Recurrent Network Mechanism of Time Integration in Perceptual Decisions. The Journal of Neuroscience, 26(4), 1314-1328.
# Probability density function
dLIM(rt = c(1.2, 0.6, 0.4), resp = c("upper", "lower", "lower"),
phi = c(0.3, 0.5, 1.0, 0.5, 1.0, 0.5, 0.0, 0.0, 1.0))
# Cumulative distribution function
pLIM(rt = c(1.2, 0.6, 0.4), resp = c("upper", "lower", "lower"),
phi = c(0.3, 0.5, 1.0, 0.5, 1.0, 0.5, 0.0, 0.0, 1.0))
# Random sampling
rLIM(n = 100, phi = c(0.3, 0.5, 1.0, 0.5, 1.0, 0.5, 0.0, 0.0, 1.0))
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