Family of closed-form solutions for two-dimensional correlated diffusion processes

Abstract

Diffusion processes with boundaries are models of transport phenomena with wide applicability across many fields. These processes are described by their probability density functions (PDFs), which often obey Fokker-Planck equations (FPEs). While obtaining analytical solutions is often possible in the absence of boundaries, obtaining closed-form solutions to the FPE is more challenging once absorbing boundaries are present. As a result, analyses of these processes have largely relied on approximations or direct simulations. In this paper, we studied two-dimensional, time-homogeneous, spatially-correlated diffusion with linear, axis-aligned, absorbing boundaries. Our main result is the explicit construction of a full family of closed-form solutions for their PDFs using the method of images (MoI). We found that such solutions can be built if and only if the correlation coefficient ρ between the two diffusing processes takes one of a numerable set of values. Using a geometric argument, we derived the complete set of ρ's where such solutions can be found. Solvable ρ's are given by ρ = − cos π k , where k ∈ Z + ∪ {+∞}. Solutions were validated in simulations. Qualitative behaviors of the process appear to vary smoothly over ρ, allowing extrapolation from our solutions to cases with unsolvable ρ's.

Direct observation of the neural computations underlying a single decision

“…We assume they share half the variance, , but the results are robust to a wide range of reasonable values. The decision bounds are allowed to collapse linearly as a function of time, such that We used the method of images van Den Berg et al (2016 ); Shan et al (2019 ) to compute the probability density of the accumulated evidence for each accumulator (which both start at zero at t = 0) as a function of time ( t ) using a time-step of 1 ms. The decision time distributions rendered by the model were convolved with a Gaussian distribution of the non-decision times, t nd , which combines sensory and motor delays, to generate the predicted RT distributions.…”

Expressions for Bayesian confidence of drift diffusion observers in dynamic stimuli tasks

“…Each accumulator corresponds to one of the choices, and the accumulators may be more or less correlated or anti-correlated with each other. It may be possible to use recently derived expressions for the state of the second accumulator at decision time, to extend the approach developed here, although it may be necessary to approximate stimuli as static prior to a decision (Shan, Moreno-Bote & Drugowitsch, 2019). Notwithstanding this consideration, we believe the DDM to be a particularly interesting case due to the normative properties of the diffusion mechanism (see Introduction).…”

“…Nevertheless, a family of alternatives to the DDM have been studied that assume not just one accumulator, but two (Bogacz et al, 2006; Moreno-Bote, 2010), which may be more or less anti-correlated with each other. It may be possible to use recently derived expressions for the state of the second accumulator at decision time, to extend the approach developed here (Shan et al, 2019), but we leave this for future research. Likewise, we leave for future research whether it is possible to extend the derivations to more general dynamic stimuli, rather than just those with normally distributed evidence fluctuations.…”

“…A key feature of our approach is that we derive expressions for the probability distribution over confidence reports (given decisions and response times) rather than focusing on first-order decisions and response-times themselves. Such expressions can be used as the basis for model fitting and parameter estimation, while avoiding the computational cost of making trial-by-trial predictions for decisions and response times (Ratcliff, 1980; Shan, Moreno-Bote & Drugowitsch, 2019; Smith, 2000; Smith & Ratcliff, 2021).…”

Self-motion perception and sequential decision-making: where are we heading?

“…In this model, evidence at each time step is drawn from a bivariate normal distribution with mean = [ μ comb , − μ comb ] and covariance=[4pt1em1ρρ1], where the two dimensions correspond to the two choice alternatives (right versus left) and the evidence for each is partially anticorrelated (i.e. ρ < −0.5 [85]). The mean μ comb is assumed to reflect the optimal weighting of evidence [45] and is therefore biased towards the more reliable cue (figure 3 a , left).…”