We study a diffusion model sampling evidence from the Gaussian distribution with constant mean μ and standard deviation σ, but with the additional flexibility of having time-varying boundaries. This model generates a decision probability p^diff and response time distributions r^diff_A and r^diff_B for the two decisions. Denoting the decision boundaries as a_A and a_B for the two decisions, where a_A and a_B are both time-dependent functions, the diffusion model can be conceived as a mapping (1)mdiff:(μ,σ,aA,aB)→(pdiff,rAdiff,rBdiff).

The mapping m^diff has been studied in the statistics literature, and an effective approach using the analysis of renewal equations has been developed (Durbin, 1971; Buonocore et al., 1987, 1990). Buonocore et al. (1990) provide an efficient algorithm to compute the response time distributions for time-varying boundaries. A summary of these methods well-suited for psychologists is given by Smith (2000). In particular, data can be generated from a diffusion model with flexible boundaries using general Markov process methods. Because (Smith, 2000) does not provide results for exactly the diffusion model we use (we use a special case of a more general one that is provided), we give explicitly the details needed to reproduce our results.

The basic idea is to specify how sample evidence paths X(t) are generated, and then use existing results that give the first passage time distributions through arbitrary boundaries that are continuously differentiable. The diffusion model we study corresponds to a Wiener process with a constant drift ξ and infinitesimal variance s².² Specifying the sample paths for this process is done by specifying the transition density (2)f(x,t∣y,τ)=ddxF(x,t∣y,τ)=12πs2(t−τ)exp                               (−(x−y−ξ(t−τ))22s2(t−τ)) where F(x, t | y, τ) is the probability of the tally being less than or equal to x at time t, given its value at an earlier time τ was y. Notice that both f and F are the densities when there is no boundary.

The first passage time densities through the time-varying absorbing boundaries, a_A and a_B, are denoted by g_A(a_A(t), t | x₀, t₀) and g_B(a_B(t), t | x₀, t₀), where x₀ and t₀ are the initial state and time. Analysis using the renewal equation (e.g., Durbin, 1971) yields the Volterra equations of the relationship between the transition density and the first passage time densities (Smith, 2000, Equation 41): (3)f(aA(t),t|x0,t0)=∫t0tgA(aA(τ),τ|x0,t0)f(aA(t),t|aA(τ),τ)dτ                                        +​∫t0t​gB(aB(τ),τ|x0,t0)f(aA(t),t|aB(τ),τ)dτf(aB(t),t|x0,t0)=∫t0tgB(aB(τ),τ|x0,t0)f(aB(t),t|aB(τ),τ)dτ                                       +​∫t0tgA(aA(τ),τ|x0,t0)f(aB(t),t|aA(τ),τ)dτ​​​​​​​​

In principle, these equations are soluble, but f(x, t | y, τ) is singular as t approaches τ, therefore Equation 3 needs to be transformed stably for practical approximation methods. A detailed description of the equation and the singularity issue can be found in Smith (2000, pp. 430–432). The kernels of the transformed equations can be found using the method developed by Buonocore et al. (1987, 1990) and detailed by Smith, (2000, pp. 441–446). By letting μ(s) = μ = constant in Equation 57 of Smith (2000), the proper function is (4)Ψ(a(t),t∣y,τ)=f(a(t),t∣y,τ)2(a′(t)−a(t)−yt−τ) where a(t) takes the form of a_A or a_B, and a′(t) denotes the first derivative of the boundary. With these results in place, diffusion model data can be produced directly from the first passage time densities, g_A and g_B, which are the same as g₁ and g₂ in Equations 47a and 47b of Smith (2000).

The inverse first passage time problem—finding the boundaries, given the evidence distribution and decision and response time distribution—is much harder than the first passage time problem. It has, however, been studied in the fields of applied mathematics and statistics (e.g., Capocelli and Ricciardi, 1972; Cheng et al., 2006; Chen et al., 2011).

Analytic expressions for the boundaries are rarely available and previous research has usually focused on developing numerical methods for computing the boundary. Theoretical work has been relatively scarce. Early work by Capocelli and Ricciardi (1972) addressed the problem of under what conditions an arbitrary density function can be interpreted as the first passage density function for a continuous one-dimensional Markov process with constant boundaries and a known starting value. Some relevant results, in the context of the types of sequential sampling models used to model human decision-making, were obtained. In particular, Capocelli and Ricciardi, (1972, corollary 2.2) found the technical conditions that guarantee the uniqueness of the solution, if it exists, for the Wiener-Lévy and the Ornstein-Uhlenbeck diffusion processes with specified initial condition.

Cheng et al. (2006) were the first to study the well-posedness—that is, the existence and uniqueness—of a specific inverse first-passage time problem close to that of interest in our study. Cheng et al. (2006) addressed the case where a diffusion model has a single boundary, so that there is only one possible decision, and the response time for that decision is being measured. For that case, they proved that for any probability density function q, there exists a unique viscosity solution to the inverse-first-passage-time problem (i.e., a unique boundary exists under weak assumptions of differentiability). Analogous results for the two-boundary case of direct interest remain an open (and active) research question in the statistics literature. To date, there is no proof that the numerical method developed in the next section of the paper always finds a unique solution.

Zucca and Sacerdote (2009) and Song and Zipkin (2011) developed numerical methods for finding time-varying boundaries in the one-boundary case. Because we are interested in diffusion models with two time-varying boundaries, we rely on the approach used by Buonocore et al. (1990). In essence, our method applies this approach, previously used as a forward method only, to the problem of finding two time-varying boundaries.

Algorithm 1 presents the main part of our numerical method for computing the time-varying boundaries as pseudo code. The aim of the algorithm is to find the two boundaries such that the first passage time densities of the process through those boundaries are equal to two desired specific density functions. The algorithm sets the interval between sampling steps to be a small value λ, and calculates the probabilities P_A,n and P_B,n that decision alternatives “A” and “B,” respectively, will be chosen after n samples. In practice, P_A,n and P_B,n can be obtained by discretizing the empirical RT distributions for the two alternatives. For the diffusion model discretized to the same sampling interval λ, and using the same Gaussian evidence distribution, the drift rate is ξ = μ/λ and the diffusion coefficient is s, where s² = σ²/λ. The first-order derivative of the boundary at step n can be approximated by a′ (n) = [a (n) − a (n − 1)]/λ. These values allow the calculation of Equations 2 and 4 above.

The algorithm finds the time-varying boundary through a point-wise approach to its construction, receiving samples from the same Gaussian evidence distribution with mean μ and standard deviation σ. Because the boundaries scale with σ without changing shape, and our assumption that the decision process starts without bias, the initial values of the boundaries can be fixed at +1 and −1, without loss of generality.

The algorithm now sets the equalities g_A (2)λ = P_{A, 2} and g_B (2)λ = P_{B, 2}, allowing for the solution of the boundaries at the second sample a_A (2) and a_B (2). These steps of the algorithm are now repeated for all of the samples, to find both boundaries in their entirety. Once a_A (1),… a_A (n), and a_B (1),… a_B (n) are available, it is possible to solve for a_A (n + 1) and a_B (n + 1) by setting the first passage time densities to be equal, so that g_A (n + 1)λ = P_{A,n + 1} and g_B (n + 1)λ = P_{B,n + 1}.

Our algorithm solves the equations at each sample using a simple grid search approach. Values between 0 and 1 are examined by a small increment l = 0.01 up to N, where N is a large number chosen such that the value of the response time distribution at Nλ is negligibly small for both decisions.

The recursive nature of the algorithm means that numerical precision errors accumulate as the sample being considered progresses. In practice, we found this sometimes necessitates a second corrective part to our numerical method. For later samples beyond a critical value, we fit the boundary a piece-wise linear curve, each segment containing 2–3 steps, minimizing the deviation between the simulated and the target first passage time distributions. The boundary that is found is thus a combination of the values returned by the algorithm up to the critical step, and brute-force piece-wise linear curve fitting.

Within the sequential sampling framework, an alternative to the class of diffusion model is the class of accumulator models (Vickers, 1970, 1979). As shown in Figure 2, accumulator models maintain two separate evidence tallies, one for each alternative decision. Each sampled piece of evidence favors one or the other decision, and only those samples that favor a decision are added to their corresponding tally. The first tally to reach the boundary results in that decision being made, and the response time is the number of samples required for this to happen.

Because of their different evidence accrual mechanisms, diffusion and accumulator model are usually regarded as being qualitatively different, and treated as competing accounts of human decision making. Empirically, the standard conclusion is that diffusion models are superior accounts of data (e.g., Ratcliff and Smith, 2004), although there are some studies that find in favor of accumulator models (e.g., Lee and Corlett, 2003). Bogacz et al. (2006) compare diffusion and accumulator models theoretically, in terms of a set of optimality properties, and conclude that accumulator models cannot be reduced to diffusion models.

Complementing this focus on the two models as competing accounts of human decision-making, a natural application of our method is to find the time-varying boundaries that make a diffusion model equivalent to an accumulator model with constant boundaries and the same Gaussian evidence distribution. This goal can be seen as a natural extension of the long-standing equivalence result presented by Pike (1968) between random-walk and race models, which are the discrete analogs, respectively, of diffusion and accumulator models. Pike, (1968, Section 4.3) showed that, when the evidence samples are unit increments or decrements, simple time-varying boundaries, decreasing one unit in each time step, make the random-walk decisions and response-time distributions equivalent to the race model.

Formally, we consider the accumulator model sampling evidence from the Gaussian distribution with mean μ and standard deviation σ, and with a fixed starting point 0 and symmetric thresholds. This model generates a decision probability p^acc for choosing decision A, and response time distributions r^acc_A and r^acc_B for the two decisions. Thus, the accumulator model can be conceived as the mapping (5)macc:(μ,σ)→(pacc,rAacc,rBacc).

Equating accumulator and diffusion models requires finding the boundaries a_A (n) and a_B (n), such that (p^acc, r^acc_A, r^acc_B) = (p^diff, r^diff_A, r^diff_B).

The mapping m^acc has been well-studied. Smith and Vickers (1988) provided an analytical expression, in the form of convolutions of the evidence distribution. For Gaussian evidence distributions, there is no closed-form solution, but a discrete approximation method is provided by Smith and Vickers (1989). In particular, we used the method detailed by Smith and Vickers, (1989, Appendix A). Their Equations A3a and A3b define P_A,n and P_B,n which are, respectively, the probability the accumulator model will choose alternative “A” or “B” after n samples.

Figure 3 shows four examples of the boundaries found by our algorithm. Each example corresponds to a different Gaussian evidence distribution, using means of μ = 0.01 and μ = 0.05 and standard deviations of σ = 0.1 and σ = 0.12. For these parameter combinations, we generated response-time distributions from an accumulator model. These distributions provided the input to our algorithm.

The boundaries found by the algorithm are shown in the main left-hand panel for each example in Figure 3. The part of the boundary found by the main algorithm is shown as a solid line, while the part found by the piece-wise approximation is shown as a broken line.³ The basic result is that the decision probabilities and response-time distributions generated by accumulator models correspond to those generated by a diffusion evidence accrual process with time-varying boundaries.

The right-hand panels in Figure 3 correspond to the two-decision alternatives, and show the accumulator and diffusion response-time distributions, as solid lines and gray histograms, respectively. These distributions are weighted by the decision probabilities, and so capture all of the aspects of model behavior that need to be equated. It is clear that the decision probabilities and response times generated by the diffusion evidence accrual process with the time-varying boundaries are very close to the target accumulator model distributions.

The four evidence distributions illustrated in Figure 3 span the interesting range of possibilities. They include cases where the response time distributions are skewed as well as symmetric, and cases where the mean response times for the two decisions are very different as well as very similar. They also include a wide range of decision probabilities, ranging from close to 50% down to about 1%.

The basic result is that diffusion models with time-varying boundaries, of the type shown in Figure 3, produce the same decisions and response time distributions as accumulator models with constant boundaries. An important aspect of this result is that the boundaries are established before any particular evidence sequence is encountered. The nature of the boundaries is not developed or changed as evidence is sampled within a trial. While establishing equivalence dynamically by adapting to current evidence is an interesting research problem in its own right (e.g., Hockley and Murdock, 1987), the current results establish a more general equivalence. They show what sorts of time-varying boundaries make the diffusion approach to evidence accrual the same as standard accumulator approaches.

An interesting aspect of the results in Figure 3 is that it is clear that the time-varying boundaries are, in general, asymmetric. For example, when the evidence distribution is a Gaussian with μ = 0.05 and σ = 0.10, the lower boundary converges to zero more quickly than the upper boundary. Figure 4 presents a follow-up analysis, exploring how important symmetry is to equate accumulator and diffusion approaches to evidence accrual. Figure 4 shows the response-time distributions for the same examples considered in Figure 3, but using a modified algorithm that constrains the boundaries to be symmetric. For the evidence distributions with mean μ = 0.01 there is still close agreement between the accumulator and diffusion response time distributions. For the more extreme examples with mean μ = 0.05, the qualitative properties of different mean response times and negative skew are preserved, but there is quantitative disagreement between the accumulator and diffusion distributions.

One of the most intuitive motivations for considering diffusion models with time-varying boundaries relates to the case of non-evidential stimuli. These are stimuli that provide equal evidence for both response alternatives, and so the expectation of the evidence distribution is zero (i.e., μ = 0). For these stimuli, constant boundaries predict at least some extremely long response times, even though there is no information to be gained from repeated sampling from the stimulus. This prediction seems problematic, both empirically and theoretically, and has even led to sequential sampling models of human decision-making being lambasted in non-psychological literatures (Lamport, 2012). Converging boundaries provide a natural mechanism for ensuring a decision is made in a reasonable time, without needing to invoke additional psychological assumptions like over-riding termination processes.

Against this background, one interesting application of our method is to find the type of boundaries consistent with behavioral data for non-evidential stimuli. We consider data collected and analyzed by Ratcliff and Rouder (1998), which have also been examined by a number of other authors (e.g., Brown and Heathcote, 2005; Vandekerckhove et al., 2008). The Ratcliff and Rouder (1998) data involve three individual subjects each doing about 8000 trials over 11 days on a brightness discrimination task, under both speed and accuracy instructions. The stimuli consist of visual arrays of black and white dots, with the number of black and white dots controlling the evidence they provide for the choice alternatives bright and dark. Of the 33 different levels of brightness considered by Ratcliff and Rouder (1998), we focus on just those stimuli with equal numbers of black and white dots that (objectively) provide no evidence for either response alternative.

To apply our algorithm to these data, we had to make a number of simplifying assumptions. First, we assumed that the drift rate was zero, because of the objective properties of the stimuli. Obviously, it is possible that psychologically the stimuli are perceived as favoring one alternative or the other, through some form of bias. Secondly, we shifted the response time distributions according to the smallest response time observed for each individual in each condition. This is a simple empirical approach that probably only roughly approximates the underlying time to encode and respond that requires the shift. Finally, because our method proved unstable with respect to the multi-modalities inherent in binned characterizations of the data, we first fit a Weibull function to the response time distributions, and applied our algorithm to samples from these distributions.

Figure 5 shows the results of our method on the Ratcliff and Rouder (1998) data, as applied to the accuracy condition.⁴ We used the Pearson's Chi-square tests standardly used in this literature⁵ to evaluate the goodness-of-fit of the Weibull distributions, binning the response times by decile, d.f. = 7. For subject “JF,” the Chi-square statistics and corresponding p-values for both alternatives are 7.14 (p = 0.41) and 13.02 (p = 0.07); for subject “KR,” they are 4.69 (p = 0.70) and 9.01 (p = 0.25); for “NH”, they are 10.02 (p = 0.19) and 10.99 (p = 0.14). The three rows in Figure 5 correspond to the three individual subjects:“JF,” “KR,” and “NH.” The main panels on the left show the boundaries found by our algorithm, with respond to discretized samples of 0.01 s duration. Here, we assume that every subject has the same evidence distribution, arbitrarily chosen to be N(0, 0.01), thus the starting values of the boundaries are now free parameters. The smaller panels on the right show the distributions of empirical response times (as gray histograms) and the distributions of response times generated by the time-varying boundaries found by our algorithm (as solid lines) for the two decision alternatives, measured in seconds. There is reasonably good agreement between these distributions, although it is better for some subjects (e.g., “JF”) than others. It is also clear that there are significant individual differences between the subjects, with “KR” taking longer to make decisions for these non-evidential stimuli.

Most interestingly, Figure 5 shows, once again, that the boundaries found are ones that converge asymmetrically. After an extended period of requiring the same level of evidence, both boundaries drop sharply toward zero and converge. They commence their descents at different times, though, with the lower boundary always converging first, but less sharply. Intuitively, when the stimulus favors neither alternative, symmetric boundaries should be able to fit the data well. We calculate the boundaries using the algorithm with the symmetry constraint in Appendix B, and find that the restricted algorithm finds boundaries close to the boundaries found by the original algorithm.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.