Abstract

Diffusion models have become essential for describing the performance and statistics of reaction times in human decision making. Despite their success, it is not known how to evaluate decision confidence from them. I introduce a broader class of models consisting of two partially correlated neuronal integrators with arbitrarily time-varying decision boundaries that allow a natural description of confidence. The dependence of decision confidence on the state of the losing integrator, decision time, time-varying boundaries, and correlations is analytically described. The marginal confidence is computed for the half-anticorrelated case using the exact solution of the diffusion process with constant boundaries and compared to that of the independent and completely anticorrelated cases.

1.  Introduction

In most interesting situations, detection of a signal involves discriminating a meaningful from a random energy fluctuation. A standard procedure employed to discriminate between these two alternatives is to integrate over time such fluctuations and decide that there is a signal embedded in the noise if the accumulated sum reaches a predetermined threshold and that there is no signal otherwise. Although this is just the way some photosensors are used to detect dim stars, humans seem to employ a remarkably similar strategy to make decisions. Human performance and reaction time statistics in two-alternative forced-choice tasks are accurately fit by models consisting of a diffusion process of an underlying variable to decision boundaries (Ratcliff, 1978; Ratcliff & Smith, 2004; Luce, 1986; Usher & McClelland, 2001; Bogacz et al., 2006; Mazurek, Roitman, Ditterich, & Shadlen, 2003; Ratcliff, Hasegawa, Hasegawa, Smith, & Segraves, 2007; Wang, 2002; Beck et al., 2008). Moreover, diffusion to bound processes seem to be neurally hardwired in several areas of the brain, where the neurons' firing rate reflects integration of noisy information for alternatives and determines choices and decision times when it reaches a fixed threshold (Hanes & Schall, 1996; Shadlen & Newsome, 1996; Platt & Glimcher, 1999; Ratcliff et al., 2007; see Schall, 2001, for a review).

Upon a decision, a crucial variable is the degree of confidence about the choice. In detecting a new particle or star, the estimated confidence will determine if we can believe in the result. Humans and animals are able to reliably estimate confidence or uncertainty about their decisions and use it to change or abort behaviors (Vickers, 1979; Juslin & Olsson, 1997; Vickers & Lee, 1998; Landy, Goutcher, Trommershuser, & Mamassian, 2007; Whiteley & Sahani, 2008; Kepecs, Uchida, Zariwala, & Mainen, 2008). Human subjects can be asked directly to report their confidence, but it is also possible to use choice behaviors in humans and animals to estimate their decision confidence. For instance, a subject can be presented with two choices: one that involves high but uncertain reward and another with low but certain reward. The confidence that the subject has about the uncertain reward at any trial will determine which option is taken, and therefore decision confidence can be estimated from its behavior at any trial. Despite the importance of decision confidence, very little is known about how it is encoded in the nervous system (but see Kepecs et al., 2008; Beck et al., 2008). Furthermore, the dependence of confidence on essential parameters of the decision process like reaction time or distance to boundary of the losing integrator has not been established. Knowing such dependencies on simple models would allow a quantitative comparison between theory and both behavioral and neurophysiological data.

This letter describes analytically decision confidence for a decision diffusion model (DDM) consisting of two neuronal integrators receiving partially correlated inputs. The main results are as follows: (1) decision confidence depends on the difference between the traveled distances by the two integrators in a general way, regardless of the correlation in the inputs and on the time dependence of the decision thresholds; (2) accumulation of evidence over time occurs without any leak or loss, that is, the final state of the integrators and elapsed time fully specify the a posteriori probability distribution over the parameters on which the decision is based; (3) marginal confidence is computed for the independent, half-anticorrelated, and completely anticorrelated cases as a function of the decision time, being at a maximum for independent integrators; and (4) the distributions of reaction times for the above cases are characterized. Models with several integrators similar to the one proposed here have been introduced before to describe qualitatively decision confidence (Vickers, 1979), but analytical expressions for confidence are in general not known for these simple models (Juslin & Olsson, 1997; Vickers & Lee, 1998; Van Zandt, 2000; Pleskac & Busemeyer, 2007). It is worth mentioning that the DDM considered here reduces to the classical DDM (Ratcliff & Smith, 2004; Luce, 1986) when the two integrators receive completely anticorrelated inputs, for which decision confidence as a function of the decision time is also completely characterized for arbitrary time-dependent boundaries.

2.  Model

I consider a decision diffusion model (DDM) consisting of two neuronal integrators with variables x1 and x2 receiving independent and cross-correlated inputs,
formula
2.1
where ηi(t), i = 1, 2, c are independent gaussian white noise processes with zero mean (〈ηi(t)〉 = 0) and unit variance (〈ηi(ti(t′)〉 = δ(tt′), where δ(tt′) is the Dirac's delta function).1 The variables x1 and x2 can correspond to averaged firing rates across a population of neurons with similar tuning, or to population-averaged voltages (Mazurek et al., 2003). The model captures the approximately linear ramping activity observed in neurons in lateral intraparietal area (LIP) and frontal eye fields (FEF) in monkeys performing simple decision tasks (Hanes & Schall, 1996; Shadlen & Newsome, 1996). The kth neuronal integrator (k = 1, 2) drifts with rate μk and accumulates input white noises with total deviation σ. The correlation coefficient, ρ ∈ [0, 1], determines the degree of correlation between the two integrators, being independent if ρ = 0 and completely correlated if ρ = 1. The correlation index, ν ∈ {−1, 1}, determines the correlation sign between them, being positively correlated if ν = 1 and anticorrelated if ν = −1. The variables x1 and x2 diffuse from the initial values a and b until one of them reaches its threshold, Θ1(t) and Θ2(t), respectively, which can depend on time.

A two-alternative forced-choice task is modeled as follows. A decision for choice k (k = 1, 2) is made whenever the variable xk reaches its threshold Θk(t) first. It can be assumed that the decision is correct if the boundary first reached corresponds to that of the integrator with the highest drift. This is a general situation that, for instance, could correspond to a task in which the goal is to detect the brighter of two lights. It also arises naturally in experiments in which the direction of motion of a random-dot motion display has to be determined during short presentations (Shadlen & Newsome, 1996; Mazurek et al., 2003). Here, the direction of motion can be modeled by the sign of the drift rates of the two integrators (i.e., μ = μ1 = −μ2>0 indicates rightward motion, while μ = μ1 = −μ2 < 0 indicates leftward motion). The difficulty of this task is controlled by the motion coherence, that is, the percentage of dots that move in the same direction as opposed to random directions, and it can be modeled by the magnitude of the drift rates, |μ| = |μ1| = |μ2|. It will be assumed that the variance of the noise and correlation coefficient remains constant across all stimulation conditions. I assume that when a decision has been made, the state of the losing integrator can be read out. Figure 1 illustrates the model for the independent (ρ = 0 in Figure 1A) and half-anticorrelated (ρ = 1/2, ν = −1, in Figure 1C) cases with time-varying thresholds.

Figure 1:

(A) Two integrators with independent white noise inputs. The system evolves until one of the integrators reaches its time-varying threshold—in this case, integrator 1. (B) The state probability density for the system in A when the boundaries are constant over time consists of three images, equation A.9, arranged on a circle (see the appendix). The two semi-lines of the absorbing boundary are indicated with l1 and l2. (C) Two integrators with time-dependent boundaries and half-anticorrelated white noise inputs. (D) The state probability density for the system in panel C with constant bounds consists of five images, equation A.8, arranged on an ellipse (see the appendix).

Figure 1:

(A) Two integrators with independent white noise inputs. The system evolves until one of the integrators reaches its time-varying threshold—in this case, integrator 1. (B) The state probability density for the system in A when the boundaries are constant over time consists of three images, equation A.9, arranged on a circle (see the appendix). The two semi-lines of the absorbing boundary are indicated with l1 and l2. (C) Two integrators with time-dependent boundaries and half-anticorrelated white noise inputs. (D) The state probability density for the system in panel C with constant bounds consists of five images, equation A.8, arranged on an ellipse (see the appendix).

The classical DDM (Ratcliff & Smith, 2004; Luce, 1986) is a special case of equation 2.1 when the integrators receive completely anticorrelated inputs (ρ = 1, ν = −1), μ = μ1 = −μ2, and the thresholds are identical and time independent, Θ1(t) = Θ2(t) = Θ. In that case, only motion parallel to the diagonal x1 = −x2 is possible, and therefore each integrator can be viewed as the antineuron of the other. Equations 2.1 reduce in this case to the system
formula
2.2
where x = x1 = −x2 and the decision boundaries lie at x = ±Θ (as before, η(t) is a gaussian white noise process). Although very relevant to fit behavioral data, the classical DDM assumes perfect anticorrelation between the integrators, and such degree of correlation seems unrealistic in the brain (Zohary, Shadlen, & Newsome, 1994).

When ρ = 0 in equations 2.1, the two integrators become independent, and then the system becomes a race model (see, e.g., Vickers, 1979; Luce, 1986). Therefore, the model defined in equations 2.1 covers a wide spectrum of classical models of decision making, ranging from the classical DDM for ρ = 1 and ν = −1, to the race model for ρ = 0, as shown in Figure 2. However, the model considered here does not include interactions or connections between the neuronal integrators, as in Usher & McClelland (2001). A discussion about these models has been recently presented in Bogacz (2007).

Figure 2:

The DDM in equation 2.1 covers continuously the range between diffusion models (classical DDM) and race models as a function of the correlation coefficient between the input noises. A mixture model corresponds, for example, to the DDM with half-anticorrelated inputs. Circles represent integrators.

Figure 2:

The DDM in equation 2.1 covers continuously the range between diffusion models (classical DDM) and race models as a function of the correlation coefficient between the input noises. A mixture model corresponds, for example, to the DDM with half-anticorrelated inputs. Circles represent integrators.

3.  Results

3.1.  Independence of the Probability Distributions over the Drift Rates on Decision Boundaries.

A decision about which drift is higher (μ12 or μ21?) in the system defined in equations 2.1 will depend on the drift estimates at decision time t given the observed trajectories of the integrators up to that time, denoted and , with initial states x1(0) = a and x2(0) = b at t = 0. The critical quantity is then the probability density of having the drifts μ1 and μ2 given the trajectories and up to time t, . Instead of working directly with this density, it is more convenient to introduce the new variables u = x1 + x2 and v = x1x2 and work with the probability density of having drifts μu = μ1 + μ2 and μv = μ1 − μ2 given the trajectories and , with initial conditions u0 = a + b and v0 = ab. In the new variables, equations 2.1 transform into two independent Wiener processes, with means μu and μv, and variances σ2u = 2σ2(1 + ρν) and σ2v = 2σ2(1 − ρν), making the calculations easier.

Let us first consider the case in which there are not boundaries. The case with boundaries will be considered immediately after this discussion. If the time is discretized in n steps of size δt, such that t = nδt, and defining ui = u(iδt) (vi = v(iδt)), i = 1, …, n, then the probability density of having a variable step δui = uiui−1vi = vivi−1) at time t = iδt follows a gaussian distribution with mean μuδtvδt) and variance σ2uδt2vδt). Since each increment is independent of the others, the probability density of the trajectory is
formula
3.1
If pu, μv) is the distribution of the drifts based on prior knowledge, Bayes' theorem states that , where the proportionality is in relation to the drift rates. If the prior distribution is uniform, pu, μv) = p0, Bayes' theorem and equation 3.1 give after normalization that the free (i.e., without boundaries) distribution over the drift rates is
formula
3.2
where (u, v) is the ending point of the trajectory at time t. Then the a posteriori probability distribution over the drift in the variable u (v) is a gaussian distribution with mean (uu0)/t ([vv0]/t) and variance σ2u/t2v/t). Hence, the mean drift is just the slope of the trajectory, computed as the ratio between the distance from initial to ending points, and elapsed time. The variance of the probability distribution decays as a function of time, so a longer time results in a narrower distribution and therefore in a better estimation of the drifts. Importantly, the identity between and pfreeu, μvuu0, vv0, t) shows that all the information in relation to the drift rates available in the trajectories is preserved in the distances traveled by the integrators (i.e., their final state) and elapsed time. This is just the description of the fact that the traveled distances and elapsed time are sufficient statistics for the distribution of drift rates.

Now I consider the case in which there are boundaries Θk(t) (k = 1, 2). In this case, only trajectories that do not cross the boundaries at any time before t (i.e., such that u + v < 2Θ1(τ) and uv < 2Θ2(τ) for all τ < t) are realizable, while the trajectories that cross the boundaries before time t are absorbed. In what follows, I show the central result that the probability distribution of the drift rates given the traveled distances by the integrators and the elapsed time when boundaries are present is identical to the probability distribution of the drift rates when there are not boundaries, equation 3.2.

Two steps will be followed in the proof. First, it is shown that the probability density of having the state (u, v) at time t when arbitrary boundaries are present can be factorized in two terms: one that depends on the drift rates but not on the boundaries and another that depends on the boundaries but not on the drift rates. Second, it is shown that this factorization and Bayes' theorem naturally lead to equation 3.2 for uniform a priori distributions over the drift rates. This proof is similar to that derived by Beck et al. (2008) for Poisson-like distributions.

The probability density of observing the state (u, v) of the integrators at time t given initial conditions (u0, v0) and drift rates μu and μv can be expressed as the integral of the probability of having the trajectory , equation 3.1, over all possible trajectories with initial point (u0, v0) and final point (u, v), which do not cross the boundaries at any time before t,
formula
3.3
where and (remember that a trajectory is defined in terms of the differential increments (δui, δvi) for i = 1, …, n in time steps δt = t/n), and the region Ω is defined as the set of trajectories that satisfies u0 + v0 + ∑ji=1ui + δvi) < 2Θ1(jδt) and u0v0 + ∑ji=1ui − δvi) < 2Θ2(jδt) for j = 1, …, n (i.e., they do not cross the boundaries before time t), along with the constraint that ∑ni=1δui = uu0 and ∑ni=1δvi = vv0 (i.e., the initial and final points are (u0, v0) and (u, v), respectively). By expanding the squared terms in the exponents of equation 3.3 and using that ∑ni=1δt = t, ∑ni=1δui = uu0, and ∑ni=1δvi = vv0, it is possible to extract out from the integral the terms that depend on the drift rates as
formula
3.4
where
formula
3.5
Note that R(u,v) depends on the boundaries but not on the drift rates.
Since the quantity of interest is the probability distribution of the drift rates given the final state of the integrators, pboundu, μvu, v, t), and this is proportional to equation 3.4 through Bayes' theorem, then only the dependencies of equation 3.4 on the drift rates are explicitly considered. This means, interestingly, that to determine the distribution over the drift rates, it is not required to compute explicitly the complicated integral in equation 3.5, a task achievable only in particular cases. Therefore, after using Bayes' theorem pboundu, μvu, v, t) ∝ pbound(u, v, t ∣ μu, μv) (proportionality is in relation to the drift rates; here a uniform a priori distribution over the drifts rates is assumed, but a similar result holds for any arbitrary a priori distribution) and normalizing the expression, it is found that
formula
3.6
regardless of the boundary conditions, which is identical to equation 3.2. This proves that pboundu, μvuu0, vv0, t) = pfreeu, μvuu0, vv0, t).

Finally, one needs to show that all information contained in the trajectory is preserved in the final state of the integrators and elapsed time in the presence of boundaries. Since for any trajectory that does not cross the bound at any time before t, , one has , then using equations 3.2 and 3.6, . Therefore, the integrators accumulate evidence with any leak or loss. This interestingly means that, as in the case without boundaries, the shape of the trajectory does not need to be remembered, since the final state of the integrators and elapsed time contain all necessary information in order to perform inference about the drift rates. More formally, the distances traveled by the integrators along with the elapsed time are sufficient statistics for the drift rates.

The equivalence between probability distributions with and without boundaries can be understood intuitively as follows. Note, first, that any trajectory that does not cross the boundaries at any time before t and has the same initial and final points leads to the same probability distribution over the drift rates, equation 3.2. Note, second, that the effect of the boundaries is to reduce the number of trajectories that can reach a given ending point from the same initial condition compared to the case with no boundaries. However, since any of the remaining realizable trajectories (i.e., those that do not cross the boundaries) with the same initial and ending points lead to the same probability distribution of the drift rates, the shape of the boundaries does not affect the information available about the drifts at time t.

The result that the probability distribution over the drift rates does not depend on the temporal profile of the decision boundaries before the decision time is nevertheless surprising. Let us assume, for instance, that the decision boundaries drop sharply to very negative values at short times and then recover back to some finite value and that the final state of one integrator is positive, x1>0. Then one could argue that the a posteriori probability distribution over the drift rate μ1 of that integrator has to be skewed to very large negative values (i.e., negative drifts are more likely than positive ones) because the trajectory of the integrator first needed to reach very low values before ending at the final state x1>0. Then, in general, one would conclude that the probability distribution over the drift rates depends not only on the final state of the integrators, but also on the temporal shape of the boundaries. Equation 3.6 shows that the above reasoning is incorrect and that in fact the probability distribution over the drift μ1 has a larger probability mass at positive values. The paradox can be resolved by noticing that the traveled distances by the integrators and elapsed time are sufficient statistics for the drift rates μk (k = 1, 2) in the decision process, while the shape of the paths traveled provides no additional information about the drift rates. Therefore, the state of the integrators, along with the elapsed time, summarizes all information one needs to know to make a decision that involves any inequality over their drift rates.

One can check that this is true in very simple cases, where analytical solutions are known for the whole probability distribution over the final state of the integrators. Let us consider, for instance, the diffusion of a single integrator, which obeys equation 2.2, and let us assume that there is only a time-dependent boundary at Θ(t) = A + Bt. Then the probability distribution at time t over the state x of the integrator with starting point x = 0 and drift rate μ follows,
formula
3.7
with D = 2A(B − μ)/σ2. (It is easy to show that p(x, t ∣ μ) vanishes at the time-dependent threshold, and since it is composed of a sum of two free solutions of the associated Fokker-Planck equation, it is the solution with a time-dependent absorbing boundary Θ(t); Cox & Miller, 1965.) Finally, using Bayes' theorem with uniform a priori distribution over the drift rate, p(μ ∣ x, t) ∝ p(x, t ∣ μ), one finds that
formula
3.8
that is, the a posteriori probability distribution over the drift rate μ is a gaussian with mean x/t and variance σ2/t, and it does not depend on the parameters A and B that define the time-dependent boundary. (Note that equation 3.8 is a particular case of equation 3.6 when restricted to the dynamics of a single integrator.) The same result holds when there are two side time-independent boundaries, a case for which analytical expressions are also known (Cox & Miller, 1965), and in fact, as it has been shown before, equation 3.8 is valid for any decision boundaries.

Figure 3 shows the probability density function over the drift rate μ computed from numerical simulations of a single integrator that obeys equation 2.2 in the presence of a nonlinear boundary Θ(t), a case for which there are no generally valid analytical solutions. Sample trajectories with a broad distribution of drift rates were generated, and from the trajectories that ended at threshold at times within a short window t and t + Δt, the distribution of drift rates at decision time was computed. As the theory predicts, equation 3.8 should be valid in the presence of any threshold, and therefore the distribution of drift rates should be a gaussian with mean Θ(t)/t and variance σ2/t when the decision is made at time t regardless of the shape of the boundary. The simulations are in perfect agreement with the theory.

Figure 3:

Distribution of the drift rate for a one-variable diffusion process, equation 2.4, with σ2 = 0.5s−1 from numerical results (thin line) and from equation 3.8 (thick line) at decision time t = 200 ms. The boundary is defined as Θ(t) = A + Bt2 with A = 1.3 and B = −5s−2. The mean and variance obtained from the simulations are 5.57s−1 and 2.50s−1, respectively, close to the predicted values 5.50s−1 and 2.50s−1. Sample trajectories were generated using drift rates uniformly distributed in the range between −200s−1 and 200s−1. The trajectories that reached the threshold Θ(t) within the time window t = 200 ms and t= 201 ms were used to compute the distribution of the drift rate.

Figure 3:

Distribution of the drift rate for a one-variable diffusion process, equation 2.4, with σ2 = 0.5s−1 from numerical results (thin line) and from equation 3.8 (thick line) at decision time t = 200 ms. The boundary is defined as Θ(t) = A + Bt2 with A = 1.3 and B = −5s−2. The mean and variance obtained from the simulations are 5.57s−1 and 2.50s−1, respectively, close to the predicted values 5.50s−1 and 2.50s−1. Sample trajectories were generated using drift rates uniformly distributed in the range between −200s−1 and 200s−1. The trajectories that reached the threshold Θ(t) within the time window t = 200 ms and t= 201 ms were used to compute the distribution of the drift rate.

A special feature of the DDM considered in equations 2.1 is that all information is readily available in the final state of the integrators and the elapsed time, equation 3.6. Indeed, this property is violated in neuronal integrator models with leak terms (negative linear xks), as considered, for example, in Usher and McClelland (2001). It can be shown that in these models, the a posteriori probability density distribution over the drift rates depends not only on the final states of the integrators but also on the mean value of the trajectories. Furthermore, if expressed only as a function of the final state of the integrators (along with the initial state and elapsed time), the probability density over the drift rate depends on the temporal profile of the decision boundaries Θk(t). It is interesting to note, then, that brain areas like LIP and FEF show ramping activity that resembles perfect, leakless integration of evidence.

3.2.  Decision Confidence.

Upon a decision, confidence can be expressed as the estimated probability that the highest signal has been correctly detected given the observed trajectories of the integrators up to decision time t. Given the final state of the integrators, the join probability density of the drift rates is given by equation 3.6. From this probability density, it is easy to compute whether the probability of μ12 is larger or lower than the probability of μ21, and the choice μ12 or μ21 will be made accordingly. In order to associate uniquely one decision boundary with one choice, the initial states of the integrators and the bounds are taken to be identical for the two integrators, a = b and Θ(t) = Θ1(t) = Θ2(t). Therefore, if the boundary of integrator k = 1 is reached first, then at decision time, x1(t) = Θ(t)>x2(t). This implies that the case μ12 has higher probability, and hence it is chosen (and, conversely, if integrator k = 2 reaches its bound first). Without loss of generality, in the following, integrator k = 1 is assumed to reach its boundary first, x1 = Θ(t), and then the choice μ12 has been made. Decision confidence can then be written using equation 3.6 as
formula
3.9
where Δ(x2) = Θ(t) − x2 and x2 is the final state of integrator 2 at decision time t. Equation 3.9 is a cumulative normal gaussian with the argument being proportional to the ratio between the difference in distance traveled by the two integrators and the squared root of the decision time. The same expression holds if the prior distribution of the drifts is p1, μ2) = p1)δ(μ1 − μ2), where p1) = p0 (p0 is a constant), which states that μ1 is distributed uniformly along with the constraint μ1 = −μ2. Decision confidence depends, then, on the difference between the states of the two integrators at decision time. The correlation between the integrators affect only σv, and hence the slope of the confidence function, but it does not change the functional dependence on the states of the integrators and elapsed time.

Figure 4A shows decision confidence as a function of decision time for a fixed state of the losing integrator and as a function of the state of the losing integrator, x2, for fixed decision time. Constant bounds, Θ(t) = Θ, for the integrators are used in this example. As a function of decision time, confidence decays monotonically from one to one-half. In the limit of long t, equation 3.9 converges to . The monotonic decay is due to the fact that fast responses are likely to happen in trials in which large drift rates are used, which leads to better performance and, hence, increased confidence. As a function of the state of the losing integrator, decision confidence decreases from one at large and negative x2 and approaches one-half when x2 is close to threshold.

Figure 4:

(A) Confidence for the decision μ12 as a function of decision time for x2 = 0.5 and as a function of x2 for t = 1 s for independent integrators using equation 3.9. (B) Marginal decision confidence as a function of decision time for independent (top, equations 3.20 and 3.22), half-anticorrelated (middle, equations 3.20 and 3.21), and completely anticorrelated (classical DDM; bottom, equation 3.11) integrators. Lines correspond to analytical expressions and data points to simulations. For all cases, σ2 = 3.8s−1, Θ1 = Θ2 = 1 and a = b = 0.

Figure 4:

(A) Confidence for the decision μ12 as a function of decision time for x2 = 0.5 and as a function of x2 for t = 1 s for independent integrators using equation 3.9. (B) Marginal decision confidence as a function of decision time for independent (top, equations 3.20 and 3.22), half-anticorrelated (middle, equations 3.20 and 3.21), and completely anticorrelated (classical DDM; bottom, equation 3.11) integrators. Lines correspond to analytical expressions and data points to simulations. For all cases, σ2 = 3.8s−1, Θ1 = Θ2 = 1 and a = b = 0.

It is possible to compute decision confidence for the classical DDM employing the general expression in equation 3.9. Here I use the prior distribution p1, μ2) = p1)δ(μ1 − μ2) with p1) = p0 (p0 is a constant), which ensures the constraint μ = μ1 = −μ2; in addition, constant bounds Θ(t) = Θ are used. Then the state of one integrator is determined by the state of the remaining integrator, since when integrator k = 1 reaches its threshold, x1 = Θ, integrator k = 2 is at x2 = −Θ. At decision time, the probability distribution over the drift rate μ can be written using equation 3.6 as
formula
3.10
and from this equation or from the general expression in equation 3.9 with σv = 2σ, one gets that decision confidence is
formula
3.11
which depends on only decision time (the dependence on the threshold and noise variance is not made explicit). For long decision times, decision confidence converges to . Experimental measurements of decision confidence have been typically fit by a/t + b functions (Vickers, 1979), which are mathematically different from the dependences reported here. However, the two expressions are not very different numerically one from each other when they are used in a limited range of decision times. It should be noted that distinguishing between the two expressions requires large amounts of data for very long reaction times, which are not typically available. Nevertheless, it would be desirable to design experiments that allow the estimation of confidence for a wide range of decision times to test the analytical expressions found in this letter.
The classical DDM can be extended to the case in which the boundaries are time dependent, Θ(t). In this case, the probability distribution over the drift rate μ at decision time is written using equation 3.6 as
formula
3.12
and decision confidence then becomes
formula
3.13
Considering again the general DDM model, equation 2.1, if the drift rates are not uniformly distributed but can take only two possible values μ1 = −μ2 = ±μ0 with probability one-half,
formula
3.14
then following similar steps as in equations 3.6 and 3.9, it can be shown that
formula
3.15
At decision time, decision confidence depends on the difference between the traveled distances by the integrators, Δ(x2) = Θ(t) − x2, but not explicitly on decision time. As in equations 3.6 and 3.9, confidence is independent of the boundaries used to generate decisions. For the classical DDM with time-dependent boundaries, Θ(t), decision confidence becomes
formula
3.16
which depends on only decision time through the boundary time dependence. It is interesting to note that for this case, decision confidence equals the expression for the probability of reaching first the correct boundary (Cox & Miller, 1965). Therefore, confidence and performance are identical here. Although this result can be shown to be rather general, it is not true for all boundary conditions.

Experimentally, human verbal reports of confidence are close but typically lower than the actual performance shown by the subjects (Vickers, 1979); in general, the slope of the confidence versus difficulty curve is lower than that of the probability correct versus difficulty curve. One could argue that verbal reports provide biased estimates of the true internal confidence because of their discretized nature. A second, and more interesting, possibility is that subjects' belief does not fully correspond to their performance because the subjects have not completely learned the task's contingencies. In either case, confidence would not correspond to the probability of being correct. The last situation can be described with the DDM presented here if, for instance, the variance σ in equation 2.1 is only approximately known by the subject and is taken to be larger than the real one. For the problem corresponding to equation 3.16, probability correct and confidence will be correlated and have the same functional dependences on μ0 (which measures the difficulty of each trial), but the slope of the performance versus difficulty curve will be higher than that of the confidence-versus-difficulty curve.

3.3.  Marginal Confidence.

Decision confidence depends in general on both the decision time and the state of the losing integrator. However, for comparison with behavioral data, it is necessary to know the marginal decision confidence where the state of the losing integrator has been integrated out, since the latter is not typically available. Marginalization requires knowing the probability density of xk (k = 1, 2) at decision time, which satisfies a Fokker-Planck equation (FPE) with absorbing boundaries. The FPE associated with equations 2.1 is
formula
3.17
where is the joint probability density of finding the integrators in state at time t given that the initial condition at time t = 0 was . The FPE has to be solved along with the condition that at the boundaries, defined as the semi-lines x1 = Θ(t) for all x2 ⩽ Θ(t) (l1) and x2 = Θ(t) for all x1 ⩽ Θ(t) (l2), P has to be zero at all times. This condition defines the semi-lines l1 and l2 as absorbing boundaries. Solutions to the FPE, equation 3.17, are known for the independent and completely anticorrelated (classical DDM) cases with time-independent boundaries (Cox & Miller, 1965). In the appendix, I provide details for the derivation of the solution for two half-anticorrelated integrators with time-independent boundaries. The above three solutions allow a direct comparison of marginal confidence between different cases. In the following, Θ = 0 and a < 0. Then |a| measures the initial distance to threshold.
In order to determine the marginal confidence (one needs to derive the joint probability density that integrator k reaches its threshold at time t and integrator jk is at state xj (xj ⩽ 0) at that time, along with the condition that no integrator has crossed its threshold before, given the initial condition . This quantity, denoted , is written as
formula
3.18
which can be obtained from the probability flux vector expression associated with the FPE (see equation 3.17; Risken, 1989). The dependence on the drift rates has been made explicit for later convenience. Its integral over xj is the conditional first passage times distribution for integrator k. For instance, using equation A.9 in the appendix, for two independent integrators equation 3.18 takes the expression
formula
3.19
and similarly for .
Marginal confidence is defined as the confidence in a decision made at time t when the state of the losing integrator and the values of the drift rates are unknown or, more formally,
formula
3.20
where decision confidence p12x2, t) is given in equation 3.9 and is the probability distribution of x2 at time t given that the first integrator has reached threshold at time t averaged over the distribution of drift rates. The latter is defined as , where C(t) is a normalization constant; , which includes the average over the distribution of drift rates; and is the a priori distribution over the drift rates. One could also define marginal confidence for a fixed set of values of drift rates, as they can be controlled experimentally, but here I focus on the averaged quantity. For two half-anticorrelated integrators, using a uniform prior along with the constraint that μ1 = −μ2, and equations A.8 and 3.16, it is found that the probability density of finding the second integrator at state x2 given that the first integrator has reached its boundary at time t takes the expression
formula
3.21
where the proportionality is in relation to the variable x2. For two independent integrators, using equations A.9 and 3.17, with the constraint that μ1 = −μ2, it is obtained
formula
3.22

Figure 4B shows that marginal confidence (computed using equations 3.20 to 3.22, for two independent and half-anticorrelated integrators and using equation 3.11 for the classical DDM with the same threshold and noise values in the three cases) is at a maximum for the whole range of decision times when the integrators are independent, nearly so when they are half-anticorrelated, and at a minimum for the classical DDM. Therefore, decision confidence in integrators with even strong correlations is close to that in independent integrators and higher than that in the classical DDM. The large difference in marginal confidence between the classical DDM and the half-anticorrelated or independent integrators is due to the fact that as decision time increases, the losing integrator in the latter two cases separates from its boundary (a reminder that the losing integrator has negative drift). Using the fact that confidence increases as the distance between the integrators increases, equation 3.9, it is now clear why marginal confidence becomes higher for the independent and half-anticorrelated cases than for the classical DDM.

3.4.  Reaction Time Distributions.

The time evolution of the probability density function of finding the neuronal integrators in a given state can be used to gain additional insight into the decision process. Figure 5 shows three snapshots of the time evolution of the probability density distribution over for two independent (see Figure 5A) and two half-anticorrelated integrators (see Figure 5B). In both cases, I use μ12 ⩾ 0. The distribution broadens over time and moves toward the boundaries. When the distributions get close to the boundaries, their peaks remain approximately stationary. For the parameter values used, the distributions approach the boundary l1 more rapidly. Therefore, most of the trajectories will escape through the semi-line l1. However, many others will cross l2 first. It can be proved that one of the two boundaries is crossed at some finite time with probability one if the drift of at least one integrator is larger than zero. The distribution spreads circularly for two independent integrators, while for two half-anticorrelated integrators, the distribution spreads more along the nonprincipal diagonal.

Figure 5:

Time evolution of the probability density distribution over for two integrators with independent (A) and half-anticorrelated (B) white noise inputs. The gray-coded distributions are plotted at three different times, using the solutions in equations A.9 and A.8. Parameters for the independent integrators (ρ = 0) are a = −1.5, b = −1, μ1 = 0.2s−1, μ2 = 0.1s−1, and σ2 = 1s−1. Parameters for the half-anticorrelated integrators (ρ = 1/2, ν = −1) are a = b = −1, μ1 = 0.75s−1, μ2 = 0s−1, and σ2 = 0.5s−1. Besides, Θ1 = Θ2 = 0.

Figure 5:

Time evolution of the probability density distribution over for two integrators with independent (A) and half-anticorrelated (B) white noise inputs. The gray-coded distributions are plotted at three different times, using the solutions in equations A.9 and A.8. Parameters for the independent integrators (ρ = 0) are a = −1.5, b = −1, μ1 = 0.2s−1, μ2 = 0.1s−1, and σ2 = 1s−1. Parameters for the half-anticorrelated integrators (ρ = 1/2, ν = −1) are a = b = −1, μ1 = 0.75s−1, μ2 = 0s−1, and σ2 = 0.5s−1. Besides, Θ1 = Θ2 = 0.

The first passage times-state joint probability distribution, equation 3.18, can be used to describe the state of the losing integrator given that a decision has been made. Figure 6A shows a typical example of the joint probability distribution. The mean value of x2 (the white line) diverges monotonically from threshold as a function of the crossing time of integrator k = 1. This is because, as shown in Figure 5A, trajectories tend to disperse at long times. A different picture emerges when the two integrators are half-anticorrelated, equations 3.18 and A.8, as shown in the right panel. The mean value of x2 approaches threshold at short decision times, while it diverges from threshold at long decision times. The first rise is due to the presence of negative correlations in the inputs to the two integrators and can be understood as follows. If integrator k = 1 reaches its threshold at a late time, this delay is likely due to a negative input fluctuation to that integrator. Since the two integrators have anticorrelated inputs, the integrator k = 2 has likely experienced a positive input fluctuation, explaining why the mean value of x2 approaches the threshold at short decision times. At long decision times, the diffusion of trajectories due to noise dominates over the previous mechanism, and therefore the mean value of x2 diverges from threshold in that regime.

Figure 6:

(A). First passage times-state joint probability distribution, , for two independent (left) and two half-anticorrelated (right) integrators (see equation 3.19, and equations 3.18 and A.8, respectively). White lines are the mean value of x2 as a function of decision time. (B) First passage times probability density distributions, gk(t), k = 1, 2, for independent (left) and half-anticorrelated (right) integrators. Full and dashed lines correspond to the integrator with higher and lower drift, respectively. Fluctuating lines correspond to numerical simulations of equations 2.1. Parameters are as in Figure 5.

Figure 6:

(A). First passage times-state joint probability distribution, , for two independent (left) and two half-anticorrelated (right) integrators (see equation 3.19, and equations 3.18 and A.8, respectively). White lines are the mean value of x2 as a function of decision time. (B) First passage times probability density distributions, gk(t), k = 1, 2, for independent (left) and half-anticorrelated (right) integrators. Full and dashed lines correspond to the integrator with higher and lower drift, respectively. Fluctuating lines correspond to numerical simulations of equations 2.1. Parameters are as in Figure 5.

Figure 6B shows the probability distributions of first passage times for each integrator along with the condition that it crossed its threshold first. This first passage time distribution is computed by integrating equation 3.18 with the probability distribution of the losing integrator. The integrator with the higher drift crosses first the threshold with a probability larger than one-half (full lines). The distributions are positively skewed, similar to the reaction times distributions typically found in two-alternative forced-choice tasks (Ratcliff & Smith, 2004; Luce, 1986).

4.  Discussion

A decision diffusion model (DDM) consisting of two neuronal integrators receiving independent as well as correlated inputs has been studied. Decision confidence has been shown to depend on the difference between the traveled distances of the two integrators at decision time. Importantly, leakless integrators accumulate evidence without loss, since their final state and elapsed time fully specify the probability distribution over the signals, regardless of the stopping rule used to make decisions. Different behaviors are obtained from leak integrators, since the probability distribution over the drift rates depends not only on the final states of the integrators and elapsed time. These results can have important consequences about neuronal coding and integration of information in lateral intraparietal area (LIP) and frontal eye fields (FEF), where almost linear, perfect integration of evidence is observed.

Previous models have proposed that decision confidence is encoded in the difference between the state of two integrators that accumulate evidence in favor of two alternatives (Vickers, 1979; Juslin & Olsson, 1997; Vickers & Lee, 1998; Van Zandt, 2000), like the DDM studied here. However, it was not known how confidence was mathematically related to the difference of the states (balance of evidence) of the integrators. As equation 3.9 shows, decision confidence is a cumulative gaussian whose argument is proportional to the balance-of-evidence term. In addition, it explicitly shows time dependences that were not anticipated by these preliminary works. Pleskac and Busemeyer (2007) have considered a model in which confidence is computed after the decision has been made. However, an interesting alternative, as proposed by the model presented here, is that that when a decision is made, there is immediate access to decision confidence and the evaluation of confidence does not require additional deliberation. Ratcliff and Starns (2009) recently presented a model about how verbal reports of decision confidence can be generated by human subjects. The model requires multiple additional integrators (one for each confidence rating—for example, high, medium, or low), whose role is to determine which rating will be reported. The DDM model that I have described here makes the interesting prediction that decision confidence is represented in the brain as a continuous value: if verbal reports about confidence are not forced, subjects might use confidence only implicitly without the necessity of discretizing it.

Biophysically inspired neuronal network models for decision making have been studied in Wang (2002) and Beck et al. (2008). These attractor models show almost linear ramping activity at low firing rates, like the DDM studied here, but they display nonlinearities that become evident when the firing rate approaches the decision boundary. Consistent with these models, Huk and Shadlen (2005) and Wong, Huk, Shadlen, and Wang (2007) have shown that brief motion pulses (100 ms) injected at different times from the onset of a dynamic random dot display affect the firing rate of LIP neurons in a time-dependent manner, which earlier pulses producing larger changes in firing rate than later pulses. As Huk and Shadlen (2005) pointed out, this result is also consistent with a perfect integrator model of inputs with time-decaying boundaries, like the DDM considered in this letter. Further experiments are required to test which is the correct model. These experiments would likely involve injecting motion pulses at different times conditioned to the same firing rate of the LIP neurons for all times. In this hypothetical experiment, the attractor model predicts time-shift invariance of the induced changes in firing rate, while the DDM with time-decaying boundaries predicts its violation. Further theoretical work is also required to connect realistic neuronal attractor networks with simplified integrator models and to determine to what extent nonlinearities that arise near the decision boundaries can be well approximated by hard decision boundaries (see Roxin & Ledberg, 2008).

Decision confidence has been shown to display two apparently contradictory behaviors (Vickers, 1979). First, confidence decays as a function of decision time when subjects are free to choose the stopping time (Henmon, 1911). In contrast, confidence increases as a function of decision time when subjects are externally forced to make a decision at an experimentally controlled time (Vickers, 1979). The first situation corresponds directly to the case that I have considered in Figure 4B: since trials for which the difference between the drift rates is large lead typically to short decision times, confidence will be higher at shorter than at longer decision times. The second situation can also be described with the DDM introduced here. This can be seen by noting that in this case, equation 3.9 becomes
formula
4.1
where Δ(x1, x2) = x1x2, x1, and x2 are the states of the integrators at time t, and it is assumed that the two integrators are allowed to evolve freely until the stopping time is externally forced (i.e., bounds are not present, as in Vickers, 1979, and the state of the integrators is read out at the forced stopping time). Since the difference between the states of the integrators grows in mean linearly with time and proportionally to the difference in their drift rates, that is, 〈Δ(x1, x2)〉> = (μ1 − μ2)t, the argument of equation 4.1grows in mean as a function of time, and therefore confidence is expected to increase. Figure 7 shows that decision confidence decreases as a function of decision time for the classical DDM when decision time is internally generated, equation 3.11, and that decision confidence in the same model without boundaries for any given drift rate μv = μ1 − μ2 = 2μ0>0, approximated as
formula
4.2
increases as a function of the externally forced stopping time. As expected, the curves move downward with increasing input noise σ, since the difficulty of the task increases with it. Mathematically, this is because both equation 3.11 and equation 4.2 are decreasing functions of σ.
Figure 7:

Decision confidence decreases with decision time when the stopping time is internally determined by a threshold crossing (continuous lines) but increases with time if the decision time is externally forced and there is no threshold (dashed lines). In both cases, the classical DDM is considered (i.e., a DDM with completely anticorrelated fluctuations), with bounds at Θ = 1 when the decision time is internally generated and no bounds when the stopping time is externally forced (for the the latter case, μ0 = 2s−1). The noise variance is σ2 = 3s−1 and 30s−1 for the thick and thin lines, respectively.

Figure 7:

Decision confidence decreases with decision time when the stopping time is internally determined by a threshold crossing (continuous lines) but increases with time if the decision time is externally forced and there is no threshold (dashed lines). In both cases, the classical DDM is considered (i.e., a DDM with completely anticorrelated fluctuations), with bounds at Θ = 1 when the decision time is internally generated and no bounds when the stopping time is externally forced (for the the latter case, μ0 = 2s−1). The noise variance is σ2 = 3s−1 and 30s−1 for the thick and thin lines, respectively.

In summary, I have provided analytical tools to describe decision confidence and uncertainty in decision making using the DDM. The analytical expressions for decision confidence in partially correlated neuronal integrators found in this letter might allow estimating confidence from multirecording neurophysiological data in decision areas like LIP and FEF (Hanes & Schall, 1996; Shadlen & Newsome, 1996; Platt & Glimcher, 1999) and compare it with behaviorally measured confidence values. The results can be extended to multiple integrators and to more general forms of noise, therefore opening a window for studying how decision uncertainty is represented in neuronal populations in the brain.

Appendix:  Solution for a DDM with Half-Anticorrelated Integrators

This appendix provides details for the derivation of the solution of the FPE, equation 3.17, for the case of half-anticorrelated integrators. It also gives the derivation for the case in which the integrators are independent.

The strategy is to find the solution of the FPE, equation 3.17, with boundary conditions as specified in the main text as a linear superposition of free solutions,
formula
A.1
with initial condition , using the well-known method of the images (Cox & Miller, 1965; Risken, 1989). Note that a free solution is a bivariate gaussian with independent variables u = x1 + x2 and v = x1x2. The free solutions satisfy the FPE, equation 3.17, for any initial condition. However, none of them satisfies the vanishing boundary condition at the semi-lines l1 and l2.
As a preliminary step, the FPE, equation 3.17, with absorbing boundary on the line x1 = 0 is solved. I seek a solution with the form
formula
A.2
consisting of a source at the initial condition (here it is not assumed that a = b, as in section 3.3) and an image source at weighted with the coefficient r. By imposing the boundary condition P(0, x2) = 0 for all x2 (i.e., the line x1 = 0 is an absorbing boundary), I find that the image coordinates are with α = 2(1 − σ2u2v)/(1 + σ2u2v), and
formula
A.3
where u0 = a + b, v0 = ab, u1 = a1 + b1 and v1 = a1b1. The solution satisfies the initial condition, . If the absorbing boundary is on the line x2 = 0, the solution instead is
formula
A.4
with the image at and
formula
A.5
where u2 = a2 + b2 and v2 = a2b2. In both cases, the image is located on the ellipse,
formula
A.6
i = 1, 2, which includes the source at the initial condition. It is convenient to introduce the matrices
formula
which define a mapping from the initial condition to the image positions through the lines x1 = 0 and x2 = 0, respectively, , i = 1, 2. Repeated and alternated applications of Π1 and Π2 to the initial condition lead to points that lie on the same ellipse, equation A.6.
A combination of the two previous images with the source at the initial condition,
formula
A.7
is not a solution to the FPE, equation 3.17, with absorbing boundaries on the semi-lines l1 and l2, for any values of c1 and c2. This is because the source at the initial condition plus one of the images can vanish on one semi-line, but not simultaneously on the other, implying the necessity of additional images. One possibility is to augment the number of images to try to match the boundary condition at the two semi-lines simultaneously. In fact, I find that a finite number of images is enough for a DDM with two half-anticorrelated integrators, corresponding to (ρ = 1/2 and ν = −1 in equation 2.1). The solution consists of five images,
formula
A.8
with σ2u = σ2 and σ2v = 3σ2; positions , , , , and ; and coefficients , , , , and . The solution is schematically represented in Figure 1D. The images are located on an ellipse, equation A.6. Their positions are obtained by applying iteratively and alternatively the matrices Πi(α), i = 1, 2, with α = 1 to the initial condition. After defining , i = 1, 2, we have , , , , and . Images 1 and 2 are the images of the source at the initial condition through lines x1 = 0 and x2 = 0, respectively. Images 3 and 4 can be thought of as the images of the first and second images through the lines x2 = 0 and x1 = 0, respectively, and image 5 as the image of images 3 and 4 through the lines x1 = 0 and x2 = 0. The coefficients have been found by using the previous results for single line boundaries, which allow imposing for each pair of image and its image through one semi-line that the boundary condition is met on that semi-line.
A finite number of free solutions suffices too when the white noise inputs are independent across the two integrators, corresponding to ρ = 0 in equations 2.1. The solution consists of three images,
formula
A.9
with σ2u = σ2v = 2σ2; positions , , and ; and coefficients , , and c3 = c1c2. A schematic representation of the solution is shown in Figure 1B. All images and initial source lie on a circle, equation A.6. The positions of the two first images are obtained by application of the matrices Πi(0), i = 1, 2 (α = 0) to the initial condition, . Their coefficients can be obtained from the coefficients of the images required to solve the FPE, equation 3.17, with absorbing boundaries on the lines x1 = 0 or x2 = 0, equations A.4 and A.5. The third image can be thought of as the image of images 1 and 2 through the lines x2 = 0 and x1 = 0, respectively. It is uniquely defined because of the transitivity of the matrix product Π1(0)Π2(0), that is, . The coefficient of the third image has been chosen in order to match the boundary condition on both semi-lines simultaneously. The solution for this case can also be obtained by multiplying the known individual solutions for two one-dimensional Wiener processes with absorbing boundaries (Cox & Miller, 1965).

Acknowledgments

I thank Alex Pouget and Jan Drugowitsch for insightful discussions. This work has been partially supported by the Swartz Foundation.

Notes

1
In numerical simulations, equation 2.1 is solved using the Euler method with small time step δt as
formula
where δxi (i = 1, 2) are the variable increments in each step and wi (i = 1, 2, c) are independent across time and follow normal gaussians.

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