Bounded Ornstein-Uhlenbeck models for
two-choice time controlled tasks
Jiaxiang Zhang a,b 1 , Rafal Bogacz a ,
a Department
b School
1
of Computer Science, University of Bristol, Bristol BS8 1UB, UK
of Psychology, University of Birmingham, Birmingham B15 2TT, UK
Correspondence address: School of Psychology, University of Birmingham,
Birmingham B15 2TT, UK. Email: [email protected]
Preprint submitted to Elsevier
23 September 2009
Abstract
The Ornstein-Uhlenbeck (O-U) model has been successfully applied to describe
the response accuracy and response time in 2-alternative choice tasks. This paper
analyses properties and performance of variants of the O-U model with absorbing
and reflecting boundary conditions that limit the range of possible values of the
integration variable. The paper focuses on choice tasks with pre-determined response
time. The type of boundary and the growth/decay parameter of the O-U model
jointly determine how the choice is influenced by the sensory input presented at
different times throughout the trial. It is shown that the O-U models with two
types of boundary hold close relationship and can achieve the same performance
under certain parameter values. The value of the growth/decay parameter that
maximizes the accuracy of the model has been identified. It is shown that when
the boundaries are introduced, the O-U model may achieve higher accuracy than
the diffusion model. This suggests that given the limited range of the firing rates
of integrator neurons, the neural decision circuits could achieve higher accuracy
employing leaky rather than linear integration in certain tasks. We also propose
experiments that could distinguish between different models of choice in tasks with
pre-determined response time.
Key words: choice, decision, Ornstein-Uhlenbeck model, reflecting boundaries,
absorbing boundaries
2
1
1
Introduction
2
Making choices on the basis of sensory information is one of the fundamental
3
cognitive functions of intelligent animals. The studies of decision process in
4
simple 2-alternative forced choice (2AFC) tasks usually employ one of two
5
paradigms. In the time controlled (TC) paradigm, subjects are required to re-
6
spond immediately after a cue presented at an experimenter-determined time
7
(Dosher, 1976, 1984; Swensson, 1972; Yellott, 1971). Alternatively, the infor-
8
mation controlled (IC) paradigm allows subjects to respond freely whenever
9
they feel confident in their choice (Luce, 1986).
10
Several mathematical models have been developed over the last half century
11
to describe the behavioural performance as well as underlying decision process
12
in 2AFC tasks (Busemeyer and Townsend, 1992, 1993; Laming, 1968; Ratcliff,
13
1978; Ratcliff et al., 1999; Stone, 1960; Usher and McClelland, 2001; Vickers,
14
1970). These sequential sampling models share a common characteristic that
15
the sensory representation of stimuli is accumulated over time to form a choice.
16
Neural correlates of such accumulation process have been observed in certain
17
cortical areas (Gold and Shadlen, 2002; Kim and Shadlen, 1999; Schall, 2001;
18
Shadlen and Newsome, 2001), which gives further support to the sequential
19
sampling framework.
20
In the IC paradigm, the decision process depends on the amount of accumu-
21
lated evidence, i.e., a choice is made as soon as sufficient evidence in favour
22
of one alternative is accumulated. In the TC paradigm, since the time of re-
23
sponse is chosen by the experimenter, a choice can be made by simply asking,
24
at the time of response, whether the accumulated evidence supporting the
3
25
first alternative exceeds that of the second. However, according to such model
26
for the TC paradigm the accumulated evidence could take arbitrarily large or
27
small values, which could lead to unrealistic predictions (Meyer et al., 1988).
28
One approach to overcome this problem is to assume that the accumulated
29
evidence is bounded within a certain range (Feller, 1968; Ratcliff, 1988). In
30
our previous study (Zhang et al., 2009), we compared the effects of two pos-
31
sible boundary mechanisms, reflecting and absorbing boundaries, on a typical
32
sequential sampling model - the diffusion model (Laming, 1968; Ratcliff, 1978;
33
Stone, 1960), and we showed that both boundary types lead to similar perfor-
34
mance in the TC paradigm and produce similar fits to experimental data.
35
This paper extends our previous findings by investigating the dynamics and
36
performance of the Ornstein-Uhlenbeck (O-U) model (Busemeyer and Diederich,
37
2002; Busemeyer and Townsend, 1992, 1993) with two types of boundaries in
38
the TC paradigm. The O-U model has been successfully applied to a vari-
39
ety of tasks (Diederich, 1995, 1997; Smith, 1995, 2000) and can approximate
40
the same computation carried out by other biologically inspired models (e.g.,
41
Usher and McClelland, 2001) in 2AFC tasks. We show that the O-U model
42
with different types of boundaries could achieve the same performance under
43
certain parameter values, though the boundaries introduce differential weight-
44
ing of evidence. We also show that when the boundaries are introduced, the
45
accuracy of the O-U model is higher than that of the diffusion model.
46
The paper is organized as follows. Section 2 reviews the decision problem
47
and the O-U model as well as boundary mechanisms. Section 3 compares
48
the properties of the O-U models with two types of boundaries. Section 4
49
investigates the performance of the model and identifies the parameter values
50
that maximize accuracy. Section 5 discusses experimental predictions of the
4
51
O-U model with boundaries and Section 6 summarizes implications of our
52
study. Mathematical details are provided in Appendices. Preliminary results
53
of this study were presented in an abstract form (Zhang and Bogacz, 2006).
54
2
55
2.1 Neurobiology of decision
56
Numerous in vivo experiments have been performed to investigate the neural
57
correlates of decision processes in monkeys. In a well-known motion discrimi-
58
nation task used in this line of research, monkeys were instructed to fixate on
59
a random moving dots stimulus presented within a circular aperture (Britten
60
et al., 1993). A proportion of dots move coherently to the left or to the right
61
and the monkey has to respond to the coherent motion direction by making
62
a saccade to a left or right target.
63
Neural recording data from the motion discrimination task indicate that the
64
decision process involves several cortical regions. First, the neural activity
65
in sensory areas (e.g., the middle temporal (MT) area) has been shown to
66
closely correlate with the statistics of the stimulus (i.e., the motion coherence)
67
(Britten et al., 1992, 1993, 1996). Second, neurons in the lateral interparietal
68
(LIP) area that directly receive inputs from area MT gradually build up or
69
attenuate their activity within a trial (Roitman and Shadlen, 2002; Shadlen
70
and Newsome, 2001). In particular, neural activity in LIP correlates with the
71
direction of eye movements (i.e., the decision) at the time of response. These
72
results suggest that LIP neurons accumulate sensory evidence from area MT
73
to form a choice.
Sequential sampling models and boundary mechanisms
5
74
The experimental results indicate that the accumulation of sensory evidence
75
may be a general decision mechanism manifested in different brain regions
76
or species (Gold and Shadlen, 2007; Schall, 2001). Based on this assumption,
77
in this paper we investigate decision problems in 2AFC tasks formalized as
78
following (Gold and Shadlen, 2001, 2002). Two populations of sensory neu-
79
rons generate continuous noisy evidence Y1 (t) and Y2 (t) supporting the two
80
alternatives Y1 and Y2 at time t. We assume that Y1 (t) and Y2 (t) are normally
81
distributed with constant means µ1 and µ2 (µ1 , µ2 > 0). The goal of the de-
82
cision process is to identify which sensory population has higher mean based
83
on Y1 (t) and Y2 (t).
84
2.2 The Ornstein-Uhlenbeck model
85
The Ornstein-Uhlenbeck (O-U) model (Uhlenbeck and Ornstein, 1930) is a
86
stochastic process that was first introduced into psychology by Busemeyer
87
and Townsend (1992, 1993). Since then many researchers applied the O-U
88
model to a variety of studies to account for response accuracy and response
89
time in choice tasks (Busemeyer and Diederich, 2002; Diederich, 1995, 1997;
90
Heath, 1992; Smith, 1995, 1998), and the model has also been extended to
91
multi-alternative tasks (McMillan and Holmes, 2006; Usher and McClelland,
92
2001).
93
In the O-U model for 2AFC tasks a non-perfect integrator accumulates the
94
difference of sensory evidence supporting two alternatives (Y1 (t) − Y2 (t)). Let
95
X(t) denote the value of the integrator state at time t. The O-U model is
96
described by the following stochastic differential equation:
dX(t) = (λX(t) + µ) dt + σdW ,
6
(1)
97
where µ and σ denote the mean and standard deviation of the normally dis-
98
tributed quantity Y1 (t)−Y2 (t) (i.e., µ = µ1 −µ2 ), dX(t) denotes the increment
99
of evidence over a small unit of time dt, dW (t) is Gaussian noise with mean
100
zero and unit variance, and λ is a growth/decay parameter. According to the
101
decision problem proposed above, the sign of µ solely determines the correct
102
choice. More specifically, µ > 0 implies that the first alternative Y1 is cor-
103
rect (µ1 > µ2 ), and µ < 0 implies that the second alternative Y2 is correct
104
(µ1 < µ2 ). For simplicity, we hereafter set µ > 0 in this paper. On the other
105
hand, the magnitude of µ and σ determine the task difficulty, i.e., for |µ| close
106
to zero or large σ, it is difficult to distinguish which alternative (Y1 or Y2 ) has
107
higher mean. On each trial, the decision process starts at t = 0 with a staring
108
point X0 = X(0) = 0. In the IC paradigm, the integrator continuously accu-
109
mulates evidence according to Eq. 1 until X(t) reaches a decision threshold.
110
In the TC paradigm, the decision process terminates at a pre-determined time
111
tc , and the choice is made on the basis of the final integrator state X(tc ), i.e.,
112
the model selects Y1 if X(tc ) > 0, or Y2 if X(tc ) < 0. Throughout this paper
113
we focus on the TC paradigm and quantify the model’s performance by the
114
error rate P (tc ), which is the probability of making an incorrect choice when
115
the response is required at time tc . The expression for the error rate of the
116
O-U model is given by Busemeyer and Townsend (1993):

v
u

µ u 2 (eλtc − 1) 
,
P (tc ) = Φ − t
σ λ (eλtc + 1)
where Φ(u) =
Z u
−∞
v2
1
√ e− 2 dv .
2π
(2)
(3)
117
It is worth to note that the parameter λ affects the linear drift rate λX(t) + µ,
118
and as a result the O-U model could exhibit differential weighting of evidence
7
119
within a trial. That is, for λ > 0 the evidence presented early in a trial has a
120
larger influence on the decision (a primacy effect), and for λ < 0 the choice
121
mainly depends on the evidence presented later in a trial (a recency effect)
122
(Wallsten and Barton, 1982). An explicit proof of this property is available
123
in Busemeyer and Townsend (1993). In particular, for λ = 0, the weighting
124
of evidence is independent to the time of its presentation and the O-U model
125
simplifies to a diffusion model (Laming, 1968; Ratcliff, 1978; Stone, 1960). In
126
Section 3 we will investigate how the primacy and recency effects of the O-U
127
model are affected by boundaries.
128
2.3 Boundary mechanisms
129
In earlier versions of sequential sampling models, the value of the integra-
130
tor states was unconstrained. As mentioned in the Introduction, this type
131
of model would allow the integrator state to have arbitrarily large or small
132
values. This is, however, a simplification. First, it is not realistic that any bi-
133
ological system could represent an infinite amount of evidence. Second, when
134
accumulated evidence is not limited, a sequential sampling model may not fit
135
experimental data (Ratcliff, 1988). One way to eliminate the above problems
136
is to transform the integrator state through a nonlinear activation function
137
(Brown et al., 2005; Brown and Holmes, 2001; Usher and McClelland, 2001).
138
Another approach, the focus of this paper, is to apply boundary conditions to
139
the existing linear models, which provides an approximation of the nonlinear
140
activation functions (Usher and McClelland, 2001).
141
The first boundary mechanism is the absorbing boundary (Feller, 1968), which
142
was originally applied to choice tasks in which IC and TC paradigms were in8
143
termixed (Diederich and Busemeyer, 2003; Ratcliff, 1988; Ratcliff and Rouder,
144
2000). Once the integrator hits an absorbing boundary, the accumulation pro-
145
cess terminates and the integrator state is maintained afterwards. To intro-
146
duce the absorbing boundaries to the O-U model, we assume two symmetric
147
boundaries at ±b, and then, the O-U model with absorbing boundaries can
148
be described by:





X(t + dt) = b,





X(t + dt) = −b,








X(t + dt) = X(t) + dX(t),
if X(t) + dX(t) ≥ b or X(t) = b ,
if X(t) + dX(t) ≤ −b or X(t) = −b ,
(4)
otherwise ,
149
where dX(t) is given by Eq. 1. In the TC paradigm, the choice is determined
150
by which boundary the integrator hits first, or if neither boundary is reached
151
before tc , the choice is determined by the same rule as the one used for the
152
model without boundaries (see Section 2.2).
153
The second boundary mechanism is the reflecting boundary (Zhang et al.,
154
2009) (also see Diederich, 1995; Diederich and Busemeyer, 2003; Ratcliff and
155
Smith, 2004), which only restricts the amount of accumulated evidence that
156
can be represented by the integrator. In other words, the value of the integrator
157
state cannot exceed a reflecting boundary, but it may continue to change (i.e.,
158
move back) due to noise, as specified by:




X(t + dt) = b,





if X(t) + dX(t) ≥ b ,
X(t + dt) = −b,








X(t + dt) = X(t) + dx(t),
if X(t) + dX(t) ≤ −b ,
(5)
otherwise .
159
Hereafter we refer to the O-U model with absorbing and reflecting bound-
160
aries as absorbing O-U and reflecting O-U models, and the model without a
9
161
boundary as unbounded O-U model. Fig. 1 shows sample paths of integrator
162
states in the O-U models with both types of boundaries, compared with the
163
unbounded O-U model. For the absorbing O-U model, the choice can be de-
164
termined early in a trial (after 0.35s in Fig. 1). For the reflecting O-U model,
165
the preferred choice may change throughout the decision process until the end
166
of the trial.
167
In a previous study, we compared the boundary effects on one sequential sam-
168
pling model - the diffusion model (Zhang et al., 2009). We showed that the
169
diffusion model with two types of boundaries produce exactly the same error
170
rate for any given tc . Moreover, the absorbing and reflecting boundaries yield
171
a primacy and recency effect, respectively. In the next section we will show
172
that similar properties also hold for the bounded O-U model.
173
2.4 Optimal decision making theory
174
Due to evolutionary pressure, neural systems are likely to adapt to their en-
175
vironment and optimize their behaviour to achieve desired goals (Anderson,
176
1990). This assumption has been invoked in the study of perceptual decision
177
making, and it has been proposed that the brain may implement statistically
178
optimal decision-making procedures (Bogacz, 2007, 2009; Bogacz et al., 2006;
179
Gold and Shadlen, 2001, 2002, 2007). In the sequential sampling framework,
180
an optimal decision procedure would achieve the lowest error rate and short-
181
est response time compared with all possible procedures to integrate sensory
182
evidence 1 . For the TC paradigm, the optimal procedure is the one that yields
1
The optimality criterion for the IC paradigm is to consider which decision strategy
yields the fastest response time for any given error rate. The procedure that satisfies
10
183
the lowest error rate for any given response time.
184
Recall that the decision problem proposed in Section 2.1 is to identify which of
185
the two alternatives has higher mean inputs. This problem can be converted
186
to the problem of hypothesis testing in statistics (Ghosh, 1970; Lehmann,
187
1959), and the optimal procedure satisfying the above criterion is provided
188
by the Neyman and Pearson (NP) procedure. It has been proved that among
189
all hypothesis tests, the NP procedure minimizes the overall probability of
190
accepting the false hypothesis for fixed sample size (Neyman and Pearson,
191
1933), and therefore it minimizes the error rate for any response time. Bogacz
192
et al. (2006) showed that as the time intervals between samples approach
193
zero, the NP procedure becomes equivalent to the diffusion model. Hence, the
194
diffusion model is the optimal decision procedure in the TC paradigm. Since
195
the O-U model with λ = 0 can be reduced to the diffusion model, the O-U
196
model also maximizes accuracy when λ = 0. In Section 4, we will show that
197
this is no longer the case for the bounded O-U model.
198
3
199
3.1 Dynamics of bounded O-U models
200
The dynamics of the absorbing and reflecting O-U models can be illustrated by
201
the probability density p(X, t) of the integrator states X(t) at time t, which is
202
calculated in Appendix A by solving the Kolmogorov backward equation with
203
boundary conditions. The density function p(X, t) of bounded O-U models is
Properties of bounded O-U models
this criterion is provided by the Sequential Probability Ratio Test (Barnard, 1946;
Wald, 1947).
11
204
a weighted sum of an infinite series:
p(X, t) =
∞
X
e−ξj t Kj (X)φj (X0 ) ,
(6)
j=0
205
where Kj (X) is defined in Equation A.16. ξj are the non-negative eigenvalues
206
from Eqs. A.27 and A.30, and φj (X0 ) are the associated eigenfunctions defined
207
in Eqs. A.26 and A.29 as a function of the starting point X0 . The solutions of
208
the eigenvalues and eigenfunctions depend on the type of boundaries. There
209
exist infinitely many eigenvalues that can be found numerically, and an ap-
210
proximation of p(X, t) can be obtained by taking a partial sum of finite terms.
211
The more eigenvalues included in Eq. 6, the more accurate the approximation
212
of the density function p(X, t).
213
Fig. 2 compares the probability densities of the absorbing and reflecting O-
214
U models at several time instants. During the first hundred milliseconds, all
215
densities are close to a Gaussian distribution with mean close to the starting
216
point X0 and shifted towards the upper boundaries as time grows. For large t,
217
the density of the absorbing O-U model collapses to zero, while the reflecting
218
O-U model has nonzero equilibrium distributions, with a small proportion to
219
the left of the origin, corresponding to the probability of making an incorrect
220
choice (for µ > 0).
221
The difference between the stationary densities can be explained from the
222
expression of p(X, t). For any ξj > 0, the time-dependent coefficient e−ξj t in
223
Eq. 6 monotonically decreases and converges to zero as t grows. Table 1 shows
224
the first ten eigenvalues ξj . For the absorbing O-U model all eigenvalues are
225
positive, hence the density of the absorbing O-U model converges to p(X, t) =
226
0 as t → ∞. For the reflecting O-U model, one of the eigenvalues is equal to
227
zero ξ0 = 0, hence the density p(X, t) converges to K0 (X)φ0 (X0 ) as t → ∞.
12
228
It is also interesting to note that the eigenvalues ξj (j > 0) of the absorbing
229
and reflecting O-U models are symmetric by flipping the sign of λ.
230
3.2 Primacy and recency effects
231
Previous studies showed that differential weighting of evidence (i.e., the pri-
232
macy and recency effects) can be introduced by either the parameter λ of
233
the unbounded O-U model (Busemeyer and Townsend, 1993) or two types
234
of boundaries (Zhang et al., 2009). Here we investigate the primacy and re-
235
cency effects in the O-U models by applying the reverse correlation method.
236
We simulate the unbounded, absorbing, and reflecting O-U models, each with
237
positive and negative λ values. For each model considered, the time course of
238
sensory evidence is recorded and averaged only from trials resulting in correct
239
choices. The averaged evidence represents how the model weights noisy inputs
240
during the decision process. For µ > 0, the model makes correct decision if the
241
integrated evidence is positive at the end of a trial. Hence a larger averaged
242
input from correct trials indicates that the sensory input at that time point
243
has, on average, a large influence on the final choice, and a smaller averaged
244
input indicates that the choice depends to a lesser extent on the input at that
245
time.
246
The simulation results are shown in Fig. 3. First, positive and negative λ values
247
introduce primacy (Fig. 3a) and recency (Fig. 3d) effects to the unbounded
248
O-U model, which is consistent with theoretical predictions by Busemeyer and
249
Townsend (1993). Second, the absorbing O-U model exhibits a strong primacy
250
effect for λ > 0 (Fig. 3b), but no clear effect was observed for λ < 0 (Fig.
251
3e). By contrast, a positive λ value does not introduce a clear effect to the
13
252
reflecting O-U model (Fig. 3c), but the reflecting O-U model has a strong
253
recency effect for λ < 0 (Fig. 3f).
254
Therefore, the primacy/recency effect in the bounded O-U models is jointly
255
determined by two independent elements: the value of λ and the type of
256
the boundary. Given different combinations of the two elements, the pri-
257
macy/recency effect in the bounded O-U model is maintained if λ and the
258
boundary provide the same effects (Fig. 3b and 3f). On the other hand, the
259
joint effect is weakened if λ and the boundary introduce opposite effects, as
260
shown in Fig. 3c and 3e. By appropriately setting the two parameters the
261
joint primacy/recency effects introduced by the boundary and λ could cancel
262
each other. In Section 4.2 we show that this condition leads to the optimal
263
performance of the bounded O-U model.
264
4
265
4.1 The error rate of bounded O-U models
266
For the reflecting O-U model, since the density of the integrator p(X, t) has
267
unit mass (see Fig. 2), the error rate is equal to the proportion of p(X, t) that
268
lies on the ’error’ side of the origin, i.e., if λ > 0, the error rate of the reflecting
269
O-U model can be calculated by integrating the density p(X, t) from −b to 0:
Performance of bounded O-U models
P(ref) (tc ) =
Z 0
−b
p(X, tc ) dX ,
(7)
270
where (ref) stands for reflecting boundary. Therefore by estimating p(X, t) nu-
271
merically we can obtain an approximation of the error rate. For the absorbing
272
O-U model, such calculation is not available as p(X, t) converges to zero as t
14
273
grows.
274
Now let us compare the error rates of the O-U models with two types of
275
boundaries. For the unbounded O-U model, the error rate remains the same
276
if we flip the sign of λ (see Eq. 2). Fig. 4 shows the error rates of the bounded
277
O-U models with positive and negative λ at different tc . The result suggests
278
that there exists error rate symmetry in the bounded O-U models. That is, by
279
flipping the sign of λ, the absorbing and reflecting O-U models produce the
280
same error rate for any tc . This relationship can be represented as:
P(abs) (tc , −λ) = P(ref) (tc , λ) ,
(8)
281
where P(abs) (.) and P(ref) (.) denote the error rates of the absorbing and
282
reflecting O-U models with parameters specified in the brackets. We have not
283
succeeded in giving a complete proof of Eq. 8 since an analytical expression
284
for the error rate is not available. However we are able to prove a special case
285
of Eq. 8 when tc → ∞, which is given in Appendix B.
286
In summary, the absorbing and reflecting O-U models with opposite signs of λ
287
produce the same accuracy, and thus they would provide the same fit to data
288
from experiments in TC paradigm.
289
4.2 The optimal parameter
290
In the TC paradigm, the unbounded O-U model achieves the minimum error
291
rate when λ = 0 (i.e., when it reduces to the diffusion model) (Bogacz et al.,
292
2006, and also see Section 2.4). After introducing boundaries, an intriguing
293
question arises: is λ = 0 still the optimal parameter value for the bounded
294
O-U models? If not, what is the optimal λ value that yields the lowest error
15
295
rate for the bounded O-U models?
296
To answer this question, we simulate the bounded O-U models with different
297
parameter values. The error rates of the bounded O-U models with different
298
values of λ, b and tc are shown as contour plots in Fig. 5. The result suggests
299
that the bounded O-U models with λ = 0 are not always optimal, but the
300
optimal value of λ depends on the values of other parameter such as b and
301
tc . First, if b is relatively small (b < 0.2 in Fig. 5), the value of λ does not
302
significantly affect the performance of the bounded O-U models in all panels of
303
Fig. 5. This result is expected since if the interval between the two boundaries
304
is tiny, the integrator state is restricted to be around X0 , and in this case
305
the noise would dominate the choice process. Therefore, the models produce
306
large error rates (around 40%) in this region. Second, if b is sufficiently large
307
(b > 2 in Fig. 5), the integrator hardly reaches any of the boundaries before tc
308
used in the simulation, and hence the bounded O-U models are approximately
309
equivalent to the unbounded O-U model, in which λ = 0 gives the lowest error
310
rate for any tc .
311
Except for these two extreme parameter regions, the error rate is affected by
312
the values of λ, b and tc . In general, given b and tc , for the absorbing O-U
313
model, λ < 0 gives the best performance, while for the reflecting O-U model,
314
λ > 0 yields the minimum error rate. This optimal value of λ is consistent
315
with the joint primacy/recency effects of the bounded O-U models proposed in
316
Section 3.2. Recall that the recency effect introduced by the reflecting bound-
317
ary can be weakened by setting λ > 0, and vice versa. To achieve the optimal
318
performance, the primacy/recency effects of the boundaries and λ should be
319
balanced and cancelled, so that the sensory evidence from different time points
320
contribute equally to the choice, as in the optimal NP decision procedure (i.e.,
16
321
no primacy/recency effects). The simulation results in Fig. 5 accord with this
322
hypothesis.
323
Fig.5 also shows that for fixed b, the absolute value of λ that yields the optimal
324
performance increases when extending tc , i.e., the optimal λ increases for the
325
reflecting O-U model and decreases for the absorbing O-U model. To find the
326
optimal λ values, we simulate the bounded O-U model under different tc . For
327
each tc , we numerically find the value of λ that yields the minimum error
328
rate, and the results are shown in Fig. 6a. For the reflecting O-U model, the
329
optimal λ increases as tc increases, with decreasing slope. The optimal λ for the
330
absorbing O-U model is a mirror of that for the reflecting O-U model, which
331
decreases as tc grows. This result further supports the error rate symmetry of
332
the bounded O-U models (Eq. 8), because if a specific λ value is optimal for
333
the reflecting O-U model, −λ would be optimal for the absorbing O-U model.
334
To validate the optimality of λ values in Fig. 6a, we simulated the reflecting
335
O-U model with different tc . For each tc , λ is set to be the optimal value
336
shown in Fig. 6a. For comparison, we also simulated the reflecting O-U model
337
with fixed λ values λ = 3 and λ = 0. The error rates of the three models are
338
illustrated in Fig. 6b. The bounded O-U model with optimal λ values achieves
339
lower error rates than the model with other constant λ.
340
5
341
The two types of boundaries may interpret different subjects’ behaviour in
342
psychological experiments. The absorbing boundary can account for the be-
343
haviour such that the choice is made before the end of the trial and never
Experimental predictions
17
344
changed. The reflecting boundary may imply that the subjects hesitate be-
345
tween the two choices even when sufficient evidence is available, and may
346
change their preference later in the trial. Nevertheless, due to the error rate
347
symmetry, it is not trivial to differentiate between the two types of boundary
348
on the basis of experimental data from simple 2AFC tasks, as the absorbing
349
and reflecting O-U models (with opposite sign of λ) would always produce the
350
same fit.
351
To investigate primacy and recency effects in the TC paradigm, stimuli could
352
be used that differ in information they contain at different times after stimulus
353
onset (Huk and Shadlen, 2005; Usher and McClelland, 2001). For example, in
354
the motion discrimination task (see Section 2.1) the coherence of motion can
355
be increased for a short period on each trial (Usher et al., 2009). The primacy
356
effect would be present, if such increase in coherence had the highest influence
357
on accuracy when occurring after stimulus onset. The recency effect would
358
be present, if the increase had the highest influence before the response. Note
359
however, that such experiment cannot differentiate between the absorbing O-U
360
model with negative λ and the reflecting O-U model with positive λ, because
361
both models do not predict strong primacy or recency effects. Our analysis
362
suggests that these two models maximize accuracy, and given evolutionary
363
pressure for accuracy of decisions, we predict that one of these two models
364
should describe choices in the TC paradigm with long tc .
365
We propose that the absorbing O-U model with negative λ and the reflecting
366
O-U model with positive λ could be distinguished in a motion discrimination
367
experiment in which the coherence is manipulated in a similar way to that
368
recently proposed by Zhou et al. (2009). In particular, the coherence could
369
be increased by a strong pulse c1 for an interval of length t1 , and then de18
370
creased by a smaller amount c2 < c1 for a longer interval t2 > t1 (Fig. 7a). If
371
the parameters of the perturbation satisfied c1 t1 = c2 t2 , the average motion
372
coherence in a trial would remain unchanged (hence the unbounded diffusion
373
model predicts that such perturbation would have no effect on accuracy). The
374
absorbing O-U model with negative λ predicts that such perturbation would
375
increase accuracy, because the first pulse could cause the correct absorbing
376
boundary to be reached. By contrast the reflecting O-U model with positive λ
377
predicts that the perturbation would decrease accuracy, because the informa-
378
tion in the first pulse may be lost if the reflecting boundary is reached. Fig. 7b
379
verifies these predictions by simulations. Note that the effects of the pulse per-
380
turbation are not due to the primacy/recency properties of the models (as we
381
intentionally selected the boundary and λ values so that the primacy/recency
382
effects were balanced), but originate from the characteristics of the two types
383
of boundaries. However, it is worth to mention that the effect of the pulse only
384
occurs if the following two conditions are satisfied. First, the pulse needs to
385
be sufficiently large to move the integrator to the boundary. Second, the pulse
386
needs to be presented early in a trial. Otherwise the pulse will not affect the
387
performance of the absorbing O-U model (as the inputs do not affect decision
388
process once the integrator hits absorbing boundaries).
389
Our analysis also shows that the accuracy in the TC paradigm is maximized if
390
parameter λ is adapted depending on the required reaction time tc . If subjects
391
indeed adapt λ depending on tc , then they should achieve higher accuracy
392
when tc can be anticipated. This predicts that the accuracy for a given tc
393
should be higher when tc is constant within a block (and hence can be antici-
394
pated) than when it varies within a block.
19
395
6
Discussion
396
The present study compared the properties and performance of the O-U mod-
397
els with absorbing and reflecting boundaries in the TC paradigm. The main
398
results are summarized in Table 2. First, we showed how different boundaries
399
affect the dynamics of decision process by visualizing the probability density
400
of the integrator state (Fig. 2). We then showed that the primacy and re-
401
cency effects introduced by the boundaries and the parameter λ coexist in
402
the bounded O-U models, and the joint effects depend on the two indepen-
403
dent elements (Fig. 3). Moreover, the absorbing and reflecting O-U models
404
can achieve the same error rate for any tc by flipping the sign of λ (Fig. 4).
405
Finally, we showed that the reflecting O-U model with positive λ (and the
406
absorbing O-U model with negative λ) produces lower error rates than the
407
model with λ = 0 (Fig. 5), and we numerically found the optimal λ values for
408
the bounded O-U models that yield the minimum error rate (Fig. 6).
409
6.1 Optimal integration with boundaries
410
This paper has shown that when the range of possible values of integrator
411
state is constrained, the simple integration of difference in evidence is not the
412
optimal procedure, and a better performance can be yield by the O-U model,
413
especially for long tc . This suggests that given the limited range of the firing
414
rates of integrator neurons, the neural decision circuits could achieve a higher
415
accuracy in tasks with long tc employing an integration procedure similar to
416
the O-U model rather than the linear integration. In particular, if the brain
417
employs a mechanism similar to the absorbing boundaries, as suggested by
20
418
neurophysiological data of Kiani et al. (2008), the accuracy of choices in task
419
with long tc would be maximized by employing leaky integration.
420
6.2 Relationship to other studies
421
Since the O-U model is reduced to the diffusion model when λ = 0, the results
422
presented in this paper extend our previous findings on the diffusion model
423
with boundaries (Zhang et al., 2009). In particular, Zhang et al. (2009) showed
424
that the diffusion models with absorbing and reflecting boundaries achieve the
425
same error rate. Such equality is a special case of the error rate symmetry of
426
the bounded O-U models by setting λ = −λ = 0.
427
In the TC paradigm, the O-U model produces the same simulated behaviour
428
as the leaky competing accumulator (LCA) model (Usher and McClelland,
429
2001), when λ parameter of the O-U model is set to the difference between
430
inhibition and decay parameters of the LCA model. Simulations of the LCA
431
model with attracting boundaries showed that it maximizes the accuracy in
432
the TC paradigm when the inhibition is lower than the leak (Bogacz et al.,
433
2007). These simulations are consistent with the results of this paper, because
434
for such parameter values the LCA model corresponds to the O-U model with
435
λ < 0.
436
Acknowledgement
437
This work was supported by the ORS grant and the EPSRC grant EP/C514416/1.
21
j
Absorbing
Reflecting
λ = −1
λ=1
λ = −1
λ=1
0
-
-
0
0
1
1.26
2.26
2.26
1.26
2
5.08
6.08
6.08
5.08
3
11.27
12.27
12.27
11.27
4
19.90
20.90
20.90
19.90
5
31.01
32.01
32.01
31.01
6
44.58
45.58
45.58
44.58
7
60.62
61.62
61.62
60.62
8
79.12
80.12
80.12
79.12
9
100.10
101.10
101.10
100.10
10
123.54
124.54
124.54
123.54
Table 1
Eigenvalues of the bounded O-U models with positive and negative λ. The eigenvalues are numerically obtained by seeking the roots of the Eqs. A.27 and A.30. The
parameters of the bounded O-U models are µ = σ = 1 s−1 , and b=1.
22
Absorbing
Reflecting
λ>0
uniform
recency
λ<0
primacy
uniform
λ=0
primacy
recency
Weighting of inputs
Error rate
λ 6= 0
symmetric
λ=0
equality
Optimality
λ<0
λ>0
Table 2
Comparison of the properties of the absorbing and reflecting O-U models.
2. 5
absorbing O−U
reflecting O−U
O−U
Integrator states
2
1. 5
1
0. 5
0
−0. 5
0
0. 2
0. 4
0. 6
Time (s)
0. 8
1
Fig. 1. Examples of trajectories of the O-U models with absorbing boundary (dashed
line), reflecting boundary (solid line), and without boundary (dotted line). Models
were simulated with µ = 0.5 s−1 , λ = 1, σ = 1 s−1 , dt = 0.001 s, and boundaries
b = 1.
23
b
a
p(X,t)
1. 2
0. 8
Time (s)
0. 1
0. 3
0. 5
0. 7
1. 2
0. 8
0. 4
0
−1
0. 4
−0. 5
0
0. 5
p(X,t)
2. 5
2
0
−1
1
c 3. 5
3
Time (s)
0. 1
0. 3
0. 5
0. 7
−0. 5
0
0. 5
1
−0. 5
0
0. 5
Integrator states
1
d
Time (s)
0. 1
0. 3
0. 5
0. 7
1. 2
0. 8
Time (s)
0. 1
0. 3
0. 5
0. 7
1. 5
1
0. 4
0. 5
0
−1
−0. 5
0
0. 5
Integrator states
1
0
−1
Fig. 2. Probability densities of integrator states in the bounded O-U models. State
values of the integrator are shown on horizontal axes. Densities are shown for parameters µ = σ = 1 s−1 , and b=1. For each panel, different curves show densities at
different time instants from 0.1s to 2s. The density functions p(X, t) are calculated
in Appendix A with eigenvalues shown in Table 1. All solutions start at X0 = 0.
(a) The absorbing O-U model with λ=1. (b) The absorbing O-U model with λ=-1.
(c) The reflecting O-U model with λ=1. (d) The reflecting O-U model with λ=-1.
24
unbounded
absorbing
reflecting
(a)
(b)
(c)
average input
0.03
0.02
0.01
0
0
0. 5
1 0
0. 5
1 0
(e)
(d)
0. 5
1
(f)
average input
0.03
0.02
0.01
0
0
0. 5
1 0
0. 5
Time (s)
1 0
0. 5
Fig. 3. Primacy and recency effects for the bounded and unbounded O-U models
with positive λ (top panels) and negative λ (bottom panels). In each panel the
O-U model is simulated for 10,000 trials. For all correct trials the input to the O-U
model at every time step is recorded and averaged. The data points show the mean
of the inputs and the error bars show the standard error. In all panels the parameter
values are µ=0.71 s−1 , σ=1 s−1 , b=0.47, tc =1s. (a) The unbounded O-U model with
λ=5.5. (b) The absorbing O-U model with λ=5.5. (c) The reflecting O-U model with
λ=5.5. (d) The unbounded O-U model with λ=-5.5. (e) The absorbing O-U model
with λ=-5.5. (f) The reflecting O-U model with λ=-5.5.
25
1
0. 5
abs O−U
ref O−U
0. 4
P(tc)
0. 3
0. 2
0. 1
0
0
1
2
3
tc(s)
Fig. 4. The error rate of the bounded O-U models at different tc . The models were
simulated with parameters: µ = σ = 1 s−1 and b=1. Every data point is averaged
from 100,000 trials. The reflecting O-U model (solid line) is simulated with λ=2,
and the absorbing O-U model (dashed line) is simulated with λ=-2.
26
0
0.1 0.2 0.3 0.4
Absorbing
Reflecting
λ (tc=0.5s)
2
0
−2
λ (tc=1s)
2
0
−2
λ (tc=4s)
2
0
−2
1
2
3
Boundary
1
2
3
Boundary
Fig. 5. The contour plots of the error rates of the bounded O-U models with different parameter values. The bounded O-U models are simulated with the following
parameters: λ in [-3,3] with step 0.1, b in [0.1,3] with step 0.1, µ = σ = 1 s−1 ,
and tc =0.5s, 1s, and 4s. For each possible combination of the parameter values, the
absorbing and reflecting O-U models are simulated for 10,000 trials to obtain the
mean error rates. The error rates are then grouped by the value of tc and the type
of boundaries, and illustrated as contour plots. The horizontal axes show the value
of b, and the vertical axes show the value of λ. Each contour plot has 20 contour
levels and the colormaps are the same for all panels. Left panels show the error rates
of the absorbing O-U model, and right panels show the error rates of the reflecting
O-U model. The panels in each row show the plots that have the same value of tc ,
as indicated on the vertical axes.
27
a
b
3
2
0. 5
λ =3
λ =0
optimal λ
0. 4
reflecting O−U
absorbing O−U
0
P (tc)
optimal λ
1
0. 2
−1
−2
−3
0. 3
0. 1
1
2
3
tc(s)
4
5
0
1
2
3
tc(s)
4
5
Fig. 6. Optimal performance of the bounded O-U models. (a) The optimal λ values
of reflecting and absorbing O-U models for different tc varying from 0.5s to 10s (in
steps of 0.01s). For each tc , the absorbing and reflecting O-U models are simulated
with parameters µ = σ = 1 s−1 , b=1, and λ varying from -100 to 100. For each
value of λ the model is simulated for 10,000 trials, and the λ that yields minimum
error rate is recorded as the optimal value at tc . (b) Error rates of the reflecting
O-U model at different tc . The model is simulated with the same parameters (µ, σ,
b) as in panel (a). Dotted line shows the error rate of the model with λ=3. Dashed
line shows the error rate of the model with λ=0 (i.e., the diffusion model). Solid
line shows the error rate of the model with optimal λ values from panel (a). Each
data point is averaged over 10,000 trials.
28
a
b
8
t1
P (tc)
Drift rate
0.06
c1
t0
t2
2
0
0
no−pulse
pulse
0.08
6
4
0.1
0. 5
1
Time (s)
c2
0.04
0.02
1. 5
0
diffusion
abs O−U
ref O−U
Fig. 7. The decision task with pulse perturbation. (a) The drift rate with pulse
perturbation. The parameter values are: µ=2 s−1 , c1 =5 s−1 , c2 =1.19 s−1 , t0 =310
ms, t1 =150 ms, t2 =630 ms. (b) The error rate of the absorbing and reflecting O-U
models, compared with that of the unbounded diffusion model. The models were
simulated with parameter values: b=0.7, λ=±5 (-5 for the absorbing O-U model
and 5 for the reflecting O-U model), c = 2 s−1 , and tc =1.8 s. Light grey bars show
the error rate of the models with constant drift rate µ=2 s−1 , and dark grey bars
show the error rate of the models with time-varying drift rate from (a). The error
bars show the standard error from 10,000 trials.
29
438
A
The probability density function for the bounded O-U model
439
This appendix calculates the probability density function p(X, t) for the O-U
440
models with absorbing and reflecting boundary conditions. Recently, a solution
441
of this problem for λ < 0 has been provided by Saphores (2005). Here we
442
reiterate the calculation and extend the solution to both positive and negative
443
λ.
444
Let us consider a general case: an O-U process defined in Equation 1 is bounded
445
over [L, R], where L and R are finite absorbing or reflecting boundaries defined
446
in Equations 4 and 5, respectively. Let p(X, X0 , t) be the probability density
447
of X(t) at time t given that X(0) = X0 . The density satisfies the Kolmogorov
448
backward equation (Karlin and Taylor, 1981):
Ã
∂ p(X, X0 , t)
∂ p(X, X0 , t)
σ2
= −(µ + λX0 )
+
∂t
∂ X0
2
!
∂ 2 p(X, X0 , t)
,
∂ X02
(A.1)
449
where the coefficients µ, λ, and σ are defined in Equation 1. Since X(t) has
450
fixed value X0 at t = 0, the initial probability distribution is a delta function:
p(X, X0 , 0) = δ(X − X0 ) .
(A.2)
451
Goel and Dyn (2003) showed that if boundaries are absorbing, p(X, X0 , t) has
452
to satisfy:
p(X, X0 , t) = 0 , for X0 = L or R ,
(A.3)
453
and reflecting boundaries at L and R imply the following no-flux boundary
454
conditions:
∂ p(x, y, t)
= 0 , for X0 = L or R .
∂y
455
(A.4)
To solve Equation A.1 we separate variables (Boyce and DiPrima, 1997) and
30
456
seek a solution as the weighted sum of elementary solutions.
p(X, X0 , t) =
∞
X
aj (t) φj (X0 ) .
(A.5)
j=0
457
For one of the elementary solutions a(t) φ(X0 ), substituting Equation A.5 in
458
A.1 and rearranging terms, we obtain:
a0 (t)
σ2
µ + λX0 0
=
φ00 (X0 ) +
φ (X0 ) .
a(t)
2φ(X0 )
φ(X0 )
(A.6)
459
Since the left side of Equation A.6 depends only on t, while the right side only
460
on X0 , they must equal to some constant, denoted by −ξ. Then solution for
461
the time-dependent coefficients a(t) is:
a(t) = Ke−ξt ,
462
(A.7)
and the function φ(X0 ) satisfies:
σ 2 00
φ (X0 ) + (µ + λX0 )φ0 (X0 ) + ξφ(X0 ) = 0 .
2
463
(A.8)
Substituting A.7 into A.5, the density p(X, X0 , t) can be represented as:
p(X, X0 , t) =
∞
X
Kj e−ξj t φj (X0 ) .
(A.9)
j=0
464
The coefficients Kj in Equation A.9 are obtained from the initial probability
465
at t = 0. Firstly multiply both side of A.8 by a function of X0 :
Ã
!
2µX0 + λX02
2
,
r(X0 ) = 2 exp
σ
σ2
466
(A.10)
then φ(X0 ) should satisfy
Ã
!
Ã
!
2µ + 2λX0
2µX0 + λX02 0
2µX0 + λX02 00
φ
(X
)
+
exp
φ (X0 ) + ξr(X0 )φ(X0 )
exp
0
σ2
σ2
σ2
"
Ã
!
#0
2µX0 + λX02 0
= exp
φ (X0 ) + ξr(X0 )φ(X0 ) = 0 .
σ2
(A.11)
31
467
Equation A.11 is the classical Sturm-Liouville equation. From the Sturm-
468
Liouville theorem (Boyce and DiPrima, 1997), there exists an infinite set
469
of non-negative eigenvalues ξj (j = 1, 2, . . . ), and associated eigenfunctions
470
φj (X0 ) that satisfy Equation A.11. Besides, the normalized eigenfunctions
471
form an orthogonal set with respect to the function r(X0 ). Hence for any two
472
eigenfunctions φi (X0 ) and φj (X0 ), there is
Z R
L
r(X0 )φj (X0 )φi (X0 ) dX0
Z R
L
= δij ,
r(X0 )φ2j (X0 ) dX0
(A.12)
473
where δij is the Kronecker delta function. For t = 0, a single elementary
474
solution of the density p(X, X0 , t) in Equation A.5 is:
p(X, X0 , t) = Kj φj (X0 ) = δ(X − X0 ) .
475
Multiplying both sides of Equation A.13 by:
Z R
L
476
r(X0 )φi (X0 )
(A.14)
,
r(X0 )φ2j (X0 ) dX0
and taking integral over the interval [L, R], we obtain:
Z R
Kj
L
Z R
r(X0 )φj (X0 )φi (X0 ) dX0
Z R
L
477
(A.13)
=
L
r(X0 )φi (X0 )δ(X − X0 ) dX0
Z R
r(X0 )φ2j (X0 ) dX0
L
, (A.15)
r(X0 )φ2j (X0 ) dX0
and the coefficients Kj is:
Kj = Z
r(X)φj (X)
R
L
.
(A.16)
r(X0 )φ2j (X0 ) dX0
478
To get the solutions of eigenvalues ξ and eigenfunctions φ(X0 ), we need to
479
solve the differential equation A.8. First assume λ < 0, and define a new
480
variable z:
z=
√
−2λ
X0 + µ/λ
.
σ
32
(A.17)
481
Substituting Equation A.17 into A.8 yields:
λφ00 (z) − λzφ0 (z) − ξφ(z) = 0 .
482
(A.18)
We seek the solution of φ(z) of the form:
φ(z) = ez
2 /4
ω(z) .
(A.19)
483
Substituting Equation A.19 in A.18, the exponential term is canceled and the
484
new function is:
Ã
!
1 ξ
1
d2 ω(z)
+
− − z 2 ω(z) = 0 .
2
dz
2 λ 4
(A.20)
485
Equation A.20 is the classical parabolic cylinder function, and there exists two
486
independent solutions, given by (Abramowitz and Stegun, 1965):

!
¶ Ã
µ
2

ξ
1
z
1

2

ω1 (z) = exp
z F
; ;
,


4
2λ 2 2







!

µ
¶ Ã



1 2
ξ 3 z2
1


,
z F
+ ; ;
ω2 (z) = z exp
4
2
2λ 2
(A.21)
2
487
where F(.) is the confluent hypergeometric function. Representing Equation
488
A.21 as a function of X0 yields:

!
Ã

1
λ
ξ

2


; ; − 2 (X0 + µ/λ) ,
E1 (X0 ) = F


2λ 2
σ







Ã
!



√

ξ 3
λ
X0 + µ/λ
1

2

F
+ ; ; − 2 (X0 + µ/λ) .
E2 (X0 ) = −2λ
σ
2 2λ 2
σ
489
For the case of λ > 0, select the substitution of X0 as:
z=
490
(A.22)
√
2λ
X0 + µ/λ
.
σ
(A.23)
Then by following the same approach, we can obtain the elementary solutions
33
491
for φ(X0 ):

Ã
! Ã
!

(µ + λX0 )2
1
ξ 1 λ

2


E1 (X0 ) = exp −
F
(X0 + µ/λ) ,
− ; ;


λσ 2
2 2λ 2 σ 2







Ã
! Ã
!


2


(µ
+
λX
)
ξ
3
λ
0


F 1 − ; ; 2 (X0 + µ/λ)2 .
E2 (X0 ) = exp −
2
λσ
2λ 2 σ
(A.24)
492
The solution of Equation A.8 is the linear combination of the two elementary
493
solutions given by A.22 or A.24 (depending on the sign of λ). For absorbing
494
boundaries, φ(X0 ) must satisfy the following boundary condition (cf. Equa-
495
tions A.3 and A.5):
φ(L) = φ(R) = 0 .
(A.25)
496
By appropriately selecting the weighting value, the solution of φ(X0 ) for ab-
497
sorbing boundary condition is:
φ(X0 ) = E1 (X0 )E2 (L) − E1 (L)E2 (X0 ) ,
(A.26)
498
and the eigenvalues ξ can be obtained numerically by seeking the roots of the
499
equation:
E1 (L)E2 (R) − E1 (R)E2 (L) = 0 .
500
If the boundaries are reflecting, the boundary condition changes to:
∂ φ(L)
∂ φ(R)
=
= 0.
∂ X0
∂ X0
501
(A.27)
(A.28)
The above condition is satisfied when:
φ(X0 ) = E1 (X0 )E20 (L) − E10 (L)E2 (X0 ) ,
34
(A.29)
502
and the associated eigenvalues are the roots of:
E10 (L)E20 (R) − E10 (R)0 E20 (L) = 0 .
(A.30)
503
Substituting Equations A.16 and A.26 (or A.29 for reflecting boundaries) in
504
A.9, we obtain a series representation of p(X, t) with starting point X0 .
505
B
506
Since the analytical solution of the error rate of the bounded O-U model is
507
not available, it is not straightforward to prove the symmetry property of the
508
bounded O-U model in Equation 8 for arbitrary tc . Instead, this appendix
509
proves a special case of Equation 8 when tc → ∞, i.e., the target equation is:
Symmetry of the error rate for the bounded O-U model
P(abs) (tc , λ)|tc →∞ = P(ref) (tc , −λ)|tc →∞ .
(B.1)
510
First let us consider the absorbing O-U model bounded between ±b. To sim-
511
plify the calculation, we make the following transformation:
µ ¶2
b
µ
b̂ ← , and µ̂ ←
µ
σ
.
(B.2)
512
For the absorbing O-U model, since the density of the integrator state p(X, t)
513
converges to zero as tc grows, X(t) will reach one of the boundaries as tc → ∞.
514
Hence, the error rate of the absorbing O-U model with infinite tc is the same
515
as the error rate of the unbounded O-U model in the IC paradigm, which has
35
516
an analytical solution given by (Bogacz et al., 2006):

s
P(abs) (tc , λ)|Tc →∞ =
517
s 
µ̂
µ̂ 
(1 + λb̂) − erf 
erf 
λ
λ
s

s
,
µ̂
µ̂
erf 
(1 + λb̂) − erf 
(1 + λb̂)
λ
λ
(B.3)
where erf(.) is the error function given by:
2 Z x −t2
e dt .
erf(x) = √
π 0
(B.4)
518
Now lets consider the error rate of the reflecting O-U model. Recall that the
519
reflecting O-U model has zero eigenvalue ξ0 = 0 (cf. Table 1). For tc → ∞,
520
the time-dependent coefficients e−ξt in Equation A.9 vanish and the stationary
521
density of X(t) is:
p(X, +∞) = φ0 (X0 ) Z
r(X)φ0 (X)
b
−b
,
(B.5)
r(X0 )φ20 (X0 ) dX0
522
where r(X) is defined in Equation A.10, and the eigenfunction φ0 is defined
523
in Equation A.29.
524
For λ < 0, the two elementary solutions of Equation A.29 are given by Equa-
525
tion A.22. In order to simplify the expression, let:
V =
526
1
X0 1
1
X
+ , and V0 =
+ = ,
µ̂
λ
µ̂
λ
λ
and the lower and upper boundaries in Equation A.22 are given by:
L=
527
(B.6)
1
1
− b̂ , and R = + b̂ .
λ
λ
(B.7)
By this transformation, the drift term µ in Equation 1 is eliminated and the
36
528
density p(X, +∞) can be calculated in the terms of V and V0 :
p(V, +∞) = φ0 (V0 ) Z
r(V )φ0 (V )
R
L
.
(B.8)
r(V0 )φ20 (V0 ) dV0
529
Substituting Equations A.6 and A.7 in Equation A.22, and setting the eigen-
530
value ξ0 = 0, the two elementary solutions of the eigenfunction φ0 (w) become:

¶
µ

1

2


,
E1 (V ) = F 0; ; −µ̂λV


2



(B.9)




µ
¶


√

1 3

2
E2 (V ) = V −2µ̂λ F
; ; −µ̂λV
.
2 2
531
The derivative of Equation B.9 is:





E10 (V ) = 0 ,
(B.10)

√



E20 (V ) = −2µ̂λ exp (µ̂λV 2 ) .
532
Substituting Equations B.9 and B.10 in A.29 yields the eigenfunction of the
533
stationary distribution:
Ã
q
φ0 (V ) =
µ̂
−2µ̂λ exp − (λb̂ − 1)2
λ
!
.
(B.11)
534
Recall that the error rate of reflecting O-U model can be represented by inte-
535
grating p(V, tc ) from lower boundary L to the start point V0 (cf. Equation 7).
536
For tc → ∞, p(V, tc ) converges to the stationary distribution in Equation B.8.
537
Hence the error rate is:
Z V0
P(ref) (tc , −λ)|tc →∞ = φ0 (V0 ) Z LR
L
37
r(V )φ0 (V ) dV
r(V0 )φ20 (V0 ) dV0
.
(B.12)
538
Substituting Equations A.10 and B.11 in B.12, the error rate can be analyti-
539
cally obtained, as:
s 
P(ref) (tc , λ)|tc →∞ =
540

s
µ̂ 
µ̂
(−1 + λb̂)
erfi 
+ erfi 
λ
λ
s

s
,
µ̂
µ̂
erfi 
(−1 + λb̂) + erfi 
(1 + λb̂)
λ
λ
(B.13)
where erfi(.) is the imaginary error function, given by:
erfi(z) = erf(zi)/i ,
(B.14)
541
where i is the imaginary unit. Substituting Equation B.14 in B.13, the ER of
542
reflecting O-U model is:
Ãs
P(ref) (tc , λ)|tc →∞
Ãs
!
!
µ̂
µ̂
erf
(1 + (−λ)b̂) − erf
−λ
−λ
=
Ãs
!
Ãs
!.
µ̂
µ̂
erf
(1 + (−λ)b̂) + erf
(1 − (−λ)b̂)
−λ
−λ
(B.15)
543
For λ > 0, the two elementary solutions in Equation A.29 are given by Equa-
544
tion A.24. By applying the same approach as B.6-B.10, the eigenfunction now
545
becomes:
Ã
q
φ0 (V ) =
µ̂
2µ̂λ exp − (λb̂ − 1)2
λ
!
.
(B.16)
546
Substituting Equations A.10 and B.16 in B.12, the eigenfunction has the same
547
expression as Equation B.15. Therefore Equation B.15 satisfies both positive
548
and negative λ.
549
Comparing Equations B.3 and B.15, it is clear that when the sign of λ in the
550
two equations changes, the error rates of the absorbing and reflecting O-U
551
models are exactly the same. i.e., the bounded O-U model satisfies Equation
552
B.1.
38
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