9.6 Precision of parameter estimates
11.2.3 Relation of the HBMi to the two state HMM
In the following we only consider the HMM with inverse Gaussian distributed dominance times. The relation of the HBMc to the one state HMM is simple as is represents only a reparametrization. Both the one state HMM and the HBMc yield independent and IG distributed dominance times.
For the intermittent case, the relation of the HBMi to the two state HMM is not as straightfor-ward. The two models are highly similar in the sense that they use two parameters to describe long and short dominance times, respectively (e.g., (µS, σS) and (bS, νS∗) for the stable state).
In the HMM, the dominance times are IG distributed, given the state with the respective parameters. In the HBMi, the dominance times are approximately IG distributed, where the minor deviation from the IG distribution originates from the minor deviation ofP from a Brownian motion with driftνS∗ (orνU∗), instead of exactly assuming drift ν0 during stimulation andνS (orνU) during blank display. However, the marginal distribution ofP at multiples of such intervalslp+lb is identical to the marginal distribution of a Brownian motion with drift νS∗ (or νU∗) at these time points, and the differences can only be observed in the meantime.
Because dominance times usually span multiple trials of durationlp+lb, the approximation is very close. To simplify comparison between the HMM and the HBMi we denote by
(µ∗S,σS∗) and (µ∗U,σU∗) the mean and standard deviation of dominance times in the HBMi in the stable and unstable state, respectively. These have analogous interpretations as (µS, σS) and (µU, σU) in the HMM and are given by
µ∗S≈2bS/νS∗ and σ∗S ≈ q
2bS/νS∗3, (11.4)
and analogously forµ∗U andσ∗U, where the approximation is again due to the minimal difference between the mean driftνS∗ and the changing driftνS+ν0 during presentation and blank display.
As another similarity, both models use additional parameters ((pSS, pU U) and (˜bS,˜bU, νB)) to describe the transition probabilities between the stable and the unstable state.
The main difference between the HBMi and the two state HMM concerns the dynamic of the state transitions between stable and unstable state. In the HMM, transition probabilities
136
11. A hierarchical Brownian motion model
are given by (1−pSS) and (1−pU U) and are independent of the duration of the previous dominance time. In contrast, in the HBMi, a transition from stable to unstable state requires that B has not reached ˜bS at the end of the respective dominance time. Therefore, the transition probability ˜pSU(di) depends on the duration di of thei-th dominance time, where shorter dominance times yield higher transition probabilities. Note that the position ofB at the end of a dominance timedi is given by an increment of a Brownian motion with driftνB∗ in the fixed time intervaldi. Therefore the position is normally distributed with meandi·νB∗ and variancedi, and the probability to remain in the stable state (which is the probability that the background process exceeds ˜bS) is given by
˜
pSS(di) :=P(Yi+1=S|Yi =S, di)≈1−Φν∗
Bdi,di
˜bS
, (11.5)
where Φµ,σ2() denotes the distribution function of the normal distribution with mean µand variance σ2 and Yi is the hidden state of the i-th dominance time. Similarly, the relation between the transition probability and the previous dominance time di is for the unstable state given by
˜
pU U(di) :=P(Yi+1=U|Yi=U, di)≈Φ−ν∗Bdi,di
˜bU
. (11.6)
Note that we use the approximate sign ”≈” because the drift ofB is not exactlyνB∗ throughout, but is assumed to change betweenν0andνB during stimulation and blank display, respectively, yielding a mean drift of νB∗. Analogously to the above explanation, differences caused by the approximation can be considered minimal.
In order to obtain quantities comparable to the transition probabilitiespSS andpU U in the HMM, we can obtain the marginal transition probabilities in the HBMi as the expected value of ˜pSS and ˜pU U by integration across all dominance times. As shown in Corollary 8.10 the positions XS and XU of B at the end of an independent stable or unstable IG distributed dominance time follow the normal-inverse Gaussian (NIG) distribution. The resulting transition probabilities in the HBMi can then be calculated as
p∗SS ≈P(XS≥˜bS) and p∗U U ≈P(XU < bU), (11.7) whereXS is NIG distributed with parameters (0,
q
νS∗2+νB∗2, νB∗,2bS) and XU is NIG dis-tributed with parameters (0,
q
νU∗2+νB∗2,−νB∗,2bU). For the parametrization, recall the proof of Corollary 8.10. In a reasonable model, not both ofp∗SS and p∗U U equal one.
One should note that due to the difference in transition probabilities of the two models, the parameters (bS, νS∗) are not direct reparametrization of (µS, σS) (and similarly for the unstable state). Furthermore, the dependence of the transition probability on the length of the previous dominance time is one important new aspect in the HBMi not described in the HMM, which will also be used in Section 14.2.1 for comparison of models and empirical observations.
Influence of the length of dominance times on the transition probabilities
As explained above (equation (11.5)), the transition probability from the stable to the unstable state ˜pSU(d) decreases with the length of the stable dominance time. This is visualized in Figure 11.8 A.
Investigating the influence of the lengthdof an unstable dominance time (which depends on bU and νU∗) on the transition probability ˜pU S(d) to the stable state, the model implication
11. A hierarchical Brownian motion model
is ambiguous which we illustrate in Figure 11.8 B and C. Non-positive borders ˜bU (panel B) cause very large transition probabilities for short unstable dominance times as there is not enough time for the background process to reach ˜bU (the state only remains unstable if the background process is below ˜bU at the end of a dominance time, recall moreover the negative drift of the background process during unstable phases). The longer the dominance time, the more time the background process has to cross ˜bU.
Positive values of ˜bU (panel C) imply generally smaller transition probabilities (due to the negative drift of the background process). Additionally, the curves have a maximum turning point as a minimum time is needed for a process with negative drift to reach a positive border with a small probability.
0.0 0.2 0.4 0.6 0.8 1.0
0 50 150 250
d[s]
p~SU(d) ●●
●
●
b~
S
30 100 170 240
A
0.0 0.2 0.4 0.6 0.8 1.0
0 5 10 15
d[s]
p~US(d)
●
●
●
●
b~
U
−3
−2
−1 0
B
d[s]
p~US(d) 0.00 0.01 0.02
0 5 10 15
●
●
●
b~
U
1 2 3
C
Figure 11.8: Transition probabilities p˜SU(d),p˜U S(d) depending on the length d of the dominance time for νB∗ = 1 and different values of ˜bS,˜bU as indicated in the legends. (A) The transition probability in the unstable state p˜SU(d) decreases with the length of the stable dominance time. (B)˜bU ≤0 leads to monotonously decreasing transition probabilitiesp˜U S(d). (C)˜bU >0 causes transition probability curves with a maximum which decreases with ˜bU increasing. The transition probabilities are much smaller than the largest
probabilities in panel B.
11.2.4 Effect of single parameter changes
How does an increase in the size of the neuronal poolbS or in the border ˜bS of the background process influence the response pattern? These and other comparable questions are important for understanding the HBMi and its interpretation properly. Therefore, we discuss the influence of single HBMi parameters on the response patterns. As the response patterns are determined by the distribution of the stable and the unstable dominance times as well as by the transition probabilities we examine the impact of each HBMi parameter onµ∗S,CV∗S, µ∗U,CV∗U, p∗SS, p∗U U. The results are summarized in Table 11.1.
138
11. A hierarchical Brownian motion model
Parameter Change µ∗S CV∗S µ∗U CV∗U p∗SS p∗U U bS
↑ ↑ ↓ → → ↑ →
↓ ↓ ↑ → → ↓ →
νS∗ ↑ ↓ ↓ → → ↓ →
↓ ↑ ↑ → → ↑ →
bU ↑ → → ↑ ↓ → Fig. 11.9
↓ → → ↓ ↑ → Fig. 11.9
νU∗ ↑ → → ↓ ↓ → Fig. 11.9
↓ → → ↑ ↑ → Fig. 11.9
˜bS ↑ → → → → ↓ →
↓ → → → → ↑ →
˜bU
↑ → → → → → ↑
↓ → → → → → ↓
νB∗ ↑ → → → → ↑ ↑
↓ → → → → ↓ ↓
Table 11.1: Effect of single parameter changes in the HBMi. The influence of changing one of the seven parametersbS, νS∗, bU, νU∗,˜bS,˜bU, νB∗ while leaving the others constant on the target parameters µ∗S,CV∗S, µ∗U,CV∗U, p∗SS, p∗U U is shown. An increase in a parameter is highlighted yellow, and a decrease is highlighted red where no influence is colored orange. The effect of bU and νU∗ on p∗U U is more complex and shown in Figure 11.9.
The explanations for the effects ofbS andνS∗ on µ∗S and CV∗S as well as for the effects ofbU
and νU∗ on µ∗U and CV∗U are as already explained similar to the effects of drift and border parameter in the HBMc (compare Section 11.1.2). IncreasingbS moreover also increases the probability to remain in the stable state p∗SS as the background process has more time to reach its border. Increasing the driftνS∗ has the opposite effect as the background process has less time to exceed the border.
A larger driftνB∗ of the background process also increases the probability to stay in the current state as the process gets faster above or below the borders ˜bS and ˜bU, respectively. Increasing
˜bS hinders the background process to pass the border and thus p∗SS decreases. Increasing ˜bU facilitates the background process to be below the border and thusp∗U U increases.
The dependence ofp∗U U on bU andνU∗ depends on the sign of ˜bU and is illustrated in Figure 11.9 and explained in the following. Panel A indicates for a positive ˜bU (orange line) a negative correlation betweenbU andp∗U U for smallbU and a positive correlation for largerbU. IncreasingbU implies longer unstable dominance times. The negative drift of B in unstable phases therefore causes for a long dominance time a large probability forB to be negative and thus belowbU. If the dominance time is short, there is not enough time for the background process to reach ˜bU and thus increasing a short dominance time by increasingbU increases the probability to reach the border and therefore has a negative impact on p∗U U. With ˜bU ≤0 (red line) there is a positive relation betweenbU and ˜pU U as longer dominance times enhance
the probability ofB to be below ˜bU.
The impact of νU∗ on p∗U U is visualized in panel B. Generally, increasing νU∗ causes shorter unstable dominance times. Assuming a positive ˜bU (orange line) only for rather long dominance times (smallνU∗) the background process has some chance to hit ˜bU. For shorter dominance times the process does not have enough time and therefore remains below ˜bU and in the
11. A hierarchical Brownian motion model
unstable state. In case of a non-positive ˜bU (red line), shorter dominance times (i.e., largerνU∗) imply thatB with negative drift has less time to reach the border and thusp∗U U approaches zero.
pUU*
A
bU
0 5 15 30
0.0 0.2 0.4 0.6 0.8 1.0
~b
U>0
~b
U≤0 pUU*
B
νU*
0 25 50
0.0 0.2 0.4 0.6 0.8 1.0
Figure 11.9: Dependence of p∗U U on bU and νU∗. In panel A we let νU∗ = 0.73 and vary bU, and in panel B νU∗ varies and bU = 1.84 is used. The orange lines code a positive
˜bU = 2.39, and the red lines code a negative˜bU =−2.39. All other parameters can be found in Table 12.2, subject C (page 162).