Dynamic neural reconfiguration for strategy switching during competitive
social interactions
Supplementary Materials
Ruihan Yang, et.al.
Contents
1 Method S1. Time-varying Granger causality with signal-dependent noise
1
2 Method S2. Numerical Simulations of time-varying systems
6
2.1 Time series with a Gaussian white noise
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.2 Time-varying model with a signal-dependent noise
. . . . . . . . . . . . . . . . . . . . . . . .
7
2.3 Simulation results for the time-varying Granger causality with Gaussian white noise
. . . . .
7
2.4 Simulation results for the time-varying Granger causality with signal-dependent noise
. . . .
8
3 Method S3. Expectation-Maximum algorithm to identify the behavioral window
9
4 Method S4. Estimate the hemodynamic response function with the GLM model
10
5 Method S5. Accounting for the possible confounding effect of the hemodynamic response
function on the findings
11
5.1 Compare the HRF delay between two brain regions
. . . . . . . . . . . . . . . . . . . . . . . .
11
5.2 Numerical simulations of the HRF delay
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
List of Tables
S1 Characteristics of the behavioral windows identified by the hidden Markov model.
. . . . . .
13
S2 Comparison of the demographics of the buyers associated with different types of behavioral
windows.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
S3 The identification of strategies when different intervals were chosen to calculate the observations.
15
Elsevier
List of Figures
S1 Detection of information flow in a system with time-varying coefficient and Gaussian white
noise.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
S2 Detection of information flow in a system with time-varying coefficient and signal-dependent
noise.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
S3 Comparison of behavioral classification between time-invariant and time-varying approaches.
18
S4 Main findings remained when the ambiguous behavior window of Subject 29 was retained.
. .
19
S5 Comparison of the information flow estimated by the classic GC and GCSDN method.
. . . .
20
S6 Behavioral correlations between the information revelation (IR) and the information flows.
. .
21
S7 Behavioral correlations between the
R
2
and the information flows.
. . . . . . . . . . . . . . .
22
S8 Results when different intervals were chosen to calculate the observations.
. . . . . . . . . . .
23
S9 Results of different thresholds of the minimum length of a stable behavioral window.
. . . . .
24
S10 Comparison of the HRFs among three ROIs.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
S11 Model performance.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
1.
Method S1. Time-varying Granger causality with signal-dependent noise
Assuming a constant effective connectivity between brain regions, the classic uses the time-series models
(both GC and GCSDN) with constant model coefficients. However, this assumption may be an oversimpli-
fication of the information processing in the brain, especially during some intensive cognitive computation
(e.g., the two-party bargaining game). To model the dynamic behavior, a more complicated model which can
better describe the dynamic characteristics is needed. Here, we first briefly discussed the consquence when
classic time-invariant model was applied to a time-varying system, and then proposed a new approach of
time-varying Granger causality with signal-dependent noise (time-varying GCSDN) to measure the dynamic
causality.
Consider the following time series model
x
t
=
ax
t
−
1
+
b
(
t
)
y
t
−
1
+
ε
t
,
where
ε
t
is a Gaussian white noise,
a
is a constant coefficient, and
b
(
t
)
is a time-varying coefficient. The
classic Granger causality can be defined as
F
y
→
x
= log
var (
b
(
t
)
y
t
−
1
+
ε
t
)
var (
ε
t
)
,
and estimated by
b
F
y
→
x
= log
P
T
−
1
s
=1
ˆ
b
2
(
s
)
y
2
s
+
P
T
−
1
s
=1
ˆ
ε
2
s
P
T
−
1
s
=1
ˆ
ε
2
s
,
where
ˆ
b
(
t
)
is the local classic estimation at each time step t and
ˆ
ε
s
is the residual process. If we ignore the
time-varying property of the model and do the conventional least square estimation for the parameters, we
can have an estimation of
̄
b
and the corresponding model residual
̄
ε
s
, and the causality can be established as
F
y
→
x
= log
P
T
−
1
s
=1
̄
b
2
y
2
s
+
P
T
−
1
s
=1
̄
ε
2
s
P
T
−
1
s
=1
̄
ε
2
s
.
For the AR model, we showed that the causality established by assuming the constant coefficient in the
model is upper bounded by the mean causality among time windows, and the condition on which the equality
holds was discussed as follows. Define
o
t
=
0
@
x
t
y
t
1
A
′
,
β
=
0
@
̄
a
̄
b
1
A
,
β
t
=
0
@
ˆ
a
ˆ
b
(
t
)
1
A
,
O
=
0
B
B
B
B
B
B
@
x
1
y
1
x
2
y
2
.
.
.
.
.
.
x
T
−
1
x
T
−
1
1
C
C
C
C
C
C
A
,
D
=
0
B
B
B
B
B
B
@
x
2
x
3
.
.
.
x
T
1
C
C
C
C
C
C
A
.
1
For the whole time series we have
O
′
D
=
O
′
Oβ
,
and for each time step
t
, the following equation holds,
o
′
t
x
t
+1
=
o
′
t
o
t
β
.
Since
P
T
−
1
t
=1
o
′
t
x
t
+1
=
O
′
D
, we get
T
−
1
X
t
=1
o
′
t
o
t
β
t
=
O
′
Oβ
.
Therefore,
β
is a weighted average of
β
t
as
β
= (
O
′
O
)
−
1
T
−
1
X
t
=1
o
′
t
o
t
β
t
.
Since what we consider here is the effect of the time-varying parameter
b
(
t
)
, we further suppose that the
constant parameter
a
= 0
, and the above formula can be simplified as
̄
b
= (
T
−
1
X
s
=1
y
2
s
)
−
1
T
−
1
X
t
=1
y
2
t
ˆ
b
(
t
)
.
Since
T
−
1
X
s
=1
̄
b
2
y
2
s
=
T
−
1
X
s
=1
y
2
s
!
−
2
T
−
1
X
t
=1
y
2
t
ˆ
b
(
t
)
!
2
T
−
1
X
s
=1
y
2
s
!
=
T
−
1
X
s
=1
y
2
s
!
−
1
T
−
1
X
t
=1
y
2
t
ˆ
b
(
t
)
!
2
≤
T
−
1
X
s
=1
y
2
s
!
−
1
T
−
1
X
t
=1
ˆ
b
2
(
t
)
y
2
t
!
T
−
1
X
t
=1
y
2
t
!
=
T
−
1
X
t
=1
ˆ
b
2
(
t
)
y
2
t
,
we have that
F
y
→
x
≤
b
F
y
→
x
,
and the equality holds only when
P
T
−
1
t
=1
ˆ
b
2
(
t
)
y
2
t
=
c
P
T
−
1
t
=1
y
2
t
with a constant c. Assuming that the local
estimation gives the exact value of the parameter, we can see that the causality established by ignoring the
time-varying property of the parameters is upper bounded by the averaged causality among local causalities
(i.e. the causality detected by each sliding window). Therefore, caution must be made when we detect the
causality for a whole time series, since the causality may exist in some time windows. Therefore, if the whole
time series can be divided into N time windows according to the task paradigm of the fMRI experiment, we
want to estimate the effective connectivity at each time window instead of the whole time series.
2
In our case, to deal with the signal-dependent noise (SDN) observed in the BOLD signal[
1
], the proposed
time-varying method is based on the Granger causality with signal-dependent noise (GCSDN) model, which
has been proved to be useful in detecting the effective connectivity previously ([
2
] and [
3
]). Suppose we
have two time series,
x
t
and
y
t
. Let
p
and
q
be the model orders,
{
A
i
(
i
= 1
,
···
, p
)
,
B
xy,j
,
B
yx,j
(
j
=
1
,
···
, q
)
,
C
xy
,
C
yx
}
be the model coefficient matrices, and
u
xy,t
,
v
yx,t
be the Gaussian white noise. The
model can be defined as follows:
0
@
x
t
y
t
1
A
=
p
X
i
=1
0
@
A
xx,i
A
xy,i
A
yx,i
A
yy,i
1
A
0
@
x
t
−
i
y
t
−
i
1
A
+
0
@
r
xy,t
r
yx,t
1
A
,
(S1)
where
r
xy,t
=
H
1
/
2
xy,t
u
xy,t
,
H
xy,t
=
C
′
xy
C
xy
+
q
X
j
=1
B
′
xy,j
0
@
x
t
−
j
y
t
−
j
1
A
x
′
t
−
j
y
′
t
−
j
B
xy,j
,
H
yx,t
=
C
′
yx
C
yx
+
q
X
j
=1
B
′
yx,j
0
@
x
t
−
j
y
t
−
j
1
A
x
′
t
−
j
y
′
t
−
j
B
yx,j
,
r
yx,t
=
H
1
/
2
yx,t
v
yx,t
.
(S2)
For the time-varying causality, we first divide the whole time series into N time windows. At each window,
the model (Eq.
S1
-
S2
) can be fit based on the direct observations and the indirect observations. Assuming
the system evolved smoothly from one time window to another, the observed time series in the current
window are considered as the direct observations of the model for this window, while the observed time-series
data in the other windows are the indirection observations. We need the indirect observations here, since
the number of data points in a given window is often too small to make reliable estimation of the model.
This is especially true for the self-paced task-fMRI experiments. In a self-paced paradigm, the number of
scans collected in one round can be as small as one or two. Here, we propose to make use of the indirect
observations by calculating the likelihood function at the
i
th
0
time window as a weighted average among all
time windows. The weight for the observations in the
i
th
window is defined as follows:
w
i,i
0
=
K