Goal-dependent modulation of effective connectivity

Supplementary Materials

Goal-dependent modulation of effective connectivity detected by Granger causality with signal-dependent noise

Qiang Luo1,†, Meghana A. Bhatt2,†, P. Read Montague3,4, and Jianfeng Feng5,6,*

1Department of Management, School of Information Systems and Management, National University of Defense Technology, Hunan 410073, P.R. China, 2Beckman Research Institute, City of Hope, Duarte, CA 91010, United States, 3Virginia Tech Carilion Research Institute, Roanoke, VA 24016, United States. 4Wellcome Trust Centre for Neuroimaging, University College London, 12 Queen Square, WC1N 3BG, United Kingdom 5Centre for Computational Systems Biology, School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R. China, 6Centre for Scientific Computing, University of Warwick, Coventry CV4 7AL, United Kingdom.

Abstract Here we present a detailed approach on how to analyze fMRI data, using Granger causality with signal dependent noise. The model stability of signal dependent noise is discussed which serves as a constraint when we fitted the model parameters. We then introduced a strict procedure to regress out all artifacts due to head movements. The processed data enables us to clearly demonstrate that noise depends on signal. With the data, we then analyzed effective connectivity circuits of seven brain regions, including the three regions in the main text as a subset.

1 Granger Causality with Signal-Dependent Noise

This novel approach to causality is a marriage of two approaches: Granger causality (C. W. J. Granger, 1969),

and the Baba-Engle-Kraft-Kroner (BEKK) model of time series with time-varying volatility (R. F. Engle and K.

F. Kroner, 1995). We have applied this approach to EEG data (Q. Luo et al., 2011), but this is the first

application to fMRI data. Signal dependent noise is widely observed in physiological experiments at the spike

level. As a consequence, we should expect that the derived measures such as LFP, EEG and the BOLD signal

should exhibit similar features (see Discussion). However, most publications related to fMRI data do not take

this key feature into account.

Here we describe the stability conditions of this model, and provide a constrained minimization optimization

for the model fitting. A Matlab package of the methods used in this paper is available on request. † Contributed to this work equally * Correspondence should be addressed to Jianfeng Feng, Fudan University, Shanghai 200433, P.R. China. E-mail: [email protected].


2 Model with Signal-Dependent Noise

2.1 Model

Suppose that we have two time series tX and tY ( 1, ,t T= ). Let ( , )t t tZ X Y= , p and q the model orders,

iA ( 1, ,i p= ), ,xy jB , ,yx jB ( 1, ,j q= ), xyC , yxC the model coefficient matrices, and ,xy tu , ,yx tv Gaussian white

noise processes. The time-series model considered is given below

, ,1

( , ) 'p

t i t i xy t yx ti

Z A Z r r−=

= +∑ , (S1)

where

1/2, , ,xy t xy t xy tr H u= , , , ,

1

q

xy t xy xy xy j t j t j xy jj

H C C B Z Z B− −=

′ ′ ′= +∑ , (S2)

and

1/2, , ,yx t yx t yx tr H v= , , , ,

1

q

yx t yx yx yx j t j t j yx jj

H C C B Z Z B− −=

′ ′ ′= +∑ . (S3)

The signal-dependent noise is modeled by the conditional variances ,xy tH and ,yx tH taking a quadratic form.

Since the proposed model of signal-dependent noise is a modification of the Baba–Engle–Kraft–Kroner (BEKK)

model (R. F. Engle and K. F. Kroner, 1995) for time-varying volatility, we refer to the time series model as the

AR-BEKK model and we refer to the method for causal inference as Granger causality with signal-dependent

noise (GC with SDN). When the coefficients in the conditional variances are all zeros, i.e. 0jB ≡ , this model

is exactly to the well-known autoregressive (AR) model.

2.2 Stability conditions

The model is stable if both the first and the second order stability conditions hold. Conditioning

on , 1, ,t iZ i p q− = ∨ , we can compute the conditional covariance on both sides of the above equation resulting


1 , ,( | ) ( , ),t p qt t xy t yx tCov Z Z diag H H− ∨

− = (S4)

where 1 1, 2,( , )t p qt t t t p qZ Z Z Z− ∨− − − − ∨ ′= . Since

21 1 1

11 1

( | ) ( | ) ( | )

( | ) ( )( ) .

t p q t p q t p qt t t t t t t

p pt p q

t t t i t i i t ii i

Cov Z Z E Z Z Z E Z Z

E Z Z Z A Z A Z

− ∨ − ∨ − ∨− − −

− ∨− − −

= =

′= −

′ ′= − ∑ ∑ (S5)

Take the expectation of 1t p qtZ − ∨− on both sides we get the unconditional expectation of ( )t tE Z Z ′ as

1 1

, , , ,1 1

( ) (( )( ) )

( ( ) , ( ) ).

p p

t t i t i i t ii i

q q

xy xy xy j t j t j xy j yx yx yx j t j t j yx jj j

E Z Z E A Z A Z

diag C C B E Z Z B C C B E Z Z B

− −= =

− − − −= =

′ ′=

′ ′ ′ ′ ′ ′+ + +

∑ ∑

∑ ∑ (S6)

For analysis of fMRI data, we usually take 1p q= = , the above equation can be rewritten as

1 1 1 1

,1 ,1

1 1

,1 ,1

1 1

( ( ))( ( ))

0 0( ( ))

0 0

( ) ( ( )) ,

t t

t t

xy xy xy xy

t t

yx yx yx yx

t t

vec E Z ZA A vec E Z Z

B B C C

vec E Z Z

B B C C

A B vec E Z Z C

− −

− −

− −

′′= ⊗

′ ′ ′⎛ ⎞ ⎛ ⎞⊗⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟′+ +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟′ ′ ′⊗⎝ ⎠ ⎝ ⎠

′≡ + +

(S7)

where ⊗ stands for Kronecker product and vec denotes the operator that stacks the columns of a matrix. The

stability conditions of the AR(1)-BEKK(1) model are given by

Condition 1: the modulus of the largest eigenvalue of the matrix A B+ is less than 1;

Condition 2: the modulus of the largest eigenvalue of the matrix 1A is less than 1.

2.3 Parameter Fitting

Based on the observations, the parameters are estimated by solving a constrained optimization problem, i.e.,

maximizing the log likelihood function while assuring the stability conditions. Let , ,( , )t xy t yx tH diag H H= , and

the log likelihood function is given by

11 1( ) ln | | , , , ,2 2t t t tl t H r H r t p q Tθ

−′= − − = ∨ (S8)


where the model parameters are

, ,{ , 1, ; , , 1, , ; , }i xy j yx j xy yxA i p B B j q C Cθ = = = .

The constrained maximum likelihood estimation (MLE) of the model parameters can be established by solving

the following optimization problem

ˆ arg max ( ),T

t p ql tθ

θθ

= ∨

= ∑ s.t. both Condition 1 and Condition 2 hold. (S9)

The optimization problem can be solved by a Matlab function ‘fmincon’ with the initial solution provided as

follows: ( 1, , )iA i p= are given by the least square estimation of 1

p

t i t i ti

Z A Z r−=

= +∑ ; the

, ,, ( 1, , )xy j yx jB B j q= and ,xy yxC C are established by assuming they are diagonal matrices, and carrying out

the least square estimations for the following models in the dimension l : 2 2 2 2, , ,

1( ) ( ) ( )

q

xy t l xy l xy j ll t j lj

r C B Z −=

= +∑ and

2 2 2 2, , ,

1( ) ( ) ( )

q

yx t l yx l yx j ll t j lj

r C B Z −=

= +∑ .

2.4 Causality detection

Let us consider the causality Y X→ . Define the following two models for tX :

1/2, , ,

1

p

t xx i t i xx t xx ti

X A X H u−=

= +∑ , (S10)

, , ,1

q

xx t xx xx xx j t j t j xx jj

H C C B X X B− −=

′ ′ ′= +∑ ; (S11)

and

1/2, , , ,

1 1

p p

t xy i t i xy i t i xy t xy ti i

X A X D Y H u− −= =

= + +∑ ∑ , (S12)

, , ,1

[ , ][ ; ]q

xy t xy xy xy j t j t j t j t j xy jj

H C C B X Y X Y B− − − −=

′ ′ ′ ′= +∑ (S13)

The first assumes no causal influence from Y to X, while the second allows Y to affect both the signal and the

variance of X. Non-causality then can be defined in the second model as some constraints of the relevant


parameters

,

, 21 , 11 ,

0, 1, , ;

[ ] 0,[ ] , 1, , .xy i

xy j xy j xx j

D i p

B B B j q

= =

= = = (S14)

This implies that

, ,1 1

trace[cov( )] trace[cov( )]p p

t xx i t i t xy i t ii i

X A X X A X− −= =

− = −∑ ∑ . (S15)

However, in the model with signal-dependent noise, the covariance of the prediction error has also been

modeled. When we compare the prediction powers of these models, we must exclude those parts of the

covariance that have been already explained by the model.

On the left side of Eq.(S15), we have

,1

, ,1

1

cov( )

( )

;

p

t xx i t ii

q

xx xx xx j t j t j xx jj

XXxx

X A X

C C B E X X B

−=

− −=

−

′ ′ ′= +

≡ Σ +Σ

∑

∑ (S16)

On the right side of Eq.(S1), if we group all terms with Y into XYΣ , we get

,1

2, ,

1

2

cov( )

( ) ( )

,

p

t xy i t ii

p

xy i t i xy ti

XY XXxy

X A X

E D Y E H

−=

−=

−

= +

≡ Σ +Σ +Σ

∑

∑ (S17)

where xx xx xxC C′Σ = and xy xy xyC C′Σ = . Under the null hypothesis of non-causality, we have

1 2trace( ) trace( )XX XXΣ = Σ and trace( ) 0XYΣ = . Therefore, the null hypothesis of non-causality can be rejected by

test the non-equality between trace( )xxΣ and trace( )xyΣ as

trace( )lntrace( )

xxY X

xy

F →

Σ=

Σ. (S18)

We can use a parametric bootstrap1 to establish a 95% confidence interval of Y XF → . If this confidence interval

does not include zero, then the causality is significant; otherwise, no significant causality can be identified.

However, since the time-series data with signal-dependent noise usually have high level noise, the significant 1 The model based bootstrap has been employed in this paper. We fit a model to the observation first, and then generate bootstrap data by simulating the fitted model many times.


causality is detected in this supplementary file only if the lower bound of the 95% confidence interval is no less

than 0.0010.

We could use the determinant (‘det’) instead of the trace to compare xxΣ and xyΣ . However, the meaning of

this comparison is less clear than its ‘trace’ counterpart. Specifically, we can not separate the terms for X and Y

in the equation 1 2det( ) det( )XX XY XXxx xyΣ +Σ = Σ +Σ +Σ , if we use ‘det’ instead of ‘trace’ in the Eq.(S1). In

addition, the numerical stability of the causality measurement defined by ‘trace’ is generally better than the ‘det’

version (C. Ladroue et al., 2009).

3 Data Analysis

In the bargaining game described in the main text, the behavioral differences displayed by the three groups were reflected in activation differences of three brain regions. Strategists showed increased activation in the right dorsolateral prefrontal cortex (DLPFC), an area implicated in cognitive control and complex decision-making (D. Knoch et al., 2006; W. Yoshida et al., 2010) and the rostral prefrontal cortex [rPFC or Brodmann area 10 (BA10)], which has been implicated in a host of computations including mentalizing and goal maintenance (N. Ramnani and A. Owen, 2004; P. Burgess et al., 2007). Finally, strategists showed a significantly different pattern of activation in response to trial-by-trial information in the temporoparietal junction (TPJ) which has long been theorized as part of a “theory of mind” network (R. Saxe and N. Kanwisher, 2003; R. Saxe and A. Wexler, 2005), but has also been implicated in more low level-computations involving reorienting to salient stimuli (J. Decety and C. Lamm, 2007). In particular, rTPJ activity was significantly increased when strategists saw cues that tended to make them switch from reputation building, to collecting rewards. The other two types showed no modulation of rTPJ activity by any external stimuli in the task. While the behavioral types are characterized by the levels of activation in these three areas, we hypothesized that the expressed behaviors might also correlate with causal (directional) changes in the effective connectivity among these areas. In order to examine effective connectivity we analyzed time-series data from the original experiment, focusing on these three regions of interest (DLPFC, rPFC, rTPJ) together with another four regions, including superior frontal gyrus, caudate, left insula, and right insula.

3.1 Preprocessing

As described in Method, we cut out the inter-trail scans and concatenated the fMRI time-series of all 60 trails

and all subjects in each group together to form a large sample of observations for the neural circuit of each

behavioral group as shown in Figure S1(A).


The standard preprocessing methods including the following four steps:

Step 1. Denoise signals by thresholding the wavelet coefficients. This method is carried out by applying the

function, mswden, provided in the wavelet toolbox in Matlab with the default settings to the signal collected for

each region;

Step 2. Detrend signals by removing the best straight-line fit from the time-series data by using a function,

detrend, provided in Matlab (Version 7.9.0.529);

Step 3. Remove temporal mean from a data matrix by calling the function, cca_rm_temporalmean, provided

in GCCA_toolbox_nov02;

Step 4. Calculate differences between adjacent elements of the time series by calling a function, diff,

provided in Matlab (Version 7.9.0.529). After the above standard preprocessing, the preprocessed data are shown in Figure S1(B). For fMRI data, the inevitable head movement of the subjects in the scanner may bring some artifacts that

would confound any statistical analysis. Therefore, we controlled for the effects of the sudden head movement on the fMRI time-series data.


Figure S1: fMRI time-series data for three ROI’s in three behavioral groups. The fMRI scans between

trails are cut out for each brain area of each subject, and the fMRI data for all subjects are concatenated together

in each behavioral group. A. The original fMRI time-series data; B. The preprocessed data; C. The data after

correcting the head movement.

A

B

C


3.2 Motion correction

We corrected for the possible effects of head motion in two steps: First we took the displacement and rotation

movements of the head from 6 parameters of head movement. Using these we defined two series as follows:

2 2 21 1 1displacement ( ) ( ) ( )t t t t t t tx x y y z z− − −= − + − + − ,

2 2 2t 1 1 1rotation ( ) ( ) ( )t t t t t tα α β β γ γ− − −= − + − + − .

The effects of sudden head movement were corrected by setting the signal back to its mean value at all time

points when either the displacement or the rotation was larger than 2 times of the corresponding standard

deviations.

Then we corrected for any remaining effects of head motion using a multiple linear regression of the resulting

fMRI signals on the movement parameters. The residuals of this regression were used for our final analysis.

The displacement and rotation series are shown in Figure S2, and signals after the motion correction are

presented in Figures S1(C). The correlation coefficients between the movements and the signals before and

after the correction are listed in Table S1. Note that the small correlation coefficient after processing (in the

scale of 10-16).

Table S1: Correlation coefficient between movement parameters and fMRI signal of ROI.

Pre-correction Post-correction(×10-16)

Group ROI Displacement Rotation Displacement Rotation

INC

BA10 -0.1293 -0.1553 0.0987 -0.1431

rDLPFC -0.1052 -0.1793 -0.0694 -0.5073

rTPJ -0.1774 -0.1489 0.1149 -0.2364

CON

BA10 -0.0990 -0.1631 -0.0347 0

rDLPFC -0.1034 -0.1641 -0.0694 0.1995

rTPJ -0.0464 -0.1004 0 0.7286

STRAT

BA10 -0.0729 -0.0111 0.0867 -0.0824

rDLPFC -0.0328 -0.0097 -0.2515 -0.2255

rTPJ -0.0015 -0.0171 -0.1518 0.1149


Figure S2: Motion parameters of head movement for three behavioral groups.

3.3 Model selection

3.3.1 Correlation observed between the squared signal and the squared noise

Considering all 7 regions, we fit an AR model for the fMRI time-series data for each behavior group and

calculated the residual process, tR . The noise level at each time point could then be measured using trace of

the squared errors, trace( )t tR R′ . Similarly, the lagged signal level was measured using the trace of the squared

signal with one time lag, 1 1trace( )t tZ Z− −′ . Outliers were filtered out using a threshold of 5σ . Although this

filtering process slightly decreased the correlation coefficients for all three behavioral groups we still found

significant correlations between lagged signal and noise as estimated by the standard AR model. The scatter

plots and correlation coefficients are shown in Figure S3.


Figure S3: Scatter plots for the squared noise process against the squared signal with one time lag. Blue

pluses represent the data points and the red lines are the linear fits.

3.3.2 Model selection by AIC

We compared the fits of the AR and AR-BEKK models using the Akaike Information Criterion (AIC):

AIC = (-2 LLF) + (2 NumParams)× × , (S19)

where ( )T

t p q

LLF l tθ= ∨

= ∑ .

The results show uniformly lower AIC’s for the AR-BEKK model as compared to the standard AR. Results are shown in Table S2.

Table S2: Comparison of AIC’s between different models for 7 ROI’s in 3 behavior groups. “Err” is the trace of the covariance of the prediction errors, “LLF” stands for the log likelihood function, “#Para” is the number of free parameters in each model, and “AIC” is short for the Akaike information criterion.

Group Model Err LLF（×104） #Para AIC（×105）

Incrementalist AR 16.9400 -8.9157 49 1.7841 AR-BEKK 9.1062 -1.5970 126 0.3219

Conservative AR 10.3502 -6.8768 49 1.3763 AR-BEKK 7.1924 -1.3285 126 0.2682

Strategist AR 10.6111 -3.8443 49 0.7698 AR-BEKK 9.0248 -0.7910 126 0.1607


4 Results

Many methods have been employed to identify the effective connectivity between brain areas. In this section, we present the directional connectivity results for the fMRI data in the bargaining game established by correlation coefficient with one time lag, classical Granger causality, and the novel adaptation, Granger causality with signal-dependent noise.

4.1 Classical Granger causality

Recently, the Granger causality has been widely used in computational neuroscience (K. Friston, 2009). The basic idea behind this method is that the causal influence from one time series to the other time series can be identified if the use of the past information of the previous time series improves the one-step-ahead prediction of the other one. The time-series model employed in this method is the autoregressive (AR) model. The partial Granger causality is one of the most popular versions of the classical Granger causality (S. Guo et al., 2008) . We apply this method to the fMRI data of our three ROI’s. The results are shown in Table S3 and Figure S4. From these results we can see that no significant causal influence can be identified by the classical Granger causality. This failure to detect any causal influence may be partially due to the incorrect assumption of constant variance in the AR model. Table S3: Causality results using classical Granger causality. The order of the AR model is 1. The medians of the causalities established by 300 bootstrap samples are reported for 42 directions, and the corresponding 95% confidence intervals are listed in brackets.

Direction 95% CI

Inc Con Strat

0.025

quantile

0.975

quantile

0.025

quantile

0.975

quantile

0.025

quantile

0.975

quantile

'BA10--->rDLPFC' 0.0000 0.0074 0.0000 0.0056 0.0000 0.0108

'rDLPFC--->BA10' 0.0000 0.0052 0.0012 0.0247 0.0000 0.0076

'BA10--->rTPJ' 0.0000 0.0110 0.0000 0.0111 0.0003 0.0222

'rTPJ--->BA10' 0.0000 0.0084 0.0000 0.0113 0.0002 0.0200

'BA10--->SuperiorFrontalGyrus' 0.0002 0.0162 0.0002 0.0180 0.0031 0.0317

'SuperiorFrontalGyrus--->BA10' 0.0000 0.0059 0.0000 0.0055 0.0000 0.0100

'BA10--->Caudate' 0.0000 0.0116 0.0000 0.0050 0.0001 0.0188

'Caudate--->BA10' 0.0000 0.0058 0.0002 0.0199 0.0001 0.0156

'BA10--->LeftInsula' 0.0000 0.0112 0.0000 0.0141 0.0000 0.0110


'LeftInsula--->BA10' 0.0000 0.0107 0.0000 0.0064 0.0000 0.0064

'BA10--->RIghtInsula' 0.0000 0.0052 0.0000 0.0151 0.0000 0.0053

'RIghtInsula--->BA10' 0.0000 0.0084 0.0000 0.0092 0.0000 0.0072

'rDLPFC--->rTPJ' 0.0000 0.0112 0.0011 0.0255 0.0133 0.0604

'rTPJ--->rDLPFC' 0.0000 0.0088 0.0000 0.0135 0.0077 0.0412

'rDLPFC--->SuperiorFrontalGyrus' 0.0000 0.0123 0.0053 0.0348 0.0148 0.0628

'SuperiorFrontalGyrus--->rDLPFC' 0.0000 0.0053 0.0000 0.0041 0.0000 0.0128

'rDLPFC--->Caudate' 0.0000 0.0143 0.0000 0.0076 0.0003 0.0193

'Caudate--->rDLPFC' 0.0000 0.0086 0.0000 0.0122 0.0000 0.0101

'rDLPFC--->LeftInsula' 0.0000 0.0072 0.0003 0.0188 0.0000 0.0067

'LeftInsula--->rDLPFC' 0.0000 0.0105 0.0000 0.0067 0.0000 0.0089

'rDLPFC--->RIghtInsula' 0.0000 0.0095 0.0000 0.0069 0.0002 0.0171

'RIghtInsula--->rDLPFC' 0.0000 0.0091 0.0002 0.0207 0.0002 0.0178

'rTPJ--->SuperiorFrontalGyrus' 0.0000 0.0057 0.0001 0.0132 0.0000 0.0092

'SuperiorFrontalGyrus--->rTPJ' 0.0000 0.0050 0.0000 0.0056 0.0000 0.0050

'rTPJ--->Caudate' 0.0023 0.0298 0.0000 0.0054 0.0000 0.0091

'Caudate--->rTPJ' 0.0000 0.0069 0.0000 0.0090 0.0000 0.0048

'rTPJ--->LeftInsula' 0.0000 0.0097 0.0006 0.0206 0.0004 0.0205

'LeftInsula--->rTPJ' 0.0000 0.0061 0.0000 0.0080 0.0038 0.0336

'rTPJ--->RIghtInsula' 0.0013 0.0256 0.0000 0.0156 0.0114 0.0536

'RIghtInsula--->rTPJ' 0.0000 0.0053 0.0000 0.0074 0.0080 0.0426

'SuperiorFrontalGyrus--->Caudate' 0.0000 0.0122 0.0000 0.0064 0.0000 0.0056

'Caudate--->SuperiorFrontalGyrus' 0.0000 0.0112 0.0022 0.0318 0.0000 0.0140

'SuperiorFrontalGyrus--->LeftInsula' 0.0000 0.0048 0.0000 0.0096 0.0000 0.0106

'LeftInsula--->SuperiorFrontalGyrus' 0.0000 0.0104 0.0000 0.0085 0.0013 0.0236

'SuperiorFrontalGyrus--->RIghtInsula' 0.0000 0.0152 0.0000 0.0092 0.0000 0.0138

'RIghtInsula--->SuperiorFrontalGyrus' 0.0000 0.0069 0.0000 0.0134 0.0056 0.0371

'Caudate--->LeftInsula' 0.0000 0.0056 0.0000 0.0129 0.0003 0.0184


'LeftInsula--->Caudate' 0.0031 0.0305 0.0000 0.0056 0.0001 0.0163

'Caudate--->RIghtInsula' 0.0000 0.0111 0.0000 0.0044 0.0002 0.0212

'RIghtInsula--->Caudate' 0.0003 0.0194 0.0000 0.0138 0.0000 0.0090

'LeftInsula--->RIghtInsula' 0.0000 0.0164 0.0000 0.0076 0.0000 0.0062

'RIghtInsula--->LeftInsula' 0.0000 0.0071 0.0000 0.0140 0.0006 0.0214

Figure S4: Causality results given by the classical Granger causality. The 95% confidence interval of the

causality is given by 300 bootstrap samples.


Figure S5: Neural circuits given by the classical Granger causality. Neural circuits between 7 regions

established by Granger causality for three different behavioral groups. The 95% confidence interval of the

causality is given by 300 bootstrap samples.

Figure S5 is the plot of the actual effective networks obtained by the classic Granger causality. Green links are

additional links due to the inclusion of more brain region areas in our analyses, in comparison with the main

text. Purple color links are the links also recovered in our results obtained using three regions. We can see that

two links between rTPJ and BA10 are missing in strategists’ brain. This is due to the different threshold used

here and in the main text.

4.2 Granger causality with signal-dependent noise

Considering the observation of significant signal-dependent noise in the AR estimates as well as the superior

model fit of the AR-BEKK model we applied Granger causality with signal-dependent noise to our data. The

results of are reported in Table S4 and Figure S6.


Table S4: Causality results by the Granger causality with signal-dependent noise. Both the model orders in the mean and in the variance are 1. The medians of the causalities established by 300 bootstrap samples are reported for 42 directions, and the corresponding 95% confidence intervals are listed in brackets.

Direction 95% CI

Inc Con Strat

0.025

quantile

0.975

quantile

0.025

quantile

0.975

quantile

0.025

quantile

0.975

quantile

'BA10--->rDLPFC' -0.0098 0.06 -0.0029 0.0316 0.0096 0.4112 'rDLPFC--->BA10' 0.0145 0.5471 -0.0032 0.0365 0.0062 0.1306

'BA10--->rTPJ' 0.0011 0.0955 -0.0022 0.0577 -0.0031 0.0446 'rTPJ--->BA10' -0.0034 0.5243 -0.0026 0.0318 0.0159 0.1619

'BA10--->SuperiorFrontalGyrus' -0.5095 0.0484 -0.0025 0.5305 -0.0027 0.0675 'SuperiorFrontalGyrus--->BA10' -0.4798 0.4847 -0.0041 0.0572 0.0037 0.1151

'BA10--->Caudate' -0.0051 0.0917 -0.0014 0.0713 -0.0052 0.0231 'Caudate--->BA10' 0.0041 0.4962 -0.0043 0.0664 -0.0019 0.0523

'BA10--->LeftInsula' -0.0037 0.0264 -0.0045 0.0289 0.0023 0.1488 'LeftInsula--->BA10' -0.0025 0.5415 0.0003 0.098 0.0074 0.1381

'BA10--->RIghtInsula' -0.0017 0.0843 -0.003 0.0525 0.0311 0.2268 'RIghtInsula--->BA10' -0.0038 0.4495 -0.0023 0.0459 -0.0041 0.0677

'rDLPFC--->rTPJ' 0.2771 0.7598 -0.0029 0.0527 -0.0042 0.0533 'rTPJ--->rDLPFC' 0.0001 0.1014 -0.0029 0.033 0.0506 0.236

'rDLPFC--->SuperiorFrontalGyrus' -0.004 0.0333 -0.0021 0.0941 -0.003 0.321 'SuperiorFrontalGyrus--->rDLPFC' -0.005 0.0439 0.0012 0.0851 -0.001 0.0874

'rDLPFC--->Caudate' 0.003 0.1208 -0.0037 0.0254 -0.0039 0.0383 'Caudate--->rDLPFC' 0.0002 0.1252 -0.004 0.0495 0.0247 0.1893

'rDLPFC--->LeftInsula' -0.0075 0.0418 -0.003 0.0354 -0.003 0.0253 'LeftInsula--->rDLPFC' -0.0044 0.0309 -0.0026 0.0638 -0.0025 0.0293

'rDLPFC--->RIghtInsula' 0.2269 0.8161 -0.0056 0.0565 -0.005 0.0446 'RIghtInsula--->rDLPFC' -0.0017 0.0722 -0.0031 0.0514 -0.0017 0.0459

'rTPJ--->SuperiorFrontalGyrus' -0.0024 0.0817 0.0281 0.2146 -0.0029 0.0383 'SuperiorFrontalGyrus--->rTPJ' 0.0278 0.2221 -0.0029 0.0495 -0.0032 0.0959

'rTPJ--->Caudate' -0.0056 0.0428 -0.003 0.0267 -0.0018 0.0458 'Caudate--->rTPJ' 0.0633 0.43 -0.0041 0.0279 -0.0026 0.0679

'rTPJ--->LeftInsula' -0.0036 0.0277 -0.0025 0.0496 -0.0011 0.0972 'LeftInsula--->rTPJ' 0.2377 0.8359 0.0245 0.2027 -0.0034 0.0686

'rTPJ--->RIghtInsula' -0.0049 0.0302 -0.0028 0.0826 -0.0025 0.0631 'RIghtInsula--->rTPJ' 0.2335 0.6755 -0.0014 0.0578 0.034 0.2194

'SuperiorFrontalGyrus--->Caudate' -0.0035 0.0437 0.0012 0.0926 -0.0035 0.0363 'Caudate--->SuperiorFrontalGyrus' 0.0214 0.1508 -0.0019 0.4501 -0.0034 0.4148

'SuperiorFrontalGyrus--->LeftInsula' 0.0001 0.1112 -0.0037 0.0225 -0.0032 0.0307


'LeftInsula--->SuperiorFrontalGyrus' 0.0643 0.2883 0.0505 0.2707 -0.0001 0.4935 'SuperiorFrontalGyrus--->RIghtInsula' -0.0026 0.039 -0.0036 0.0316 -0.0033 0.025 'RIghtInsula--->SuperiorFrontalGyrus' 0.0138 0.1433 0.0094 0.1559 -0.0022 0.0381

'Caudate--->LeftInsula' 0.0018 0.1316 -0.0022 0.0644 -0.0038 0.0233 'LeftInsula--->Caudate' -0.0041 0.0466 -0.0029 0.0467 -0.0003 0.0824

'Caudate--->RIghtInsula' -0.0011 0.1394 -0.0005 0.0991 -0.0029 0.0416 'RIghtInsula--->Caudate' 0.127 0.752 -0.0029 0.0391 -0.0033 0.0365

'LeftInsula--->RIghtInsula' 0.0174 0.1557 0.003 0.0802 0.0839 0.4 'RIghtInsula--->LeftInsula' -0.0053 0.0468 -0.0014 0.0353 -0.3538 0.2216

Figure S6: Causality results given by the Granger causality with signal-dependent noise. The 95%

confidence interval of the causality is given by 300 bootstrap samples.

Unlike standard Granger causality, this analysis revealed significant causal influences among the regions of

interest as well as differences among the three behavioral types. As shown in FigureS7, we have 7 connections

for Conservatives, 10 connections for Strategists, and 16 connections for Incrementalists.


Figure S7: Neural circuits given by the Granger causality with signal-dependent noise. Neural circuits

between 7 regions established by GC with SDN for three different behavioral groups. The 95% confidence

interval of the causality is given by 300 bootstrap samples.

In Figure S7, we plot the circuits among the seven brain regions. As mentioned in the classical Granger case,

the circuits are slightly different due to the different threshold used. In general, we see that in incrementalist,

rTPJ is served as a sink, but it acts more like a source in strategists.

5 Discussion

5.1 Possible Sources of Signal-Dependent Noise

A natural question is what the source of the signal-dependent noise in blood-oxygen-level-dependent (BOLD)

signals is. Let us agree that the BOLD signal is a convolution of the hemodynamic and the local field potential

(LFP) which is a summation of the local M synaptic inputs. Mathematically, we have

1( )

( , , , )

M

i rest ii

V V dNdLFP x y z t

M=

−=∑

, (S20)

where ( , , )x y z is the location, t is the time, Vi is the membrane potential of a neuron, Vrest is the resting

potential of the channel and Ni is the incoming spike train. We can approximate the point process Ni by a

continuous process

i i i idN dt dBλ λ= + ,


where iλ is the firing rate and Bi is the Brownian motion. Hence Eq. (S20) becomes

1 1( ) ( )

( , , , )

M M

i rest i i rest i ii i

V V dt V V dBdLFP x y z t

M

λ λ= =

− + −=∑ ∑

.

Using the law of large numbers, we have

1 1

1 [ ( ) ( ) ]

( ) (( ) ) ,

M M

i rest i i rest i ii i

i rest i i rest i

V V dt V V dBM

E V V dt V V dB

λ λ

λ σ λ= =

− + −

= − + −

∑ ∑

where ( )σ ⋅ is the standard deviation of a random variable, here the standard deviation of ( )i rest iV V λ− . After

convoluting with a hemodynamic kernel K, the BOLD signal is

*[ ( ) ] *[ (( ) )] .i rest i i rest iK E V V dt K V V dBλ σ λ− + −

Therefore, the dependency of the noise term (the second term above) on the first term (signal term) is clear.

5.2 Applicability of GC with SDN

Among three areas highlighted in the bargaining game by Bhatt et al. (2010), we hypothesized that behavior might also be characterized by the effective connectivity among these areas, but no effective connectivity between these regions was detected by any of the frequently used methodologies from the fMRI time-series data, including the simple correlation analysis and the classical Granger causality. While the former often provides good clues as to functional connectivity, it fails to yield any information about direction of effective connectivity. Conversely, Granger causality provides information about the direction of effective connectivity, but may lack sensitivity (M. Schippers et al., 2011). Neither of these methods allow for time-varying variability of a signal, which would be particularly interesting in this study given the previous findings showing an interaction between external stimuli and activation in the rTPJ. Recently, the time-varying variability and noise of the brain have gained significant attention(R. Stein et al., 2005; A. Faisal et al., 2008). Particularly, more and more evidences suggest the presence of “signal” within what was once considered to be undesirable “noise” in the BOLD signal acquired in a fMRI experiment (D. D. Garrett et al., 2011). Allowing for signal-dependent noise also allows for the signal in one region at a time point to affect the variance of another at the subsequent time steps, as might be observed if one area were responsible for maintaining a long term goal and modulating attention based on the environment as necessary.

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