Investigating a Data-Driven Deep-Learning Approachto Simulate Whole Brain Dynamics
Takuto Okuno1, Alexander Woodward1
1 Connectome Analysis Unit, RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama 351-0198, JapanSummaryD• We present a novel whole brain analytics and simulation method.
• Deep-learning based causality analysis significantly outperforms
conventional analytics methods such as Functional Connectivity,
multivariate Granger Causality, and Transfer Entropy in the large nodecase.

E𝑆2𝑆2 𝑡 = 𝑆6 𝑡 − 1F2G22444666𝑆4 𝑡 = 𝑆6 𝑡 − 1

𝑆6𝑆42Abstract

26464246Figure 2. Training result of fully random (uniform distribution) signals

To elucidate brain function and pursue pathological studies, thedevelopment of in silico simulation offers a number of opportunities to study
the properties of the brain in a non-invasive manner. In this study, we use adata driven method to analyze the causal relationship between non-linear
input and output, and we evaluate its performance over a number ofaspects. Our model comprises a fully connected network where each node
is a deep neural network. The weights within and between nodes can betrained on multi-modal data, such as fMRI BOLD, calcium imaging, etc., andour model can be easily expanded to large-scale simulation.

(A) Node network image and training conditions. (B) Left shows before training signals of 8nodes (Red: teaching signal, Blue: DLCM output). Right shows after training signals. Left
Mean Absolute Error (MAE) is 0.456. Right MAE is 0.018. DLCM was successfully trained. (C)Training time result of signal length and node count. (D) Node network image forevaluating Functional Connectivity (FC), mvGC and DLCM-GC. Node 6 signal is copied tonode 2 and 4 at next time step. (E) FC result of D. Color range is [-1, 1]. (F) mvGC result ofD. Color range is [-5𝜎, 5𝜎]. (G) DLCM-GC result of D. Color range is [-5𝜎, 5𝜎].

Methods
Deep-Learning based Causal Model (DLCM)

A

𝑐𝑖𝑗1

𝑆𝑖𝑆𝑛𝑥𝑖𝑗

𝑥𝑖1

𝑐𝑖𝑗𝑛

𝑦𝑖𝐻𝑁
𝐈𝐈
𝜔𝑖𝐻𝑁1

𝐈𝐈
𝜔𝑖1𝐼𝑁

𝐈𝐈𝜔𝑖11𝑑𝑛𝑗𝑚𝑦𝑖𝑘𝐈𝐈𝐈𝐈𝜔𝑖11⋯ 𝜔𝑖𝐻𝑁𝐼𝑁𝑾𝐈𝐈⋮⋱⋮𝑖 =𝐈𝐈𝐈𝐈𝜔𝑖𝐻𝑁1 ⋯ 𝜔𝑖𝐻𝑁𝐼𝑁
𝐈𝐈𝐈𝐈𝜷𝐈𝐈𝑖 = (𝛽𝑖1 , ⋯ , 𝛽𝑖𝐻𝑁 )T𝐈𝐈 T
𝒀𝑖 = 𝑔(𝑾𝐈𝐈𝑖 𝑿𝑖 +𝜷𝑖 )𝐈𝐈
𝜔𝑖𝐻𝑁𝐼𝑁𝑥𝑖𝐼𝑁

𝑐𝑖𝑗𝑖
𝑑𝑖1𝑚
𝑐𝑖1𝑛

𝑐𝑖11 𝑐𝑖1𝑖

𝑑𝑖𝑗𝑚
𝑐𝑖𝑗1𝑑𝑖𝑗1
𝑐𝑖𝑗𝑖 𝑐𝑖𝑗𝑛

8

𝑐𝑖𝐼𝑁1
𝑑𝑖𝐼𝑁𝑚
𝑑𝑐𝑖𝐼𝑁𝑖 𝑐𝑖𝐼𝑁𝑛𝑖𝐼𝑁1𝐈𝐈
𝑾𝐈𝑖 = (𝜔𝑖1, ⋯ , 𝜔𝑖𝐻𝑁)

I

𝑺 𝑡 + 1 = 𝒁 = (𝑧1 , ⋯ , 𝑧𝑛 )

𝑆𝑖 𝑡

𝑆𝑛 𝑡

𝐼1 𝑡

𝐼𝑚 𝑡

{𝑆𝑖 , 𝐼𝑞 , 𝑥𝑖𝑗 , 𝑦𝑖𝑘 , 𝑧𝑖 } 𝑖=1..𝑛,𝑗=1..𝐼𝑁, ∈ ℛ

Node statussignals

Exogenoussignals

𝑔 = 𝑅𝑒𝐿𝑈(A) Whole brain node status network and stimulus coming from outside brain (mainlysensory input) (B) Showing signaling framework of one node status. A Deep Neural
Network (DNN) node receives the signal state of other brain nodes and exogenous signals(outside brain stimulus). DNN unit calculates one output signal, in this case the next time
step of the signal state of a node. A DNN unit is trained by teacher signals and the input ofother node states and exogenous signals. (C) The mathematical expression of DeepLearning based Causal Model.

DLCM Granger Causality (DLCM-GC)
Granger Causality (GC)[1] is a well established analytics method for timeseries and multivariate GC (mvGC)[2] is applied for analyzing fMRI BOLDsignals. If 𝑺 𝑡 is the node vector (𝑆1 𝑡 , ⋯ , 𝑆𝑛 𝑡 ) and each node 𝑺𝑖 has timeseries of {𝑆𝑖 1 , ⋯ , 𝑆𝑖 𝐿 }. 𝑆𝑖 𝑡 is expressed by time lag 𝑝, the coefficient vector
𝑨𝑖𝑘 and a residual 𝜀𝑖 𝑡 :𝑝𝑆𝑖 𝑡 = ෍ 𝑨𝑖𝑘 𝑺 𝑡 − 𝑘 T + 𝜀𝑖 𝑡

(1)

𝑘=1

If node j relation is removed from node i,𝑝(−𝑗) (−𝑗)(−𝑗)T= ෍ 𝑨𝑖𝑘 𝑺𝑡 − 𝑘 + 𝜀𝑖𝑡
𝑘=1(2)

the mvGC of 𝑆𝑗→𝑖 is expressed by(−𝑗)𝑆𝑗→𝑖 = log 𝑣𝑎𝑟(𝜺𝑖 )ൗ𝑣𝑎𝑟(𝜺𝑖 )𝑣𝑎𝑟() denotes the statisticalvariance. For mvGC the coefficients and residuals are calculated by linear
regression. Recently, Neural Network based Granger Causality is studied inseveral literatures, such as NN-GC [3], RNN-GC [4], ES-GC [5] and DNN-GC[6]. DLCM can also calculate this type of Granger Causality. We simply use
𝑬𝒓𝒓𝑖 = 𝑺𝑖 − 𝑻𝒆𝒂𝒄𝒉𝑖 of single node i as residuals 𝜺𝑖 and removing signal from(−𝑗)
(−𝑗)
(−𝑗)
node j, 𝑬𝒓𝒓𝑖= 𝑺𝑖− 𝑻𝒆𝒂𝒄𝒉𝑖 as residuals 𝜺𝑖 . Then, DLCM-GC is defined(−𝑗)𝑣𝑎𝑟(𝑬𝒓𝒓𝑖 )ൗ𝑣𝑎𝑟(𝑬𝒓𝒓𝑖 )(4)

Importantly, removing the signal from another node can be though of as
lesion simulation. One brain region is virtually impaired and expressed asGranger Causality. Therefore, DLCM can also work as a lesion model [7].

Results
DLCM is trained by fully random signals
ABC

Hidden1 Neurons : 32
Hidden2 Neurons : 22

Training time [sec]Training Conditions
Node Number : 8
Exogenous Input : 0Signal Length : 100

True Positive Rate[-1 1]

Target node

AUC [-]

0.5

1H

Kruskal-Wallis ANOVA: P = 1.0e-06
1**

0.8

*

0.7

0.5

False Positive Rate

K

10.8

spcDCM
DLCM-GC
LINUE-TE0.6

pat1(0.05) pat2(0.11) pat3(0.16) pat4(0.27) pat5(0.36)

pat6(0.05) pat7(0.11) pat8(0.16) pat9(0.27) pat10(0.36)

Network Pattern (Density)

Network Pattern (Density of |𝑎𝑖𝑗 |>0.2)

1100000
10000
1000
FC
spcDCM
mvGC
DLCM-GC
RNN-GCLINUE-TE
NNNUE-TE

100
101
0.1
0.01
0.001
81216Node Number [count](A) Node network image pat7 - all nodes were connected and some nodes had strong
connections. (B) Adjacency matrix A of network in A, used by Spectral DCM (spcDCM) [8].
This provides a ground truth connectivity for the DCM and 8 node signals were generated
together with 8 exogenous signals. (C) ROC curve of detecting ground truth matrix, (strong
connection:|𝑎𝑖𝑗 |>0.2). Seven algorithms were compared: FC, spcDCM, mvGC, RNN-GC,DLCM-GC, LINUE-TE (Transfer Entropy) and NNNUE-TE. (D) AUC of C, Steel test: * P<0.05. (E)Node network image of pat10. (F) Adjacency matrix A of network in E. (G) ROC curve ofdetecting ground truth in F. (H) AUC of G. (I) Network density (sparse pattern) vs. AUC
result. (J) Network density (fully connected) vs. AUC result. (K) Computation time results.
spcDCM was calculated by DCM12 on MATLAB 2019b. LINUE-TE and NNNUE-TE werecalculated by MuTE [9] on MATLAB. RNN-GC was calculated using Python 3.6. Machine
spec was OS: Windows 10 Pro; CPU: AMD Ryzen Threadripper 3970X 32-Core Processor
3.70GHz, Mem:128GB. Tested node counts were 8, 12 and 16 and number of trials were
spcDCM N=20,10,4 (due to large computation time) and all others N=20,10,10.

DLCM-GC significantly outperforms mvGC and LINUETE in large node case
AB

Source node

D0E

Source node

1CFC
mvGC
pwGC
DLCM-GC
LINUE-TE

False Positive Rate

0.8
0.7
0.6

*0.8

FC
mvGC
pwGC
DLCM-GC
LINUE-TE

0.7

0.4

0.9

Kruskal-Wallis ANOVA: P = 9.45e-07

0.6

0.8
0.7
0.60.50.4

1G

0.9

0.5

F

False Positive Rate

Kruskal-Wallis ANOVA: P = 2.05e-06

110

0.9

0.5

*

*

*

*
0.4pat11(16) pat12(32) pat13(48) pat14(64) pat15(80) pat16(98)Network Pattern (Node Number)

Figure 4. Evaluation result of 5 algorithms to detect original connections.
(A) Ground truth connectome matrix C (pat12(32)) for generating node signals calculatedby the reduced wong-wang model [10] from The Virtual Brain software [11]. Matrices weregenerated from Allen's mouse connectome matrix [12][13]. Node numbers were
16,32,48,64,80,98 and all network densities (|𝑐𝑖𝑗 |>1) were 0.15. (B) ROC curve of detecting
ground truth matrix C. 5 algorithms, FC, mvGC, pwGC, DLCM-GC, LINUE-TE werecompared. (C) AUC of B, Steel test: * P<0.05. (D) Ground truth connectome matrix D(pat16(98)) (E) ROC curve of detecting ground truth matrix D. (F) AUC of E. (G) AUC resultsfor different node numbers.
This research was supported by the program for Brain Mapping by Integrated
Neurotechnologies for Disease Studies (Brain/ MINDS) from the Japan Agency for Medical
Research and Development, AMED. Grant number: JP20dm0207001.

600

309015060120180

References

4003002001000200

Training Epoch : 1000

0.7

Acknowledgement

500

Fully random
Independent signals

0.8

0.6

0.7
FCmvGCRNN-GCNNNUE-TE*

Figure 3. Evaluation result of 7 algorithms to detect ground truth connections

(3)

(−𝑗)
where 𝜺𝑖 and 𝜺𝑖
are time series of residuals,

𝑆𝑗→𝑖 = log

0.8

0.60

*

0.9

0.9

0.7

Figure 1. Definition of Deep-Learning based Causal Model

(−𝑗)
𝑆𝑖𝑡

AUC [-]

Outside brain stimulus(sensory input)

𝐈𝐈𝐈𝐈𝐈{𝑐𝑖𝑗𝑝 , 𝑑𝑖𝑗𝑞 , 𝑏𝑖𝑗 , 𝜔𝑖𝑗𝑘, 𝜔𝑖𝑘, 𝛽𝑖𝑘, 𝛽𝑖𝐈 } 𝑖=1..𝑛,𝑗=1..𝐼𝑁, ∈ ℛ
𝑘=1..𝐻𝑁,𝑝=1..𝑛,𝑞=1..𝑚J

0.9

𝑘=1..𝐻𝑁,𝑞=1..𝑚

False Positive Rate

*0.6

1

Source node

1

Target node

𝑆1 𝑡

5

4

Target node𝐼𝑚3

pat10 (density = 1.14)(density of |𝑎𝑖𝑗 |>0.2 = 0.36)

𝑧𝑖 = 𝑾𝐈𝑖 𝒀𝑖 T +𝛽𝑖𝐈

𝐼1

2

76

GFC
spcDCM
mvGC
RNN-GCDLCM-GC
LINUE-TE
NNNUE-TE

AUC [-]

𝑐𝑛𝑗𝑖𝑑𝑛𝑗1

𝐈𝜔𝑖𝐻𝑁

F11AUC [-]𝑑1𝑗𝑚

𝑩𝑖 = (𝑏𝑖1 , ⋯ , 𝑏𝑖𝐼𝑁 )𝑿𝑖 = 𝑔(𝑪𝑖 𝑺 𝑡 T + 𝑫𝑖 𝑰 𝑡 T + 𝑩𝑖 )T

E0Source node

Kruskal-Wallis ANOVA: P = 7.0e-06

0.9

Calculation time [sec]

𝑑1𝑗1

𝑦𝑖1

𝑐𝑛𝑗1

𝐈
𝜔𝑖𝑘

pat7 (density = 1.14)(density of |𝑎𝑖𝑗 |>0.2 = 0.11)

D

1

True Positive Rate

𝑐𝑛𝑗𝑛

𝑐1𝑗𝑛

𝑆1 𝑐1𝑗𝑖

Deep Neural
Network unit

54

AUC [-]

𝑐1𝑗1

𝐈
𝜔𝑖1

6

AUC [-]𝑧𝑖

𝑺 𝑡 = (𝑆1 𝑡 , ⋯ , 𝑆𝑛 𝑡 )𝑰 𝑡 = (𝐼1 𝑡 , ⋯ , 𝐼𝑚 𝑡 )AUC [-]

𝑆𝑖 𝑡 + 137𝑐𝑖11 ⋯ 𝑐𝑖1𝑛⋮⋱⋮𝑪𝑖 =𝑐𝑖𝐼𝑁1 ⋯ 𝑐𝑖𝐼𝑁𝑛𝑑𝑖11 ⋯ 𝑑𝑖1𝑚⋮⋱⋮𝑫𝑖 =𝑑𝑖𝐼𝑁𝑚 ⋯ 𝑑𝑖𝐼𝑁𝑚

Inside brain state
Training

8Target nodeCNode signal stateC

True Positive RateBB

1

True Positive RateA

DLCM-GC can detect network connections from DCM
simulated BOLD signals as well as mvGC, etc.

400

600

800

Signal Length

1000

1200

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