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A Nonlinear Circuit Network Toward Brain Voxel Modeling
Takashi Matsubara†, Hiroyuki Torikai‡,
Tetuya Shimokawa∗, Kenji Leibnitz∗, and Ferdinand Peper∗
†Graduate School of Engineering Science, Osaka University,
1-3 Machikaneyama-cho, Toyonaka-shi, Osaka 560-8531, Japan.
‡Faculty of Computer Science and Engineering, Kyoto Sangyo University,
Motoyama, Kamigamo, Kita-ku, Kyoto-shi, Kyoto 603-8555, Japan.
* Center for Information and Neural Networks (CiNet),
National Institute of Information and Communications Technology, and Osaka University,
1-4 Yamadaoka, Suita, Osaka 565-0871, Japan.
E-mails: matubara@hopf.sys.es.osaka-u.ac.jp, torikai@cse.kyoto-su.ac.jp,
shimokawa@nict.go.jp, leibnitz@nict.go.jp, and peper@nict.go.jp.
Abstract—This paper presents a nonlinear dynam-
ical model for predicting activities of voxels in a human
primary visual cortex (V1); the activities are repre-
sented by blood-oxygen-level-dependent (BOLD) sig-
nals. The prediction performance is shown to be bet-
ter than those of other traditional major models, e.g.,
general linear model and multivariable autoregressive
model.
1. Introduction
Responses and interactions of human brain regions
have been investigated from various viewpoints. In-
vestigations of the responses and the interactions of
the brain regions contribute the more accurate map-
ping of the network topology and the elucidation of
the brain functions. The retinotopic organization is
mapping of visual inputs from the retina to responded
brain regions; the visual inputs are often flickering
checkerboard stimuli with different polar angles and
eccentricities [1–5]. The responses of the brain regions
are sampled by using functional magnetic resonance
imaging (fMRI), which detects the blood-oxygen-level-
dependent (BOLD) signals; the scanned spatial unit
is called “voxel”. Traditionally, the responded vox-
els have been identified by using cross correlation [2]
and covariance [3,4] between the visual inputs and the
BOLD signals. In addition, the BOLD signals are pre-
dicted by general linear model (GLM) [5, 6]; this is on
the assumption that the BOLD signals are calculated
by summation of the visual inputs and do not have any
dynamics. On the other hand, the brain voxel dynam-
ics and their interactions represented by BOLD signals
are often modeled in a little more complicated ways:
multivariable autoregressive model (MARM) [7–9] and
dynamic causal model (DCM) [10, 11]. The linear and
bilinear models are simple and versatile models, but
not specifically designed for the brain voxel dynamics.
More appropriate and powerful model is needed. This
paper focuses on the dynamic responses of voxels in
the human primary visual cortex (V1) to the visual
inputs and models them in a more sophisticated way.
2. Integrate and Oscillate Model
This section models the dynamic responses of vox-
els in the human primary visual cortex (V1) to the
visual inputs; the model is called “integrate and oscil-
late model” (IOM) hereafter. The visual inputs consist
of some types of flickering checkerboard stimuli, which
are indexed by i∈I≡ {0,1,...}. For simplicity, the
interactions between voxels are ignored and the voxel
dynamics are assumed to be affected only by the vi-
sual inputs. A BOLD signal is scanned with a period
of T, and the n-th scanned BOLD signal of a voxel
is expressed as x(n)∈R(n= 0,1,...). A predicted
value of the n-th scanned BOLD signal x(n) is denoted
by y(n) hereafter. In addition, each visual input iis
expressed as
Ii(n) =
1if the visual input iis
applied at the time n,
0 otherwise .
This paper presents the IOM described by the follow-
ing equation:
y(n) = v(n) + A(n) cos(θ(n)) + k.
v(n) is a major component, A(n) and θ(n) are an am-
plitude and an angle of a minor component, respec-
tively, and the dynamics of which are described by
v(n+ 1) = v(n) + bPi∈IaiIi(n+ 1) −v(n),
θ(n+ 1) = θ(n) + ω+ωvv(n),
A(n+ 1) = A0+Avv(n+ 1),
(1)
where the DC bias k, the coefficients b,ai(i∈I),
the angular velocity coefficient wv, the basic radius
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2013 International Symposium on Nonlinear Theory and its Applications
NOLTA2013, Santa Fe, USA, September 8-11, 2013
visual input i= 0 visual input i= 1 visual input i= 2
Figure 1: The three types of flickering checkerboard stimuli.
visual inputs
0 150
0
1
I0I1I2
0 150
0
1
I0I1I2
fMRI voxel BOLD signal
0 150
180
205
BOLD signal
0 150
180
205
BOLD signal
Integrate and Oscillate Model
(IOM)
0 150
180
205
BOLD signal
IOM
0 150
180
205
BOLD signal
IOM
General Linear Model
(GLM)
0 150
180
205
BOLD signal
GLM
0 150
180
205
BOLD signal
GLM
Multivariable Autoregressive Model
(MARM)
0 150
180
205
BOLD signal
MARM
0 150
180
205
BOLD signal
MARM
known dataset unknown dataset
Figure 2: The visual inputs, the BOLD signal x(n) of a V1 voxel, and the predicted BOLD signals y(n)of the
IOM, the GLM, and the MARM, from top to bottom. Each left (right) figure shows the known dataset (the
unknown dataset).
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A0, and the radius coefficient Avare parameters.
In dynamical system viewpoint, vis a slow major
stimulus-induced dynamics and θis a fast minor semi-
autonomous dynamics.
fMRI Scan Dataset and Parameter Fitting:
The scan dataset is sampled with the conditions as
provided below. The visual inputs are 375 second
time length and consist of the three types of flicker-
ing checkerboard stimuli shown in Fig. 1, which are
indexed by i∈I≡ {0,1,2}from left to right. The
BOLD signals x(n) are scanned by 3 T MRI (3 mm
thick) and the scan period is T= 2.5[sec], and thus,
each recorded time series x(n) contains 150 scans. The
MNI coordinate of the scan dataset is transformed to
the Talairach coordina te [12, 13] and each voxel is la-
beled by the Talairach client [13–15]. The visual inputs
and an example of the BOLD signal of a V1 voxel are
shown in the upper left figures in Fig. 2.
The parameters k,b,ai(i∈I), wv,A0, and Av, and
the initial states v(0) and θ(0) of the IOM are initial-
ized randomly and updated by a simple 2-opt search
algorithm for minimizing the root-mean-square error
(RMS). The center left figure in Fig. 2 also shows the
predicted BOLD signal y(n) of the IOM after param-
eter fitting.
Other Traditional Modelings: General linear
model (GLM) is expressed as the following equation:
y(n) = X
i∈I
aiIi(n) + k,
where the coefficients ai(i∈I) and the DC bias kare
parameters.
Multivariable autoregressive model (MARM) is ex-
pressed as the following equation:
y(n+ 1) =
p−1
X
j=0 ajy(n−j) + X
i∈I
bi,j Ii(n−j)!,
where the degree pof the MARM, and the coefficients
ajand bi,j (i∈I, j = 0,...,p−1) are parameters. In
this paper, p= 10 is used.
Performance Comparison: For testing the perfor-
mance of predicting an unknown dataset, another scan
dataset with the same conditions as above is provided;
the first 24 scans are used for estimating the internal
states (i.e., the initial states v(0) and θ(0) and the DC
bias k) and the other 126 scans are used for perfor-
mance test. The prediction performance is evaluated
by the average of the prediction errors (i.e., the RMS)
of 50 randomly chosen V1 voxels which have high co-
variances (r≥0.5) with the summation I=Pi∈IIiof
the visual inputs. Fig. 2 shows the BOLD signal x(n)
Table 1: BOLD Signal Prediction Performance
Model Average RMS for
known dataset unknown dataset
IOM 1.90 2.45
GLM 2.29 3.78
MARM 1.23 3.21
of the unknown scan dataset, and the predicted BOLD
signals y(n) of the IOM, the GLM, and the MARM.
The average RMS are summarized in Table 1, which
implies that the IOM is the best way for predicting
the BOLD signals of V1 voxels.
3. Conclusion
This paper has presented a novel model “integrate
and oscillate model” (IOM), which predicts BOLD sig-
nals of V1 voxels responded to visual inputs. The pre-
diction error of the IOM has been shown to be small
with compared to those for the other major BOLD
signal prediction models: general linear model and
multivariable autoregressive model. That is, the IOM
has been confirmed to be a better way for predicting
BOLD signals of V1 voxels. Future work includes: (a)
modeling the interactions between brain voxels, (b) a
theoretical analysis of the IOM, (c) prediction perfor-
mance testing on the BOLD signals responded by more
complicated visual inputs, and (d) prediction perfor-
mance comparison with the other major BOLD signal
prediction models, e.g., DCM.
The authors would like to thank Yusuke Morito
of National Institute of Information and Communica-
tions Technology for providing BOLD signal data. The
authors would like to also thank Professor Toshimitsu
Ushio of Osaka University for helpful discussions. This
work was partially supported by the Support Cen-
ter for Advanced Telecommunications Technology Re-
search (SCAT), Grant-in-Aid for JSPS Fellows, KAK-
ENHI (24700225), the Telecommunications Advance-
ment Foundation (TAF), and Toyota Riken Scholar.
References
[1] R. B. Tootell et al., “Functional analysis
of primary visual cortex (V1) in humans.”
Proceedings of the National Academy of Sciences
of the United States of America, vol. 95, no. 3,
pp. 811–7, 1998.
[2] W. Chen et al., “Mapping of lateral geniculate
nucleus activation during visual stimulation in
human brain using fMRI.” Magnetic Resonance
in Medicine, vol. 39, no. 1, pp. 89–96, 1998.
- 423 -
[3] K. A. Schneider, M. C. Richter, and S. Kastner,
“Retinotopic organization and functional subdi-
visions of the human lateral geniculate nucleus:
a high-resolution functional magnetic resonance
imaging study.” The Journal of Neuroscience,
vol. 24, no. 41, pp. 8975–85, 2004.
[4] K. A. Schneider and S. Kastner, “Visual
responses of the human superior colliculus: a
high-resolution functional magnetic resonance
imaging study.” Journal of Neurophysiology,
vol. 94, no. 4, pp. 2491–503, 2005.
[5] D. Bressler, N. Spotswood, and D. Whitney,
“Negative BOLD fMRI response in the visual cor-
tex carries precise stimulus-specific information.”
PloS ONE, vol. 2, no. 5, p. e410, 2007.
[6] K. Friston, A. Holmes, and J. Poline, “Analysis of
fMRI time-series revisited,” NeuroImage, vol. 2,
no. 1, pp. 45—-53, 1995.
[7] R. Goebel et al., “Investigating directed cortical
interactions in time-resolved fMRI data using
vector autoregressive modeling and Granger
causality mapping,” Magnetic Resonance Imag-
ing, vol. 21, no. 10, pp. 1251–1261, 2003.
[8] A. Roebroeck, E. Formisano, and R. Goebel,
“Mapping directed influence over the brain
using Granger causality and fMRI.” NeuroImage,
vol. 25, no. 1, pp. 230–42, 2005.
[9] D. Marinazzo et al., “Nonlinear connectivity by
Granger causality.” NeuroImage, vol. 58, no. 2,
pp. 330–8, 2011.
[10] K. Friston, L. Harrison, and W. Penny, “Dynamic
causal modelling,” NeuroImage, vol. 19, no. 4,
pp. 1273–1302, 2003.
[11] S. M. Smith et al., “Network modelling methods
for FMRI.” NeuroImage, vol. 54, no. 2, pp.
875–91, 2011.
[12] J. Talairach and P. Tournoux, Co-Planar
Stereotaxic Atlas of the Human Brain: 3-D
Proportional System: An Approach to Cerebral
Imaging. Thieme, 1988.
[13] J. L. Lancaster et al., “Bias between MNI and
Talairach coordinates analyzed using the ICBM-
152 brain template.” Human Brain Mapping,
vol. 28, no. 11, pp. 1194–205, 2007.
[14] “talairach.org.” url: http://www.talairach.org/
[15] J. L. Lancaster et al., “Automated Talairach
atlas labels for functional brain mapping.”
Human Brain Mapping, vol. 10, no. 3, pp.
120–31, 2000.
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