a state-space model of the hemodynamic approach: … · a state-space model of the hemodynamic...
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NeuroImage 21 (2004) 547–567
A state-space model of the hemodynamic approach:
nonlinear filtering of BOLD signals
Jorge J. Riera,a,* Jobu Watanabe,a Iwata Kazuki,a Miura Naoki,a
Eduardo Aubert,b Tohru Ozaki,c and Ryuta Kawashimaa
aAdvanced Science and Technology of Materials, NICHe, Tohoku University, Sendai 980-8579, JapanbNeurophysics Department, Cuban Neuroscience Center, CNIC, CubacThe Institute of Statistical Mathematics of Tokyo, Japan
Received 13 May 2003; revised 11 September 2003; accepted 25 September 2003
In this paper, a new procedure is presented which allows the estimation
of the states and parameters of the hemodynamic approach from blood
oxygenation level dependent (BOLD) responses. The proposed method
constitutes an alternative to the recently proposed Friston [Neuroimage
16 (2002) 513] method and has some advantages over it. The procedure
is based on recent groundbreaking time series analysis techniques that
have been, in this case, adopted to characterize hemodynamic
responses in functional magnetic resonance imaging (fMRI). This
work represents a fundamental improvement over existing approaches
to system identification using nonlinear hemodynamic models and is
important for three reasons. First, our model includes physiological
noise. Previous models have been based upon ordinary differential
equations that only allow for noise or error to enter at the level of
observation. Secondly, by using the innovation method and the local
linearization filter, not only the parameters, but also the underlying
states of the system generating responses can be estimated. These states
can include things like a flow-inducing signal triggered by neuronal
activation, de-oxyhemoglobine, cerebral blood flow and volume.
Finally, radial basis functions have been introduced as a parametric
model to represent arbitrary temporal input sequences in the
hemodynamic approach, which could be essential to understanding
those brain areas indirectly related to the stimulus. Hence, thirdly, by
inferring about the radial basis parameters, we are able to perform a
blind deconvolution, which permits both the reconstruction of the
dynamics of the most likely hemodynamic states and also, to implicitly
reconstruct the underlying synaptic dynamics, induced experimentally,
which caused these states variations. From this study, we conclude that
in spite of the utility of the standard discrete convolution approach
used in statistical parametric maps (SPM), nonlinear BOLD phenom-
ena and unspecific input temporal sequences must be included in the
fMRI analysis.
D 2003 Elsevier Inc. All rights reserved.
Keywords: Hemodynamic; Synaptic dynamics; State-space model
1053-8119/$ - see front matter D 2003 Elsevier Inc. All rights reserved.
doi:10.1016/j.neuroimage.2003.09.052
* Corresponding author. Advanced Science and Technology of
Materials, NICHe, Tohoku University, Aoba 10, Aramaki, Aobaku, Sendai
980-8579, Japan. Fax: +81-22-217-4088.
E-mail addresses: [email protected], [email protected]
(J.J. Riera).
Available online on ScienceDirect (www.sciencedirect.com.)
Introduction
The elucidation of the temporal dynamic of neuronal activation
in specific brain areas, associated with particular motor, sensorial
or cognitive events, constitutes a current problem of great interest
in neuroimaging. Considerable progress has been possible due to
the use of the functional magnetic resonance imaging (fMRI)
modality, which permits the exploration of hemodynamic changes
with excellent spatial resolution using the blood oxygenation level
dependent (BOLD) signal. A standard way to analyze the BOLD
response, which has almost become the norm since the statistical
parametric maps (SPM) came into use, consists of approaching the
whole brain as a black box characterized by its transfer response
function (Hemodynamic Response Function, HRF). Based on this
construct, a specific stimulus sequence, used to define the exper-
imental design matrix, discretely convolutes with the HRF to
produce predictors of the BOLD response. This model assumes
a stationary and linear correspondence between the BOLD signal
and the underlying physiological processes. There are ever more
reasons to suspect that HRFs vary from subject to subject, from
region to region and from task to task (Aguirre et al., 1998;
Buckner et al., 1998; Duann et al., 2002). Hence, in the last few
years, a great deal of attention has been paid to the development of
methods for the estimation of the voxel-dependent HRF from
BOLD data from a priori specification of the design matrix for
particular experimental paradigms, an identification problem that is
badly conditioned. These methods range from the nonlinear basis
functions pioneers used (Friston et al., 1994, 1998; Lange and
Zeger, 1997; Rajapakse et al., 1998) to the most recent ones that
apply to the nonparametric bayessian framework (Burock and
Dale, 2000; Goutte et al., 2000; Marrelec et al., 2003).
However, it has been recently recognized that hemodynamic
changes result from a complex nonlinear dynamical system very
sensitive to electrophysiological episodes (i.e., neuronal synaptic
activity), which are of special interest in making inferences from
BOLD signals. Many previous works have shown the presence of
nonlinear phenomena in the fMRI signals. Examples are the
nonlinear dose– responses curves with prominent hysteresis
reported by Berns et al. (1999) during a finger opposition task;
hemodynamic refractoriness (Friston et al., 1998; Huettel and
J.J. Riera et al. / NeuroImage 21 (2004) 547–567548
McCarthy, 2000); and the spatial heterogeneity of the nonlinear
characteristics of BOLD signals (Birn et al., 2001; Huettel and
McCarthy, 2001). In addition to the multiphasic nature of the
BOLD signal, it is well known that fMRI systems have very
limited sampling rates, which limits its ability to determine the time
varying course of synaptic activation that occurs on a different
temporal scale.
Fortunately, the physiological mechanisms underlying the rela-
tionship between synaptic activation and vascular/metabolic con-
trolling systems have been widely reported (Iadecola, 2002;
Magistretti and Pellerin, 1999). Hence, some authors have attemp-
ted to model the BOLD signal at the macroscopic level by
differential equations systems, relating the hemodynamical varia-
tions to relative changes in a set of variables with physiological
sense. The Balloon approach, based on the mechanically compel-
ling model of an expandable venous compartment (Buxton et al.,
1998) and the standard Windkessel theory (Mandeville et al.,
1999), has become an established idea. Friston et al. (2000) have
extended the Balloon approach, named in this paper simply the
hemodynamic approach, to include interrelationships between
physiological (i.e., neuronal synaptic activity and a flow-inducing
signal) and hemodynamic processes. In the hemodynamic ap-
proach, a set of four nonlinear and nonautonomous ordinary
differential equations governs the dynamics of the intrinsic varia-
bles: the flow-inducing signal, the Cerebral Blood Flow (CBF), the
Cerebral Blood Volume (CBV) and the total de-oxyhemoglobine
(dHb). This dynamic system is, in effect, nonautonomous due to
the time-varying dependence of the synaptic activity, which will be
referred to henceforth as the input sequence.
Though this theoretical model could have a tremendous impact
on fMRI analysis, there is little work done in fitting and validating
it from actual data. The most important attempt to date has been
presented by Friston (2002) using a Volterra series expansion to
capture nonlinear effects on the output of the model produced by
predefined input sequences. In that work, the Volterra kernels were
explicitly computed for the hemodynamic approach, after a set of
assumptions that forced the original deterministic and continuous
differential equations system to have a bilinear form. An EM
implementation of a Gauss–Newton search method, in the context
of maximum a posteriori mode estimation, was used to determine
the hemodynamic parameters. Even though this methodology
theoretically allows the computation of Volterra kernels of any
order, in practice, a finite truncation of the series must be carried
out, limiting the representation of higher order nonlinear dynamics.
In this work, Friston (2002) assumes that the uncertainty of the
model originated solely from instrumental errors; and considered
the particular case of a known input sequence (directly associated
with the stimulus), which differs for each of the brain voxels only
in terms of the scaling factor (i.e., the neuronal efficacies).
Therefore, the results obtained by this method depend heavily
upon prior knowledge about the experimental design and those
responses to neuronal events not under experimental control,
which, therefore, cannot be detected. Even so, the most significant
disadvantage of this method is the properness of the quasi-
analytical integration scheme proposed to evaluate numerically
the partial derivatives J = Bh(gh/y(n))/Bh and also to reconstruct the
temporal dynamics of any of the intrinsic variables in the hemo-
dynamic approach, which are of special interest because of their
imminent physiological interpretability. The idea as originally
proposed in Friston (2002) for integrating ordinary differential
equations needs to be evaluated in terms of numerical performance.
A basic obstacle for overcoming the drawbacks mentioned thus
far, seems to be the general difficulty of translating the continuous
differential equations systems into discrete time series models that
also include inherent physiological noise contributions. A
‘‘bridge’’ between Stochastic Differential Equations (SDEs) and
time series models, originally inspired by the pioneering work of
Ozaki (1985), has been recently formalized into a theoretical basis
by Biscay et al. (1996). The general idea is based on the local
linearization (LL) method, using random measures to integrate
SDE near discretely and regularly distributed time instants, and
assuming a local piecewise linearity. Therefore, the LL formalism
permits the conversion of a SDE system into a state-space equation
with a background Gaussian noise, where a stable reconstruction of
the trajectories of the state-space variables is obtained by a one-
step straightforward prediction (i.e., a nonlinear autoregressive
model). The resultant theory permits the generalization of the LL
methodology to the case of nonlinear and nonautonomous systems
with an additive/multiplicative Wiener force driving the dynamics
(Jimenez and Ozaki, 2003).
Moreover, in a more general state-space model, the actual data
relate to a subset of state-space variables by an observation
equation, which, in the least desirable case, as is the case of BOLD
signals, is nonlinear and largely polluted by instrumental errors,
with some missing values as well (i.e., the integrating step is
smaller than the sampling rate). In general, the estimation of the
state of a continuous SDE system from noisy discrete observations
can be performed based on the nonlinear filter theory, initially
formulated (Kalman filter theory) to provide a sequential and
computationally efficient solution to the linear filtering and pre-
diction problems. However, finding the optimal nonlinear system
identification method (i.e., the estimation of the model’s parame-
ters and the trajectories of nonobservable states) remains an active
area of research. A viable approach of dealing with such nonlinear
identification problems involves the use of the LL filter and the
innovation method (Ozaki, 1992, 1993, 1994). The LL filter and
the innovation method are essentially a sizeable extension of the
basic Kalman filter theory and a quasi-likelihood method, respec-
tively. Recently, Jimenez and Ozaki (2003) have generalized the
original LL filter scheme in the context of nonlinear and nonau-
tonomous dynamical systems with an additive/multiplicative Wie-
ner process. It, combined with the innovation method, yields a
plausible optimization algorithm based on the maximization of the
likelihood of the innovational process (i.e., representing the data
not explained by the model) (Technical Report: Jimenez and Ozaki,
2002). Therefore, a test of Gaussianity (follows Kolmogorov–
Smirnov test) applied to the innovation process could be used as a
criterion for gauging how well the model fits the data.
In this paper, the hemodynamic approach was reformulated into a
state-space model by applying the LL methodology. This nonlinear
autoregressive model was used to simulate the trajectories of the
state-space variables, and hence, the BOLD signal. The observation
equation was assumed to be as was originally presented by Buxton et
al. (1998), with an additional Gaussian instrumental error. In the
context of the LL filter theory, the equations for the evolution of the
conditional mean and the covariance matrix of the state-space
variables were properly defined for the hemodynamic approach.
Equations for differences, which depend on the LL filter gain, were
also reported and during the estimation procedure, they were only
evaluated on those time instants where data were available, as
defined in the LL filtering strategy for the case of missing values.
Additionally, the use of regular spaced Radial Basis Functions
J.J. Riera et al. / NeuroImage 21 (2004) 547–567 549
(RBFs) was proposed for alternatively considering the arbitrary
dynamics in the input sequence of the hemodynamic approach.
Our results are presented in three parts. In the first part,
simulated BOLD data are used to demonstrate the nonlinearity of
the hemodynamic approach. This simply involves integrating the
SDE via the LL method to produce synthetic BOLD responses to
a variety of experimental inputs. To demonstrate the nonlinearity,
those responses are compared with the equivalent output based
upon a discrete convolution approach. The discrete convolution
approach was obtained by convolving the input sequence with
the first-order Volterra kernel implied by the model parameters
used in the full model. In this section, we are not trying to
estimate anything but to simply demonstrate, phenomenally, the
nonlinear behavior of the system and how noise at either the
physiological or instrumental level can determine the observed
response. In the second section, we move on to the estimation of
both the model parameters and the hidden states-space variables
from sparsely simulated BOLD data using two distinctive input
cases: time-locked infrequent single events and a prolonged train
of frequent events. This section is presented to illustrate the
robustness of the estimation given that parameters are predefined;
consequently, the true evolution of states-space variables can be
predicted.
In the final and third section, we apply our procedure to real
fMRI data obtained from a standard motor task (i.e., closing/
opening the right hand) carried out by five healthy subjects during
a block design paradigm. In this case, the stimulus sequence was
assumed to have voxel-dependent amplitudes, allowing the pre-
dictors to be a priori defined (i.e., via standard SPM analysis) and
Fig. 1. The diagram of the hemodynamic approach. A train of action potentials ar
release to the synaptic clefts (neuronal synaptic activity u(t)). As a conseque
dendrites of post-synaptic neurons due to the activation of (chemically gated chan
the cellular membrane. The neurotransmitters are rapidly terminated by a re-u
gradients are restored by ATPase transport mechanisms. Hence, a metabolic and o
Some sort of signal-inducing sub-system directly related to u(t) will cause an inc
now recognized that nitric oxide (NO) plays an important role to re-activate th
variables associated with the BOLD signal, change dynamically, obeying mecha
extraction mechanisms, respectively.
was also assumed to be proportional to the synaptic activation.
Therefore, the model was not a well suited to those brain areas
considered likely to be removed from the neuronal synaptic
activation directly related to that stimulus sequence. For those
areas, contributions coming from other functional regions may well
be important, and hence, should also be considered in the input of
the hemodynamic approach. The local dependent amplitudes of the
synaptic activity (via RBFs) were estimated from both simulated
BOLD time series and also actual recordings obtained from a
champion subject performing a motor experiment with an event-
related paradigm. The utility of being able to make inferences
about the underlying synaptic activity that sometimes does, and at
other times does not, conform to the experimental manipulations,
was demonstrated.
Methods
Experimental tasks
Five right-handed, normal volunteers (three males and two
females) aged 24–37 years (32 F 5 years (mean F SD)) were
used in this study. The subject’s handednesses were evaluated by
the Edinburgh questionnaire (Oldfield, 1971). In accordance with
the guidelines approved by the Declaration of Human Rights at
Helsinki in 1964, the subjects were requested to give their written
informed consent. The detailed brain anatomy of each subject was
determined using a spoiled gradient-echo sequence (recovery time
TR = 9.7 ms, echo time TE = 4 ms, flip angle FA = 12j) with a 1.5-
riving to the pre-synaptic terminal buttons induces the neurotransmitters to
nce, excitatory and/or inhibitory electric potentials are originated in the
nels) ionic currents ICurrent that create an electrochemical disequilibrium in
ptake mechanism in the astrocytes processes, while the electrochemical
xygen demand will appear in the neighborhood of the activated brain area.
rease of blood flow in the arterioles before enter to the capillary bed. It is
is sub-system. The cerebral blood volume and total de-oxyhemoglobine,
nical deformation laws for expandable venous compartments and oxygen
J.J. Riera et al. / NeuroImage 21 (2004) 547–567550
T scanner (Siemens Vision, Erlangen, Germany) consisting of 96
slices with a voxel size of 1.25 � 0.9 � 1.92 mm.
Subjects were supine on the MRI scanner bed during the
fMRI sessions. The subjects were asked by visual cues to
perform right hand movement tasks. Instructions to start and
stop the task and visual cues were video projected on a screen
positioned on the head coil in the MRI gantry during the fMRI
study. For each set of measurements, the data from the first 10
scans were discarded to eliminate transient measurements taken
before the achievement of dynamic equilibrium, and the remain-
ing scans were submitted for analysis. All subjects performed the
motor tasks after a period of 60 s rest (i.e., see below for a
description of the blocked design and event-related paradigms).
During the moving condition, a small circle at the center of the
screen was used as a cue for indicating to the subject to close its
hand and a cross indicated to open it. The cues always lasted for
200 ms at this condition. At resting condition, a fixation cross
Fig. 2. The time series of the state-space variables of the hemodynamic approach f
by another with a delay. (a) Shows the BOLD responses to both events, when pre
Shows the residue, which remains after removing the sum of the BOLD responses
stimuli. The shaded area can be used as an indicator of the size of the refractorin
different duration (axis) are used to represent the non-linear dose– response relati
was displayed in the center of the screen. Subjects were requested
to keep their hands open whenever they were resting and not to
move their bodies except for executing the closing/opening
movements of the right hand. All movements of the right hand
and of the whole body were monitored using a video tape
recorder.
Each subject’s head was fixed using ear fixation blocks. The
inter-scan interval was TR = 1.2 s. In each scan, eight horizontal
slices of T2-weighted gradient-echo echo-planer images (TE = 60
ms, FA = 90j) covering the whole brain were collected with a
voxel size of 3 � 3 mm in plane, and 10 mm thick with a 5-mm
gap. The individual fMRI images were realigned to remove
movement-related artifacts, and the slice timing was adjusted to
that of the 5th slice. The T1 anatomical and fMRI images were co-
registered and spatially normalized to the Talairach coordinate
system using both linear and nonlinear parameters. The fMRI
images were processed with the help of the statistical parametric
or a stimulus, consisting of a single event of a very short duration followed
sented separately (continuous thin lines) and consecutively (dotted line). (b)
to the separate stimuli from the BOLD response obtained by the consecutive
ess effect (c) as a function of the timing of the second event. Stimuli with
onship (d).
Fig. 2 (continued).
J.J. Riera et al. / NeuroImage 21 (2004) 547–567 551
mapping software (SPM99 toolbox, Welcome Department of
Cognitive Neurology, London, UK).
Block design paradigm
The five subjects performed a task consisting of nine blocks
of 60 s moving conditions and a 60 s resting condition. During
the moving condition, the circles/cross cues appeared at regularly
spaced intervals at a frequency of 1.6 Hz (see Fig. 4 top and a
more detailed view at Fig. 6, top). Note that a Gaussian of around
200 ms of duration was used to represent the synaptic activity
originating from a single moving episode. Each measured session
lasted 19 min. A mean image was created for each subject. An
appropriate design matrix was obtained by using the discrete
convolution approach that involves HRF and the stimulus se-
quence, as provided by the general linear model in the SPM99
toolbox. The theory of Gaussian fields was used to obtain the
significance level used in a statistical SPM99 t test (P < 0.05,
corrected for multiple comparison). The most significant brain
areas were obtained after clustering the SPM99 t test. The
Talairach coordinates of hot-spots in each of these brain areas
were reported. A time series of 950 scans was obtained from each
of the hot-spots of all the subjects. The parameters of the
hemodynamic approach were fitted from the various time series
using the methodology detailed in the next sub-section. The
extent to which hemodynamic approach fit the data were evalu-
ated by testing the Gaussian distributions of the histogram for the
innovation process using the Kolmogorov–Smirnov test. The
subject showing a higher significance level in that test for M1
area was used as the champion data throughout the paper.
Event-related paradigm
The champion subject was asked to perform a slightly more
complicated motor task, based on an event-related paradigm. For
a total of 20 times, this subject repeats the task of closing/
opening the right hand, but now obeying cues of circles/crosses at
an irregular time distribution during the moving condition,
J.J. Riera et al. / NeuroImage 21 (2004) 547–567552
followed by 24 s of rest (see Fig. 8, top). In the same way,
Gaussians were used to represent the synaptic activity emerging
during movements. The measured session lasted 15 min. A time
series of 750 scans for the hot-spot of the M1 brain area was
obtained from this study. This time series was used to estimate
not only the parameters of the hemodynamic approach but also to
reconstruct the temporal sequence of its input for this particular
Fig. 3. The effects that nonlinear properties of the hemodynamic approach produc
(a); the auto-correlation functions (b); and the 3rd-order cumulants (c) are presen
subject. A sub-section below will give details about how to use
RBFs for that purpose.
Hemodynamic approach
The hemodynamic approach can be mathematically formulated
by a nonlinear and nonautonomous ordinary differential equations
e on a white noise input are explored. The time varying mean and variance
ted for both approaches.
J.J. Riera et al. / NeuroImage 21 (2004) 547–567 553
system that relates several variables, with physiological meaning,
to neuronal synaptic activity u (t) (see Fig. 1). The time varying
vector x (t) = (x1 (t), x2 (t), x3 (t), x4 (t))t summarizes the dynamic of
the system, where the normalized intrinsic variables x1 (t), x2 (t),
x3 (t) and x4 (t) are the flow-inducing signal, CBF, CBV and dHb,
respectively. This approach can be generalized to include a scalar
Wiener process x (t) representing an additive physiological system
noise, with a vector g = {gi} defining the strength of randomness
for each variable (i.e., Eq. (1)).
dxðtÞ ¼ f ðx; uÞdt þ gdxðtÞ ð1Þ
The vector function f (x, u) is defined by the equations (see Friston
et al., 2000 for excellent discussions about these equations):
f1ðx; uÞ ¼ euðtÞ � a1x1ðtÞ � a2ðx2ðtÞ � 1Þ ð2Þ
Fig. 4. Time series of the state-space variables of hemodynamic approach for the c
(1) is clearly shown.
f2ðx; uÞ ¼ x1ðtÞ ð3Þ
f3ðx; uÞ ¼1
a3ðx2ðtÞ � x3ðtÞ
1a4 Þ ð4Þ
f4ðx; uÞ ¼1
a3
x2ðtÞa5
1� ð1� a5Þ1
x2ðtÞh i
� x4ðtÞx3ðtÞ1�a4a4
� �ð5Þ
Eq. (1) defines a nonlinear SDE, which is also considered nonau-
tonomous because of the explicit temporal dependence given by
the input function u(t), only affecting the first equation of the
system. The vector x (0) = (0,1,1,1)t represents the initial values for
the intrinsic variables. The parameter e can be associated with the
neuronal efficacy. Note the fact that there is an extreme singularity
when x2!0. Despite this limited case being highly improbable due
to the normalization of the variables, the mentioned limit exists and
ase of a prolonged train of frequent events. The saturation effect of the SDE
Fig. 5. The BOLD signals obtained for a prolonged train of frequent events. The BOLD responses obtained from using discrete convolution and hemodynamic
approaches overlap. The similarity between the BOLD responses for both approaches justifies the use of the linear model for this particular paradigm.
J.J. Riera et al. / NeuroImage 21 (2004) 547–567554
it can be theoretically evaluated by the expression:
limx2!0
f4ðx; uÞ ¼ � x4ðtÞx3ðtÞ1�a4a4
a3, for the parameters 0V a5 < 1, which
correspond to the actual physiological range. The 2nd and 3rd
equations in Eq. (1) were originally reported by Buxton et al.
(1998) and Mandeville et al. (1999). The use of a Wiener process
in Eq. (1) originates from the general idea that Markov’s sequences
represent fluctuations of physical magnitudes in biophysical sys-
tems very well (i.e., in some cases associated with a Brownian
motion). Any Markov’s process with a continuous path has a
Langevan type representation where a differential equation prop-
erly describes the dynamics and the driving force of the system is a
random Gaussian white noise (Feller, 1971).
The parameters {a1, a2, : : :, a5} can be interpreted in terms of
the physical properties of the vascular system. The two first SDEs
in Eq. (1) describe electro/vascular coupling, which physically
represent a damped oscillator with an external force eu (t) and a
resonance frequency of - ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 þ 4a2
p� �=4p. By definition a1 =
1/ss, where ss is the time constant of signal decay, or the
elimination component; and a2 = 1/sf is defined in terms of the
time constant of the feedback auto-regulatory mechanism sf. Theparameter a3 = s0 is the mean transit time in the post-capillary
venous compartment, and can be calculated by the ratio of the
volume of the compartment and the blood flow getting into it
during baseline condition. This parameter has been also interpreted
for steady-state conditions as the time constant of an equivalent
analogical RC circuit. Therefore, changes in a3 will produce
alterations in the time scale of the BOLD signal, slowing down
its dynamic with respect to the CBF when a3 is increased. It is
generally believed that the mismatch between CBF and CBV is due
to variations in this parameter. The Windkessel theory, based on
mechanical pressure compensation laws, establishes the interrela-
tionship between outflow and volume dynamics during post-
capillary venous inflation, where the evolution of the mean transit
time determines the underlying physiology. The parameter a4 = a is
the stiffness exponent or Grubb’s parameter 1/a4 = c + b. Theparameters c = 2 and b > 1 represent the laminar flow and
diminished volume reserve at high pressure, respectively. This
parameter is closely related to the flow–volume relationship;
hence, it determines the degree of nonlinearity of the BOLD
response. The CBF and CVB will change dynamically following
SDE (1); therefore this parameter will also vary. However, in a
steady-state condition the CBF and CBV both appeared to reach a
plateau level. The value reported from animal studies a4 = 0.38 F0.10 (Mandeville et al., 1999) seems to be very stable during
steady-state stimulation condition. This result is congruent to that
obtained previously by Grubb et al. (1974) using PET measure-
ments during hypercapnia. The resting net oxygen extraction
fraction by the capillary bed is represented by the parameter a5 =
E0. The values reported for this parameter are in the range of 0.20
V a5 V 0.55 (Friston et al., 2000). The BOLD response will
depend considerably on a5 in the case of transient stimulus. For
instance, the brief initial dip, rarely observed in BOLD signals, is
attributed to a very light increase in this parameter. However, from
simulations performed by our group, we can assert that increasing
this parameter in a block design paradigm only produces a scaling
factor effect. In this paper, we assumed that parameters a4 and a5are known. Friston et al. (2000) have experimentally estimated
these parameters using the Volterra kernels associated with the
hemodynamic approach in an experiment in which a subject has to
listen to monosyllabic or bisyllabic concrete nouns. They reported
that the histograms of the parameters, for values obtained in
different voxels, show Gaussian distributions with mean values
(0.65, 0.40, 0.98, 0.33, 0.34).
State-space model, LL filter and the innovation method
An analytic solution of SDE (1) is not available as a result of
the highly nonlinear dependences of the intrinsic variables. There-
fore, a numerical method to approximate its solution is desirable.
Jimenez et al. (1999) have extended the original LL integrator to
the general case of multivariate, nonlinear and nonautonomous
SDE with additive system noise. This method, which will be used
in this paper, can be summarized in the following steps: (a) the
local linearization of the drift coefficient in each interval [t, t + D]
by a truncated Ito–Taylor expansion of f (x, u), (b) the analytic
computation of the solution of the resulting linear SDE and (c) the
approximation of Ito’s integral involved in the solution obtained in
step (b) by the composite trapezoidal rule. It was demonstrated
Fig. 10. Glass images of the SPM99 t test are plotted for the champion subject (top). The most significant brain regions M1, Cerebellum, CMA and SMA are labeled. 3D views of the images overlap with the
individual T1 anatomical image from different slices (bottom-right). The time course of the 1st eigenvalue for the hot-spot of the M1 area shows a drift component (bottom-left).
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using simulations with different integrating steps that the accuracy
and order of convergence of this numerical method are better than
those shown by the other feasible schemes (Biscay et al., 1996;
Jimenez et al., 1999). The value D = 0.1 s is an acceptable
integrating step for SDE (1). The SDE (1) after applying the LL
method, results in the state-space equation (i.e., representing a
nonlinear autoregressive model):
xtþD ¼ xt þ r0ðJ xf ðxt; utÞ;DÞf ðxt; utÞ þ ðDr0ðJ xf ðxt ; utÞ;DÞ
� r1ðJ xf ðxt; utÞ;DÞÞJtf ðxt ; utÞ þ nt ð6Þ
The matrix functions rnðM ; aÞ ¼ ma
0eMuundu can be computed
using the Lemma 1 presented by van Loan (1978) (see Jimenez,
2002 for more details). The mean and covariance of stochastic
process nt were discussed in Jimenez et al. (1998).
Fig. 6. The superposition of trajectories of the state-space variables reconstructed f
(dots) and the true parameters used in the simulation (narrow lines). A detailed de
space variables were x0 = (0,1,1,1)t.
The magnitudes Jfx (x, u) and Jf
t (x, u) are the Jaccobians matrix
(with respect to the state-space variables x) and the derivative (with
respect to time t) of function f (x, u), respectively.
J xf ðx; uÞ¼
�a1 �a2 0 0
1 0 0 0
01
a3� x
1�a4a4
3
a3a40
01
a3a51� 1� a5ð Þ
1x2 þ log 1� a5ð Þ 1� a5ð Þ
1x2
x2
" #� 1� a4ð Þx4x
1�2a4a4
3
a3a4� x
1�a4a4
3
a3
8>>>>>>>>>>>>><>>>>>>>>>>>>>:
9>>>>>>>>>>>>>=>>>>>>>>>>>>>;
Note that same indetermination for Jfx (x, u) exists when
x2!0. In this particular case, the limit can be theoretically
evaluated by: lim J xf ðx; uÞ ¼ 1=ða3a5Þ. The vector function
x2!0 4;2rom the LL method in the case of using both: the estimated state parameters
scription of a single block is shown on the top. The initial values for state-
J.J. Riera et al. / NeuroImage 21 (2004) 547–567 557
Jtf ðx; uÞ ¼
eduðtÞdt
; 0; 0; 0� �t
can be evaluated without difficulty if it
is of differentiable input functions.
Eq. (2) describes the discrete dynamic of the state-space
variables xt. However, the state-space variables are related to the
BOLD signal ys by a nonlinear observation equation ys = h (xts) + es(s = 1, : : :, NS, where NS is the number of scans). In general, the
instrumental error is assumed i.i.d. es = N (0, r2). Buxton et al.
(1998) deduced a direct relationship between the BOLD signals
and two of the intrinsic variables (i.e., CBV x3 and dHb x4), given
by the function:
hðxÞ ¼ V0 k1ð1�x4Þþk21�x4
x3
� �þk3ð1� x3Þ
� �ð7Þ
The constant V0 represents the resting blood volume fraction. The
parameters k1, k2 and k3 are dimensionless and correspond to how
the (extra/intra) vascular systems and the changing balance effect
contribute to the BOLD response, respectively. Their values
depend on the characteristic of the fMRI system. In the particular
case of a 1.5-T scanner with TE around 40 ms, they can be
evaluated empirically by the expressions: k1 i 7a5, k2 i 2 and
k3 i 2a5 � 0.2 (Boxerman et al., 1995; Ogawa et al., 1993).
Fig. 7. The simulated BOLD signal from the hemodynamic approach after model i
the innovation process (top). The observations of the BOLD signal are given at a v
innovation process (left) and the surface of the optimization functional obtained fr
and bottom-left are obtained using the parameters corresponding to the minimum
The observation of the BOLD signal normally takes place
with a very low sampling rate. Assuming that the total time
interval is T (in s), then NS = T/TR. Note that NS b N, where N
= T/D represents the number of time instants in which the state-
space variable xmust be evaluated. In general, N = (TR/D) NS; and
in the case of this particular fMRI study, N = 12 NS (i.e., TR = 1.2 s
and D = 0.1 s). However, the reconstruction of trajectories of
the state-space variables, from the poorly observed BOLD
signal ys additionally contaminated by normally distributed ins-
trumental noise es, can be performed in the context of recursive
filters.
It is well known that classical recursive filters are computa-
tionally unstable in the case of diverse types of nonlinear
problems. This instability is obvious as a numerical explosion
in the computation of the prediction or filtering estimates at
certain instants of time. To overcome the computational instabil-
ity of the classical recursive filters, in Ozaki (1993), an alterna-
tive recursive nonlinear filter was introduced, being the use of LL
method to discretize the prediction equation the essential approx-
imation. This recursive nonlinear filter has been named the LL
filter. The high stability of LL filters is the key to their success in
the solution of nonlinear filtering problems for which other
filtering algorithms fail (Ozaki, 1993, Ozaki et al., 2000).
dentification by using the non-linear recursive algorithm is superimposed on
ery discrete set of time instants (dots). On the bottom, the histogram of the
om varying two of the parameters (right) are shown. The graphs on the top
value of the optimization functional.
Fig. 8. The event related paradigm (top). The superposition of trajectories of the state-space variables reconstructed from the LL method in the case of using:
(a) the estimated state parameters (hence, input u(t)) (dots) and (b) the true state parameters, used in the simulation (thin lines). It was assumed u(t) = 0 as the
initial value for input temporal sequence in the optimization problem, corresponding with trajectories of state-space variables close to zero as shown in this
figure. x0 = (0,1,1,1)t.
J.J. Riera et al. / NeuroImage 21 (2004) 547–567558
Recently, in Jimenez and Ozaki (2003), the estimation of the
unobserved states has been extended to more general state-space
models, which has made it possible to apply the methodology to
the type of hemodynamic model considered in this paper. On the
other hand, the estimation of a set of state parameters, defined by
the vector q = {a1: : :3, e, r, gi = 1: : :4}, is also of interest. In the
construction of a practical algorithm that permits the identifica-
tion of the state parameters q, the idea of using a maximum
likelihood criterion defined from certain innovational processes
has been essential (Ozaki, 1992, 1994). The LL filter and
innovation method have been successfully applied in the identi-
fication of a neural mass from actual EEG data (Valdes et al.,
1999) and for the identification of a model for the HIV-AIDS
epidemic from actual data (Pedroso et al., 2003).
The application of this recursive algorithm to identify the state-
space model (i.e., estimation of q and xt), in the particular case of
the hemodynamic approach that contain some missing values,
performs in the following way:
1. The equations for the evolution of the conditional mean xt/t and
covariance matrix Pt/t of the state-space variables.
xtþD=t ¼ xt=tþ r0ðJ xf ðxt=t; utÞ;DÞf ðxt=t; utÞ þ ðDr0ðJ xf ðxt=t; utÞ;DÞ� r1ðJ xf ðxt=t ; utÞ;DÞÞJt
f ðxt=t; utÞ
PtþD=t ¼ eJxfðxt=t ;utÞDPt=te
J xt
fðxt=t ;utÞDþ
Z D
0
eJxfðxt=t ;utÞsgg te
J xt
fðxt=t ;utÞsds
2. The zero mean innovation process ms = ys � h (xts/ts � D), with
variance defined by:
r2s ¼ cðxts=ts�DÞtPts=ts�Dcðxts=ts�DÞ þ r2
J.J. Riera et al. / NeuroImage 21 (2004) 547–567 559
3. The following difference equations must be satisfied for each
data value:
xts=ts ¼ xts=ts�D þ ksms
Pts=ts ¼ Pts=ts�D � ðkscðxts=ts�DÞtÞPts=ts�D
where the gain of the LL filter is defined by:
ks ¼Pts=ts�Dcðxts=ts�DÞ
r2s
Hence, it can be defined the optimization functional to be mini-
mized with respect to the state parameters h (i.e., the minus log
likelihood of the data y).
�logðS ðy1; . . . ; yNSÞÞ ¼ �logð2pÞNS þ
XNS
s¼1
log r2s
�� ��þ m2sr2s
� �
The entries of the vector function cðxÞ ¼ 0; 0;Vo k2x4x23
� k3
� �;
�Vo k2
1x3� k1
� ��t
are the derivatives of h (x) with respect to the
state-space variables x (i.e., Bx h (x)). Steps (2) and (3) only performwhen data are available.
The estimation of the parameters of the hemodynamic approach
given a fixed stimulus sequence could be inappropriate in several
voxels, particularly in those brain regions nondirectly time locked
to the stimulus, which also may receive contributions from com-
plementary domains. Therefore, it is of interest to estimate not only
the parameters of the hemodynamic approach but also the temporal
sequence representing its input. This is an ill-posed inverse prob-
Fig. 9. The simulated BOLD signal from the hemodynamic approach after model i
in this figure, the BOLD signal is poorly recorded in a set of 35 time instants (d
lem; hence, a priori information must be added to constrain the
number of possible solutions. The RBFs uk (t) = exp (�(t � lk)2/
2qk2) will be used in this paper as a parametrization of the input u (t),
which can be interpreted as a smoothness criterion of the temporal
variations of the synaptic activity, given by the general waveform
uðtÞ ¼PN
k¼1 ekukðtÞ. The parameters lk and qk represent the centerand the thickness of the RBFs, which are given a priori depending
on the length of the time windows, while the amplitude ek must be
estimated for each RBF from BOLD signals. In this paper, a regular
grid with TR as the RBF inter-distance (i.e., lk = kTR) will be used.
The thickness of the RBFs will be assumed to be qk = TR/2. In this
particular case, the nonlinear filter algorithm performs in the same
way as explained above, but the state parameters are redefined as
q = {a1: : :3, r, gi = 1: : :4, ek}.The procedures, used for applying the LL methods and optimi-
zation algorithm to actual and simulated data, were implemented in
Matlab 5.3. Additional Matlab modules were coded for the nonlin-
ear state-space and observation equations for the hemodynamic
approach.
Results
Analysis of nonlinearities in the hemodynamic approach
In SPM analysis, to specify the brain response of a complex
stimulus, a discrete, stationary and linear convolution of the
dentification is superimposed on the innovation process (top). As illustrated
ots). The true and estimated input temporal sequences overlap (bottom).
Table 1
Talairach coordinates of the hot-spots for each significant brain area
Subject
nos.
M1 Cerebellum CMA SMA
1 �32, �24, 52
(37.39)
18, �56, �28
(30.05)
�10, �8, 42
(15.50)
�10, �16, 76
(14.70)
2 �40, �22, 64
(45.77)
20, �60, 52
(11.30)
�4, �34, 50
(14.72)
�6, �12, 72
(19.27)
3 �38, �20, 64
(45.84)
18, �56, �26
(22.11)
�4, �6, 58
(18.39)
NO
4 �38, �26, 56
(35.19)
22, �56, �32
(29.51)
�6, �22, 42
(8.76)
�2, �6, 60
(11.17)
5 �38, �14, 60
(33.26)
18, �62, �24
(17.19)
�4, �10, 46
(10.06)
NO
Note. The values of the SPM99 t test are shown below.
J.J. Riera et al. / NeuroImage 21 (2004) 547–567560
HRF and the stimulus sequence describing the specific exper-
imental paradigm must be computed. In this section, a compar-
ison of the time series obtained via discrete convolution and
hemodynamic approaches is performed for two particular stim-
uli, time locked infrequent single events and a prolonged train
of frequent events. Obviously, these two kinds of stimulus are
associated with event-related and block design experimental
paradigms, respectively. In the simulations, the mean values
reported by Friston et al. (2000) for the parameters of the
hemodynamic approach were used. The realizations of the
trajectories of the state-space variables of the hemodynamic
approach were obtained using the LL method (Eq. 2). The
discrete convolution approach was obtained by convolving the
input sequence with the first-order Volterra kernel (see Discus-
sion for details).
Infrequent single events
Hemodynamic refractoriness is the most common nonlinear
property of BOLD signals, expressed as interactions among stimuli
that can lead to the suppression and increased latency/amplitude of
response to a stimulus incurred by preceding stimuli. A stimulus,
constituted by a single event of a very short duration followed by
another (i.e., with a temporal latency s s), represents the simplest
example to illustrate some important aspects of the refractoriness.
Fig. 2a shows the two infrequent events (top), separated by the
interval s. Continuous thin lines show the responses to both events,
when presented separately. The whole response of the hemody-
namic approach to the two consecutive events is shown by the bold
dotted line. It can be easily noted that the BOLD response to the
second event is different to the one corresponding to it without a
preceding stimulus. Fig. 2b shows the residue which remains after
removing the BOLD responses when the stimuli are presented
separately from the one produced by consecutively stimulation.
The total shaded area can be used as an indicator of the size of the
refractoriness effect. Fig. 2c shows the dependency of the area as a
function of the latency s of the second event for the BOLD signal
simulated from both approaches (the hemodynamic ‘‘dotted-line’’
and discrete convolution ‘‘asterisk-line’’). The size of the refrac-
toriness effect for the hemodynamic approach increases while the
latency between both events diminishes and it reduces rapidly for
Fig. 11. (a) The left column shows three time series for the hot-spot in each regio
BOLD signal ‘‘thick lines’’ and the innovation process ‘‘narrow lines’’). The co
window of stimulation is also illustrated. (b) The dynamic of the state-space vari
distant events. However, there is a local minimum reach at
approximately 6 ms for which 90% of the BOLD response is
recovered. This theoretical result is in accordance with experimen-
tal data, i.e., see Figure 4 in Huettel and McCarthy (2000).
Furthermore, the effect of the refractoriness slightly increases again
due to the undershoot phase in the BOLD response (reaching a
local maximum at s = 7 s). This secondary effect disappears totally
for second events delayed for more than 11 s. The BOLD response
to the first event is shown symbolically to evaluate how the size of
the refractoriness effect depends on what HRF phase is present at
the timing of the second event (continuous thin line). As a
consequence of the refractoriness, the use of a more complicated
sequence of single events will show a departure of simulated
BOLD signals obtained using discrete convolution and hemody-
namic approaches. An example of this is that a stimulus sequence
consisting of a set of frequent singles events of the same amplitude
is followed by one weaker infrequent event. In this example, the
post-stimulus undershooting after the train of frequent events and
the sustained positive BOLD response to the retarded infrequent
event, in case of the hemodynamic approach, are much larger than
that predicted by the linear discrete convolution approach.
An additional nonlinear effect is related to the dose–response
relationship reported from experimental fMRI studies. Fig. 2d
shows the nonlinear dependence of the BOLD response (i.e., the
amplitude of the peak) while using constant stimulus with different
durations. It is known that using a global scaling factor can
accommodate the magnitude of the BOLD response obtained by
the discrete convolution approach. However, scaling the BOLD
signal will not eliminate that nonlinear effect if stimuli with
different durations are given in the same experimental paradigm.
In this case, the BOLD signal obtained by the discrete convolution
approach was scaled to have the same amplitude of that obtained
by the hemodynamic approach for the case of the stimulus with a
duration of 6 s. The maximum values of the BOLD signal were
relative compared to cases of shorter stimulations. The comparison
was performed for both cases: discrete convolution (black rhom-
bus) and hemodynamic (black circles) approaches. This theoretical
result corroborates the experimental finding reported by Birn et al.
(2001), where shorter duration stimuli produced visual and motor
BOLD signals larger than expected from the linear discrete
convolution approach.
The use of the LL method also permits the inclusion of
background system and observational noises, the values used for
this simulation being g = (0.05,0,0,0)t and r2 = 0.015, respectively.
Fig. 3 shows the effect that nonlinear properties of the hemody-
namic approach produce on a white noise input. The plots on the
left correspond to simulations performed by using the hemody-
namic approach, and on the right side are those using the discrete
convolution approach. In the particular case of BOLD signals
obtained by using the hemodynamic approach, the following facts
are illustrated: (a) the variance of the signal changes along the time,
which reflects a nonstationary property; (b) the auto-correlation
function is higher than when the discrete convolution approach is
used; and (c) the 3rd-order cumulant, a magnitude associate with
nonlinear characteristics of the system, shows a strong dependency
with signal lags.
n (the average BOLD signal ‘‘asterisks’’, a smooth version of the average
lumn on the right shows the histograms for the innovation processes. The
ables for the M1 area is shown.
J.J. Riera et al. / NeuroImage 21 (2004) 547–567 561
oImage 21 (2004) 547–567
A prolonged train of frequent events
The design in blocks constitutes another interesting and useful
experimental paradigm, where brain areas are explored by the
application of a certain sustained train of single events at regularly
spaced time intervals. Fig. 4 shows the behavior of the state-space
variables of the hemodynamic approach for the specified stimulus
sequence (Fig. 4 top: stimulus ‘‘black blocks’’ and resting ‘‘blank
spaces’’ and Fig. 6 top: details of a single ‘‘black block’’). The
same values were used for parameters as in the previous simula-
tion. A steady-state condition is reached; a fact probably attribut-
able to a sort of dynamic saturation phenomena in SDE (1) (see
Discussion). Fig. 5 compares the BOLD signals simulated from the
hemodynamic and discrete convolution approaches for this partic-
ular experimental paradigm.
The fact that BOLD signals obtained from linear and nonlinear
models are very similar in case of this type of stimulus can be
noted from Fig. 5. The only differences between them are in terms
of the scaling factors. This could justify the use of the discrete
convolution approach for block design experimental paradigms in
SPM analysis, where the scaling factors are easily accommodated
in any of the proposed methods for estimating the HRF. However,
its use in event-related paradigms seems to be more questionable
and a more detailed study must be performed from model-specific
experiments. The next sub-section is a discussion of how the state
parameters of the hemodynamic approach and trajectories of state-
space variables are determined from simulated data by the use of
the LL filter and innovation method. The fMRI data were simu-
lated for both types of experimental paradigms.
Identification of the state-space model
In the first part of this sub-section, the state parameters of the
hemodynamic approach and trajectories of the state-space variables
were estimated given a priori the input sequence in a block design
paradigm. The optimization method based on the nonlinear recur-
sive algorithm, as described in Methods, was used to identify the
state-space model from simulated recordings. Fig. 6 shows results
obtained from a particular block of the simulated fMRI data shown
in Fig. 4 (top). As illustrated at the top of this figure, each frequent
single motor event was simulated by a sharp Gaussian function of
200 ms in duration to approximate each single motor episode.
Though BOLD data show very slow oscillations (see Fig. 7
top), the flow-inducing signal exhibits oscillatory activity with
high-order frequencies (Fig. 6 bottom), which illustrate that the
hemodynamic approach behaves as a low-pass rectifier filter. As
illustrated in Fig. 6, which superimposes the trajectories of the
state-space variables reconstructed from the LL method after
estimating the state parameters by the recursive algorithm (dots)
upon the true trajectories used in the simulation (thin line), it is
feasible to successfully identify the state-space model from con-
taminated BOLD observations recorded with a very low sampling
rate. However, this visual criterion was not used as a conclusive
measure of the degree to which the model described the simulated
data correctly. Rather, the quality of the fit should be measured by
analyzing the characteristics of the innovation process, which
represents the data not explained from the hemodynamic approach.
For almost all simulations, the variance of the innovation process
was less than 2.5% of the total signal power. The Kolmogorov–
Smirnov test to evaluate the departure of the Gaussian distribution
of the histogram for the innovation process can be used as a proper
criterion of how well the model fits the data (Fig. 7, left-bottom).
J.J. Riera et al. / Neur562
The test of Gaussian distribution of the histogram of innovation
process was always accepted for simulated BOLD signals. How-
ever, it is important to clarify that this fact is not evident in the case
of simulations because the low sampling rate equates to a lack of
information which may have a strong effect on such a test.
To qualify the suitability of the nonlinear recursive algorithm, the
hyper-surface constructed from evaluating the optimization func-
tional �log (S (y1, : : :, yNS)) for different values of the state param-
eters in the physiological range is shown in Fig. 7 (right-bottom). In
this case, only the state parameters a1 and a2 were varied; however,
the same property was observed while examining the rest of the state
parameters. Despite the existence of both a low sampling rate and
pollution due to a large background system and observational noises
used in that simulation, the optimization functional shows a very
smooth shape with a local minimum, which guarantees convergence
and numerical stability. The temporal dynamic of the innovation
process corresponding to the minimum of the functional, for this
particular example, is shown in Fig. 7 (top).
In the second part of this sub-section, the state parameters of the
hemodynamic approach (i.e., which also includes the RBF para-
metric model for the input u (t)) were estimated for an event-related
paradigm (shown in the top of Fig. 8). Vertical bars represent the
infrequent single events and the flat areas correspond to the resting
condition. The realization of the state-space variables computed
using the LL method after identifying the hemodynamic approach
(dots) from simulated BOLD data almost exactly coincide with that
obtained from the true hemodynamic parameters and input time
series (thin line) used previously to create the simulation (see Fig.
8). The trajectories of the state-space variables corresponding to the
use of the initial values of the state parameters are also presented in
the figure (which represent oscillations around x (0)).
Fig. 9 (top) shows the superposition of BOLD simulated data
with the innovation process, later interpreted as the residue left
over after explaining the data with the hemodynamic approach. It
was shown that the innovation process represents a white Gaussian
process, which exemplifies the robustness of using the novel
recursive algorithm even when a high order of missing values
exists. As explained in the methods, 12 values were lost between
each observation, and those observations were highly contaminated
by instrumental errors. The superposition of true and estimated
input time series (thin line and dots) is presented in Fig. 9 (bottom).
The robustness of the procedure for model identification is clear
from this simulation. However, the number of RBFs was fixed a
priori and regularly distributed on the window of analysis (i.e., one
function centered for each observation value). Hence, the method
must be extended to estimate the number of locally supported
RBFs by using some sort of model selection criterion.
Experimental tasks
Block design paradigm
The SPM99 t test was applied to the fMRI data for each subject
using an ‘‘On vs. Off’’ contrast. It should be clarified that though
the SPM uses the discrete convolution approach as a time series
linear model, most of the results obtained from that toolbox are
very useful in practice, especially in the detection of areas directly
associated to the stimulus in the case of block design paradigms. In
our experiment, the most significant brain regions were M1,
Cerebellum, Cingulate Motor Area (CMA) and Supplementary
Motor Area (SMA). Table 1 summarizes the statistic analysis for
each subject.
Table 2
The value of the estimated parameters for the area M1
Parameter Subject
no. 1
Subject
no. 2
Subject
no. 3
Subject
no. 4
Subject
no. 5
a1 2.19 3.10 0.95 1.97 2.03
a2 2.85 2.97 2.97 2.99 2.89
a3 4.20 6.16 3.66 5.93 3.89
- (Hz) 0.10 0.09 0.12 0.10 0.10
r2 0.0023 0.0009 0.0021 0.0036 0.0037
The parameters a1, a2 and a3 are given in s.
J.J. Riera et al. / NeuroImage 21 (2004) 547–567 563
As expected, for each subject, the activation in the M1 area was
the most significant. The cerebellum was the second most activated
area (exception for subject no. 2). The activations in CMA and
SMAwere a little less significant; with the fact that activity in CMA
was lightly larger than at SMA. There was no activation in the SMA
for two subjects. Subject no. 1 was chosen as the champion data
because the test used to gauge how well the hemodynamic approach
fit the data was most significant for the BOLD signal recorded from
its M1 area compared to the other subjects. The data to be presented
Fig. 12. For the event-related motor task the following plots are shown: the avera
(thick line) after identifying the state-space model (a); the estimates of the state-sp
innovation process (d). The array of RBFs used for model identification (e) and
in this paper correspond to the champion subject, but results are
consistent for all of them.
Fig. 10 shows the SPM99 t test using ‘‘glass images’’ (maxi-
mum intensity projections) from the SPM99 toolbox (top). The
four activated brain regions are labeled below. The 3D rendering of
the SPM99 t test overlaps with the individual T1 anatomical image
from different slices. A time series of the 1st eigenvalue in the hot-
spot of the M1 area shows a drift that was modeled for the four
brain areas in each subject with piecewise linear functions (left-
bottom). The time series of hot-spots of the brain regions of interest
for each subject was pooled over blocks to reduce instrumental
error. The state parameters of the hemodynamic approach were
estimated from each time series of the averaged BOLD signal. Fig.
11a shows the average BOLD signal for each region (asterisks), a
smooth version of the data obtained using a 4-order moving
average with a rectangular window (the thick line) and the time
series of the innovation process (the narrow line). The histograms
for the innovation process of each region are also plotted (Fig. 11
(a), right). It should be noted that the time series of these four brain
regions are quite different. The test was accepted only for M1 areas
in all of the subjects, which reflects the fact that the other three
ge BOLD signal (black circles) and the reconstruction from the LL method
ace variables (b); the input temporal sequence (c); and the histogram of the
a particular RBF (f) are also shown.
J.J. Riera et al. / NeuroImage 21 (2004) 547–567564
brain regions are less directly related to the stimulus and could
have received complementary influences from cortical or sub-
cortical brain structures. The estimated dynamics corresponding
to the state-space variables for M1 area are shown in Fig. 11b.
The state parameters obtained by identifying the state-space
model at M1 area for each of the subjects are presented in Table 2.
The values of parameters a1 and a2 are in the range reported by
Friston et al. (2000), but the parameter a3 is considerably larger. The
state parameters gi = 1: : :4 characterizing the amplitude of system
noise was estimated, though the value for each subject was around
20 times smaller than the variance introduced by instrumental error
r2. The mean values of these parameters across the subject are: a1 =
1.79 s, a2 = 2.94 s (hence - = 0.104 Hz) and a3 = 4.76 s.
Event-related paradigm
The champion subject was requested to perform the more
complex motor task, described in Experimental tasks. The identifi-
cation problem in this case includes the estimation of the parameters
of the RBFs; hence, the input temporal sequence u (t). The raw
BOLD data from the M1 hot-spot were also pooled over blocks to
reduce instrumental error. The average BOLD signal (black circles)
and that which was reconstructed from the LL method after
identifying the state-space model by the recursive algorithm (the
thick line) are shown in Fig. 12a. The reconstructions of the state-
space variables, which have an immediate physiological interpreta-
tion, are also plotted (b) and finally, the input temporal sequence,
with a high-frequency dynamic is presented (c). It can be noted that
the peaks of activation of u (t) almost coincide with the timing of the
infrequent single motor events. However, there are two electrophys-
iological episodes marked by arrows that do not correspond with the
Fig. 13. Simulated BOLD signals resulting from using time constants varying from
stimulation paradigm, which could be due to misestimations. Even
so, the test of adequacy of the hemodynamic approach to the BOLD
signal was accepted. The Gaussian distribution of the histogram of
the innovation process is also plotted (d). Finally, the RBFs used in
this case for the identification of input temporal sequence are shown
(e and f).
Conclusion
In this paper, the SPM99 toolbox was used to identify brain
regions directly related to a motor task consisting of closing and
opening the right hand in case of a block design paradigm for five
healthy subjects. The regions M1, Cerebellum, CMA and SMA
were involved in the performance of the task with order of SPM99
t test significance as presented above. The time series of the hot-
spots for each region varied considerably. The recursive algorithm,
based on reformulating the hemodynamic approach as a state-space
model, was used to successfully identify parameters of the model.
The performance of such a methodology was illustrated using
simulated BOLD signals obtained for this particular experimental
paradigm. Region M1 was the only area that exhibited a dynamic
in correspondence with the predefined stimulus sequence used in
the hemodynamic approach, a fact that was proved using the
criterion to determine how well the model fit (the Kolmogorov–
Smirnov test for Gaussian distribution of the innovations). It was
also illustrated that the optimization functional used in the inno-
vation method showed a very smooth dependency when the
parameters were varied, even for the worst case of missing data
contaminated by large instrumental error.
2 to 7 s at 1-s steps (the effect of increasing s0 is indicated by the arrows).
J.J. Riera et al. / NeuroImage 21 (2004) 547–567 565
The time constants for the signal decay (or elimination com-
ponent) and feedback auto-regulatory mechanism were, for all five
subjects, in concordance with the physiologically acceptable range
reported. The resonance frequency computed from these values
was around - = 0.104 Hz, which can be considered to be the
frequency of vasomotor signals. However, the mean transit time
was consistently larger than that stated in other comparative
studies. In the equivalent analogical RC circuit, the time constant
determines the slope of the increasing and decreasing phases of the
BOLD signal. Fig. 13 shows differences in the simulated BOLD
signals resulting from using time constants varying from 2 to 7 s at
1-s steps. The actual BOLD signal at M1 (i.e., the smoothed
version represented by the thick line, Fig. 11a (top)), exhibits a
dynamic that corresponds to a time constant s0 c 4 s (see side by
side with Fig. 13). For every subject, the actual BOLD signal at M1
showed a less-pronounced slope than that expected from simula-
tions using typical transit times. This may be justified in that during
the tight stimulation conditions of our experiment, the hypotheses
implicit in the hemodynamic approach need to be reconsidered
(i.e., see Zheng et al. (2002) for an extension of the hemodynamic
approach to include a more general coupling between the flow and
the oxygen delivery mechanism). However, an alternative expla-
nation could be that pooling over blocks may introduce an
undesirable smoothing effect on the BOLD signals.
It was also shown by the use of simulated data, that in the case
of block design paradigm, the hemodynamic and discrete convo-
lution approaches produce very similar BOLD signals, disregard-
ing scaling factors and also the effect of system and observational
noises. This fact can be interpreted as a steady-state behavior
reached for a BOLD signal that could result from the saturation
property of SDE (1) when sustained frequent events are used as its
input. Hence, the predictors defined by the SPM99 toolbox can be
easily interpreted in this experimental paradigm, especially for
those stimulus-related regions.
However, in cases of infrequent events the nonlinear effect of
the BOLD hemodynamic response must be considered carefully.
This assertion calls into question the use of the discrete convolu-
tion approach in cases of event-related paradigms. In the particular
case of such experimental paradigms, the result of analysis using
the SPM99 toolbox must be cautiously interpreted. In such
situations, we propose the use of RBFs that allows not only the
identification of parameters of the hemodynamic approach, but
also the reconstruction of the dynamics of the input, closely related
to the synaptic activation. This is also an adequate analysis for
those brain regions of interest not directly associated with the
stimulus by primary pathways.
Discussion
The value of the t test in SPM analysis is sensitive to two
factors: (1) the size of the effect, for example, the degree of
activation, and (2) the standard error associated with the estimator
of this effect. Furthermore, these estimators depend upon a priori
knowledge about what caused the activation (i.e., a linear combi-
nation of predictors of interest, which are a priori established from
stimulus sequence). Because of this fact, we can almost assert that
those significant regions have a very strong BOLD signal and an
additional high temporal correlation with BOLD predictors. How-
ever, we cannot distinguish between these two factors in the
SPM99 t test. Hence, those regions receiving influence from other
brain structures may not have a dynamic significantly correlated to
the predictors, even when they have a large BOLD signal. These
regions will exhibit a nonsignificant value of the t test. A method
has been recently presented to identify those significant brain
regions by exploring the differences of phase synchronization with
a task-dependent reference function, using a nonparametric per-
mutation test (Laird et al., 2002). Furthermore, several methods
using parametric and nonparametric models have now been intro-
duced in the SPM analysis to estimate the voxel dependent HRFs.
The uses of such analysis will improve the accuracy of the SPM
analysis. The procedure presented in this paper does not necessar-
ily require defining a priori the predictors.
Though it is not practical due to the large computational costs
involved, the identification method based on the hemodynamic
approach could be of interest for those particular brain regions that
we have already tested using the SPM analysis or that we expect,
from physiological knowledge, to be involved in the brain activa-
tion circuit.
The hemodynamic approach as originally presented by Friston
et al. (2000) has two main disadvantages: (1) it assumes that the
uncertainty of the model originates solely from instrumental errors
(in the hemodynamic approach presented in this paper it is
equivalent to consider g = (0,0,0,0)t in Eq. (1)); and (2) it assumes
the stimulus sequence is known, but in fact it differs for each brain
voxel in terms of scaling factors only (i.e., the neuronal efficacies).
Therefore, the results will significantly depend upon the authen-
ticity of the BOLD signal predictors.
Additionally, the method proposed by Friston (2002) to esti-
mate the state parameters using the Volterra kernels expansion
presents several critical limitations. It uses a Taylor expansion of
x (t) around x (0) and u (t) = 0, which permits the transformation of
Eq. (1) into a bilinear form dx V(t) = (A + u(t)B) x V(t)dt, where thenew vector of state-space variables is defined by x V(t) = (1, x (t)).
The matrices
A ¼ 0 0
�J xf ðxð0Þ; 0Þxð0Þ J xf ðxð0Þ; 0Þ
� �
and
B ¼ 0 0
e 0
� �
have constant entries. It was considered a fact that:
f ðxð0Þ; 0Þ ¼ ð0; 0; 0; 0Þt Bf ðxð0Þ; 0ÞBu
¼ ðe; 0; 0; 0Þt
BJ xf ðxð0Þ; 0ÞBu
¼ 0
For this case, The Volterra kernel jiðt; k1; . . . ; kiÞ ¼ BiyðtÞ
Buðk1Þ...BuðkiÞcanbe evaluated theoretically for any order. However, in practice a
finite truncation of the series must be carried out, limiting the
representation of higher order nonlinear dynamics.
ys ¼ j0ðtsÞþXli¼1
Z. . .
Zjiðts; k1; . . . ; kiÞuðk1Þ . . . uðkiÞdk1 . . . dki
By definition:
viðt; k1; . . . ; kiÞ ¼ hðexpððt � kiÞAÞB expððki � ki�1ÞAÞB . . . exp
� ððk2 � k1ÞAÞB expðk1AÞxVð0ÞÞ
J.J. Riera et al. / NeuroImage 21 (2004) 547–567566
The Volterra kernel expansion can be evaluate (i.e., up to second
order) by:
j0ðtÞ ¼ hðv0Þ
j1ðt; k1Þ ¼Bhðv0ðtÞÞBXV
v1ðt; k1Þ
j2ðt; k1; k2Þ¼Bhðv0ðtÞÞ
BXVv2ðt;k1;k2Þ þ vt1ðt; k1Þ
B2hðv0ðtÞÞBXV2
v1ðt; k2Þ
The method proposed in Friston (2002) to carry out the
numerical integration of the Eq. (1) uses two sequential approaches:
(a) to obtain the bilinear form mentioned above; and (b) to compute
the linear exact solution in the interval [t, t + D], assuming that the
function u (t) is constant: xt + DV c exp(D(A + utB))xtV.However, the authors have doubts about the performance of that
method to integrate ordinary differential equation. An evaluation of
several indicators will guarantee that the state-space variables after
integration preserve most of the qualitative properties of the original
dynamic system. The LL method is applied directly to the original
nonlinear continuous time series model, so that when D!0, the
discretized time series approaches the true continuous solution. It
can be noticed by simple visual inspection that, in Friston’s
approach, the discretized time series approaches the solution of
the bilinear model; hence, it will never approach the true solution.
The performance of the LL method has been rigorously evaluated in
recent years using several of those indicators. Examples are the
following: the use of the LL method to compute Lyapunov
exponents of dynamical systems (Carbonell et al., 2002); the study
of general conditions under which this numerical scheme preserves
the stationary points and periodic orbits of the ordinary differential
equations and the local stability at the steady states (Jimenez et al.,
2002); and the order of uniform strong convergence (Jimenez and
Biscay, 2002). Additionally, simulations of SDEs have been used to
compare the performance of the LL method and the most popular
numerical strategies (Jimenez et al., 1999).
A detailed discussion about the nonlinear coupling between the
BOLD signal and the CBF based on simulations using the
hemodynamic approach has been presented in Mechelli et al.
(2001). In that paper, the authors reviewed the impact on nonlinear
hemodynamic responses produced by changing the stimuli char-
acteristics: the a priori stimulus, the epoch length, the presentation
rate, the onset asynchrony, etc. In a very interesting paper, the
optimal inter-stimulus interval, given the fixed stimulus duration,
was reported during a bilateral finger tapping, demonstrating
discordance between linear predictors and actual BOLD data
(Bandenttini and Cox, 2000).
In our opinion, the hemodynamic approach represents the first
approximation to explain the nonlinear interactions between CBF
and CBV based on a very simple and reliable mathematical
construct. However, the physiological mechanisms from where
synaptic activation is translated into the language of CBF changes
are still in the process of being modeled. In our paper, a simple
linear link between the flow-inducing signal and the CBF was
assumed. However, the hemodynamic approach has been recently
extended to include more realistic electrophysiological and hemo-
dynamical coupling in which metabolism processes also play a
very important role (Aubert and Costalat, 2002). Unfortunately,
this extended model is far from practical for use in situations for
actual BOLD, Positron Emission Tomography (PET) and/or elec-
trophysiological recordings due to its mathematical complexity. A
straightforward extension of the hemodynamic approach has been
recently proposed by Zheng et al. (2002), which permits the
interpretation of the modulatory effects of changes in tissue oxygen
concentration. The authors are now reviewing those more general
approaches for inclusion in the nonlinear time series analysis,
which will allow us to perform a fusion of neuroimaging multi-
modalities from a biophysics framework.
The authors are very interested in the exploration of the
response of the hemodynamic approach to nonlinear synaptic
activity. However, this is beyond the scope of the actual paper.
We are now implementing a method that uses a Poisson random
process linearly filtered to represent more complex dynamics of
the input.
Acknowledgments
The authors would like to thank Dr. J.C. Jimenez from ‘‘El
Instituto de Cibernetica, Matematica y Fısica’’, Havana, Cuba for
useful discussions and suggestions during the preparation of this
manuscript. This study has been supported by JST/RISTEX, R and
D promotion scheme for regional proposals promoted by TAO, and
the 21st Century Center of Excellence (COE) Program (Ministry of
Education, Culture, Sports, Science and Technology) entitled ‘‘A
Strategic Research and Education Center for an Integrated
Approach to Language and Cognition’’ (Tohoku University).
References
Aguirre, G.K., Zarahn, E., D’Esposito, M., 1998. The variability of human
BOLD hemodynamic responses. NeuroImage 8, 360–369.
Aubert, A., Costalat, R., 2002. A model of the coupling between brain
electrical activity, metabolism, and hemodynamics: application to the
interpretation of functional neuroimaging. NeuroImage 17, 1162–1181.
Bandenttini, P.A., Cox, R.W., 2000. Event-related fMRI contrast using
constant enterstimulus interval: theory and experiment. Magn. Reson.
Med. 43, 540–548.
Berns, G.S., Song, A.W., Mao, H., 1999. Continuous functional magnetic
resonance imaging reveals dynamic nonlinearities of dose– response
curves for finger opposition. J. Neurosci. 19, RC17.
Birn, R.M., Saad, Z.S., Bandettini, P.A., 2001. Spatial heterogeneity of
nonlinear dynamics in the fMRI BOLD response. NeuroImage 14,
817–826.
Biscay, R., Jimenez, J.C., Riera, J.J., Valdes, P.A., 1996. Local linearization
method for the numerical solution of stochastic differential equations.
Ann. Inst. Stat. Math. 48 (4), 631–644.
Boxerman, J.L., Bandettini, P.A., Kwong, K.K., Baker, J.R., Davis, T.L.,
Rosen, B.R., Weisskoff, R.M., 1995. The intravascular contribution to
fMRI signal change: Monte Carlo modeling and diffusion-weighted
studies in vivo. Magn. Reson. Med. 34, 4–10.
Buckner, R.L., Koutstaal, W., Schacter, D.L., Dale, A.M., Rotte, M.,
Rosen, B.R., 1998. Functional –anatomic study of episodic retrieval
using fMRI (II). Selective averaging of event-related fMRI trials to test
the retrieval success hypothesis. NeuroImage 7, 163–175.
Burock, M.A., Dale, A.M., 2000. Estimation and detection of event-related
fMRI signals with temporally correlated noise: a statistically efficient
and unbiased Approach. Hum. Brain Mapp. 11, 249–260.
Buxton, R.B., Wong, E.C., Frank, L.R., 1998. Dynamics of blood flow and
oxygenation changes during brain activation: the balloon model. Magn.
Reson. Med. 39, 855–864.
Carbonell, F., Jimenez, J.C., Biscay, R., 2002. A numerical method for the
computation of the Lyapunov exponents of nonlinear ordinary differ-
ential equations. Appl. Math. Comput. 131, 21–37.
J.J. Riera et al. / NeuroImage 21 (2004) 547–567 567
Duann, J., Jung, T., Kuo, W., Yeh, T., Makeig, S., Hsieh, J., Sejnowski,
T.J., 2002. Single-trial variability in event-related BOLD signals. Neu-
roImage 15, 823–835.
Feller, W., 1971. Probability theory and its applications, Second ed., vol. II
Wiley, NY.
Friston, K.J., 2002. Bayesian estimation of dynamical systems: an applica-
tion to fMRI. NeuroImage 16, 513–530.
Friston, K.J., Jezzard, P., Turner, R., 1994. Analysis of functional MRI time
series. Hum. Brain Mapp. 1, 153–171.
Friston, K.J., Josephs, O., Rees, G., Turner, R., 1998. Nonlinear event-
related responses in fMRI. Magn. Reson. Med. 39, 41–52.
Friston, K.J., Mechelli, A., Turner, R., Price, C.J., 2000. Nonlinear re-
sponses in fMRI: the balloon model, volterra kernels, and other hemo-
dynamics. NeuroImage 12, 466–477.
Goutte, C., Nielsen, F.A., Hansen, L.K., 2000. Modeling the hemodynamic
response in fMRI using smooth FIR filters. IEEE Trans. Med. Imag. 19
(12), 1188–1201.
Grubb, R.L., Phelps, M.E., Eichling, J.O., 1974. The effects of vascular
changes in PaCO2 on cerebral blood volumen, blood flow and vascular
mean transit time. Stroke 5, 630–639.
Huettel, S.A., McCarthy, G., 2000. Evidence for a refractory period in the
hemodynamic response to visual stimuli as measured by MRI. Neuro-
Image 11, 547–553.
Huettel, S.A., McCarthy, G., 2001. Regional differences in the refractory
period of the hemodynamic response: an event-related fMRI study.
NeuroImage 14, 967–976.
Iadecola, C., 2002. Intrinsic signals and functional brain mapping: caution,
blood vessel at work. Cerebral Cortex 12, 223–224 (CC Commentary).
Jimenez, J.C., 2002. A simple algebraic expression to evaluate the local
linearization schemes for stochastic differential equations. Appl. Math.
Lett. 15, 775–780.
Jimenez, J.C., Biscay, R., 2002. Approximation of continuous time stochas-
tic processes by the local linearization method revisited. Stoch. Anal.
Appl. 20 (1), 105–121.
Jimenez, J.C., Ozaki, T., 2002a. Linear estimation of continuous-discrete
linear state space model with multiplicative noise. Syst. Control Lett.
47, 91–101.
Jimenez, J.C., Ozaki, T., 2002b. An approximate innovation method for the
estimation of diffusion processes from discrete data. Technical Report
2002-184, Instituto de Cibernetica, Matematica y Fisica, La Habana,
Cuba.
Jimenez, J.C., Ozaki, T., 2003. Local linearization filters for non-linear
continuous-discrete state space models with multiplicative noise. Int.
J. Control 76 (12), 1159–1170.
Jimenez, J.C., Valdes, P.A., Rodriguez, L.M., Riera, J.J., Biscay, R., 1998.
Computing the noise covariance matrix of the local linearization
scheme for the numerical solution of stochastic differential equations.
Appl. Math. Lett. 11 (1), 19–23.
Jimenez, J.C., Shoji, I., Ozaki, T., 1999. Simulation of stochastic differ-
ential equations through the local linearization method. A comparative
study. J. Stat. Phys. 94 (3/4), 587–602.
Jimenez, J.C., Biscay, R., Mora, C., Rodriguez, L.M., 2002. Dynamic
properties of the local linearization method for initial-value problems.
Appl. Math. Comput. 126, 63–81.
Laird, A.R., Rogers, B.P., Carew, J.D., Arfanakis, K., Moritz, C.H.,
Meyerand, M.E., 2002. Characterizing instantaneous phase relation-
ship in whole-brain fMRI activation data. Hum. Brain Mapp. 16,
71–80.
Lange, N., Zeger, S.L., 1997. Nonlinear Fourier time series analysis for
human brain mapping by functional magnetic resonance imaging. J.R.
Stat. Soc. Appl. Stat. 46, 1–29.
Magistretti, P.J., Pellerin, L., 1999. Cellular mechanisms of brain energy
metabolism and their relevance to functional brain imaging. Phil.
Trans.R. Soc. Lond. B 354, 1155–1163.
Mandeville, J.B., Marota, J.J.A., Ayata, C., Zaharchuk, G., Moskowitz,
M.A., Rosen, B.R., Weisskoff, R.M., 1999. Evidence of cerebrovascu-
lar postarteriole windkessel with delayed compliance. J. Cereb. Blood
Flow Metab. 19 (6), 679–689.
Marrelec, G., Benali, H., Ciuciu, P., Pelegrini-Issac, M., Poline, J.B., 2003.
Robust Bayesian estimation of the hemodynamic response function in
event-related BOLD MRI using basic physiological information. Hum.
Brain Mapp. 19, 1–17.
Mechelli, A., Price, C.J., Friston, K.J., 2001. Nonlinear coupling between
evoked rCBF and BOLD signals: a simulation study of hemodynamic
responses. NeuroImage 14, 862–872.
Ogawa, S., Menon, R.S., Tank, D.W., Kim, S.G., Merkle, H., Ellerman,
J.M., Ugurbil, K., 1993. Functional brain mapping by blood oxygen-
ation level-dependent contrast magnetic resonance imaging: a compa-
rison of signal characteristics with a biophysical model. Biophys. J.
64, 803–812.
Oldfield, R.C., 1971. The assessment and analysis of handedness: the
Edinburgh Inventory. Neuropsychologia 9 (1), 97–113.
Ozaki, T., 1985. Nonlinear time series models and dynamical systems. In:
Hannan, E.J., Krishnaiah, P.R., Rao, M.M. (Eds.), Handbook of Statis-
tics, vol. 5. North-Holland, Amsterdam, pp. 25–83.
Ozaki, T., 1992. Identification of nonlinearities and non-Gaussianities in
time series. In: Brillinger, D.R., et al. (Eds.), New Directions in Time
Series Analysis, Part I. IMAVolumes in Mathematics and Its Applica-
tion, vol. 45. Springer Verlag, NY, pp. 227–264.
Ozaki, T., 1993. A local linearization approach to nonlinear filtering. Int. J.
Control 57, 75–96.
Ozaki, T., 1994. The local linearization filter with application to non-
linear system identification. In: Bozdogan (Ed.), Proceedings of the
first US/Japan Conference on the Frontiers of Statistical Modeling:
An Informational Approach. Kluwer Academic Publishers, Dordrecht,
pp. 217–240.
Ozaki, T., Jimenez, J.C., Haggan-Ozaki, V., 2000. Role of the likelihood
function in the estimation of chaos models. J. Time Ser. Anal. 21,
363–387.
Pedroso, L.M., Marrero, A., de Arazoza, H., 2003. Nonlinear parametric
model identification using genetic algorithms. Lecture Notes in Com-
puter Science 2687. Springer-Verlag, Heidelberg, pp. 473–480.
Rajapakse, J.C., Kruggel, F., von Cramon, D.Y., 1998. Modeling hemody-
namic response for analysis of functional MRI time-series. Hum. Brain
Mapp. 6, 283–300.
Valdes, P., Jimenez, J.C., Riera, J., Biscay, R., Ozaki, T., 1999. Nonlinear
EEG analysis based on a neural mass model. Biol. Cybern. 81, 415–424.
van Loan, C.F., 1978. Computing integrals involving the matrix exponen-
tial. IEEE Trans. Automat. Contr. AC-23, 395–404.
Zheng, Y., Martindale, J., Johnston, D., Jones, M., Berwick, J., Mayhew, J.,
2002. A model of hemodynamic response and oxygen delivery to brain.
NeuroImage 16, 617–637.