www.elsevier.com/locate/ynimg

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: riera@idac.tohoku.ac.jp, riera_jorge@hotmail.com

(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).

J.J.Riera

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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;2

rom 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).

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