Identification and comparison of stochastic metabolic/hemodynamic models (sMHM) for the generationof the BOLD signal

Roberto C. Sotero & Nelson J. Trujillo-Barreto &

Juan C. Jiménez & Felix Carbonell &Rafael Rodríguez-Rojas

Received: 18 July 2007 /Revised: 19 December 2007 /Accepted: 26 February 2008 / Published online: 3 October 2008# Springer Science + Business Media, LLC 2008

Abstract This paper extends a previously formulateddeterministic metabolic/hemodynamic model for the gener-ation of blood oxygenated level dependent (BOLD)responses to include both physiological and observationstochastic components (sMHM). This adds a degree offlexibility when fitting the model to actual data byaccounting for un-modelled activity. We then show howthe innovation method can be used to estimate unobservedmetabolic/hemodynamic as well as vascular variables of thesMHM, from simulated and actual BOLD data. Theproposed estimation method allowed for doing modelcomparison by calculating the model’s AIC and BIC. Thismethodology was then used to select between different

neurovascular coupling assumptions underlying sMHM.The proposed framework was first validated on simulationsand then applied to BOLD data from a motor taskexperiment. The models under comparison in the analysisof the actual data considered that vascular response wascoupled to: (I) inhibition, (II) excitation, (III) bothexcitation and inhibition. Data was best described by modelII, although model III was also supported.

Keywords Model comparison . fMRI .

Stochastic differential equations . Biophysical model

1 Introduction

The development of functional neuroimaging techniquesrepresented a breakthrough in the study of the living brain.Among these, functional magnetic resonance imaging(fMRI) that records magnetization changes due to varia-tions in blood oxygenation is one of the most popular dueto its non-invasive properties and its high spatial resolution.Nevertheless, despite the widespread use of this technique,the physiological mechanisms underlying the couplingbetween neuronal activity and the blood oxygenation leveldependent (BOLD) signal are still under study.

In this context, the development of computationalmodels has proven to be a useful way of testing hypothesisand making predictions about those processes whereexperimental data is insufficient. In recent years biologi-cally informed dynamic causal models (DCM) (Friston etal. 2003) have been used to answer key questions about theneuronal architecture and biophysical parameters responsi-ble for generating imaging time-series in both fMRI andelectrophysiology. In this work, we use one of the more

J Comput Neurosci (2009) 26:251–269DOI 10.1007/s10827-008-0109-3

Action Editor: Barry J. Richmond

R. C. Sotero (*)Brain Dynamics Department, Cuban Neuroscience Center,Avenue 25, Esq 158, #15202, Cubanacán, Playa, P.O. Box 6412,Havana, Cubae-mail: rcarlos@cneuro.edu.cu

N. J. Trujillo-BarretoBrain Dynamics Department,Cuban Neuroscience Center,Avenue 25, Esq 158, #15202, Cubanacán, Playa,P.O. Box 6412, Havana, Cubae-mail: trujillo@cneuro.edu.cu

J. C. Jiménez : F. CarbonellDepartament of Interdisciplinary Mathematics,Institute of Cybernetics, Mathematics and Physics,Havana, Cuba

R. Rodríguez-RojasInternational Center for Neurological Restoration (CIREN),Havana, Cuba

advanced models of coupling between neuronal activity andhemodynamic responses, as observed with fMRI (Soteroand Trujillo-Barreto 2007). Our key contribution is the useof Bayesian model selection to answer an importantquestion about the neurophysiology of hemodynamicsignals; namely, are hemodynamic responses caused pre-dominantly by the activity of excitatory neurons, inhibitoryneurons or a combination. We will show that this questioncan be answered, with a high degree of confidence, usingcompletely non-invasive data. Not only does this approachfurnish quantitative measures of neurophysiology, in termsof metabolism and neuronal activity, but it also placesimportant constraints on future biophysically informedmodels of imaging time-series.

As said before, the solution of such a model inferencetask presuppose the inversion of generative models of thefMRI signal. These models have evolved from thosedescribing the dynamic changes of hemodynamic andvascular variables and their connection to the BOLD signal(Buxton et al. 1998), to more sophisticated ones, whichattempt to link these variables to the neuronal responses(Friston et al. 2000; Buxton et al. 2004). In a previouscommunication we extended these models further toincorporate the role played by excitatory and inhibitoryneuronal activities in the generation of the BOLD signal, bytaking the metabolic dynamics into account (Sotero andTrujillo-Barreto 2007). This is the generative model used inthis paper and from now on will be denoted as MHM(metabolic/hemodynamic model). Additionally, we increaseour MHM by taking into account two stochastic compo-nents. These components correspond to both physiologicaland measurement noises, that are present in the recordedBOLD signal. Measurement noise can arise from thermallygenerated random noise from the scanner electronics and isusually treated as observation errors. On the other hand thephysiological noise is due to stochastic fluctuations inmetabolic and vascular responses (Biswal et al. 1995;Krüger and Glover 2001). This type of error clearly affectsthe dynamics of the system (dynamic noise) and thus needsto be taken into account. In the context of the Friston’sextended Balloon model (Friston et al. 2000), measurementnoise was taken into account at the level of the observationequation (Friston 2002), and more recently Riera et al.(2004) extended this to include physiological noise at thelevel of the states equation.

The new stochastic model will be fitted to simulated andactual data (model inversion or model identification), byusing the innovation method (Ozaki 1994, Jiménez et al.2006), which has been successfully used in differentcontexts. In Valdés et al. (1999) for the simultaneousestimation of the unknown parameters and the unobservedcomponents of a neural mass model for EEG data, inPedroso et al. (2003) for the estimation of a HIV-AIDS

epidemic model from actual data, and in a biophysicalmodel of the EEG-fMRI fusion (Riera et al. 2007).

Based on our stochastic MHM we then reformulate theproblem of assessing the neurophysiology of the hemody-namic signal as a model inference task. That is, a keyassumption of the MHM was that cerebral blood flow(CBF) is related to excitatory and not to inhibitory activity(Sotero and Trujillo-Barreto 2007). Although this issuggested by recent experimental findings, the mechanismsunderlying the neurovascular coupling are matter of currentdebate (Attwell and Iadecola 2002; Lauritzen 2005).Additionally, the impact of inhibition on the generation ofthe BOLD signal might vary across subjects and acrossexperimental paradigms. This raises the question about thevalidity of the assumption. To address this problem thepresent paper proposes a model comparison (MC) approachthat is used to select among competing models that differon the type of neuro-vascular coupling assumed. This isdone by calculating the Akaike Information Criterion(AIC) and the Bayesian Information Criterion (BIC)corresponding to each model.

The remainder of the paper is structured as follows. InSection 2 a modified version of the previous MHM isbriefly reviewed and generalized as a SDE system. In thissame section it is described how the innovation method canbe used for estimating the parameters of the model. Theapplication of this approach to both simulated andexperimental data is shown in Section 3. Finally, inSection 4 some limitations of the present approach as wellas possible extensions are analyzed.

2 Methods

2.1 Modified deterministic metabolic/haemodynamic model(MHM)

The deterministic MHM under consideration is a modifica-tion of the original model presented in detail in Sotero andTrujillo-Barreto (2007). The goal of that work was topropose a biophysical model that couples inhibitory andexcitatory neuronal activity to the resulting BOLD re-sponse. This model capitalized on experimental findingsshowing evidence for a different impact of inhibition andexcitation on brain metabolism and hemodynamics.

An overall view of this MHM is depicted in Fig. 1. Allthe variables are normalized to baseline values. As seen inthe figure, changes in excitatory (ue(t)) and inhibitory (ui(t))neuronal activities are linked to the corresponding changesin glucose consumption (ge(t) and gi(t)) by means of lineardifferential equations. Then, the total glucose consumptionis calculated as a weighted average of the excitatory andinhibitory contributions. The glucose variables were then

252 J Comput Neurosci (2009) 26:251–269

directly related to the metabolic rates of oxygen forexcitatory (me(t)) and inhibitory (mi(t)) activities, as wellas to the total oxygen consumption (m(t)).

The equations describing the metabolic model werederived from the following considerations (see Sotero andTrujillo-Barreto 2007 for a detailed discussion on this).First, the metabolism of inhibition was modelled to match aconstant oxygen-to-glucose index (OGI) OGI|i=6. Thisvalue results from assuming that glycolysis is the mainprocess that fuels inhibitory activity. This also means thatall the lactate produced by glycolysis in the astrocytespasses to the neuron where it is completely oxidized(Pellerin and Magistretti 1994). Excitatory synapses in turnare more abundant and less efficient than inhibitorysynapses (Attwell and Iadecola 2002). Thus, we assumethat during high excitation levels, the increase in the ATPdemand is such that a faster (although less efficient)metabolic mechanism is needed to provide enough energyfor neurotransmitter recycling. Consequently, an increasingfraction x of the glucose undergoes a glycogenoliticpathway, according to the Glycogen shunt model (Shulmanet al. 2001). As a result, a fraction of lactate accumulatesbecause it is lost to the oxidative metabolism occurring inthe neuron. Notice, however, that glycogenolysis may not

be the only mechanism responsible for the incompleteoxidation of glucose. Thus, in this paper the interpretationof the fraction x is generalized to account for the glucoselost to the oxidative pathway due to all possible non-oxidative processes that may take place. Following Soteroand Trujillo-Barreto (2007), this fraction is modelled as asigmoid function:

x tð Þ ¼ 1

1þ e�c ge tð Þ�dð Þ ð1Þ

where the parameters c and d control the steepness andposition of the threshold, respectively. This means that thefraction x of glucose following the glycogenolitic pathwayis an increasing function of the metabolic rate of glucosedue to excitatory activity. That is, for low metabolic rate ofglucose (low excitatory activity) the predominant process isglycolisis (x≈0; OGI≈6), while for a higher glucoseconsumption (higher excitation levels), glycogenolisisbecomes increasingly important (x>0; OGI=6−3x). Atpresent there are very few experimental data about thedetailed functional dependence between x and glucoseconsumption for excitatory activity. Shulman et al. (2001)for example used very scarce data to derive approximateempirical values of the relationship between lactate efflux

Fig. 1 Graphical representation of the metabolic/hemodynamic model (MHM) used in the paper. This is a modification of the original modelproposed in Sotero and Trujillo-Barreto (2007)

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across the blood–brain barrier and OGI values. In thepresent paper on the contrary we use the non-linear function(1) with the free parameters c and d that will be estimatedfrom the data. Notice that by changing these parameters, awide range of monotonically increasing functions can bemodelled (concave, linear or convex) depending on therange of variation of the glucose consumption.

Other modifications of the previous MHM are related tothe coupling between excitatory neuronal activity andnormalized CBF. In Sotero and Trujillo-Barreto (2007),the mathematical expressions for this, were given by theneuro-vascular coupling model by Friston et al. (2000). Inthe present paper these expressions will be slightlymodified to consider a simplified case in which timeconstants for signal decay (ts) and for autoregulatoryfeedback (t f) are related as t f ¼ 4t2s ¼ k2

s (Fig. 1). Byassuming this fixed relationship between signal and auto-regulatory time constants we are effectively convolving theexcitatory synaptic activity with a non-negative alphafunction. This leads to a parsimonious and stable model.However, it is a simplification of Friston’s original neuro-vascular coupling model in which the signal decay andauto-regulatory feedback time constants are given anarbitrary relationship. When treated as free parameters, theformer corresponds roughly to the decay of extra cellularsignals such as nitric oxide, whereas the latter has a timeconstant that is consistent with 0.1 Hz oscillatory V (vaso-active) signals in optical recordings and the low-frequencyfluctuations seen in resting state fMRI (Friston et al. 2000).This is roughly the case for our simplified version;however, the damped oscillatory response that models theCBF undershoot is attenuated.

Within the context of Friston’s model this transientbehaviour of CBF signal is critical since it is directlyresponsible for the post-stimulus undershoot of the BOLDresponse. Nevertheless, the exact nature of this phenomenonis a matter of current debate. While there are supporters ofthe idea that the CBF undershoot might actually be involvedin the generation of the BOLD post-stimulus undershoot(Hoge et al. 1999), other authors suggest the possibility thatthe later might be the result of a delayed cerebral bloodvolume (CBV) compliance (Mandeville et al. 1999; Kida etal. 2007) or due to a continued elevation of oxygenmetabolism after vascular recovery (Lu et al. 2004). Thissecond point of view has led to the formulation of modelsin which either CBV or oxygen consumption return tobaseline more slowly than CBF (Mandeville et al. 1999;Valabregue et al. 2003; Buxton et al. 2004; Kong et al.2004), creating an undershoot of the BOLD signal withoutan undershoot of the CBF signal. In our model we takethese two effects into account. First, the CBV compliance isconsidered by using non-zero values for the viscoelasticparameter t (see Fig. 1) that controls how fast CBV adjusts

to changes in CBF (Buxton et al. 2004). Second, theprolonged oxygen consumption during the post-stimulusperiod is easily incorporated by capitalizing on the explicitdissociation of vascular response and energy metabolism inour model (see Fig. 1). This indicates the underlyingassumption that increased metabolic demand need not perse cause a concomitant increase in blood flow (Attwell andIadecola 2002; Lu et al. 2004; Sotero and Trujillo-Barreto2007). We will go back to this point in the Section 4.

On the other hand, in the previous MHM the normalizedCBF ( f ) was related to excitation and not to inhibition.Although there is recent experimental support for thisassumption (see Sotero and Trujillo-Barreto 2007 for adetailed discussion), the actual impact of inhibition on thegeneration of the BOLD signal might vary across subjectsand across experimental paradigms. Thus, in the presentpaper this assumption is relaxed and the equation for f ismodified to take into account both excitation and inhibition:

s:f tð Þ ¼ e ue tð Þ � 1ð Þ þ m ui tð Þ � 1ð Þ � 2kssf tð Þ � k2s f tð Þ � 1ð Þ: ð2Þ

Coefficients e and μ control the relative contribution ofeach type of activity. This will be used later to exploredifferent types of neurovascular coupling models.

Finally, we use the Balloon model (Buxton et al. 1998,2004) as a bridge for linking the output of the metabolicand vascular models, to normalized CBV (v) and deoxy-hemoglobin content (q). The measured BOLD signal is thencalculated by using a linear observation equation as inBuxton et al. (2004). The MHM in Fig. 1 is then describedby a system of eight nonlinear and non autonomousordinary differential equations and one linear observationequation. This can be written in a compact form by usingvector notation:

dy tð Þ ¼ a t; y; qð Þdtz tið Þ ¼ q23 q24 1� y7 tið Þð Þ � q25 1� y8 tið Þð Þð Þ ¼ c qð ÞT y tið Þ � 1ð Þ

ð3Þwhere z(ti) is the observed BOLD signal at the i-th timepoint, y(t) is the vector of states given by:

y ¼ y1 tð Þ; y2 tð Þ:::::; y8 tð Þ½ �T¼ ge; se; gi; si; f ; sf ; q; v� �T ð4Þ

with initial values:

y 0ð Þ ¼ 1; 0; 1; 0; 1; 0; 1; 1½ �T ð5Þθ is the vector of model parameters (see Table 1), 1 denotesa column vector of ones with length 8 and the constantvector c(θ) is:

c qð Þ ¼ 0; 0; 0; 0; 0; 0;�q23q34; q23q25½ �T ð6Þ

The detailed expression for vector a(t, y, θ) is given inAppendix A.

254 J Comput Neurosci (2009) 26:251–269

As seen in Fig. 1, the inputs to the MHM are thenormalized excitatory and inhibitory neuronal activitieswhich will be modeled by using periodic Bell functions:

ue tð Þ ¼ 1þ q14

1þ t�ntT�q18q16

��� ���2q17ui tð Þ ¼ 1þ q15

1þ t�ntT�q21q19

��� ���2q20ð7Þ

where nt ¼ maxn

n ¼ 0; 1 . . . : t � nT < Tf g and period T.Here, parameters θ14 and θ15 are respectively the amplitudechanges in excitatory and inhibitory activities with respectto baseline; θ18 and θ21 denote the temporal localization ofthe center of the curve; θ16 and θ19 represent the full with athalf maximum of the activities; and θ17 and θ20 control theraising and decaying steepness. Note that this parameter-isation of the input is periodic. This means that we canmodel a burst of inhibitory and excitatory activity at theonset of any event or epoch. Critically, because theparameters of this input function are free, we are effectivelyestimating the inputs to the system, as well as the hiddenstates (see below). We will see examples of these estimatedinput functions later in the simulations and empiricalexamples.

2.2 Stochastic MHM (sMHM)

In this section, the MHM is generalized to incorporatestochastic components representing physiological/system

and observation noises (sMHM). The new model isexpressed as a system of eight stochastic differentialequations (SDE) describing the dynamics of the unobservedstates, and one observation equation for the BOLDresponse. This system can be written as a continuous-discrete state-space model of the form:

dy tð Þ ¼ a t; y; qð Þdt þ b qð Þdw tð Þz tið Þ ¼ c qð ÞTy tið Þ � c qð ÞT1þ e tið Þ ð8Þ

where the square of the elements of the vector b(θ) denotesthe variances of the physiological noise and a(t, y, θ) hasthe same expression as in Eq. (3) (see also Appendix A)and is known as drift. In this case, the vector θ containstwo more parameters representing the observation andstates noise variances (see Table 1).

Note that Eq. (8) differs from Eq. (3) for the MHM intwo ways. First, a scalar Wiener process w(t) wasintroduced in the states equation, which represents theadditive physiological noise. This is based on studiesshowing fluctuations in local extracellular glucose concen-tration coupled to neuronal events (Hu and Wilson 1997;McNay and Gold 2002). In the same way, it has beenreported that CBV fluctuates spontaneously while subjectsare in a physical and mental state of rest (Eke and Hermán1999). Similar spontaneous oscillations in CBV have beenalso reported by Elwell et al. (1999) and Schroeter et al.(2004) and were related to the activity of vascular smoothmuscle cells and the physical properties of the cerebral

Table 1 Interpretation and values of the parameters used in the simulations

Parameter Interpretation Value

θ1(aeke) Efficacy of glucose consumption response to excitation 1θ2(ke) Inverse of the time constant for glucose consumption response to excitation 1.5θ3(aiki) Efficacy of glucose consumption response to inhibition 1θ4(ki) Inverse of the time constant for glucose consumption response to inhibition 1.5θ5(c) Steepness of the sigmoid function 2.5θ6(d) Position of the threshold of the sigmoid function 1.6θ7(g) Baseline ratio of excitatory to inhibitory synaptic activity in the voxel 5θ8(e) Efficacy of blood flow response to excitation 0.5θ9(ks) Inverse of the time constant for CBF response 0.6θ10(t0) Transit time through the balloon 1θ11(t) Time constant that controls how fast CBV adjusts to changes in CBF 10θ12(a) Coefficient of the steady state flow–volume relationship 0.4θ13(σ) Noise variance for the state variables 0.001θ14(ne) Increase with respect to baseline in excitatory activity 0.5θ15(ni) Increase with respect to baseline in inhibitory activity 0.6θ16, θ17, θ18 (xe1, xe2, xe3) Parameters related to the shape of excitatory activity 3,10,3θ19, θ20, θ21 (xi1, xi2, xi3) Parameters related to the shape of inhibitory activity 3,10,3θ22(σobs) Noise variance for the observations 0.001θ23 (V0) Baseline blood volume 0.03θ24 (a1) Weight for deoxyhemoglobin change 3.4θ25 (a2) Weight for blood volume change 1θ26 (μ) Efficacy of blood flow response to inhibition 0

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vessel wall. Additionally, CBF changes as measured bylaser-Doppler flowmetry shows stochastic fluctuations too(Hermán and Eke 2006). These can be induced by severalfactors such as blood pressure variations, changing vasculardiameters, and those arising from the rheological propertiesof flowing blood (Hermán and Eke 2006). Taking this intoaccount, noise was added to the glucose consumption andCBF inducing signals as well as to CBV and deoxy-hemoglobin concentration changes, that is:

b ¼ 0; q13; 0; q13; 0; q13; q13; q13½ �T : ð9ÞFor the sake of simplicity we have assumed the same

variance for all components.The second difference between the sMHM presented

here and the previous MHM is that measurement noise e(ti)was added to the observation equation (see Eq. (8)). Thisnoise is assumed to distribute normally with zero mean andvariance θ8 (e(ti∼N(0,θ8)), and can be interpreted asobservation errors that might arise during data recordingdue to instrumental noise as well as environmental factors.This is also important from the modeling point of viewwhen fitting the model to actual data, since this term canaccount for un-modeled activity. For simplicity, the inter-sect −c(θ)T1 in the observation equation can be absorbed inthe noise term and without loosing generality, the Eq. (8)can be rewritten as:

dy tð Þ ¼ a t; y; qð Þdt þ bdw tð Þz tið Þ ¼ c qð ÞTy tið Þ þ e tið Þ ð10Þ

where:

e tið Þ � N �c qð ÞT1; q8� �

: ð11Þ

2.3 Model identification: the innovation methodand the local linearization filter

Typically DCMs based upon ordinary differential equations(and known inputs) do not have to worry about estimatingthe hidden states. However, our model is based upon morerealistic stochastic differential equations (see Eq. (10)). Thismeans that we have to infer both the parameters of themodel and the underlying hidden states. This is a dual-estimation problem requiring inference on both parametersand states. This problem can be solved using an iterativeapproach, whereby we optimise the parameters assumingthe states are fixed; we then optimise the states assumingthe parameters are fixed. These iterated steps can beregarded as a form of recursive maximum likelihood or,in the context of priors, iterated conditional modes. It is notunrelated to expectation maximisation, although we do notconsider uncertainty in the states when estimating theparameters or vice-versa. To estimate the states, given the

parameters, we use a technique called local linearization(LL). This can be regarded as a form of extended Kalmanfiltering that is formulated for continuous state-spacemodels. When optimising the parameters, we simply takethe maximum likelihood based upon the prediction error orestimated innovations. This is known as the innovationmethod (Ozaki 1994).

This approach is not the only option in the literature.That is, the problem in which the parameters of a diffusionprocess (of the type described by Eq. (10)) must beestimated from discrete and noisy observations, have beenstudied with several other methods, such as maximumlikelihood (Schweppe 1965), prediction errors recursions(Solo 1980) and quasi-likelihood (Nolsoe et al. 2000)methods (see Jiménez et al. 2006 for a recent review).Nevertheless, the innovation method based on the LL filteris the most suitable for solving our estimation problemsince it can be applied to nonlinear and non autonomousstate-space models with low computational cost (see forexample Riera et al. 2004).

For the state-space model (10) the innovation estimator bqis defined (Ozaki 1994, Jiménez and Ozaki 2006) by:

bq ¼ minq

�L qð Þf g ð12Þ

where the log-likelihood L(θ) is calculated as:

L qð Þ ¼ �n ln 2pð Þ �Xni¼1

ln det b@ϑti=ti�1

� �� �� bϑT

tib@ϑti=ti�1

� ��1bϑti

ð13Þwhere bϑT

tiand b@ϑ

ti=ti�1are approximations to the discrete

innovation ϑTti¼ z tið Þ � c qð ÞTyti=ti�1

þ c qð ÞT1 and its vari-ance

Pϑti=ti�1

¼ c qð ÞTPtiþ1=tic qð Þ þ q8. Here yti=ti�1

andPtiþ1=ti

denotes predictions of y and its variance at t= ti.Estimated values bϑT

tiand b@ϑ

ti=ti�1are obtained by means of

the LL filter as described in Appendix B. The goodness-of-fit of the model is evaluated by testing whether thedistribution of the fitted innovation deviates from aGaussian distribution (Jiménez and Ozaki 2006).

In order to solve the nonlinear optimization problem inEq. (12), a sequential quadratique programming (SQP)method as implemented in the MATLAB OptimizationToolbox was used (Gill et al. 1981). This method allows forspecifying constraints in the form of upper and lowerbounds for the parameters to be estimated. In the presentcase, these bounds were chosen to cover the range ofbiophysically plausible values of each free parameter.

As SQP is a local optimizer, it may converge to localminima. To avoid this, the following procedure wasapplied. First, 10 equally spaced initial parameter valuesθ0 were taken from the respective ranges of variation of theparameters and used to feed the SQP algorithm. Then, theoptimal set of parameters was obtained by choosing those

256 J Comput Neurosci (2009) 26:251–269

corresponding to the greatest likelihood, provided the fittedinnovations were Gaussian. This procedure emulates aglobal optimizer by obtaining solutions which are relativelyindependent of the parameters’ initial values θ0.

2.4 Model comparison: AIC and BIC

The MC framework has been used to great effect in themodelling of fMRI data but in a different context. Penny etal. (2004) for example, used MC to compare DynamicCausal Models for fMRI (Friston et al. 2003). This allowedfor guiding choices about model structure, both concerningthe intrinsic functional connectivity pattern and the contex-tual modulation of individual connections. Penny et al.(2007) used Bayesian MC as a framework for principledselection of signal and noise models within the context ofthe general linear model. In the present paper MC will beused for testing the appropriateness of some of theassumptions of our sMHM. Specifically, we are interestedin the model inference task of selecting the best amongcompeting models that differ on the type of neuro-vascularcoupling assumed for a given data set.

From a Bayesian perspective, model inference is basedon Bayes rule whereby, after observing data z, prior beliefs(p(H)) about model structure are updated to posteriorbeliefs (p(H|z)). This update requires the evidence p(z|H).MC is then implemented by picking the model thatmaximizes the posterior probability p(H|z). If the modelpriors p(H) are uniform then this is equivalent to pickingthe model with the highest evidence. In this case, modelswith high evidence optimally trade-off the two conflictingrequirements of a good model, its simplicity and itscapability for data fitting (MacKay 1992). Nevertheless,the computation of the model evidence requires a priorprobability for the parameters of the model to be specified.In our case we are not using such probabilities, in fact, in aBayesian interpretation we can assume that our approachconsiders non-informative uniform prior probabilities for allthe parameters involved in the model. In this context,alternative approximations to the model log-evidence canbe used instead (Penny et al. 2004; Kass and Raftery 1993).These are the AIC (Akaike 1973, 1983) and the SchwarzCriterion, which in the limit of large sample sizescorresponds to the BIC (Schwarz 1978).

The AIC advocates that given a class of competingmodels for a data set, one chooses the model that minimizes

AIC ¼ �2L qð Þ þ 2p ð14Þwhere p is the number of parameters and L(θ) is the log-likelihood (see Eq. (13)). The justification for this criterionis primarily based on predictive arguments. That is, if thepredictive distribution of a future data is conditional on asingle model and on its estimated parameters, the AIC

picks the model which gives the best approximationasymptotically, in a Kullback–Leibler sense (Akaike1973). However such a predictive distribution does nottake into account the uncertainty about parameters valuesand model form (Aitchison and Dunsmore 1975), leadingto over-parameterized models (Shibata 1976; Katz 1981).A second argument by Akaike (1983) is Bayesian. It statesthat MCs based on AIC are asymptotically equivalent tothose based on Bayes factors (see below). This is true,however, only if the precision of the prior is comparable tothat of the likelihood, but not in the more usual situationwhere the prior information is small compared to theinformation provided by the data. In this later case,Schwarz criterion indicates that the model with highestposterior probability is the one that minimizes

BIC ¼ �2L qð Þ þ p log Nobsð Þ ð15Þ

where Nobs is the number of observations.Empirically, BIC is observed to be biased towards

simple models and AIC to complex models. In fact,comparing Eqs. (14) and (15) indicates that for values ofp and Nobs typical for the data examine here, BIC pays aheavier parameter penalty than AIC. For a more detaileddiscussion on the pros and cons of the two approximationsthe interested reader may refer to Kass and Raftery (1993).In the present paper the two criteria are reported.

On the other hand, individual AIC or BIC values are notinterpretable as they contain arbitrary constants and aremuch affected by sample size. Thus, pair-wise MCs will bebased on the Bayes factors. These are ratios of evidences(or differences between log-evidences) and can be inter-preted as a summary of the evidence provided by the datain favour of one scientific theory, represented by astatistical model, as opposed to another. Additionally, ifthere are more than two models to compare then we chooseone of them as a reference model and calculate Bayesfactors relative to that reference. The Bayes’ factors (in log-values) between model Hi and the reference model H0

(logBi0) corresponding to AIC and BIC approximations canbe expressed as:

logBi0 AICð Þ � � 1

2$AICi0 ¼ � 1

2AICi � AIC0ð Þ ð16Þ

and

logBi0 BICð Þ � � 1

2$BICi0 ¼ � 1

2BICi � BIC0ð Þ ð17Þ

respectively. These approximations to the log Bayes factor(or more simply differences in log-evidence) have quanti-tative interpretations. That is, a difference in log-evidenceor log-marginal- of e corresponds roughly to a Bayes factorof exp(e)∼20. This means that the marginal likelihood ofthe better model is twenty times greater than the alternate

J Comput Neurosci (2009) 26:251–269 257

model (i.e., we can be 95% confident that one model isbetter than the other). In this paper, we will always compareour models to the best model considered. This forces thelog Bayes factor of the best model to have a value of zeroand the remaining comparisons to be less than zero.Additionally we will follow the suggestion of Penny et al.(2004) that a model selection can be made if the twomeasures Eqs. (16) and (17) provide consistent evidence.

3 Results

In this section numerical simulations are used to study thedynamic properties of the sMHM, as well as to validate theestimation procedure and the model comparison frame-work. The application of the proposed approach to actualfMRI data recorded during a motor task is also shown.

3.1 Validation of the model identification methodfrom simulated data

The simulated BOLD data was generated from our sMHMby integration of the SDE in Eq. (10). For this, the LLscheme (Jiménez et al. 1999, Jiménez 2002) was used withan integration step size of 0.1 s. The values of theparameters used in the simulation are shown in Table 1.The inputs were train pulses representing inhibitory andexcitatory neuronal activities, which were generated usingEq. (7). Each train consisted of eight pulses with intervalbetween pulses of 31 s. With these settings, 100 realizationsof the BOLD signal z were generated. Each realizationconsisted of 336 s of fMRI activity with a temporalresolution of 0.1 s. The data was then down-sampled by afactor of 30 to give a low temporal resolution of 3 s.

The dynamics of all the variables involved in the modelfor one realization of the BOLD signal are summarized inFig. 2. For illustrative purposes the results were displayedin the form of a flow chart (similar to that in Fig. 1)showing the transformations undergone by the inhibitoryand excitatory neuronal activity at the different stages of thesMHM. Note that in this simulation CBF changes areinduced by the excitatory activity only (μ=0).

Given the simulated data, our generative model was theninverted for each realization of z separately. For this, someof the parameters of the model were fixed to the valuesused in the forward calculations, which were taken fromoverall values reported in the literature (see Table 1). Thesewere the parameters related to the BOLD observationequation (θ23,θ24,θ25) (taken from Buxton et al. 2004),the parameter θ10 representing the transit time through theBalloon (from Friston et al. 2000), the parameter θ12 for thesteady-state flow-volume relation (Buxton et al. 2004) andthe parameter θ7 that denotes the baseline ratio of excitatory

to inhibitory synaptic activity in the voxel (Abeles 1991;Tamura et al. 2004). In the case of θ7 we also capitalized onprevious results demonstrating that simulated BOLD signalshowed little variation over a wide rage of values of thisparameter (Sotero and Trujillo-Barreto 2007). Additionally,the parameters θ17 and θ20 were also set to guarantee a fastonset and offset of the excitatory and the inhibitory synapticactivity. The remaining 17 parameters were taken as freeparameters and estimated from the data. In all cases, thetolerances for constraints violations and for log-likelihoodand estimated parameter convergences were all the sameand equal to 10−16.

The numerical results are reported in Table 2 thatsummarizes the mean of the estimated parameters acrossrealizations and the 95% confidence interval. This showsthat all the parameters are accurately estimated, except forthe noise variance of the state variables θ13 which wasslightly overestimated. This conclusion can also be drawnfrom Fig. 3 where the histograms of the estimatedparameters are shown. In this figure the small arrowindicates the true value of the parameter in each case. Therobustness of the estimation method to variations in theobserved signal given by both the observation and systemnoises is appreciated from the small variability of theestimated parameters around the true values.

The actual and reconstructed BOLD signals, as well asthe histogram of the fitted innovations for one realization ofz (the one depicted in Fig. 2) are shown in Fig. 4. As can beseen, the original signal is satisfactorily recovered and thefitted innovations do not deviate from a Gaussian distribu-tion. This last condition was tested for by using theLilliefors test (Lilliefors 1967) with a significant level of5%.

The corresponding temporal dynamics of the recon-structed states by as well as of other variables involved in themodel are shown in Figs. 5 and 6, respectively. Thevariables in Fig. 5 were calculated from the estimated by andbq. The results are in agreement with the true dynamicsshown in Fig. 2.

3.2 Validation of the model comparison frameworkfrom simulated data

As stated above, the MC framework offers a principled wayof making inferences about different aspects of a givenmodel. In our case we are interested on making inferencesabout the type of neurovascular coupling assumed in theMHM for a given data set. For this, three possible versions

Fig. 2 Temporal evolution of all haemodynamics and metabolicvariables of the sMHM for one realisation of z. This represents theresponse of the model to a pulses train of inhibitory and excitatoryneuronal activities

b

258 J Comput Neurosci (2009) 26:251–269

J Comput Neurosci (2009) 26:251–269 259

the CBF equation in the MHM were used to defined threedifferent models to be compared:

I : s:f tð Þ ¼ m ui tð Þ � 1ð Þ � 2kssf tð Þ � k2

s f tð Þ � 1ð ÞII : s

:f tð Þ ¼ e ue tð Þ � 1ð Þ � 2kssf tð Þ � k2

s f tð Þ � 1ð ÞIII : s

:f tð Þ ¼ e ue tð Þ � 1ð Þ þ m ui tð Þ � 1ð Þ � 2kssf tð Þ � k2

s f tð Þ � 1ð Þ:

ð18ÞThese models assume that CBF changes depend on (I)

inhibitory activity solely, (II) excitatory activity solely or(III) both excitatory and inhibitory activity with differentrelative contributions (e and μ). Notice that model II is theone we have been using in the simulations shown so far.

In order to validate the use of the MC framework in ourcontext, three different simulated data sets were analysed.Each data set was generated as one realization of the BOLDsignal z under one of the three assumptions in Eq. (18). Thesame simulation settings as in the previous section wereused in all cases, and the inhibitory and excitatorycontributions were set to e=0.5 and μ=0.3, respectively.Then models I, II and III were fitted to each data setseparately and the corresponding AIC and BIC werecomputed.

The results of the calculations are shown in Table 3. Asexpected, the zero values for ΔAICi0, ΔBICi0 are located in

Table 2 True and estimated values of the model parameters computedfrom 100 realizations of z

Parameter Truevalues

Mean estimated values(95% confidence interval)

Search range(LB, UB)

θ1 1 0.9974±0.1268 (0.1, 1.2)θ2 1.5 1.4989±0.0366 (1, 2)θ3 1 0.9952±0.0518 (0.1, 1.2)θ4 1.5 1.5091±0.1203 (1, 2)θ5 2.5 2.4957±0.0964 (2, 3)θ6 1.5 1.5035±0.1977 (1.1, 1.9)θ8 0.4 0.3998±0.0184 (0.1, 0.8)θ9 0.6 0.6484±0.1729 (0.2, 1)θ11 10 10.0151±0.6036 (5, 15)θ13 0.01 0.0151±0.0101 (0, 0.06)θ14 0.6 0.5655±0.0810 (0, 1)θ15 0.4 0.4442±0.0846 (0, 1)θ16 10 10.5045±2.1124 (2, 14)θ18 30 31.1636±3.3802 (20, 40)θ19 10 10.091±2.5605 (2, 14)θ21 30 29.8764±3.0362 (20, 40)θ22 0.002 0.0021±0.0018 (0.001, 0.01)

Fig. 3 Histograms of the estimated parameters for 100 realisations of z. The arrows indicate the actual values of the parameters

260 J Comput Neurosci (2009) 26:251–269

Fig. 4 Fitting of the sMHM tosimulated data. Left panel: Sim-ulated and reconstructed BOLDsignal. Right panel: Histogramof the fitted innovations

Fig. 5 Temporal dynamics of the reconstructed states by of the sMHM for the simulated data

J Comput Neurosci (2009) 26:251–269 261

the diagonal of the table, showing that in all cases evidencein favour of the correct model was obtained.

3.3 Estimation from actual data

The proposed methodology was then tested on actual data.For this, fMRI time series were obtained from a singlesubject at 1.5 T using a Magnetom Symphony scanner(Siemens, Erlangen, Germany) equipped with a fastgradient system for echoplanar imaging (EPI). A standardradiofrequency head coil was used with foam padding to

comfortably restrict head motion and partially suppressscanner noise. High-resolution 3D sagittal T1-weightedimages were obtained in every subject. BOLD data werecollected using a gradient-echo, echoplanar sequence (TR=3,000 ms, TE=48 ms). Each scanning run comprised 126volumes (nine rest-activation blocks) with 24 sagital slices.The spatial resolution was 2×2 mm in plane, and 5 mmthrough plane, with no skip in between planes.

The subject performed a motor task, comprising fingeropposition sequential movement (five repetitions) followedby repetitive flexion-extension movements of the hand.

Fig. 6 Variables of the sMHM calculated using the estimatedparameters and the reconstructed states from the simulated data.These are excitatory (ue) and inhibitory (ui) neuronal activities, total

glucose (g) and oxygen (m) consumptions, and the fraction of glucosex consumed in non-oxidative processes

Table 3 Validation of the use of the MC framework for three different sMHM variants

Data simulated with variant sMHM variant

I II III

logBi0 (AIC) logBi0 (BIC) logBi0 (AIC) logBi0 (AIC) logBi0 (BIC) logBi0 (AIC)

I 0 0 −23.12 −23.12 −18.75 −73.55II −16.25 −16.25 0 0 −5.38 −60.38III −65.30 −10.30 −66.12 −11.12 0 0

262 J Comput Neurosci (2009) 26:251–269

Each block consisted of rest state (i.e., no stimulus)alternated with activation. Subjects were instructed to keepthe amplitude and frequency of the motor task constant.

The data were pre-processed with SPM2 (WellcomeDepartment of Imaging Neuroscience, http://www.fil.ion.ucl.ac.uk/spm). The time-series were realigned, correctedfor movement-related effects, and spatially normalized intothe standard Talairach space (Talairach and Tournoux1988). Three regions of interest (ROI) were selected (seeFig. 7) from the areas activated during the experiment:thalamus, supplementary motor area (SMA) and primarymotor area (M1). The BOLD signal z for each ROI wassummarized by using the corresponding principal eigen-variates, as implemented in SPM2.

We then considered the inference task of selecting theoptimal type of neuro-vascular coupling for the signal ateach ROI by using the MC framework described in thesimulations. The corresponding ΔAICi0 and ΔBICi0 areshown in Table 4. The results show that model II in whichCBF depends only on excitatory activity is the one that bestexplains the data. Model I, it was the worst model. In thatcase the innovations were not even Gaussian. Althoughmodel III did received support from the data according toΔAICi0, only the best model was selected for the rest of theanalysis.

The reconstructed and actual BOLD signals in the caseof model II are compared in Fig. 8 for the three selectedROIs. The right panels show the histograms of the fittedinnovations.

The estimated parameters for the three brain areasanalysed are reported in Table 5. Notice that parametervalues for M1 and SMA are very similar, and that theposition of the threshold of the sigmoid function (θ6), theinverse of the time constant for CBF response (θ9), andthe time constant that controls how fast CBV adjusts tochanges in CBF (θ11), have close values for the three areas.

Using the estimated parameters, the temporal dynamicsof the state variables by were reconstructed for all areas andthe results are shown in Fig. 9. Excitatory and inhibitory

neuronal activities, glucose consumed for excitatory activ-ity, total glucose and oxygen consumptions, and thefraction of glucose x consumed in non oxidative processare also calculated for each area and displayed in Fig. 10.According to the results, the increase in excitatory activitywas greater in the thalamus (37%) than in SMA and M1(13% in both cases). On the contrary, inhibitory activity hadlarger increases in SMA and M1 (98% and 91% respec-tively) than in the thalamus (79%). Interestingly, in thethalamus both excitatory and inhibitory activity increasesbegan at the same time, whereas in SMA and M1 excitatoryactivity preceded inhibitory activity for about 3 s. Inaddition, the time duration of excitatory activity pulses inthe three areas was shorter than the duration of inhibitoryactivity pulses. That is, after excitation ended, there wasstill inhibition for about 15 s in the thalamus and for about6 s in SMA and M1.

On the other hand, CBF showed increases of about 42%in SMA and M1 and of about 21% in the thalamus. Theglucose consumed for excitatory activity increased in 16%in the thalamus and only between 2% and 3% in SMA andM1. On the contrary, glucose consumed for inhibitoryactivity increased in the thalamus in 10% and had largerincreases of almost 100% in SMA and M1. As a result,total glucose consumption had approximately the sameincrease of 19% in all areas, but lasted longer in SMA andM1. For the case of total oxygen consumption, it increasedmore in SMA and M1 (18–19%) than in the thalamus(15%).

4 Discussion

In this paper a previous biophysical generative model of theBOLD signal (Sotero and Trujillo-Barreto 2007) wasgeneralized to include both physiological and measurementnoises as well as to consider relative contributions ofexcitatory and inhibitory activity to the observed signal.The new model was then described by a system of

Fig. 7 Regions of interest se-lected from the brain areas acti-vated during the motor taskexperiment. White arrows showfrom left to right: primary motorarea (M1), supplementary motorarea (SMA) and thalamus

J Comput Neurosci (2009) 26:251–269 263

stochastic differential equations. This was inverted byusing the innovation method based on the LL filter (Ozaki1994; Jiménez and Ozaki 2003, 2006). The rationale forchoosing this method was that it provides an efficient andstable numerical scheme for the integration and estimationof stochastic differential equations, which has also beenused successfully in several previous works (Valdés et al.1999; Pedroso et al. 2003; Riera et al. 2004, 2007).Simulations showed that it is possible to estimate theunknown parameters and the unobserved states of thestochastic model that is, cerebral blood volume, deoxy-hemoglobin content and cerebral blood flow changes, aswell as total oxygen and glucose consumptions. Further-

more, the model can also estimate the specific contributionsof excitation and inhibition to oxygen and glucoseconsumption as well as the levels of excitatory andinhibitory activities. Thus, the model proposed here couldbe helpful for exploring the relationship between excitatory,inhibitory, metabolic and hemodynamic activities in healthyand disease brains.

At this point, it is worth discussing some importantissues about our modeling of both physiological andobservation noise. In the present paper we have used aparsimonious model involving independent Gaussian vari-ables for discrete observations and Gaussian white noise(Wiener diffusion process) for the stochastic differential

Table 4 ΔAICi0 and ΔBICi0 values calculated for each sMHM variant and for each ROI

Data simulated with variant sMHM variant

I II III

logBi0 (AIC) logBi0 (BIC) logBi0 (AIC) logBi0 (AIC) logBi0 (BIC) logBi0 (AIC)

Thalamus −12.26 −12.26 0 0 −1.34 −56.35M1 −8.85 −8.85 0 0 −1.90 −56.90SMA −16.84 −16.84 0 0 −1.42 −56.40

Fig. 8 Fitting of the sMHM to the motor task data. Left panels: Comparison of the reconstructed and actual BOLD signals for the three brainareas selected. Right panels: Histograms of the fitted innovations are plotted

264 J Comput Neurosci (2009) 26:251–269

equations describing the states dynamics. However, inpractice, these two sources of noise will typically showpronounced serial correlations because they are not gener-ated by simple diffusions. We want to discuss some cases in

which the model and inference tools presented here can stillbe applied by introducing very few modifications.

As pointed out by Worsley et al. (2002), the simplestmodel of the temporal correlation structure in BOLDsignals is a first order autoregressive AR(1) model.Following this idea, we can proceed by pre-whitening thedata and then modeling the residuals (after fitting an AR(1)model) in the same way as in Eq. (10) (Worsley et al.2002). In the case of the physiological noise, one couldthink of some kind of “colored noise”, as opposed to theusual white noise Wiener process for modeling stochasticdifferential equations. Two different strategies could befollowed. Firstly, it is well-known that Gaussian whitenoise is a good approximation to a colored noise processwhen the time scale of the deterministic system is muchlarger than the time scale of the noise correlation. In suchcircumstances, a Gaussian white noise could be a validassumption. A more realistic scenario is to assume acolored system noise explicitly, which means that we needto adapt all the inference tools to that problem. In that case,a very promising approach is to use a variational treatmentof our dynamic model (see Friston et al. 2008). However,the inference method presented here can still be applied ifwe model the system noise as an Ornstein–Uhlembechprocess. This is a colored noise with a well-known

Table 5 Estimated values of the parameters for the three ROIs(thalamus, M1 and SMA) selected in the motor task experiment

Parameter Thalamus M1 SMA Search range (LB, UB)

θ1 0.57 0.20 0.12 (0.1, 1.2)θ2 1.05 1.22 1.19 (1, 2)θ3 0.54 1.10 1.05 (0.1, 1.2)θ4 1.91 1.15 1.20 (1, 2)θ5 2.21 2.70 2.90 (2, 3)θ6 1.80 1.81 1.80 (1.1, 1.9)θ8 0.23 0.79 0.77 (0.1, 0.8)θ9 0.52 0.50 0.51 (0.2, 1)θ11 13.81 14.57 14.01 (5, 15)θ13 0.02 0.02 0.05 (0, 0.06)θ14 0.37 0.13 0.13 (0, 1)θ15 0.79 0.91 0.98 (0, 1)θ16 4.40 12.00 12.00 (2, 14)θ18 31.60 34.68 34.69 (20, 40)θ19 12.00 12.00 12.00 (2, 14)θ21 39.62 39.45 39.42 (20, 40)θ22 0.05 0.05 0.1 (0.001, 0.1)

The search range of the parameters is reported

Fig. 9 Temporal dynamics of the reconstructed states by for the ROIs selected from the thalamus, SMA and M1

J Comput Neurosci (2009) 26:251–269 265

correlation structure and is related to a Wiener process by asimple stochastic differential equation. Therefore, themodification of the innovation approach is straightforward(Carbonell et al. 2007).

The key contribution of the present paper is theintroduction and validation of the Bayesian model compar-ison framework as a tool for making inferences about thetype of neurovascular coupling assumed in the generativemodel. That is, in a previous contribution (Sotero andTrujillo-Barreto 2007) CBF changes were considered todepend on excitatory activity only. However, at present welack of a complete understanding of the processes linkingneuronal activity with vascular responses. While is recog-nized that glutamate release and the interaction of glutamatewith its receptors on neurons are necessary for theproduction of CBF responses (Lauritzen 2005), the roleplayed by inhibition is still obscure. It was established forexample that in the cerebellar cortex, GABA is notinvolved in activation-induced increases of CBF (Attwelland Iadecola 2002). In hippocampus and neocortex,however, it has been found that exogenous GABA dilatesprecapillary microvessels (Fergus and Lee 1997). In the

present paper the model comparison framework was used tomake inference about this type of hypothesis. That is, theparameters and the unobserved states were estimated fromthe data for three different models that were defined byassuming that CBF changes depend on (I) inhibitoryactivity solely, (II) excitatory activity solely or (III) bothexcitatory and inhibitory activity with different relativecontributions. The corresponding AIC and BIC were thencalculated and used to select the best model.

The methodology was first validated with simulationsand then applied to actual data collected during a motortask experiment. For this data, model I was the best,although model II received support too. It is interesting tonote that the model assuming that inhibition was the onlycause of vascular responses was the worst model. This isconsistent with the idea that excitation is necessary forinducing CBF changes. According to these results, despitethe complex processes relating CBF changes and neuronalactivity (Iadecola 2004), at the voxel spatial scale it is agood approximation to neglect the inhibitory contributions.Nevertheless, it should be noted that these results are fromdata of one subject only.

Fig. 10 Temporal dynamics of the different variables of the sMHMfor the three ROIs (thalamus, SMA and M1) selected in the motor taskexperiment. These are variables are excitatory (ue) and inhibitory (ui)

neuronal activities, total glucose (g) and oxygen (m) consumptions,and the fraction of glucose x consumed in non-oxidative processes

266 J Comput Neurosci (2009) 26:251–269

From the physiological point of view, the resultsobtained with the best model in this specific data set werein broad agreement with recent experimental and theoreticaladvances in the knowledge of the basal ganglia functioning.Our results suggest an extended inhibition of the thalamicneurons followed by stimulation in a smaller neuralpopulation. In a short temporal window compared to theduration of the stimulus, the excitatory signal is suppressedwhile inhibitory signal remain during motor task. In thelight of the recently introduced “dynamic model” of thebasal ganglia function (Nambu 2005; Miyachi et al. 2006)this sequential information processing in the thalamus canbe responsible for selection of the specific motor programmediated by thalamic and cortical neurons, while othercompeting motor programs in the surrounding areas aresuppressed.

Other interesting result of our model is the anticipationof the excitatory signal in the SMA. Previous neurophys-iological (Escola et al. 2003) and imaging (Sabatini et al.2000; Haslinger et al. 2001; Mattay et al. 2002; Ceballos-Baumann 2003; Rodríguez-Rojas et al. 2005) studies havesuggested the implication of the SMA in the planning andexecution of complex motor tasks. In this context, thepreceding excitatory signal could be mediated by cortico-cortical connections, while the basal ganglia–thalamic–cortical pathway heavily mediate inhibitory signal.

Although we have focused our analysis on assessing thenature of the neuro-vascular coupling as regards the type ofneuronal activity (inhibition or excitation or both) respon-sible for CBF changes at the voxel level, the same analysis

could be used for testing other assumptions of the sMHMmodel. In previous sections for example, we discussedabout the nature of the BOLD post-stimulus undershoot.This is actually a matter of current debate and at least threepossible explanations have been proposed in the literature.Either (1) this phenomenon is directly the consequenceof a CBF post-stimulus undershoot as in Friston et al.(2000), or (2) it is the result of a delayed CBV compliance(Mandeville et al. 1999; Kida et al. 2007), or (3) of acontinued elevation of oxygen metabolism after vascularrecovery (Lu et al. 2004). In the present paper we have onlyconsidered mechanisms (2) and (3) while mechanism (1)was omitted by assuming a fixed relationship between thesignal decay and autoregulatory time constants (see Fig. 1).This assumption can then be relaxed by assuming thesetime constants as free parameters and thus the threemechanisms can be taken into account. Then the MCapproach can be used to select between models relaying oneach of the three mechanisms or on any combination ofthem. In this sense, the model comparison frameworkproposed here gives our modelling approach a great degreeof flexibility since it could be used as a principled way fortesting the appropriateness of the assumptions of the modelfor specific experimental paradigms.

Appendix A: Expression for the drift

The drift term of system (8) is given by:

a ¼

y2 tð Þq1 ue tð Þ � 1ð Þ � 2q2y2 tð Þ � q22 y1 tð Þ � 1ð Þ

y4 tð Þq3 ui tð Þ � 1ð Þ � 2q4y4 tð Þ � q24 y3 tð Þ � 1ð Þ

y6 tð Þq8 ue tð Þ � 1ð Þ þ q26 ui tð Þ � 1ð Þ � 2q9y6 tð Þ � q29 y5 tð Þ � 1ð Þ

1q10 q7þ1ð Þ y3 tð Þ þ q7

l� 1

1�e�q5 y1 tð Þ�q6ð Þ

l� 1

1�e�q5 1�q6ð Þ

y1 tð Þ� �

�q10 y8 tð Þð Þ

1q12þq11y5 tð Þ

� �y7 tð Þ

q10 q10þq11ð Þy8 tð Þ

1q10þq11

y5 tð Þ � y8 tð Þð Þ 1q12

� �

0BBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCA

ð19Þ

where l ¼ 2� 11�eq5q6

Appendix B: Local linearization filter algorithm

Let a state-space model be defined by the continuous stateequation:

dy tð Þ ¼ a t; y tð Þ; qð Þdt þ b qð Þdw tð Þ; for t � t0 ð20Þ

and the discrete observation equation:

ztk ¼ c qð ÞTy tkð Þ þ etk ; for k ¼ 0; 1; ::::;N : ð21Þ

Assuming that θ is given and starting with byt0=t0 ¼ yt0=t0 ,bPt0=t0 ¼ Pt0=t0 , the LL filter for the model (20)–(21) is

J Comput Neurosci (2009) 26:251–269 267

defined by the recursive computation of the followingestimates (Jiménez and Ozaki 2003; Riera et al. 2007):

For all k=0,...,M−1.

1. Prediction: For j=0,…,M−1,(M≥1):

sj ¼ tk þ jtkþ1 � tk

Mð22Þ

bysjþ1=tk ¼ bysj=tk þZsjþ1�sj

0

eDsj sjþ1�sj�sð Þ a sj;bysj=tk ; q� �þ dsj s

� �ds

ð23Þ

bPsjþ1=tk ¼ eDsj sjþ1�sj�sð ÞbPsj=tk eDT

sjsjþ1�sj�sð Þ þ

Zsjþ1�sj

0

eDsj sbbTeDT

sjsds

ð24Þ2. Innovation:

bϑtkþ1 ¼ ztkþ1 � cTbytkþ1=tk ð25Þ

b@ntkþ1=tk

¼ cTbPtkþ1=tk cþ q8 ð26Þ3. Filter:

bytkþ1=tkþ1¼ bytkþ1=tk þ ktkþ1

bϑtkþ1 ð27Þ

bPtkþ1=tkþ1¼ bPtkþ1=tk � ktkþ1c

TbPtkþ1=tk ð28Þ

where ktkþ1 ¼ bPtkþ1=tkcTðb@ϑ

tkþ1Þ�1 is the filter gain. In the

above algorithm M=1 denotes the number of pointsbetween a pair of consecutive observations tk,tk+1 usedto compute the predictions bytkþ1=tk and bPtkþ1=tk , Dsj ¼@@y aðsj; ysj=tk ; qÞ and dsj ¼ @

@s aðsj; ysj=tk ; qÞ. Explicit expres-sions in terms of exponential matrices for integrals (23)and (24) are given in Jiménez and Ozaki, (2003). Thus,the numerical computation of the conditional momentsbytkþ1=tk and

bPtkþ1=tk is reduced to use a convenient algorithmto compute matrix exponentials. This was achieved inthe present paper by using Krylov subspace methods(Hochbruck and Lubich 1997).

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