487DAVID Biol Res 40, 2007, 487-502Biol Res 40: 487-502, 2007 BRDynamic Causal Models and Autopoietic Systems

OLIVIER DAVID

Inserm, U836, Grenoble Institut des Neurosciences, University Hospital, Bât. EJ Safra, BP 170, 38042Grenoble Cedex 9France, Université Joseph Fourier, Grenoble, France.CTRS-IDEE, University Hospital, Lyon, France.

ABSTRACT

Dynamic Causal Modelling (DCM) and the theory of autopoietic systems are two important conceptualframeworks. In this review, we suggest that they can be combined to answer important questions about self-organising systems like the brain. DCM has been developed recently by the neuroimaging community toexplain, using biophysical models, the non-invasive brain imaging data are caused by neural processes. Itallows one to ask mechanistic questions about the implementation of cerebral processes. In DCM theparameters of biophysical models are estimated from measured data and the evidence for each model isevaluated. This enables one to test different functional hypotheses (i.e., models) for a given data set.Autopoiesis and related formal theories of biological systems as autonomous machines represent a body ofconcepts with many successful applications. However, autopoiesis has remained largely theoretical and hasnot penetrated the empiricism of cognitive neuroscience. In this review, we try to show the connections thatexist between DCM and autopoiesis. In particular, we propose a simple modification to standardformulations of DCM that includes autonomous processes. The idea is to exploit the machinery of thesystem identification of DCMs in neuroimaging to test the face validity of the autopoietic theory applied toneural subsystems. We illustrate the theoretical concepts and their implications for interpretingelectroencephalographic signals acquired during amygdala stimulation in an epileptic patient. The resultssuggest that DCM represents a relevant biophysical approach to brain functional organisation, with apotential that is yet to be fully evaluated.

Key terms: Dynamic Causal Modelling, brain functional organization, plasticity, autonomous systems,autopoiesis.

I. INTRODUCTION

Cognitive experiments in neuroimaging relymainly upon two techniques: functionalMagnetic Resonance Imaging (fMRI) detectschanges in cerebral blood flow, volume andthe ensuing changes in concentration ofdeoxyhemoglobin (Attwell and Iadecola,2002; Logothetis and Wandell, 2004). Thesemeasurements are acquired in each voxel ofthe brain volume, i.e. every 3 mm or so, inrelation to a given stimulus or cognitive task.On the other hand, electroencephalography(EEG) (Nunez and Srinivasan, 2005) andmagnetoencephalography (MEG)(Hamalainen et al., 1993) measure, on thescalp, fluctuations of the electric potentialand magnetic field, respectively, emitted by

underlying neuronal populations. In recentyears, research teams have developedapproaches for the fusion of fMRI/EEG/MEG data. Such efforts are motivated by theobservation that combining the hightemporal resolution of MEG/EEG and thehigh spatial resolution of fMRI should leadto the optimal technique for functionalneuroimaging. For instance, the sourcelocalisation of MEG/EEG signals can beconstrained by fMRI activation maps andprofit from the localisation power of fMRI(Dale et al., 2000). Although most fusionmethods are perfectly tenable from a signalprocessing point of view, they are notgrounded in a detailed analysis of thebiophysical mechanisms generating data; forexample, it is still unclear how fMRI/EEG/

Corresponding author: Inserm U836, Grenoble Institut des Neurosciences (former Inserm U594), CHU Grenoble - Pav B -BP 217, Grenoble, F-38043 France, Tel: +33 476765978, Fax: +33 476765896, Email odavid@ujf-grenoble.fr

Received: May 11, 2007. Accepted: March 3, 2008

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MEG signals are related to underlying neuralnetworks.

To better understand the relationshipsbetween neuronal ensembles andneuroimaging data, a research initiative hasemerged recently. It is predicated on thedevelopment of biophysical, or generativemodels, for neuroimaging data (Buxton etal., 1998; David et al., 2005; David et al.,2006b; David and Friston, 2003; Friston etal., 2000; Poznanski and Riera, 2006; Rieraet al., 2004; Riera et al., 2006; Riera et al.,2007; Robinson et al., 2001; Stephan et al.,2004; Vazquez et al., 2006) (Figure 1).Basically, the idea is to relate neuronalvariables (synaptic time constants andefficacies, inhibition/excitation, neuralconnectivity, etc.) to macroscopic data (localfield potentials, scalp MEG/EEG, fMRI).Here, researchers face two problems: (i) aforward problem, which corresponds to themapping from biophysical phenomena tomeasured data (fMRI or MEG/EEG); (ii) andan inverse problem which corresponds to theinversion of the forward model; in otherwords to the estimation of forward modelparameters, given a data set and some knownstimuli. Because they are biophysicallygrounded, generative models represent aprincipled and mechanistic basis for fMRI/EEG/MEG data fusion. Inferences are madeon neuronal parameters estimated from fMRIand/or MEG/EEG, or on unobservedneuronal states. These quantities are the truecommon denominator of any neuroimagingdata and transcend modality-specific aspects.

In this review, the focus is on theformalism developed for Dynamic CausalModelling (DCM) (David et al., 2006a;Friston et al., 2003; Garrido et al., 2007;Kiebel et al., 2006; Penny et al., 2004;Stephan et al., 2005). DCM is a genericapproach for analysing fMRI and EEG/MEG data using generative models. Itimposes constraints on the mathematicalstructure of generative models so that theycan be inverted easily using Bayesianestimation procedures. In brief, thesemodels are usually deterministic input-output systems, which can be decomposedinto a differential state equation and anonlinear output or observer function.Following discussions which animated aworkshop “Networks in Cognitive Systems/ Trends and Challenges in Biomedicine:From Cerebral Process to MathematicalTools Design” held at the ValparaisoInstitute of Complex Systems in December2006, I show here how this first generationof DCMs can be adapted to embed moreautonomous modulatory mechanisms. Thegoal is to show that these models can beadapted to get closer to the self-organisedand dissipative dynamics of living systems,as covered by formal theories used inbiology such as autopoiesis (Varela et al.,1974). As an illustration, intracerebral EEGdata, recorded in an epileptic patient duringneurostimulation, will be used to illustratehow important questions about autonomousdynamics at the level of neuronalconnections can be posed and addressed.

Figure 1: Generative models are biophysical models, which try to explain neuroimaging data(forward problem). The inverse problem consists of identifying the biophysical parameters of thesemodels from the measured data. Dynamic Causal Modelling estimates the parameters of a givengenerative model (fMRI or MEG/EEG) using a Bayesian scheme.

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II. DYNAMIC CAUSAL MODELLING (DCM)

II.1 Concept

The main idea behind DCM (David et al.,2006a; Friston et al., 2003; Kiebel et al.,2006) is to treat the brain as a deterministicnonlinear dynamical system that is subjectto inputs, and produces outputs. Effectiveconnectivity, i.e. the influence that oneregion exerts on another, is parameterizedin terms of coupling among unobservedbrain states, i .e. neuronal activity indifferent regions. Coupling is estimated byperturbing the system and measuring theresponse. In other words, the principal aimof DCM is to explain evoked brainresponses as deterministic responses tosome perturbations, i.e. stimuli, in terms ofcontext-dependent coupling, which allowsfor differences in the shape of responses.These perturbations elicit changes inunobserved neuronal activity simulated inneural networks, which is transformed intoobserved macroscopic neuroimaging datausing a modality-specific forward model(Figure 2).

DCM was developed first for fMRI(Friston et al., 2003) and can be used for anytype of experimental design, as long as thedata are acquired sequentially (DCM being adynamical model, it necessitates continuoustime-series). Here, the neuronal activity of

each brain region participating in a DCM issummarised by one state variable, coined“synaptic activity”. Interactions betweenregions are modelled simply using a bilinearmodel that allows for input-dependentmodulation of connectivity over time. Thismeans that the neural dynamics generatedare very simple (basically mono-exponentialresponses) and the relationships between realneuronal activity and modelled “synapticactivity” are quite obscure. However, it isnot possible to estimate complicated neuraldynamics from fMRI signals because theyhave intrinsically slow time constants (theycan be considered as the output of a low-passfilter embodied by hemodynamic processes)and are sampled sparsely (every second orso, i.e. much slower than neural processes).The role of the neural model in DCM forfMRI is simply to estimate a summary ofneural interactions, i.e. the strength ofdirected neuronal connections. The synapticactivity is estimated from the BloodOxygenated Level Dependent (BOLD)signals by the means of a hemodynamicmodel (Friston et al., 2000) (Figure 3). Theparameters of the hemodynamic model areestimated in each region to take into accountspatial variability of hemodynamicresponses. Inverting this model to estimatecausal interactions at the neuronal levelmeans the estimates are, in theory, notsensitive to this hemodynamic variability.

Figure 2: General concept of DCM. Brain activity is modelled with neural networks using a modelof interactions (connectivity between different brain regions and/or neuronal populations). Theneural states generate macroscopic data through a hemodynamic model for fMRI or an electricalmodel for MEG/EEG. The estimation of the parameters of the models allows one to estimateneuronal interactions, either from fMRI or from MEG/EEG. The fusion between fMRI/MEG/EEGdata is implemented via the generative models at the level of neural networks.

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DCM for EEG relies on a neuronalmodel of interactions that is more plausiblethan the one used for fMRI. EEG signalsare the macroscopic result of the activity ofmillions of neurons and DCMs for EEG useneural-mass models, which assumedynamics can be modelled by randomfluctuations around population dynamicswith a point mass (David and Friston,2003). DCM for EEG has been developedas a generic tool to analyse evokedpotentials obtained at the scalp level for anykind of neuropsychological or cognitiveexperiment. The generative model of DCMfor EEG (David et al., 2005) is based on theJansen model (Jansen and Rit, 1995), aneural-mass model developed originally forexplaining visual responses. It is combinedwith rules of cortical-cortical connectivityderived from the analysis of connectionsbetween the different cortical layers in thevisual cortex of the monkey (Crick andKoch, 1998). In the Jansen model, a corticalarea, understood here as an ensemble ofstrongly interacting macro-columns, ismodelled by a population of excitatorypyramidal cells, receiving (i) inhibitory andexcitatory feedback from local (i .e.

intrinsic) interneurons and (ii) excitatoryinput from neighbouring or remote (i.e.extrinsic) areas. It is composed of threesubpopulations: a population of excitatorypyramidal (output) cells receives inputsfrom inhibitory and excitatory populationsof interneurons, via intrinsic connections(intrinsic connections are confined to thecortical sheet). Within this model,excitatory interneurons can be regarded asspiny stellate cells found predominantly inlayer four and receiving forwardconnections (Miller, 2003). Excitatorypyramidal cells and inhibitory interneuronsare considered to occupy agranular layersand receive backward and lateral inputs.The resulting model (David et al., 2005)including intrinsic and extrinsic cortical-cortical connections, is a set of differentialequations describing interactions betweendifferent inhibitory and excitatory neuronalpopulations (Figure 3). It can be specifiedeasily to embed any hierarchical cortical-cortical network using forward, backwardand lateral connections. The largecytoarchitectonic variability of theneocortex and of other brain structures suchas the hippocampus and amygdala makes

Figure 3: Schematic of the state equations f (Eq. 1) and output equations g (Eq. 2) used in fMRIand MEG/EEG. The state equation is more complex in MEG/EEG than in fMRI, whereas the outputequation is simpler in MEG/EEG than in fMRI.

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the plausibility of a generic modelquestionable. However, the crucial pointhere is that the main purpose of a forwardmodel, in the context of DCM, is toconstrain dynamics in a neuronallyplausible way (time constants, propagationdelay, directionality of the informationtransfer, etc.). This is exactly what ageneric model can do, maintaining anappropriate balance between complexity,plausibility and modularity.

II.2 Theory

Because DCMs are not restricted to linearor instantaneous systems, they generallydepend on a large number of freeparameters. However, because they arebiologically grounded, parameter estimationis constrained. A natural way to embodythese constraints is within a Bayesianframework. Consequently, DCMs areestimated using Bayesian inversion andinferences about particular connections aremade using their posterior or conditionaldensity. The full set of equations for DCMspecification and Bayesian parameterestimation can be found in the originalpapers (David et al., 2005; David et al.,2006a; Friston et al., 2003; Kiebel et al.,2006; Penny et al., 2004). The key steps aresummarised below.

II.2.1 Model specification

A DCM is a dynamical system. It isspecified in terms of a state equation and anoutput equation. The state equation can bewritten as

x = f (x,u,θ) (1)

where x are the neuronal states, u are theextrinsic inputs and θ are the modelparameters. The output equation links theunobserved neuronal states x to themeasured data y using a nonlinearinstantaneous function g:

y = g(x,θ). (2)

Equations (1) and (2) completely specifythe forward model, that is how to link

neuronal states x and their extrinsicperturbations u to the macroscopic data y. Inother words, the functions f and g arespecific to the modality used. In fMRI, f isfairly simple (because there is noinformation about detailed neural dynamicsin BOLD signals) and approximatesneuronal interactions with a bilinear model.The function g is much more complex,because it models the different biophysicalprocesses at the origins of the BOLD effect:(i) the synaptic activity triggers avasodilatory signal which induces changes inblood flow; (ii) according to the Balloonmodel (Buxton et al., 1998), changes inblood flow lead to changes in blood volumeand in deoxyhemoglobin concentration. Incomparison, in EEG, the f function is rathercomplex (nonlinear differential delayedequations) because one examines muchricher neural dynamics than in fMRI. Incontrast, the g function is extremely simple.It is the standard forward head model usedfor source localisation, namely a linearproduct of the pyramidal cell depolarisation(part of the hidden neural states, which areestimated via f) by the lead field of eachregion of the DCM. Figure 3 summarises thedifferent equations. In fMRI, the parametersθ are the coupling parameters (connectivity)and hemodynamic parameters which controlthe dynamics of changes in blood flow,blood volume and deoxyhemoglobin content.For MEG/EEG, they are inhibitory andexcitatory synaptic time constants andefficacies, intrinsic and extrinsicconnectivity, and propagation delay.

II.2.2 Estimation of model parameters

DCM uses a Bayesian scheme forestimating model parameters based onExpectation-Maximisation (Friston et al.,2002). The outputs of the parameterestimation procedure are posteriorprobabilities of model parameters p(θ|y)which are a combination of the likelihood(or confidence in the data) p(θ|y) and priorexpectations about the parameters (forexample, synaptic time constants areexpected to be around 5-10 ms) p(θ):

p(θ|y) ∝p(y|θ)p(θ). (3)

.

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Hyperparameters tune the relative influenceof the data and of prior expectations. Theyare estimated from the data using arestricted Maximum Likelihood. The mostimportant aspect is that inferences about themodel parameters, and particularly aboutconnectivity parameters, can be performeddirectly from the posterior distribution ofthose parameters (under Gaussianassumptions, one estimates and uses theconditional or posterior mean andcovariance of the parameters).

II.2.3 Model comparison

The main advantage of DCM is that itallows one to test competing functionalhypotheses. For each functional hypothesis,a model m is specified in terms ofanatomical connections between regionsand possibly the modulation of someconnections by experimental context. Thisis equivalent to constructing a specificfunction f (Eq. 1) for each model. After theestimation of parameters of each competingmodel, the models are compared to find themost plausible model, or functionalhypothesis. This is done using Bayesianmodel selection where the evidence of eachmodel is used to quantify the modelplausibility (Penny et al., 2004). Theevidence of model m is given by

p(y|m) = ∫ p(y|θ,m)p(θ,m)∝ p(y|θ)dθlog(p(y|m)) = accuracy(m)-complexity(m)

The log-evidence can be decomposed into adifference between two components: anaccuracy term, which quantifies the data fit,and a complexity term, which penalizesmodels with a large number of parameters.Therefore, the evidence embodies the twoconflicting requirements of a good model,that it explains the data and is as simple aspossible. The most likely model is the onewith the largest log-evidence. Conventionally,strong evidence in favour of one modelrequires the difference in log-evidence to bethree or more with other models.

Assuming each data set is independentof the others, the best model at the grouplevel is obtained by multiplying themarginal likelihoods or equivalently, by

adding the log-evidences from each subject(Garrido et al., 2007):

1n p(y1,…, yn|m) = 1n p(yj|m) (5)

where n is the number of subjects. Note thatthe evidence can only be approximatedunder some assumptions. To obtain aconsistent model comparison, one can usethe Akaike Information Criterion (AIC) orthe Bayesian Information Criterion (BIC)(Penny et al., 2004) to get bounds on theevidence and to select a model if theinference obtained with AIC and BIC isconcordant.

III. AUTOPOIETIC SYSTEMS

Autopoietic theory, or autopoiesis, is aformal attempt to describe living systems asphysical open (dissipative) systems, butwith a degree of autonomy (Varela, 1979).Autonomy is a general framework tounderstand their fundamental organisation.It is particularly useful when consideringthe individuality of living systems atdifferent scales. It relies on circularcausality (Figure 4), which is the centralaspect of an autopoietic system (Letelier etal., 2003; Maturana and Varela, 1980):

“an autopoietic system is organised as abounded network of processes ofproduction, transformation anddestruction of components which (i)through their interactions andtransformations continuously regenerateand realise the network of processes thatproduced them; (ii) constitute the systemas a concrete entity in the space in whichthe components exist by specifying thetopological realisation of the system assuch a network”.

In other words, an autopoietic systemproduces a unity that is topographically andfunctionally segregated from its background.The operational closure (processes whichproduce components that are reinserted inthe original processes by the means of otherprocesses) of autopoietic systems is ageneral principle of organisation, which can

· (4)

∑n

j=1

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be applied in many contexts; such asecosystems, artificial intelligence andartificial life, social sciences, linguistics,economics and so on. In fact, autopoieticsystems are a special case of a larger class oforganisationally closed systems (Varela,1979). This class includes (M,R) systems(Letelier et al., 2003; Rosen, 1958).

It is clear that the formalisms of DCMand autonomous systems such asautopoietic systems are not very different.The next section addresses theirconnections. This will lead to a simplemodification of the standard DCMs to allowdeterministic autonomous activity to begenerated from perturbations, thusincluding autopoietic systems in theformalism of DCM.

IV. DCM AND AUTOPOIETIC SYSTEMS

In neurodynamics, there are two classes ofeffects: dynamic effects and structuraleffects (David et al. , 2006b). Thedistinction arises from a simple view of

Figure 4: Partial representation of an autopoietic system at different scales: (A) Abstract level; (B)Cellular level; (C) Animal body level. The diagrams show the circular causality, or operationalclosure, which defines the autonomy of living systems. The system configuration (membraneboundaries/sensory-motor coupling) specifies a network of processes (metabolic network/nervoussystem) which in turn determines the system configuration or the dynamics of similar processes.Modified from (Rudrauf et al., 2003).

neuronal responses, as the response of aninput-state-output system, such as a DCMdefined by Eq. (1-2), to perturbations. FromEq. (1), it is immediately clear that thestates x, and implicitly the system’sresponse y, can only be changed byperturbing the extrinsic inputs u or theparameters, θ. We refer to these as dynamicand structural effects respectively. Thisdistinction arises in a number of differentcontexts. From a purely dynamical point ofview, transients elicited by dynamic effectsare the system’s response to input changes;for example, presentations of a stimulus inan Event Related Potential (ERP) study.The duration and form of the resultingdynamic effect depends on the dynamicalstability of the system to perturbations ofits states (i.e. how the system’s trajectorieschange with the state). Structural effectsdepend on structural stability (i.e. how thesystem’s trajectories change with theparameters). Systematic changes in theparameters can produce systematic changesin the response, even in the absence ofinput. For systems that show autonomous

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( i .e. periodic or chaotic) dynamics,changing the parameters is equivalent tochanging the attractor manifold, whichinduces a change in the system’s states(Breakspear et al., 2003; Friston, 1997). Forsystems with fixed points and Volterrakernels, changing the parameters isequivalent to changing the kernels andtransfer functions. This changes the spectraldensity relationships between the inputs andoutputs. As such, structural effects areclearly important in the genesis of inducedoscillations because they can producefrequency modulation of ongoing activitythat does not entail phase-locking to anyevent. More generally, they play a criticalrole in short-term plasticity mechanismsobserved in neuroimaging, for instancesubject’s habituation after repetitivestimulation. Activity-dependent changes insynaptic activity are an important exampleof a structural effect that is induced bydynamic effects. This coupling of structuraland dynamic mechanisms is closely relatedto the circular causality that characterisesautopoietic systems. In fact, we will focuson activity or time-dependent changes inconnectivity in the empirical example later.

At the neurobiological level, thedistinction between dynamic and structuralinputs speaks immediately of the differencebetween drivers and modulators (Shermanand Guillery, 1998). In sensory systems, adriver ensemble can be identified as thetransmitter of receptive field properties. Forinstance, neurons in the lateral geniculatenuclei drive primary visual area responsesin the cortex, so that retinotopic mapping isconserved. Modulatory effects areexpressed as changes in certain aspects ofinformation transfer, by the changingresponsiveness of neuronal ensembles in acontext-sensitive fashion. A commonexample is attentional gain. Other examplesinvolve extra-classical receptive fieldeffects that are expressed beyond theclassical receptive field. Generally, theseare thought to be mediated by backwardand lateral connections. In terms ofsynaptic processes, it has been proposedthat the post-synaptic effects of drivers arefast (e.g. ionotropic receptors), whereasthose of modulators are slower and more

enduring (e.g. metabotropic receptors). Themechanisms of action of drivers refer toclassical neuronal transmission, eitherbiochemical or electrical, and are wellunderstood. Conversely, modulatory effectscan engage a complex cascade of highlynonlinear cellular mechanisms (Turrigianoand Nelson, 2004). Modulatory effects canbe understood as transient departures fromhomeostatic states, lasting hundreds ofmilliseconds, due to synaptic changes in theexpression and function of receptors andintracellular messaging systems. Classicalexamples of modulatory mechanismsinvolve voltage- dependent receptors, suchas NMDA receptors. These receptors do notcause depolarisation directly (i.e. a dynamiceffect) but change the units sensitivity todepolarisation (i.e. a structural effect).

In short, the distinction betweendeterministic input-output systems, such asDCMs, and autonomous systems asformulated in theories such as autopoiesisand (M,R) systems is how dynamic andstructural effects are instantiated and howthey are coupled. In the standardinterpretation, an autopoietic system createsan autonomous web of (molecular) processesthat maintain autopoietic self-organisation(c.f., self-assembly in chemical systems).This means that it does not have structuralinputs. In other words, the environment doesnot define the internal dynamics. Theenvironment only perturbs the system’sdynamics. Here, there is no distinctionbetween DCMs and autopoietic systems:both receive dynamic inputs, which act astransient perturbations. However, inautopoietic systems, the dynamic inputstrigger internal changes, or structural effects,which are defined by the very organisationof the autopoietic system itself. Incontradistinction, the current formulation ofDCM does not specify such operationalclosure. Instead, structural changes arespecified as explicit and direct consequencesof particular dynamic inputs. For instance,the changes in the dynamics of a DCM areusually defined by the modulation ofinterregional effective connectivity by anexternal modulatory input (i.e. the bilinearterm in fMRI or the distinction betweenexperimental conditions in MEG/EEG).

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Therefore, there is no operational closure inthe sense that an extrinsic input has to beadded to initiate structural effects.

However, the operational closure ofautopoietic systems is simple to specify in thecontext of DCM. In abstract form, the internalprocesses which realise transient structuralmodifications, triggered by dynamic input,can be defined as a generalised convolution

θ = h(x,θ) (6)

where h can be any function and x and θ arethe past history of the neuronal states x (inautopoietic terms: network of processes)and parameters θ (in autopoietic terms:molecular configuration). In summary,autopoietic systems can be definedoperationally with a small set of equations,which extend the analytical formalism ofDCM:

x = f (x,u,θ)θ = h(x,θ) . (7)y = g(x,θ)

The first two equations embedoperational closure: perturbations u initiatechanges in the states (processes) x thatdepend on the parameters θ (configuration).In return, the system’s configuration, orstructure, is a function of the history of itsconfiguration and of its processes. This isthe basis of an autonomous system, whichgenerates intrinsic structural changestriggered by external inputs. The lastequation is simply the output equationthrough which the states of the system aretransformed into measurable variables y. Inneuroimaging, these are the BOLD signalsand scalp MEG/EEG. Figure 5 places Eq.(7) into an autopoietic scheme.

Now that the formalism of a DCM forautopoietic systems has been established,we will test the face-validity of thisapproach using experimental data fromdeep brain stimulation in an epilepticpatient. We will then discuss the benefits ofincluding autonomous dynamics in thecontext of DCM in comparison to itsstandard formulation.

Figure 5: Autopoietic interpretation of Eq. (7). The parameters θ play the role of the configuration.The processes are the neural states x. They are specified by the state equation f and their pasthistory and the past history of the parameters. In turn, these determine a new configuration usingthe function h. Perturbations u is the equivalent of energy inflow. The macroscopic data y are theoutput of an observer equation g; hence they do not play an explicit role in the intrinsic dynamicsof the system.

~ ~

~ ~

~~

.

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V. AN ILLUSTRATION: SHORT-TERM PLASTICITY

DURING PRE-SURGICAL NEUROSTIMULATION

Epilepsy is a common chronic neurologicaldisorder characterised by recurrentspontaneous epileptic seizures. A corticalimbalance between excitatory andinhibitory mechanisms is likely to be thepathophysiological basis for human partialepilepsy. In addition to long-lastingsusceptibility to epileptic discharges,transient modifications of neural networksproperties, such as those induced byelectrical stimulation, can also lead to theoccurrence of epileptic events in patients(Chauvel et al., 1993; Kahane et al., 1993;Kahane et al., 2004; Kalitzin et al., 2005;Schulz et al., 1997; Valentin et al., 2002;Wilson et al., 1998). In particular, severalstudies have noted that short-term plasticityof evoked responses (Wilson et al., 1998;David et al. , 2008) or of oscillatoryresponses (Kalitzin et al., 2005) is inducedeasily by repetitive stimulations in theepileptogenic regions, without causing asystematic seizure. Knowing whether thesefast changes in evoked responses conformto autonomous dynamics is an importantissue which can be addressed explicitly bythe theoretical considerations of theprevious section.

The full description of the clinical andscientific context of pre-surgicalneurostimulation and of our data acquisitionprotocol and patient characteristics can befound elsewhere (David et al., 2008). Herewe summarise those elements needed tounderstand the DCM treatment. The patientincluded in this study was suffering from

temporal lobe epilepsy. She had beenselected for resective surgery and hadundergone standard pre-surgical clinicalevaluations, including 1 Hz intracerebralelectrical stimulation (Kahane et al., 1993;Kahane et al., 2004). The patient was fullyinformed and gave her consent before beingimplanted and stimulated. Intracerebralrecordings were performed using an audio-video-EEG monitoring system (Micromed,Treviso, Italy) that recorded up to 128contacts simultaneously, so that a largerange of mesial and cortical areas weresampled. Stimulation at 1 Hz (pulse width 3milliseconds) was applied to the amygdalabetween two contiguous contacts. The goalsof the stimulation were the reproduction ofthe aura, the induction of an electro-clinicalseizure, and/or the localization of eloquentcortical areas to be spared during surgery.Bipolar stimuli were delivered using aconstant current rectangular pulse generatordesigned for a safe diagnostic stimulationof the human brain, using parametersproved to produce no structural damage.The intensity used was 3mA. Stimulationlasted 34 seconds (34 brief stimulations)and evoked responses were recorded in theamygdala, anterior hippocampus, temporalpole and fusiform gyrus. After stimulation25, some irregular spiking and fastoscillations were observed reflecting theelectro-clinical signs of the forthcomingseizure. The first clinical symptomsoccurred around stimulation 31 (i.e., after31 seconds). The anterior hippocampus wasthe candidate for an epileptic focus andshort-term plasticity was expressed most inthis structure (Figure 6).

Figure 6: Evoked responses in the anterior hippocampus during 1Hz stimulation of the amygdala.Note the increasing amplitude of responses up to stimulation 25 (24 s). After stimulation 25,responses become irregular with fast activity indicating a non-physiological (epileptic) behaviour.

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For simplicity, we isolated the anteriorhippocampus for a DCM study andmodelled it with a cortical macro-columncomposed of inhibitory and excitatoryneuronal populations (Jansen and Rit,1995). The Jansen model is certainly notthe optimal neuronal model for the anteriorhippocampus but the objective was not todetail the activity of the perforant pathway,dentate gyrus, CA1, CA2, CA3 and so on.The Jansen model is simply a way tosummarise the complex neuronalinteractions in a neural mass model thatcaptures the basic dynamics of observedmacroscopic EEG. Isolating thehippocampus from the rest of the brainimposes formal topological constraintswhich do not necessarily exist (thehippocampus is embedded in moreextended neural networks). However, itallows us to deal with a simple neuralmodel, in which all inputs from otherstimulated regions (direct connections withthe amygdala but also potential relay withother structures, see (David et al., 2008) fora more complete analysis) are pooled undera single exogenous input. This means wemade the implicit assumption that theanterior hippocampus is an autonomoussystem, itself nested in a bigger system (thebrain, or the human body, or society, or theuniverse). This is justified by the fact thatneural activity can be recorded inhippocampal slices in vitro.

To explain the changes of hippocampalresponses y shown in Figure 6, weconsidered the following competing modelsor hypotheses (Figure 7):

Figure 7: The different DCMs of the hippocampus tested to explain the data shown in Figure 6.The bold arrows correspond to the modulated connections. Models a and b are standard DCMs.Models c incorporates autonomous dynamics on the parameters. Note that the loop betweenexcitation and inhibition captures the concept of interactions between excitatory and inhibitoryneuronal populations but does not exactly reflect the architecture of the Jansen model (see Fig. 3).

a. Model a: Changes in y (responses) are adirect consequence of changes in u(input). In other words, plasticity hasbeen expressed outside the hippocampusby neural networks linking the amygdalato the hippocampus (monosynaptic orpolysynaptic connections). In this modelwe modelled a different input strengthfor each stimulation.

b. Model b: Input u is stable overstimulations and short-term plasticityobserved in y corresponds to amodulation of the excitatory efficacy ofintrinsic connections within thehippocampus, the dynamics of which areset by an exogenous modulatory input.This is a structural mechanism explainedwithin the standard formulation of DCM(Eq. 1-2). In this model there are twoinputs; a dynamic input, which is nowfixed for each stimulation and a structuralinput that changed the connectivity and isspecific to each stimulation.

c. Model c: Input u is stable overstimulations and short-term plasticityobserved in y corresponds to anautonomous modulation of excitatoryefficacy of intrinsic connections withinthe hippocampus. This corresponds to anautonomous DCM (Eq. 7). Here, weconsider a simple linear autoregressivemodel for the structural dynamicsconcerning excitatory synaptic efficacies.Thus the structural input of model b isreplaced by Eq. (6) which reduces to:

He = aiHe (8)(n) ∑1

i=1

(n-i)

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where He is the excitatory synapticefficacy at stimulation n and I is themodel order; in other words, the horizonbelow which past activity has an effecton the current structure. This model hasfewer parameters than the precedingmodels because the number ofautoregression coefficients is less thanthe number of stimulations. However, aswe will see below it can model equally,if not more, complex dynamics. Besidesknowing whether model c can explainthe stimulation-induced dynamics, thesize of the memory effect in theautonomous dynamics is i tselfinteresting, i .e. what is the mostplausible I. To assess this, we performeda Bayesian model comparison among themodels constructed for each value of Ibetween 1 and N-1 (N=25 is the maximalnumber of stimulations before thebeginning of epileptic activity). Eachmodel is noted model cI.

The parameters for each model wereestimated from the data y shown in Figure6. Pre- processing comprised: (i) band-passfiltering between 5 and 40 Hz and (ii)concatenation of the first 25 evokedresponses between 0 and 150 ms. Theresults are shown in Figure 8. Oncomparing the log-evidences of thedifferent models, it transpired that model ais the most likely. Indeed, this modelreproduces the observed time series with aremarkable fidelity. This suggests that thetime- dependent responses, expressed in theanterior hippocampus, are most probablydue to changes in its extrinsic inputs (orchanges in sensitivity to inputs as mediatedby the modulation of the expression ofNMDA receptors). One point is importantto stress when looking at the time series ofmodel b (and of c1 and c10): the changes insynaptic efficacies are related directly tothe maximum amplitude of evokedresponses, and are therefore highlycorrelated with the extrinsic input in modela. However, model b fits the first positivecomponent of the evoked responses wellbut not the second negative component. Infact, the excitatory efficacy has an effectnot only on the amplitude of responses but

also on their shape and, implicitly, theirfrequency content. Because model a, whichestimates an excitatory efficacy overstimulations, is able to fit the data for anykind of response amplitude, these resultssuggest that the efficacy of connectionsintrinsic to the hippocampus is more or lessconstant. This is the main reason whymodel a is much more plausible than theothers. To conclude, the first interestingaspect of this DCM analysis is that theshort-term changes in hippocampalresponses to stimulation of the amygdalaare more likely to be caused by plasticity ineffective connectivity between thehippocampus and other brain regions, asopposed to some modulation of intrinsichippocampal susceptibility. This calls for aDCM analysis extended to other brainregions, which can be found in (David etal., 2008).

Nonetheless, let us continue thediscussion of the results by focusing onmodel b and its autonomous formulation(models c). We will describe the constraintsand the advantages of using an autonomousformulation of a DCM. For models c, thelog-evidences indicate that model c10 is themost plausible, given the data. Thiscorresponds to a model order of ten, for theautoregressive evolution of synapticefficacy. For comparison, we show inFigure 8 the time series for this model andalso those for the simplest model (modelc1). Both models show a gradual increase inexcitatory 1 efficacy with the repetition ofthe stimulation. The dynamics generated arevery simple (monotonous) for model c1 andfairly more complex for model c10, where apattern lasting ten stimulations is repeatedapproximately. In other words, there is aconstraint on the model dynamics, which isgiven by the structural equation (Eq. 6 or8). If this is too strict the model will notadjust to the data in comparison to whenthere is no such constraint (model b).

Results obtained with models a and bwere very interesting because they showedit was possible to estimate the modulationof the extrinsic input to the hippocampus(model a), or of the excitatory synapticefficacy within the hippocampus (model b).Besides the ability to reproduce data, a

(n)

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good model is also characterised by itsability to make predictions. For models a andb, this is no prediction because these modelshave no autopoietic memory. Incontradistinction, predictions can be madewith an “autonomous DCM” because there isan explicit modelling of the rate of changesof structural effects over time; this generatesautonomous structural dynamics, which hasconsequences for neural responses. Thevisual inspection of the evolution for modelsc1 and c10 shows the inertia of theautonomous dynamics characterised by theautoregressive model. It is easy to imaginethat the system will continue to diverge ifthe stimulations were to be repeated advitam eternam. This is exactly what we havesimulated in Figure 9 for model c10: weadded two stimulations (stimulations 26 and27) and let the system predict the dynamics.According to the parameters of theautoregressive model estimated fromprevious stimulations, the DCM predicted an

increase of excitatory efficacy as expectedfrom Figure 8. The corresponding timeseries are somewhat more interesting: theyshow a catastrophic divergence, indicatingthat the Jansen model is approaching aphase-transition or bifurcation. Moreprecisely, the system manifold is no longer amass point attractor centred on zero. Onemight interpret this as the hippocampusentering an epileptic regime because of anincrease of excitatory efficacy, which iswhat actually happened. The fascinatingaspect is that the autonomous DCM hasestimated, from the pre-ictal regime, a set ofexcitatory efficacies at the limit of the pointof bifurcation between a stable and divergentdynamics. This indicates that physiologicalbrain dynamics could be at the limit ofstability and particularly prone to generateoscillations and complex nonlineardynamical behaviours such as chaoticitinerancy (Tsuda, 2001) and epilepticseizures in pathological circuits.

Figure 8: On top, the log-evidences (under BIC assumption) of the different models clearly showthat model a is the most plausible. Model b ranks second. Among “autonomous” models, model c10

is the best. Below, on the left hand side, time series are shown (original: grey, adjusted: black) formodels a, b, c1 and c10. Corresponding variables modulated over stimulations (input power formodel a, excitatory synaptic efficacy for models b, c1 and c10) are shown on the right hand side.

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VI. CONCLUSION

This review attempts a synthesis, in simpleterms, of two important conceptualframeworks: Dynamic Causal Modelling(Friston et al., 2003) and the theory ofautopoietic systems (Varela et al., 1974).DCM has been developed recently by theneuroimaging community to explain, usingbiophysical models, how fMRI/MEG/EEGdata are related to neural processes. Theclassical approach in neuroimaging is toexplore a data set with the followingquestion: Where is a given processimplemented in the brain? Standardstatistical maps are then constructed toreveal regional effects and variousstatistical tests can be performed toestablish the regional specificity ofdifferent experimental manipulations. DCMgoes further by asking: How are theresponses implemented in mechanisticterms? This question represents theopportunity to rethink the design of

cognitive experiments in functionalneuroimaging and to appreciate theunderlying neural mechanisms. Theparameters of biophysical models areestimated from the measured data. Differentfunctional hypotheses can therefore betested explicitly. DCM represents a relevantbiophysical approach to exploring braindata with a potential which has yet to befully evaluated.

Since the 1970s, autopoiesis and relatedformal theories of living systems asautonomous machines has had manysuccessful applications in various arenasoutside biology (Letelier et al., 2003). Butautopoiesis, though acclaimed by theoristsin many disciplines (Mingers, 1995), hashad a limited practical impact because ofthe difficulties in applying theoretical ideas,such as wholeness, to experimental data.Here, we have tried to disclose theconnections between DCM and autopoiesis.In particular, we have proposed a simplemodification to the standard formulation of

Figure 9: When adding (artificial) additional stimulations, the parameters of model c10 predict anincrease of excitatory efficacy (white bars). This corresponds to diverging responses, which can beinterpreted as the beginning of an induced seizure. This is a prediction which is possible onlybecause structural changes are specified autopoietically within the model.

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DCM that accommodates a simple model ofautonomy. The idea was to exploit theinferential machinery of the systemidentification with DCMs in neuroimagingto test the face validity of the autopoietictheory applied to neural subsystems. Thisexciting field of research is still essentiallyunexplored and we hope to have advancedthe feasibility of this approach.

ACKNOWLEDGEMENTS

I am much indebted to Francisco Varelaand Karl Friston, who have initiated mostof the ideas developed here, for theirsupport during my doctoral and post-doctoral trainings. I also thank Chileanstudents for their stimulating questionsthroughout the ISCV workshop. This workwas funded by INSERM.

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