Klaas E Stephan
email: k.stephan@fil.ion.ucl.ac.uk

Lee M Harrison

Stefan J Kiebel

O David

W D Penny

Karl J Friston

Dr Klaas Enno

Dynamic causal models of neural system dynamics: current state and future extensions

Keywords: fMRI, EEG, event-related potentials, effective connectivity, neural system, dynamic causal modelling

Complex processes resulting from the interaction of multiple elements can rarely be understood by analytical procedures; instead, mathematical models of system dynamics are required. This insight, which disciplines like physics have embraced for a long time already, is gradually gaining importance in the study of cognitive processes by functional neuroimaging. In this field, causal mechanisms in neural systems are described in terms of effective connectivity. Recently, Dynamic Causal Modelling (DCM) was introduced as a generic method to estimate effective connectivity from neuroimaging data in a Bayesian fashion. One of the key advantages of DCM over previous methods is that it distinguishes between neural state equations and modality-specific forward models that translate neural activity into a measured signal. Another strength is its natural relation to Bayesian Model Selection (BMS) procedures. In this article, we review the conceptual and mathematical basis of DCM and its implementation for functional magnetic resonance imaging data and event-related potentials. After introducing the application of BMS in the context of DCM, we conclude with an outlook to future extensions of DCM. These extensions are guided by the long-term goal of using dynamic system models for pharmacological and clinical applications, particularly with regard to synaptic plasticity.

Introduction

Modern cognitive neuroscience uses a variety of non-invasive techniques for measuring brain activity. These techniques include electrophysiological methods, e.g. electroencephalography (EEG) and magnetoencephalograpy (MEG), and functional imaging methods, e.g. positron emission tomography (PET) and functional magnetic resonance imaging (fMRI). Two intertwined concepts, functional specialization and functional integration, have been guiding neuroimaging applications over the last decades (Friston 2002a). Functional specialization assumes a local specialization for certain aspects of information processing but allows for the possibility that this specialization is anatomically segregated across different cortical areas. Most current functional neuroimaging experiments use this perspective and interpret the areas that are activated by a certain task component as the elements of a distributed system. However, this explanation is somewhat speculative and clearly incomplete as long as one does not characterize how the local computations are bound together by context-dependent interactions among these areas. This binding is the functional integration within the system which can be characterized in two ways, namely in terms of functional connectivity and effective connectivity. While functional connectivity describes statistical dependencies between data, effective connectivity rests on a mechanistic model of the causal effects that generated the data [START_REF] Friston | Functional and effective connectivity in neuroimaging: a synthesis[END_REF]. This article focuses exclusively on a recently established technique for determining the effective connectivity in neural systems of interest on the basis of measured fMRI and EEG/MEG data: Dynamic Causal Modelling (DCM; [START_REF] Friston | Dynamic causal modelling[END_REF]. We review the conceptual and mathematical basis of DCM and Bayesian Model Selection (BMS; Penny et al. 2004a) and demonstrate some applications, using empirical and simulated data. We also touch on some future extensions of DCM that are driven by the long-term goal of using DCM for pharmacological and clinical applications, particularly with regard to questions about synaptic plasticity.

Effective connectivity and a general state equation for neural systems

The term effective connectivity has been defined by various authors in convergent ways. A general definition is that effective connectivity describes the causal influences that neural units exert over another [START_REF] Friston | Functional and effective connectivity in neuroimaging: a synthesis[END_REF]). More specifically, other authors have proposed that "effective connectivity should be understood as the experiment-and timedependent, simplest possible circuit diagram that would replicate the observed timing relationships between the recorded neurons" [START_REF] Aertsen | Dynamics of activity and connectivity in physiological neuronal Networks[END_REF]. Both definitions emphasize that determining effective connectivity requires a causal model of the interactions between the elements of the neural system of interest. Before we describe the specifics of the model on which DCM rests, let us derive a general mathematical form of models of effective connectivity. For this purpose, we choose deterministic differential equations with time-invariant parameters as a mathematical framework. Note that these are not the only possible mathematical representation of systems; in fact, many alternatives exist, e.g. state space models or iterative maps. The underlying concept, however, is quite universal: a system is defined by a set of elements with n time-variant properties that interact with each other. Each time-variant property x i (1 ≤ i ≤ n) is called a state variable, and the n-vector x(t) of all state variables in the system is called the state vector (or simply state) of the system at time t:

          = ) ( ) ( ) ( 1 t x t x t x n M (1)
Taking an ensemble of interacting neurons as an example, the system elements would correspond to the individual neurons, each of which is represented by one or several state variables. These state variables could refer to various neurophysiological properties, e.g. postsynaptic potentials, status of ion channels, etc. Critically, the state variables interact with each other, i.e. the evolution of each state variable depends on at least one other state variable. For example, the postsynaptic membrane potential depends on which and how many ion channels are open; vice versa, the probability of voltage-dependent ion channels opening depends on the membrane potential. Such mutual functional dependencies between the state variables of the system can be expressed quite naturally by a set of ordinary differential equations that operate on the state vector:

) ( ) ,..., ( ) ,..., ( 1 1 1 x F x x f x x f dt dx n n n =           = M (2)
However, this description is not yet sufficient. First of all, the specific form of the dependencies f i needs to be specified, i.e. the nature of the causal relations between state variables. This requires a set of parameters θ which determine the form and strength of influences between state variables. In neural systems, these parameters usually correspond to time constants or synaptic strengths of the connections between the system elements. The Boolean nature of θ, i.e. the pattern of absent and present connections, and the mathematical form of the dependencies f i represent the structure of the system. And second, in the case of non-autonomous systems (and these are the ones of interest to biology) we need to consider the input into the system, e.g. sensory information entering the brain. We represent the set of all m known inputs by the m-vector function u(t). Extending Eq. 2 accordingly leads to a general state equation for non-autonomous deterministic systems ) , , (

θ u x F dt dx =
(3) A model whose form follows this general state equation provides a causal description of how system dynamics results from system structure, because it describes (i) when and where external inputs enter the system and (ii) how the state changes induced by these inputs evolve in time depending on the system's structure. Given a particular temporal sequence of inputs u(t) and an initial state x(0), one obtains a complete description of how the dynamics of the system (i.e. the trajectory of its state vector x in time) results from its structure by integration of Eq. 3:

∫ + = τ θ τ 0 ) , , ( ) 0 ( ) ( dt u x F x x
(4) Equation 3 therefore provides a general form for models of effective connectivity in neural systems. As described elsewhere [START_REF] Friston | Dynamic causal modelling[END_REF][START_REF] Stephan | On the role of general system theory for functional neuroimaging[END_REF]), all established models of effective connectivity can be related to this general equation or, in the case of regression-like models, its static equivalent

) , , ( θ u x F x =
. In the next sections, we show how DCM models neural population dynamics using a bilinear implementation of this general form. This is combined with a forward model that translates neural activity into a measured signal.

Before we proceed to DCM, it is worth pointing out that we have made two main assumptions in this section to simplify the exposition to the general state equation. First, it is assumed that all processes in the system are deterministic and occur instantaneously. Random components (noise) and delays could be accounted for by using stochastic differential equations and delay differential equations, respectively. Second, we assume that we know the inputs that enter the system. This is a tenable assumption in neuroimaging because the inputs are experimentally controlled variables, e.g. changes in stimuli or instructions. It may also be helpful to point out that using time-invariant dependencies f i and parameters θ is not a restriction. Although the mathematical form of f i per se is static, the use of time-varying inputs u allows for dynamic changes in what components of f i are "activated". For example, input functions that can only take values of one or zero and that are multiplied with the different terms of a polynomial function can be used to induce time-dependent changes from nonlinear to linear behaviour (e.g. by "switching off" all higher order terms in the polynomial) or vice versa. Also, there is no principled distinction between states and time-invariant parameters. Therefore, estimating time-varying parameters can be treated as a state estimation problem.

Principles of Dynamic Causal Modelling (DCM)

An important limitation of previous methods for determining effective connectivity from functional imaging data, e.g. structural equation modelling [START_REF] Büchel | Modulation of connectivity in visual pathways by attention: cortical interactions evaluated with structural equation modelling and fMRI[END_REF][START_REF] Mcintosh | Structural equation modeling and its application to network analysis in functional brain imaging[END_REF] or multivariate autoregressive models [START_REF] Goebel | Investigating directed cortical interactions in time-resolved fMRI data using vector autoregressive modeling and Granger causality mapping[END_REF][START_REF] Harrison | Multivariate autoregressive modeling of fMRI time series[END_REF], is that they operate at the level of the measured signals. This is a serious problem because the causal architecture of the system that we would like to identify is expressed at the level of neuronal dynamics which is not directly observed using non-invasive techniques. In the case of fMRI data, for example, previous models of effective connectivity were fitted to the measured time series which result from a haemodynamic convolution of the underlying neural activity. Since classical models do not include the forward model linking neuronal activity to the measured haemodynamic data, analyses of inter-regional connectivity performed at the level of haemodynamic responses are problematic. For example, different brain regions can exhibit marked differences in neurovascular coupling, and these differences, expressed in different latencies, undershoots, etc., may lead to false inference about connectivity. A similar situation is seen with EEG data where there is a big difference between signals measured at each electrode and the underlying neuronal activity: changes in neural activity in different brain regions lead to changes in electric potentials that superimpose linearly. The scalp electrodes therefore record a mixture, with unknown weightings, of potentials generated by a number of different sources.

Therefore, to enable inferences about connectivity between neural units we need models that combine two things: (i) a parsimonious but neurobiologically plausible model of neural population dynamics, and (ii) a biophysically plausible forward model that describes the transformation from neural activity to the measured signal. Such models make it possible to fit jointly the parameters of the neural and of the forward model such that the predicted time series are optimally similar to the observed time series. This combination of a model of neural dynamics with a biophysical forward model is a core feature of DCM. Currently, DCM implementations exist both for fMRI data and event-related potentials (ERPs) as measured by EEG/MEG. These modalityspecific implementations are briefly summarized in the next sections.

DCM for fMRI DCM for fMRI uses a simple model of neural dynamics in a system of n interacting brain regions. It models the change of a neural state vector x in time, with each region in the system being represented by a single state variable, using the following bilinear differential equation:

( ) Cu x B u A u x F dt dx m j j j n +         + = = ∑ =1 ) ( , , θ (5)
Note that this neural state equation follows the general form for deterministic system models introduced by Equation 3, i.e. the modelled state changes are a function of the system state itself, the inputs u and some parameters ) ( n θ that define the functional architecture and interactions among brain regions at a neuronal level. The neural state variables represent a summary index of neural population dynamics in the respective regions. The neural dynamics are driven by experimentally controlled external inputs that can enter the model in two different ways: they can elicit responses through direct influences on specific regions (e.g. evoked responses in early sensory cortices; the C matrix) or they can modulate the coupling among regions (e.g. during learning or attention; the B matrices). Given this bilinear state equation, the neural parameters

} , , { ) ( C B A n = θ can be expressed as partial derivatives of F: 0 2 ) ( 0 = = ∂ ∂ = ∂ ∂ ∂ = ∂ ∂ = x j j u u F C u x F B x F A (6)
As can be seen from these equations, the matrix A represents the effective connectivity among the regions in the absence of input, the matrices ) ( j B encode the change in effective connectivity induced by the j th input u j , and C embodies the strength of direct influences of inputs on neuronal activity. Figure 1 summarises this bilinear state equation and shows a specific example model.

[Figure 1 near here] DCM for fMRI combines this model of neural dynamics with an experimentally validated haemodynamic model that describes the transformation of neuronal activity into a BOLD response. This so-called "Balloon model" was initially formulated by Buxton et al. (1998) and later extended by [START_REF] Friston | Nonlinear responses in fMRI: the Balloon model, Volterra kernels, and other hemodynamics[END_REF]. Briefly, it consists of a set of differential equations that describe the relations between four haemodynamic state variables, using five parameters ) ( h θ . More specifically, changes in neural activity elicit a vasodilatory signal that leads to increases in blood flow and subsequently to changes in blood volume and deoxyhaemoglobin content. The predicted BOLD signal is a nonlinear function of blood volume and deoxyhaemoglobin content. This haemodynamic model is summarised by Figure 2 and described in detail by [START_REF] Friston | Nonlinear responses in fMRI: the Balloon model, Volterra kernels, and other hemodynamics[END_REF].

[Figure 2 near here]

The combined neural and haemodynamic parameter set

} , { ) ( ) ( h n θ θ θ =
is estimated from the measured BOLD data, using a fully Bayesian approach with empirical priors for the haemodynamic parameters and conservative shrinkage priors for the coupling parameters. Details of the parameter estimation scheme, which rests on a gradient ascent procedure embedded into an expectation maximization (EM) algorithm and uses a Laplace (i.e. Gaussian) approximation to the true posterior, can be found in Friston (2002b).

Once the parameters of a DCM have been estimated from measured BOLD data, the posterior distributions of the parameter estimates can be used to test hypotheses about connection strengths. Due to the Laplace approximation, the posterior distributions are defined by their posterior mode or maximum a posteriori (MAP) estimate and their posterior covariance. Usually, the hypotheses to be tested concern context-dependent changes in coupling. A classical example is given by Figure 3. Here, DCM was applied to fMRI data from a single subject, testing the hypothesis that in a hierarchical system of visual areas (c.f. Figure 1) attention to motion enhanced the backward connections from the inferior frontal gyrus (IFG) onto superior parietal cortex (SPC) and from SPC onto V5, respectively. Other examples of single-subject analyses can be found in [START_REF] Mechelli | A dynamic causal modeling study on category effects: bottom-up or top-down mediation[END_REF], Penny et al. (2004b) and [START_REF] Stephan | Investigating the functional role of callosal connections with dynamic causal models[END_REF]. For statistical inference at the group level, various options exist. The simplest approach is to enter the conditional estimates of interest into a classical second-level analysis; for examples see [START_REF] Bitan | Shifts of effective connectivity within a language network during rhyming and spelling[END_REF] and [START_REF] Smith | Task and content modulate amygdala-hippocampal connectivity in emotional retrieval[END_REF]. A more coherent approach may be to use Bayesian analyses at the group level as well (e.g. [START_REF] Garrido | Dynamic causal modelling of the mismatch negativity[END_REF]).

[Figure 3 near here] Fitted to regional fMRI time series, a given DCM explains how local brain responses were generated from the interplay of the three mechanisms described by the state equation (Eq. 5): inter-regional connections, their contextual modulation and driving inputs. Figure 4 provides a simple fictitious example that is based on simulated data. In this example we fix the parameters and use DCM as a model to generate synthetic data, as opposed to its usual use, i.e. estimating parameter values from empirical data. Let us imagine we are dealing with a 2x2 factorial experiment (Fig. 4A) where one experimental factor controls sensory stimulation (stimulus S 1 vs. stimulus S 2 ) and a second factor controls task requirements (task T 1 vs. task T 2 ). Let us further imagine that, using conventional statistical parametric mapping, we had found a main effect of sensory stimulation in a particular brain area x 1 (with observed time series y 1 ; see Fig. 4B, upper panel) and a stimulus-by-task interaction in area x 2 (with observed time series y 2 ). This interaction means that the difference between stimulus S 1 and stimulus S 2 is larger during task T 1 than during task T 2 (see Fig. 4B, lower panel). We can generate the (noise-free) data shown in Fig. 4B using the DCM displayed by Fig. 4C. The stimulus main effect in area x 1 results from the driving inputs to x 1 being much stronger for stimulus S 1 than for stimulus S 2 . This differential effect is then conveyed onto area x 2 by the connection from x 1 to x 2 . Critically, the strength of this connection is strongly enhanced during task T 1 , but only marginally influenced during task T 2 . This difference in modulation causes the interaction in area x 2 (note that this model would have produced an interaction in area x 1 as well if we had chosen a stronger back-connection from x 2 to x 1 ).

Usually, of course, DCM is applied in the reverse fashion, i.e. to estimate the parameters

} , , { ) ( C B A n = θ
from measured fMRI data as in Fig. 4B. The goal is to infer the neuronal mechanisms that have shaped local brain responses, e.g. the presence of main effects or interactions. Simulations like the one described above can also be used to explore the robustness of parameter estimation in DCM. For example, one can generate data multiple times, adding observation noise (see Fig. 4D), and then trying to re-estimate the parameters from the noisy data.

[Figure 4 near here]

We are currently working on various extensions to DCM for fMRI. For example, one extension is to represent each region in the model by multiple state variables, e.g. populations of excitatory and inhibitory neurons (Marreiros et al., in preparation; see also [START_REF] Harrison | Stochastic models of neuronal dynamics[END_REF]. A similar approach has already been implemented in DCM for ERPs which is described in the following section. Another extension for DCM for fMRI is to augment the state equation by additional non-linear, e.g. quadratic, terms (Stephan et al., in preparation):

          + +         + = ∑ = x D x x D x Cu x B u A dt dx n T T m j j j ) ( ) 1 ( 1 ) ( M (7)
This extension enhances the kind of dynamics that DCM can capture and enables the user to implement additional types of models. For example, beyond modelling how connection strengths are modulated by external inputs, one can now model how connection strengths change as a function of the output from areas. This ability is critical for various applications, e.g. for marrying reinforcement learning models with DCM (c.f. [START_REF] Stephan | On the role of general system theory for functional neuroimaging[END_REF]. In a neural system model of descriptive learning theories like temporal difference learning, the prediction error, encoded by the activity of a particular neural unit, determines the change of connection strength between other neural units that encode properties of conditional and unconditional stimuli (see [START_REF] Schultz | Neuronal coding of prediction errors[END_REF]. Figure 5A shows a simulation example where the connection from an area x 1 to another area x 2 is enhanced multiplicatively by the output from a third region x 3 , i.e. . Critically, x 3 is not only driven by external inputs, but also receives an input from x 2 . This means that for an excitatory connection from x 2 to x 3 , a positive reinforcement effect results: the higher activity in x 3 , the more strongly inputs from x 1 to x 2 will be enhanced, leading to higher activity in x 2 , which, in turn, drives x 3 even further. Figure 5B shows an example of this effect, using simulated data. Such a model, of course, lives on the brink of stability and is prone to runaway excitation. In contrast, an inhibitory connection from x 2 to x 3 makes the model extremely stable because the higher the activity in x 3 , the higher the response in x 2 to x 1 inputs and thus the stronger the inhibitory feedback onto x 3 (not shown here).

[Figure 5 near here] DCM for ERPs ERPs as measured with EEG or MEG have been used for decades to study electrophysiological correlates of cognitive operations. Nevertheless, the neurobiological mechanisms that underlie their generation are still largely unknown. DCM for ERPs was developed as a biologically plausible model to understand how event-related responses result from the dynamics in coupled neural ensembles. It rests on a neural mass model which uses established connectivity rules in hierarchical sensory systems to assemble a network of coupled cortical sources [START_REF] David | A neural mass model for MEG/EEG: coupling and neuronal dynamics[END_REF][START_REF] David | Modelling event-related responses in the brain[END_REF]. These rules characterise connections with respect to their laminar patterns of origin and termination and distinguish between (i) forward (or bottom-up) connections originating in agranular layers and terminating in layer 4, (ii) backward (or top-down) connections originating and terminating in agranular layers, and (iii) lateral connections originating in agranular layers and targeting all layers. These inter-areal cortico-cortical connections are excitatory, using glutamate as neurotransmitter, and arise from pyramidal cells.

[Figure 6 near here] Each region or source is modelled as a microcircuit in which three neuronal subpopulations are combined and assigned to granular and supra-/infragranular layers. A population of excitatory pyramidal (output) cells receives inputs from inhibitory and excitatory populations of interneurons via intrinsic (intra-areal) connections. Within this model, excitatory interneurons can be regarded as spiny stellate cells found predominantly in layer 4 and in receipt of forward connections. Excitatory pyramidal cells and inhibitory interneurons are considered to occupy infra-and supragranular layers and receive backward and lateral inputs.

The neural state equations are summarized in Figure 7. To perturb the system and model event-related responses, the network receives inputs via input connections. These connections are exactly the same as forward connections and deliver input u to the spiny stellate cells in layer 4. Input u represents afferent activity relayed by subcortical structures and are modelled as two parameterized components, a gamma density function (representing an event-related burst of input that is delayed and dispersed by subcortical synapses and axonal conduction) and a discrete cosine set (representing fluctuations in input over peristimulus time). The influence of this input on each source is controlled by a parameter vector C. Overall, the DCM is specified in terms of the state equations shown in Figure 7 and a linear forward model

ε θ + = = 0 ) , , ( Lx y u x f dt dx (8)
where 0

x represents the transmembrane potential of pyramidal cells and L is a lead field matrix coupling electrical sources to the EEG channels. In comparison to DCM for fMRI, the forward model is a simple linearity as opposed to the nonlinear haemodynamic model in DCM for fMRI. In contrast, the state equations of DCM for ERPs are much more complex and realistic (compare Figures 1 and7). As an example, the state equation for the inhibitory subpopulation is τ represent lumped rate constants (i.e. lumped across dendritic spines and the dendritic tree). The second operator S transforms the average potential of each subpopulation into an average firing rate. This is assumed to be instantaneous and is a sigmoid function. Intra-areal interactions among the subpopulations depend on constants 4 ... 1 γ which control the strength of intrinsic connections and reflect the total number of synapses expressed by each subpopulation. In Equation 9, the top line expresses the rate of change of voltage as a function of current. The second line specifies how current changes as a function of voltage, current and presynaptic input from extrinsic and intrinsic sources. For simplification, our description here has omitted the fact that in DCM for ERPs all intra-and inter-areal connections have conduction delays. This is implemented by delay differential equations.

[Figure 7 near here] Just as with DCM for fMRI, the DCM for ERPs is usually used to investigate whether coupling strengths change as a function of experimental context. Figure 8 shows an example of a DCM applied to EEG data from a single subject performing an auditory oddball task [START_REF] David | Dynamic causal modeling of evoked responses in EEG and MEG[END_REF]): forward and backward connections between primary auditory and orbitofrontal cortex are stronger during processing of oddball stimuli compared to standard stimuli.

[Figure 8 near here] Several extensions of DCM for ERPs are planned or already under way. For example, [START_REF] Kiebel | Dynamic causal modelling of evoked responses in EEG/MEG with lead-field parameterization[END_REF] demonstrated that one does not necessarily have to assume known lead field parameters (L in Eq. 8) for the forward model. Instead, it is possible to estimate lead-field and coupling parameters simultaneously and thus use DCM for ERPs as a source reconstruction approach with physiologically informed constraints. Future efforts will concentrate on further enhancing the biological realism of the model. One approach may be to free the coupling parameters between the neuronal populations within regions, thus enabling a characterisation of local interactions between excitatory and inhibitory neuronal populations, for example. Another and more long-term goal will be to represent the effects of particular neurotransmitters, like dopamine or acetylcholine, on neuronal population activity. This will be particularly important for potential clinical applications of DCM (see below).

Bayesian Model Selection (BMS)

A generic problem encountered by any kind of modelling approach is the question of model selection: given some observed data, which of several alternative models is the optimal one? This problem is not trivial because the decision cannot be made solely by comparing the relative fit of the competing models. One also needs to take into account the relative complexity of the models as expressed, for example, by the number of free parameters in each model. Model complexity is important to consider because there is a trade-off between model fit and generalisability (i.e. how well the model explains different data sets that were all generated from the same underlying process). As the number of free parameters is increased, model fit increases monotonically whereas beyond a certain point model generalisability decreases. The reason for this is "overfitting": an increasingly complex model will, at some point, start to fit noise that is specific to one data set and thus become less generalisable across multiple realizations of the same underlying generative process. (Generally, in addition to the number of free parameters, the complexity of a model also depends on its functional form; see [START_REF] Pitt | When a good fit can be bad[END_REF]. This is not an issue for DCM, however, because here competing models usually have the same functional form.)

Therefore, the question "What is the optimal model?" can be reformulated more precisely as "What is the model that represents the best balance between fit and complexity?" In a Bayesian context, the latter question can be addressed by comparing the evidence, Here, the number of free parameters (as well as the functional form) are considered by the integration. Unfortunately, this integral cannot usually be solved analytically, therefore an approximation to the model evidence is needed.

In the context of DCM, one potential solution could be to make use of the Laplace approximation, i.e. to approximate the model evidence by a Gaussian that is centered on its mode. As shown by Penny et al. (2004a) C is the error covariance, p θ is the prior mean of the parameters, and p C is the prior covariance):

( ) ( ) ( ) ( )       - - + - -       - - - - = - = - - p y p T p y y p y T y C C C u h y C u h y C m complexity m accuracy m y p θ η θ η η η θ θ θ θ ε θ ε | 1 | | | 1 | 2 1 ln 2 1 ln 2 1 , ( , ( 2
1 ln 2 1 ) ( ) ( ) | ( ln (12)
This expression properly reflects the requirement, as discussed above, that the optimal model should represent the best compromise between model fit (accuracy) and model complexity. We use it routinely in the context of DCM for ERPs (compare [START_REF] David | Dynamic causal modeling of evoked responses in EEG and MEG[END_REF]).

In the case of DCM for fMRI, a complication arises. This is due to the complexity term which depends on the prior density, for example, the prior covariance of the intrinsic connections. This is problematic in the context of DCM for fMRI because the prior covariance is defined in a model-specific fashion to ensure that the probability of obtaining an unstable system is very small. (Specifically, this is achieved by choosing the prior covariance of the intrinsic coupling matrix A such that the probability of obtaining a positive Lyapunov exponent of A is p < 0.001; see [START_REF] Friston | Dynamic causal modelling[END_REF] for details.) Consequently, one cannot easily compare models with different numbers of connections. Therefore, alternative approximations to the model evidence are useful for DCMs of this sort.

Suitable approximations, which do not depend on the prior density, are afforded by the Bayesian Information Criterion (BIC) and Akaike Information Criterion (AIC), respectively. As shown by Penny et al. (2004a), for DCM these approximations are given by

θ θ d m accuracy AIC N d m accuracy BIC - = - = ) ( ln 2 ) ( (13)
where θ d is the number of parameters and N is the number of data points (scans). If one compares the complexity terms of BIC and AIC, it becomes obvious that BIC pays a heavier penalty than AIC as soon as one deals with 8 or more scans (which is virtually always the case for fMRI data):

7.39 ln 2 2 ≈ > ⇒ > e N d N d θ θ (14)
Therefore, BIC will be biased towards simpler models whereas AIC will be biased towards more complex models. This can lead to disagreement between the two approximations about which model should be favoured. In DCM for fMRI, we have therefore adopted the convention that, for any pairs of models m i and m j to be compared, a decision is only made if AIC and BIC concur (see below); the decision is then based on that approximation which gives the smaller Bayes factor (BF): BMS plays a central role in the application of DCM. The search for the best model, amongst several competing ones, precedes (and is often equally important to) the question which parameters of the model represent significant effects. Several studies have used BMS successfully to address complex questions about the architecture of neural systems. For example, Penny et al. (2004) investigated which connections in a system of hierarchically connected visual areas were most likely to underlie the modulatory effects of attention to motion that were observed in the BOLD responses of area V5. They found, using data from a single subject, that the best model was one in which attention enhanced V5 responses to V1 inputs. In another single-subject study, [START_REF] Stephan | Investigating the functional role of callosal connections with dynamic causal models[END_REF] systematically derived 16 different models that could have explained BOLD activity in visual areas during lateralised presentation of visual word stimuli. They found evidence that, in this subject, inter-hemispheric connections served task-dependent information transfer from the non-dominant to the dominant hemispherebut only when the stimulus was initially received by the non-dominant hemisphere. Finally, [START_REF] Garrido | Dynamic causal modelling of the mismatch negativity[END_REF] extended the previous work by [START_REF] David | Dynamic causal modeling of evoked responses in EEG and MEG[END_REF] and applied BMS in the context of an auditory oddball study, measured with EEG, to find the most likely explanation, in terms of coupling changes, for the well-known mismatch negativity potential. They found that their group of healthy controls was divided into two subgroups characterised by different optimal models. Re-examining the ERPs of these subgroups separately revealed a significant difference in the expression of mismatchrelated responses that would have been missed in conventional ERP analyses. This example highlights that BMS may also be of considerable interest for defining clinical populations for whom biological markers are presently lacking. This issue is taken up in the next and final section.

Outlook to future applications of DCM DCM is currently the most advanced framework for inferring the effective connectivity in neural systems from measured functional neuroimaging data. Our hope is that over the next years, the generic framework of DCM and the ongoing developments, some of which were briefly described in this article, will contribute to a more mechanistic understanding of brain function. Of particular interest will be the use of neural system models like DCM (i) to understand the mechanisms of drugs and (ii) to develop models that can serve as diagnostic tools for diseases linked to abnormalities of connectivity and synaptic plasticity, e.g. schizophrenia.

Concerning pharmacology, many drugs used in psychiatry and neurology change synaptic transmission and thus functional coupling between neurons. Therefore, their therapeutic effects cannot be fully understood without models of drug-induced connectivity changes in particular neural systems. So far, only relatively few studies have studied pharmacologically induced changes in connectivity (e.g. [START_REF] Honey | Dopaminergic drug effects on physiological connectivity in a human cortico-striato-thalamic system[END_REF]. As highlighted in a recent review by [START_REF] Honey | Human pharmacological MRI[END_REF], an exciting possibility for the future is to use system models at the early stage of drug development to screen for substances that induce desired changes of connectivity in neural systems of interest with a reasonably well understood physiology. The success of this approach will partially depend on developing models that include additional levels of biological detail (e.g. effects of different neurotransmitters, see above) while being parsimonious enough to ensure mathematical identifiability and physiological interpretability; see [START_REF] Breakspear | Modulation of excitatory synaptic coupling facilitates synchronization and complex dynamics in a biophysical model of neuronal dynamics[END_REF], [START_REF] Harrison | Stochastic models of neuronal dynamics[END_REF], [START_REF] Jirsa | Connectivity and dynamics of neural information processing[END_REF] and [START_REF] Robinson | Prediction of electroencephalographic spectra from neurophysiology[END_REF] for examples that move in this direction.

Another important goal is to explore the utility of models of effective connectivity as diagnostic tools [START_REF] Stephan | On the role of general system theory for functional neuroimaging[END_REF]). This seems particularly attractive for psychiatric diseases whose phenotypes are often very heterogeneous and where a lack of focal brain pathologies points to abnormal connectivity (dysconnectivity) as the cause of the illness. Given a pathophysiological theory of a specific disease, connectivity models might allow one to define an endophenotype of that disease, i.e. a biological marker at intermediate levels between genome and behaviour, which enables a more precise and physiologically motivated categorization of patients (Gottesman & Gould 2003). Such an approach has received particular attention in the field of schizophrenia research where a recent focus has been on abnormal synaptic plasticity leading to dysconnectivity in neural systems concerned with emotional and perceptual learning [START_REF] Friston | The disconnection hypothesis[END_REF][START_REF] Stephan | Synaptic plasticity and dysconnection in schizophrenia[END_REF]. A major challenge will be to establish neural systems models which are sensitive enough that their connectivity parameters can be used reliably for diagnostic classification and treatment response prediction of individual patients. Ideally, such models should be used in conjunction with paradigms that are minimally dependent on patient compliance and are not confounded by factors like attention or performance. Given established validity and sufficient sensitivity and specificity of such a model, one could use it in analogy to biochemical tests in internal medicine, i.e. to compare a particular model parameter (or combinations thereof) against a reference distribution derived from a healthy population [START_REF] Stephan | On the role of general system theory for functional neuroimaging[END_REF]). Alternatively, if different clinical subgroups exhibit different "fingerprints" of dysconnectivity, each represented by a particular DCM, model selection could provide a powerful approach to classify patients (compare the study by [START_REF] Garrido | Dynamic causal modelling of the mismatch negativity[END_REF] on healthy subjects). Such procedures could help to decompose current psychiatric entities like schizophrenia into more well-defined subgroups characterized by common pathophysiological mechanisms and may facilitate the search for genetic underpinnings.

Figure legends

Figure 1 A. The bilinear state equation of DCM for fMRI. B. An example of a DCM describing the dynamics in a hierarchical system of visual areas. This system consists of areas V1 and V5 and the superior parietal cortex (SPC). Each area is represented by a single state variable (x 1 ...x 3 ). Black arrows represent connections, grey arrows represent external inputs into the system and thin dotted arrows indicate the transformation from neural states into haemodynamic observations (thin boxes; see Fig. 2 for the haemodynamic forward model). In this example, visual stimuli (photic) drive activity in V1 which is propagated to V5 and SPC through the connections between the areas. The V1→V5 connection is allowed to change whenever the visual stimuli are moving, and the SPC→V5 connection is modulated whenever attention is directed to motion. The state equation for this particular example is shown on the right.

Figure 2 Summary of the haemodynamic model used by DCM for fMRI. Neuronal activity induces a vasodilatory and activity-dependent signal s that increases blood flow f. Blood flow causes changes in volume and deoxyhaemoglobin (v and q). These two haemodynamic states enter the output nonlinearity which results in a predicted BOLD response y. Adapted, with permission by Elsevier Ltd., from [START_REF] Friston | Dynamic causal modelling[END_REF]. A. DCM applied to data from a study on attention to visual motion by [START_REF] Büchel | Modulation of connectivity in visual pathways by attention: cortical interactions evaluated with structural equation modelling and fMRI[END_REF]. The model is similar to the one shown in Fig. 1, except for the addition of another area, the inferior frontal gyrus (IFG). The most interesting aspect of this model concerns the role of motion and attention in exerting bilinear effects on connections in the model. The presence of motion in the visual stimulation enhances the connection from area V1 to the motion sensitive area V5. The influence of attention is to enable backward connections from the IFG to the superior parietal cortex (SPC) from SPC to V5. Dotted arrows connecting regions represent significant bilinear affects in the absence of a significant intrinsic coupling. Inhibitory self-connections are not displayed for clarity. B. Fitted responses based upon the conditional estimates and the adjusted data. The insert shows the approximate location of the regions. Adapted, with permission by Elsevier Ltd., from [START_REF] Friston | Dynamic causal modelling[END_REF].

Figure 4 A. Summary of a fictitious 2x2 factorial experimental design, comprising task and stimulus factors. B. Simulated BOLD responses of two areas, y 1 and y 2 (without observation noise). The first area shows a main effect of stimulus and the second area additionally shows a stimulus-by-task interaction. The red and green bars denote when task 1 and task 2 are performed, respectively. C. The DCM which was used to generate the noise-free responses shown in panel B. As shown schematically, all inputs were boxcar functions. +++ denotes strongly positive and + denotes weakly positive inputs and connection strengths, -denotes negative connection strengths. The different strengths of the driving inputs induce a main effect of stimulus in the first area, x 1 . This effect is conveyed onto the second area, x 2 , by means of the x 1 →x 2 connection. Critically, the strength of this connection varies as a function of which task is performed. This bilinear modulation induces a stimulus-by-task interaction in x 2 (c.f. panel B). D. Data generated from the model shown in panel C but with additional observation noise (signal-to-noise ratio of unity).

Figure 5 A. Example of a DCM with second-order terms in the state equation. In this example, the third area modulates the connection from the first to the second area. The first area is driven by two different stimuli (stim1, stim2; randomly mixed events, represented as delta functions, 4 seconds apart) and the third area is driven by some inputs representing cognitive set (mod1, mod2; alternating blocks of 15 s duration, shown as grey boxes in Fig. 5B). Note that the third area is not only driven by external input but also receives an input from the second area. +++ denotes strongly positive and + denotes weakly positive inputs and connection strengths, -denotes negative connection strengths. B. Simulated responses of this system (note that all inputs and connections were given positive weights in this simulation). From top to bottom, the plots show the neural (x) and haemodynamic (y) responses in alternating fashion. The x-axis denotes time (for haemodynamic responses in seconds, for neural responses in time bins of 4 ms), the y-axis denotes arbitrary units. It can be seen easily that evoked activity in the first area only causes a significant response in the second area if the third area shows a high level of activity and thus enables the x 1 →x 2 connection. Furthermore, due to the excitatory x 2 →x 3 connection, a positive reinforcement effect results. Both mechanisms lead to obvious nonlinearities in the generated data (see thick arrows for an example). Note that this model, similar the one in Fig. 4, also generates a "stim × mod" interaction in the second area. This is harder to see by eye than in Fig. 4 because here the driving inputs are randomly mixed events and additionally, strong non-linear effects occur.

Figure 6 A schema of the neural populations which are modelled separately for each region in DCM for ERPs. Different regions are coupled by forward, backward and lateral connections, all of which originate from excitatory pyramidal cells but target specific populations. The figure shows a typical hierarchical network composed of three cortical areas. Extrinsic inputs evoke transient perturbations around the resting state by acting on a subset of sources, usually the lowest in the hierarchy. Reproduced with permission by Elsevier Ltd. from [START_REF] David | Dynamic causal modeling of evoked responses in EEG and MEG[END_REF]. Schematic of the neural model in DCM for ERPs. This schema shows the state equations describing the dynamics of a microcircuit representing an individual region (source). Each region contains three subpopulations (pyramidal, spiny stellate and inhibitory interneurons) that are linked by intrinsic connections and have been assigned to supragranular, granular and infragranular cortical layers. Different regions are coupled.through extrinsic (long-range) excitatory connections as described in Fig. 7. Reproduced with permission by Elsevier Ltd. from [START_REF] David | Dynamic causal modeling of evoked responses in EEG and MEG[END_REF].

Figure 8 DCM for ERPs measured during an auditory oddball paradigm. Left: Predicted (thick) and observed (thin) responses in measurement space. These are a projection of the scalp or channel data onto the first three spatial modes or eigenvectors of the channel data. The predicted responses are based on the conditional expectations of the DCM parameters and show very good agreement with the measured data. Right: Graph depicting the sources and connections of a DCM in which both forward and backward connections were allowed to change between odddball and standard trials. The relative strength of coupling strengths for oddball relative to standard stimuli are shown alongside the connections. The percent conditional confidence that this difference is greater than zero is shown in brackets. Only changes with 90% confidence or more (solid lines) are shown numerically. In all connections the coupling was stronger during oddball processing, relative to standards. A1: primary auditory cortex, OF: orbitofrontal cortex, PC: posterior cingulate cortex, STG: superior temporal gyrus. Reproduced with permission by Elsevier Ltd. from [START_REF] David | Dynamic causal modeling of evoked responses in EEG and MEG[END_REF].

  backward and lateral connections respectively. Within each subpopulation, the dynamics of neural states are determined by two operators. The first transforms the average density of presynaptic inputs into the average postsynaptic membrane potential. This is modelled by a linear transformation with excitatory (e) and inhibitory (i) kernels parameterized

  can be considered as a normalization constant for the product of the likelihood of the data and the prior probability of the parameters, therefore ∫

  , this yields the following expression for the natural logarithm (ln) of the model evidence (

  have developed for using p-values in frequentist statistics, there are conventions for the use of Bayes factors. For example, Raftery (1995) suggests interpretation of Bayes factors as providing weak (BF < 3), positive (3 ≤ BF < 20), strong (20 ≤ BF < 150) or very strong (BF ≥ 150) evidence for preferring one model over another.

Figure 3

 3 Figure 3

Figure 7

 7 Figure 7

Acknowledgments

This work was supported by the Wellcome Trust.


Martin Aranguren
email: martin.aranguren@cnrs.fr

Hijab-wearing women face interactional discrimination in everyday encounters: two field experiments

The document introduces the idea of interactional discrimination and puts it to test through two interlocking field experiments. As a first approximation, interactional discrimination covers part of the phenomena usually designated with the more informal label of "everyday discrimination". It is also close to other more formal ideas such as "interpersonal discrimination", "micro-aggresion" and "assault on selfworth". The latter three ideas are predicated on an opposition between economic and non-economic outcomes of discrimination. In contrast, the defining feature of interactional discrimination is the specific context in which it occurs: a spatio-temporally bounded episode of social interaction. The two reported field experiments show that women who wear an Islamic headscarf in France are exposed to interactional discrimination on the part of men. The data indicate that in the course of a face-to-face engagement with a hijab-wearing woman, men avert gaze, and that on average this pattern of gaze aversion causes strong negative emotions or stress in the recipient. The study shows that stress represents a specific form of harm that interactional discrimination causes to women who wear the hijab in public. It is argued that pre-existing ideas of everyday discrimination, owing to their focus on the conceptual experience of indignity, are not well-suited for accommodating the often unreflective experience of stress.

Interactional discrimination: justification and definition

Before dealing with its interactional variety, it is useful to start with a definition of discrimination in general. It may be said that actor X discriminates actor Y when X treats Y differently from Z, the cause of the difference in treatment is the fact that Y and Z are (seen as) members of different social groups, and the consequence of the difference in treatment is an outcome less favorable to Y than it would had been the case had X treated Y as Z (adapted from Lippert-Rasmussen 2006). Discrimination is a causal and comparative concept. Its causal story is made up of two stages: first, a difference in group membership (e.g. Muslim vs non-Muslim) causes a difference in treatment; second, the difference in treatment causes a relative harm or disadvantage.

Comparison is necessary at each stage: first, to establish a contrast between the way in which Y and Z are respectively treated; second, to establish an ordering (in terms of more and less, better and worse) between the consequences of that difference in treatment for Y and Z.

The typical "correspondence study" in labor economics and other related fields perfectly conforms to this definition [START_REF] Bertrand | Field Experiments on Discrimination[END_REF]. To investigate discrimination in the job market, the researcher creates at least two CVs that are identical in all respects save for a piece of information that allows the employer to assign the (fake) applicant to one social group or another.

To assess if the two candidates are treated differently, the researcher records whether a given CV sent to a given employer (the experimental stimulus) prompts a reply, e.g. an invitation to a job interview (the response). By aggregating these dichotomous responses across the two experimental conditions, the researcher obtains a callback rate by condition -more rarely, researchers go on to see whether the callback translates in a job offer. If these turn out to differ, the researcher concludes that discrimination is at work.

Correspondence studies have been remarkably successful in demonstrating discrimination on various grounds across a variety of contexts. But one unintended consequence of their success, I surmise, has been to lead researchers and the public to restrict the practical applicability of the concept of discrimination to domains in which the detrimental outcome of receiving unequal treatment can be inferred from the act of applying, as in correspondence studies. Thus the concept of discrimination has been extended to other areas in which, analogous to the labor market, the obtention of a desired object goes through an application process, such as access to housing, credit or education (Pager and Shepherd 2008). Because the objects for which applications are relevant tend to coincide with goods and services that can be assigned a monetary value, another unintended bias present in public debates as much as in scientific practice has been to restrict the outcomes that may be affected by discrimination to the sole economic dimension (Lamont et al. 2016).

Given this unintended but unnecessary restriction of the concept of discrimination, stress research is well poised to widen the view. Stress is generally undesirable. If stress is the consequence of a difference in treatment caused by assignment to different social groups, it follows that stress must be admitted, alongside economic utility, as an independent dimension of life upon which discrimination may impinge.

However, to the best of my knowledge in the area of stress the objectivistic definition of discrimination that I have spelled out has not been used. In its lieu, what has been measured is perceived discrimination, as self-reported [START_REF] Williams | Understanding How Discrimination Can Affect Health[END_REF]. Perceived discrimination raises the vexing methodological problems of unreliable recall, absence of an external criterion and confounding with the outcome. At the time of responding, does the respondent remember all the experieced occurrences of discrimination? Did discrimination really occur? Is the respondent's report of perceived discrimination an indication that discrimination occurred or a symptom of the respondent's mental health condition? Discrimination may be said to be interactional when its constituent components (assignment to a group, difference in treatment, detrimental outcome) occur within a spatially and temporally bounded unit of social interaction. The idea of interactional discrimination operates entirely at the "micro" level of interpersonal encounters and leaves the nature of the adverse consequence unspecified. The detrimental outcome may manifest itself on the economic dimension, on that of stress, on both or on any other. Further, interactional discrimination primarily designates an objective episode of social exchange, a pattern of observable behavior, and only secondarily a subjective report of that episode.

In this sense, the idea of interactional discrimination addresses two mostly tacit limitations of contemporary research practice in the field of discrimination. The first limitation is conceptual.

Although not necessarily in theory, in practice most research on objective (vs. perceived) discrimination has restricted the application of this concept to the economic realm, focusing on access to employment, housing, credit, etc. The second limitation is methodological. Again, although not necessarily expressing a theoretical choice, in practice most research that has inquired into the non-economic adverse consequences of discrimination (e.g. stress) has done so on the basis of self-reported measures of perceived discrimination. The concept of interactional discrimination is at the same time neutral with regard to the adverse outcome of differential treatment (vs. a restricted focus on economic disadvantage) and designates primarily an observable episode of social interaction (vs. a possible confusion with the analytically distinct experience of being discriminated against).

In sum, what makes discrimination interactional is not any particular form of harm or disadvantage, economic, emotional or other, but the particular unit or context within which the discriminating occurs: a spatio-temporally delimited episode of social exchange in conditions of copresence -that particular social unit whose classical sociological analysis can be found in the writings of Erving [START_REF] Goffman | Behavior in Public Places: Notes on the Social Organization of Gatherings[END_REF]. In this sense, the cognate ideas of microaggresion [START_REF] Sue | Racial Microaggressions in Everyday Life: Implications for Clinical Practice[END_REF]) and assault on self-worth (Lamont et al. 2016) point to a specific variety of interactional discrimination characterized by a particular type of harmful consequence, namely the experience of indignity. But the variety of interactional discrimination documented in the present article is different. Its harmful consequence is stress, whether or not the stressful confrontation is interpreted as an assault on dignity.

Muslims and interactional discrimination

The available cross-national evidence from Europe indicates that a sizable proportion of non-Muslims hold anti-Muslim views and, at the other end, that Muslims feel discriminated against.

Depending on the survey, the measure and the country, in the ten European states with the largest proportion of Muslim residents the incidence of anti-Muslim attitudes oscillates between 5% and 38% of the population (European Commission 2019; European Values Study 2017), and the part of Muslim respondents who report incidents of discrimination ranges from 9% to 37% (FRA 2017;Pew Research Center 2006).

Moving from subjective perceptions to objective measures of discrimination, the results depend on gender, the context of social activity and the stimulus used to signal Islamic affiliation.

In the context of hiring, and typically using the mention of a religious volunteering antecedent in a fake CV as the signal, correspondence studies conducted in France [START_REF] Adida | Identifying Barriers to Muslim Integration in France[END_REF]Pierné 2013;[START_REF] Valfort | Anti-Muslim Discrimination in France: Evidence from a Field Experiment[END_REF]) but also in other European countries [START_REF] Stasio | Muslim by Default or Religious Discrimination? Results from a Cross-National Field Experiment on Hiring Discrimination[END_REF][START_REF] Koopmans | Taste or Statistics? A Correspondence Study of Ethnic, Racial and Religious Labour Market Discrimination in Germany." Ethnic Race Relations for the Twenty-First Century[END_REF] consistently show that Muslim applicants have a lower probability of receiving a job callback. In the context of everyday contacts in public places, typically using a perceivable sign of Islamic religiosity as the stimulus, the research design allows to know more about the perpetrators and the act of discrimination, but the results are less clear-cut.

A series of field experiments in European cities have examined the effects of the Islamic headscarf or hijab on helping behavior. Some have found that people are less likely to offer assistance to a woman if she wears the hijab [START_REF] Aidenberger | Unveiling Everyday Discrimination. Two Field Experiments on Discrimination against Religious Minorities in Day-to-Day Interactions[END_REF][START_REF] Choi | Linguistic Assimilation Does Not Reduce Discrimination against Immigrants: Evidence from Germany[END_REF] but others indicate no difference in helping between the presence and the absence of the headscarf (MASKED; Diekmann, Jann, and Näf 2014). One of these field experiments, performed with the same procedure in Brussels, Paris and Vienna, also included measures of more subtle cues of interpersonal involvement, finding a mix of warmth and coldness in response to the hijab depending on the city and the gender of the participant (MASKED). Another field experiment in Paris, using a similar procedure to assess whether people treated a bearded man differently when he insinuated that he was going to a mosque, indicated that in this "mosque condition" participants offered help less often and showed lower interpersonal involvement (MASKED). The mentioned differences in treatment observed in the context of interpersonal encounters in public places are candidate instances of interactional discrimination against Muslims. Depending on how they are interpreted, they may cause stress in the recipient. Using the Daily Life Experiences subscale of a checklist instrument named RaLES [START_REF] Harrell | The Racism and Life Experiences Scales (RaLES): Daily Life Experiences (Racial Hassles)[END_REF]) as a catalogue of acknowledged stressful experiences, not being helped when in need or being the recipient of inadequate interpersonal involvement in day-to-day contacts with strangers could be experienced as instances of "being treated rudely or disrespectfully", "others reacting to you as if they were afraid or intimidated", "being treated as if you were stupid", "having your ideas ignored" or "not being taken seriously". Even an intensification of interpersonal involvement could be stressful if the recipient interprets the increase in warmth as an instance of "being treated in an 'overly' friendly or superficial way".

In this sense, there is some evidence that European Muslims regard instances of inadequate nonverbal involvement as offenses. According to the EU-MIDIS survey, nearly 40% of Muslim female respondents who indicated that they wear religious attire in public (mostly, the hijab) reported that they were the target of "inappropriate staring" or "offensive gestures" because they did so (FRA 2017). To the best of my knowledge, the connections between Islamic faith, interactional discrimination, stress and mental health have not been investigated in previous work.

Hypotheses

The studies that follow investigate the hypothesis that women who wear the Islamic headscarf are exposed to interactional discrimination. This hypothesis implies two different predictions: P1: In face-to-face encounters, women who wear the hijab are treated differently from women who do not wear it.

P2: The differences in treatment to which hijab-wearing women are exposed in face-to-face encounters cause them a disadvantage.

To anticipate, a first field experiment shows that the difference in treatment takes the specific shape of a deficit in visual contact in the course of a face-to-face conversation. The second field experiment shows that the average woman responds to this deficit in visual contact with strong negative emotions or stress. In other words, a surplus in stress arises as one of the penalties that interactional discrimination imposes on women who wear the hijab in public.

Method Field Experiment 1

Experimental design. The experiment follows a 2 (headscarf vs. uncovered hair) x 2 (female vs. male passengers) x 6 (metro stations) between-subjects design with random assignment of passengers to conditions and with equal sampling time devoted to each of the 24 unique factor combinations.

Sampling. Data collection proceeded between May 6 and June 21, 2019. The sampling technique consisted in approaching the first passenger arriving to the platform after the departure of the last train and before the arrival of the following one, alternating between the sexes to ensure equal representation.

Procedure. Experimental assays took place on metro platforms and involved a team of three:

Experimenter 1, Experimenter 2 and a female confederate. Experimenter 1 recruited passengers as they arrived at the platform, inviting them to participate in an experiment "on decision making." If the passenger accepted, her or she was told that he or she would be playing an ultimatum game [START_REF] Camerer | Behavioral Game Theory: Experiments in Strategic Interaction[END_REF] with another passenger. While Experimenter 1 explained the rules of the game to the recruited passenger, Experimenter 2 approached discreetly with the confederate, pretending to be filling a questionnaire. Experimenter 1 then introduced the passenger to Experimenter 2 and greeted the confederate, presented as the second player of the ultimatum game. At this point, players were told that before the roles in the game were randomly assigned and the corresponding decisions made, the procedure required them to have a short discussion about the game. With the passenger's (and the confederate's pretended) consent, the ensuing interaction was videotaped with two cameras. The confederate's contributions to the conversation followed a script.

Experimental treatment. In one condition, the confederate wore an Islamic headscarf or hijab. In the other, she appeared with uncovered hair. The rest of the clothing was identical between the conditions. The confederate is the same woman in both conditions.

Measurements. The demographic variables probed with the questionnaire were age, educational achievement, income, and religion. The outcome variables included, on the one hand, the passenger's response to the game and, on the other, a set of nonverbal behaviors indicative of interpersonal involvement, including eye contact. The responses to the game were directly recorded on the tablet as they were given but the nonverbal measures were taken in the laboratory on the basis of the collected video and audio materials, using to this effect the program Elan Linguistic Annotator [START_REF] Sloetjes | Annotation by Category -ELAN and ISO DCR[END_REF]. Regarding the nonverbal outcomes, intercoder reliability was assessed by computing Krippendorff's alpha [START_REF] Hayes | Answering the Call for a Standard Reliability Measure for Coding Data[END_REF] for interval data for a subsample of independently coded assays (at least 20%), yielding in all cases coefficients higher than 0.7.

Eye contact : measurement details. The intensity of eye contact was operationalized as a proportion known as the "gaze rate", which equals the time spent looking at the other's eyes or face divided by the total observation period. Two observation periods were considered in this connection, namely the time spent in the role of the listener and alternatively the time spent in the role of the speaker [START_REF] Exline | Visual Behavior as an Aspect of Power Role Relationships[END_REF]. The script is organized in such a way that the confederate speaks at length twice but the passenger once. Consequently, each passenger contributes two measures of gaze rate while listening and one measure of gaze rate while speaking. In this experiment, eye contact is used to examine differences in treatment between conditions (not whether differences in eye contact are stressful, which is the aim of the second experiment).

Statistical analyses. The outcomes were analyzed using regression models estimated with Bayesian inference. The gaze rate measurements, but restricted to the interval [0, 1],

were treated with a beta-distributed logistic regression. The dichotomous choice in the game was analyzed with a linear probability model.

Field Experiment 2

Experimental design. The experiment follows a 2 (experimental condition) x 6 (metro stations) between-subjects design with random assignment of passengers to conditions and with equal sampling time devoted to each of the 12 unique factor combinations.

Pilot work. During October 2019 a series of field visits were scheduled to calibrate the procedure.

The experimenters served as confederates and played a simplified version of the script that a professional actor was to perform in full-fledged version for the main study. This pilot work confirmed that most women spontaneously framed the ultimatum game in terms of an "obvious" egalitarian division. It also helped to design an adequate stimulus, i.e. a credible and effective imitation of the pattern of gaze that men had directed to the hijab-wearing confederate in Experiment 1. Last, it contributed to phrase in relevant everyday terms the questions meant to capture women's experience of this look.

Sampling. Data collection proceeded between November 4 and December 4, 2019. The sampling technique was identical to the one used in the previous experiment, except that only female passengers were approached.

Procedure. With minor differences, the unfolding of the task was identical to the procedure used in the previous experiment.

Experimental treatment. In the control condition, the male confederate was instructed to look at the interaction partner in a "normal" way when he was in the role of the listener. More precisely, he had to gaze continuously, with minor interruptions such as occasional blinks, while the passenger spoke to him. In the treatment condition, his gaze followed a "program", i.e. an explicit set of rules, which were active as long as the passenger held the floor. The goal of the program was to generate a natural-looking imitation of the pattern of gaze of the men who had interacted with the hijabwearing woman in Experiment 1. The program was not a fixed repetitive routine but a set of responses contingent on the passenger's visual and verbal behavior, in such a manner that its performance did not result in an awkward "robotic" gaze, partly because the underlying rules were not easy to infer. To assess stimulus equivalence and adequacy, the visual behavior of the confederate was measured for all assays. In performance of the program, the conderate's gaze rate turned out to fall within the second quartile of passengers' original gaze rates in the hijab condition in Experiment 1. The term "hijab-gaze" refers to the pattern of visual behavior that resulted from the performance of this program.

Measurements. The demographic variables measured with the questionnaire were age, educational achievement and income. The outcomes were measured using 9-point Likert-type questions concerned with i) stress operationalized as negative affect (four items), ii) attributions of interpersonal involvement (two items) and iii) judgments of conformity with rules of politeness (two items). The negative affect items inquired into the degree to which the confederate's gaze elicited negatively-valenced emotional states (mal à l'aise, gênée, agacée, perturbée). The involvement items sought to capture the extent to which the confederate's gaze indicated that he was attentive to, and interested in, the participant's speech (attentif, intéressé). The politeness items asked about the degree to which the confederate's gaze was judged to be correct and respectful (correct, respectueux). Internal reliability was satisfactory for all sets of questions related to the same construct, as assessed by Cronbach's alpha (negative affect, alpha=0.82; involvement, alpha=0.86; politeness, alpha=0.83).

Statistical analyses. The answers to the 9-point scale questions were analyzed with ordered probit regressions estimated with Bayesian inference.

Results

Field Experiment 1: female confederate wears the hijab On average, passengers gaze at the confederate [78%, 83%] of the time. Confirming P1 for men, in the hijab condition male passengers reduce by [-12%, -2%] the intensity of eye contact.

Additionally, a treatment*gender interaction in the interval [-18%, -3%] indicates that the effect of the hijab differs by gender, suggesting opposite effects of the garb for men and women. In contrast, the responses to the game do not appear to differ between the experimental conditions, neither overall nor within gender.

Field Experiment 2: male confederate replays the "hijab-gaze"

Building on the results of Field Experiment 1, P2 reads more precisely: The pattern of gaze that male passengers directed to the female confederate when she was wearing the Islamic headscarf is stressful for the recipient.

Regardless of the experimental condition, on a 9-point scale the average score is [0.17, 1.28] for negative effect, [6.25, 7.22] for involvement and [7.78, 8.75] for politeness. Confirming P2, being the target of the hijab-gaze increases women's negative affect by [0.81, 2.63], with effect size d= [0.42, 1.37]. Similarly, the hijab-gaze leads to a [-1.94, -0.08] decrease in the level of interpersonal involvement attributed to the confederate, with d= [-1.56 , -0.06]. Although only credible at a less stringent alpha=0.10, participants similarly reported that the hijab-gaze was on average less polite than the control gaze.

Discussion

Using a version of the "ultimatum game" in a public place of Paris, Experiment 1 assessed whether a female confederate is treated differently in a brief face-to-face exchange with a stranger when she signals high Islamic religiosity by wearing a Replicating the result of another experiment (MASKED), the study found that male passengers do diminish the intensity of eye contact in response to the headscarf.

Using the same procedure in the same setting, Experiment 2 investigated the average effect, for women, of being the target of this less intense pattern of gaze, as performed by a male confederate. The results revealed that this look causes stress and the effect size statistic (whose most probable value nears one standard deviation) underscored that this seemingly harmless pattern of gaze actually functions to elicit a powerful negative reaction in the recipient.

Three analytical benefits of the concept of interactional discrimination

Whenever types of discrimination have been distinguished in the past, there has been a tendency to dichotomize the phenomenal field into two categories, setting discrimination in the labour market (especially as it can be measured with correspondence studies) as the point of reference. The other category has been traditionally constructed as a residue, that is to say, as the type of discrimination that is left once discrimination in the labor (or credit, or real estate, etc.) market has been taken out.

This presupposes a prior definition of what is essential to this paradigmatic type of discrimination, and the usual approach has been to locate this essence in the economic nature of the disadvantage at stake. By opposition, the residual category has tended to cover forms of discrimination whose detrimental consequence is non-economic. Some prominent instances of discrimination types constructed with this residual logic are the ideas of "interpersonal discrimination" [START_REF] Michelle | Formal and intepersonal discrimination: a field study of bias toward homosexual applicants[END_REF], "microaggression" [START_REF] Sue | Racial Microaggressions in Everyday Life: Implications for Clinical Practice[END_REF] and "assault on self-worth" (Lamont et al. 2016). The latter two in particular are centrally concerned with the experience of indignity, as the counterpart of economic disadvantage.

The notion of interactional discrimination is not predicated on the opposition between the economic and the non-economic. It arises rather from the acknowledgement of a simple fact: a significant portion of the phenomena covered by those residually constructed categories manifests in the realm of face-to-face interaction. Whatever else may mark out these phenomena, they are also characterized by the setting or context within which the act of discrimination is embedded, namely a spatio-temporally bounded episode of social interaction.

When the criterion for constructing types of discrimination is shifted from the nature of the adverse consequence (economic vs. non-economic) to the setting or context of discrimination (interactional vs. non-interactional), two implications follow. First, nothing prevents an act of discrimination embedded in an episode of social interaction from being economic in its detrimental consequence. Bargaining, for instance, represents a form of face-to-face interaction where the occurrence of discrimination may clearly impose economic disadvantage on the recipient. Second, the form of economic disadvantage widely documented by correspondence studies is notoriously non-interactional: when the decision is made not to call back the discriminated applicant, the decision is obviously not made in the presence of the applicant.

But in the research reported here the form of disadvantage under scrutiny, namely stress, is non-economic. Why bother to introduce new concepts if already established categories such as "micro-aggresion" or "assault on self-worth" can surely do an equivalent job? The answer is that they actually do not. These categories are not meant to cover every form of disadvantage arising in the course of an interactional encounter, but only a quite specific sub-class: acts by others that victims interpret as attacks on their dignity. This is a very demanding definition indeed. It is not difficult to argue that a person may well be discriminated in an interpersonal encounter without realizing it, or experience the discriminatory act yet without invoking the concept of dignity.

In the realm of experience, stress manifests as negative emotion. Interpreting others' behavior as an attack on one's dignity is not necessary for that behavior to arouse a negative emotion. In other words, the conceptually mediated experience of indignity is not necessary for stress to occur, and more generally conceptual thought is not necessary for emotions to be elicited (Lambie and Marcel 2002). The essentially non-reflective experience of negative emotion does not necessarily have a correlate in the essentially reflective experience of indignity.

The concept of interactional discrimination thus brings three immediate benefits:1) it subsumes "micro-aggressions" and "assaults on self-worth" as specific sub-types concerned with lived indignity, 2) it makes theoretical room for discrimination that is interactional and economic (e.g. bargaining), and 3) it carves out an analytical niche for stress as a detrimental consequence of discrimination that is neither economic nor necessarily concerned with the experience of indignity (e.g. the stressful experience of the "hijab-gaze").

These three types of interactional discrimination have been empirically investigated in the research presented above. The "hijab-gaze" could have been construed as a micro-aggression if it had been rated by its recipients as impolite, disrespectful, or improper. But it was not. After bargaining, participants in the first experiment could have discriminated the hijab-wearing confederate on the economic dimension by taking advantage of their privilege as "dictators" in the simplified ultimatum game. But they did not. Nonetheless, male participants did discriminate the hijab-wearing confederate in that they looked at her in a way that on average elicits strong negative emotions in the recipient. Without the concept of interactional discrimination, this subtle differential treatment and its powerful detrimental outcome would have gone unnoticed.

Acknowledgements

The analyses presented in this paper are preliminary and have not been peer-reviewed. The data from Paris and Brussels have been collected as part of the MIDI project, funded by MITI-Centre National de la Recherche Scientifique with a Momentum grant to Martin Aranguren. The funder had no role in the conduct of this research.Interactional discrimination: field experiments

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