Causal graphs illustrating the effect of interventions. A : Graph showing a case where the statistical dependence between Y and X is (partly) due to a causal interaction from Y to X . B : Graph showing a case where the statistical dependence between Y and X is induced solely by the confounding variable Z . C : Graph corresponding to the intervention do ( Y ~ y ’ ) in the causal graph shown in A . D : Graph corresponding to the intervention do ( Y ~ y ’ ) in the causal graph shown in B . doi:10.1371/journal.pone.0032466.g001 

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... to a given value but that the mechanisms generating this variable are perturbed. We will see below that the variability in the intervention can be captured introducing a probability distribution of interventions P ( do ( V k ~ v )) . The effect of an intervention on the joint distribution is most easily seen when the joint distribution is factorized according to a causal graph. In this case the intervention do ( V k ~ v ) corresponds to deleting the term corresponding to V k in the factorization and setting V k ~ v in all other terms depending on V k . This truncated factorization of the joint distribution is referred to as the postinterventional distribution , that is, the distribution resulting from an intervention. In Supporting Information S1 we give a formal definition of the effect of an intervention and some examples. Graphically, the effect of an intervention is particularly illuminating: intervening in one variable corresponds to the removal of all the arrows pointing to that variable in the causal graph. This represents the crucial aspect of interventions mentioned above: intervention ‘disconnects’ the intervened variable from the rest of the system. Figure 1A and B illustrate two different scenarios for how a statistical dependence between Y and X could come about. If only Y and X are observed, these scenarios are indistinguishable without intervening. The causal graphs corresponding to the intervention do ( Y ~ y ’ ) are shown in Figure 1C and D. It is clear that only the graph shown in Figure 1C implies a statistical relation between Y and X illustrating how the intervention helps to distinguish between causal and spurious (non- causal) associations. Indeed, interventions can generally be used to infer the existence of causal connections. Conditions and measures to infer causal connectivity from interventions have been studied for example in [16,17]. Given this calculus of interventions, the causal effect of the intervention of a variable Y on a variable X is defined as the postinterventional probability distribution p ( x j do ( Y ~ y )) (see Definition 3.2.1 in [10] and Supporting Information S1). This definition understands the causal effect as a function from the space of Y to the space of probability distributions of X . In particular, for each intervention do ( Y ~ y ’ ) , p ( x j do ( Y ~ y ’ )) denotes the probability distribution of X given this intervention. In Supporting Information S1 we show how p ( x j do ( Y ~ y ’ )) can be computed from a given factorization of the joint distribution of X and Y . Note that this definition of causal effect is valid also if X and Y are multivariate. This definition of causal effects is very general and in practice it is often desirable to condense this family of probability distributions (i.e. one distribution per intervention) to something lower- dimensional. Often the field of study will suggest a suitable measure of the causal effect. Consider the example of the two neurons introduced above, and let do ( Y ~ y ’ ) denote the intervention corresponding to making neuron Y emit an action potential. To not introduce new notation we let X and Y stand for both the identity of the neurons as well as their membrane potentials. Then a reasonable measure of the causal effect of Y on X could be That is, the difference between the expected values of the postinterventional distribution p ( X j do ( Y ~ y ’ )) and the marginal distribution p ( X ) (c.f. [18]). In other words, the causal effect would be quantified as the mean depolarization induced in X by an action potential in Y . Clearly this measure does not capture all possible causal effects, for example, the variability of the membrane potential could certainly be affected by the intervention. Intervening one variable is similar to conditioning on this variable, this is illustrated both in the notation and also in the effect of an intervention on the joint distribution. However, there is a very important difference in that an intervention actually changes the causal structure whereas conditioning does not. As mentioned above, it is this aspect of the intervention that makes it a key tool in causal analysis. Formally, this difference is expressed in that p ( X j do ( Y ~ y ’ )) in general differs from p ( X j Y ~ y ’ ) . Consider for example the case when X is causing Y but not the other way around, i.e. X ? Y , then P ( X j do ( Y ~ y ’ )) ~ P ( X ) whereas, in general, P ( X j Y ~ y ’ ) = P ( X ) . A very important and useful aspect of this definition of casual effect is that if all the variables in the system are observed the causal effect can be computed from the joint distribution over the variables in the observed non-intervened system. That is, even if the causal effect is formulated in terms of interventions, we might not need to actually intervene in order to compute it. See Supporting Information S1 for details of this procedure and S2 for the calculation of causal effects in the graphs of Figure 1. On the other hand, if there are hidden (non-observed) variables, physical intervention is typically required to estimate the causal effect. The definition of causal effects stated above is most useful when studying the effect of one or a few singular events in a system, that is, events isolated in time that can be thought of in terms of interventions. However, in neuroscience the interest is often in functional relations between different subsystems over an extended period of time (say, during one trial of some task). Furthermore, the main interest is not in the effect of perturbations, but in the interactions that are part of the brains natural dynamics. Consider for example the operant conditioning experiment, a very common paradigm in systems neuroscience. Here a subject is conditioned to express a particular behavior contingent upon the sensory stimuli received. Assume we record the simultaneous activity of many different functional ‘units’. Then a satisfactory notion of causal effect of one unit on another should quantify how much of the task-related activity in one unit can be accounted for by the impact of the causal connections from the other, and not the extent to which it would be changed by an externally imposed intervention. Of course, there are other cases where the effect of an intervention is the main interest, such as for example in studies of deep brain stimulation (e.g. [19]). In these cases the interventional framework is readily applicable and we will consequently not consider these cases further. We will instead focus on the analysis of natural brain dynamics which is also where DCM and Granger causality typically have been applied. These considerations indicate some requirements for a definition of causal effects in the context of natural brain dynamics. First, causal effects should be assessed in relation to the dynamics of the neuronal activity. From a modeling point-of-view this implies that the casal effect can typically not be identified with parameters in the model. Second, the causal effects should characterize the natural dynamics, and not the dynamics that would result from an external intervention. This is because we want to learn the impact of the causal connections over the unaltered brain activity. We will refer to causal effects that fulfill these requirements as natural causal effects between dynamics . We will now see that it is possible to derive a definition of natural causal effects between dynamics from the interventional definition of causal effects. We start by examining when natural causal effects between variables exist and in the following section we consider natural causal effects between dynamics. As explained above, a standard way to define causal effects is in terms of interventions. Yet, many of the most pertinent questions in neuroscience cannot be formulated in terms of interventions in a straightforward way. Indeed, workers are often interested in the ‘the influence one neural system exerts over another’ in the unperturbed (natural) state [8]. In this section we will state the conditions for when the impact of causal connections from one subsystem to another (as quantified by the conditional probability distribution) can be given such a cause-and-effect interpretation. We first consider the causal effect of one isolated intervention. For a given value y of the random variable Y , we define the natural causal effect of y on the random variable X to be p ( x j Y ~ y ) if and only if In words, if and only if conditioning on y is identical to intervening to Y ~ y , the influence of y on X is a causal effect that we call a natural causal effect . Since the observed conditional distribution is equal to the postinterventional distribution given Y ~ y , we interpret this as the intervention naturally occurring in the system. Note that this definition implies that if Eq. 2 does not hold, then the natural causal effect of y on X does not exist. Next we formulate the natural causal effect between two (sets of) random variables. The natural causal effect of Y on X is given by if and only if Eq. 2 holds for each value of Y . Note that Eq. 3 is a factorization of the joint distribution of X and Y . Indeed we have This means that if Eq. 2 holds for each value of Y then the natural causal effect of Y on X is given by the joint distribution of X and Y . At first glance it might seem strange that the factor p ( Y ~ y ) appears in the definition of the natural causal effect (Eq. 3). After all, the effects of the causal interactions are ‘felt’ only by X , and we think of Y as being the cause. However, it is clear that the conditional distributions p ( x j Y ~ y ) will in general depend on y which means that to account for the causal effects of Y on X we need to consider all the different values of Y according to the distribution with which they are observed. This means that we must consider how often the different single natural interventions Y ~ y happen, ...

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... the dynamics. DCM addresses question Q2 about the mechanisms aiming to provide a realistic model of how the dynamics are generated. In this regard, the analysis of the causal effects considered above cannot substitute the DCM approach. However, considering that only in some cases the natural causal effects between brain regions exist helps to bound the meaning of effective connectivity, generally understood as the influence one region exert on another. The coupling parameters can be related to causal effects of external interventions [14] but do not quantify natural causal effects occurring in the recorded dynamics. Furthermore, the analysis we suggested above can straightforwardly be extended to examine the causal effect of a change in a coupling parameter on some aspects of the dynamics, something which is not easy to evaluate from the comparison of the coupling parameters across conditions. We will illustrate this point further when analyzing causal effects in an example system below. We now examine a model system to illustrate the distinction between the inference of causality and the analysis of the causal effects. Here we focus on a simple example of a stationary Markov binary process. In Supporting Information S1 we study the case of linear Gaussian stationary stochastic processes. We note that in these examples all the measures used are calculated analytically (see Methods) to isolate the fundamental properties of the measures from issues related to estimation from data. We consider the transfer entropy T Y ? X and two measures related to Eqs. 13 and 16, respectively. In particular, instead of the Kullback-Leibler divergence used in these equations we calculate the Jensen- Shannon Divergence (JSD) (see Methods), since it is well defined for probabilities with different domains, as the ones resulting from the Markov process explained below. So consider a stationary bivariate Markov binary process of order 1 . Both X and Y take only values 0 and 1 . The process is completely determined by the transition probabilities and by the condition of stationarity: Furthermore, we assume that only unidirectional causal connections from Y to X exist. Accordingly, the transition probabilities can be separated as the product p ( x i z 1 , y i z 1 j x i , y i ) ~ p ( x i z 1 j x i , y i ) p ( y i z 1 j y i ) . In particular, we let the transition probabilities for Y be p ( Y i z 1 ~ y j Y i ~ y ) ~ d , that is, d is the probability that the same value is taken at subsequent steps. The transition probabilities for X 1 z g are such that p ( X i z 1 ~ y j Y i ~ y ) ~ , independently of the 2 value of X i . Therefore g determines the strength of the connection from Y i to X i z 1 , and there is a causal connection from Y to X for g w 0 . For g ~ 0 , X i z 1 takes value 0 or 1 with equal probability and independently of X i and Y i . In the case d ~ 0 , when Y deterministically alternates between 0 and 1 , this example corresponds to one already discussed in Kaiser and Schreiber (2002) [46]. Here we present results for nonzero values of d with different degree of stochasticity. We calculate the measures using 3 time lags for the past X i and Y i , since this is enough for convergence and for higher lags the values obtained do not differ significantly. In fact, given that causal connections are only of order 1 , one time lag is enough when conditioning on Y i . However, since in the transfer entropy (Eq. 15) the conditional entropy H ( X i z 1 j X i ) appears, where there is no conditioning on Y i , one has to consider all the information about Y i that exists in the past X i . First we examine how the transfer entropy T Y ? X depends on g and d (Figure 4A). Supporting its use for the inference of causality from Y to X , the transfer entropy T Y ? X is zero if and only if g ~ 0 . In the opposite direction T X ? Y is always zero for this example (results not shown). In the Granger causality approach the transfer entropy is used also as a measure of the strength of the causal connection. From the Figure it is clear that the relation between the coupling parameter g and T Y ? X depends strongly on d . In fact, for low values of d , T Y ? X is nonmonotonic with g . In words, the Granger causality measure is nonmonotonic with the parameter that determines the strength of the connection. This can be understood taking into account that transfer entropy quantifies the extra reduction of uncertainty that results from considering the past of Y after considering the past of X . For low d the dynamics of Y are almost deterministic, and thus when g increases the dynamics of X become also more and more deterministic. For such, almost deterministic, dynamics the remaining uncertainty of X i z 1 after conditioning on X i is already very small, and thus the extra reduction given Y i decreases with high g . In fact, for the extreme values d ~ 0,1 ( Y completely deterministic), the nonmonotonicity leads to T Y ? X ~ 0 for g ~ 1 see Figure 1 in Kaiser and Schreiber (2002) [46]. In such extreme cases transfer entropy cannot be used even to infer causal interactions. In fact, this limitation of transfer entropy in the inference of causality for strongly synchronized systems is well known e.g. [17,47,48]. In Figure 4B we show the Jensen-Shannon Divergence (JSD) for distributions of the type p ( x i z 1 j Y i ~ y i ) . Since there is only unidirectional causality from Y to X this distribution fulfills the criterion of existence for natural causal effects (Eq. 2), and thus the JSD can be used to quantify the changes in the natural causal effects from Y to X when g changes. This corresponds to the type of analysis described in relation to Eq. 13. Here the different configurations are identified by the value of g . In particular we take the distribution p ( x i z 1 j Y i ~ y i ) obtained for g ~ 0 , for which there is no causal connection, as a reference to compare the natural causal effects to. Notice that for this Markov process, since by construction there is no causal connection from X i to X i z 1 , we have that p ( x i z 1 j Y i ~ y i ) ~ p ( x i z 1 j X i ~ x i , Y i ~ y i ) . This means that the natural causal effects p ( x i z 1 Y ~ y ) correspond to the distribution appearing in the numerator of the logarithm in the definition of the transfer entropy (Eq. 15). What is different with respect to the transfer entropy is the probability distribution used as a reference for comparison. Now the natural causal effects are compared across configurations. We see that the changes in the natural causal effect p ( x i z 1 j Y i ~ y i ) monotonically increase with g and are independent of d . The independence of d results from the particular generation 1 of the process, since p ( Y i ~ 0) ~ p ( Y i ~ 1) ~ independently of d 2 and since the causal interactions are of order 1 we have that p ( x i z 1 j Y i ~ y i ) ~ p ( x i z 1 j Y i ~ y i ) . The monotonic divergence with respect to the distribution obtained for g ~ 0 demonstrating that in this case there is a monotonic relation between g (the strength of the connection), and the impact of the causal connection (the natural causal effects). This should be contrasted to the results using transfer entropy described above. In Figure 4C we show the Jensen-Shannon divergences for distributions of the type p ( x i ) (Eq. 16). In contrast to the probability distributions p ( x i z 1 j Y i ~ y i ) , these distributions do not represent a natural causal effect that occurs in the dynamics. However, since the change of g can be seen in itself as an external intervention of the system, we can compare p ( x i ) in dependence on g as a way to quantify the causal effect of this change in the model. As before we take as a reference the distribution obtained for g ~ 0 . For d = 1 = 2 , JSD( p à ( x i ), p ( x i )) increases monotonically with g indicating that an increase in the strength of the connection renders the distributions more different. However, for d ~ 1 = 2 a constant zero value is obtained. Importantly, this should not be seen as a limitation of JSD( p à ( x i ) ; p ( x i )) to quantify the causal effect, on the contrary it indicates that, with respect to the distribution p ( x i ) , the changes in the effective connectivity ( g ) have no effect for d ~ 1 = 2 . This illustrates how focusing on the value of a coupling parameter may be insufficient in order to describe the impact that a change has in a particular aspect of the dynamics. In this work we have analyzed the applicability of the cause- and-effect framework to the study of natural dynamics of systems consisting of interacting subsystems. Our main result is that it is generally not possible to characterize the effects of the interactions for each subsystem separately. That is, the effect of causal interactions can typically not be described in terms of the effect of one subsystem over another. Rather, the interactions unifies the subsystems and creates a dynamics that transcends the limits posed by the individual systems. This result is generic in the sense that it only depends on the causal structure (i.e. on the topology of the causal connections) and not on the details of the system under study. Our work suggests that analyzing the effect of interactions in the natural dynamics in terms of cause-and-effect is of limited use, in particular in systems where the functional units tend to be heavily interconnected, such as the brain. We emphasize that our contribution should not be seen as a new method to substitute other approaches to causal analysis. The conditions of existence of natural causal effects indicate that inference of causal connections and analysis of causal influences should be considered different types of analysis with different requirements. When natural causal effects do not exist, they can not be quantified, no matter what measure is used. Our analysis therefore has important implications for all approaches aiming at ...