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Complex-system causality in large-scale brain networks Comment on “Foundational perspectives on causality in large-scale brain networks” by M. Mannino and S.L. Bressler Luiz Pessoa a,∗ , Mahshid Najafi b a Department of Psychology, University of Maryland, College Park, MD 20742, USA b Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA

Received 5 October 2015; accepted 12 October 2015 Available online 23 October 2015 Communicated by L. Perlovsky

Keywords: Causality; Complex systems; Networks; Modularity; Functional MRI

Mannino and Bressler [1] discuss foundational issues related to understating causality in a complex system such as the brain. We largely agree with their main point that standard versions of causality, such as those espoused in classical physics, provide an inadequate basis to support the understanding of complex systems. In a nutshell, instead of thinking that one event causes another, it is more fruitful to think that the occurrence of one event changes the probability of occurrence of other events. Such probabilistic notion of causation is, we believe, an important step in attempting to unravel the workings of the brain. Although we strongly support the approach advocated by Mannino and Bressler, we would prefer referring to it as “complex system causality” instead of “probabilistic causality.” This is because, even though a probabilistic account should still be used, neutral terminology could help fend off inevitable counter reactions related to the “inherent” nature of the brain. As Mannino and Bressler state, their formalism is agnostic with respect to the question of whether the brain operates deterministically or stochastically. In any case, below, we briefly discuss our views on understanding causality in the brain. 1. Causation To a great extent, the mission of neuroscience is to understand the nature of signals observed in different parts of the brain, and to attempt to disentangle the potential contributions to those signals. But here lies the problem, too. The problem can be illustrated by considering a type of reasoning prevalent in neuroscience, what can be called the billiard ball causal model (Fig. 1A). In this model, force applied to a ball leads to its movement on the table until it hits the DOI of original article: http://dx.doi.org/10.1016/j.plrev.2015.09.002. * Corresponding author at: Department of Psychology, Biology–Psychology Building, University of Maryland, College Park, MD 20742, USA.

E-mail address: [email protected] (L. Pessoa). http://dx.doi.org/10.1016/j.plrev.2015.10.009 1571-0645/© 2015 Elsevier B.V. All rights reserved.

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Fig. 1. Schematic representation of causal frameworks. (A) Billiard ball scheme. Complex system scheme with two (B) or many “particles” (C). In complex systems like the brain, standard causation conceptualizations fail to capture the interdependent nature of signals across regions.

target ball. In this case, the reason the target ball moves is obvious: the first ball hits it, and via the force applied to the target ball, the target ball moves. But this mode of thinking, which has been very productive in the history of science, is too impoverished when complex systems – the brain for one – are considered. To illustrate why, consider two brain “systems,” such as emotion and cognition. One possibility is that these systems are decomposable. Emotion processes (and brain circuits) operate separately from cognitive processes (and brain circuits). The separation may be partial; for example, anatomical connections may link the two. But the overall system with emotion and cognition is decomposable in that each subsystem operates according to its own intrinsic principles, independently of the other [2]. A second scenario can be called nondecomposable, because the interrelatedness in the brain of the subsystems (via extensive anatomical connectivity) is such that they are no longer isolable – interfering with one, will influence the other, and vice versa. Of course, in between these two extremes lies a continuum of possible organizations [3]. The contention that we make is that brain “systems” are not isolable from one another. This is not to say that the associated mental processes are so interrelated as to become one and the same thing. But when systems are not isolable, understanding the interrelatedness between “subsystems” means that we should consider interactions between systems and integration of signals as the central elements to be unraveled [3]. This is where standard frameworks of causation do not provide useful intuition. Let’s return to the billiard-ball model discussed above. Its simplicity lies in the existence of two spatially separate billiard balls that make simple contact with each other. We can think here of typical diagrams seen in neuroscience papers that place mental processes in separate boxes (like billiard balls) that can affect each other in direct, simple ways (like a ball hitting another), as diagrammed by arrows connecting the boxes. But this analogy will not be helpful in nondecomposable systems – like the brain. Whereas thinking of causation in complex systems is much more challenging, consider the modification illustrated in Fig. 1B. Here, the two balls are connected by a spring, and the goal of explanation is not to explain where ball 2 ends up. Instead, when the initial force is applied to ball 1, the goal is to understand the evolution of the ball1 –ball2 system as the two balls interact with each other. More generally, a series of springs with different coupling properties links the multiple elements in the system, and we are interested in understanding the evolution of a large “multi-particle” system (Fig. 1C). 2. Dynamic brain networks The upshot is that simple ways of reasoning about causation are inadequate when unraveling the workings of a complex system such as the brain. Therefore, we suggest that, instead of focusing on causation as the inherent goal of explanations in neuroscience, a fruitful research avenue is to develop formal tools that describe the multivariate covariance structure of brain data. In other words, we are interested in describing the joint state of a set of brain regions, and how this joint state evolves through time. Consider the set of activation strengths for a set of brain

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Fig. 2. Dynamic covariance estimation. Simulated functional MRI data were generated for two brain regions (signal-to-noise ratio: 1). Actual covariance values are shown in black. The red line shows the estimated covariance according to the Bayesian Multivariate Dynamic Covariance Model [4] and the results of a simple windowed procedure are shown in green. Note that, unlike correlation, covariance values are not normalized between −1 and 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

regions: x1 , x2 , · · · , xn . The vector x describes the current state of the system and x(t) describes the trajectory across time. A major goal in understanding the system is to understand how groups of regions dynamically coalesce into coherent units and how they dissolve when their assembly is no longer needed to meet processing demands. Understanding the joint state of a set of regions thus calls for estimations of the covariance (or correlation) structure in the data. However, the covariance structure is not stationary; instead, it fluctuates across time as a function of dynamic processes that are engaged by the brain. This is, of course, the reason behind the growth in analyses of functional connectivity that are dynamic, as well as dynamic network analysis. But covariance is a property that is challenging to estimate as a function of time. Among other reasons, this is because the covariance between two variables, cov(x, y), is measured by using all of the data (that is, time points) in x and y – the temporal dimension is effectively lost by collapsing the entire data into a single estimate. A trivial solution is, therefore, to “window” the data and consider only the segments of interest to compute the covariance. Thus, one could determine the covariance at time t by centering the window at t and considering k past and k future data points. However, to make the estimation sensitive to local changes in covariance, the window should be relatively short; but doing so, causes the estimation to be highly susceptible to noise. Thus, better methods are needed to estimate covariance than simple sliding window methods. Fortunately, recent years have witnessed considerable advances in methods for covariance estimation, in particular in the domain of Bayesian techniques. We briefly illustrate one of these methods here, called the Bayesian Multivariate Dynamic Covariance Model [4]. The time-varying covariance matrix (where each matrix entry corresponds to the covariance of the associate pair of brain regions), t , is estimated as follows:   = C Tt C t + B t x t−1 x Tt−1 B t + At At t

t−1

where the matrices A, B, and C are matrices of coefficients to be estimated given the data. Like in other Autoregressive and Moving Average (ARMA) settings, the t − 1 term shows the dependence on the immediate past (which can be  extended further into the past if desired). In this formulation, the latent covariance matrix is dependent on both its t  most recent past value t−1 and the previous time-series observation x t−1 . The dynamic parameters can be estimated via a diffusion process in which the values at t − 1 are obtained by adding a small perturbation to the parameters at time t [4]. To briefly illustrate the method, we simulated functional MRI data in a blocked design. As shown in Fig. 2, the Bayesian method does a good job of tracking covariance, although it is relatively slow at catching up when the covariance changes abruptly. For comparison, we also show results of a windowed procedure; its behavior is rather poor and noisy. To conclude, understanding how modern state-of-the-art dynamic covariance models behave with functional MRI data, and how MRI data can be improved to allow the effective application of these models, are important goals for future research.

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Acknowledgement Research funded in part by the National Institute of Mental Health (MH071589). References [1] Mannino M, Bressler SL. Foundational perspectives on causality in large-scale brain networks. Phys Life Rev 2015;15:107–23. http://dx.doi.org/10.1016/j.plrev.2015.09.002 [in this issue]. [2] Bechtel W, Richardson RC. Discovering complexity: decomposition and localization as strategies in scientific research. 2nd ed. Cambridge, MA: MIT Press; 2010. [3] Pessoa L. The cognitive-emotional brain: from interactions to integration. Cambridge: MIT Press; 2013. [4] Wu Y, Hernández-Lobato JM, Ghahramani Z. Dynamic covariance models for multivariate financial time series. In: 30th international conference on machine learning. 2013. p. 558–66.

Complex-system causality in large-scale brain networks: Comment on "Foundational perspectives on causality in large-scale brain networks" by M. Mannino and S.L. Bressler.

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