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Physics of Life Reviews 15 (2015) 124–127
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Comment
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.
L. Pessoa, M. Najafi / Physics of Life Reviews 15 (2015) 124–127
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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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L. Pessoa, M. Najafi / Physics of Life Reviews 15 (2015) 124–127
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.
L. Pessoa, M. Najafi / Physics of Life Reviews 15 (2015) 124–127
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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.