Signal Processing - May 2016 - 24
amplitudes of these eigenmodes or patterns correspond to the
order parameters described in the "State-Space Modeling and
Effective Connectivity" section. The (negative) inverse of the
Lyapunov exponent corresponds to the characteristic time con-
stant of each mode, where each mode with a small exponent
(large time constant) corresponds to an intrinsic brain network
or resting-state network.
Causal modeling of neuronal dynamics
structural causal modeling. This is because the arrow of time
can be used to convert a directed cyclic graph into an acy-
clic graph when the nodes are deployed over successive time
points. This leads to SEM with time-lagged data and related
autoregression models, such as those employed by Granger
causality described previously. As established in the previous
section, these can be regarded as discrete time formulations of
DCMs in continuous time.
The past decade has seen the introduction of graph theory to
Structural and dynamic causal modeling
brain imaging. Graph theory provides an important formula-
As already established, in relation to the modeling of fMRI
tion for understanding dynamics on structure. Developments
time series, DCM refers to the (Bayesian) inversion and com-
in this area have progressed on two fronts:
parison of models that cause observed data.
understanding connections between graphs
These models are usually state-space mod-
Neural pathways are
and probability calculus and the use of
els expressed as (ordinary, stochastic, or
flexible, adaptable,
probabilistic graphs to resolve causal inter-
random) differential equations that govern
connected, and moldable
actions. The probabilistic graph frame-
the motion of hidden neurophysiological
work goes beyond classical constructs by
states. These models are generally equipped
by changes in our
providing powerful symbolic machinery
with an observer function that maps from
environment or by
and notational convenience (e.g., the use
hidden states to observed signals [see (1)].
injury or disease.
of dependency graphs to resolve Simpson's
The basic idea behind DCM is to formulate
paradox; see "Simpson-Yule Paradox").
one or more models of how data are caused
Within this enterprise, one can differentiate at least two
in terms of a network of distributed sources. These sources talk
streams of work: one based on Bayesian dependency graphs or
to each other through parameterized connections and influ-
graphical models called structural causal modeling [99] and
ence the dynamics of hidden states that are intrinsic to each
the other based on causal influences over time, which we con-
source. Model inversion provides estimates of their parameters
sider under DCM. Structural causal modeling originated with
and the model evidence.
SEM [47] and uses graphical models (Bayesian dependency
We have introduced DCM for fMRI using a simple state-
graphs or Bayes nets) in which direct causal links are encoded
space model based on a bilinear approximation (extensions
by directed edges. These tools have been largely developed by
to, for example, nonlinear [103] and two-state [104] DCM,
Pearl [22] and are closely related to the ideas in [100]-[102].
among others, are also available and are in use) to the underly-
An essential part of network discovery in structural causal
ing equations of motion that couple neuronal states in differ-
modeling is the concept of intervention: eliminating connec-
ent brain regions [32]. Most DCMs consider point sources for
tions in the graph and setting certain nodes to given values.
both fMRI and EEG/MEG data (cf. equivalent current dipoles)
Structural causal modeling lends a powerful and easy-to-use
and are formally equivalent to the graphical models used in
graphical method to show that a particular model specifica-
structural causal modeling. However, in DCM, they are used
tion identifies a causal effect of interest. Moreover, the results
as explicit generative models of observed responses. Inference
derived from structural causal modeling do not require spe-
on the coupling within and between nodes (brain regions) is
cific distributional or functional assumptions, such as multi-
generally based on perturbing the system and trying to explain
variate normality, linear relationships, and so on. However,
the observed responses by inverting the model. This inversion
it is not the most suitable framework to understand coupled
furnishes posterior or conditional probability distributions
dynamical systems because it is limited in certain respects.
over unknown parameters (e.g., effective connectivity) and the
Crucially, it deals only with conditional independencies in
model evidence for model comparison [105]. The power of the
DAGs. This is problematic because brains perform computa-
Bayesian model comparison in the context of DCM has become
tions on a directed and cyclic graph. Every brain region is
increasingly evident. This now represents one of the most
connected reciprocally (at least polysynaptically), and every
important applications of DCM and allows different hypoth-
computational theory of brain function rests on some form of
eses to be tested, where each DCM corresponds to a specific
reciprocal or reentrant message passing. Another drawback is
hypothesis about functional brain architectures [106]-[112].
that the causal calculus of structural causal modeling ignores
DCM has been used mostly for (task-based) fMRI and electro-
time. Pearl argued that a causal model should rest on function-
physiological dynamics (EEG/MEG/LFPs), but the most recent
al relationships between variables. However, these functional
advances have focused on the modeling of intrinsic brain net-
relationships cannot deal with (cyclic) feedback loops. Pearl
works in the absence of exogenous influence, known as resting-
[14] argued for DCMs when attempting to identify hysteresis
state fMRI [74]. In the remainder of this section, we briefly
effects, where causal influences depend on the history of the
review these developments and discuss these new mathematical
system. Interestingly, the DAG restriction can be finessed
models. We also showcase some of their clinical applications to
by considering dynamics and temporal precedence within
neurodegenerative diseases, such as Parkinson's disease.
24
IEEE Signal Processing Magazine
|
May 2016
|
Table of Contents for the Digital Edition of Signal Processing - May 2016
Signal Processing - May 2016 - Cover1
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