Network: Computation in Neural Systems
March–December 2011; 22(1–4): 148–153
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Viewpoint
The challenge of non-ergodicity in network
neuroscience
JOHN D. MEDAGLIA1, DEEPA M. RAMANATHAN1,
UMESH M. VENKATESAN1, & FRANK G. HILLARY1,2
1
2
Department of Psychology, Pennsylvania State University, University Park, PA and
Department of Neurology, Hershey Medical Center, Hershey, PA
(Received 11 April 2011; accepted 11 July 2011)
Abstract
Ergodicity can be assumed when the structure of data is consistent across individuals and
time. Neural network approaches do not frequently test for ergodicity in data which holds
important consequences for data integration and intepretation. To demonstrate this
problem, we present several network models in healthy and clinical samples where there exists
considerable heterogeneity across individuals. We offer suggestions for the analysis,
interpretation, and reporting of neural network data. The goal is to arrive at an
understanding of the sources of non-ergodicity and approaches for valid network modeling
in neuroscience.
Keywords: functional neuroimaging, traumatic brain injury, network modeling, neuroscience
Statistical analyses applied to group data are predominately used for hypothesis
testing in many social sciences and the neurosciences. However, it is widely
acknowledged that group data may not fully represent the individual and in extreme
cases, a group model may not be a valid representation of individual processes. For
example, in neuroimaging research, results are frequently averaged together from
Correspondence: Frank G. Hillary, Associate Professor, Psychology Department, Pennsylvania State University,
223 Moore Building, University Park, PA 16802. E-mail: [email protected]
ISSN 0954-898X print/ISSN 1361-6536 online/01/01-04000148–153 ß 2011 Informa Healthcare Ltd.
DOI: 10.3109/09638237.2011.639604
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The challenge of non-ergodicity in network neuroscience
149
many individuals using parametric brain maps, or in terms of networks, by
concatenating timeseries data or averaging covariance matrices (e.g., as in
Independent Components Analysis (ICA), graph theory, structural equation
modeling, etc.) (see Kim et al. 2007; Bullmore & Sporns 2009; Calhoun et al.
2009; Gates et al. 2011; Smith et al. 2011). However, for results to be valid,
statistical models must be equivalent between the level of the group and the level of
the individual. More formally, the data structures must be ergodic for a group model
to validly represent individual processes. Models based on interindividual variation
only validly apply to individuals in certain stringent conditions, and when these
conditions are not met, person-specific techniques are necessary for valid results
(Molenaar & Campbell 2009). To be ergodic, a construct or process of interest
must satisfy conditions of homogeneity (conformity with the same statistical model)
and stationarity (invariance of data structure across time). Non-ergodicity suggests
qualitatively different processes across individuals and time, and is distinct from
mere error variance (Molenaar 2004). If a goal of in vivo neural network research is
to accurately characterize brain processes, proper exploration of ergodicity under
controlled experimental conditions is critical.
As an illustration of this problem in functional networks, we present visual
representations of extended unified structural equation models estimated in the
individual (euSEM; Gates et al. 2011; see Note for the euSEM equation). This
technique is a powerful approach to modeling individual data structures from
multiple functional timeseries. It estimates contemporaneous and lagged relationships in neuroimaging data between regions of interest, and how those relationships
are modulated by the presence or absence of a task condition via regression with
bilinear terms. Figure 1 presents three euSEM models estimated for fMRI data
collected during performance of the 2-back task in healthy individuals (see Hillary
et al. 2011 for full details of the original sample and task).
Subjects were close in age, education level, and behavioral performance in the
scanner (greater than 90% accuracy, 792–916 millisecond reaction times). Despite
these similarities, considerable variation in the number, strength, and direction of
connections are observable. An important observation is that differences between
individual models may be strongly accounted for by even minor differences in
subject demographics or overt behavioral performance. However, the untested
possibility exists that these differences are, in fact, due to real process differences in
human brains despite otherwise equivalent demographics. It should also be noted
that changes within individuals are not widely examined, and it may be the case the
model structures are highly variable even within one run of certain kinds of cognitive
tasks due to maturational processes. Error in estimation likely does not contribute
Note: indicates the ROI time series, u a one-vector input series (which may be expanded to
include more inputs) convolved with a hemodynamic response function, A the contemporaneous relations among ROIs, the lagged associations, input effects, the bilinear
associations, and z error assumed to be a white noise process (see Gates et al. 2011).
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J. D. Medaglia et al.
much to model variance as the euSEM has been found to be robust in simulations
where stationarity is assumed to be high (Gates et al. 2011).
Non-ergodicity may be even more prevalent in clinical contexts. Clinical
syndromes often vary greatly in terms of etiology and symptomatology. Moreover,
how two different brains react to relatively similar pathology may also be highly
individualized. In the following three cases (see Figure 2; see also Hillary et al. 2011
for full characteristics of the task and original sample), individuals that suffered from
moderate to severe traumatic brain injury (TBI) are matched for reaction time
(range ¼ 793–878 ms average), as it is known that reaction time is a strong predictor
of overall fMRI signal in healthy individuals and TBI (Rypma et al. 1999; Rypma
and D’Esposito 2000; Rypma et al. 2001; Rypma et al. 2002; Rypma et al. 2006;
Sweet et al. 2006; Hillary et al. 2010). One might argue that variation in age, gender,
and education level could account for some differences between individuals.
Specific lesions and time since injury might also predict some functional variance.
However, some particular features of these functional networks will be similar to
healthy cases, where it is unclear that controlling for these predictors accounts for all
network non-ergodicity. In clinical samples, it is imperative to implement personspecific techniques, as clinical interpretation and basic questions about how neural
Figure 1. Three euSEM effective connectivity models in healthy young adults.
Note: Red regions represent the bilateral Brodmann’s area (BA) 46. Blue regions represent
the bilateral BA 39. Light green regions represent the anterior BA 32/24. Arrow thickness
represents the strength of relationships between regions (beta weights). Note: for purposes of
illustration only contemporaneous relationships are displayed.
Figure 2. Three euSEM effective connectivity models in individuals following moderate to
severe TBI.
Note: Red regions represent the bilateral BA 46. Blue regions represent the bilateral BA 39.
Light green regions represent the anterior BA 32/24. Only contemporaneous relationships are
depicted. Arrow thickness represents the strength of relationships between regions (beta
weights).
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The challenge of non-ergodicity in network neuroscience
151
systems adapt to neuropathology will present unique challenges to classical
aggregate statistical approaches. Of greatest concern is that the degree of nonergodicity in networks in clinical populations is unknown.
Full understanding of non-ergodicity in neuroscience networks will require direct
investigation of the variance in models across individuals. We suggest network
approaches have particular implicit strengths that may begin to help solve this
problem. Timeseries data from recordings such as fMRI or EEG often involve
hundreds or thousands of data points. Approaches such as structural equation
modeling, ICA, graph theory, and other timeseries network approaches can easily be
applied to neuroimaging datasets at the individual level in addition to examining
change within individuals across time. One initial consideration concerns statistical
methods. The current euSEM procedures use an automatic search based on
LaGrange multiplier tests to determine best fitting models. Some algorithmic
choices in the euSEM framework may cause heterogeneity if there is no clear choice
between paths in the multiplier test. Algorithms that minimize the possibility of
random path selection in a model will help to avoid this issue. Additionally,
researchers applying data-driven strategies are encouraged to attempt to fit alternate
viable models (e.g., theoretically informed models or models from other individuals
in the sample) to individual data to determine if differences are true or statistical
artifacts. In non-ergodic conditions due to factors beyond statistical methodology, a
researcher may be challenged with the following question after estimating many
individual models: ‘‘How can inferences be drawn from the individual level when
everyone appears different?’’ We propose that there are a few simple starting points
for reconciling general observations with highly heterogeneous individuals:
(1) Characterizing hetereogeneity (the degree of non-ergodicity) is a potential
problem of unknown scale in neuroimaging network data. Therefore,
explicitly examining and reporting metrics of heterogeneity is important. For
example, within structural equation modeling approaches, group models can
be estimated from aggregate timeseries data and compared to each individual
dataset in a confirmatory way. Metrics such as chi square, root-mean square
error of approximation (RMSEA), standardized root mean square residual
(SRMR), non-normed fit index (NNFI), and comparative fit index (CFI) all
provide straightforward metrics of fit that are easily reportable. The presence
or absence of specific relationships and degrees of relationships across
models are also clear, reportable metrics. ICA approaches have similar
significance parameters and goodness of fit indices. Qualities such as
numbers of components and what components encompass are concrete
indicators of heterogeneity. Finally, frameworks such as graph theory afford
the opportunity to characterize heterogeneity in numerous ways, including
centrality of nodes within the network, degree distributions, and strengths of
specific relationships.
(2) After analysis of individual models and their variety, theoretical positions can
be advanced in a straightforward manner by assessing the generality of
specific network features. One can observe whether a feature in the sample
(and under certain conditions) is completely unique, shared by some
individuals, or shared by all individuals. Only features shared by all
individuals are truly generalizable in the original nomothetic sense (cf.,
Lamiell 1998).
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(3) Finally, specific research can be conducted to determine sources of nonergodicity. For features of networks that are either unique in samples or
observed some of the time, the following question is critical to examine: ‘‘Is
network variance random, due to unseen contributions of a mediating
variable, or simply a different solution to a networking problem under certain
conditions?’’ While not examined in the previously discussed euSEM
examples, non-stationarity in neural networks will require considerable
exploration as a potential primary contributor to non-ergodicity.
Note that these proposed steps toward a true understanding of non-ergodicity
depend explicitly on research that characterizes individual timeseries data. It will
likely be the case that some processes are completely ergodic, and no variance in
important network structures occurs. However, there may be an unknown number
of conditions where non-ergodicity exists and has important consequences for the
interpretation of data. It is very likely that network models ranging from disease to
attention, working memory, motor control, executive functions, and many complex
cognitive or behavioral systems have qualitative variability that would yield
important information about generalizable findings and unique occurrences in
natural neural systems. Until individual techniques become a routine approach to
analyzing network data, a currently unknown percentage of results generalized from
aggregate approaches will be at least partially invalid. We suggest that complete
understanding of when and why non-ergodicity occurs will rely on analysis of
multiple levels of neural organization, dynamic internal properties,developmental
and environmental processes, and stochastic processes. Explicit examinations of
ergodicity are the only means for clarifying and solving this problem in network
neuroscience.
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