WORKSHOP ON DIRECTED
FUNCTIONAL CONNECTIVITY
ANALYSIS USING WIENERGRANGER CAUSALITY
Dr. Steven Bressler
Cognitive Neurodynamics Laboratory
Center for Complex Systems & Brain Sciences
Department of Psychology
Florida Atlantic University
http://www.ccs.fau.edu/~bressler/
THE PROBLEM
To understand the neural basis of
cognition
 To
understand the neural basis of
cognition requires understanding the
influences exerted by neuronal groups in
the cortex on one another.

Stimulation

Ablation

Statistical time series analysis
DIRECTED FUNCTIONAL
CONNECTIVITY

Directed Functional
Connectivity (DFC)
refers to a class of
metrics used to infer
causal influence
(effective connectivity)
between neuronal
groups.

DFC estimates directionally
oriented functional
relations between neuronal
groups.
THE CONCEPT OF CAUSALITY
David Hume

“an object, followed by
another, and where all
objects similar to the
first are followed by
objects similar to the
second”
Bertrand Russell
“…to exhibit certain
confusions, especially in
regard to teleology and
determinism, which appear
to me to be connected with
erroneous notions as to
causality"
INTERPRETING CAUSALITY
Deterministic Causality

A change in the
activity of one group
(A) necessarily
produces a change in
that of another group
(B).
P(B|A) = 1
Probabilistic Causality

A change in the
activity of A
contributes an
influence that changes
the likelihood of a
change occurring in
the activity of B.
P(B|A) > P(B|~A)
CAUSALITY IN THE CORTEX
[MANNINO & BRESSLER, IN PREP]
The complex large-scale anatomical
connectivity of the cerebral cortex is
characterized by bi-directionality and
heavy convergence.
 The influence from one cortical neuronal
group (A) does not necessarily produce a
measurable effect in another group (B),
and cortical causality is nondeterministic.
 Group A may change the likelihood of a
change in B, so causal influences in the
cortex are probabilistic.

DFC & CAUSALITY
Metrics for DFC should reflect probabilistic
causality rather than deterministic causality
between neuronal groups.
 Time series DFC metrics should quantify the
degree of predictability of the time series of one
group from that of another.
 Granger causality is a probabilistic concept of
causality that is based on predictability.
 For two simultaneous time series, one series is
called ‘Granger causal’ to the other if we can
better predict the second series by incorporating
knowledge of the first one.

STATISTICAL CAUSALITY
Norbert Wiener
Clive Granger
WIENER CAUSALITY
Wiener considered 2 functions, f1(α) and f2(α). He
proposed a “nonnegative quantity not exceeding 1
which measures the additional effectiveness of the
past values of f2(α) in helping to determine the
present value of f1(α), and may be considered as a
measure C of the causality effect of f2(α) on
f1(α).”
Norbert Wiener
The Theory of Prediction, 1956
GRANGER CAUSALITY
“In the early 1960's I was considering a pair of
related stochastic processes which were clearly
inter-related and I wanted to know if this
relationship could be broken down into a pair of
one way relationships. It was suggested to me to
look at a definition of causality proposed by a very
famous mathematician, Norbert Weiner, so I
adapted this definition (Wiener 1956) into a
practical form and discussed it.”
Clive Granger’s
personal account in:
Seth A. (2007) Granger causality,
Scholarpedia 2(7):1667.
GRANGER CAUSALITY
 “Applied
economists
found the definition
understandable and
useable and
applications of it
started to appear.”
Clive Granger’s
personal account in:
Seth A. (2007) Granger causality,
Scholarpedia 2(7):1667.

“However, several
writers stated that ‘of
course, this is not real
causality, it is only
Granger causality.’
Thus, from the
beginning,
applications used this
term to distinguish it
from other possible
definitions.”
UNIVARIATE
AUTOREGRESSIVE MODEL
xt= [a1xt-1 + a2xt-2 + a3xt-3 + … + amxt-m] + εt
where xt is a zero-mean stationary stochastic process, ai are model
coefficients, m is the model order, and εt is the residual error.
DEFINITION OF GRANGER
CAUSALITY [BRESSLER & SETH, NEUROIMAGE,
2011]
 Granger
(1969): Let
x1, x2, …, xt
and
y1, y2, …, yt
represent two time
series.

Compare two autoregressive
models:
xt = a1xt-1 + … + amxt-m + 1t
and
restricted
xt = b1xt-1 + … + bmxt-m
+ c1yt-1 + … + cmyt-m + 2t unrestricted
Let 1= var(1t)
and
2= var(2t).
 If 2 < 1, then the y time
series has a Granger casual
influence on the x time
series.

TIME DOMAIN GRANGER
CAUSALITY MEASURE
The time domain Granger
causality from y to x is:
Fy  x
1
 ln
2
TOTAL INTERDEPENDENCE
[DING ET AL, HANDBOOK OF TIME SERIES ANALYSIS, 2006]
 The
total interdependence between time
series xt and yt can be decomposed into
three components:
Fx,y = Fx®y + Fy®x + Fx·y
where the 3rd component, called the instantaneous causality, may be
due to factors exogenous to the (x, y) system, such as a common
driving input.
4 POSSIBLE RELATIONS
BETWEEN 2 TIME SERIES

Unidirectional GC only from x to y

Unidirectional GC only from y to x

Bidirectional GC between x and y

Independence of x and y
BIVARIATE VOXELWISE GC
FROM FMRI [BRESSLER ET AL, J NEUROSCI, 2008]

Granger Causality
was computed as an
F-statistic for every
voxel pair of a pair of
Regions of Interest
(ROIs).
F _value =
RSSr - RSSur / m
(
RSS
T = # observations = ~700
m = model order = 1
Distributions of F-statistics in top-down and
bottom-up directions for ROIs RaIPS & RV3A.
)
RSSur / T - 2m -1
The fraction of voxel pairs having significant GC
from one ROI to another was greater in topdown than bottom-up direction.
60 ROI pairs
from 8 ROIs in
each cerebral
hemisphere of 6
subjects were
analyzed.
VoxelRandomized
Control
By subject
By ROI pair
TrialRandomized
Control
MULTIVARIATE VOXELWISE GC
WITH LASSO[TANG ET AL, PLOS COMP BIOL, 2012]

The MVAR Model:
m
Zt = å Bk Zt-k + Et
k=1
The b-coefficients represent GC:
The Least
Absolute
Shrinkage and
Selection Operator
(LASSO) makes
model estimation
feasible when too
few observations
are available to
otherwise estimate
the model
coefficients.
LASSO GC VS.
CORRELATION

LASSO GC shows directional
asymmetry between ROIs by
use of summary statistic W.

Connectivity between FEF
and VP is more sparse for
LASSO GC (A) than for
Correlation (B), indicating
greater spatial specificity.
CONDITIONAL GC OF DAN & VOC
IN VISUAL SPATIAL ATTENTION
Positive W
Negative W
Modified method after Garg et al 2011, FARM using fMRI
SPECTRAL GRANGER
CAUSALITY[DING ET AL, HANDBOOK OF
TIME SERIES ANALYSIS, 2006]

MVAR Model: Xt = A1Xt-1 +
The
 + AmXt-m + Εt
Spectral Matrix is defined as:
S( f ) = <X (f ) X (f )*> = H(f )  H*(f )
where * denotes matrix transposition & complex
conjugation;  is the covariance matrix of Et ; and
1
m

is the transfer function of the
2 i k f 
H(f )    Ak e

 k 0

system.
SPECTRAL GRANGER
CAUSALITY[DING ET AL, HANDBOOK OF
TIME SERIES ANALYSIS, 2006]

Geweke (1982) found a
spectral representation
of the time domain
Granger causality:
()
Iy®x f = ln
()
Sxx f
()
()
*
H xx f S2H xx
f
where Sxx(f) is the spectral power of x,
Hxx(f) is an element of the transfer
matrix H(f)=A-1(f), and * denotes matrix
transposition & complex conjugation.
X(f)=H(f)E(f); X(f)=A-1(f), where
the first
p
component of A is: a jk (f ) = 1- åa jk e-2p ikf
k=1

Geweke showed that:
Fy  x
 1  1
 ln   
  2  2

 I  f df
yx

SPECTRAL GRANGER
CAUSALITY OF MACAQUE
LFPS

Coefficient matrices
are obtained by
solving the
multivariate YuleWalker equations (of
size mp2), using the
Levinson, Wiggens,
Robinson algorithm,
as implemented by
Morf et al. (1978).
Repeated trials are
treated as realizations
of a stationary
stochastic process.
 The model order is
determined by
parametric testing.

SPECTRAL GRANGER
CAUSALITY OF MACAQUE
LFPS

Spectral GC is bidirectional in the
frequency domain.

It shows predictability
in 2 directions as a
function of frequency.
Coherence between PAR1 and PAR2
GC from PAR1 to PAR2 (black)
GC from PAR2 to PAR1 (orange)
Model order = 15
SENSORIMOTOR GC IN
MOTOR CONTROL [BROVELLI ET AL,
PNAS, 2004]
GC pattern consistent with:
• Anatomical connectivity
• MacKay functional loop
• Clinical deafferentation
evidence
EXTRASTRIATE-V1 GC IN
VISUAL EXPECTATION [BRESSLER
ET AL, STAT MED, 2007]
GC pattern consistent with:
• Descending anatomical connectivity
• Visual anticipation
BIVARIATE VERTEXWISE
SPECTRAL GC FROM
MEG[MAHALINGAM ET AL, CNS, 2008]
WORKING MEMORY GC IN
FRONTOPARIETAL
SYSTEM[SALAZAR ET AL, SCIENCE, 2012]
GC spectra
from PPC to
PFC (red) and
from PFC to
PPC (blue)
during late
delay period
Directional GC
difference
(solid).
Significance
level from
surrogate
difference
distribution
(dashed)
PPCPFC GC > PFCPPC GC
CONCLUSIONS
Granger Causality is a useful tool to investigate
influences between neural groups.
 It can be applied to multivariate time series of
different neural signal types.
 When applied to fMRI BOLD data, it can be
useful to non-invasively study inter-regional
interactions in humans under normal
physiological conditions.
 When applied to LFP data, it can be useful to
study inter-regional interactions with high
spatial, temporal, and frequency precision.
