Table Of ContentConsolidating a Link Centered Neural
Connectivity Framework with Directed Transfer
5 Function Asymptotics
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0
2
Luiz A. Baccala´ Daniel Y. Takahashi Koichi Sameshima
n
a
January 26, 2015
J
3
2
Abstract
]
C We present a unified mathematical derivation of the asymptotic be-
N haviour of three of the main forms of directed transfer function (DTF)
complementingrecentpartialdirectedcoherence(PDC)results[3]. Based
.
o ontheseresultsandnumericalexamplesweargueforanewdirected‘link’
i centeredneuralconnectivityframeworktoreplacethewidespreadcorrela-
b
tionbasedeffective/functionalnetworkconceptssothatdirectednetwork
-
q influences between structures become classified as to whether links are
[ active inadirect orinanindirect waytherebyleadingtothenewnotions
of Granger connectivity and Granger influenciability which are more de-
1
scriptive than speaking of Granger causality alone.
v
6
3
1 Introduction
8
5
0 Introduced as a frequency domain characterization of the interaction between
1. multiple neural structures directed transfer function (DTF) [16] can be thought
0 asafactorinthecoherencebetweenpairsofobservedtimeseries[5]. Ahistorical
5 perspectiveonDTFbytheirauthorscanbefoundin[17]togetherwithitsmany
1
variants.
:
v Onaparwithit,standspartialdirectedcoherence (PDC)[5]asitsdualmea-
i sure. ThechiefdistinctionbetweenthemisthatPDCcapturesactive immediate
X
directional coupling between structures whereas DTF, in general, portrays the
r
a existence of directional signal propagation even if it is only indirect, by going
through intermediate structures rather through immediate direct causal influ-
ence [6]. DTF, therefore, represents signal ‘reachability’ in a graph theoretical
sense whereas PDC is akin to an adjacency matrix description [13].
Since DTF’s introduction, we examined two of its closely related variants
(a) directed coherence (DC) [2] which is DTF’s scale invariant form (and dual
to generalized PDC (gPDC)[8]) and (b) information DTF (ιDTF) which is an
informationtheoreticgeneralizationofDTF,dualtoinformationPDC(ιPDC),
both of which provide accurate size effect information [26, 27, 28].
1
In this paper, we derive and illustrate inference results for the above DTF
variants from a unified perspective closely paralleling the inference results in
[3] and further illustrated in [22] for PDC and its variants. The importance of
accurate asymptotics for DTF is that jointly DTF and PDC allow extending
the current paradigm of effective/function connectivity to a more general and
informative context [7].
AfterbrieflyreviewingDTF’sformulations(Sec. 2)togetherwithasummary
oftheunifiedasymptoticresults(Sec. 3),numericalillustrations(Sec. 4)discuss
some implications of the results as further elaborated in Sec. 5 with their
implications for the new connectivity analysis paradigm we proposed in [7].
For reader convenience, mathematical details are left to the Appendix whose
implementation is to appear in the next release of the AsympPDC package [3].
2 Background
ThedeparturepointfordefiningallDTFrelatedvariantsisanadequatelyfitted
multivariate autoregressive time series (i.e. vector time series) model to which
a multivariate signal x(n) made up by x (n), k = 1,...,K simultaneously
k
acquired time series conforms to
p
(cid:88)
x(n)= A(l)x(n−l)+w(n), (1)
l=1
where w(n) stands for a zero mean white innovations process of with Σ =
w
[σ ] as its covariance matrix and p is the model order. The a (l) coefficients
ij ij
composing each A(l) matrix describe the lagged effect of the j-th on the i-th
series, wherefrom one can also define a frequency domain representation of (1)
via the A¯(f) matrix whose entries are given by
p
1− (cid:80)aij(l)e−j2πλl, if i=j
A¯ (λ)= l=1 (2)
ij p
−(cid:80)aij(l)e−j2πλl, otherwise
l=1
√
where j= −1, so that one may define
H(λ)=A¯−1(λ) (3)
with H (λ) entries and rows denoted h . This leads to a general expression
ij i
sH¯ (λ)
γ (λ)= ij (4)
ij (cid:112)
hH(λ)Sh (λ)
i i
summarizingalltheformsofDTFformjtoiconsideredherein. Thesuperscript
H denotes the usual Hermitian transpose. The reader should be forewarned to
usetheadequateexpressionforsandStoobtaineachDTFvariantin(4)using
Table 1.
2
Table 1: Defining variables according to DTF type in (4)
variable DTF DC ιDTF
s 1 σ1/2 σ1/2
ii ii
S I (I (cid:12)Σ ) Σ
K K w w
3 Result Overview
The statistical behaviour of (4) in terms of the number of times series data
points(n )canbeapproximatedinvokingthedelta method[30]consistingofan
s
appropriateTaylorexpansionofthestatistics,leading,undermildassumptions,
to the following results:
3.1 Confidence Intervals
Inmostapplications,becausen islarge,usuallyonlythefirstTaylorderivative
s
suffices. In the present context, parameter asymptotic normality implies that
DTF’s point estimate will also be asymptotically normal, i.e.
√
n (|γ (λ)|2−|γ (λ)|2)→d N(0,γ2(λ)), (5)
s (cid:98)ij ij
whereγ2(λ)isafrequencydependentvariancewhosefulldisclosurerequiresthe
introduction of further notation and is postponed to the Appendix.
3.2 Null Hypothesis Test
Under the null hypothesis,
H : |γ (λ)|2 =0 (6)
0 ij
γ2(λ)vanishesidenticallysothat(5)nolongerappliesandthenextTaylorterm
becomesnecessary[24]. Thenextexpansiontermisquadraticintheparameter
vector and corresponds to one half of DTF’s Hessian at the point of interest
with an O(n−1) dependence.
s
Theresultingdistributionisthatofasumofatmosttwoproperlyweighted
independentχ2 variableswheretheweightsdependonfrequency. Explicitcom-
1
putation is left to the Appendix given the need of specialized notation, but can
be summarized as
q
n (hH(λ)Sh (λ))(|γ (λ)|2−|γ (λ)|2)→d (cid:88)l (λ)χ2 (7)
s i i (cid:98)ij ij k 1
k=1
where l (λ) are at most two frequency dependent eigenvalues (q ≤ 2) coming
k
from a matrix that depends on DTF’s values. Note that implicit dependence
3
also comes from the left side of (7) on DTF’s denominator. See further details
in Proposition 2. Brief comments and explicit computational methods relating
sums of χ2 variables may be found in [20], [14] and [25].
1
These results closely parallel PDC ones in [3], the main difference lying in
howthefrequencydependentcovarianceoftheparametervectorsarecomputed.
4 Numerical Illustrations
In the examples that follow dashed lines indicate threshold values and gray
shades stand for point confidence intervals around significant points. Unless
stated otherwise, innovations noise is zero mean, unit variance and mutually
uncorrelated. The frequency domain graphs are displayed in the standard form
ofanarraywheregrayeddiagonalpanelscontaintheestimatedtimeseriespower
spectra.
Example 1 Consider the connectivity from x →x whose dynamics is repre-
1 2
sented by an oscillator which influences another structure without feedback:
√
x (n) = 0.95 2x (n−1)−0.9025x (n−2)+w (n)
1 1 1 1
x (n) = −0.5x (n−1)+0.5x (n−1)+w (n) (8)
2 1 2 2
As in all bivariate cases, DTF and PDC coincide numerically, yet because
DTF computation requires actual matrix inversion in the general case, its null
hypothesis threshold limits are affected by the spectra (top panel in Fig. 1) (the
x (n) in this case) casting decision doubts at n = 50 points (mid panel) as
1 s
opposed to the PDC case (bottom panel).
At n = 500, DTF is above threshold for x → x throughout the frequency
s 1 2
interval (Fig. 2). Further comparison is provided in Fig. 3a where the actually
observed DTF values for n = 50 are more spread than those in Fig. 3b for
s
n =500. In the x →x direction, (7) behaviour is readily confirmed.
s 2 1
Even though bivariate DTF and PDC numerically coincide, the need to take
into account A¯(λ) inversion under the DTF’s null hypothesis can lead to overly
conservativethresholdsandconsequentfailuretoproperlyrejectH ifn issmall
0 s
as in Fig. 1.
Example 2 This example shows an oscillator x (n) whose influence travels
1
back to itself through a loop containing the x (n) → x (n) link in the feedback
2 3
loop pathway (Fig. 4) and whose dynamics follows:
√
x (n) = 0.95 2x (n−1)−0.9025x (n−2)
1 1 1
+0.35x (n−1)+w (n)
3 1
x (n) = 0.5x (n−1)+0.5x (n−1)+w (n)
2 1 2 2
x (n) = x (n−1)−0.5x (n−1)+w (n) (9)
3 2 3 3
4
x x
1 2
1 1
]
.
u
.
a
[
a
r
t
c
e
p
S
0 0
0 .5 0 .5
l
Frequency( )
1 2 2 1
0.5
2
2
|
F
T
D 1
|
0 0
0 .5 0 .5
1 2 2 1
1 0.4
2
|
C
D
P
|
0 0
0 .5 0 . 5
l
Frequency( )
Figure 1: Comparison between DTF (middle row) and PDC (bottom row) for
Ex. 1 showing the effect of the existing resonance (time series spectra top row)
onthresholddecisionlevels(dashedcurves)usingn =50simulateddatapoints.
s
The effect of increasing n can be appreciated in Fig. 2. Gray shades describe
s
95% confidence levels when above threshold.
5
1 2 2 1
1 0.1
2
|
F
T
D
|
0 0
0 .5 0 .5
1 0.1
2
|
C
D
P
|
0 0
0 .5 0 .5
l
Frequency( )
Figure 2: For n =500 a DTF single trial realization is safely above threshold -
s
compareittousingn =50inFig. 1(middlepanelrow). Grayshadesdescribe
s
95% confidence levels when above threshold.
6
a
ns = 50
1 2 2 1
e
.999 til
n
e .990 ua.999
ntil .950 e q
qua .750 quar.990
al -s
m .250 hi
r c.950
No .050 d
e
t
.010 h.750
g
.001 ei.500
W.001
0.4 0.6 0.8 0 0.1 0.2 0.3 0.4
|DTF(l=.15)|2 |DTF(l=.15)|2
b
ns = 500
1 2 2 1
e
.999 ntil
.990 ua.999
e q
til .950 e
n r
qua .750 qua.990
mal .250 hi-s
or d c.950
N .050 e
t
.010 gh.750
.001 ei.500
W.001
0.75 0.8 0.85 0.9 0 0.005 0.01 0.015 0.02
|DTF(l=.15)|2 |DTF(l=.15)|2
Figure 3: Quantile distribution behaviour showing the distribution goodness of
fit improvement as with sample size for Ex. 1 (n = 50 (a) versus n = 500
s s
(b)). Statistical spread decrease in the non-existing link x → x is evident as
2 1
the improved normal fit of the x → x existing connection. For each value of
1 2
n , m=2000 simulations were performed.
s
7
with the covariance matrix of w given by
1 5 0.3
Σw = 5 100 2 , (10)
0.3 2 1
ensuring that x (n) contributes a large amount of innovation power to the loop.
2
Because ιDTF deals well with unbalanced innovations, it was used with n =
s
500 (Fig. 5) and n =2000 (Fig. 6) points leading to the following features: (a)
s
thelarge|ιDTF |2 above1forsomefrequenciesareduetothelargeinnovations
i2
associated with x (n) in (10); (b) except for low ιDTF values that require more
2
points for reliable estimation, calculations confirm that signals originating at
any structure reach all other structures; and (c) because of the much smaller
relative power originating from x (n), its influence is much harder to detect.
3
The allied ιPDC, also shown, confirms which immediate links are directionally
active even for n =500.
s
It is interesting to observe that |ιDTF |2 has a peak around the x reso-
23 1
nance frequency which manifests itself because the innovations originating in x
3
(w (n)) are filtered by passing through the resonant filter represented by struc-
3
turex beforereachingx . Thissametypeofinfluenceisnotsoreadilyapparent
1 2
(clear only at n = 2000) in |ιDTF |2 because the power contributed by x is
s 13 3
smallwithrespecttothatofothersourcesreachingx aroundthatsameresonant
1
frequency.
The sharp jump in |ιDTF |2 is a byproduct of the fast phase shift that takes
32
place around x ’s resonance as x ’s signal travels through it to reach x .
1 2 3
A glimpse of the ensemble ιDTF’s behaviour can be appreciated in Fig. 7
showing how difficult it is to detect it if its values are low.
2
1
3
Figure4: Ex2loopconnectivitystructure. Signalsfromanystructurereachall
other structures.
Example 3 The next example comes from [5], whose direct connections are
8
a
1 0.1
1
1
=
i
0 0 0
1 1 0.1
2|F a.u.]
DT i = 2 ectra [
|i Sp
0 0 0
1 1
1
3
=
i
0 0 0
0 .5 0 .5 0 .5
j = 1 j = 2 j = 3
l
Frequency( )
b
1 0.1 1
1
=
i
0 0 0
1 1 .01
2|C a.u.]
PD i = 2 ectra [
|i Sp
0 0 0
0.1 1 1
3
=
i
0 0 0
0 .5 0 .5 0 .5
j = 1 j = 2 j = 3
l
Frequency( )
Figure 5: Ex. 2 information ιDTF more widely spread results (a) contrasted
to ιPDC (b) results for n = 500 and α = 0.05. Time series spectra are dis-
s
played along the main panel diagonal (gray backgrounds). Sources are marked
j (columns) and targets i (rows).
9
a
0.1
1
1
=
i
0 0
1 0.1
2
|
F 2
T =
D i
i
|
0 0
0 .5
1
1
3
=
i
0 0
0 .5 0 .5
j = 1 j = 2 j = 3
l
Frequency( )
b
0.01 1
1
=
i
0 0
1 0.01
2
|
C
D = 2
P i
i
|
0 0
0 .5
0.01 1
3
=
i
0 0
0 .5 0 .5
j = 1 j = 2 j = 3
l
Frequency( )
Figure 6: Improvement of connectivity estimates under n = 2000 over Fig. 5
s
single trial behaviour using ιDTF (a) and ιPDC (b). Time series spectra are
omitted but can be appreciated from Fig. 5. Sources are marked j and targets
i.
10
