Model-based and model-free connectivity methods for electrical neuroimaging

Directed dynamical connectivity in electrical
neuroimaging: which tools should I use?
A very partial and personal overview, in good faith but still
Daniele Marinazzo
Department of Data Analysis, Faculty of Psychology and Educational Sciences,
Ghent University, Belgium
@dan marinazzo
http://users.ugent.be/~dmarinaz/
Daniele Marinazzo Directed connectivity in electrical neuroimaging

At least two distinct ways one can think of causality
Temporal precedence, i.e. causes precede their consequences
Physical inﬂuence (control), i.e. changing causes changes their
consequences

Two classes of methods
Assume independent measurements at each node
Inference of networks from temporally correlated data (dynam-
ical networks)

Using temporal dynamics
We model a dynamical system at each node
Two main approaches:
Dynamic Bayesian networks (Hidden Markov Models)
Model-free and model-based investigation of temporal correla-
tion

What to expect from ”causality” measures in neuroscience
Causal measures in neuroscience should reflect effective con-
nectivity, i.e. the underlying physiological influences exerted
among neuronal populations in different brain areas. → Dy-
namic Causal Models

namic Causal Models
Diﬀerent but complementary goal: to reﬂect directed dynam-
ical connectivity without requiring that the resulting networks
recapitulate the underlying physiological processes. → Granger
Causality, Transfer Entropy

namic Causal Models
Diﬀerent but complementary goal: to reﬂect directed dynam-
ical connectivity without requiring that the resulting networks
recapitulate the underlying physiological processes. → Granger
Causality, Transfer Entropy
The same underlying (physical) network structure can give rise
to multiple distinct dynamical connectivity patterns
In practice it is always unfeasible to measure all relevant vari-
ables
Bressler and Seth 2010

Basic idea of Dynamic Causal Models
We have several neural populations ..

.. with interactions among and within them

What we see and what we don’t

Forward model

Bayesian framework

Model inference
Prior: what connections are included in the model
Likelihood: Incorporates the generative model and prediction
errors
Model evidence: Quantiﬁes the goodness of a model (i.e.,
accuracy minus complexity). Used to draw inference on model
structure.
Posterior: Probability density function of the parameters given
the data and model. Used to draw inference on model param-
eters.

Inference on model structure
Which model (or family of models) has highest evidence?

Which model (or family of models) has highest evidence?
Inference on model parameters
Which parameters are statistically signiﬁcant, and what is their
size/sign?

A necessary step, unless strong prior knowledge about structure
Bayesian model comparison (BMS) compares the (log) model
evidence of diﬀerent models (i.e., probability of the data given
model)
log model evidence is approximated by free energy

model)
The Kullback - Leibler divergence between the real and approx-
imate conditional density minus the log-evidence

model)
A Bayesian Expectation Maximization

model)
A Bayesian Expectation Maximization
ok, a model ﬁt

Inference on model parameters
Often a second step in DCM studies
Inference on the parameters of the clear winning model (if there
is one)
If no clear winning model (or if optimal model structure diﬀers
between groups) then Bayesian model averaging (BMA) is
an option
Final parameters are weighted average of individual model pa-
rameters and posterior probabilities

Group level inference
Diﬀerent DCMs are ﬁtted to the data for every subject.
Group inference on the models (or groups of models: in DCM
terminology families of models e.g. all models with input to
region A vs. input to region B, or vs. both, three families):
Bayesian model selection
Winning model/family is the one with highest exceedance prob-
ability
Group inference on model parameter: Either on the winning
model or Bayesian model averaging (BMA) across models (within
a winning family or all models when BMS reveal no clear win-
ner)
(BMA) Parameter(s) of interest are harvested for every subject
and subjected to frequentist inference (e.g. t-test)

DCM for ERPs/ERFs
Bottom-up: connection from low to high hierarchical areas
top-down: connection from high to low hierarchical areas (Felle-
man 1991)
Lateral: same level in hierarchical organization (e.g. interhemi-
spheric connection)
Prior on connection: forward → backward → lateral
Layers within regions interact via intrinsic connections

DCM inference: summary

Inﬂuences in multivariate datasets
We must condition the measure to the eﬀect of other variables

Inﬂuences in multivariate datasets
We must condition the measure to the eﬀect of other variables
The most straightforward solution is the conditioned approach,
starting from Geweke et al 1984

Beyond conditioning: joint information

Transfer entropy and Markov property
Absence of causality: generalized Markov property
p(x|X, Y ) = p(x|X)
Transfer Entropy
Transfer entropy (Schreiber 2000) quantiﬁes the violation of the
generalized Markov property
T(Y → X) = p(x|X, Y ) log
p(x|X, Y )
p(x|X)
dx dX dY
T measures the information ﬂowing from one series to the other.

Transfer entropy and regression
Risk functional
The minimizer of the risk functional
R [f ] = dX dx (x − f (X))2
p(X, x)
represents the best estimate of x given X, and corresponds to the
regression function
f ∗
(X) = dxp(x|X) x

Markov property for uncorrelated variables
The best estimate of x, given X and Y is now:
g∗
(X, Y ) = dxp(x|X, Y ) x
p(x|X, Y ) = p(x|X) ⇒ f ∗
(X) = g∗
(X, Y )
and the knowledge of Y does not improve the prediction of x

Transfer entropy (entropy rate)
SX = − dx dX p(x, X) log[p(x|X)]
SXY = − dx dX dY p(x, X, Y ) log[p(x|X, Y )]
Regression
EX = dx dX p(x, X) (x − dx p(x |X) x )2
EX,Y = dx dX dY p(x, X, Y ) (x − dx p(x |X, Y ) x )2

Granger causality and Transfer entropy
GC and TE are equivalent for Gaussian variables and other
quasi-Gaussian distributions
(Barnett et al 2009, Hlavackova-Schindler 2011, Barnett and
Bossomaier 2012)
In this case they both measure information transfer.

Granger causality and Transfer entropy
GC and TE are equivalent for Gaussian variables and other
quasi-Gaussian distributions
(Barnett et al 2009, Hlavackova-Schindler 2011, Barnett and
Bossomaier 2012)
In this case they both measure information transfer.
Uniﬁed approach (model based and model free)
Mathematically more treatable
Allows grouping variables according to their predictive content
(Faes et al. 2014)

Joint information

Joint information
Let’s go for an operative and practical deﬁnition

Joint information
Let’s go for an operative and practical deﬁnition
Relation (B and C) → A
synergy: (B and C) contributes to A with more information
than the sum of its variables
redundancy: (B and C) contributes to A with less information
than the sum of its variables
Stramaglia et al. 2012, 2014, 2016

Generalization of GC for sets of driving variables
Conditioned Granger Causality in a multivariate system
δX(B → α) = log
(xα|X B)
(xα|X)

δX(B → α) = log
(xα|X B)
(xα|X)
Unnormalized version
δu
X(B → α) = (xα|X B) − (xα|X)

δX(B → α) = log
(xα|X B)
(xα|X)
Unnormalized version
δu
X(B → α) = (xα|X B) − (xα|X)
An interesting property
If {Xβ}β∈B are statistically independent and their contributions in
the model for xα are additive, then δu
X(B → α) =
β∈B
δu
X(β → α).
This property does not hold for the standard deﬁnition of GC, neither
for entropy-rooted quantities, because logarithm.

Question from the audience:
What does it ever mean to have an unnormalized measure of
Granger causality?
Don’t you lose any link with information?

Deﬁne synergy and redundancy in this framework
Synergy
δu
X(B → α) >
β∈B δu
XB,β(β → α)

Synergy
δu
X(B → α) >
β∈B δu
XB,β(β → α)
Redundancy
δu
X(B → α) <
β∈B δu
XB,β(β → α)

Synergy
δu
X(B → α) >
β∈B δu
XB,β(β → α)
Redundancy
δu
X(B → α) <
β∈B δu
XB,β(β → α)
Stramaglia et al. IEEE Trans
Biomed. Eng. 2016
Pairwise syn/red index
ψα(i, j) = δu
Xj(i → α) − δu
X(i → α)
= δu
X({i, j} → α) − δu
X(i → α) − δu
X(j → α)

Do it yourself!
Statistical Parametric Mapping - DCM http://www.fil.ion.
ucl.ac.uk/spm/
MVGC (State-Space robust implementation) http://users.
sussex.ac.uk/~lionelb/MVGC/
BSmart (Time-varying, Brain-oriented) http://www.brain-smart.
org/
MuTE (Multivariate Transfer Entropy, GC in the covariance
case) http://mutetoolbox.guru/
emVAR (Frequency Domain) http://www.lucafaes.net/emvar.
html
ITS (Information Dynamics) http://www.lucafaes.net/its.
html

Thanks
Hannes Almgren, Ale Montalto and Frederik van de Steen (UGent)
Sebastiano Stramaglia (Bari)
Pedro Valdes Sosa (CNeuro and UESTC)
Laura Astolﬁ and Thomas Koenig

References
David et al., 2006: Dynamical causal modelling of evoked reponses in EEG and MEG (NI)
Stephan et al., 2010: Ten simple rules for dynamic causal modeling (NI)
Penny et al., 2004: Comparing Dynamic causal models (NI)
Litvak et al., 2008: EEG and MEG Data Analysis in SPM8 (CIN)
Bressler and Seth, 2010: Wiener-Granger causality, a well-established methodology (NI)
Montalto et al., 2014: MuTE: A MATLAB Toolbox to Compare Established and Novel Estimators of the
Multivariate Transfer Entropy (PLOS One)
Bastos and Schoﬀelen, 2016: A Tutorial Review of Functional Connectivity Analysis Methods and Their
Interpretational Pitfalls (Front N Sys)
Stramaglia et el. 2106: Synergetic and Redundant Information Flow Detected by Unnormalized Granger
Causality (IEEE TBME)

Model-based and model-free connectivity methods for electrical neuroimaging

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