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Model-based and model-free connectivity methods for electrical neuroimaging
1. 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
2. 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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
3. 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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
4. Two classes of methods
Assume independent measurements at each node
Inference of networks from temporally correlated data (dynam-
ical networks)
Daniele Marinazzo Directed connectivity in electrical neuroimaging
5. 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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
6. 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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
7. 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
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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
8. 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
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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
9. Basic idea of Dynamic Causal Models
We have several neural populations ..
Daniele Marinazzo Directed connectivity in electrical neuroimaging
10. Basic idea of Dynamic Causal Models
.. with interactions among and within them
Daniele Marinazzo Directed connectivity in electrical neuroimaging
11. Basic idea of Dynamic Causal Models
What we see and what we don’t
Daniele Marinazzo Directed connectivity in electrical neuroimaging
12. Basic idea of Dynamic Causal Models
Forward model
Daniele Marinazzo Directed connectivity in electrical neuroimaging
13. Basic idea of Dynamic Causal Models
Bayesian framework
Daniele Marinazzo Directed connectivity in electrical neuroimaging
14. Basic idea of Dynamic Causal Models
Bayesian framework
Daniele Marinazzo Directed connectivity in electrical neuroimaging
15. Basic idea of Dynamic Causal Models
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.
Daniele Marinazzo Directed connectivity in electrical neuroimaging
16. Basic idea of Dynamic Causal Models
Inference on model structure
Which model (or family of models) has highest evidence?
Daniele Marinazzo Directed connectivity in electrical neuroimaging
17. Basic idea of Dynamic Causal Models
Inference on model structure
Which model (or family of models) has highest evidence?
Inference on model parameters
Which parameters are statistically significant, and what is their
size/sign?
Daniele Marinazzo Directed connectivity in electrical neuroimaging
18. Inference on model structure
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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
19. Inference on model structure
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
The Kullback - Leibler divergence between the real and approx-
imate conditional density minus the log-evidence
Daniele Marinazzo Directed connectivity in electrical neuroimaging
20. Inference on model structure
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
The Kullback - Leibler divergence between the real and approx-
imate conditional density minus the log-evidence
A Bayesian Expectation Maximization
Daniele Marinazzo Directed connectivity in electrical neuroimaging
21. Inference on model structure
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
The Kullback - Leibler divergence between the real and approx-
imate conditional density minus the log-evidence
A Bayesian Expectation Maximization
ok, a model fit
Daniele Marinazzo Directed connectivity in electrical neuroimaging
22. 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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
23. 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)
Daniele Marinazzo Directed connectivity in electrical neuroimaging
24. 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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
27. Influences in multivariate datasets
We must condition the measure to the effect of other variables
Daniele Marinazzo Directed connectivity in electrical neuroimaging
28. 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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
29. Beyond conditioning: joint information
Daniele Marinazzo Directed connectivity in electrical neuroimaging
30. 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.
Daniele Marinazzo Directed connectivity in electrical neuroimaging
31. 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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
32. Transfer entropy and regression
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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
33. Transfer entropy and regression
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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
34. 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.
Daniele Marinazzo Directed connectivity in electrical neuroimaging
35. 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)
Daniele Marinazzo Directed connectivity in electrical neuroimaging
37. Joint information
Let’s go for an operative and practical definition
Daniele Marinazzo Directed connectivity in electrical neuroimaging
38. 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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
39. Generalization of GC for sets of driving variables
Conditioned Granger Causality in a multivariate system
δX(B → α) = log
(xα|X B)
(xα|X)
Daniele Marinazzo Directed connectivity in electrical neuroimaging
40. Generalization of GC for sets of driving variables
Conditioned Granger Causality in a multivariate system
δX(B → α) = log
(xα|X B)
(xα|X)
Unnormalized version
δu
X(B → α) = (xα|X B) − (xα|X)
Daniele Marinazzo Directed connectivity in electrical neuroimaging
41. Generalization of GC for sets of driving variables
Conditioned Granger Causality in a multivariate system
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.
Daniele Marinazzo Directed connectivity in electrical neuroimaging
42. 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?
Daniele Marinazzo Directed connectivity in electrical neuroimaging
43. 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?
Daniele Marinazzo Directed connectivity in electrical neuroimaging
44. Define synergy and redundancy in this framework
Synergy
δu
X(B → α) >
β∈B δu
XB,β(β → α)
Daniele Marinazzo Directed connectivity in electrical neuroimaging
45. Define synergy and redundancy in this framework
Synergy
δu
X(B → α) >
β∈B δu
XB,β(β → α)
Redundancy
δu
X(B → α) <
β∈B δu
XB,β(β → α)
Daniele Marinazzo Directed connectivity in electrical neuroimaging
47. 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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
48. 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
Daniele Marinazzo Directed connectivity in electrical neuroimaging
49. 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)
Daniele Marinazzo Directed connectivity in electrical neuroimaging