footprints of causality - training course in fmri · on causality i: sampling and noise. in 2007...

Effective Connectivity

Luis Hernandez-Garcia, Ph.D.UM FMRI course

Hernandez-Garcia , FMRI Course 2018

Motivation Brain as a network

Inputs, outputs, causal relationships

FMRI can localize brain activity FMRI can identify connectivity (networks) Can FMRI be used to characterize the

hierarchy of the network? Top Down or Bottom-Up ?

Does activity in A cause activity in B?


Today’s outline Networks and FMRI Granger Causality Dynamic Causal Modeling Ascendency Evaluation of methods. … but what is Causality?


Elements of Brain Networks

• Nodes• Edges: driving

inputs• Endogenous

connections• Exogenous

Inputs• Modulation• Feedback

z1

z2

z3

u1

u2

a21

b21

c11

a32

a23

c22

a33a22

a11


Temporal Resolution in FMRI We can image quite fast (<50ms./slice)

with new multi-band techniques you can do a whole brain in less than 500 ms.

just barely enough to catch neuronal shifts in activation

HOWEVER - we observe much slower vascular effects, not neuronal ones.

In a linear system, a shift in neuronal events corresponds to the same shift in vascular events… Is the BOLD effect linear enough? Vascular responses are variable throughout the brain


Temporal Resolution in FMRI

High temporal resolution: mental chronometry

Menon, R. S., Luknowsky, D. C., & Gati, J. S. (1998). Mental chronometry using latency-resolved functional MRI. Proc NatlAcad Sci U S A, 95(18), 10902-07.


Temporal Resolution in FMRI

Sensitivity of statistical tests to temporal shifts

Hernandez, L., Badre, D., Noll, D., & Jonides, J. (2002). Temporal sensitivity of event-related fMRI. Neuroimage, 17(2), 1018-26.


Granger Causality

A test to determine the directionality of a connection between a pair of nodes.

Disclaimer: “Granger-causality” is not “Causality”


Autoregressive systems(background)

Part of the “current” sample can be predicted from previous samples (history)

Examples: drifts in FMRI scans, price of oil …


Granger Causality What if the current sample of a given

process (A) can also be predicted in part by the history of another process (B)?

Roebroeck, A., Formisano, E., & Goebel, R. (2005). Mapping directed influence over the brain using Granger causality and fMRI. Neuroimage, 25(1), 230-242.


Granger Causality Are the cross-terms significant?

1. Fit the models with and without the cross terms

2. Use the residuals to compute F statistics

Inference through “boot-strapping”


Multivariate Granger Causality

Multiple time courses Multiple influences of previous samples

Multiple ROI’s Time-shifted ROI’s

Influence coefficientsDeshpande, G., et al., Multivariate Granger causality analysis of fMRI data. Human Brain Mapping, 2009. 30(4): p. 1361-1373.


Variations on the theme

Add instantaneous terms into the formulation. “SEM - Granger hybrid”

Kim, J., Zhu, W., Chang, L., Bentler, P. M. & Ernst, T. Unified structural equation modeling approach for the analysis of multisubject, multivariate functional MRI data. Hum Brain Mapp 28, 85-93 (2007).

Partial Directed Coherence. Similar analysis - frequency domain.

Baccala, L. A. & Sameshima, K. Partial directed coherence: a new concept in neural structure determination. Biol Cybern 84, 463-474 (2001).


Limitations of Granger Causality

Different HRF’s in different parts of the brain could appear as false Granger Causal relationships.

David, O., Guillemain, I., Saillet, S., Reyt, S., Deransart, C., Segebarth, C., Depaulis, A., 2008. Identifying Neural Drivers with Functional MRI: An Electrophysiological Validation. PLoS Biol 6, e315.

Handwerker, D.A., Ollinger, J.M., D'Esposito, M., 2004. Variation of BOLD hemodynamic responses across subjects and brain regions and their effects on statistical analyses. Neuroimage 21, 1639-1651.

Deshpande G, Sathian K, Hu X..Effect of hemodynamic variability on Granger causality analysis of fMRI Neuroimage. 2010 Sep;52(3):884-96. Epub 2009 Dec 11

Heavily dependent on sampling and noise characteristics.

Solo, V. On causality I: Sampling and noise. in 2007 46th IEEE Conference on Decision and Control. 2007.

Adding noise to one channel can “reverse” the direction of the causality.


Limitations of Granger Causality

Stokes, P. A., & Purdon, P. L.(2017).A study of problems encountered inGranger causality analysis from aneuroscience perspective.

Proceedings of the NationalAcademy of Sciences, 201704663.

http://doi.org/10.1073/pnas.1704663114


Dynamic Causal Modeling

Friston, K. J., Harrison, L., & Penny, W. (2003). Dynamic causal modelling. Neuroimage, 19(4), 1273-302.

Accounts for :

Endogenous connections

Modulation

Direct Inputs

Feedback

z1

z2

z3

u1

u2

a21

b21

c11

a32

a23

c22

a33a22

a11



Fit of a priori network model: “state space” models include

feedback, external modulation Vascular models …Lots of parameters to

estimate! (Bayesian Estimation needed)


Dynamic Causal ModelingState-Space Modeling in a nutshell:

Let z be the values in the different nodes. Z is a vector with a value for each node

Z is a function of time

Those values will change as a function of the inputs from other nodes and from external inputs [u(t)] … etc.



Z1(t)

Z2(t)

Z3(t)

u1(t)

u2(t)

a21

b21

c11

a32

a23

c22

a33a22

a11



Further complications: we don’t observe z(t) directly, but rather z(t)*hrf(t) hrf(t) is the hemodynamic response, modeled using

the “Balloon model” (Buxton, Magn Reson Med 39, 855-864 (1998).)


“How does brain activity, z, change over time?”

The Neural Model

V5

V1z1

z2

a11

a22

c11

Driving input u1

a21u2

u1

z1

z2

b21

Attention u2

Could model be used to model a main effect and interaction

(Slide by Peter Zeidman from the UCL web site)

Dynamic Causal Modeling The whole thing can be expressed as a bi-

linear model

Our mission is to estimate θ


Estimation in DCM

Lots of parameters to estimate!

Estimate hemodynamic parameters and DCM parameters separately.

Estimation: Maximum Likelihood / Likelihood Ratio. Bayesian Estimation


Detour… Maximum Likelihood Estimation

“Likelihood” is the probability of observing the data, given a set of parameters for a model.

P(Y | M ) We calculate this probability by

Assuming a particular distribution for the data Sticking the “given” parameters into the “given” model This lets you calculate a THEORETICAL distribution of the data

and it’s residuals Search for the parameters that maximize the probability of the

data.


Detour… Maximum Likelihood Estimation

Example - imagine a linear model and assume

that the observed residuals follow a normal distribution.

We can write an equation for the probability of observing the data, given an arbitrary set of parameters.

If we guess the parameters at random, the calculated residuals will follow Gaussian distribution.

If we guess them correctly, we should get the highest probability.


Maximizing the Likelihood Sometimes it’s a simple, analytical max/min

problem: figure out derivative, find the zeros, … etc.

Usually … it’s not so simple. The structure of the residuals is often not known.

Use iterative techniques to find the solution that minimizes the residuals, maximizes likelihood function.


Bayesian Estimation Begin with definition of conditional probability

P(x|y) = P(xy) / P(y)P(y|x) = P(xy) / P(x)

... algebra, blah , blah...

P(x|y) = P(y|x)P(y) / P(x)

Voila: Bayes’ Theroem


Bayesian EstimationBayes Theorem:

P(M|y) = P(y|M) P(M) / P(y)

Find M that maximizes posterior probability P(M|y)

“prior”

“posterior”“evidence”

“likelihood”


Inference We already obtained probabilities as the output! Model Comparison

Try several models Simplify (remove elements of the model) Which model give the best fit?

Gaussianity of the residual error. Akaike Information Criterion (AIC):

AIC = 2k – 2 ln (L)

Bayesian Information Criterion (BIC) BIC = -2 ln (ML) + k ln(N)


Some practical points about DCM implementation (SPM)

Outputs are the estimated distributions of the parameters. Parameter estimates Likelihood of parameters Covariance of the errors

Need at least ten data points per parameter to estimate (heuristic) .(Penny, W., Roberts, S., 1999. Bayesian neural network for classification: how useful is the evidence framework? Neural Netw. 12, 877–892.)

Model comparison is a powerful tool for inference. Trade-off between complexity and accuracy.

Look at posterior covariance matrices. High off-diagonal terms indicate that some parameters are correlated,

redundant?Hernandez-Garcia , FMRI Course 2018

Limitations of DCM

Very complete model, but … Many parameters. Can you trust it? Confounds between hemodynamic and neural

parameters. Dependence on sampling rate. Priors are important!

(see S. Frässle et al. Test-retest reliability of dynamic causal modelingfor fMRI . NeuroImage 117 (2015) 56–66 )


Examples of Clinical Applications of DCMs

Major depression Almeida, J.R., Versace, A., Mechelli, A., Hassel, S., Quevedo, K., Kupfer, D.J., Phillips, M.L., 2009. Abnormal

amygdala-prefrontal effective connectivity to happy faces differentiates bipo lar from major depression. Biol. Psychiatry 66, 451–459.

Schlösser, R.G., Wagner, G., Koch, K., Dahnke, R., Reichenbach, J.R., Sauer, H., 2008. Fronto-cingulate effective connectivity in major depression: a study with fMRI and dynamic causal modeling. NeuroImage 43, 645–655.

Autism Grèzes, J., Wicker, B., Berthoz, S., de Gelder, B., 2009. A failure to grasp the affective meaning of actions in autism

spectrum disorder subjects. Neuropsychologia 47, 1816–1825. Radulescu, E., Minati, L., Ganeshan, B., Harrison, N.A., Gray, M.A., Beacher, F.D., Chatwin, C., Young, R.C.,

Critchley, H.D., 2013. Abnormalities in fronto-striatal connectivity within language networks relate to differences in grey-matter heterogeneity in Asperger syndrome. NeuroImage Clin. 2, 716–726.

Schizophrenia Deserno, K., Sterzer, P., Wüstenberg, T., Heinz, A., Schlagenhauf, F., 2012. Reduced prefrontal-parietal effective

connectivity and working memory deficits in schizophrenia.J. Neurosci. 32, 12–20. Brodersen, K., Penny, W., Harrison, L., Daunizeau, J., Ruff, C., Duzel, E., Friston, K., Stephan, K., 2008. Integrated

Bayesian models of learning and decision making for saccadic eye movements. Neural Netw. 21, 1247–1260. Roiser, J.P., Wigton, R., Kilner, J.M., Mendez, M.A., Hon, N., Friston, K.J., Joyce, E.M., 2013. Dysconnectivity in the

frontoparietal attention network in schizophrenia. Front.Psychol. 4, 176.


Ascendancy Looks into the questions of sufficiency and

necessity General idea:

Identify neuronal events in a set of nodes Is one node active every time the other one

is? A subset of the time? Is that statistically significant?


Ascendancy

Patel, R. S., Bowman, F. D., & Rilling, J. K. (2006). A Bayesian approach to determining connectivity of the human brain. Hum Brain Mapp, 27(3), 267-76.


Ascendancy Begin by identifying events: indicator function (very

tricky stuff !!!) Hernandez-Garcia, L. and M.O. Ulfarsson, Neuronal event detection in fMRI time series using iterative deconvolution techniques. Magnetic Resonance Imaging, 2011. 29(3): p. 353-364.

Look for coincidences in events (Relax timing requirements – it’s all blurred anyway)

Is activity in one node a subset of the activity in another node? Ascendant.

You can think of this as a measure of “necessity”


Use Bayesian statistics to infer significance (Dirichlet prior distributions)

Ascendancy


Limitations of Ascendancy

Event detection, timing

Physiological confounds

Interpretation: we’re only looking at two

nodes at a time

Bayesian statistics require assumptions


Comparing Network Analysis Strategies

Smith et al: NeuroImage 54 (2011) 875–891

Strategies: Coherence. Lags: Granger, DCM. Bayes Net: Graph Theory. Conditional Probability.


Comparing StrategiesSmith et al: NeuroImage 54 (2011) 875–891


Comparing Strategies


about that comparison ...

• Preprocessing involved high pass filters that forced data to be Gaussian....

• Several of the Bayesian Network analysis algorithms test for non-Gaussianity.

• ... They actually perform really well without the filter.

• Ramsey, J.D., Hanson, S.J., Glymour, C., 2011. Multi-subject search correctly identifies causal connections and most causal directions in the DCM models of the Smith et al. simulation study. NeuroImage 1–11.


About Causality …How do we infer Cause and Effect ?

Big picture: Philosophy, Cognition, Social Sciences, Policy, Artificial Intelligence…

We infer causal relationships by association measures and exploring “counterfactuals”.

Experimentally: Interact with the system. What changes? Is function preserved? (TMS, ablations, modulation …)

"… we may define a cause to be an object, followed by

another, and where all objects, similar to the first,

are followed by objects similar to the second. Or in other words, where, if the

first object had not been, the second never had existed

…" — David Hume, An Enquiry Concerning

Human Understanding.


“Footprints” of Causal Relationship

Causality in time series (signal processing) Correlation / Statistical Dependence Timing: Effect always after cause

Science in general uses “counterfactuals” Necessity ? Sufficiency ?


Conclusions Tread carefully DCM is most popular (requires a priori network) Many approaches: None of these methods are perfect.

(In fact, they’re all pretty lousy for FMRI data)

Importance of good ROIs ! Correlation based and DAG methods are very good at identifying

structures. Ascendancy is very good at identifying pairwise directionality. Lag based methods break down very easily in FMRI

Open discussion?


How do we infer Cause and Effect ?



Newton Leibnitz Fourier

TaylorDescartes Fermat

Einstein




Simmons

Other references Valdes-Sosa, P.A., Roebroeck, A., Daunizeau, J., Friston, K., 2011. Effective

connectivity: influence, causality and biophysical modeling. NeuroImage 58, 339–361. Ramsey, J.D., Hanson, S.J., Hanson, C., Halchenko, Y.O., Poldrack, R.A., Glymour,

C., 2010. Six problems for causal inference from fMRI. NeuroImage 49 (2), 1545–1558.

Ramsey, J.D., Hanson, S.J., Glymour, C., 2011. Multi-subject search correctly identifies causal connections and most causal directions in the DCM models of the Smith et al. simulation study. NeuroImage 1–11.


footprints of causality - training course in fmri · on causality i: sampling and noise. in 2007...

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