In Search of a Common Scale for Causal Fusion in Neuroimaging

Sergey Plis

joint work with David Danks, Vince Clark, Andy Mayer, Terran Lane,
Eswar Damaraju, Mike Weisend, Tom Eichele, Jianyu Yang, and Vince Calhoun

The Mind Research Network

definitions
dangers of unimodal analysis
a model-based solution
towards a graph space model fusion

definitions

neuroimaging

studies brain function
produces immense amounts of data
data comes in various forms
each with their strength and weaknesses
our goal: use that data to infer causal relations among brain networks

functional MRI

measures Blood Oxygenation Level Dependent (BOLD) response
produces 4D data
Strength: relatively well localized
Weakness: slow sampling rate

Magnetoencephalography

measures magnetic field
Strength: instant reflection of underlying activity
Weakness: uncertain spatial location

the same underlying phenomena

Bayesian Network

graph characterization

dangers of unimodal analysis

processing pipeline

result

result up close

a model-based solution

dynamic Bayesian net

Results

dynamic Bayesian net

Causal Structure from Overlapping Variable Sets

theory

the situation

Causal timescale $\ne$ Measurement timescale

$\approx100ms$

$???$

$\approx2s$

What causal inferences can be made in this situation?

two challenges

Forwards inference: Given a causal structure at causal timescale, what is the implied structure at (undersampled) measurement timescale?
Backwards inference: Given inferred causal structure at measurement timescale (with unknown undersampling), what structures at causal timescale are possible?

representation

Dynamic Bayesian Network (DBN)
- first order Markov
- causally sufficient

undersampling effect

(better) representation

undersample

A quick look at behavior

DAG

A quick look at behavior

Superclique SCC loop

A quick look at behavior

Oscillating SCC loop

algorithm

Virtual nodes

Conflict persistence

conflicts persist

${\cal{G}}^1 \text{ conflicts with } {\cal{H}}^u \implies \forall {\cal G} \supseteq {\cal G}^1 {\cal{G}} \text{ conflicts with } {\cal{H}}^u$

Search Tree

The MSL algorithm is correct and complete

Unfortunately it is slow

computational complexity

$\{ m_1 , \dots , m_l \}$ - the sets of virtual node identifications for each of the edges or edge-pairs
$\prod_i^{l} \text{len}(m_i)$ - computational complexity of using edge-pairs
$\text{len}$ - the number of possible identifications for that particular edge or edge-pair
Computational advantage, expressed as a log-ratio: $$ \log{r} = \sum_i^{l} \log{\text{len}(m_i)} - e\log{n}. $$

still not fast enough

pre-computed $O(n^2)$ pruning data-structure
employed additional constraints and observations

A virtual node $V$ in $S \xrightarrow{V} E$ cannot be identified with node $X$ if any of the following holds:

$\exists W \in \chg{1}{S}\setminus X$ s.t. $\nexists W \leftrightarrow X$ in ${\cal{H}}^2$
$\exists W\in \chg{1}{X}\setminus E$ s.t. $\nexists W \leftrightarrow E$ in ${\cal{H}}^2$
$\exists W\in \chg{1}{X}$ s.t. $\nexists S \rightarrow W$ n ${\cal{H}}^2$
$\exists W \in \chg{1}{E}$ s.t. $\nexists X \rightarrow W$ in ${\cal{H}}^2$
$\exists W\in \pag{1}{S}$ s.t. $\nexists W \rightarrow X$ in ${\cal{H}}^2$
$\exists W \in \pag{1}{X}$ s.t. $\nexists W \rightarrow E$ in ${\cal{H}}^2$

A virtual node pair $V_1, V_2$ for a fork $E_1 \xleftarrow{V_1} S \xrightarrow{V_2} E_2$ cannot be identified with nodes $X_1, X_2$ if any of the following holds:

$V_1$ in $E_1 \xleftarrow{V_1} S$ cannot be identified with $X_1$
$V_2$ in $S \xrightarrow{V_2} E_2$ cannot be identified with $X_2$
$V_1 \equiv E_2 \wedge V_2 \not\in pa_{{\cal{H}}^2}(E_1)$
$V_2 \equiv E_1 \wedge V_1 \not\in pa_{{\cal{H}}^2}(E_2)$
$S \equiv V_2 \wedge V_1 \ne V_2 \wedge V_1 \ne E_2$ and $\nexists E_2 \leftrightarrow V_1$ in ${\cal{H}}^2$
$S \equiv V_1 \wedge (V_1 \equiv V_2 \vee V_2 \equiv E_2) $ and $\nexists E_1 \leftrightarrow E_2$ in ${\cal{H}}^2$
$S \equiv V_1 \wedge (V_1 \equiv V_2 \vee V_2 \equiv E_2) $ and $\nexists E_1 \leftrightarrow E_2$ in ${\cal{H}}^2$
$S \equiv V_2 \wedge (V_1 \equiv V_2 \vee V_1 \equiv E_1)$ and $\nexists E_1 \leftrightarrow E_2$ in ${\cal{H}}^2$
$V_1 \equiv V_2$ and $\nexists E_1 \leftrightarrow E_2$ in ${\cal{H}}^2$

A virtual node pair $V_1, V_2$ for two-edge sequence $S \xrightarrow{V_1} M \xrightarrow{V_2} E$ cannot be identified with $X_1, X_2$ if any of the following holds:

$V_1$ in $S \xrightarrow{V_1} M$ cannot be merged to $X_1$
$V_2$ in $M \xrightarrow{V_2} E$ cannot be merged to $X_2$
$V_1 \equiv V_2 \wedge ( M \not\in pa_{{\cal{H}}^2}(M) \vee V_1 \not\in pa_{{\cal{H}}^2}(V_2) \vee S \not\in pa_{{\cal{H}}^2}(E))$