Dear Branislava,
I have one important question
regarding the core of meaning of the DCM effective connectivities
directions, which I thought it would be better
to direct to you rather than to the whole mailing list.
The coeficients A (a12, a21 ...) are directional effective connectivities
as has been stated in numerous publications.
The base for this is part of the formula ( whether bilinear or nonlinear
)
dz/dt = Az+ ..... ( I present here a simplified version )
which could be rewriten as dz1/dt = a12z2 + .... etc.
where z1 and z2 are two regions of interest.
From this formula, I can only conclude that the change of activation in
z1 is proportional to the activity in z2 (with a coefficient of
proportionality
a12). I do not see the causality nor directionality. How do we obtain the
directionality in DCM from this, or furthermore the causality?
Causality is induced by the use of the differential equation dz1/dt =
a12z2 +. In other words,
z2 causes a change in z1. Causality is implicit in the generative
model. This contrasts with
models of statistical dependencies used for Granger causality and other
mutual
information measures, which are not generative models.
Causal is used here in its physics sense (hence the dependence on the
equations of motion)
and more specifically in a control theory sense (motion is caused by
inputs (u) and leads to a
response or effect that is caused). It can also be regarded in terms of
casual calculus (in the
sense of directed Bayesian graphs) if we invoke a probabilistic version
of the generative model
dz/dt = f(z,u) + e
which means p(dz/dt | z,u) = N(f(z,u),cov(e)); I.e., the motion
dz/dt is caused by z and u.
I hope this helps.
With very best wishes,
Karl
