Spatio-Temporal Clustering

W.D. Penny and K. Friston

Studies of functional specialisation aim to relate cognitive function
to anatomical location. In neuroimaging this mapping is
inferred from time-series of high resolution images containing
many thousands of voxels. We propose that inference
can be naturally enabled within the confines of a 
probabilistic generative model that describes how data are generated
from a mixture of spatially localised, temporally coherent, or 
`active' components and spatially distributed, temporally
incoherent, or `null' components. Inferences concerning
functional specialisation can then be made on the basis of the 
inferred active components. This has the 
benefit that inferences are made, not at the `voxel level' - the 
high dimensional space in which the data collected - but at 
the `cluster level' - a more natural 
level of description that reflects the underlying functional
anatomy.

Our model is related to functional clustering analyses of
neuroimaging data [1] which
aim to identify clusters of 
voxels with similar time-courses. In the present work, 
we aim to identify clusters of voxels with
similar time-courses that are in similar anatomical locations, hence the 
name Spatio-Temporal Clustering (STC). 
Critically, the time course of a cluster is generated by 
explanatory variables that embody the experimental design.
This explicitly ties together the two aspects of functional specialisation; 
function and anatomy.

Specifically, data are viewed as 
being generated from a K-component mixture model as follows.
For a voxel at location v, mixture component k is selected
with probability p(k|v). Typically k=1 will correspond to 
a null component and other values of k to active components. 
The probability p(k|v) defines a spatial prior and is related to the
likelihood p(v|k) via Bayes rule. 
For active components [2]
 we define p(v|k) to be a Gaussian with mean m(k) and
covariance C(k). The corresponding spatial prior p(k|v) selects
nearby components with high probability.
A time series for that voxel, y, is then selected with 
probability p(y|k) which is defined via a
 General Linear Model (GLM) with design matriz X and 
having regression coefficients w(k). 

We have shown [2] that the model parameters
 can be estimated using a Generalised Expectation Maximisation (GEM) 
algorithm. More recently we have realised that an equivalent
probability model exists in the joint space of time series and
voxel positions ie. s=(y,v). Estimation of parameters in this
space can then be implemented using standard 
Expectation Maximisation (EM), which is much faster than GEM.
 
In the limit of the number of clusters being equal 
to the number of voxels we recover the mass-univariate 
approach underlying Statistical Parametric Mapping (SPM) with
a single GLM for each voxel.
We also note that
modelling at the cluster level allows us to pool information
from nearby voxels for highly efficient 
parameter and variance-component estimation and
STC provides a mathematically principled way of deciding which voxels to
put in which pool. 

References

[1] E. Rostrup, F.A. Nielsen, C. Goutte, P. Toft and L.K. Hansen (1999).
On Clustering fMRI time series. Neuroimage, 9:298-310. 

[2] W.D. Penny and K.J. Friston (2002) Mixtures of General Linear Models
for Functional Neuroimaging. Submitted.