Descriptive Analysis and Motivation for the Mesh Model
In this study, BOLD signals 
are measured at time instants
, 
, at voxel coordinates 
, 
where 
is the number of time samples, and 
is the number of voxels. The data set 
consists of the voxels 
, which are distributed in brain in three dimensions. Therefore, the position 
of a voxel 
at time instant 
is a three dimensional vector. At each time instant
, the participant is processing (either encoding or retrieving) a word belonging to a cognitive process. Therefore, the samples 
has an object label at each time instance. In Mesh Learning (Ozay, Öztekin, Öztekin, & Vural, 2012), the cognitive states are modeled by local meshes for each individual voxel, called seed voxel 
, which is defined in a neighborhood system 
(see Figure 1). In this mesh, voxel 
is connected to 
-nearest neighboring voxels 
by the arcs with weights 
. Therefore, the relationship among the BOLD signals measured at each voxel, are represented by the arc weights. 
-nearest neighbors, 
, are defined as the spatially-nearest neighbors to the seed voxel, where the distances between the voxels are computed using Euclidean distances between the spatial coordinates 
of the voxels in brain. The arc weights 
of the mesh are estimated by the following linear regression equation:
, (1)
Figure 1.The Star Mesh, which represents the voxel intensity values
at the center and its 4-nearest neighbors. Blue node represents the center voxel and the orange nodes represent the surrounding voxels.
where
indicates the error of voxel
at time instant
, which is minimized for estimating the arc weights
. This procedure is conducted by minimizing the expected square error defined as follows,
(2) where
is the set of
p-nearest neighbors of the j
th voxel at
.
Minimizing Equation (2) with respect to 
is accomplished by employing Levinson-Durbin recursion (Vaidyanathan, 2007), where 
is the expectation operator. The arc weights 
, which are computed for each seed voxel at each time instant 
, is used to form the mesh arc vector 
. Furthermore, a mesh arc matrix 
is constructed by concatenating the mesh arc vectors at each time instant, 
. Finally, feature matrix 
which represents the Mesh Arc Descriptor (MAD), is constructed. The feature matrix, extracted during both memory encoding and retrieval stages is further used in training and testing phases in the classification of cognitive processes, respectively. For the details of the mesh learning algorithm see Fırat et al. (2012) and Özay, Öztekin, Öztekin, and Yarman Vural (2011).
The motivation of representing voxels in the brain by local meshes can be validated by analyzing an individual voxels’ intensity change and the change of the sum of squared difference of intensities 
in the neighborhood of that voxel in time. Individual voxel intensity values, which are measured at each time instant, do not possess any discriminative information as illustrated in Figure 2 with red line. Note that the signal intensity value for a voxel is almost constant for each time instant. Since the measurements along the time axis correspond to separate cognitive processes, in most of the problems, it is unlikely to discriminate them by using multi-voxel pattern analysis (MVPA) methods, which classify the voxel intensity values by a machine learning tool. On the contrary, there is a slight variation of the sum of squared distances of intensity values in differing neighbor sizes. The above observation shows that the relationships among voxels carry more information than individual voxel intensity values, at each time instant.
Figure 2. Sum of squared difference,
, of intensity values for a voxel and its N-nearest neighboring voxels over time in log space. The time axis indicates the fMRI measurements from 10 semantic categories.