How neurons influence each other's firing depends on the strength of synaptic connections among them. Motivated by the highly interconnected structure of the brain, in this study, we propose a computational model to estimate the relationships among voxels and employ them as features for cognitive state classification. We represent the sequence of functional Magnetic Resonance Imaging (fMRI) measurements recorded during a cognitive stimulus by a set of local meshes. Then, we represent the corresponding cognitive state by the edge weights of these meshes each of which is estimated assuming a regularized linear relationship among voxel time series in a predefined locality. The estimated mesh edge weights provide a better representation of information in the brain for cognitive state or task classification. We examine the representative power of our mesh edge weights on visual recognition and emotional memory retrieval experiments by training a support vector machine classifier. Also, we use mesh edge weights as feature vectors of inter-subject classification on Human Connectome Project task fMRI dataset, and test their performance. We observe that mesh edge weights perform better than the popular fMRI features, such as, raw voxel intensity values, pairwise correlations, features extracted using PCA and ICA, for classifying the cognitive states.