Self-Organizing Maps (SOM) can be utilized for the purpose of discovering underlying structures in a distributed system. In the proposed study, SOM is employed to track neuronal pathways in brain Diffusion Tensor Magnetic Resonance (DTMR) images. DTMR Imaging (DTMRI) provides local directional information along the nerve fiber bundles in a vector form. The directional information is obtained by solving for the eigensystem of the tensors produced by the diffusion of the water molecules during the imaging process in MRI. The idea is to map a network of connected nodes of a SOM structure onto the investigated neuronal tracks. First, the imaging matrix is simulated with random eigenvectors and SOM is implemented and verified on these synthetic diffusivity tracks. Then the analysis follows on real DTMR images. The aim of the study is to propose an alternate method for the DTMRI tractography with SOM. Preliminary results on two dimensional synthetic diffusion images are encouraging to pursue the idea in 3D real images. In this study we discuss the proposed method in detail. (4 pages)