The interdisciplinary field of neuroscience has made significantprogress in recent decades, providing the scientific community in gen-eral with a new level of understanding on how the brain works beyondthe store-and-fire model found in traditional neural networks. Mean-while, Machine Learning (ML) based on established models has seena surge of interest in the High Performance Computing (HPC) com-munity, especially through the use of high-end accelerators, such asGraphical Processing Units (GPUs), including HPC clusters of same.In our work, we are motivated to exploit these high-performance com-puting developments and understand the scaling challenges for new–biologically inspired–learning models on leadership-class HPC resources.These emerging models feature sparse and random connectivity pro-files that map to more loosely-coupled parallel architectures with alarge number of CPU cores per node. Contrasted with traditional MLcodes, these methods exploit loosely-coupled sparse data structures asopposed to tightly-coupled dense matrix computations, which benefitfrom SIMD-style parallelism found on GPUs. In this paper we introducea hybrid Message Passing Interface (MPI) and Open Multi-Processing(OpenMP) parallelization scheme to accelerate and scale our computa-tional model based on the dynamics of cortical tissue. We ran compu-tational tests on a leadership class visualization and analysis cluster atArgonne National Laboratory. We include a study of strong and weakscaling, where we obtained parallel efficiency measures with a minimumabove 87% and a maximum above 97% for simulations of our biologicallyinspired neural network on up to 64 computing nodes running 8 threadseach. This study shows promise of the MPI+OpenMP hybrid approachto support flexible and biologically-inspired computational experimen-tal scenarios. In addition, we present the viability in the application ofthese strategies in high-end leadership computers in the future.