White matter microstructure underpins cognition and function in the human brain through the facilitation of neuronal communication, and the non-invasive characterization of this structure remains an elusive goal in the neuroscience community. Efforts to assess white matter microstructure are hampered by the sheer amount of information needed for characterization. Current techniques address this problem by representing white matter features with single scalars that are often not easy to interpret. Here, we address these issues by introducing tools from soft matter for the characterization of white matter microstructure. We investigate structure on a mesoscopic scale by analyzing its homogeneity and determining which regions of the brain are structurally homogeneous, or ``crystalline" in the context of materials science. We find that crystallinity is a reliable metric that varies across the brain along interpretable lines of anatomical difference. We also parcellate white matter into ``crystal grains," or contiguous sets of voxels of high structural similarity, and find overlap with other white matter parcellations. Our results provide new means of assessing white matter microstructure on multiple length scales, and open new avenues of future inquiry.

Firing-rate (FR) or neural-mass models are widely used for studying computations performed by neural populations. Despite their success, classical firing-rate models do not capture spike timing effects on the microscopic level such as spike synchronization and are difficult to link to spiking data in experimental recordings. For large neuronal populations, the gap between the spiking neuron dynamics on the microscopic level and coarse-grained FR models on the population level can be bridged by mean-field theory formally valid for infinitely many neurons. It remains however challenging to extend the resulting mean-field models to finite-size populations with biologically realistic neuron numbers per cell type (mesoscopic scale). In this talk, I present a mathematical framework for mesoscopic populations of generalized integrate-and-fire neuron models that accounts for fluctuations caused by the finite number of neurons. To this end, I will introduce the refractory density method for quasi-renewal processes and show how this method can be generalized to finite-size populations. To demonstrate the flexibility of this approach, I will show how synaptic short-term plasticity can be incorporated in the mesoscopic mean-field framework. On the other hand, the framework permits a systematic reduction to low-dimensional FR equations using the eigenfunction method. Our modeling framework enables a re-examination of classical FR models in computational neuroscience under biophysically more realistic conditions.