Alzheimer’s puzzle through the lens of mathematics

Alzheimer’s disease remains one of the most formidable challenges in neurology, both in terms of its growing prevalence and the complexity of its underlying mechanisms. While amyloid plaques and tau tangles are well-established pathological markers, a fundamental question persists: why do certain brain regions deteriorate early, while others show remarkable resistance? This selective vulnerability and resilience is not a trivial detail—it shapes clinical progression and influences therapeutic strategies.

To shed light on this mystery, a research team at the University of California, San Francisco adopted an original approach, using mathematical models to map how tau pathology spreads and to compare it with the brain’s genetic susceptibility landscape.

Decoding disease with numbers

For over three decades, scientists have known that tau protein does not spread randomly but follows a predictable path—from the hippocampus and entorhinal cortex to other areas. However, some regions, such as primary sensory cortices, remain largely spared, suggesting that not all neurons are equally vulnerable.

Traditional explanations have focused on metabolism or resistance to oxidative stress. While helpful, they don’t fully explain the remarkably consistent and predictable trajectory of tau accumulation. To go beyond these limitations, Chaitali Anand and her colleagues at UCSF introduced a novel tool into Alzheimer’s research: mathematics.

By applying an advanced model called the extended network diffusion model (eNDM), the researchers simulated how tau travels along white matter tracts between brain regions—while also accounting for local tendencies of neurons to accumulate or clear the protein. The core idea was to compare two maps: the predicted one generated by equations, and the actual one derived from tau PET imaging in 196 patients. Any mismatch between the two becomes a crucial clue pointing to additional influences—particularly genetic ones. This method, blending brain imaging, connectivity, and mathematical modeling, marks a pivotal shift. It allows scientists to distinguish between degeneration driven by communication networks and that driven by region-specific biology.

The genetic footprint of brain connections

The results show that the mathematical model accurately captures most of the observed tau distribution, confirming that brain connectivity plays a major role in the disease’s progression. However, certain regions—especially in the temporal and orbitofrontal lobes—accumulated more tau than predicted, indicating that genetic factors also play a significant role.

By cross-referencing imaging data with the expression of 100 known risk genes, the researchers identified four distinct profiles: network-dependent vulnerability, network-independent vulnerability, network-dependent resilience, and network-independent resilience.

In other words, some genes influence disease spread through neural connections, while others affect the intrinsic vulnerability of specific cells. The first group includes genes like MAPT, which encodes tau, and TSPOAP1, involved in inflammation. In contrast, genes like PRNP and JAZF1 increase risk independently of connectivity, whereas BACE1 and FOXF1 appear to have a protective effect.

Functional analysis supports this division: network-related genes are mostly associated with neuronal death and stress responses, while network-independent ones are linked to amyloid metabolism and immune processes.

This integrated approach offers a solution to a long-standing paradox: why doesn’t the map of genetic risk always match the map of actual pathology? The answer lies in the coexistence of two dynamics—one shaped by brain circuitry, the other by the biology of individual regions.

Practically speaking, this distinction opens the door to more targeted therapies: either disrupting disease propagation along networks or reinforcing local cellular defenses. More broadly, it demonstrates how mathematics, genetics, and imaging can come together to transform our understanding of Alzheimer’s and suggest new preventive strategies.

The study by Anand and her team highlights the value of bridging traditionally separate fields. By merging equations with gene expression data, they have produced an unprecedented map of brain vulnerability—one that distinguishes between the forces of connectivity and local biology. While further validation is needed, this integrated approach paves the way for a comprehensive model of Alzheimer’s—one that unites genes, networks, and imaging to better understand how the brain breaks down, and how it may be protected.