Significance Alzheimer’s disease (AD) affects 10% of the elderly population. The disease remains poorly understood with no cure. The main symptom is memory loss, but other symptoms might include impaired executive function (ability to plan and accomplish goals; e.g., grocery shopping). The severity of behavioral symptoms and brain atrophy (gray matter loss) can vary widely across patients. This variability complicates diagnosis, treatment, and prevention. A mathematical model reveals distinct brain atrophy patterns, explaining variation in gray matter loss among AD dementia patients. The atrophy patterns can also explain variation in memory and executive function decline among dementia patients and at-risk nondemented participants. This model can potentially be applied to understand brain disorders with varying symptoms, including autism and schizophrenia.