A research article published in Neurology found that people in their seventies who were more physically active showed less brain shrinkage over the course of three years. Unfortunately, socially and mentally stimulating situations didn't seem to have an effect, but something as simple as going for a walk a few times a week lead to the people retaining more grey matter in their brain, as well as having less damage to their white matter.
Simultaneously, a paper has been released in the Archives of Neurology looking at patients over the age of 90. They found that those with poor physical performance were significantly more likely to suffer from dementia. In the paper, they say:
Our cross-sectional study found a strong dose-dependent association between poor physical performance and dementia in the oldest old, with higher odds of dementia associated with poorer physical performance.The results reveal that even modest declines in physical performance are associated with increased odds of dementia. The strongest association is seen with gait slowing, followed by five chair stands, grip strength and standing balance.
The authors saw a 30 fold increase in likelihood of dementia between those that couldn't walk, and those that walked the fastest.