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Brain aging can be viewed as having 2 parts: chronological age (normally  the brain grey matter volume slowly shrinks with advancing age) and a lifetime of exposures - which can be negative from unhealthy lifestyle and injuries, and positive from a healthy lifestyle and enriched environments. That's why after a lifetime there can be wide variation in the physiological age of our brains. These differences in the  brain (in the grey matter) can be measured with magnetic resonance imaging (MRIs).

The researchers in this study used the concept of physiological age - the difference between the chronological age and predicted age, as a marker of brain health. They looked at adults of varying ages,and found that the more flights of stairs a person climbs daily, and the more years of school a person had completed, the "younger" their brain physically appears.  This study was a cross-sectional study and so shows an association rather than a definite cause, but interestingly other forms of exercise did not show this link (walking/hiking, jogging, running, bicycling, aerobic exercise, lap swimming, tennis.squash/racquetball, low intensity exercise). From Science Daily:

Want a younger brain? Stay in school -- and take the stairs

Taking the stairs is normally associated with keeping your body strong and healthy. But new research shows that it improves your brain's health too -- and that education also has a positive effect. In a study recently published in the journal Neurobiology of Aging, researchers led by Jason Steffener, a scientist at Concordia University's Montreal-based PERFORM Centre, show that the more flights of stairs a person climbs, and the more years of school a person completes, the "younger" their brain physically appears.

The researchers found that brain age decreases by 0.95 years for each year of education, and by 0.58 years for every daily flight of stairs climbed -- i.e., the stairs between two consecutive floors in a building.

For the study, Steffener and his co-authors used magnetic resonance imaging (MRI) to non-invasively examine the brains of 331 healthy adults who ranged in age from 19 to 79. They measured the volume of grey matter found in participants' brains because its decline, caused by neural shrinkage and neuronal loss, is a very visible part of the chronological aging process. Then, they compared brain volume to the participants' reported number of flights of stairs climbed, and years of schooling completed. 

Results were clear: the more flights of stairs climbed, and the more years of schooling completed, the younger the brain. "This study shows that education and physical activity affect the difference between a physiological prediction of age and chronological age, and that people can actively do something to help their brains stay young," he says.