Here's an interesting report from US News depicting how a higher body fat percentage actually translates into less brain tissue, thus resulting in Alzheimer's, dementia, and other cognitive impairments. 

...new research that found that elderly individuals who were obese or overweight had significantly less brain tissue than individuals of normal weight...

"...This is the first study to show physical evidence in the brain that connects overweight and obesity and cognitive decline," said Thompson, who is professor of neurology at UCLA and a member of the UCLA Laboratory of Neuro Imaging...

...Clinically obese people had 8 percent less brain tissue, while the overweight had 4 percent less brain tissue compared to normal-weight individuals...

...Obesity affects every system in your body. The body can't be splintered. It's completely linked. We are what we eat and we eat too much," he said. "The bottom line is that an obese, sedentary person is going to have a breakdown of every organ system, and that includes a greater chance of impotence and infertility and other things that people don't generally think are directly related to obesity."

So, what does this all mean and how does it come about? What is the relation between obesity and our brains? Nora over at PrimalBlog explains:

think back to your physiology 101 class (the one you were made to forget in order to pass your pharmacology 101 class): How is insulin defined in your textbooks? Allow me to refresh your memories: “Insulin is the fat storage hormone”.

Still confused?

Let’s take a trip down memory lane to the Textbook of Medical Physiology, which states: “All body fat is made from glucose”. (–Not dietary fat, mind you, but G-L-U-C-O-S-E:  i.e., sugar)

We getting warmer?

Do you think…just MAYBE that there could be a possible relationship between sugar/starch consumption (something hardly anyone in this culture consumes at all, right?), obesity AND brain glycation, leading to neurodegenerative processes (i.e., shrinking brains)? Alzheimer’s disease, after all, is, put simply, a state of advanced brain neuropathy. What causes neuropathy (think back to the diabetic model)? Glycation. What causes glycation again? …Oh yeah—SUGAR. Where do we get excessive blood sugar? –Sweets, sodas, juices, alcohol, cereal grains, corn, potatoes, legumes (yes, legumes are about 60% starch), low fat dairy, rice, pasta, crackers, bread, snack chips… After all, even so-called “complex carbs” (other than fiber) ultimately become sugar once they are digested and enter the bloodstream.  –And how much dietary carbohydrate do we actually require as humans?  Answer:  ZERO...

...neurogenic inflammation and neurodegenerative processes, enhanced catabolic (degenerating) cortisol and damaging insulin production due to gluten sensitivity?? A study using frozen blood samples (taken from US Air Force recruits) 50 years ago found that the prevalence of gluten sensitivity today is fully four times more common than it was 50 years ago. Celiac disease may impact as many as 1 in every 33 people in today’s society—known to be profoundly neurodegenerative. –And guess what? Food sensitivities are also provoking of weight gain and obesity. –Oh, and gluten containing foods are also—you guessed it—starch based (read: sugar)... 

...It doesn’t take a rocket scientist to figure out that gluten and starchy grains are not “health foods” to anyone and all of us are fundamentally better off without them. For some, however, they are especially deadly and destructive in chronic, progressive and insidious ways...

...A protein found in connection with gluten consumption called zonulin is known to enhance intestinal permeability—the very thing that commonly leads to antibody reactions to foods leading to sensitivities, allergies and even (eventually) autoimmune disorders…and worse. Microglia cells in our brains are what act as immune cells in the brain and respond to foreign compounds. Microglial activation is known to cause neurogenic inflammation and brain degeneration. Moreover, there is a direct link between gastrointestinal inflammation (commonly generated by irritants such as gluten), microglial activation (as a result of infiltrating foreign compounds with enhanced intestinal permeability) and brain degeneration.

There you have it. Less sugar and grains = less fat and more brain power!