Olfactory Mathematics

The lack of adequate equipment to study the motion of the planets in the 16th Century did not prevent Nicolaus Copernicus from accurately demonstrating that the sun was not, in fact, at the center of the cosmos. Through mathematics, the brilliant astronomer was able to disprove the convoluted logic of his contemporaries by mapping the trajectories of the planets correctly if and only if the sun - not the earth - was placed at the center.

Astronomical computation in the 21st Century is, of course, far less enigmatic. Oddly enough, the workings of the planets outside our own atmosphere can now prove easier to comprehend than the workings of the olfactory bulbs inside our own heads. But, thanks to a team of researchers at the Friedrich Miescher Institute for Biomedical Research in Switzerland, the simple perfection of complex mathematics may again be providing the answers.

According to a new study published in Nature Neuroscience and detailed at PhysOrg.com, the small team of neurobiologists and mathematicians first studied decorrelation in the olfactory system and found that it makes representations of odors in the brain more distinct. Without pattern decorrelation - a fundamental computation in the brain which also has many applications in computer science and engineering - discriminating odors would be much more cumbersome. However, the mechanisms underlying decorrelation were complicated by non-linearity and recurrent connectivity through which neurons can influence each other in a complex fashion. Thus, the meaning of this arrangement was not understood.

Martin Wiechert, the mathematician in Rainer Friedrich’s group, has now cracked this problem. His mathematical theory, published in Nature Neuroscience, shows that decorrelation emerges naturally from two features of recurrent neuronal circuits: sparse connections and high spontaneous activity.
“We have known for a while from observations that activity is high and connectivity sparse within the olfactory bulb. But we did not know why this is the case,” explains Rainer Friedrich, FMI Group Leader and lead author of the study. “Our mathematical results allow us to explain not only the structure of neuronal circuits in the olfactory bulb but also to understand better what happens when we smell.”

Biologically relevant processes are difficult to explain mathematically not merely because of their complexity, but also because the data necessary to test the mathematical principles is often missing. “Our laboratory is uniquely predisposed for such a study. We have been able to measure activity in individual neurons very efficiently with optical methods for a while now. Such data is crucial to develop mathematical circuit models,” says Friedrich. “As the project progressed, experimentation and mathematical model development cross-fertilized each other. Based on the mathematical studies we devised experiments and were able to provide the data to confirm or rebut mathematical reasoning. In exchange, we received mathematically validated information on how neuronal circuits work.”

While mechanisms of decorrelation are now understood, other neuronal processes such as learning or memory still await mathematical explanation. But no worries: Rainer Friedrich’s lab is perfectly equipped to tackle these questions as well.