VOLUME 12, NUMBER 14 • JANUARY 26, 2001

Scientists propose mathematical answers to biological questions

Associate Professor Marcelo Magnasco came to The Rockefeller University in 1992 as a postdoctoral fellow. One year later he was the first faculty member recruited to Rockefeller’s Center for Studies in Physics and Biology.

Marcelo Magnasco and his colleagues are using mathematical analysis to learn more about biological systems.

Magnasco’s work at Rockefeller expands upon seeds planted by then-President Torsten Wiesel and Toyota Professor Mitchell Feigenbaum when they created the Center. Former NIH Director Harold Varmus (now president of Memorial Sloan-Kettering Cancer Center) reiterated the concept of a stronger physics and biology collaboration in a centennial lecture to the American Physical Society in 1999, identifying the "need to transport intellects across artificial disciplinary boundaries in attempting to open borders that have been traditionally hard to cross." True to this unwitting mandate, Magnasco collaborates with Professor James Hudspeth, with whom he has an established publication record and with whom he published a recent paper that unifies seemingly disparate biological observations about the workings of the ear.

It is notable that Magnasco’s graduate students, Guillermo Cecchi (now a postdoctoral fellow) and Mariano Sigman, have followed a similar trajectory and will soon co-publish a paper with Magnasco and another Rockefeller biology-based collaborator, Professor Charles Gilbert (Sigman is a graduate student of both Magnasco and Gilbert). This time the topic is identifying the structure shared by all natural images, the basic geometric units of which images are composed, and understanding how this relates to the function of the visual system. Magnasco, Cecchi and Sigman, in cooperation with their biology-based colleagues, have focused on mathematical analysis to explain the physics of neurosensory organs.

What is also interesting, and perhaps radical in its simplicity, is that both projects use well-known mathematical concepts to fundamentally describe what biologists deem complex phenomenon-observations heretofore considered too random to be characterized by one equation or theory. Magnasco’s research on the ear considers the biological theories that have attempted to characterize the dynamics of the aural system. For example, the cochlea, the organ inside the ear encased in a spiral minaret of bone, was long thought to function like a musical instrument such as a piano or a harp, whereby incoming sounds could make the strings vibrate at varying frequencies.

A succession of theories, starting with astrophysicist Tommy Gold’s in 1948, have proved otherwise, though not without considerable skepticism; Gold’s argument, though largely dismissed at the time, was that without a feedback mechanism incoming sounds would simply drown or dissipate because of the fluid in the ear. It wasn’t until the 1960s and 1970s, respectively, when Hungarian physiologist Georg Von Bekesy and American physiologist William Rhode hinted at the presence of "biological amplifiers" in the ear, that scientists started altering their thinking. Magnasco’s work provides an overview and consideration of this neuroscientific history as introduction. But little more was understood about ear amplification until the 1980s when biologists David Corey and Hudspeth fleshed out the theory by suggesting that stereocilia, or hair cells in the ear, are connected via a spring mechanism to tiny channels that, when pulled open, admit calcium ions through the membranes of the hair cells. This influx of ions triggers the nerve signal.

What Magnasco and colleagues have done is to provide a model for the existence of a "trapdoor amplifier." In other words, the ear tunes its response to acoustic stimulus in order to optimize its sensitivity. Because the ear is known through anecdotal and empirical observations to "play tricks," Magnasco and colleagues tried to figure out some of these tricks.

Why, for example, does the ear sometimes hear pitches that are not actually present, or succeed at compressing loud sounds to minimize damage to the system? Magnasco and colleagues propose that some of these so-called strange properties of our hearing apparatus are due to the fact that it operates at a delicate threshold, like a balance poised to tip one way or the other. The threshold has a mathematical representation, the Hopf bifurcation. A Hopf bifurcation is like "a sound technician adjusting the volume of an amplifier to the loudest possible setting before feedback oscillation ensues," says Magnasco.

Sigman’s and Cecchi’s research on the structure of visual perception in the eye likewise shows that something that a priori was complex like the common structure to all scenes can be explained using the very simple geometric rule of cocircularity.

If one assumes the prevailing analogy that the eye functions like a camera, it is relatively easy to discern the basic work of the organ. Light passes through the cornea and is reflected on the receptor cells of the retina.

The retina’s reception of light signals the optic nerve and sends messages to the "projection area" of the visual cortex in the brain-as easy as a point-and-shoot camera.

However, seeing involves much more than just creating a pixel-by-pixel description of an image. It involves recognizing faces and trees, and knowing that a face is always the same when seen from different views or with other objects occluding it. In fact, one of the most difficult problems (in terms of its computational complexity) that the visual system encounters is to group different elements of a scene into individual objects. In spite of this complexity, this process seems spontaneous and effortless to us: We open our eyes and we immediately understand the scene as a whole.

Neuroscientists invest much labor in understanding precisely how the human visual system analyzes images. But what about the structure of the visual landscape itself? Does the world organize itself in a particular manner and does the visual system "know" this organization and make use of it?

Sigman, Cecchi and their colleagues at Rockefeller have asked whether or not natural images display consistent statistical properties that the eye recognizes that set them apart from random light and shadow displays. To attempt to answer this question they used 4,000 black and white pictures of natural scenes from a public database to study the geometric regularities of edges or line segements (oriented elements).

What they found is that there are correlations among objects in the whole visual field, and that their arrangement can be predicted through the simple geometric rule of cocircularity. In other words, the many different images we see during our lives–faces, animals, trees, buildings–have a common organization. They are made of contours that are smooth and that are, most of the time, very close to being an arc of a circle. Circles are the common "skeleton" of all images. But of course, the world is not only circles. The regions where images part from these regularities, like corners and junctions, are important singularities that also characterize forms.

Sigman, Cecchi, Magnasco and their colleagues even identify similarities in their findings about the visual field and previous physiological and psychophysical studies, suggesting that the human visual system "reads" according to recognized geometries of natural scenes.