One very interesting area of application for ideas and techniques from nonlinear dynamics is the study of biological cycles. The circadian rhythm, the internal 24 hour clock that a very large number of organisms have, has been a particular object of study over the years. Naturwissenschaften‘s list of 100 most cited papers includes a classic paper by Aschoff and Pohl on phase relations between a circadian rhythm and a zeitgeber (a stimulus that entrains the clock). The most prominent zeitgeber is of course the day-night cycle, but other stimuli can reset your circadian clock, including meal times and social interaction. In this particular paper, the authors examine the relationship between the circadian phase (e.g. the time of maximum observed activity relative to start of day) and the day length. Studies like these often use ideas from nonlinear dynamics on the entrainment of oscillators to derive insights into the workings of the clock based on how the phase changes as the difference between the natural frequency of the clock and the entraining (zeitgeber) frequency increases. In this case however, the authors focused on quantitative differences between the phase responses of different groups of organisms. We now know that there are several, evolutionarily distinct circadian oscillators operating in different groups of organisms to which the results of Aschoff and Pohl could likely be correlated.

Coupled oscillators are a recurring theme in nonlinear dynamics, and the Naturwissenschaften list also includes a paper by Hübler and Lüscher on the possibility of controlling one oscillator using a driving signal derived from a second oscillator. Although this is very much a fundamental study, this kind of work has found a number of applications over the years. I have already mentioned the use of such studies to understand biological oscillators. Coupled oscillators show up all over the place, both in natural and in engineered systems. One example that has been the focus of a lot of research is the use of coupled oscillators in secure communications. The problem here is that you want an authorized receiver to get your message, but you don’t want anyone else to be able to eavesdrop. I’m not sure what the current status of this research is, but there have been a number of proposals over the years to use coupled chaotic oscillators for this purpose. The original idea was (relatively) simple: If two chaotic oscillators have the same parameters, they can be made to synchronize by introducing a driving signal that increases with the difference between the transmitted signal and the computed signal at the receiver. Even small differences in parameters are enough to ruin the synchronization because of the sensitive dependence property of chaotic systems. If you add a low-amplitude signal to the transmitted signal, the receiver will still synchronize to the transmitted chaotic “carrier”. The computed chaotic trajectory at the receiver can be subtracted from the incoming signal, the difference then being the superimposed message. The key to make this work is to share a set of parameters for the chaotic system via a private channel. Easy in principle, but there are lots of technical conditions that have to be met to make this work, and lots of variations to be explored to find the most secure means of encoding the message within the transmitted signal.

The control of a process by an oscillator sometimes also has spatial manifestations. An example of this is provided in the paper on slime-mold aggregation by Gerisch in the Naturwissenschaften top-100 list. When they are well fed, slime molds live as single cells. Starve them, and they aggregate, form a fruiting body, and disperse spores. How do they know where to go during the aggregation process? The answer turns out to involve periodic cyclic AMP (cAMP) signaling. Starving cells put out periodic pulses of cAMP. The cells don’t all signal at the same rate, and they tend to synchronize to and move toward the fastest signaler. Note again the importance of coupled oscillators: This works in part because the cells “listen” to each other’s cAMP signals and adjust the frequency of their own oscillator to match the fastest frequency they “hear”.

Well, those are the things that caught my attention in the Naturwissenschaften list. There is lots of other wonderful stuff in there. I would love to hear what caught everyone else’s fancy.