Biological oscillators

Having abandoned the unreachable goal of surveying the field, even superficially, we have opted to present brief looks at a handful of oscillatory phenomena in biology. Even here, our treatment will only scratch the surface. We suspect that, for the expert, this chapter will be the least satisfying in the book. Nonetheless, we have included it because it may also prove to be the most inspiring chapter for the novice. 

The range of periods of biological oscillators is considerable, as shown in Table 13.1. In this chapter, wc focus on three examples of biological oscillation the activity of neurons polymerization of microtubules and certain pathological conditions, known as dynamical diseases, that arise from changes in natural biological rhythms. With the possible exception of the first topic, these are not 

One of the most widespread forms of biological oscillation involves the behavior of calcium ions. Oscillations in (Ca ) can arise either spontaneously or as the result of stimulation by hormones or neurotransmitters. Although their function has not been completely elucidated, it appears that calcium oscillations and 

Nerve cells, or neurons, are the primary means by which signaling and information transfer occur in living organisms (Levitan and Kaczmarek, 1997). The signals that neurons generate and transmit are electrical in nature moreover, they are often oscillatory. In this section, we take a brief look at how neurons work and how some of the ideas that we have developed for studying chemical oscillators can usefully be applied to understanding neural oscillations. 

The concentration of inside the neuron is significantly greater than the concentration of this species in the surrounding medium, while Na , Ca , and Cl are present in higher concentrations outside the cell than inside. As a result of these concentration differences, which are maintained by a complex pumping machinery, there is a potential, across the membrane. The electrical activity of a neuron depends upon the movement of ions—primarily cations—across the cell membrane. This movement is made possible by the existence of ion channels, specialized proteins embedded in the membrane that allow the transport of ions down the concentration gradient across the otherwise impermeable cell membrane. 

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Until the 1950s, the rare periodic phenomena known in chemistry, such as the reaction of Bray [1], represented laboratory curiosities. Some oscillatory reactions were also known in electrochemistry. The link was made between the cardiac rhythm and electrical oscillators [2]. New examples of oscillatory chemical reactions were later discovered [3, 4]. From a theoretical point of view, the first kinetic model for oscillatory reactions was analyzed by Lotka [5], while similar equations were proposed soon after by Volterra [6] to account for oscillations in predator-prey systems in ecology. The next important advance on biological oscillations came from the experimental and theoretical studies of Hodgkin and Huxley [7], which clarified the physicochemical bases of the action potential in electrically excitable cells. The theory that they developed was later applied [8] to account for sustained oscillations of the membrane potential in these cells. Remarkably, the classic study by Hodgkin and Huxley appeared in the same year as Turing s pioneering analysis of spatial patterns in chemical systems [9]. 

Glass, L., Guevara, M., Belair, J. and Shrier, A., 1984, Global bifurcations of a periodically forced biological oscillator. Phys. Rev. A 29,1348-1357. 

As an extension of the contribution of Dr. Stucki, I would like to bring attention to a rather complex biological oscillator. It deals with the case of invertebrate euryhaline species, clearly a case of integration of a biological system into its environment, that is, what the biologists call adaptation. 

Biology (oscillating neurons, firefly flashing rhythm, human sleep-wake cycle)... 

IIIF) Chay, T. R. A Model for Biological Oscillations. Proc. Natl. Acad. Sci. U.S.A. 78,... 

All the complex behavior described so far in this Chapter arises from the diffusive coupling of the local dynamics which in the homogeneous case have simple fixed points as asymptotic states. If the local dynamics becomes more complex, the range of possible dynamic behavior in the presence of diffusion becomes practically unlimited. It is clear that coupling chaotic subsystems could produce an extremely rich dynamics. But even the case of periodic local dynamics does so. Diffusively coupled chemical or biological oscillators may become synchronized (Pikovsky et ah, 2003), or rather additional instabilities may arise from the spatial coupling. This may produce target waves, spiral patterns, front instabilities and several different types of spatiotemporal chaos or phase turbulence (Kuramoto, 1984). 

R. Mirollo and S. Strogatz. Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math., 50 1645-1662, 1990. 

I. Ozden, S. Venkataramani, M. A. Long, B. W. Connors, and A. V. Nur-mikko. Strong coupling of nonlinear electronic and biological oscillators Reaching the amplitude death regime. Phys. Rev. Lett, 93 158102, 2004. 

A systematic determination of the phase delay (A< <0) or phase advance (A > 0) as a function of the phase of the oscillations yields the phase response curves of fig. 2.19. The two curves were established at different levels of the perturbation in product y. In both cases, a discontinuity occurs as a delay abruptly transforms into a phase advance. In the terminology of Winfree (1980), such phase response curves are of type 0. If the magnitude of the product perturbation were sufficiently small, a very weak phase shift would occur at all phases

phase response curve would then be of type 1. The phase response curves of fig. 2.19 are reminiscent of those observed for a large number of biological oscillations, including circadian rhythms (Winfree, 1980). 

Ca oscillations moreover, the simulations predict that there exists a critical phase where a pulse of precise magnitude can transiently suppress oscillations. Such a transient suppression of the oscillations by a critical perturbation is a general property of chemical and biological oscillators (Winfree, 1980). 

The peroxidase reaction provides another prototype for periodic behaviour and chaos in an enzyme reaction. As noted by Steinmetz et al. (1993), in view of its mechanism based on free radical intermediates, this reaction represents an important bridge between chemical oscillations of the Belousov-Zhabotinsky type, and biological oscillators. In view of the above discussion, it is noteworthy that the model proposed by Olsen (1983), and further analysed by Steinmetz et al. (1993), also contains two parallel routes for the autocatalytic production of a key intermediate species in the reaction mechanism. As shown by experiments and accounted for by theoretical studies, the peroxidase reaction possesses a particularly rich repertoire of dynamic behaviour (Barter et al, 1993) ranging from bistability (Degn, 1968 Degn et al, 1979) to periodic oscillations (Yamazaki et al, 1965 Nakamura et al, 1969 ... 

Friesen, W.O., G.D. Block C.G. Hocker. 1993. Formal approaches to understanding biological oscillators. Annu. Rev. Physiol. 55 661-81. 

Pavlidis, T. 1973. Biological Oscillators Their Mathematical Analysis. Academic Press, New York. 

Natural frequencies of biological oscillations can be found by finding the roots of certain... 

Grasman, J. 1984. The mathematical modeling of entrained biological oscillators. Bull Math. Biol, 46 ... 

It has also been suggested that the assembly of biological oscillators may be modeled as parameter tunnel diode memory system with couplings. The frequencies of such a system can include values very much lower than the design frequency, which thus supports our prediction of mode softening. 

Self-excited oscillations are quite relevant in the context of biological oscillator. Typical self-excited oscillations observed in the case of (i) density oscillator, (ii) liquid-liquid interface oscillator and (iii) liquid-vapour interface oscillator, the mechanism of... 

Table 15.3. Typical Biological Oscillators and their functions.

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