### Biological Oscillators: their Mathematical Analysis

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## Qualitative similarities between the behavior of coupled oscillators and circadian rhythms

Free Shipping Free global shipping No minimum order. Preface Acknowledgments Chapter 1. Fundamentals of the Mathematical Theory of Oscillators 1. Examples of Biological Rhythms 2. Entrainment of Oscillators by External Inputs 4. The Dynamics of Circadian Oscillators 5. Populations of Interacting Oscillators 7. Biological Phenomena Attributable to Populations of Oscillators 8. Powered by. You are connected as. Connect with:. Use your name:. Thank you for posting a review!

We value your input. Share your review so everyone else can enjoy it too. However, even in the absence of noise, the introduction of random perturbations allows the extraction of information about the deterministic system. This is done by letting the perturbation amplitudes go to zero. To investigate the dependence of the synchronization frequency on the frequencies of the coupled oscillators, and the different coupling parameters, we assume synchronization, which reduces the FPE equation to an equation in two variables.

We divide this section in two parts. First, we consider a simple deterministic system in which the effect of coupling can be understood. In the second part, we randomly perturb a more general version of the previous model to show that the FP equation provides an approximation for the syncrhonization frequency, and obtain some insights on the effect of noise.

Let us then consider a system similar to the one already studied. This system has the advantage of allowing direct calculations around the limit cycle, which can be written explicitly. The resulting linear system is. This assumption may not always be biologically realistic, but it allows us to obtain the common synchronization frequency in a simple way.

Later on we consider the general case and recover this formula as a particular case. This provides an estimate for the synchronization frequency of. That is, when. Otherwise, exponentially large growth can be expected. Notice that unless the a ij are equal, the previous reasoning is not consistent and no conclusion can be drawn. We claim that introducing random perturbations and using the FP equation allows us to circumvent this difficulty and analyze the general case.

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This is the content of what follows. First of all, we write the system in the form. The probability density, u x 1 , …, x n , y 1 , …, y n , t of the system being in the state x 1 , …, x n , y 1 , …, y n at time t satisfies the FP Equation. Looking for stationary solutions, i. In turn, interpreting these as curves in the phase portrait, the resulting solutions would correspond to periodic trajectories with frequency.

Importantly, it shows that the synchronization frequency decreases with the coupling strength. In particular, formula Equation 12 can be used to study how coupled ultradian oscillations can give rise to circadian oscillations Figure 4. Figure 4. It is reasonable to conjecture from this result, that there is a functional limit in the coupling strength for oscillating tissues in nature above which the tissue oscillations dies. To the best of our knowledge, it is the first time that diffusive coupling has been shown to be able to induce such oscillator death.

We have also derived an estimation for the synchronization frequency of a linearly coupled network of non-linear oscillators in terms of the oscillator natural frequencies and the coupling parameters [Equation 12 ]. The presented results are indeed local, that is, the synchronous oscillation is only locally asymptotically stable. The oscillators are not synchronized at the beginning of the simulations, but their spread in state space is very small to ensure convergence to the synchronous oscillation.

For a larger spread, a more complex behavior is observed. We believe that these results constitute predictions that, although possibly difficult to test experimentally, would be worth verifying in light of the existing evidence about the joint frequency modulation of activity between different tissues during the day [ 23 , 45 ]. The results we have presented thus far emphasize the importance of simple mathematical models in understanding situations where synchronization of multiple oscillating populations appears. The results presented here may help to shed light on both physiological and pathological phenomena involving synchronization of oscillators in different tissues Parkinson's disease [ 46 , 47 ], epilepsy [ 48 , 49 ].

The other way around, it is also of potential importance to unravel mechanisms underlying the disappearance of coordinated oscillatory regimes. In a future publication, we plan to formally justify our estimations, and further, integrate the analysis of oscillations in the cellular and network levels of biological organization, to build up on our understanding of coupling oscillators at the tissue level.

## Biological Oscillators in Nanonetworks—Opportunities and Challenges

Two important extensions of the current models that we are studying are, the full characterization in higher codimension of the bifurcation structures of the system 1a , and also, replacements of the van der Pol dynamics with biophysical models of excitable cells [ 50 ]. This last extension may prove useful to explain possible compensatory mechanisms that take place during the beginning of a pathology [ 51 ].

All authors wrote the paper. All authors performed the analysis. AF and MH-V performed simulations. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The authors would like to thank Beatriz Fuentes-Pardo for her input in discussions about the modeling and its connections with the experimental work she has done. We would also like to thank Carolina Barriga-Montoya for her work on the Fokker-Planck equations and her support with the modeling.

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## Biological Oscillators in Nanonetworks—Opportunities and Challenges

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In: Guckenheimer J, Holmes P, editors. New York, NY: Springer , p. Theory and Applications of Hopf Bifurcation. Cambridge: CUP Archive Singularities and Groups in Bifurcation Theory.