In a network of coupled oscillators (diffusive coupling) while it is intuitive to expect that the magnitude of coupling increases with increase in concentration of coupling factor (provided the diffusion rates and other factors are held constant)....
1. Would this increase be linear, sigmoid or does the coupling strength follow an inverted U shaped curve with an increase initially and eventual drop?
2. Does higher concentration of coupling factor lead to unstable network, or in other words will the network attain local stability but is easily susceptible to mutual desynchronization by an external agent?
3. Also, is this trend different for mean field coupling?
Fascinating questions. Not sure I can answer them all directly but I can recommend some reading if you are interested in the topic. It turns out there are many people within the circadian field who are interested in questions such as these and have used mathematical models of the circadian system to try and answer them.
1) I would say that it is sigmoid, at least based on mathematical models that test increased coupling strength. This should hold true for your question If you assume that increasing the concentration of a diffusible factor indeed increases coupling between the oscillators (up to the saturation limit of the receptors for that signalling molecule). I am not aware of a scenario where it would be an inverted U, unless there was some weird biphasic response to the molecule itself. Say if there were two receptors, one with high affinity which increased coupling and one with low affinity that decreased or blocked it. I am unaware of such a system for neuropeptide signalling but that doesn't mean one doesn't exist.
2) Theoretically, increased coupling always leads to greater stability and synchonization of the oscillators. However it all depends on zeitgeber strength. If the coupling agent is a weaker zeitgeber and some other external factor is a stronger one, then it is likely that the entire system would be more easily shifted by the external factor. In the case of strong coupling between oscillators, all oscillators should still remain synchronized to each other even if they are perturbed by an outside force, i.e. they would all move together or not at all.The opposite would be true for a weakly coupled system where a strong outside force on some (but not all) cells could desynchronize the system. I think it is safe to say that, in general, increased coupling will always convert local stability into global stability (assuming all cells are equally coupled and are equally perturbed by outside forces of course). The situation changes very quickly and dynamically if you have multiple diffusible synchronizing factors released from different subpopulations within the network. This is much more like the case of the suprachiasmatic nucleus, with multiple diffusible factors all released from spatially distinct subpopulations of cells with both local and long-range connectivity (think small-world models). In general though, this more complicated scheme allows for greater stability and synchronization across the network while still allowing phase-resetting stimuli to entrain the system in a predictable and controlled way.
Abraham, U., Granada, A. E., Westermark, P. O., Heine, M., Kramer, A., & Herzel, H. (2010). Coupling governs entrainment range of circadian clocks. Mol Syst Biol, 6.
Achermann, P., & Kunz, H. (1999). Modeling Circadian Rhythm Generation in the Suprachiasmatic Nucleus with Locally Coupled Self-Sustained Oscillators: Phase Shifts and Phase Response Curves. Journal of Biological Rhythms, 14(6), 460-468.
Ananthasubramaniam, B., Herzog, E. D., & Herzel, H. (2014). Timing of Neuropeptide Coupling Determines Synchrony and Entrainment in the Mammalian Circadian Clock. PLoS Comput Biol, 10(4), e1003565.
Bordyugov, G., Granada, A. E., & Herzel, H. (2011). How coupling determines the entrainment of circadian clocks. The European Physical Journal B, 82(3-4), 227-234.
DeWoskin, D., Geng, W., Stinchcombe, A. R., & Forger, D. B. (2014). It is not the parts, but how they interact that determines the behaviour of circadian rhythms across scales and organisms. Interface Focus, 4(3), 20130076.
Kori, H., Kawamura, Y., & Masuda, N. (2012). Structure of cell networks critically determines oscillation regularity. Journal of Theoretical Biology, 297, 61-72.
Vasalou, C., Herzog, E. D., & Henson, M. A. (2009). Small-World Network Models of Intercellular Coupling Predict Enhanced Synchronization in the Suprachiasmatic Nucleus. Journal of Biological Rhythms, 24(3), 243-254.
Gonze, D., Bernard, S., Waltermann, C., Kramer, A., & Herzel, H. (2005). Spontaneous synchronization of coupled circadian oscillators. Biophys J, 89, 120 - 129.
Fascinating questions. Not sure I can answer them all directly but I can recommend some reading if you are interested in the topic. It turns out there are many people within the circadian field who are interested in questions such as these and have used mathematical models of the circadian system to try and answer them.
1) I would say that it is sigmoid, at least based on mathematical models that test increased coupling strength. This should hold true for your question If you assume that increasing the concentration of a diffusible factor indeed increases coupling between the oscillators (up to the saturation limit of the receptors for that signalling molecule). I am not aware of a scenario where it would be an inverted U, unless there was some weird biphasic response to the molecule itself. Say if there were two receptors, one with high affinity which increased coupling and one with low affinity that decreased or blocked it. I am unaware of such a system for neuropeptide signalling but that doesn't mean one doesn't exist.
2) Theoretically, increased coupling always leads to greater stability and synchonization of the oscillators. However it all depends on zeitgeber strength. If the coupling agent is a weaker zeitgeber and some other external factor is a stronger one, then it is likely that the entire system would be more easily shifted by the external factor. In the case of strong coupling between oscillators, all oscillators should still remain synchronized to each other even if they are perturbed by an outside force, i.e. they would all move together or not at all.The opposite would be true for a weakly coupled system where a strong outside force on some (but not all) cells could desynchronize the system. I think it is safe to say that, in general, increased coupling will always convert local stability into global stability (assuming all cells are equally coupled and are equally perturbed by outside forces of course). The situation changes very quickly and dynamically if you have multiple diffusible synchronizing factors released from different subpopulations within the network. This is much more like the case of the suprachiasmatic nucleus, with multiple diffusible factors all released from spatially distinct subpopulations of cells with both local and long-range connectivity (think small-world models). In general though, this more complicated scheme allows for greater stability and synchronization across the network while still allowing phase-resetting stimuli to entrain the system in a predictable and controlled way.
Abraham, U., Granada, A. E., Westermark, P. O., Heine, M., Kramer, A., & Herzel, H. (2010). Coupling governs entrainment range of circadian clocks. Mol Syst Biol, 6.
Achermann, P., & Kunz, H. (1999). Modeling Circadian Rhythm Generation in the Suprachiasmatic Nucleus with Locally Coupled Self-Sustained Oscillators: Phase Shifts and Phase Response Curves. Journal of Biological Rhythms, 14(6), 460-468.
Ananthasubramaniam, B., Herzog, E. D., & Herzel, H. (2014). Timing of Neuropeptide Coupling Determines Synchrony and Entrainment in the Mammalian Circadian Clock. PLoS Comput Biol, 10(4), e1003565.
Bordyugov, G., Granada, A. E., & Herzel, H. (2011). How coupling determines the entrainment of circadian clocks. The European Physical Journal B, 82(3-4), 227-234.
DeWoskin, D., Geng, W., Stinchcombe, A. R., & Forger, D. B. (2014). It is not the parts, but how they interact that determines the behaviour of circadian rhythms across scales and organisms. Interface Focus, 4(3), 20130076.
Kori, H., Kawamura, Y., & Masuda, N. (2012). Structure of cell networks critically determines oscillation regularity. Journal of Theoretical Biology, 297, 61-72.
Vasalou, C., Herzog, E. D., & Henson, M. A. (2009). Small-World Network Models of Intercellular Coupling Predict Enhanced Synchronization in the Suprachiasmatic Nucleus. Journal of Biological Rhythms, 24(3), 243-254.
Gonze, D., Bernard, S., Waltermann, C., Kramer, A., & Herzel, H. (2005). Spontaneous synchronization of coupled circadian oscillators. Biophys J, 89, 120 - 129.
@Blum: Thank you very much for the detailed answer!! I am aware of the papers you have suggested and in fact some of those studies are the basis for my question.
It is true that in general the increased coupling does stabilize the circadian network and that the dynamics would differ as a function of zeitgeber and other factors, but for simplicity lets consider all else being equal. And for the same reason that you pointed out, I also expect the coupling strength to be a sigmoidal function of coupling factor concentration. For instance, reduced concentration of a certain neuropeptide in drosophila (and of course other mammalian studies) has been shown to increase the incidence of mutual desynchrony in circadian network leading to complex behaviors or deterministic chaos as some would like to call it. Interestingly, I have also come across a study in drosophila where over-expression of the same neuropeptide results in pretty similar phenotypes as that for hypomorphic or null mutants. This is why I was wondering if it can be a an inverted U shaped function.
Of course, the non-homogeneous nature of coupling within a similar tissue like the SCN further complicates the issue. But given the universal nature in certain aspects of coupled systems, I was wondering if one could generalize certain dynamics based on our knowledge from oscillation theory or control systems theory.
I think the key there might be to try and understand what over expression is doing to the signalling. Increased concentration might mean increased function/signalling under most conditions however this drosophila systems can sometimes have weird artifacts, I.e. the protein is so highly expressed it disrupts signalling or otherwise effects cell health. I think you would need to separately increase signalling in some other manner to be sure, e.g receptor over expression or introducing a hypermorphic mutation.
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