48
B. Pietras and A. Daffertshofer
cf. the phase histograms of the final conditions (insets in Fig. 3.10). But for the smallworld and Hagmann networks, these two techniques converge to the same resulting
behavior. For details about the simulations see the Appendix.
In a nutshell, we can conclude that topology effects overcome otherwise precise
predictions of the phase model such that even the least accurate direct averaging
method does not perform worse than the other techniques.
3.6 Conclusion
Phase reduction is a powerful tool to simplify the dynamics of complex networks of
neural oscillators. The reduced phase model allows a reliable prediction of various
network behavior. Not only the transition between an incoherent state and a fully
synchronized state can be revealed, but, by taking higher harmonics of the reduced
phase interaction function into account, also non-trivial collective behavior can be
forecast such as cluster states or slow-switching between clusters.
There does not, however, exist “the” phase reduction, but one has to choose from
a variety of phase reduction techniques. The different reduction methods can broadly
be classified as numerical and analytical phase reduction techniques. We compared
different phase reduction techniques of oscillatory neural networks and showed that
close to a particular bifurcation boundary, all reduction techniques retrieve the same
qualitative results. Further away from bifurcation boundaries, different techniques
start to diverge and one has to pay careful attention to whether, e.g., an analytically
reduced phase model indeed captures the correct network behavior. We advocate a
combination of numerical and analytical approaches to ensure the sought-for accuracy of the phase model while, at the same time, allowing for a direct mapping
between the parameters of the neural network model with those of the reduced phase
model. By this, one can identify key parameters of the neural network that have major
influence on the synchronization properties of the network.
Furthermore, we showed that phase reduction has important limitations when facing strong coupling and realistic connectivity structure. Although augmented phase
reduction and phase-amplitude reduction techniques for single oscillators have seen
strong advances in recent years [57, 67], their extension to oscillatory networks has
yet to be achieved. We illustrated some peculiar characteristics of coupling-induced
behavior, such as birth and death of oscillations, and highlighted how insights about
the dynamics of two strongly-coupled oscillators can be used to explain network
effects such as clustering into groups of various sizes and quasiperiodic dynamics
on a network level. For small coupling strengths, predictions by the reduced phase
model remain valid. For stronger coupling, the validity of the reduced phase model
breaks down. A reasonably good proxy for a critical coupling strength beyond which
the phase model loses validity, can be obtained from the dynamics of two coupled
identical systems. We hypothesize that this critical coupling strength is exceeded
once the coupling-induced behavior becomes more complex, that is, when identical
initial conditions result in distinct oscillatory dynamics of the two oscillators.
B. Pietras and A. Daffertshofer
cf. the phase histograms of the final conditions (insets in Fig. 3.10). But for the smallworld and Hagmann networks, these two techniques converge to the same resulting
behavior. For details about the simulations see the Appendix.
In a nutshell, we can conclude that topology effects overcome otherwise precise
predictions of the phase model such that even the least accurate direct averaging
method does not perform worse than the other techniques.
3.6 Conclusion
Phase reduction is a powerful tool to simplify the dynamics of complex networks of
neural oscillators. The reduced phase model allows a reliable prediction of various
network behavior. Not only the transition between an incoherent state and a fully
synchronized state can be revealed, but, by taking higher harmonics of the reduced
phase interaction function into account, also non-trivial collective behavior can be
forecast such as cluster states or slow-switching between clusters.
There does not, however, exist “the” phase reduction, but one has to choose from
a variety of phase reduction techniques. The different reduction methods can broadly
be classified as numerical and analytical phase reduction techniques. We compared
different phase reduction techniques of oscillatory neural networks and showed that
close to a particular bifurcation boundary, all reduction techniques retrieve the same
qualitative results. Further away from bifurcation boundaries, different techniques
start to diverge and one has to pay careful attention to whether, e.g., an analytically
reduced phase model indeed captures the correct network behavior. We advocate a
combination of numerical and analytical approaches to ensure the sought-for accuracy of the phase model while, at the same time, allowing for a direct mapping
between the parameters of the neural network model with those of the reduced phase
model. By this, one can identify key parameters of the neural network that have major
influence on the synchronization properties of the network.
Furthermore, we showed that phase reduction has important limitations when facing strong coupling and realistic connectivity structure. Although augmented phase
reduction and phase-amplitude reduction techniques for single oscillators have seen
strong advances in recent years [57, 67], their extension to oscillatory networks has
yet to be achieved. We illustrated some peculiar characteristics of coupling-induced
behavior, such as birth and death of oscillations, and highlighted how insights about
the dynamics of two strongly-coupled oscillators can be used to explain network
effects such as clustering into groups of various sizes and quasiperiodic dynamics
on a network level. For small coupling strengths, predictions by the reduced phase
model remain valid. For stronger coupling, the validity of the reduced phase model
breaks down. A reasonably good proxy for a critical coupling strength beyond which
the phase model loses validity, can be obtained from the dynamics of two coupled
identical systems. We hypothesize that this critical coupling strength is exceeded
once the coupling-induced behavior becomes more complex, that is, when identical
initial conditions result in distinct oscillatory dynamics of the two oscillators.