40
B. Pietras and A. Daffertshofer
of oscillation death for a chain of Wilson-Cowan neural masses [30], see also the
work by Daffertshofer and van Wijk on a (heterogeneous) network of Wilson-Cowan
neural masses [23]. Those coupling-induced effects only occur for reasonably large
coupling strengths, and a straightforward identification of the phase dynamics as
within the theory of weak coupling is not possible. While sufficiently weak coupling
ensures that the shape and the frequency of the limit-cycle orbits remain almost
unchanged, strong coupling leads to non-negligible amplitude effects. These can
destabilize synchronized states, quench oscillations, or cause collective chaos, and
a phase reduction has only been proposed for quite restrictive assumptions; see [52]
and the references therein. Hence, phase-amplitude reductions [19, 78, 87], see also
[57, 67] for reviews, have to be employed that also take interactions between phase
and amplitude dynamics into account. The theory of weakly coupled oscillators
additionally requires that the actual trajectories of the oscillators are always close to
the isolated limit-cycle solution. While reduction methods exist that allow for a phase
reduction farther away from the underlying periodic orbit, see, e.g., [57, 67, 89], we
here try to answer the question whether appropriate conventional phase models can
still capture coupling-induced collective dynamics.
Oscillation birth and clustering
To investigate coupling-induced behavior, it appears illustrative to start with two
coupled identical Wilson-Cowan neural masses. In Fig. 3.4 we show the bifurcation
diagram with respect to the coupling strength. Without coupling, κ = 0, the dynamics Eq. (3.1) feature only one stable stationary solution (black solid line). Increasing
the coupling strength induces oscillations through a (double) Hopf bifurcation (red
dot). The critical coupling strength κ H can also be determined analytically, see the
Appendix but also [4]. In our example, it is considerably small with κ H = 0.00531
(note that we did not rescale the coupling by a factor 1/N ). In this coupling-induced
oscillatory regime, the initial conditions can have a major impact on the resulting
dynamics. For small coupling strengths κ < 0.6 (see green dot), the two WilsonCowan neural masses evolve from any initial conditions either into the same limit
cycles or into the low activity resting state (blue solid curve). For larger coupling
strengths, however, only identical initial conditions result into the same (red) limit
cycles. Different initial conditions for the two coupled neural masses may still lead
to stable oscillations, but the respective limit cycles of each neural mass can differ
in amplitude and shape (cf. the green curves in Fig. 3.4). Moreover, these distinct
oscillations that resulted from distinct initial conditions are stable beyond a critical
coupling strength at which those oscillations from identical initial conditions have
ceased to exist (through a fold bifurcation of limit cycles, see the inset in Fig. 3.4).
From the point of view of oscillation quenching mechanisms [50], the onset of (identical) limit-cycle oscillations of the two Wilson-Cowan neural masses for small coupling strengths is a mechanism inverse to amplitude death. Larger coupling induces
a symmetry breaking from two identical to two distinct limit cycles, and thus drives
the system into an oscillation death-related regime. Note, however, that the WilsonCowan neural masses keep oscillating around the same, spatially uniform center,
B. Pietras and A. Daffertshofer
of oscillation death for a chain of Wilson-Cowan neural masses [30], see also the
work by Daffertshofer and van Wijk on a (heterogeneous) network of Wilson-Cowan
neural masses [23]. Those coupling-induced effects only occur for reasonably large
coupling strengths, and a straightforward identification of the phase dynamics as
within the theory of weak coupling is not possible. While sufficiently weak coupling
ensures that the shape and the frequency of the limit-cycle orbits remain almost
unchanged, strong coupling leads to non-negligible amplitude effects. These can
destabilize synchronized states, quench oscillations, or cause collective chaos, and
a phase reduction has only been proposed for quite restrictive assumptions; see [52]
and the references therein. Hence, phase-amplitude reductions [19, 78, 87], see also
[57, 67] for reviews, have to be employed that also take interactions between phase
and amplitude dynamics into account. The theory of weakly coupled oscillators
additionally requires that the actual trajectories of the oscillators are always close to
the isolated limit-cycle solution. While reduction methods exist that allow for a phase
reduction farther away from the underlying periodic orbit, see, e.g., [57, 67, 89], we
here try to answer the question whether appropriate conventional phase models can
still capture coupling-induced collective dynamics.
Oscillation birth and clustering
To investigate coupling-induced behavior, it appears illustrative to start with two
coupled identical Wilson-Cowan neural masses. In Fig. 3.4 we show the bifurcation
diagram with respect to the coupling strength. Without coupling, κ = 0, the dynamics Eq. (3.1) feature only one stable stationary solution (black solid line). Increasing
the coupling strength induces oscillations through a (double) Hopf bifurcation (red
dot). The critical coupling strength κ H can also be determined analytically, see the
Appendix but also [4]. In our example, it is considerably small with κ H = 0.00531
(note that we did not rescale the coupling by a factor 1/N ). In this coupling-induced
oscillatory regime, the initial conditions can have a major impact on the resulting
dynamics. For small coupling strengths κ < 0.6 (see green dot), the two WilsonCowan neural masses evolve from any initial conditions either into the same limit
cycles or into the low activity resting state (blue solid curve). For larger coupling
strengths, however, only identical initial conditions result into the same (red) limit
cycles. Different initial conditions for the two coupled neural masses may still lead
to stable oscillations, but the respective limit cycles of each neural mass can differ
in amplitude and shape (cf. the green curves in Fig. 3.4). Moreover, these distinct
oscillations that resulted from distinct initial conditions are stable beyond a critical
coupling strength at which those oscillations from identical initial conditions have
ceased to exist (through a fold bifurcation of limit cycles, see the inset in Fig. 3.4).
From the point of view of oscillation quenching mechanisms [50], the onset of (identical) limit-cycle oscillations of the two Wilson-Cowan neural masses for small coupling strengths is a mechanism inverse to amplitude death. Larger coupling induces
a symmetry breaking from two identical to two distinct limit cycles, and thus drives
the system into an oscillation death-related regime. Note, however, that the WilsonCowan neural masses keep oscillating around the same, spatially uniform center,