32
B. Pietras and A. Daffertshofer
Fig. 3.1 Network of three
coupled Wilson-Cowan
neural masses. Each node
contains excitatory and
inhibitory populations, E k
and I k , that are internally
coupled with strengths c i j ,
i, j ∈ {E, I }. Interaction
between two neural masses
k = j occurs via their
respective excitatory
populations, where C k j
denotes the connectivity
whether node k receives
input from node j
within a stable limit cycle) from the actual firing rate E = E(t), we avoid spurious
contributions from other neural masses when they are all synchronized. In general, the interaction within and between different nodes may be time-delayed due
to finite signal transmission of, in particular, long-range connections. Allowing for
time delays between nodes, or for spatial interaction kernels, yields more intricate
coupling dynamics, see, e.g., [47, 73]. For the sake of legibility we here restrict our
analysis to instantaneous interactions. We note, however, that phase reduction can
also be employed in face of time delays [67, Sect. 10.3.2].
Depending on the choice of parameters, the Wilson-Cowan model Eq. (3.1) can
exhibit rich dynamics such as self-sustained oscillations and multi-stability, see,
e.g., [14, 45, 67, 88]. Here, we restrict the parameter values (see Appendix) to the
dynamical regime in which every isolated (κ = 0) node displays stable limit cycle
oscillations. Each point on this stable limit cycle can, in general, be described in terms
of a phase and an amplitude. If the attraction towards the limit cycle is sufficiently
fast, it is possible to ignore the amplitude dynamics so that the one-dimensional phase
variable φ k reliably describes the state of the oscillator not only on the limit cycle,
but also in its close vicinity. Moreover, assuming weak coupling between nodes, we
can then capitalize on the theory of weakly coupled oscillators [32, 45, 67] to extract
the phase dynamics of each node k = 1, . . . , N in form of
˙
φ k = ω k +
κ
N
N
j=1
C k j k − φ j )
(3.2)
with a natural frequency term ω k and a phase interaction function that depends
on the phase difference φ k − φ j between two nodes k = j. The phase interaction
function is typically periodic in ψ and can thus be expanded in a Fourier series:
= a 0 + a 1 cos(ψ) + b 1 sin(ψ) + a 2 cos(2ψ) + b 2 sin(2ψ) + · · ·
(3.3)
B. Pietras and A. Daffertshofer
Fig. 3.1 Network of three
coupled Wilson-Cowan
neural masses. Each node
contains excitatory and
inhibitory populations, E k
and I k , that are internally
coupled with strengths c i j ,
i, j ∈ {E, I }. Interaction
between two neural masses
k = j occurs via their
respective excitatory
populations, where C k j
denotes the connectivity
whether node k receives
input from node j
within a stable limit cycle) from the actual firing rate E = E(t), we avoid spurious
contributions from other neural masses when they are all synchronized. In general, the interaction within and between different nodes may be time-delayed due
to finite signal transmission of, in particular, long-range connections. Allowing for
time delays between nodes, or for spatial interaction kernels, yields more intricate
coupling dynamics, see, e.g., [47, 73]. For the sake of legibility we here restrict our
analysis to instantaneous interactions. We note, however, that phase reduction can
also be employed in face of time delays [67, Sect. 10.3.2].
Depending on the choice of parameters, the Wilson-Cowan model Eq. (3.1) can
exhibit rich dynamics such as self-sustained oscillations and multi-stability, see,
e.g., [14, 45, 67, 88]. Here, we restrict the parameter values (see Appendix) to the
dynamical regime in which every isolated (κ = 0) node displays stable limit cycle
oscillations. Each point on this stable limit cycle can, in general, be described in terms
of a phase and an amplitude. If the attraction towards the limit cycle is sufficiently
fast, it is possible to ignore the amplitude dynamics so that the one-dimensional phase
variable φ k reliably describes the state of the oscillator not only on the limit cycle,
but also in its close vicinity. Moreover, assuming weak coupling between nodes, we
can then capitalize on the theory of weakly coupled oscillators [32, 45, 67] to extract
the phase dynamics of each node k = 1, . . . , N in form of
˙
φ k = ω k +
κ
N
N
j=1
C k j k − φ j )
(3.2)
with a natural frequency term ω k and a phase interaction function that depends
on the phase difference φ k − φ j between two nodes k = j. The phase interaction
function is typically periodic in ψ and can thus be expanded in a Fourier series:
= a 0 + a 1 cos(ψ) + b 1 sin(ψ) + a 2 cos(2ψ) + b 2 sin(2ψ) + · · ·
(3.3)