34
B. Pietras and A. Daffertshofer
into a phase model Eq. (3.2). As the form of the Wilson-Cowan model Eq. (3.1)
does not comply with that of Eq. (3.5), it is compulsory to first transform the original dynamics appropriately. Only then we can employ standard methods [67] to
retrieve the phase model Eq. (3.2). We note that the “pre-processing” step can be
achieved either numerically or analytically [67, Sects. 4 and 6], giving rise to classifying different phase reduction approaches as either analytic or numerical reduction
techniques.
Numerical approaches tend to be more accurate. But their software implementation often computes the necessary properties for phase reduction internally, which
leaves the link between the original and the phase model parameters unclear. By contrast, analytic approaches build on subsequent algebraic transformations that yield a
rigorous representation of the phase model parameters in terms of the original parameters, although such a representation may become convoluted. The pre-processing of
the full dynamics Eq. (3.1) into Eq. (3.5) is based on the idea that close to a particular
bifurcation, different models exhibit similar dynamics. Given that the Wilson-Cowan
model Eq. (3.1) exhibits oscillatory behavior close to a Hopf bifurcation, we thus
aim at transforming Eq. (3.1) into the simplest model that captures the essence of
the dynamics close to a Hopf bifurcation point. This simplest model is called Hopf
normal form and we can obtain it with a so-called normal form, or center manifold, reduction [45, 59, 67]. There is, however, a caveat. Since we consider not only
a single isolated neural oscillator but a network of coupled neural oscillators, we
require a network Hopf normal form: That is, not only the uncoupled part f (x) in
Eq. (3.5) has to be brought into Hopf normal form, but also the coupling function
g k (x 1 , . . . , x N ) has to be identified accordingly subject to all symmetry constraints
that are inherent to multiple Hopf bifurcations. Although there is a mathematical
proof for such a network Hopf normal form, to the best of our knowledge, no general
and exact algorithm for deriving it is at hand. Instead, two approximative schemes
have proven fruitful to retain the simplified network Hopf normal form: Kuramoto’s
reductive perturbation approach and Poincaré’s nonlinear transform approach, for
details we refer to [67].
Stepping over these often laborious algebraic transforms, there is an alternative
analytic approach which becomes exact for weakly coupled oscillators that follow a
circular limit cycle: Haken’s reduction via averaging [41, 67], see also [13, 40]. For
planar oscillatory dynamics close to a Hopf bifurcation, the Jacobian of the uncoupled
Wilson-Cowan dynamics Eq. (3.1) evaluated at the unstable fixed point (E
0
k , I
0
k ) has
a pair of complex conjugate eigenvalues λ ± = μ ± iω with positive real part, μ > 0,
which corresponds to the distance to the Hopf bifurcation point.
1 We then express
the dynamics in terms of the deviations x k =
E k − E
0
k (μ), I k − I
0
k (μ)
around the
unstable fixed points. Approximating the sigmoidal activation function S up to third
order and applying some laborious algebraic transforms [67], one can derive a fairly
generic form of the dynamics Eq. (3.1) that reads
1 Typically, one measures this distance in parameter space, e.g., in parameter E such that μ =
E − H
E , where the Hopf bifurcation occurs at E = H
E .
B. Pietras and A. Daffertshofer
into a phase model Eq. (3.2). As the form of the Wilson-Cowan model Eq. (3.1)
does not comply with that of Eq. (3.5), it is compulsory to first transform the original dynamics appropriately. Only then we can employ standard methods [67] to
retrieve the phase model Eq. (3.2). We note that the “pre-processing” step can be
achieved either numerically or analytically [67, Sects. 4 and 6], giving rise to classifying different phase reduction approaches as either analytic or numerical reduction
techniques.
Numerical approaches tend to be more accurate. But their software implementation often computes the necessary properties for phase reduction internally, which
leaves the link between the original and the phase model parameters unclear. By contrast, analytic approaches build on subsequent algebraic transformations that yield a
rigorous representation of the phase model parameters in terms of the original parameters, although such a representation may become convoluted. The pre-processing of
the full dynamics Eq. (3.1) into Eq. (3.5) is based on the idea that close to a particular
bifurcation, different models exhibit similar dynamics. Given that the Wilson-Cowan
model Eq. (3.1) exhibits oscillatory behavior close to a Hopf bifurcation, we thus
aim at transforming Eq. (3.1) into the simplest model that captures the essence of
the dynamics close to a Hopf bifurcation point. This simplest model is called Hopf
normal form and we can obtain it with a so-called normal form, or center manifold, reduction [45, 59, 67]. There is, however, a caveat. Since we consider not only
a single isolated neural oscillator but a network of coupled neural oscillators, we
require a network Hopf normal form: That is, not only the uncoupled part f (x) in
Eq. (3.5) has to be brought into Hopf normal form, but also the coupling function
g k (x 1 , . . . , x N ) has to be identified accordingly subject to all symmetry constraints
that are inherent to multiple Hopf bifurcations. Although there is a mathematical
proof for such a network Hopf normal form, to the best of our knowledge, no general
and exact algorithm for deriving it is at hand. Instead, two approximative schemes
have proven fruitful to retain the simplified network Hopf normal form: Kuramoto’s
reductive perturbation approach and Poincaré’s nonlinear transform approach, for
details we refer to [67].
Stepping over these often laborious algebraic transforms, there is an alternative
analytic approach which becomes exact for weakly coupled oscillators that follow a
circular limit cycle: Haken’s reduction via averaging [41, 67], see also [13, 40]. For
planar oscillatory dynamics close to a Hopf bifurcation, the Jacobian of the uncoupled
Wilson-Cowan dynamics Eq. (3.1) evaluated at the unstable fixed point (E
0
k , I
0
k ) has
a pair of complex conjugate eigenvalues λ ± = μ ± iω with positive real part, μ > 0,
which corresponds to the distance to the Hopf bifurcation point.
1 We then express
the dynamics in terms of the deviations x k =
E k − E
0
k (μ), I k − I
0
k (μ)
around the
unstable fixed points. Approximating the sigmoidal activation function S up to third
order and applying some laborious algebraic transforms [67], one can derive a fairly
generic form of the dynamics Eq. (3.1) that reads
1 Typically, one measures this distance in parameter space, e.g., in parameter E such that μ =
E − H
E , where the Hopf bifurcation occurs at E = H
E .