3 Reduced Phase Models of Oscillatory Neural Networks
31
one always ought to keep in mind the limitations and range of applicability of an
accurately reduced phase model.
In the following, we will demonstrate these aspects in more detail. We will guide
our presentation along the example of a network of interacting Wilson-Cowan neural
masses, which will be introduced in Sect. 3.2. We will present different phase reduction techniques for a network of weakly coupled oscillators in Sect. 3.3 and show
that they may indeed result in different predictions about the collective dynamics.
We reported these results previously in an extensive review on network dynamics
and phase reduction techniques [67]. Here we add to this by highlighting possible
limitations of phase reduction for oscillatory networks in Sects. 3.4 and 3.5. Particular focus lies on network topologies, especially when considering a realistic brain
network connectivity structure, and on coupling strengths beyond the weakly perturbed paradigm. We demonstrate why in these cases a reduced phase model may
(not) provide valuable information about the actual network dynamics.
3.2 Networks of Wilson-Cowan Neural Masses
Considering large-scale oscillatory brain networks, the elementary network components, or nodes, can be assumed to be neural populations consisting of a large number
of neurons. From the variety of neural population, or neural mass, models, the seminal Wilson-Cowan neural mass model [88] serves as an exquisite example to derive
the phase dynamics in great detail. The Wilson-Cowan model describes the dynamics of the mean firing rates of neuronal populations. At every node k = 1, . . . , N of
the network, we placed properly balanced pairs of excitatory and inhibitory populations with mean firing rates E k = E k (t) and I k = I k (t), respectively. The nodes
are coupled to other nodes through the connections between their excitatory populations [21, 23, 76]. The connection weigths are typically given by a coupling matrix
C = {C jk } j,k=1,...,N . We illustrate the basic structure of this network in Fig. 3.1.
The dynamics at node k takes on the form
˙
E k = −E k + S
⎡
⎣ a E
⎛
⎝ c E E E k − c E I I k − E +
κ
N
N
j=1
C k j g(E j )
⎞
⎠
⎤
⎦
˙
I k = −I k + S [a I (c I E E k − c I I I k − I )] .
(3.1)
The function S[x] = (1 + e
−x
)
−1 is a sigmoid function with thresholds E and
I that need to be exceeded by the total input into neural mass k to elicit firing; the parameters a E and a I describe the slopes of the sigmoids. The constants
c E E , c E I , c I E , c I I quantify the coupling strengths within each (E/I ) node and κ 1
scales the coupling between different nodes. Pairwise interaction between different nodes is mediated through the coupling function g(E), which we choose as
g(E) = E(t) − E
0 . By subtracting an average E
0 (typically, the unstable fixed point
31
one always ought to keep in mind the limitations and range of applicability of an
accurately reduced phase model.
In the following, we will demonstrate these aspects in more detail. We will guide
our presentation along the example of a network of interacting Wilson-Cowan neural
masses, which will be introduced in Sect. 3.2. We will present different phase reduction techniques for a network of weakly coupled oscillators in Sect. 3.3 and show
that they may indeed result in different predictions about the collective dynamics.
We reported these results previously in an extensive review on network dynamics
and phase reduction techniques [67]. Here we add to this by highlighting possible
limitations of phase reduction for oscillatory networks in Sects. 3.4 and 3.5. Particular focus lies on network topologies, especially when considering a realistic brain
network connectivity structure, and on coupling strengths beyond the weakly perturbed paradigm. We demonstrate why in these cases a reduced phase model may
(not) provide valuable information about the actual network dynamics.
3.2 Networks of Wilson-Cowan Neural Masses
Considering large-scale oscillatory brain networks, the elementary network components, or nodes, can be assumed to be neural populations consisting of a large number
of neurons. From the variety of neural population, or neural mass, models, the seminal Wilson-Cowan neural mass model [88] serves as an exquisite example to derive
the phase dynamics in great detail. The Wilson-Cowan model describes the dynamics of the mean firing rates of neuronal populations. At every node k = 1, . . . , N of
the network, we placed properly balanced pairs of excitatory and inhibitory populations with mean firing rates E k = E k (t) and I k = I k (t), respectively. The nodes
are coupled to other nodes through the connections between their excitatory populations [21, 23, 76]. The connection weigths are typically given by a coupling matrix
C = {C jk } j,k=1,...,N . We illustrate the basic structure of this network in Fig. 3.1.
The dynamics at node k takes on the form
˙
E k = −E k + S
⎡
⎣ a E
⎛
⎝ c E E E k − c E I I k − E +
κ
N
N
j=1
C k j g(E j )
⎞
⎠
⎤
⎦
˙
I k = −I k + S [a I (c I E E k − c I I I k − I )] .
(3.1)
The function S[x] = (1 + e
−x
)
−1 is a sigmoid function with thresholds E and
I that need to be exceeded by the total input into neural mass k to elicit firing; the parameters a E and a I describe the slopes of the sigmoids. The constants
c E E , c E I , c I E , c I I quantify the coupling strengths within each (E/I ) node and κ 1
scales the coupling between different nodes. Pairwise interaction between different nodes is mediated through the coupling function g(E), which we choose as
g(E) = E(t) − E
0 . By subtracting an average E
0 (typically, the unstable fixed point