3 Reduced Phase Models of Oscillatory Neural Networks
51
tion of the attracting high-activity resting state, and of the attracting limit-cycles,
respectively, and found the same behavior for a range of different random selections of initial conditions. We simulated Eq. (3.1) over T = 1000s with a time-step
dt = 0.0005s using an Euler-forward scheme. Simulations using a Runge-Kutta
fourth order scheme resulted in the same dynamics. We extracted the phases from
each Wilson-Cowan node (E k , I k ) using the function θ k = atan2(x k , y k ), where
x k = E k − E
0 and y k = I k − I
0 denote the deviations from the unstable fixed point.
The degree of synchronization is measured as phase coherence in terms of the realvalued Kuramoto order parameter, |z(t)| = |
k exp(iθ k (t))|. Both the phase values
as well as the Kuramoto order parameter have been extracted on a coarser time scale
with dt = 0.1s.
For the network simulations in Fig. 3.9, we simulated the dynamics Eq. (3.1) for
N = 66 identical Wilson-Cowan neural masses with coupling strength κ = 0.15 over
T = 2000s with a time step of dt = 0.0005s using an Euler-forward scheme, and
extracted the phases and the Kuramoto order parameter as described above. We used
a moving average of 20s to better compare the evolution of the degree of synchronization for the different networks. We also simulated the corresponding phase dynamics
Eq. (3.2) of the reduced phase models on the different network structures and in the
three dynamical regimes. The Fourier coefficients for the respective phase models
are listed in Tables 3.3, 3.4 and 3.5. Since amplitude effects cannot occur in the phase
model, we set the coupling strength to κ = 0.25, to accelerate possible synchronization transitions and to better identify transient dynamics. Moreover, we increased the
network size for full and small-world network connectivity to N = 200 to reduce
finite-size effects. The phase dynamics were simulated for T = 10000s with a time
step dt = 0.001s using an Euler-forward scheme. We computed the Kuramoto order
parameter for each time step dt = 0.1s and showed the final phase distribution in a
histogram plot with 31 bins.
References
1. G. Ansmann, R. Karnatak, K. Lehnertz, U. Feudel, Extreme events in excitable systems and
mechanisms of their generation. Phys. Rev. E 88, 052911 (2013)
2. G. Ansmann, K. Lehnertz, U. Feudel, Self-induced switchings between multiple space-time
patterns on complex networks of excitable units. Phys. Rev. X 6, 011030 (2016)
3. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Synchronization in complex networks. Phys. Rep. 469(3), 93 (2008)
4. D.G. Aronson, G.B. Ermentrout, N. Kopell, Amplitude response of coupled oscillators. Phys.
D 41(3), 403 (1990)
5. P. Ashwin, S. Coombes, R. Nicks, Mathematical frameworks for oscillatory network dynamics
in neuroscience. J. Math. Neurosci. 6(1), 2 (2016)
6. M. Barahona, L.M. Pecora, Synchronization in small-world systems. Phys. Rev. Lett. 89(5),
054101 (2002)
7. A. Barrat, M. Barthélemy, A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University Press, Cambridge, 2008)
8. D.S. Bassett, E.T. Bullmore, Small-world brain networks revisited. Neuroscientist 23(5), 499
(2016)
51
tion of the attracting high-activity resting state, and of the attracting limit-cycles,
respectively, and found the same behavior for a range of different random selections of initial conditions. We simulated Eq. (3.1) over T = 1000s with a time-step
dt = 0.0005s using an Euler-forward scheme. Simulations using a Runge-Kutta
fourth order scheme resulted in the same dynamics. We extracted the phases from
each Wilson-Cowan node (E k , I k ) using the function θ k = atan2(x k , y k ), where
x k = E k − E
0 and y k = I k − I
0 denote the deviations from the unstable fixed point.
The degree of synchronization is measured as phase coherence in terms of the realvalued Kuramoto order parameter, |z(t)| = |
k exp(iθ k (t))|. Both the phase values
as well as the Kuramoto order parameter have been extracted on a coarser time scale
with dt = 0.1s.
For the network simulations in Fig. 3.9, we simulated the dynamics Eq. (3.1) for
N = 66 identical Wilson-Cowan neural masses with coupling strength κ = 0.15 over
T = 2000s with a time step of dt = 0.0005s using an Euler-forward scheme, and
extracted the phases and the Kuramoto order parameter as described above. We used
a moving average of 20s to better compare the evolution of the degree of synchronization for the different networks. We also simulated the corresponding phase dynamics
Eq. (3.2) of the reduced phase models on the different network structures and in the
three dynamical regimes. The Fourier coefficients for the respective phase models
are listed in Tables 3.3, 3.4 and 3.5. Since amplitude effects cannot occur in the phase
model, we set the coupling strength to κ = 0.25, to accelerate possible synchronization transitions and to better identify transient dynamics. Moreover, we increased the
network size for full and small-world network connectivity to N = 200 to reduce
finite-size effects. The phase dynamics were simulated for T = 10000s with a time
step dt = 0.001s using an Euler-forward scheme. We computed the Kuramoto order
parameter for each time step dt = 0.1s and showed the final phase distribution in a
histogram plot with 31 bins.
References
1. G. Ansmann, R. Karnatak, K. Lehnertz, U. Feudel, Extreme events in excitable systems and
mechanisms of their generation. Phys. Rev. E 88, 052911 (2013)
2. G. Ansmann, K. Lehnertz, U. Feudel, Self-induced switchings between multiple space-time
patterns on complex networks of excitable units. Phys. Rev. X 6, 011030 (2016)
3. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Synchronization in complex networks. Phys. Rep. 469(3), 93 (2008)
4. D.G. Aronson, G.B. Ermentrout, N. Kopell, Amplitude response of coupled oscillators. Phys.
D 41(3), 403 (1990)
5. P. Ashwin, S. Coombes, R. Nicks, Mathematical frameworks for oscillatory network dynamics
in neuroscience. J. Math. Neurosci. 6(1), 2 (2016)
6. M. Barahona, L.M. Pecora, Synchronization in small-world systems. Phys. Rev. Lett. 89(5),
054101 (2002)
7. A. Barrat, M. Barthélemy, A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University Press, Cambridge, 2008)
8. D.S. Bassett, E.T. Bullmore, Small-world brain networks revisited. Neuroscientist 23(5), 499
(2016)