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Phase Model
Eugene M. Izhikevich and Bard Ermentrout (2008), Scholarpedia, 3(10):1487. | doi:10.4249/scholarpedia.1487 | revision #129938 [link to/cite this article] |
Coupled oscillators interact via mutual adjustment of their amplitudes and phases. When coupling is weak, amplitudes are relatively constant and the interactions could be described by phase models.

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Phase of oscillation
Many physical, chemical, and biological systems can produce rhythmic oscillations (Winfree 2001), which can be represented mathematically by a nonlinear dynamical system x' = f(x)
The phase of oscillation can also be defined outside \gamma using the notion of isochrons. The change of variables x(t) = \gamma(\vartheta(t)) transforms the nonlinear system in a neighborhood of \gamma into an equivalent but simpler phase model \vartheta' = 1.
The phase of oscillation could also be defined for chaotic systems (Pikovsky et al. 2001) using the observation that many chaotic attractors, as in Rossler oscillator, look like smeared limit cycles.

Weak forcing
The same change of variables transforms a weakly forced oscillator x' = f(x) + \varepsilon s(t)
- WinfreeQ(\vartheta)is a normalized phase response curve (PRC) to infinitesimal pulsed perturbations. That is, one measures PRC of the oscillator x'=f(x) by perturbing each component of state vector x with brief pulses of small amplitude with area of the pulse A\ , and then takes Q=PRC/A in the limit A\rightarrow 0 \ .
- KuramotoQ(\vartheta) = grad\Theta(x)\ , where \Theta(x) is the isochron function defined in a neighborhood of the periodic orbit \gamma\ . That is, one starts from every point x in a neighborhood of \gamma and determines its asymptotic phase, \Theta(x)\ , relative to the phase of the solution starting with x_0\ .
- MalkinQ(\vartheta)is the solution to the adjoint problem dQ/d\vartheta=-\{Df(\gamma(\vartheta))\}^\top Q\;, with the normalization condition Q(\vartheta) \cdot f(\gamma(\vartheta))=1 for any \vartheta\ . That is, one determines the Jacobian matrix Df along the periodic orbit and then solves, usually numerically, the adjoint problem.
Malkin's condition, though least intuitive, is the most useful in applications.
Examples of reduction
The infinitesimal PRC function Q(\vartheta) can be found analytically in a few simple cases.
Phase oscillators
A nonlinear phase oscillator \dot{x} = f(x) with periodic phase variable x \in [0, 1] and f>0 has Q(\vartheta) = 1/f(\gamma(\vartheta))\ . Indeed, the function can be found from Malkin's normalization condition Q(\vartheta) \cdot f(\gamma(\vartheta))=1\ .
SNIC oscillators
A system near saddle-node on invariant circle (SNIC) bifurcation has Q(\vartheta) proportional to 1-\cos^2 \vartheta\ .
Andronov-Hopf oscillators
A system near supercritical Andronov-Hopf bifurcation has Q(\vartheta) proportional to \sin (\vartheta-\psi)\ , where \psi is a constant phase shift.
Other interesting cases
Izhikevich (2000) derived the phase model for weakly coupled relaxation oscillators. Brown et al. (2004) consider other interesting cases, including homoclinic oscillators. Coupled bursters are considered by Izhikevich (2007). Pulse coupled oscillators provide many other analytically solvable examples.
Weakly coupled oscillators
Let us treat s(t) in x' = f(x) + \varepsilon s(t) as the input from the network, and consider weakly coupled oscillators x_i' = f_i(x_i) + \varepsilon \sum_{j=1}^n g_{ij}(x_i, x_j).
The implicit assumption of weak coupling is that the relative position (phase) of the oscillators changes slowly with respect to their motion around the limit cycle (absolute phase). This implies slow convergence to a steady state phase locking. If the coupling is not sufficiently weak but is pulsatile in nature, the methods for pulse-coupled oscillators can be utilized, otherwise there are no general methods.
Phase model
Introducing phase deviation variables \vartheta_i = t + \varphi_i\ , one can transform the system above into the form \varphi_i' = \varepsilon \sum_{j=1}^n h_{ij}(t + \varphi_i, \ t + \varphi_j).
Computational neuroscience provides an important application of phase models. In this case, the state variables x_i and x_j describe activities of the post-synaptic (forced) and pre-synaptic (forcing) periodically firing neurons, and the function g_{ij} describes the time course of synaptic input. The phase variables \varphi_i and \varphi_j describe the timings of firings of the neurons, and the function H_{ij} describes normalized phase resetting curve (Netoff et al 2005).
Analysis
Two coupled oscillators
Consider two mutually coupled oscillators with nearly identical periods \varphi'_1 = 1 + \varepsilon \omega_1 + \varepsilon H_{12}(\varphi_2-\varphi_1)
where \omega = \omega_2 - \omega_1
All equilibria of this system are solutions to H(\chi) = -\omega\ , provided that \omega is small enough. Geometrically, the equilibria are intersections of the horizontal line -\omega with the graph of H\ . They are stable if the slope of the graph is negative at the intersection, i.e., H'(\chi)<0\ . If oscillators are identical, then H(\chi) is an odd function (i.e., H(-\chi)=-H(\chi)), and \chi=0 and \chi=\pi are always equilibria, possibly unstable, corresponding to the in-phase and anti-phase synchronized solutions. The in-phase synchronization of coupled oscillators in the figure is stable because the slope of H (dashed curves) is negative at \chi=0\ . The max and min values of the function H determine the tolerance of the network to the frequency mismatch \omega\ , since there are no equilibria outside this range.
Chains of oscillators
The behavior of chains of phase models is considerably more complex than that of pairs, even for nearest neighbor coupling. The reason for this is that when coupling is local, oscillators at the ends get different inputs from those in the middle so that phase locking may not even exist. However, in a large class of models, chains can be analyzed either by direct calculation or by letting the size of the chains tend to infinity. In the former case, Cohen et al. (1982) examined a linear chain of nearest neighbor oscillators with a frequency gradient: \theta_i' = \omega_i + \sin (\theta_{i+1}-\theta_i) + \sin(\theta_{i-1}-\theta_i).
Networks of oscillators that are not arranged in a ring have "boundaries" that can lead to patterns of phase difference that look like waves. If the coupling is isotropic, the waves take the form of one-dimensional target waves, either originating at the center and propagating symmetrically to the edges, or starting at the edges and propagating to the center. For example, an array of nearest neighbor-coupled oscillators of the form \theta_i' = \sin(\theta_{i+1}-\theta_i - \alpha) + \sin(\theta_{i-1}-\theta_i -\alpha) \quad i=0,\ldots,N
Linear arrays of oscillators
Now consider a network of n>2 weakly all to all coupled oscillators. To determine the existence and stability of synchronized states in the network, we need to study equilibria of the corresponding phase model \varphi_i' = \varepsilon \omega_i + \varepsilon \sum_{j\neq i}^n H_{ij}(\varphi_j-\varphi_i).
Vector \phi=(\phi_1,\ldots,\phi_n) is an equilibrium when 0 = \omega_i + \sum_{j\neq1}^n H_{ij}(\phi_j-\phi_i)
In general, determining the stability of equilibria is a difficult problem. Ermentrout (1992) found a simple sufficient condition. If
- a_{ij} = H_{ij}'(\phi_j-\phi_i) \geq 0\ , and
- the directed graph defined by the matrix a = (a_{ij}) is connected, (i.e., each oscillator is influenced, possibly indirectly, by every other oscillator),
then the equilibrium \phi is neutrally stable, and the corresponding limit cycle x(t+\phi) of the phase model is asymptotically stable.
Another sufficient condition was found by Hoppensteadt and Izhikevich (1997). If the phase model satisfies
- \omega_1=\cdots = \omega_n = \omega (identical frequencies)
- H_{ij}(-\chi) = - H_{ji}(\chi) (pair-wise odd coupling)
for all i and j\ , then the network dynamics converge to a limit cycle. On the cycle, all oscillators have equal frequencies 1+\varepsilon\omega and constant phase deviations. The proof follows from the observation that the phase model is a gradient system in a rotating coordinate system.
2D Arrays of oscillators
Two-dimensional arrays provide a much richer class of dynamics. Consider the two-dimensional analogues of the one-dimensional chain of nearest-neighbor coupled oscillators: \theta_{i,j}' = H_N(\theta_{i+1,j}-\theta_{i,j}) + H_S(\theta_{i-1,j}-\theta_{i,j}) + H_E(\theta_{i,j+1}-\theta_{i,j}) + H_W(\theta_{i,j-1}-\theta_{i,j})
Beyond these cross product patterns, it is possible to find non-trivial spatial patterns which are not a consequence of boundary effects. For example, consider the simple sinusoidal array \theta_{i,j}' = \omega + \sum_{k,l} \sin(\theta_{k,l}-\theta_{i,j})
References
- Brown E., Moehlis J., and Holmes P. (2004) On the phase reduction and response dynamics of neural oscillator populations. Neural Computation, 16:673-715.
- Cohen, A.H., Holmes, P.J. and Rand, R.H., (1982) The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model, J. Mathematical Biology 13:345-369.
- Ermentrout, G. B. (1986) Losing amplitude and saving phase. Lecture Notes in Biomath., 66, Springer, Berlin-New York.
- Ermentrout G. B. (1992) Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators. SIAM Journal on Applied Mathematics 52:1665-1687.
- Glass L. and Mackey M.C. (1988) From Clocks to Chaos. Princeton University Press.
- Hoppensteadt F.C. and Izhikevich E.M. (1997) Weakly Connected Neural Networks. Springer-Verlag, NY
- Izhikevich E.M. (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. The MIT Press.
- Izhikevich E.M. (2000) Phase Equations For Relaxation Oscillators. SIAM Journal on Applied Mathematics, 60:1789-1805
- Kopell, N. and Ermentrout G.B. (1986) Symmetry and phaselocking in chains of weakly coupled oscillators. Comm. Pure Appl. Math. 39: 623-660.
- Kopell N. and Ermentrout G.B. (1990) Phase transitions and other phenomena in chains of coupled oscillators. SIAM J. Appl. Math. 50:1014-1052
- Kuramoto Y. (1984) Chemical Oscillations, Waves, and Turbulence. Springer-Verlag, New York.
- Netoff T. I., Acker C. D., Bettencourt J.C. and White J.A. (2005) Beyond two-cell networks: Experimental measurement of neuronal responses to multiple synaptic inputs. J. Comp. Neurosci. 18:287-295
- Pikovsky A., Rosenblum M., Kurths J. (2001) Synchronization: A Universal Concept in Nonlinear Science. CUP, Cambridge.
- Paullet J. E. and Ermentrout G. B. (1994) Stable rotating waves in two-dimensional discrete active media. SIAM J. Appl. Math. 54: 1720-1744
- Ren L. and Ermentrout G. B. (1998) Monotonicity of phaselocked solutions in chains and arrays of nearest-neighbor coupled oscillators. SIAM J. Math. Anal. 29:208-234
- Winfree A. (2001) The Geometry of Biological Time. Springer-Verlag, New York, second edition.
Internal references
- Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.
- John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
- John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
- Jonathan E. Rubin (2007) Burst synchronization. Scholarpedia, 2(10):1666.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
- Bard Ermentrout (2008) Ermentrout-Kopell canonical model. Scholarpedia, 3(3):1398.
- Eugene M. Izhikevich and Richard FitzHugh (2006) FitzHugh-Nagumo model. Scholarpedia, 1(9):1349.
- Kresimir Josic, Eric T. Shea-Brown, Jeff Moehlis (2006) Isochron. Scholarpedia, 1(8):1361.
- Rodolfo Llinas (2008) Neuron. Scholarpedia, 3(8):1490.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Carmen C. Canavier (2006) Phase response curve. Scholarpedia, 1(12):1332.
- Carmen C. Canavier and Srisairam Achuthan (2007) Pulse coupled oscillators. Scholarpedia, 2(4):1331.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
- Emmanuil E. Shnol (2007) Stability of equilibria. Scholarpedia, 2(3):2770.
- Arkady Pikovsky and Michael Rosenblum (2007) Synchronization. Scholarpedia, 2(12):1459.
- Frank Hoppensteadt (2006) Voltage-controlled oscillations in neurons. Scholarpedia, 1(11):1599.
See Also
Ermentrout-Kopell canonical model, Isochron, Kuramoto model, Periodic orbit, Phase response curve, Pulse coupled oscillators, Relaxation oscillator, Synchronization, Voltage-controlled oscillations in neurons