Phase model

From Scholarpedia

This article has not been peer-reviewed or accepted for publication yet; It may be unfinished, contain inaccuracies, or unapproved changes.

Author: Dr. Eugene M. Izhikevich, The Neurosciences Institute, San Diego, California
Author: Dr. Bard Ermentrout, Dept of Mathematics, Univ Pittsburgh, Pittsburgh PA

Figure 1: Phase of oscillation of the FitzHugh-Nagumo model with I=0.5.

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.

1 Phase of oscillation
2 Weak forcing
3 Examples of reduction
4 Weakly coupled oscillators
5 Phase model
6 Analysis
7 History
8 References
9 See Also

[edit]

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

$x' = f(x)$

having a periodic orbit $\gamma$ . Let $x_0$ be an arbitrary point on $\gamma$ , then any other point on the periodic orbit can be characterized by the time, $\vartheta$ , since the last passing of $x_0$ ; see Fig.1. The variable $\vartheta$ is called phase of oscillation, and it is bounded by the period of oscillation $T$ . The phase is often normalized by $T$ or $T/2\pi$ , so that it is bounded by $1$ or $2\pi$ , respectively.

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

$\vartheta' = 1.$

Such a change of variables removes the amplitude but saves the phase of oscillation (Ermentrout 1986). It is often convenient to assume that the phase $\vartheta$ is defined on the unit circle $S^1$ .

Figure 2: Examples of function $Q=(Q_1, Q_2)$ for Andronov-Hopf oscillator $z'=(1+i)z-z|z|^2$ and van der Pol oscillator $x'=x-x^3-y$ , $y'=x$ ; modified from Izhikevich (2007).

[edit]

Weak forcing

The same change of variables transforms a weakly forced oscillator

$x' = f(x) + \varepsilon s(t)$

into the phase model of the form

$\vartheta' = 1 + \varepsilon Q(\vartheta) \cdot s(t),$

where, the term $\varepsilon s(t)$ denotes a weak time-dependent (and possibly $x$ -dependent) input, e.g., from other oscillators in a network; the dot, " $\cdot$ ", denotes the scalar (dot) product of two vectors; The function $Q(\vartheta)$ , illustrated in the Fig.2, is called linear response function, sensitivity function, or infinitesimal PRC. It satisfies three equivalent conditions (Izhikevich 2007):

Winfree: $Q(\vartheta)$ 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 $A$ , and then takes $Q=$ PRC $/A$ in the limit $A\rightarrow\infty$ .
Kuramoto: $Q(\vartheta) =$ 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 $x_0$ .
Malkin: $Q(\vartheta)$ 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.

[edit]

Examples of reduction

The infinitesimal PRC function $Q(\vartheta)$ can be found analytically in a few simple cases.

[edit]

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 $Q(\vartheta) \cdot f(\gamma(\vartheta))=1$ .

[edit]

SNIC oscillators

A system near saddle-node on invariant circle (SNIC) bifurcation has $Q(\vartheta)$ proportional to $1-\cos \vartheta$ .

[edit]

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.

[edit]

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).

[edit]

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 corresponding phase model

$\vartheta_i' = 1 + \varepsilon \, Q_i(\vartheta_i) \cdot \sum_{j=1}^n g_{ij}(\gamma_i(\vartheta_i), \gamma_j(\vartheta_j)),$

has the form

$\vartheta_i' = 1 + \varepsilon \sum_{j=1}^n h_{ij}(\vartheta_i, \vartheta_j),$

where $h_{ij} = Q_i g_{ij}$ describes the influence of phase of the j-th oscillator on the i-th oscillator, and each $\vartheta_i$ has its own period $T_i$ .

[edit]

Phase model

Introducing phase deviation variables $\vartheta_i = t + \varphi_i$ , one can transform the system above into the form

$\varphi_i' = \varepsilon \sum_{j=1}^n h_{ij}(t + \varphi_i, \ t + \varphi_j).$

This system can be averaged to the phase model

$\varphi_i' = \varepsilon \sum_{j=1}^n H_{ij}(\varphi_j-\varphi_i)\;,$

where each function

$H_{ij}(\chi) = \lim_{T\rightarrow \infty} \frac{1}{T} \int_0^T h_{ij}(t, \ t +\chi)\, dt$

describes the interaction between oscillators. This function is just a constant unless the oscillators have nearly resonant periods, i.e., the ratio $T_i/T_j$ is $\varepsilon$ -close to a low-order rational number $p/q$ ( $p+q$ is small). Since the dynamics of two coupled non-resonant oscillators is described by an uncoupled phase model ( $H=$ const), such oscillators do not interact. That is, the phase of one of them cannot change the phase of the other one even on the long time scale of order $1/\varepsilon$ .

[edit]

Analysis

[edit]

Two coupled oscillators

Consider two mutually coupled oscillators with nearly identical periods

$\varphi'_1 = 1 + \varepsilon \omega_1 + \varepsilon H_{12}(\varphi_2-\varphi_1)$

$\varphi'_2 = 1 + \varepsilon \omega_2 + \varepsilon H_{21}(\varphi_1-\varphi_2)\;,$

where $\omega_i = H_{ii}(0)$ are small frequency deviations. Let $\chi = \varphi_2-\varphi_1$ denote the phase difference between the oscillators, then

$\chi' = \varepsilon \omega + \varepsilon H(\chi),$

Figure 3: Examples of the connection function H; modified from Izhikevich (2007).

where

$\omega = \omega_2 - \omega_1$

and $H(\chi) = H_{21}(-\chi)-H_{12}(\chi),$ is the frequency mismatch and the anti-symmetric part of the coupling, respectively, illustrated in the Fig.3, dashed curves. A stable equilibrium of this system corresponds to a stable limit cycle of the phase model.

All equilibria of this system are solutions to $H(\chi) = -\omega$ , and they 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. 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.

[edit]

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:

$\theta_i' = \omega_i + \sin (\theta_{i+1}-\theta_i) + \sin(\theta_{i-1}-\theta_i).$

As long as the differences in the frequencies are small enough, there will be a phase-locked solution. Interestingly, if the length of the chain is $N$ and the frequency gradient is linear with slope $b$ then $b = O(1/N)$ as $N\to\infty$ . That is, nearest neighbor chains can support very small gradients when the coupling is sinusoidal (and, in fact, any odd periodic function). However, if the coupling function contains any even components (that is, replace $sin\theta$ with $\sin (\theta+\beta)$ , then frequency gradients as that are $O(1)$ can be supported in nearest neighbor chains of coupled phase oscillators. Kopell & Ermentrout (1986,1990) derived a set of continuum equations from which general phase-locked solutions could be found.

[edit]

Linear arrays of oscillators

Now consider a network of $n>2$ weakly coupled oscillators. To determine the existence and stability of synchronized states in the network, we need to study equilibria of the corresponding phase model

$\varphi_i' = \varepsilon \omega_i + \varepsilon \sum_{j\neq i}^n H_{ij}(\varphi_j-\varphi_i).$

Existence of one equilibrium of the phase model above implies the existence of the entire circular family of equilibria, since translation of all $\varphi_i$ by a constant phase shift does not change the phase differences $\varphi_i-\varphi_j$ and hence the form of the phase model. This family corresponds to a periodic orbit, on which all oscillators have equal frequencies and constant phase shifts, i.e., they are synchronized, possibly out-of-phase.

Vector $\phi=(\phi_1,\ldots,\phi_n)$ is an equilibrium when

$0 = \omega_i + \sum_{j\neq1}^n H_{ij}(\phi_j-\phi_i)$ for all $i$ .

It is stable when all eigenvalues of the linearization matrix (Jacobian) at $\phi$ have negative real parts, except one zero eigenvalue corresponding to the eigenvector along the circular family of equilibria ( $\phi$ plus a phase shift is a solution too since the phase shifts $\phi_j-\phi_i$ are not affected).

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.

[edit]

2D Arrays of oscillators

[edit]

History

[edit]

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. Ermentrout, G.B., Symmetry and phaselocking in chains of weakly coupled oscillators. Comm.-Pure-Appl.-Math. 39 (1986),623-660.
Kopell, N.; Ermentrout, G.B., Phase transitions and other phenomena in chains of coupled oscillators. 1990 SIAM-J.-Appl.-Math. 50 (1990),1014-1052
Kuramoto Y. (1984) Chemical Oscillations, Waves, and Turbulence. Springer-Verlag, New York.
Pikovsky A., Rosenblum M., Kurths J. (2001) Synchronization: A Universal Concept in Nonlinear Science. CUP, Cambridge.
Winfree A. (2001) The Geometry of Biological Time. Springer-Verlag, New York, second edition.

[edit]

Phase model

From Scholarpedia

Contents

Phase of oscillation

Weak forcing

Examples of reduction

Phase oscillators

SNIC oscillators

Andronov-Hopf oscillators

Other interesting cases

Weakly coupled oscillators

Phase model

Analysis

Two coupled oscillators

Chains of oscillators

Linear arrays of oscillators

2D Arrays of oscillators

History

References

See Also

Views

Personal tools

Scholarpedia

Encyclopedia of

For readers

For authors

Toolbox