The theory of weakly coupled oscillators [5, 213] is now a standard tool of dynamical systems theory and has been used by many authors to study oscillatory neural networks; see for example [213–217]. The book by Hoppensteadt and Izhikevich provides a very comprehensive review of this framework [43], which can also be adapted to study networks of relaxation oscillators (in some singular limit) [146, 218].
Consider, for illustration, a system of interacting limit-cycle oscillators (8). Following the method in Sect. 5.5, similar to (29) we obtain the network’s phase dynamics in the form
$$ \frac{\mathrm{d}}{\mathrm{d}t} {\theta_{i}} = \omega_{i} + \epsilon \sum_{j=1}^{N} w_{ij} \bigl\langle Q_{i}\bigl(u_{i}(\theta_{i})\bigr) , G \bigl(u_{j}(\theta_{j})\bigr) \bigr\rangle , $$
(30)
where the frequency \(\omega_{i}\) allows for the fact that oscillators are not identical and, for this reason, we will assume that \(\theta_{i}\in [0,2\pi)\). Precisely this form of network model was originally suggested by Winfree to describe populations of coupled oscillators. The Winfree model [219] assumes a separation of time scales so that an oscillator can be solely characterised by its phase on cycle (fast attraction to cycle) and is described by the network equations,
$$ \frac{\mathrm{d}}{\mathrm{d}t} {\theta}_{i} = \omega_{i} + \epsilon R( \theta _{i}) \frac {1}{N} \sum_{j=1}^{N} P(\theta_{j}) , $$
(31)
describing a globally coupled network with a biologically realistic PRC R and pulsatile interaction function P. Using a mixture of analysis and numerics Winfree found that with large N there was a transition to macroscopic synchrony at a critical value of the heterogeneity of the population. Following this, Kuramoto [5] introduced a simpler model with interactions mediated by phase differences, and showed how the transition to collective synchronisation could be understood from a more mathematical perspective. For an excellent review of the Kuramoto model see [220] and [221].
The natural way to obtain a phase-difference model from (30) is, as in Sect. 5.5, to average over one period of oscillation. For simplicity let us assume that all the oscillators are identical, and \(\omega_{i}=\omega\) for all i, in which case we find that
$$ \frac{\mathrm{d}}{\mathrm{d}t} {\theta}_{i} = \omega+ \epsilon\sum _{j=1}^{N} w_{ij} H(\theta_{j}- \theta_{i}), $$
(32)
where
$$ H(\psi) = \frac{1}{2\pi}\int_{0}^{2\pi} \bigl\langle Q \bigl(u(s)\bigr), G \bigl(u(\psi + s)\bigr) \bigr\rangle \,\mathrm{d}s . $$
(33)
The 2π-periodic function H is referred to as the phase interaction function (or coupling function). If we write complex Fourier series for Q and G as
$$Q(t)=\sum_{n\in \mathbb {Z}} Q_{n} \mathrm {e}^{i n t} \quad \mbox{and}\quad G(t)=\sum_{n\in \mathbb {Z}} G_{n} \mathrm {e}^{ i n t}, $$
respectively, then
$$ H(\psi) = \sum_{n\in \mathbb {Z}} H_{n} \mathrm {e}^{i n \psi} $$
(34)
with \(H_{n} = \langle Q_{-n}, G_{n} \rangle\). Note that a certain caution has to be exercised in applying averaging theory. In general, one can only establish that a solution of the unaveraged equations is ϵ-close to a corresponding solution of the averaged system for times of \(O(\epsilon^{-1})\). No such problem arises in the case of hyperbolic fixed points corresponding to phase-locked states.
When describing a piece of cortex or a central pattern generator circuit with a set of oscillators, the biological realism of the model typically resides in the phase interaction function. The simplest example is \(H(\psi)=\sin(\psi)\), which when combined with a choice of global coupling defines the well-known Kuramoto model [5]. However, to model realistic neural networks one should calculate (33) directly, using knowledge of the single neuron iPRC and the form of interaction. As an example consider a synaptic coupling, described in Sect. 2.5, that can be written in the form \(G(u(\psi)) = \sum_{m} \eta(\psi+m 2\pi)\), and a single neuron model for which the iPRC in the voltage variable v is given by R (say experimentally or from numerical investigation). In this case
$$ H(\psi) = \int_{0}^{\infty}R (2 \pi s-\psi) \eta(2 \pi s) \,\mathrm {d}s . $$
(35)
If instead we were interested in diffusive (gap junction) coupling then we would have
$$ H(\psi) = \frac{1}{2 \pi}\int_{0}^{2 \pi} R(s) \bigl[v(s+\psi)-v(s)\bigr] \,\mathrm{d}s .$$
For the HH model \(R(t)\) is known to have a shape like \(-\sin(t)\) for a spike centred on the origin (see Fig. 13). Making the further choice that \(\eta(t)=\alpha^{2} t \mathrm {e}^{-\alpha t}\) then (35) can be evaluated as
$$ H(\psi) = \frac{[1-(1/\alpha)^{2}] \sin(\psi)-2/\alpha\cos(\psi)}{2 \pi[1+(1/\alpha)^{2}]^{2}}. $$
(36)
In the particular case of two oscillators with reciprocal coupling and \(\omega=1\) then
$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} {\theta}_{1} &= 1 + \epsilon H(\theta _{2}-\theta_{1}) , \\ \frac{\mathrm{d}}{\mathrm{d}t} {\theta}_{2} &= 1 + \epsilon H(\theta _{1}-\theta_{2}) , \end{aligned}$$
and we define \(\varphi:=\theta_{2}(t)-\theta_{1}(t)\). A phase-locked solution has a constant phase difference ϕ that is a zero of the (odd) function
$$K(\varphi)=\epsilon\bigl[H(-\varphi)-H(\varphi)\bigr]. $$
A given phase-locked state is then stable provided that \(K'(\varphi)<0\). Note that by symmetry both the in-phase (\(\varphi=0\)) and the anti-phase (\(\varphi =\pi\)) states are guaranteed to exist. For the form of the phase interaction function given by (36), the stability of the synchronous solution is governed by the sign of \(K'(0)\):
$$ \operatorname {sgn}K'(0) = \operatorname {sgn}\bigl\{ -\epsilon H'(0)\bigr\} = \operatorname {sgn}\bigl\{ -\epsilon\bigl[\bigl(1-(1/\alpha)\bigr)^{2}\bigr] \bigr\} . $$
Thus for inhibitory coupling (\(\epsilon< 0\)) synchronisation will occur if \(1/\alpha> 1\), namely when the synapse is slow (\(\alpha\rightarrow0\)). It is also a simple matter to show that the anti-synchronous solution (\(\varphi =\pi \)) is stable for a sufficiently fast synapse (\(\alpha \rightarrow\infty\)). It is also possible to develop a general theory for the existence and stability of phase-locked states in larger networks than just a pair.
Phase, Frequency and Mode Locking
Now suppose we have a general population of \(N\geq2\) coupled phase oscillators
$$ \frac{\mathrm{d}}{\mathrm{d}t} {\theta}_{j}=f_{j}(\theta_{1}, \ldots ,\theta_{N}), $$
described by phases \(\theta_{j}\in \mathbb {R}/2\pi \mathbb {Z}\). For a particular continuous choice of phases \(\theta(t)\) for the trajectory one can define the frequency of the jth oscillator as
$$\varOmega_{j}=\lim_{T\rightarrow\infty} \frac{1}{T} \bigl[ \theta _{j}(T)-\theta_{j}(0)\bigr]. $$
This limit will converge under fairly weak assumptions on the dynamics [222], though it may vary for different attractors in the same system, for different oscillators j and in some cases it may vary even for different trajectories within the same attractor.
We say two oscillators j and k are phase locked with ratio \((n:m)\) for \((n,m)\in \mathbb {Z}^{2}\setminus(0,0)\) with no common factors of n and m, if there is an \(M>0\) such that
$$\bigl\vert n\theta_{j}(t)-m\theta_{k}(t)\bigr\vert < M , $$
for all \(t>0\). The oscillators are frequency locked with ratio \((n:m)\) if
$$n\varOmega_{j}-m\varOmega_{k}=0. $$
If we say they are simply phase (or frequency locked) without explicit mention of the \((n:m)\) ratio, we are using the convention that they are \((1:1)\) phase (or frequency) locked. The definition of \(\varOmega_{j}\) means that if two oscillators are phase locked then they are frequency locked. The converse is not necessarily the case: two oscillators may be frequency locked but not phase locked if the phase difference \(n\theta_{j}(t)-m\theta_{k}(t)\) grows sublinearly with t.
For the special case of globally coupled networks (\(w_{ij}=1/N\) for the system (32)), the system is \(S_{N} \times \mathbb {T}\) equivariant. By topological arguments, maximally symmetric solutions describing synchronous, splay, and a variety of cluster states exist generically for weak coupling [118]. The system (32) with global coupling is in itself an interesting subject of study in that it is of arbitrarily high dimension N but is effectively determined by the single function \(H(\varphi)\) that is computable from a single pair of oscillators. The system (and variants thereof) have been productively studied by thousands of papers since the seminal work of Kuramoto [5].
Dynamics of General Networks of Identical Phase Oscillators
The collective dynamics of phase oscillators have been investigated for a range of regular network structures including linear arrays and rings with uni- or bi-directional coupling e.g. [118, 120, 213, 223], and hierarchical networks [224]. In some cases the systems can be usefully investigated in terms of permutation symmetries of (32) with global coupling, for example \(\mathbb {Z}_{N}\) or \(\mathbb {D}_{N}\) for uni- or bi-directionally coupled rings. In other cases a variety of approaches have been developed and adapted to particular structures though these have not in all cases been specifically applied to oscillator networks; some of these approaches are discussed in Sect. 3.3
We recall that the form of the coupling in (32) is special in the sense that it assumes the interactions between two oscillators are independent of any third—this is called pairwise coupling [118, 120]. If there are degeneracies such as
$$ \sum_{k=0}^{m-1} H \biggl( \varphi+\frac{2\pi k}{m} \biggr)=0 , $$
(37)
which can appear when some of the Fourier components of H are zero, this can lead to degeneracies in the dynamics. For example [225], while Theorem 7.1 in [118] shows that if H satisfies (37) for some \(m\geq2\) and N is a multiple of m then (32), with global coupling, will have m-dimensional invariant tori in phase space that are foliated by neutrally stable periodic orbits. This degeneracy will disappear on including either non-pairwise coupling or introducing small but nonzero Fourier components in the expansion of H but as noted in [226] this will typically be the case for the interaction of oscillators even if they are near a Hopf bifurcation.
We examine in more detail some of the phase-locked states that can arise in weakly coupled networks of identical phase oscillators described by (32). We define a \(1:1\) phase-locked solution to be of the form \(\theta_{i}(t)=\varphi_{i}+ \varOmega t\), where \(\varphi_{i}\) is a constant phase and Ω is the collective frequency of the coupled oscillators. Substitution into the averaged system (32) gives
$$ \varOmega= \omega+{\epsilon}\sum_{j=1}^{N} w_{ij} H(\varphi_{j} - \varphi_{i}) $$
(38)
for \(i=1,\ldots,N\). These N equations determine the collective frequency Ω and \(N-1\) relative phases with the latter independent of ϵ.
It is interesting to compare the weak-coupling theory for phase-locked states with the analysis of LIF networks from Sect. 4.3. Equation (13) has an identical structure to that of Eq. (38) (for \(I_{i} = I\) for all i), so that the classification of solutions using group theoretic methods is the same in both situations. There are, however, a number of significant differences between phase-locking equations (38) and (13). First, Eq. (13) is exact, whereas Eq. (38) is valid only to \(O(\epsilon)\) since it is derived under the assumption of weak coupling. Second, the collective period of oscillations Δ must be determined self-consistently in Eq. (13).
In order to analyse the local stability of a phase-locked solution \(\varPhi=(\phi_{1},\ldots,\phi_{N})\), we linearise the system by setting \(\theta_{i}(t)={\varphi}_{i}+\varOmega t+ \widetilde{\theta}_{i}(t)\) and expand to first-order in \(\widetilde{\theta}_{i}\):
$$ \frac{\mathrm{d}}{\mathrm{d}t} \widetilde{\theta}_{i}= {\epsilon} \sum _{j=1}^{N}\widehat {\mathcal {H}}_{ij}(\varPhi) \widetilde{\theta}_{j} , $$
where
$$ \widehat{\mathcal{H}}_{ij}(\varPhi)=w_{ij}H'( \varphi_{j}-\varphi_{i}) -\delta_{i,j}\sum _{k=1}^{N} w_{ik}H'( \varphi_{k}-\varphi_{i}) , $$
and \(H'(\varphi) = \mathrm{d}H(\varphi)/\mathrm{d}\varphi\). One of the eigenvalues of the Jacobian \(\widehat{\mathcal {H}}\) is always zero, and the corresponding eigenvector points in the direction of the uncoupled flow, that is, \((1,1,\ldots,1)\). The phase-locked solution will be stable provided that all other eigenvalues have a negative real part. We note that the Jacobian has the form of a graph-Laplacian mixing both anatomy and dynamics, namely it is the graph-Laplacian of the matrix with components \(-w_{ij}H'(\varphi_{j}-\varphi_{i})\).
Synchrony
Synchrony (more precisely, exact phase synchrony), where \(\theta _{1}=\theta_{2}= \cdots= \theta_{N-1}=\theta_{N} = \varOmega t+ t_{0}\) for some fixed frequency Ω, is a classic example of a phase-locked state. Substitution into (32), describing a network of identical oscillators, shows that Ω has symmetry \(S_{N}\) and must satisfy the condition
$$ \varOmega= \omega+ \epsilon H(0)\sum_{j=1}^{N} w_{ij} \quad \forall i . $$
One way for this to be true for all i is if \(H(0)=0\), which is the case, say, for \(H(\theta)=\sin(\theta)\) or for diffusive coupling, which is linear in the difference between two state variables so that \(H(0)=0\). The existence of synchronous solutions is also guaranteed if \(\sum_{j=1}^{N} w_{ij}\) is independent of i. This would be the case for global coupling where \(w_{ij}=1/N\), so that the system has permutation symmetry.
If the synchronous solution exists then the Jacobian is given by \(-\epsilon H'(0)\mathcal{L}\) where \(\mathcal{L}\) is the graph-Laplacian with components \(\mathcal{L}_{ij}=\delta_{ij}\sum_{k} w_{ik}-w_{ij}\). We note that \(\mathcal{L}\) has one zero eigenvalue, with eigenvector \((1,1,\ldots,1,1)\). Hence if all the other eigenvalues of \(\mathcal{L}\) lie on one side of the imaginary axis then stability is solely determined by the sign of \(\epsilon H'(0)\). This would be the case for a weighted connectivity matrix with all positive entries since the graph-Laplacian in this instance would be positive semi-definite. For example, for global coupling we have \(\mathcal{L}_{ij} = \delta_{ij}-N^{-1}\), and the (\(N-1\) degenerate) eigenvalue is +1. Hence the synchronous solution will be stable provided \(\lambda= -\epsilon H'(0)<0\).
Asynchrony
Another example of a phase-locked state is the purely asynchronous solution whereby all phases are uniformly distributed around the unit circle. This is sometimes referred to as a splay state, discrete rotating wave with \(\mathbb{Z}_{N}\) symmetry or splay-phase state and can be written \({\mathrm {d}\theta_{i}}/{\mathrm {d}t}=\varOmega\) with \(\theta_{i+1}-\theta_{i}=2\pi/N\) ∀i. Like the synchronous solution it will be present but not necessarily stable in networks with global coupling, with an emergent frequency that depends on H:
$$ \varOmega= \omega+ \epsilon\frac{1}{N}\sum_{j=1}^{N} H \biggl(\frac {2\pi j}{N} \biggr). $$
In this case the Jacobian takes the form
$$ \widehat{\mathcal{H}}_{nm}(\varPhi)= \frac{\epsilon}{N} [A_{n-m} -\delta_{nm} \varGamma ] , $$
where \(\varGamma=\sum_{k} H'(2\pi k/N)\) and \(A_{n}=H'(2\pi n/N)\). Hence the eigenvalues are given by \(\lambda_{p} = \epsilon[\nu_{p} -\varGamma]/N\), \(p=0,\ldots,N-1\) where \(\nu_{p}\) are the eigenvalues of \(A_{n-m}\): \(\sum_{m} A_{n-m} a _{m}^{p} = \nu_{p} a_{n}^{p}\), where \(a_{n}^{p}\) denote the components of the pth eigenvector. This has solutions of the form \(a_{n}^{p} = \mathrm {e}^{-2 \pi i np/N}\) so that \(\nu_{p} = \sum_{m} A_{m} \mathrm {e}^{2 \pi i m p/N}\), giving
$$ \lambda_{p} = \frac{\epsilon}{N} \sum_{m=1}^{N} H' \biggl(\frac{2\pi m}{N} \biggr) \bigl[\mathrm {e}^{2 \pi i m p/N}-1 \bigr] , $$
and the splay state will be stable if \(\operatorname {Re}(\lambda_{p}) < 0\)
\(\forall p \neq0\).
In the large N limit \(N \rightarrow\infty\) we have the useful result that (for global coupling) network averages may be replaced by time averages:
$$ \lim_{N \rightarrow\infty} \frac{1}{N} \sum _{j=1}^{N} F \biggl(\frac {2\pi j}{N} \biggr) = \frac{1}{2\pi} \int_{0}^{2\pi} F(t) \,\mathrm{d}t = F_{0} , $$
for some 2π-periodic function \(F(t)=F(t+2\pi)\) (which can be established using a simple Riemann sum argument), with a Fourier series \(F(t) = \sum_{n} F_{n} \mathrm {e}^{i n t}\). Hence in the large N limit the collective frequency of a splay state (global coupling) is given by \(\varOmega= \omega+ \epsilon H_{0}\), with eigenvalues
$$ \lambda_{p} = \frac{\epsilon}{2\pi} \int _{0}^{2\pi} H'(t) \mathrm {e}^{ i p t} \, \mathrm{d}t = -2 \pi i p \epsilon H_{-p} . $$
Hence a splay state is stable if \(-p \epsilon \operatorname {Im}H_{p} < 0\), where we have used the fact that since \(H(\theta)\) is real, then \(\operatorname {Im}H_{-p}=- \operatorname {Im}H_{p}\). As an example consider the case \(H(\theta) = \theta(\pi-\theta )(\theta -2 \pi)\) for \(\theta\in[0,2\pi)\) and \(\epsilon>0\) (where H is periodically extended outside \([0,2\pi)\)). It is straightforward to calculate the Fourier coefficients (34) as \(H_{n}=6i/n^{3}\), so that \(-p \epsilon \operatorname {Im}H_{p} = - 6 \epsilon/p^{2} < 0\)
\(\forall p \neq0\). Hence the asynchronous state is stable. If we flip any one of the coefficients \(H_{m} \rightarrow- H_{m}\) then \(\operatorname {Re}\lambda_{m} >0\) and the splay state will develop an instability to an eigenmode that will initially destabilise the system in favour of an m-cluster, and see Fig. 14.
Clusters for Globally Coupled Phase Oscillators
For reviews of the stability of cluster states (in which subsets of the oscillator population synchronise, with oscillators belonging to different clusters behaving differently) we refer the reader to [118, 120, 227]; here we use the notation of [228]. If a group of N oscillators is neither fully synchronised nor desynchronised it may be clustered. We say \(\mathcal {A}=\{ A_{1},\ldots,A_{M}\}\) is an M-cluster partition of \(\{1,\ldots,N\}\) if
$$ \{1,\ldots,N\}=\bigcup_{p=1}^{M} A_{p} , $$
where \(A_{p}\) are pairwise disjoint sets (\(A_{p}\cap A_{q}=\emptyset\) if \(p \neq q\)). Note that if \(a_{p}=\vert A_{p}\vert \) then
$$ \sum_{p=1}^{M} a_{p} = N . $$
One can refer to this as a cluster of type \((a_{1},\ldots,a_{M})\). It is possible to show that any clustering can be realised as a stable periodic orbit of the globally coupled phase oscillator system [228] for suitable choice of phase interaction function; more precisely, there is a coupling function H for the system (32), with global coupling, such that for any N and any given M-cluster partition \(\mathcal {A}\) of \(\{1,\ldots, N\}\) there is a linearly stable periodic orbit realising that partition (and all permutations of it). See also [229], where the authors consider clustering in this system where \(H(\varphi)=\sin M\varphi\). More generally, there are very many invariant subspaces corresponding to spatio-temporal clustering that we can characterise in the following theorem.
Theorem 6.1
(Theorem 3.1 in [118])
The subsets of
\(\mathbb {T}^{N}\)
that are invariant for (32), with global coupling, because of symmetries of
\(S_{N}\times \mathbb {T}\)
correspond to isotropy subgroups in the conjugacy class of
$$\varSigma_{k,m}:=(S_{k_{1}}\times\cdots\times S_{k_{\ell}})^{m} \times_{s} \mathbb {Z}_{m}, $$
where
\(N=mk\), \(k=k_{1}+\cdots+k_{\ell}\)
and
\(\times_{s}\)
denotes the semidirect product. The points with this isotropy have
ℓm
clusters that are split into
ℓ
groups of
m
clusters of the size
\(k_{i}\). The clusters within these groups are cyclically permuted by a phase shift of
\(2\pi/m\). The number of isotropy subgroups in this conjugacy class is
\(N!/[m(k_{1}!\cdots k_{\ell}!)]\).
It is a nontrivial problem to discover which of these subspaces contain periodic solutions. Note that the in-phase case corresponds to \(\ell =m=1\), \(k_{1}=N\) while splay phase corresponds to \(\ell=k_{1}=1\), \(m=N\). The stability of several classes of these solutions can be computed in terms of properties of \(H(\varphi)\); see for example Sect. 6.2.1 and Sect. 6.2.2 and for other classes of solution [118, 120, 228].
Generic Loss of Synchrony in Globally Coupled Identical Phase Oscillators
Bifurcation properties of the globally coupled oscillator system (32) on varying a parameter that affects the coupling \(H(\varphi)\) are surprisingly complicated because of the symmetries present in the system; see Sect. 3.6. In particular, the high multiplicity of the eigenvalues for loss of stability of the synchronous solution means:
-
Path-following numerical bifurcation programs such as AUTO, CONTENT, MatCont or XPPAUT need to be used with great care when applying to problems with \(N\geq3\) identical oscillators—these typically will not be able to find all solutions branching from one that loses stability.
-
A large number of branches with a range of symmetries may generically be involved in the bifurcation; indeed, there are branches with symmetries corresponding to all possible two-cluster states \(S_{k} \times S_{N-k}\).
-
Local bifurcations may have global bifurcation consequences owing to the presence of connections that are facilitated by the nontrivial topology of the torus [118, 230].
-
Branches of degenerate attractors such as heteroclinic attractors may appear at such bifurcations for \(N\geq4\) oscillators.
Hansel et al. [231] consider the system (32) with global coupling and phase interaction function of the form
$$ H(\varphi)=\sin(\varphi-\alpha)-r\sin(2\varphi) , $$
(39)
for \((r,\alpha)\) fixed parameters; detailed bifurcation scenarios in the cases \(N=3,4\) are shown in [232]. As an example, Fig. 15 shows regions of stability of synchrony, splay-phase solutions and robust heteroclinic attractors as discussed later in Sect. 7.
Phase Waves
The phase-reduction method has been applied to a number of important biological systems, including the study of travelling waves in chains of weakly coupled oscillators that model processes such as the generation and control of rhythmic activity in central pattern generators (CPGs) underlying locomotion [233, 234] and peristalsis in vascular and intestinal smooth muscle [213]. Related phase models have been motivated by the observation that synchronisation and waves of excitation can occur during sensory processing in the cortex [235]. In the former case the focus has been on dynamics on a lattice and in the latter continuum models have been preferred. We now present examples of both these types of model, focusing on phase wave solutions [236].
Phase Waves: A Lattice Model
The lamprey is an eel-like vertebrate which swims by generating travelling waves of neural activity that pass down its spinal cord. The spinal cord contains about 100 segments, each of which is a simple half-centre neural circuit capable of generating alternating contraction and relaxation of the body muscles on either side of body during swimming. In a seminal series of papers, Ermentrout and Kopell carried out a detailed study of the dynamics of a chain of weakly coupled limit-cycle oscillators [213, 237, 238], motivated by the known physiology of the lamprey spinal cord. They considered N phase oscillators arranged on a chain with nearest-neighbour anisotropic interactions, as illustrated in Fig. 16, and identified a travelling wave as a phase-locked state with a constant phase difference between adjacent segments. The intersegmental phase differences are defined as \(\varphi_{i} = \theta_{i+1} - \theta_{i}\). If \(\varphi_{i} < 0\) then the wave travels from head to tail whilst for \(\varphi_{i}>0\) the wave travels from the tail to the head. For a chain we set \(W_{ij} = \delta_{i-1,j}W_{-} + \delta_{i+1,j}W_{+}\) to obtain
$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} {\theta}_{1}&= \omega_{1} + W_{+} H( \theta_{2} -\theta_{1}) , \\ \frac{\mathrm{d}}{\mathrm{d}t} {\theta}_{i} &= \omega_{i} + W_{+} H( \theta _{i+1} -\theta_{i}) + W_{-} H (\theta_{i-1}- \theta_{i}),\quad i=2,\ldots,N-1 , \\ \frac{\mathrm{d}}{\mathrm{d}t} {\theta}_{N} &= \omega_{N} + W_{-} H(\theta _{N-1} -\theta_{N}) , \end{aligned}$$
where \(\theta_{i}\in \mathbb {R}/2\pi \mathbb {Z}\). Pairwise subtraction and substitution of \(\varphi_{i} = \theta_{i+1} - \theta_{i}\) leads to an \(N-1\)-dimensional system for the phase differences,
$$ \frac{\mathrm{d}}{\mathrm{d}t} {\varphi}_{i} = \Delta\omega_{i} + W_{+} \bigl[H(\varphi_{i+1})- H(\varphi_{i})\bigr] + W_{-} \bigl[H (- \varphi_{i})-H (-\varphi_{i-1})\bigr], $$
for \(i=1, \ldots, N-1\), with boundary conditions \(H(-\varphi_{0}) = 0 = H(\varphi_{N+1})\), where \(\Delta\omega_{i} = \omega_{i+1} - \omega_{i}\). There are at least two different mechanisms that can generate travelling wave solutions.
The first is based on the presence of a gradient of frequencies along the chain, that is, \(\Delta\omega_{i}\) has the same sign for all i, with the wave propagating from the high frequency region to the low frequency region. This can be established explicitly in the case of an isotropic, odd interaction function, \(W_{\pm}=1\) and \(H(\varphi) = -H(-\varphi)\), where we have
$$ \frac{\mathrm{d}}{\mathrm{d}t} {\varphi}_{i} = \Delta\omega_{i} + H( \varphi _{i+1}) + H (\varphi_{i-1}) - 2 H( \varphi_{i}) . $$
The fixed points \(\varPhi=(\varphi_{1},\ldots,\varphi_{N})\) satisfy the matrix equation \({ H} ( \varPhi) = -{ A}^{-1} { D}\), where \({ H} (\varPhi) = (H(\varphi_{1}), \ldots, H(\varphi_{N}))^{\top}\), \({ D} = (\Delta\omega_{1}, \ldots, \Delta\omega_{N})^{\top}\), and A is a tridiagonal matrix with elements \(A_{ii}=-2\), \(A_{i,i+1}=A_{1+1,i}=1\). For the sake of illustration suppose that \(H(\varphi) = \sin(\varphi+\sigma)\). Then a solution Φ will exist if every component of \({ A}^{-1} { D}\) lies between ±1. Let \(a_{0} = \max\{ | ({ A}^{-1} { D})_{i} | \}\). If \(a_{0} < 1\) then for each \(i=1,\ldots,N\) there are two distinct solutions \(\varphi_{i}^{\pm}\) in the interval \([0, 2\pi]\) with \(H'(\varphi_{i}^{-} ) > 0\) and \(H'(\varphi_{i}^{+} ) < 0\). In other words, there are \(2^{N}\) phase-locked solutions. Linearising about a phase-locked solution and exploiting the structure of the matrix A, it can be proven that only the solution \(\varPhi^{-}=(\varphi_{1}^{-}, \ldots,\varphi_{N}^{-})\) is stable. Assuming that the frequency gradient is monotonic, this solution corresponds to a stable travelling wave. When the gradient becomes too steep to allow phase locking (i.e. \(a_{0} > 1\)), two or more clusters of oscillators (frequency plateaus) tend to form that oscillate at different frequencies. Waves produced by a frequency gradient do not have a constant speed or, equivalently, constant phase lags along the chain.
Constant speed waves in a chain of identical oscillators can be generated by considering phase-locked solutions defined by \(\varphi_{i} = \varphi\) for all i, with a collective period of oscillation Ω determined using \({\mathrm {d}\theta_{1}}/{\mathrm {d}t} = \varOmega\) to give \(\varOmega=\omega_{1} + W_{+} H(\varphi_{1})\). The steady state equations are then \(\Delta\omega_{1} +W_{+} H(-\varphi) =0\), \(\Delta\omega_{N-1} -W_{-}H(\varphi) =0 \) and \(\Delta\omega_{i} =0\), for \(i=2,\ldots, N-2\). Thus, a travelling wave solution requires that all frequencies must be the same except at the ends of the chain. One travelling solution is given by \(\Delta \omega_{N-1}=0\) with \(\Delta\omega_{1} = -W_{-} H(-\varphi)\) and \(H(\varphi) = 0\). For the choice \(H(\varphi)=\sin(\varphi+\sigma)\) we have \(\varphi =-\sigma\) and \(\Delta\omega_{1} = -W_{-} \sin(2 \sigma)\). If \(2 \sigma< \pi\) then \(\Delta\omega_{1} =\omega_{2} - \omega_{1}<0\) and \(\omega_{1}\) must be larger than \(\omega_{2}\) and hence all the remaining \(\omega_{i}\) for a forward travelling wave to exist. Backward swimming can be generated by setting \(\omega_{1}=0\) and solving in a similar fashion.
Phase Waves: A Continuum Model
There is solid experimental evidence for electrical waves in awake and aroused vertebrate preparations, as well as semi-intact and active invertebrate preparations, as nicely described by Ermentrout and Kleinfeld [235]. Moreover, these authors argue convincingly for the use of phase models in understanding waves seen in cortex and speculate that they may serve to label simultaneously perceived features in the stimulus stream with a unique phase. More recently it has been found that cortical oscillations can propagate as travelling waves across the surface of the motor cortex of monkeys (Macaca mulatta) [239]. Given that to a first approximation the cortex is often viewed as being built from a dense reciprocally interconnected network of corticocortical axonal pathways, of which there are typically 1010 in a human brain it is natural to develop a continuum phase model, along the lines described by Crook et al. [240]. These authors further incorporated axonal delays into their model to explain the seemingly contradictory result that synchrony is stable for short range excitatory coupling, but unstable for long range. To see how a delay-induced instability may arise we consider a continuum model of identical phase oscillators with space-dependent delays
$$ \frac{\partial}{\partial t} \theta(x,t) = \omega+ \epsilon\int_{D} W(x,y) H\bigl(\theta (y,t)-\theta(x,t) - \tau(x,y)\bigr) \,\mathrm{d}y , $$
(40)
where \(x \in D \subseteq \mathbb {R}\) and \(\theta\in \mathbb {R}/2\pi \mathbb {Z}\). This model is naturally obtained as the continuum limit of (32) for a network arranged along a line with communication delays \(\tau(x,y)\) set by the speed of an action potential v that mediates the long range interaction over a distance between points in the tissue at x and y. Here we have used the result that for weak coupling delays manifest themselves as phase shifts. The function W sets the anatomical connectivity pattern. It is convenient to assume that interactions are homogeneous and translation invariant, so that \(W(x,y) = W(|x-y|)\) with \(\tau(x,y)=|x-y|/v\), and we either assume periodic boundary conditions or take \(D=\mathbb {R}\).
Following [240] we construct travelling wave solutions of Eq. (40) for \(D=\mathbb {R}\) of the form \(\theta(x,t) = \varOmega t + \beta x\), with the frequency Ω satisfying the dispersion relation
$$ \varOmega= \omega+\epsilon\int_{-\infty}^{\infty}\, \mathrm{d}y W\bigl(\vert y\vert \bigr) H \bigl(\beta y - \vert y\vert /v \bigr) . $$
When \(\beta=0\), the solution is synchronous. To explore the stability of the travelling wave we linearise (40) about \(\varOmega t + \beta x\) and consider perturbations of the form \(\mathrm {e}^{\lambda t} \mathrm {e}^{ipx}\), to find that travelling phase waves solutions are stable if \(\operatorname {Re}\lambda(p) <0\) for all \(p \neq0\), where
$$ \lambda(p)= \epsilon\int_{-\infty}^{\infty}W\bigl(\vert y\vert \bigr) H'\bigl(\beta y - \vert y\vert /v\bigr) \bigl[ \mathrm {e}^{i p y} -1\bigr] \,\mathrm{d}y . $$
Note that the neutrally stable mode \(\lambda(0) =0\) represents constant phase shifts. For example, for the case that \(H(\theta)=\sin\theta\) then we have
$$ \operatorname {Re}\lambda(p)=\pi\epsilon\bigl[\varLambda(p, \beta_{+}) + \varLambda(-p, \beta_{-}) \bigr] , $$
where
$$ \varLambda(p,\beta) = W_{c}(p+\beta) +W_{c}(p-\beta)-2 W_{c}(\beta),\quad W_{c}(p) = \int_{0}^{\infty}W(y) \cos(p y) \,\mathrm{d}y , $$
and \(\beta_{\pm}= \pm\beta- 1/(v)\). A plot of the region of wave stability for the choice \(W(y)=\exp(-\vert y\vert )/2\) and \(\epsilon>0\) in the \((\beta,v)\) plane is shown in Fig. 17. Note that the synchronous solution \(\beta=0\) is unstable for small values of v. For a discussion of more realistic forms of phase interaction, describing synaptic interactions, see [241].
Phase Turbulence
For appropriate choice of the anatomical kernel and the phase interaction function, continuum models of the form (40) can also support weak turbulent solutions reminiscent of those seen in the Kuramoto–Sivashinsky (KS) equation. The KS equation generically describes the dynamics near long wavelength primary instabilities in the presence of appropriate (translational, parity and Galilean) symmetries, and it is of the form
$$ \theta_{t} = -\alpha\theta_{xx}+\beta( \theta_{x})^{2} -\gamma\theta _{xxxx} , $$
(41)
where \(\alpha,\beta,\gamma>0\). For a further discussion of this model see [76]. For the model (40) with decaying excitatory coupling excitation and purely sinusoidal phase coupling, simulations on a large domain show a marked tendency to generate phase slips and spatio-temporal pattern shedding, resulting in a loss of spatial continuity of \(\theta(x,t)\). However, Battogtokh [242] has shown that a mixture of excitation and inhibition with higher harmonics in the phase interaction can counteract this tendency and allow the formation of turbulent states. To see how this can arise consider an extension of (40) to allow for a mixing of spatial scales and nonlinearities in the form
$$ \frac{\partial}{\partial t} \theta(x,t) = \omega+ \sum_{\mu=1}^{M} \epsilon_{\mu}\int_{\mathbb {R}} W_{\mu}(x-y) H_{\mu}\bigl(\theta(y,t)-\theta(x,t)\bigr) \,\mathrm{d}y , $$
(42)
where we drop the consideration of axonal delays. Using the analysis of Sect. 6.3 the synchronous wave solution will be stable provided \(\lambda(p) < 0\) for all \(p \neq0\) where
$$ \lambda(p) = \sum_{\mu=1}^{M} \epsilon_{\mu}H_{\mu}'(0) \bigl[ \widehat {W}_{\mu}(p)-\widehat{W}_{\mu}(0) \bigr],\quad \widehat{W}_{\mu}(p) = \int_{\mathbb {R}} W_{\mu}\bigl(\vert y\vert \bigr) \mathrm {e}^{i p y} \,\mathrm{d}y . $$
After introducing the complex function \(z(x,t) = \exp(i \theta(x,t))\) and writing the phase interaction functions as Fourier series of the form \(H_{\mu}(\theta) = \sum_{n} H_{n}^{\mu} \mathrm {e}^{i n \theta}\) then we can rewrite (42) as
$$ z_{t} = i z \Biggl\{ \omega+ \sum_{\mu=1}^{M} \sum_{n \in \mathbb {Z}} \epsilon _{\mu}H_{n}^{\mu}z^{-n} \psi_{n}^{\mu}\Biggr\} , $$
(43)
where
$$ \psi_{n}^{\mu}(x,t) = \int_{\mathbb {R}} W_{\mu}(x-y) z^{n} (y,t) \,\mathrm{d}y \equiv W_{\mu}\otimes z^{n} . $$
The form above is useful for computational purposes, since \(\psi_{n}^{\mu}\) can easily be computed using a fast Fourier transform (exploiting its convolution structure). Battogtokh [242] considered the choice \(M=3\) with \(H_{1}(\theta)=H_{2}(\theta)=\sin(\theta+ \alpha)\), \(H_{3}=\sin(2\theta+ \alpha)\) and \(W_{\mu}(x)=\gamma_{\mu}\exp(-\gamma_{\mu}|x|)/2\) with \(\gamma _{2}=\gamma_{3}\). In this case \(\widehat{W}_{\mu}(p) = \gamma_{\mu}^{2}/(\gamma _{\mu}^{2}+p^{2})\), so that
$$ \lambda(p) = -p^{2} \cos(\alpha) \biggl(\frac{\epsilon_{1}}{\gamma _{1}^{2}+p^{2}}+ \frac{\epsilon_{2}+2 \epsilon_{3}}{\gamma_{2}^{2}+p^{2}} \biggr). $$
By choosing a mixture of positive and negative coupling strengths the spectrum can easily show a band of unstable wave-numbers from zero up to some maximum as shown in Fig. 18. Indeed this shape of spectrum is guaranteed when \(\sum_{\mu}\epsilon _{\mu}H_{\mu}'(0)>0\) and \(\sum_{\mu}\epsilon_{\mu}H_{\mu}'(0)/\gamma_{\mu}^{2}<0\). Similarly the KS equation (41) has a band of unstable wave-numbers between zero and one (with the most unstable wave-number at \(1/\sqrt{2}\)). For the case that all the spatial scales \(\gamma _{\mu}^{-1}\) are small compared to the system size then we may develop a long wavelength argument to develop local models for \(\psi _{n}^{\mu}\). To explain this we first construct the Fourier transform \(\widehat{\psi}_{n}^{\mu}(p,t) = \widehat{W}_{\mu}(p) f_{n}(p,t)\), where \(f_{n}\) is the Fourier transform of \(z^{n}\) and use the expansion \(\widehat {W}_{\mu}(p) \simeq1-\gamma_{\mu}^{-2} p^{2} +\gamma_{\mu}^{-4} p^{4} + \cdots\). After inverse Fourier transforming we find
$$ \psi_{n}^{\mu}\simeq \bigl[1 + \gamma_{\mu}^{-2} \partial_{xx} - \gamma _{\mu}^{-4} \partial_{xxxx} + \cdots \bigr] z^{n} . $$
Noting that \(H^{1}_{1}=H_{2}^{1}=H_{3}^{2}=\exp(i \alpha)/(2i) \equiv\varGamma\) with all other Fourier coefficients zero then (43) yields
$$\begin{aligned} \theta_{t} ={}& \varOmega+ 2 \operatorname {Re}\varGamma\sum _{\mu=1,2} \epsilon _{\mu} \mathrm {e}^{-i \theta} \bigl( \gamma_{\mu}^{-2}\partial_{xx} - \gamma_{\mu}^{-4}\partial _{xxxx} + \cdots \bigr) \mathrm {e}^{i \theta} \\ &{}+ \epsilon_{3} \mathrm {e}^{-2 i \theta} \bigl(\gamma_{3}^{-2} \partial_{xx} - \gamma_{3}^{-4} \partial_{xxxx} + \cdots \bigr) \mathrm {e}^{2 i \theta} , \end{aligned}$$
(44)
where \(\varOmega=\omega+ \sum_{\mu} \epsilon_{\mu}H_{\mu}(0)\). Expanding (44) to second order gives \(\theta_{t} = \varOmega -\alpha\theta_{xx} +\beta(\theta_{x})^{2}\), where \(\alpha=-\sum_{\mu}\epsilon_{\mu}H_{\mu}'(0) \gamma_{\mu}^{-2}\) and \(\beta =-\sum_{\mu}\epsilon_{\mu}H_{\mu}''(0) \gamma_{\mu}^{-2}\). Going to higher order yields fourth-order terms and we recover an equation of KS type with the coefficient of \(-\theta_{xxxx}\) given by \(\gamma=\sum_{\mu}\epsilon_{\mu}H_{\mu}'(0) \gamma_{\mu}^{-4}\). To generate phase turbulence we thus require \(\alpha>0\), which is also a condition required to generate a band of unstable wave-numbers, and \(\beta,\gamma>0\). A direct simulation of the model with the parameters for Fig. 18, for which \(\alpha,\beta,\gamma>0\), shows the development of a phase turbulent state. This is represented in Fig. 19 where we plot the absolute value of the complex function \(\varPsi= (\epsilon_{1} W_{1}+\epsilon_{2} W_{2} )\otimes z + \epsilon_{3} W_{3} \otimes z^{2}\).
