Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

2.1 The Hodgkin–Huxley Model and Its Planar Reduction

The work of Hodgkin and Huxley in elucidating the mechanism of action potentials in the squid giant axon is one of the major breakthroughs of dynamical modelling in physiology [50], and see [51] for a review. Their work underpins all modern electrophysiological models, exploiting the observation that cell membranes behave much like electrical circuits. The basic circuit elements are (1) the phospholipid bilayer, which is analogous to a capacitor in that it accumulates ionic charge as the electrical potential across the membrane changes; (2) the ionic permeabilities of the membrane, which are analogous to resistors in an electronic circuit; and (3) the electrochemical driving forces, which are analogous to batteries driving the ionic currents. These ionic currents are arranged in a parallel circuit. Thus the electrical behaviour of cells is based upon the transfer and storage of ions such as potassium (K⁺) and sodium (Na⁺).

Our goal here is to illustrate, by exploiting specific models of excitable membrane, some of the concepts and techniques which can be used to understand, predict, and interpret the excitable and oscillatory behaviours that are commonly observed in single cell electrophysiological recordings. We begin with the mathematical description of the Hodgkin–Huxley model.

The standard dynamical system for describing a neuron as a spatially isopotential cell with constant membrane potential V is based upon conservation of electric charge, so that

$$ C \frac{\mathrm{d}}{\mathrm{d}t} V = I_{\mathrm{ion}} +I , $$

where C is the cell capacitance, I the applied current and \(I_{\mathrm{ion}}\) represents the sum of individual ionic currents:

$$ I_{\mathrm{ion}} = -g_{\mathrm{K}} (V-V_{\mathrm{K}}) - g_{\mathrm {Na}} (V-V_{\mathrm{Na}}) - g_{L} (V-V_{L}). $$

In the Hodgkin–Huxley (HH) model the membrane current arises mainly through the conduction of sodium and potassium ions through voltage-dependent channels in the membrane. The contribution from other ionic currents is assumed to obey Ohm’s law (and is called the leak current). The \(g_{\mathrm{K}}\), \(g_{\mathrm{Na}}\) and \(g_{L}\) are conductances (reciprocal resistances) that can be interpreted as gating variables. The great insight of Hodgkin and Huxley was to realise that \(g_{\mathrm {K}}\) depends upon four activation gates: \(g_{\mathrm{K}} = \overline{g}_{\mathrm {K}} n^{4}\), whereas \(g_{\mathrm{Na}}\) depends upon three activation gates and one inactivation gate: \(g_{\mathrm{Na}} = \overline{g}_{\mathrm{Na}} m^{3} h\). Here the gating variables all obey equations of the form

$$ \frac{\mathrm{d}}{\mathrm{d}t} {X} = \frac{X_{\infty}(V)-X}{\tau _{X}(V)},\quad X \in\{ m,n,h\}. $$

The conductance variables described by X take values between 0 and 1 and approach the asymptotic values \(X_{\infty}(V)\) with time constants \(\tau_{X}(V)\). These six functions are obtained from fits with experimental data. It is common practice to write \(\tau_{X}(V) = 1/(\alpha_{X}(V) + \beta _{X}(V))\), \(X_{\infty}(V) =\alpha_{X}(V) \tau_{X}(V)\), where α and β have the interpretation of opening and closing channel transition rates, respectively. The details of the HH model are provided in the Appendix for completeness. A numerical bifurcation diagram of the model in response to constant current injection is shown in Fig. 1, illustrating that oscillations can emerge in a Hopf bifurcation with increasing drive.

The mathematical forms chosen by Hodgkin and Huxley for the functions \(\tau_{X}\) and \(X_{\infty}\), \(X \in\{m,n,h\}\), are all transcendental functions (i.e. involve exponentials). Both this and the high dimensionality of the model make analysis difficult. However, considerable simplification is attained with the following observations: (i) \(\tau_{m}(V)\) is small for all V so that the variable m rapidly approaches its equilibrium value \(m_{\infty}(V)\), and (ii) the equations for h and n have similar time-courses and may be slaved together. This has been put on a more formal footing by Abbott and Kepler [52] by expressing both n and h as functions of a scalar quantity \(U(t)\). They obtain a reduced planar model (of the full Hodgkin–Huxley model) in \((V,U)\) co-ordinates under the replacement \(m \rightarrow m_{\infty}(V)\) and \(X = X_{\infty}(U)\) for \(X \in\{ n , h \}\) with a prescribed choice of dynamics for U (such that \(\mathrm{d}^{2} V/ \mathrm{d} t^{2}\) is similar in the two models for a fixed V). The phase plane and nullclines for this system are shown in Fig. 2.

For zero external input the fixed point is stable and the neuron is said to be excitable. When a positive external current is applied the low-voltage portion of the V nullcline moves up whilst the high-voltage part remains relatively unchanged. For sufficiently large constant external input the intersection of the two nullclines falls within the portion of the V nullcline with positive slope. In this case the fixed point is unstable and the system may support a limit cycle. If an emergent limit cycle is stable then a train of action potentials will be produced and the system is referred to as being oscillatory. Action potentials may also be induced in the absence of an external current for synaptic stimuli of sufficient strength and duration. This simple planar model captures all of the essential features of the original HH model yet is much easier to understand from a geometric perspective. Indeed the model is highly reminiscent of the famous FHN model, in which the voltage nullcline is taken to be a cubic function. Both models show the onset of repetitive firing at a nonzero frequency as observed in the HH model (when an excitable state loses stability via a subcritical Hopf bifurcation). However, unlike real cortical neurons they cannot fire at arbitrarily low frequency. This brings us to consider modifications of the original HH formalism to accommodate bifurcation mechanisms from excitable to oscillatory behaviours that can respect this experimental observation.

2.2 The Cortical Model of Wilson

Many of the properties of real cortical neurons can be captured by making the equation for the recovery variable of the FHN equations quadratic (instead of linear). We are thus led to the cortical model of Wilson [53]:

$$\begin{aligned} C \frac{\mathrm{d}}{\mathrm{d}t} {v} & = f(v) - w +I,\quad f(v) = v(a-v) (v-1) , \\ \frac{\mathrm{d}}{\mathrm{d}t} {w} & = \beta(v-v_{1}) (v-v_{2})-\gamma w, \end{aligned}$$

where \(0< a<1\) and \(\beta,\gamma, C>0\). Here v is like the membrane potential V, and w plays the role of a gating variable. In addition to the single fixed point of the FHN model it is possible to have another pair of fixed points, as shown in Fig. 3(left). As I increases two fixed points can annihilate at a saddle node on an invariant circle (SNIC) bifurcation at \(I=I_{c}\) [48]. In the neighbourhood of this global bifurcation the firing frequency scales like \(\sqrt{I-I_{c}}\). For large enough I there is only one fixed point on the middle branch of the cubic, as illustrated in Fig. 3(right). In this instance an oscillatory solution occurs via the same mechanism as for the FHN model.

2.3 Morris–Lecar with Homoclinic Bifurcation

A SNIC bifurcation is not the only way to achieve a low firing rate in a single neuron model. It is also possible to achieve this via a homoclinic bifurcation, as is possible in the Morris–Lecar (ML) model [54]. This was originally developed as a model for barnacle muscle fibre. Morris and Lecar introduced a set of two coupled ODEs incorporating two ionic currents: an outward, non-inactivating potassium current and an inward, non-inactivating calcium current. Assuming that the calcium currents operate on a much faster time scale than the potassium current one, they formulated the following two-dimensional system:

$$\begin{aligned} C\frac{\mathrm{d}}{\mathrm{d}t}V &= g_{L}(V_{L}-V) +g_{\mathrm{K}} w (V_{\mathrm{K}}-V) + g_{\mathrm{Ca}}m_{\infty}(V) (V_{\mathrm{Ca}}-V) + I , \\ \frac{\mathrm{d}}{\mathrm{d}t}w &= \lambda(V) \bigl(w_{\infty}(V)-w\bigr), \end{aligned}$$

with \(m_{\infty}(V) = 0.5(1+\tanh[(V-V_{1})/V_{2}])\), \(w_{\infty}(V) = 0.5(1+ \tanh[(V-V_{3})/V_{4}])\), \(\lambda(V) = \phi\cosh[(V-V_{3})/(2V_{4})]\), where \(V_{1}, \ldots, V_{4} \) and ϕ are constants. Here w represents the fraction of K⁺ channels open, and the Ca²⁺ channels respond to V so rapidly that we assume instantaneous activation. Here \(g_{L}\) is the leakage conductance, \(g_{\mathrm{K}}\), \(g_{\mathrm{Ca}}\) are the potassium and calcium conductances, \(V_{L}\), \(V_{\mathrm{K}}\), \(V_{\mathrm{Ca}}\) are corresponding reversal potentials, \(m_{\infty}(V)\), \(w_{\infty}(V)\) are voltage-dependent gating functions and \(\lambda(V)\) is a voltage-dependent rate. Referring to Fig. 4, as I decreases the periodic orbit grows in amplitude, it comes closer to a saddle point and the period increases such that near the homoclinic bifurcation, where the orbit collides with the saddle at \(I=I_{c}\), the frequency of oscillation scales as \(-1/\log(I-I_{c})\).

2.4 Integrate-and-Fire

Although conductance-based models like that of Hodgkin and Huxley provide a level of detail that helps us understand how the kinetics of channels (with averaged activation and inactivation variables) can underlie action-potential generation, their high dimensionality is a barrier to studies at the network level. The goal of a network-level analysis is to predict emergent computational properties in populations and recurrent networks of neurons from the properties of their component cells. Thus simpler (lower-dimensional and hopefully mathematically tractable) models are more appealing—especially if they fit single neuron data.

2.5 Neuronal Coupling

At a synapse presynaptic firing results in the release of neurotransmitters that cause a change in the membrane conductance of the postsynaptic neuron. This postsynaptic current may be written \(I_{s}(t) = g_{s} s(t) (V_{s} - V(t))\) where \(V(t)\) is the voltage of the postsynaptic neuron, \(V_{s}\) is the membrane reversal potential and \(g_{s}\) is a constant conductance. The variable s corresponds to the probability that a synaptic receptor channel is in an open conducting state. This probability depends on the presence and concentration of neurotransmitter released by the presynaptic neuron. The sign of \(V_{s}\) relative to the resting potential (which without loss of generality we may set to zero) determines whether the synapse is excitatory (\(V_{s} >0\)) or inhibitory (\(V_{s} < 0\)).

Interestingly even purely inhibitory synaptic interactions between non-oscillatory neurons can create oscillations at the network level, and can give rise to central pattern generators of “half-centre” type [59]. To see this we need only consider a pair of (reduced) Hodgkin–Huxley neurons with mutual reciprocal inhibition mediated by an α-function synapse with a negative reversal potential. The phenomenon of anode break excitation (whereby a neuron fires an action potential in response to termination of a hyperpolarising current) can underlie a natural anti-phase rhythm, and is best understood in terms of the phase plane shown in Fig. 2. In this case inhibition will effectively move the voltage nullcline down, and the system will equilibrate to a new hyperpolarised state. Upon release of inhibition the fixed point will move to a higher value, though to reach this new state the trajectory must jump to the right hand branch (since now \({\mathrm {d}V}/{\mathrm {d}t}>0\)). An example is shown in Fig. 5.

Gap junctions differ from chemical synapses in that they allow for direct communication between cells. They are typically formed from the juxtaposition of two hemichannels (connexin proteins) and allow the free movement of ions or molecules across the intercellular space separating the plasma membrane of one cell from another. As well as being found in the neocortex, they occur in many other brain regions, including the hippocampus, inferior olivary nucleus in the brain stem, the spinal cord, and the thalamus [60]. Without the need for receptors to recognise chemical messengers, gap junctions are much faster than chemical synapses at relaying signals. The synaptic delay for a chemical synapse is typically in the range 1–100 ms, while the synaptic delay for an electrical synapse may be only about 0.2 ms.

It is common to view the gap junction as nothing more than a channel that conducts current according to a simple ohmic model. For two neurons with voltages \(v_{i}\) and \(v_{j}\) the current flowing into cell i from cell j is given by \(I_{\mathrm{gap}}(v_{i},v_{j})=g (v_{j}-v_{i})\), where g is the constant strength of the gap-junction conductance. They are believed to promote synchrony between oscillators (e.g. see [61]), though the story is more subtle than this as we shall discuss in Sect. 4.

2.6 Neural Mass Models

As well as supporting oscillations at the single neuron level, brain tissue can also generate oscillations at the tissue level. Rather than model this using networks built from single neuron models, it is has proven especially useful to develop low-dimensional models to mimic the collection of thousands of near identical interconnected neurons with a preference to operate in synchrony. These are often referred to as neural mass models, with state variables that track coarse grained measures of the average membrane potential, firing rates or synaptic activity. They have proven especially useful in the description of human EEG power spectra [62], as well as resting brain state activity [63] and mesoscopic brain oscillations [64].

In many neural population models, such as the well-known Wilson–Cowan model [65], it is assumed that the interactions are mediated by firing rates rather than action potentials (spikes) per se. To see how this might arise we rewrite (3) in the equivalent form using a sum of Dirac δ-functions,

$$ \biggl(1 + \frac{1}{\alpha} \frac{\mathrm{d}}{\mathrm{d}t} \biggr) \biggl(1 + \frac {1}{\beta } \frac{\mathrm{d}}{\mathrm{d}t} \biggr) s = \sum_{m \in \mathbb {Z}} \delta(t-T_{m}). $$

(4)

Identifying the right hand side of (4) as a train of presynaptic spikes motivates the form of a phenomenological rate model in the form

$$ Q s = f , $$

(5)

with f identified as a firing rate and Q identified as the differential operator \((1+\alpha^{-1} \mathrm{d}/\mathrm {d}t)(1+\beta^{-1} \mathrm{d}/\mathrm{d} t)\). At the network level it is then common practice to close this system of equations by specifying f to be a function of presynaptic activity. A classic example is the Jansen–Rit model [66], which describes a network of interacting pyramidal neurons (P), inhibitory interneurons (I) and excitatory interneurons (E), and has been used to model both normal and epileptic patterns of cortical activity [67]. This can be written in the form

$$Q_{E} s_{P} = f(s_{E} -s_{I}) , \qquad Q_{E} s_{E} = C_{2}f(C_{1} s_{P}) +A ,\qquad Q_{I}s_{I} = C_{4} f(C_{3} s_{P}) , $$

which is a realisation of the structure suggested by (5), with the choice

$$ f(v) = \frac{\nu}{1+\mathrm {e}^{-r(v-v_{0})}}, $$

and \(Q_{a} = (1+\beta_{a}^{-1}\mathrm{d}/\mathrm{d}t)^{2} \beta_{a}/A_{a}\) for \(a \in \{E,I\}\). Here A is an external input. When this is a constant we obtain the bifurcation diagram shown in Fig. 6. Oscillations emerge via Hopf bifurcations and it is possible for a pair of stable periodic orbits to coexist. One of these has a frequency in the alpha band and the other is characterised by a lower frequency and higher amplitude. Recently a network of such modules, operating in the alpha range and with additive noise, has been used to investigate mechanisms of cross-frequency coupling between brain areas [68]. Neural mass models have also previously been used to model brain resonance phenomena [69], for modelling of epileptic seizures [70–72], and they are very popular in the neuroimaging community for model driven EEG/fMRI fusion [73].

Now that we have introduced some oscillator models for neurons and neural populations it is appropriate to consider the set of tools for analysing their behaviour at the network level.

3.1 Synchrony and Asynchrony

One of the most important observations concerning the collective dynamics of coupled nonlinear systems relates to whether the collection behaves as one or not—whether there is an attracting synchronous state, or whether more complex spatio-temporal patterns such as generalised synchrony (also called clustering) appear. There is a very large literature, even restricting to the case of applications of synchrony, and one where we cannot hope to do the whole area justice. We refer in particular to [75, 76]. Various applications of synchrony of neural models are discussed, for example, in [77–85] while there is a large literature (e.g. [17]) discussing the role of synchrony in neural function. Other work looks for example at synchronisation of groups of networks [86] and indeed synchrony can be measured experimentally [87] in groups of neurons using dynamic patch clamping.

3.2 Clusters, Exact and Generalised Synchrony

If one has a notion of synchrony between the systems of (7), it is possible to discuss certain generalised forms of synchrony, including clustering according to mutual synchrony. Caution needs to be exercised whenever discussing synchrony—there are many distinct notions of synchrony that may be appropriate in different contexts and, in particular, synchrony is typically a property of a particular solution at a particular point in time rather than a property of the system as a whole.

An important case of synchrony is exact synchrony: we say \(x_{i}(t)\) and \(x_{j}(t)\) are exactly synchronised if \(x_{i}(t)=x_{j}(t)\) for all t. Generalised synchrony is, as the name suggests, much more general and corresponds to there simply being a functional relationship of the form \(x_{i}(t)=F(x_{j}(t))\). Another related notion is that of clustering, where different groups of oscillators are exactly synchronised but there is no exact synchrony between the groups. For oscillators, phases can be used to define additional notions such as phase and frequency synchrony: see Sect. 6.1.

3.3 Networks, Motifs and Coupled Cell Dynamics

It is interesting to try and understand the effect of network structure on synchrony, so we briefly outline some basic graph theoretic measures of network structure. The in-degree of the node i is the number of incoming connections (i.e. \(d_{\mathrm{in}}(i)=\sum_{j} A_{ij}\)), while the out-degree is the number of outgoing connections (i.e. \(d_{\mathrm{out}}(i)=\sum_{j} A_{ji}\)) and the distribution of these degrees is often used to characterise a large graph. A scale-free network is a large network where the distribution of in (or out) degrees scales as a power of the degree. This can be contrasted with highly structured homogeneous networks (for example on a lattice) where the degree may be the same at each node. Other properties commonly examined include the clustering properties and path lengths within the graph. There are also various measures of centrality that help one to determine the most important nodes in a graph—for example the betweenness centrality is a measure of centrality that is the probability that a given node is on the shortest path between two uniformly randomly chosen nodes [90]. As expected, the more central nodes are typically most important if one wishes to achieve synchrony in a network.

Other basic topological properties of networks that are relevant to their dynamics include, for example, the following, most of which are mentioned in [75, 90]: The network is undirected if \(A_{ij}=A_{ji}\) for all i, j, otherwise it is directed. We say nodes j and i in the network \(A_{ij}\) are path-connected if for some n there is a path from j to i, i.e. \((A^{n})_{ij}\neq0\) for some n. The network is strongly connected if for each i, j it is path-connected in both directions while it is weakly connected if we replace \(A_{ij}\) by \(\max (A_{ij},A_{ji})\) (i.e. we make the network undirected) and the latter network is strongly connected.In the terminology of artificial neural networks, a strongly connected network is recurrent while one that is not strongly connected must have some feedforward connections between groups of nodes. There is a potential source of confusion in that strong and weak connectivity are properties of a directed network—while strong and weak coupling are properties of the coupling strengths for a given network.

The diameter of a network is the maximal length of a shortest path between two points on varying the endpoints. Other properties of the adjacency matrix are discussed for example in [93] where spectral properties of graph Laplacians are linked to the problem of determining stability of synchronised states. Other work we mention is that of Pecora et al. [94, 95] on synchronisation in coupled oscillator arrays (and see Sect. 4.1), while [96] explores the recurrent appearance of synchrony in networks of pulse-coupled oscillators (and see Sect. 4.2).

Finally, we mention network motifs—these are subgraphs that are “more prevalent” than others within some class of graphs. More precisely, given a network one can look at the frequency with which a small subgraph appears relative to some standard class of graphs (for example Erdös–Rényi random graphs) and if a certain subgraph appears more often than expected, this characterises an important property of the graph [97]. Such analysis has been used in systems biology (such as transcription or protein interaction networks) and has been applied to study the structure in neural systems (see for example [98, 99]) and the implications of this for the dynamics. They have also been used to organise the analysis of the dynamics of small assemblies of coupled cells; see for example [100, 101].

3.4 Weak and Strong Coupling

Continuing with systems of the form (7) or (8), if the coupling parameter ϵ is, in some sense small, we refer to the system as “weakly coupled”. Mathematically, the weak-coupling approximation is very helpful because it allows one to use various types of perturbation theory to investigate the dynamics [43]. For coupling of limit-cycle oscillators it allows one to greatly reduce the dimension of phase space. Nonetheless, many dynamical effects (e.g. “oscillator death” where the oscillations in one or more oscillators are completely suppressed by the action of the network [102]) cannot occur in the weak-coupling limit, and, moreover, real biological systems often have “strong coupling”. We will return to this topic to discuss oscillator behaviour in Sect. 4.3. One can sometimes use additional structure such as weak dissipation and weak coupling of the oscillators to perform a semi-analytic reduction to phase oscillators; see for example [103, 104].

3.5 Synchrony, Dynamics and Time Delay

Nonetheless, much can be learned about stability, control and bifurcation of dynamically synchronous states in the presence of delay; for example [84, 105–108], and the volume [109] include a number of contributions by authors working in this area. There are also well-developed numerical tools such as DDE-BIFTOOL [110, 111] that allow continuation, stability and bifurcation analysis of coupled systems with delays. For an application of these techniques to the study of a Wilson–Cowan neural population model with two delays we refer the reader to [112].

3.6 A Short Introduction to Symmetric Dynamics

Although no system is ever truly symmetric, in practice many models have a high degree of symmetry.¹ Indeed many real-world networks that have grown (e.g. giving rise to tree-like structures) are expected to be well approximated by models that have large symmetry groups [113].

3.7 Permutation Symmetries and Oscillator Networks

We review some aspects of the equivariant dynamics that have proven useful in coupled systems that are relevant to neural dynamics—see for example [118, 119]. In doing so we mostly discuss dynamics that respects some symmetry group of permutations of the systems. The full permutation symmetry group (or simply, the symmetric group) on N objects, \(S_{N}\), is defined to be the set of all possible permutations of N objects. Formally it is the set of permutations \(\sigma:\{1,\ldots,N\}\rightarrow\{1,\ldots,N\}\) (invertible maps of this set). To determine effects of the symmetry, not only the group must be known but also its action on phase space. If this action is linear then it is a representation of the group. The representation of the symmetry group is critical to the structure of the stability, bifurcations and generic dynamics that are equivariant with the symmetry.

The presence of symmetries means that solutions can be grouped together into families—given any x the set \(\varGamma x:= \{ gx : g\in\varGamma\}\) is the group orbit of x and all points on this group orbit will behave in dynamically the same way.

3.8 Invariant Subspaces, Solutions and Symmetries

Given a point \(x\in \mathbb {R}^{N}\) (for simplicity we consider the case \(d=1\) below) we define the isotropy subgroup (or simply the symmetry) of x under the action of Γ on \(\mathbb {R}^{N}\) to be

$$\varSigma_{x}:=\{ g\in\varGamma: gx=x\}. $$

This is a subgroup of Γ, and the set of these groups forms a lattice (the isotropy lattice) by inclusion of subgroups. They are dynamically important in that for any trajectory \(x(t)\) we have \(\varSigma_{x(0)}=\varSigma_{x(t)}\) for all t. A converse problem is to characterise the set of all points with a certain symmetry. If H is a subgroup (or more generally, a subset) of Γ then the fixed-point space of H is defined to be

$$\operatorname {Fix}(H):=\bigl\{ x\in \mathbb {R}^{N} : gx=x \mbox{ for all }g\in H\bigr\} . $$

Because all \(x\in \operatorname {Fix}(H)\) have symmetry H these subspaces are dynamically invariant. See Fig. 7 that illustrates this principle. Typical points \(x\in \operatorname {Fix}(H)\) have \(\varSigma_{x}=H\), however, some points may have more symmetry; more precisely, if \(H\subset K\) are isotropy subgroups then \(\operatorname {Fix}(H)\supset \operatorname {Fix}(K)\); and the partial ordering of the isotropy subgroups corresponds to a partial ordering of the fixed-point subspaces.

Identifying symmetries up to conjugacy allows for a considerable reduction of the number of cases one needs to consider; note that conjugate subgroups must have fixed-point subspaces of the same dimension where essentially the same dynamics will occur.

The fixed-point subspaces are often used (implicitly or explicitly) to enable one to reduce the dimension of the system and thus to make it more tractable. As an example, to determine the existence of an exactly synchronised solution one only needs to suppose \(x_{i}(t)=x(t)\) and determine whether there is such a solution \(x(t)\) for the system (7).

One way of saying this is that the only possible spatio-temporal symmetries of periodic orbits are cyclic extensions of isotropy subgroups. Further theory, outlined in [116], shows that one can characterise possible symmetries of chaotic attractors; these may include a much wider range of spatio-temporal symmetries \((H,K)\) including some that do not satisfy the hypotheses of Theorem 3.1. This means that the symmetries of attractors may contain dynamical information about the attractor.

3.9 Symmetries and Linearisation

Fortunately, there is a well-developed theory that enables one to exactly characterise the structure of the Jacobians of such maps—this involves splitting the action into a number of isotypic components according to irreducible representations that are the most trivial invariant subspaces under the action of the group. We do not have space here to go into this in detail, but refer the reader to [116] and references therein. This characterisation can be extended to nonlinear terms of vector fields, and more general invariant sets (such as periodic orbits) in addition to equilibria.

3.10 Bifurcations with Symmetry and Genericity

3.11 Robust Heteroclinic Attractors, Cycles and Networks

The presence of symmetries in a dynamical system can cause highly nontrivial dynamics even away from bifurcation points. Of particular interest are robust invariant sets that consist of networks of equilibria (or periodic orbits, or more general invariant sets) connected via heteroclinic connections that are preserved under small enough perturbations that respect the symmetries [128]. These structures may be cycles or more generally networks. They can be robust to perturbations that preserve the symmetries and indeed they can be attracting [116, 129]. We are particularly interested in the attracting case in which case we call these invariant sets heteroclinic attractors and trajectories approaching such attractors show a typical intermittent behaviour—periods that are close to the dynamics of an unstable saddle-type invariant set, and switches between different behaviours.

In higher-dimensional systems, heteroclinic attractors may have subtle structures such as “depth two connections” [130], “cycling chaos” where there are connections between chaotic saddles [116, 131, 132] and “winnerless competition” [133, 134]. Related dynamical structures are found in the literature in attractors that show “chaotic itinerancy” or “slow switching”. Such complex attractors can readily appear in neural oscillator models in the presence of symmetries and have been used to model various dynamics that contribute to the function of neural systems; we consider this, along with some examples, in Sect. 7.

3.12 Groupoid and Related Formalisms

Some less restrictive structures found in some coupled dynamical networks also have many of the features of symmetric networks (including invariant subspaces, bifurcations that appear to be degenerate, and heteroclinic attractors) but without necessarily having the symmetries.

One approach [135] has been to use a structure of groupoids—these are mathematical structures that satisfy some, but not all, of the axioms of a group and can be useful in understanding the constraints on the dynamics of coupled cell systems of the form (6). A groupoid is similar to a group except that the composition of two elements in a groupoid is not always defined, and the inverse of a groupoid element may only be locally defined. This formalism can be used to describe the permutations of inputs of cells as in [135, 136].

Given a coupling structure of this type, an admissible vector field is a vector field on the product space of all cells that respects the coupling structure, and this generalises the idea of an equivariant vector field in the presence of a symmetry group acting on the set of cells. The dynamical consequences of this have a similar flavour to the consequences one can find in symmetric systems except that fewer cases have been worked out in detail, and there are many open questions.

To illustrate, consider the system of three cells,

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}x_{1} &= g(x_{1},x_{2},x_{3}) , \\ \frac{\mathrm{d}}{\mathrm{d}t}x_{2} &= g(x_{2},x_{1},x_{3}) , \\ \frac{\mathrm{d}}{\mathrm{d}t}x_{3} &= h(x_{3},x_{1}) , \end{aligned}$$

where \(g(x,y,z)=g(x,z,y)\); this is discussed in Sect. 5 in [136] in some detail. This has a coupling structure as shown in Fig. 8. In spite of there being no exact symmetries in the system there is a nontrivial invariant subspace where \(x_{1}=x_{2}\). In the approach of [136] the dynamically invariant subspaces that can be understood in terms of the balanced colourings of the graph where the cells are grouped in such a way that the inputs are respected—this corresponds to an admissible pattern of synchrony.

The invariant subspaces that are forced to exist by this form of coupling structure have been called polydiagonals in this formalism, which correspond to clustering of the states. For every polydiagonal one can associate a quotient network by identifying cells that are synchronised, to give a smaller network. As in the symmetric case the existence of an invariant subspace does not guarantee that it contains any attracting solutions. Some work has been done to understand generic symmetry breaking bifurcations in such networks—see for example [138], or spatially periodic patterns in lattice networks [139]. Variants of this formalism have been developed to enable different coupling types between the same cells to be included.

Periodic orbits in such networks can also have interesting structures associated with the presence of invariant subspaces. The so-called rigid phase conjecture [136, 140], recently proved in [141], states that if there is a periodic orbit in the network such that two cells have a rigid phase relation between them (i.e. one that is preserved for all small enough structure-preserving perturbations) then this must be forced by either a \(\mathbb {Z}_{n}\) symmetric perturbation of the cells in the network, or in some quotient network.

An alternative formalism for discussing asymmetric coupled cell networks has been developed in [137, 142–144] that also allows one to identify invariant subspaces. Each cell has one output and several inputs that may be of different types. These papers concentrate on the questions: (a) When are two-cell networks formally equivalent (i.e. when can the dynamics of one cell network be found in the other, under suitable choice of cell)? (b) How can one construct larger coupled cell systems with desired properties by “inflating” a smaller system S, such that the larger system has S as a quotient? (c) What robust heteroclinic attractors exist in such systems?

4.1 Stability of the Synchronised State for Complex Networks of Identical Systems

For a ring of identical (or near identical) coupled periodic oscillators in which the connections have randomly heterogeneous strength, Restrepo et al. [160] have used the MSF method to determine the possible patterns at the desynchronisation transition that occurs as the coupling strengths are increased. Interestingly they demonstrate Anderson localisation of the modes of instability, and show that this could organise waves of desynchronisation that would spread to the whole network. For a further discussion as regards the use of the MSF formalism in the analysis of synchronisation of oscillators on complex networks we refer the reader to [75, 161], and for the use of this formalism in a non-smooth setting see [162]. This approach has recently been extended to cover the case of cluster states by making extensive use of tools from computational group theory to determine admissible patterns of synchrony [163] (and see also Sect. 3.12) in unweighted networks.

4.2 Pulse-Coupled Oscillators

Another example of a situation in which analysis of network dynamics can be carried out without the need for any reduction or assumption is that of pulse-coupled oscillators, in which interactions between neurons are mediated by instantaneous “kicks” of the voltage variable.

In the absence of coupling each oscillator has period Δ and there is a natural phase variable \(\phi(t) =t/\Delta\mod1\) such that \(\phi=0\) when \(v=0\) and \(\phi=1\) when \(v=1\). Mirollo and Strogatz further assume that the dynamics of each (uncoupled) oscillator is governed by \(v(t)=f(\phi)\) where f is a smooth function satisfying \(f(0)=0\), \(f(1)=1\), \(f^{\prime}(\phi)>0\) and \(f^{\prime\prime}(\phi)<0\) for all \(\phi\in[0,1]\). Because of these hypotheses on f, it is invertible with inverse \(\phi=g(v)\).

If an oscillator is pulled up to firing threshold due to the coupling and firing of a group of m oscillators which have already synchronised then the oscillator is ‘absorbed’ into the group and remains synchronised with the group for all time. (Here synchrony means firing at the same time.) Since there are now more oscillators in the synchronised group, the effect of the coupling on the remaining oscillators is increased and this acts to rapidly pull more oscillators into synchronisation. Mirollo and Strogatz [164] proved that for pulsatile coupling and f satisfying the conditions above, the set of initial conditions for which the oscillators do not all become synchronised has zero measure. Here we briefly outline the proof for two pulse-coupled oscillators. See Mirollo and Strogatz [164] for the generalisation of this proof to populations of size N.

Consider two oscillators labelled A and B with \(\phi_{A}\) and \(v_{A}\) denoting, respectively, the phase and state of oscillator A and similarly for oscillator B. Suppose that oscillator A has just fired so that \(\phi_{A}=0\) and \(\phi_{B}=\phi\) as in Fig. 9(a). The return map \(R(\phi)\) is the phase of B immediately after A next fires. It can be shown that the return map has a unique, repelling fixed point:

It can be shown that almost all initial conditions eventually become synchronised since (i) R has a unique fixed point \(\overline{\phi} \in(\delta, h^{-1}(\delta))\) and (ii) this fixed point is unstable (i.e. \(\vert R^{\prime}(\overline{\phi})\vert >1\)). To see that R has a unique fixed point, observe that fixed points ϕ̅ are roots of \(F(\phi)\equiv\phi-h(\phi)\). Now \(F(\delta)=\delta-1<0\) and \(F(h^{-1}(\delta))= h^{-1}(\delta )-\delta>0\) so F has a root in \((\delta, h^{-1}(\delta))\) and this root is unique since \(F^{\prime}(\phi)=1-h^{\prime}(\phi)>2\).

Extensions to the framework of Mirollo and Strogatz include the introduction of a time delay in the transmission of pulses and the consideration of inhibitory coupling. It has been observed that delays have a dramatic effect on the dynamics in the case of excitatory coupling. Considering first a pair of oscillators, Ernst et al. [165] demonstrate analytically that inhibitory coupling with delays gives stable in-phase synchronisation while for excitatory coupling, synchronisation with phase lag occurs. As the number of globally coupled oscillators increases, so does the number of attractors which can exist for both excitatory and inhibitory coupling.

In the presence of delays many different cluster state attractors can coexist. The dynamics settle down onto a periodic orbit with clusters reaching threshold and sending pulses alternately [165–167]. Under the addition of weak noise when the coupling is inhibitory, the dynamics stay near this periodic orbit indicating that all cluster state attractors are stable [167]. However, the collective behaviour shows a marked difference when the coupling is excitatory. In this case, weak noise is sufficient to drive the system away from the periodic orbit and results in persistent switching between unstable (Milnor) attractors.

These dynamics are somewhat akin to heteroclinic switching and the relationship between networks of unstable attractors and robust heteroclinic cycles has been addressed by a number of authors [168–170]. In particular, Broer et al. [170] highlight a situation in which there is a bifurcation from a network of unstable attractors to a heteroclinic cycle within a network of pulse-coupled oscillators with delays and inhibitory coupling. They note that the model used in previous work [165–167] is locally noninvertible since the original phase of an oscillator cannot be recovered once it has received an input which takes it over threshold causing the phase to be reset. Kirst and Timme [170] employ a framework in which supra-threshold activity is partially reset, such that \(v_{j}(t^{+})= \mathcal{R}(v_{j}(t)-1)\) if \(v_{j}>1\) with a reset function \(\mathcal {R}(v)=cv\), \(c \in[0,1]\), which ensures that the flow becomes locally time invertible when \(c>0\). They demonstrate that for \(c=0\) (where the locally noninvertible dynamics is recovered), the system has a pair of periodic orbits \(A_{1}\) and \(A_{2}\), which are unstable attractors enclosed by the basin of each other. When \(c>0\), \(A_{1}\) and \(A_{2}\) are non-attracting saddles with a heteroclinic connection from \(A_{1}\) to \(A_{2}\). Furthermore, there is a continuous bifurcation from the network of two unstable attractors when \(c=0\) to a heteroclinic two cycle when \(c>0\).

For an interesting dynamical systems perspective on the differences between “kick” synchronisation (in pulsatile coupled systems) and “diffusive” synchronisation [171] and the lack of mathematical work on the former problem see [172]. For example, restrictions on the dynamics of symmetrically coupled systems of oscillators when the coupling is time-continuous can be circumvented for pulsatile coupling leading to more complex network dynamics [173].

In the real world of synaptic interactions, however, pulsatile kicks are probably the exception rather than the rule, and the biology of neurotransmitter release and uptake is better modelled with a distributed delay process, giving rise to a postsynaptic potential with a finite rise and fall time. For spike-time event driven synaptic models, described in Sect. 2.5, analysis at the network level is hard for a general conductance-based model (given the usual expectation that the single neuron model will be high-dimensional and nonlinear), though far more tractable for LIF networks, especially when the focus is on phase-locked states [174–176]. Indeed in this instance many results can be obtained in the strongly coupled regime [177], without recourse to any approximation or reduction.

4.3 Synaptic Coupling in Networks of IF Neurons

As an explicit example let us consider the synchronous state (\(\phi _{i}=\phi\) for all i). From (13) we see that this is guaranteed to exist for the choice \(\sum_{j} W_{ij}=\gamma\) and \(I_{i}=I[1-\epsilon K(0) \gamma]\) for some constant \(I>1\), which sets the period as \(\Delta= \ln[I/(I-1)]\). Using the result that \(K'(\phi )=-\Delta K(\phi) + \Delta P(\phi\Delta)/I\) the spectral problem for the synchronous state then takes the form

$$ \bigl[ (\lambda-1) \bigl(I-1 + \epsilon I \gamma K'(0)/\Delta \bigr) + \epsilon\gamma K'(0)/\Delta \bigr] \delta_{i}= \epsilon G(0,\lambda) \sum_{j=1}^{N} W_{ij} \delta_{j} . $$

We may diagonalise this equation in terms of the eigenvalues of the weight matrix, denoted by \(\nu_{p}\) with \(p=1,\ldots, N\) (and we note that γ is the eigenvalue with eigenvector \((1,1,\ldots,1)\) corresponding to a uniform phase shift). Looking for nontrivial solutions then gives the set of spectral equations \(\mathcal{E}_{p}(\lambda)=0\), where

$$ \mathcal{E}_{p}(\lambda) = (\lambda-1) \bigl(I-1+\epsilon I \gamma K'(0)/\Delta \bigr) + \epsilon\gamma K'(0)/\Delta- \epsilon\nu_{p} G(0,\lambda). $$

(15)

We may use (15) to determine bifurcation points defined by \(\vert \lambda \vert =1\). For sufficiently small ϵ, solutions to (15) will either be in a neighbourhood of the real solution \(\lambda=1\) or in a neighbourhood of one of the poles of \(G(0,\lambda)\). Since the latter lie in the unit disc, the stability of the synchronous solution (for weak coupling) is determined by \(\epsilon K'(0)[\operatorname {Re}\nu_{p}-\gamma] <0\). For strong coupling the characteristic equation has been solved numerically in [177] to illustrate the possibility of Hopf bifurcations (\(\lambda=\mathrm {e}^{i \theta}\), \(\theta\neq0\), \(\theta \neq\pi\)) with increasing \(\vert \epsilon \vert \), leading to oscillator death in a globally coupled inhibitory network for a sufficiently slow synapse and bursting behaviour in asymmetrically coupled networks. Bifurcation diagrams illustrating these phenomenon are shown in Fig. 10. To see how these phenomena can occur from a more analytical perspective it is useful to consider the Fourier representation \(J(t) = (2 \pi )^{-1} \int_{-\infty}^{\infty}\mathrm{d}\omega\widetilde{J}(\omega ) \mathrm {e}^{i \omega t}\), where \(\widetilde{J}(\omega)=\alpha^{2}/(\alpha+i \omega )^{2}\), so that G may easily be evaluated with \(\lambda=\mathrm {e}^{z}\) as

$$ G\bigl(0,\mathrm {e}^{z}\bigr) = \frac{1}{\Delta} \sum _{n \in \mathbb {Z}} \frac{\widetilde{J}(\omega _{n}-iz/\Delta) (i \omega_{n} + z/\Delta) (\mathrm {e}^{z}-\mathrm {e}^{-\Delta}) }{1+i \omega _{n} + z/\Delta},\quad \omega_{n} = 2 \pi n/\Delta, $$

(16)

from which it is easy to see a pole at \(z=-\alpha\Delta\). This suggests writing z in the form \(z=-\alpha(1+\kappa_{p}) \Delta\) and expanding the spectral equation in powers of α to find a solution. For small α we find from (16) that \(G(0,\mathrm {e}^{z}) = -\alpha(1+\kappa_{p}) (1-\mathrm {e}^{-\Delta })/(\kappa_{p}^{2} \Delta)\). Balancing terms of order α then gives \(\kappa^{2}_{p}=\epsilon\nu_{p} (1-\mathrm {e}^{-\Delta})/(\Delta^{2}(I-1))\), where we use the result that \(G(0,\mathrm {e}^{0}) = O(\alpha^{2})\). Thus for small α, namely slow synaptic currents, we have \(K'(0)=0\), so that a weak-coupling analysis would predict neutral stability (consistent with the notion that a set of IF neurons with common constant input would frequency lock with an arbitrary set of phases). However, our strong coupling analysis predicts that the synchronous solution will only be stable if \(\operatorname {Re}z^{\pm}_{p}<0\) with \(z^{\pm}_{p}=[-1\pm\kappa_{p}] \alpha \Delta\). Introducing the firing rate function \(f=1/\Delta\) then \(z^{\pm}_{p}\) can be written succinctly as

$$ z^{\pm}_{p}= \bigl[-1 \pm\sqrt{\epsilon\nu_{p} f'(I)} \bigr] \alpha \Delta. $$

Thus for an asymmetric network (with at least one complex conjugate pair of eigenvalues) it may occur that as \(\vert \epsilon \vert \) is increased a pair of eigenvalues determined by \(z^{\pm}_{p}\) may cross the imaginary axis to the right hand complex plane signalling a discrete Hopf bifurcation in the firing times. For a symmetric network with real eigenvalues an instability may occur as some \(\kappa_{p} \in \mathbb {R}\) increases through 1, signalling a breakdown of frequency locking. The above results (valid for slow synapses) can also be obtained using a firing rate approach, as described in [177].

The results above, albeit valid for strong coupling, are only valid for LIF networks. To obtain more general results for networks of limit-cycle oscillators it is useful to consider a reduction to phase models.

5.1 Isochronal Coordinates

There are very few instances where the isochrons can be computed in closed form (though see the examples in [179] for plane-polar models where the radial variable decouples from the angular one). Computing the isochron foliation of the basin of attraction of a limit cycle is a major challenge since it requires knowledge of the limit cycle and therefore can only be computed in special cases or numerically.

One computationally efficient method for numerically determining the isochrons is backward integration, however, it is unstable and in particular for strongly attracting limit cycles the trajectories determined by backwards integration may quickly diverge to infinity. See Izhikevich [48] for a MATLAB code which determines smooth curves approximating isochrons. Other methods include the continuation-based algorithm introduced by Osinga and Moehlis [180], the geometric approach of Guillamon and Huguet to find high-order approximations to isochrons in planar systems [181], quadratic- and higher-order approximations [182, 183], and the forward integration method using the Koopman operator and Fourier averages as introduced by Mauroy and Mezić [184]. This latter method is particularly appealing and given its novelty we describe the technique below.

The Koopman operator approach for constructing isochrons for a T-periodic orbit focuses on tracking observables (or measures on a state space) rather than the identification of invariant sets. The Koopman operator, K, is defined by \(K = z \circ\varPhi_{t}(x)\), where \(z: \mathbb {R}^{n} \rightarrow \mathbb {R}\) is some observable of the state space and \(\varPhi_{t}(x)\) denotes the flow evolved for a time t, starting at a point x. The Fourier average of an observable z is defined as

$$ \widehat{z}(x;\omega) = \lim_{T \rightarrow\infty} \frac {1}{T}\int _{0}^{T} (z \circ\varPhi_{t}) (x) \mathrm {e}^{-i \omega t} \,\mathrm{d}t . $$

(18)

For a fixed x, (18) is equivalent to a Fourier transform of the (time-varying) observable computed along a trajectory. Hence, for a dynamics with a stable limit cycle (of frequency \(2\pi/T\)), it is clear that the Fourier average can be nonzero only for the frequencies \(\omega_{n}= 2 \pi n/ T\), \(n \in \mathbb {Z}\). The Fourier averages are the eigenfunctions of K, so that

$$ K \widehat{z}(x;\omega_{n}) = \mathrm {e}^{i \omega_{n} t} \widehat{z}(x;\omega _{n}),\quad n \in \mathbb {Z}. $$

Perhaps rather remarkably the isochrons are level sets of \(\widehat {z}(x;\omega_{n})\) for almost all observables. The only restriction being that the first Fourier coefficient of the Fourier observable evaluated along the limit cycle is nonzero over one period. An example of the use of this approach is shown in Fig. 11, where we plot the isochrons of a Stuart–Landau oscillator.

5.2 Phase–Amplitude Models

An alternative (non isochronal) framework for studying oscillators with an attracting limit cycle is to make a transformation to a moving orthonormal coordinate system around the limit cycle where one coordinate gives the phase on the limit cycle while the other coordinates give a notion of distance from the limit cycle. It has long been known in the dynamical systems community how to construct such a coordinate transformation; see [186] for a discussion. The importance of considering the effects of both the phase and the amplitude interactions of neural oscillators has been highlighted by several authors including Ermentrout and Kopell [187] and Medvedev [188], and that this is especially pertinent when considering phenomenon such as oscillator death (and see Sect. 3.4). Phase–amplitude descriptions have already successfully been used to find equations for the evolution of the energies (amplitudes) and phases of weakly coupled weakly dissipative networks of nonlinear planar oscillators (modelled by small dissipative perturbations of a Hamiltonian oscillator) [103, 189, 190]. Lee et al. [191] use the notion of phase and amplitudes of large networks of globally coupled Stuart–Landau oscillators to investigate the effects of a spread in amplitude growth parameter (units oscillating with different amplitudes and some not oscillating at all) and the effect of a homogeneous shift in the nonlinear frequency parameter.

Some caution must be exercised when applying this transformation as it will break down when the determinant of the Jacobian of the transformation vanishes. This never occurs on cycle (where \(\rho=0\)) but it may do so for some \(|\rho|=k>0\), setting an upper bound on how far from the limit cycle these phase–amplitude coordinates can be used to describe the system. In [192] it is noted that for the planar ML model the value of k can be relatively small for some values of ϑ, but that breakdown occurs where the orbit has high curvature. In higher-dimensional systems this issue would be less problematic.

Medvedev [188] has employed this phase–amplitude description to determine conditions for stability of the synchronised state in a network of identical oscillators with separable linear coupling. Medvedev [194] has also used the framework to consider the effects of white noise on the synchronous state, identifying the types of linear coupling operators which lead to synchrony in a network of oscillators provided that the strength of the interactions is sufficiently strong.

5.3 Dynamics of Forced Oscillators: Shear-Induced Chaos

Since phase–amplitude coordinates can capture dynamics a finite distance away from the limit cycle (and additionally have the advantage over isochronal coordinates of being defined outside of the basin of attraction of the limit cycle), they can be used to model dynamical phenomena in driven systems where the perturbations necessarily push the dynamics away from the limit cycle. There is no need to make any assumptions about the strength of the forcing ϵ.

The phase–amplitude description of a forced oscillator is able to detect the presence of other structures in the phase space. For example if the system were multistable, phase–amplitude coordinates would track trajectories near these other structures and back again, should another perturbation return the dynamics to the basin of attraction of the limit cycle. These coordinates would also detect the presence of other non-attracting invariant structures such as saddles in the unperturbed flow. Orbits passing near the saddle will remain there for some time and forcing may act to move trajectories near this saddle before returning to the limit cycle. It may also be the case that the forcing acts to create trapping regions if the forcing is strong compared to the attraction to the limit cycle.

Another dynamical phenomenon which phase–amplitude coordinates are able to capture is the occurrence of shear-induced chaos. Shear in a system near a limit cycle Γ is the differential in components of the velocity tangent to the limit cycle as one moves further from Γ in phase space. Shear forces within the system act to speed up (or slow down) trajectories further away from the limit cycle compared to those closer to it. This phenomenon is illustrated in Fig. 12.

As we show below, when an impulsive force is applied (the system is kicked) a ‘bump’ in the image of Γ is produced. If there is sufficient shear in the system then the bump is folded and stretched as it is attracted back to the limit cycle. Such folding can potentially lead to the formation of horseshoes and strange attractors. However, if the attraction to the limit cycle is large compared to the shear strength or size of the kick then the bumps will dissipate before any significant stretching occurs.

Shear-induced chaos is most commonly discussed in the context of discrete time kicking of limit cycles. Wang and Young [195–197] prove rigorous results in the case of periodically kicked limit cycles with long relaxation times. Their results provide details of the geometric mechanism for producing chaos. Here we briefly review some of these results. More detailed summaries can be found in [198] and [199].

For a more general system with periodic forcing the shear may not appear explicitly as a parameter. To elucidate what kind of kicks may cause shear induced chaos in this case we appeal to the isochrons of the system. Suppose, as illustrated in Fig. 12, that a section \(\varGamma _{0}\) of the limit cycle is kicked upwards with the end points held fixed and assume \(\tau= np\) for some \(n,p \in\mathbb{Z}^{+}\). Since the isochrons are invariant under the action of \(\varPhi_{T}\), during relaxation the flow moves each point of the kicked curve \(\kappa(\varGamma_{0})\) back towards Γ along the isochrons. In Fig. 12 we can clearly see the effect of the shear with a fold forming.

From Fig. 12 one can see that kicks along isochrons or in directions roughly parallel to the isochrons will not produce strange attractors, nor will kicks that carry points from one isochron to another. The cause of the stretching and folding is the variation in how far points \(x \in\varGamma\) are moved by κ in the direction transverse to the isochrons (i.e. the ordering of points in terms of asymptotic phase is altered by the action of the kick). Lin and Young [198] emphasise that the occurrence of shear induced chaos depends on the interplay between the geometries of the kicks and the dynamical structures of the unforced system.

In the case of the linear shear model above, given by Eq. (23), the isochrons of the unforced system are simply the lines with slope \(-\lambda/\sigma\) in \((\vartheta, y)\) coordinates. Variation in kick distances in directions transverse to these isochrons is guaranteed with any non-constant function H, with greater variation given by larger values of \(\sigma/\lambda\) and A.

Beyond the rigorous results proved by Wang and Young [195–198] concerning periodically kicked limit cycles of the linear shear model and for supercritical Hopf bifurcations, Ott and Stenlund [193] prove that shear-induced chaos may exist near general limit cycles. In addition, Lin and Young [198] have carried out numerical studies including random kicks at times given by a Poisson process and systems driven by white noise. They also consider forcing of a pair of coupled oscillators. In all cases, shear-induced chaos occurs when the shearing and amplitude of the forcing are large enough to overcome the effects of damping.

Lin et al. [200] demonstrate that the ML model can exhibit shear-induced chaos near the homoclinic bifurcation when periodically forced, by plotting images of the periodic orbit under successive applications of the kick map and calculating the maximum Lyapunov exponent. They also emphasise that the phenomenon of shear-induced chaos cannot be detected by the perturbative techniques such as those outlined in Sect. 5.4 and Sect. 5.5 below.

The analysis of the behaviour of generic networks of oscillators within a phase–amplitude framework is a challenging open problem but such a description would allow for greater accuracy (compared to the phase-only methods traditionally used and described below) in elucidating a richer variety of the complex dynamics of oscillator networks.

5.4 Phase Oscillator Models

The iPRC at a point on cycle is equal to the gradient of the (isochronal) phase at that point. Writing this vector quantity as Q means that weak forcing \(\epsilon g(t)\) in the original high-dimensional models transforms as \({\mathrm {d}\vartheta}/{\mathrm {d}t}=1+\epsilon \langle Q, g \rangle\) where \(\langle\cdot,\cdot\rangle\) defines the standard inner product and ϵ is a small parameter. For periodic forcing such equations can be readily analysed, and questions relating to synchronisation, mode-locking and Arnol’d tongues can be thoroughly explored [76]. Moreover, this approach forms the basis for constructing models of weakly interacting oscillators, where the external forcing is pictured as a function of the phase of a firing neuron. This has led to a great deal of work on phase-locking and central pattern generation in neural circuitry and see for example [43].

However, the assumption that phase alone is enough to capture the essentials of neural response is one made more for mathematical convenience than being physiologically motivated. Indeed for the popular type I ML firing model with standard parameters, direct numerical simulations with pulsatile forcing show responses that cannot be explained solely with a phase model [200], as just highlighted in Sect. 5.3 (since strong interactions will necessarily take one away from the neighbourhood of a cycle where a phase description is expected to hold).

5.5 Phase Response

In general, Eq. (26) must be solved numerically to obtain the iPRC, say, using the adjoint routine in XPPAUT [127] or MatCont [209]. However, for the case of a nonlinear IF model, defined by (1), the PRC is given explicitly by

$$ Q(\vartheta) = \frac{1}{I+f \circ\varPsi^{-1}(\vartheta)},\quad \varPsi (v) = \tau\int_{v_{{R}}}^{v(t)} \frac{\mathrm{d}v'}{I+f(v')} , $$

(27)

where the period of oscillation is found by solving \(\varPsi(v_{\mathrm{th}})=T\) (and the response function is valid for finite perturbations). For example, for the quadratic IF (QIF) model, obtained from (1) with the choice with \(f(v)=v^{2}\), and taking the limit \(v_{\mathrm{th}} \rightarrow\infty\) and \(v_{{R}} \rightarrow-\infty\) we find \(Q(\vartheta) = \sin^{2}(\vartheta\pi/T)/I\) with \(T=\pi\tau/\sqrt{I}\), recovering the iPRC expected of an oscillator close to a SNIC bifurcation [210, 211]. For a comprehensive discussion of iPRCs for various common oscillator models see the excellent survey article by Brown et al. [201]. The iPRC for planar piece-wise linear IF models can also be computed explicitly [58], although we do not discuss this further here. In Fig. 13 we show the numerically computed iPRCs for the Hodgkin–Huxley model, the Wilson cortical model and the Morris–Lecar model previously discussed in Sect. 2. For completeness it is well to note the iPRC for a planar oscillator close to a supercritical Hopf bifurcation has an adjoint with a shape given by \((-\sin(2 \pi t/T),\cos(2 \pi t/T))\) for an orbit shape \((\cos (2 \pi t/T), \sin(2 \pi t/T))\).

6.1 Phase, Frequency and Mode Locking

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].

6.2 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

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).

6.2.2 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.

6.2.4 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.

6.3 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].

6.3.1 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 \(\varPsi= (\epsilon_{1} W_{1}+\epsilon_{2} W_{2} )\otimes z + \epsilon_{3} W_{3} \otimes z^{2}\).

7.1 Robust Heteroclinic Attractors for Phase Oscillator Networks

Hansel et al. [231] considered the dynamics of (32) with global coupling and phase interaction function of the form (39) for \((r,\alpha)\) fixed parameters. For large N, they find an open region in parameter space where typical attractors are heteroclinic cycles that show slow switching between states where the clustering is into two clusters of macroscopic size. This dynamics is examined in more depth in [244] where the simulations for typical initial conditions show a long and intermittent transient to a two-cluster state that, surprisingly, is unstable. This is a paradox because only a low-dimensional subset of initial conditions (the stable manifold) should converge to a saddle. The resolution of this paradox is a numerical effect: as the dynamics approaches the heteroclinic cycle where the connection is in a clustered subspace, there can be numerical rounding into the subspace. For weak perturbation of the system by additive noise, [244] find that the heteroclinic cycle is approximated by a roughly periodic transition around the cycle whose approximate period scales as the logarithm of the noise amplitude.

The bifurcations that give rise to heteroclinic attractors in this system on varying \((r, \alpha)\) are quite complex even for small N. As discussed in [232] one can only find attracting robust heteroclinic attractors in (32), (39) for \(N\geq4\): in this case Fig. 15 shows a region where robust heteroclinic attractors of the type illustrated in Fig. 21 appear. A trajectory approaching such a network will spend much of its time near a cluster state with two groups of two oscillators each. Each time there is a connection between the states, one of the groups will break clustering for a short time, and over a longer period the clusters will alternate between breaking up and keeping together. Qualitatively similar heteroclinic attractors can be found for example in coupled Hodgkin–Huxley type limit-cycle oscillators with delayed synaptic coupling as detailed in [251] and illustrated in Fig. 22.

The heteroclinic attractors that appear for \(N>4\) can have more complex structures. For \(N=5\) this is investigated in [253, 254] for (32), (39) and in [252] for the phase interaction function

$$ H(\varphi)=\sin(\varphi+\alpha)-r\sin(2\varphi+\beta) , $$

(45)

where α, β and r are parameters. Figure 23 illustrates a trajectory of (32) with global coupling and phase interaction function (45) as a raster plot for five phase oscillators with parameters \(r=0.2\), \(\alpha=1.8\), \(\beta=-2.0\) and \(\omega=1.0\) and with addition of noise of amplitude 10⁻⁵. Observe there is a dynamic change of clustering over the course of the time-series with a number of switches taking place between cluster states of the type

$$(\theta_{1},\ldots,\theta_{5})=\varOmega t (1,1,1,1,1)+(y,y,g,b,b), $$

where y, g and b represent different relative phases that are coloured “yellow”, “green” or “blue” in Fig. 24 to other symmetrically related cluster states. One can check that the group orbit of states with this clustering gives 30 symmetrically related cluster states for the system; details of the connections are shown in Fig. 24. All cluster states connect together to form a single large heteroclinic network that is an attractor for typical initial conditions [252]. Figure 23 illustrates the possible switchings between phase differences near one particular cluster state for the heteroclinic network in this case. The dynamics of this system can be used for encoding a variety of inputs, as discussed in [255] where it is shown that a constant heterogeneity of the natural frequencies between oscillators in this system leads to a spatio-temporal encoding of the heterogeneities. The sequences produced by the system can be used by a similar system with adaptive dynamics to learn a spatio-temporal encoded state [228].

Robust heteroclinic attractors also appear in a range of coupled phase oscillator models where the coupling is not global (all-to-all) but such that it still preserves enough invariant subspaces for the connections to remain robust. For example, [222] study the dynamics of a network “motif” of four coupled phase oscillators and find heteroclinic attractors that are “ratchets”, i.e. they are robust heteroclinic networks that wind preferentially around the phase space in one direction—this means that under the influence of small perturbations, phase slips in only one direction can appear.

7.2 Winnerless Competition and Other Types of Heteroclinic Attractor

Analogous behaviour has been found in a range of other coupled systems, for example [137] or delayed pulse-coupled oscillators [167–170, 245, 258]. Recent work has also considered an explicit constructive approach to heteroclinic networks to realise arbitrary directed networks as a heteroclinic attractor of a coupled cell system [247] or as a closely-related excitable network attractor that appears at a bifurcation of the heteroclinic attractor [259].

More complex but related dynamical behaviour has been studied under the names of “chaotic itinerancy” (see for example [260]), “cycling chaos” [131], “networks of Milnor attractors” [261] and “heteroclinic cycles for chaotic oscillators” [262]. It has been suggested that these and similar models are useful for modelling of the functional behaviour of neural systems [263].

Because heteroclinic attractors are quite singular in their dynamical behaviour (averages of observables need not converge, there is a great deal of sensitivity of the long-term dynamics to noise and system heterogeneity), it is important to consider the effect of noise and/or heterogeneities in the dynamics. This leads to a finite average transition time between states determined by the level of noise and/or heterogeneity (which may be due to inputs to the system) and the local dynamics—see for example [264]. Another useful feature of heteroclinic attractors is that they allow one to model “input–output” response of the system to a variety of inputs.

8.1 Phase Reduction of a Planar System with State-Dependent Gaussian White Noise

8.2 Phase Reduction for Noise with Temporal Correlation

9.1 Ott–Antonsen Reduction for the Winfree Model

Recent work by Montbrió et al. [293] has shown that QIF networks can also be analysed without having to invoke the OA ansatz. Indeed they show that choosing a Lorentzian distribution for the voltages also leads to a reduced mean-field description that tracks the population firing rate and the mean membrane voltage. Interestingly they also relate these variables back to the complex Kuramoto order parameter via a conformal mapping. A treatment of QIF networks with gap-junction coupling has also recently been achieved by Laing [294], using the OA ansatz, showing how this can destroy certain spatio-temporal patterns, such as localised “bumps”, and create others, such as travelling waves and spatio-temporal chaos. The OA ansatz has also proved remarkably useful in understanding nontrivial solutions such as chimera states (where a sub-population of oscillators synchronises in an otherwise incoherent sea).

9.2 Chimera States

Phase or cluster synchronised states in systems of identical coupled oscillators have distinct limitations as descriptions of neural systems where not just phase but also frequency clearly play a part in the processing, computation and output of information. Indeed, one might expect that for any coupled oscillator system that is homogeneous (in the sense that any oscillators can be essentially replaced by any other by a suitable permutation of the oscillators), the only possible dynamical states are homogeneous in the sense that the oscillators behave in either a coherent or an incoherent way. This expectation, however, is not justified—there can be many dynamical states that cannot easily be classified as coherent or incoherent, but that seem to have a mixture of coherent and incoherent regions. Such states have been given the name “chimera state” by Abrams and Strogatz [295, 296] and have been the subject of intensive research over the past five years. For reviews of chimera state dynamics we refer the reader to [297, 298].

Kuramoto and Battogtokh [299, 300] investigated continuum systems of oscillators of the form

$$ \frac{\partial}{\partial t}\theta(x,t) = \omega- \epsilon\int_{D} G \bigl(x-x'\bigr) \sin\bigl(\theta(x,t)-\theta\bigl(x',t \bigr) +\alpha\bigr) \,\mathrm{d}x' , $$

(62)

where θ represent phases at locations \(x \in D \subseteq \mathbb {R}\), the kernel \(G(u)=\kappa\exp(-\kappa \vert u\vert )/2\) represents a non-local coupling and ω, α, κ are constants. Interestingly this model is precisely in the form presented in Sect. 6.3 as Eq. (40) for an oscillatory model of cortex, although here there are no space-dependent delays and the interaction function is \(H(\theta) = \sin(\theta-\alpha)\). Kuramoto and Battogtokh found for a range of parameters near \(\alpha=\pi/2\), and carefully selected initial conditions that the oscillators can split into two regions in x, one region which is frequency synchronised (or coherent) while the other region shows a nontrivial dependence of frequency on location x. An example of a chimera state is shown in Fig. 25.

However, it seems that chimera states are much more “slippery” in finite oscillator systems than in the continuum limit. In particular, Wolfrum and Omel’chenko [302] note that for finite approximations of the ring (62) by N oscillators, with a mixture of local and nearest R-neighbour coupling corresponding to (63) with a particular choice of coupling matrix \(K_{ij}\), chimera states apparently only exist as transients. However, the lifetime of the typical transient apparently grows exponentially with N. Thus, at least for some systems of the form (63), chimeras appear to be a type of chaotic saddle. This corresponds to the fact that the boundaries between the regions of coherent and incoherent oscillation fluctuate apparently randomly over a long time scale. These fluctuations lead to wandering of the incoherent region as well as change in size of the region. Eventually these fluctuations appear to result in typical collapse to a fully coherent oscillation [302].

Although this appears to be the case for chimeras for (63), there are networks such as coupled groups of oscillators; [303] or two-dimensional lattices [304] where chimera attractors can appear. It is not clear what will cause a chimera to be transient or not, or indeed exactly what types of chimera-like states can appear in finite oscillator networks. A suggestion of [305] is that robust neutrally stable chimeras may be due to the special type of single-harmonic phase interaction function used in (62), (63).

More recent work includes investigations of chimeras (or chimera-like states) in chemical [306] or mechanical oscillator networks [307]; chimeras in systems of coupled oscillators other than phase oscillators have been investigated in many papers; for example in Stuart–Landau oscillators [299, 308, 309], Winfree oscillators [290] and models with inertia [310]. Other recent work includes discussion of feedback control to stabilise chimeras [311], investigations of chimeras with multiple patches of incoherence [312], multicluster and travelling chimera states [313].

In a neural context chimeras have also been found in pulse-coupled LIF networks [314], and hypothesised to underly coordinated oscillations in unihemispheric slow-wave sleep, whereby one brain hemisphere appears to be inactive while the other remains active [315].

10.1 Functional and Structural Connectivity in Neuroimaging

Functional connectivity (FC) refers to the temporal synchronisation of neural activity in spatially remote areas. It is widely believed to be significant for the integrative processes in brain function. Anatomical or structural connectivity (SC) plays an important role in determining the observed spatial patterns of FC. However, there is clearly a role to be played by the dynamics of the neural tissue. Even in a globally connected network we would expect this to be the case, given our understanding of how synchronised solutions can lose stability for weak coupling. Thus it becomes useful to study models of brain like systems built from neural mass models (such as the Jansen–Rit model of Sect. 2.6), and ascertain how the local behaviour of the oscillatory node dynamics can contribute to global patterns of activity. For simplicity, consider a network of N globally coupled identical Wilson–Cowan [65] neural mass models:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} {x}_{i} &= -x_{i}+P+c_{1} f(x_{i})-c_{2} f(y_{i}) +\frac {\epsilon }{N} \sum _{j=1}^{N} f(x_{j}) , \\ \frac{\mathrm{d}}{\mathrm{d}t} {y}_{i} &= -y_{i}+Q+c_{3} f(x_{i})-c_{4} f(y_{i}) , \end{aligned}$$

for \(i=1,\ldots, N\). Here \((x_{i},y_{i}) \in \mathbb {R}^{2}\) represents the activity in each of a pair of coupled neuronal population models, f is a sigmoidal firing rate given by \(f(x) = 1/(1+\mathrm {e}^{-x})\) and \((P,Q)\) represent external drives. The strength of connections within a local population is prescribed by the coefficients \(c_{1},\ldots, c_{4}\), which we choose as \(c_{1} = c_{2} = c_{3} = 10\) and \(c_{4} = -2\) as in [43]. For \(\epsilon=0\) it is straightforward to analyse the dynamics of a local node and find the bifurcation diagram in the \((P,Q)\) plane as shown in Fig. 26. Moreover, for \(\epsilon\ll1\) we may invoke weak-coupling theory to describe the dynamics of the full network within the oscillatory regime bounded by the two Hopf curves shown in Fig. 26. From the theory of Sect. 6.2.1 we would expect the synchronous solution to be stable if \(\epsilon H'(0) >0\). Taking \(\epsilon>0\) we can consider \(H'(0)\) as a proxy for the robustness of synchrony. The numerical construction of this quantity, as in [317], predicts that there will be regions in the \((P,Q)\) plane associated with a breakdown of FC (where \(H'(0) <0\)), as indicated by points a and c in Fig. 26. This highlights the role that local node dynamics can have on emergent network dynamics. Moreover, we see that simply by tuning the local dynamics to be deeper within the existence region for oscillatory solutions we can, at least for this model, enhance the degree of FC. It would be interesting to explore this simple model further for more realistic brain like connectivities, along the lines described in [318]. Moreover, given that this would preclude the existence of the synchronous state by default (since we would neither have \(H(0)=0\) nor would \(\sum_{j} w_{ij}\) be independent of i) then it would be opportune to explore the use the recent ideas in [163, 319] to understand how the system could organise into a regime of remote synchronisation whereby pairs of nodes with the same network symmetry could synchronise. For related work on Wilson–Cowan networks with some small dynamic noise see [320], though here the authors construct a phase oscillator network by linearising around an unstable fixed point, rather than use the notion of phase response.

10.2 Central Pattern Generators

CPGs are (real or notional) neural subsystems that are implicated in the generation of spatio-temporal patterns of activity [321], in particular for driving the relatively autonomous activities such as locomotion [322–324] or for driving involuntary activities such as heartbeat, respiration or digestion [325]. These systems are assumed to be behind the creation of the range of walking or running patterns (gaits) that appear in different animals [326]. The analysis of phase locking provides a basis for understanding the behaviour of many CPGs, and for a nice overview see the review articles by Marder and Bucher [327] and Hooper [328].

In some cases, such as the leech (Hirudo medicinalis) heart or Caenorhabditis elegans locomotion, the neural circuitry is well studied. For more complex neural systems and in more general cases CPGs are still a powerful conceptual tool to construct notional minimal neural circuitry needed to undertake a simple task. In this notional sense they have been extensively investigated to design control circuits for actuators for robots; see for example the review [329]. Recent work in this area includes robots that can reproduce salamander walking and swimming patterns [330]. Since the control of motion of autonomous “legged” robots is still a very challenging problem in real-time control, one hope of this research is that nature’s solutions (for example, how to walk stably on two legs) will help inspire robotic ways of doing this.

CPGs are called upon to produce one or more rhythmic patterns of actuation; in the particular problem of locomotion, a likely CPG is one that will produce the range of observed rhythms of muscle actuation, and ideally the observed transitions between then. For an early discussion of design principles for modelling CPGs, see [46]. This is an area of modelling where consideration of symmetries as in Sect. 3.6 has been usefully applied to constrain the models. For example [331] examine models for generating the gaits in a range of vertebrate animals, from those with two legs (humans) through those with four (quadrupeds such as horses have a wide range of gaits—walk, trot, pace, canter, gallop—they may use) or larger numbers of legs (myriapods such as centipedes). Insects make use of six legs for locomotion while other invertebrates such as centipedes and millipedes have a large number of legs that are to some extent independently actuated. As an example, [332] consider a schematic CPG of 2n oscillators for animals with n legs, as shown in Fig. 27(a). The authors use symmetry arguments and Theorem 3.1 to draw a number of model-independent conclusions from the CPG structure.

One can also view CPGs as a window into more fundamental problems of how small groups of neurons coordinate to produce a range of spatio-temporal patterns. In particular, it is interesting to see how the observable structure of the connections influences the range and type of dynamical patterns that can be produced. For example, [333] consider a simple three-cell “motif” networks of bursters and classify a range of emergent spatio-temporal patterns in terms of the coupling parameters. Detailed studies [334] investigate properties such as multistability and bifurcation of different patterns and the influence of inhomogeneities in the system. This is done by investigating return maps for the burst timings relative to each other.

The approach of [135, 136] discussed in Sect. 3.12 provides an interesting framework to discuss CPG dynamics in cases where the connection structure is given but not purely related to symmetries of the network. For example, [141] use that formalism to understand possible spatio-temporal patterns that arise in lattices or [100] that relates synchrony properties of small motif networks to spectral properties of the adjacency matrix.

10.3 Perceptual Rivalry

Many neural systems process information—they need to produce outputs that depend on inputs. If the system effectively has no internal degrees of freedom then this will give a functional relationship between output and input so that any temporal variation in the output corresponds to a temporal variation of the input. However, this is not the case for all but the simplest systems and often outputs can vary temporally unrelated to the input. A particularly important and well-studied system that is a model for autonomous temporal output is perceptual rivalry, where conflicting information input to a neural system results, not in a rejection or merging of the information, but in an apparently random “flipping” between possible “rival” states (or percepts) of perception. This nontrivial temporal dynamics of the perception appears even in the absence of a temporally varying input. The best studied example of this type is binocular rivalry, where conflicting inputs are simultaneously made to each eye. It is widely reported by subjects that perception switches from one eye to the other, with a mean frequency that depends on a number of factors such as the contrast of the image [335]. More general perceptual rivalry, often used in “optical illusions” such as ambiguous figures—the Rubin vase, the Necker cube—show similar behaviour with percepts shifting temporally between possible interpretations.

Various approaches [336] have been made to construct nonlinear dynamical models of the generation of a temporal shifting between possible percepts such as competition models [337], bifurcation models, ones based on neural circuitry [338], or conceptual ones [339] based on network structures [340] or on heteroclinic attractors [341].