Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience
 Peter Ashwin^{1}Email author,
 Stephen Coombes^{2} and
 Rachel Nicks^{3}
DOI: 10.1186/s1340801500336
© Ashwin et al. 2016
Received: 7 July 2015
Accepted: 30 October 2015
Published: 6 January 2016
Abstract
The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.
Keywords
Central pattern generator Chimera state Coupled oscillator network Groupoid formalism Heteroclinic cycle Isochrons Master stability function Network motif Perceptual rivalry Phase oscillator Phase–amplitude coordinates Stochastic oscillator Strongly coupled integrateandfire network Symmetric dynamics Weakly coupled phase oscillator network Winfree model1 Introduction
Coupled oscillator theory is now a pervasive part of the theoretical neuroscientist’s toolkit for studying the dynamics of models of biological neural networks. Undoubtedly this technique originally arose in the broader scientific community through a fascination with understanding synchronisation in networks of interacting heterogeneous oscillators, and can be traced back to the work of Huygens on “an odd kind of sympathy” between coupled pendulum clocks [1]. Subsequently the theory has been developed and applied to the interaction between organ pipes [2], phaselocking phenomena in electronic circuits [3], the analysis of brain rhythms [4], chemical oscillations [5], cardiac pacemakers [6], circadian rhythms [7], flashing fireflies [8], coupled Josephson junctions [9], rhythmic applause [10], animal flocking [11], fish schooling [12], and behaviours in social networks [13]. For a recent overview of the application of coupled phase oscillator theory to areas as diverse as vehicle coordination, electric power networks, and clock synchronisation in decentralised networks see the recent survey article by Dörfler and Bullo [14].
Given the widespread nature of oscillations in neural systems it is no surprise that the science of oscillators has found such ready application in neuroscience [15]. This has proven especially fruitful for shedding light on the functional role that oscillations can play in feature binding [16, 17], cognition [18], memory processes [19], odour perception [20, 21], information transfer mechanisms [22], interlimb coordination [23, 24], and the generation of rhythmic motor output [25]. Neural oscillations also play an important role in many neurological disorders, such as excessive synchronisation during seizure activity in epilepsy [26, 27], tremor in patients with Parkinson’s disease [28] or disruption of cortical phase synchronisation in schizophrenia [29]. As such it has proven highly beneficial to develop methods for the control of (de)synchronisation in oscillatory networks, as exemplified by the work of Tass et al. [30, 31] for therapeutic brain stimulation techniques. From a transformative technology perspective, oscillatory activity is increasingly being used to control external devices in brain–computer interfaces, in which subjects can control an external device by changing the amplitude of a particular brain rhythm [32].
Neural oscillations can emerge in a variety of ways, including intrinsic mechanisms within individual neurons or by interactions between neurons. At the single neuron level, subthreshold oscillations can be seen in membrane voltage as well as rhythmic patterns of action potentials. Both can be modelled using the Hodgkin–Huxley conductance formalism, and analysed mathematically with dynamical systems techniques to shed light on the mechanisms that underly various forms of rhythmic behaviour, including tonic spiking and bursting (see e.g. [33]). The high dimensionality of biophysically realistic single neuron models has also encouraged the use of reduction techniques, such as the separation of time scales recently reviewed in [34, 35], or the use of phenomenological models, such as FitzHugh–Nagumo (FHN) [36], to regain some level of mathematical tractability. This has proven especially useful when studying the response of single neurons to forcing [37], itself a precursor to understanding how networks of interacting neurons can behave. When mediated by synaptic interactions, the repetitive firing of presynaptic neurons can cause oscillatory activation of postsynaptic neurons. At the level of neural ensembles, synchronised activity of large numbers of neurons gives rise to macroscopic oscillations, which can be recorded with a microelectrode embedded within neuronal tissue as a voltage change referred to as a local field potential (LFP). These oscillations were first observed outside the brain by Hans Berger in 1924 [38] in electroencephalogram (EEG) recordings, and have given rise to the modern classification of brain rhythms into frequency bands for alpha activity (8–13 Hz) (recorded from the occipital lobe during relaxed wakefulness), delta (1–4 Hz), theta (4–8 Hz), beta (13–30 Hz) and gamma (30–70 Hz). The latter rhythm is often associated with cognitive processing, and it is now common to link large scale neural oscillations with cognitive states, such as awareness and consciousness. For example, from a practical perspective the monitoring of brain states via EEG is used to determine depth of anaesthesia [39]. Such macroscopic signals can also arise from interactions between different brain areas, the thalamocortical loop being a classic example [40]. Neural mass models (describing the coarse grained activity of large populations of neurons and synapses) have proven especially useful in understanding EEG rhythms [41], as well as in augmenting the dynamic causal modelling framework (driven by large scale neuroimaging data) for understanding how eventrelated responses result from the dynamics of coupled neural populations [42].
One very influential mathematical technique for analysing networks of neural oscillators, whether they be built from single neuron or neural mass models, has been that of weakly coupled oscillator theory, as comprehensively described by Hoppensteadt and Izhikevich [43]. In the limit of weak coupling between limitcycle oscillators, invariant manifold theory [44] and averaging theory [45] can be used to reduce the dynamics to a set of phase equations in which the relative phase between oscillators is the relevant dynamical variable. This approach has been applied to neural behaviour ranging from that seen in small rhythmic networks [46] up to the whole brain [47]. Despite the powerful tools and widespread use afforded by this formalism, it does have a number of limitations (such as assuming the persistence of the limit cycle under coupling) and it is well to remember that there are other tools from the mathematical sciences relevant to understanding network behaviour. In this review, we encompass the weakly coupled oscillator formalism in a variety of other techniques ranging from symmetric bifurcation theory and the groupoid formalism through to more “physicsbased” approaches for obtaining reduced models of large networks. This highlights the regimes where the standard formalism is applicable, and provides a set of complementary tools when it does not. These are especially useful when investigating systems with strong coupling, or ones for which the rate of attraction to a limit cycle is slow.
In Sect. 2 we review some of the key mathematical models of oscillators in neuroscience, ranging from single neuron to neural mass, as well as introduce the standard machinery for describing synaptic and gapjunction coupling. We then present in Sect. 3 an overview of some of the more powerful mathematical approaches to understanding the collective behaviour in coupled oscillator networks, mainly drawn from the theory of symmetric dynamics. We touch upon the master stability function approach and the groupoid formalism for handling coupled cell systems. In Sect. 4 we review some special cases where it is either possible to say something about the stability of the globally synchronous state in a general setting, or that of phaselocked states for strongly coupled networks of integrateandfire neurons. The challenge of the general case is laid out in Sect. 5, where we advocate the use of phase–amplitude coordinates as a starting point for either direct network analysis or network reduction. To highlight the importance of dynamics off cycle we discuss the phenomenon of shearinduced chaos. In the same section we review the reduction to the standard phaseonly description of an oscillator, covering the wellknown notions of isochrons and phase response curves. The construction of phase interaction functions for weakly coupled phase oscillator networks is covered in Sect. 6, together with tools for analysing phaselocked states. Moreover, we go beyond standard approaches and describe the emergence of turbulent states in continuum models with nonlocal coupling. Another example of something more complicated than a periodic attractor is that of a heteroclinic attractor, and these are the subject of Sect. 7. The subtleties of phase reduction in the presence of stochastic forcing are outlined in Sect. 8. The search for reduced descriptions of very large networks is the topic of Sect. 9, where we cover recent results for Winfree networks that provide an exact meanfield description in terms of a complex order parameter. This approach makes use of the Ott–Antonsen ansatz that has also found application to chimera states, and which we discuss in a neural context. In Sect. 10 we briefly review some examples where the mathematics of this review have been applied, and finally in Sect. 11 we discuss some of the many open challenges in the field of neural network dynamics.

The basics of nonlinear differential equation descriptions of dynamical systems such as linear stability and phaseplane analysis.

Ideas from the qualitative theory of differential equations/dynamical systems such as asymptotic stability, attractors and limit cycles.

Generic codimensionone bifurcation theory of equilibria (saddle node, Hopf) and of periodic orbits (saddle node of limit cycles, heteroclinic, torus, flip).
2 Neurons and Neural Populations as Oscillators
Nonlinear ionic currents, mediated by voltagegated ion channels, play a key role in generating membrane potential oscillations and action potentials. There are many ordinary differential equation (ODE) models for voltage oscillations, ranging from biophysically detailed conductancebased models through to simple integrateandfire (IF) caricatures. This style of modelling has also proved fruitful at the population level, for tracking the averaged activity of a near synchronous state. In all these cases bifurcation analysis is especially useful for classifying the types of oscillatory (and possibly resonant) behaviour that are possible. Here we give a brief overview of some of the key oscillator models encountered in computational neuroscience, as well as models for electrical and chemical coupling necessary to build networks.
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.
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 lowvoltage portion of the V nullcline moves up whilst the highvoltage 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
2.3 Morris–Lecar with Homoclinic Bifurcation
2.4 IntegrateandFire
Although conductancebased 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 actionpotential generation, their high dimensionality is a barrier to studies at the network level. The goal of a networklevel analysis is to predict emergent computational properties in populations and recurrent networks of neurons from the properties of their component cells. Thus simpler (lowerdimensional 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 (\(V_{s} < 0\)).
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 gapjunction 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 lowdimensional 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].
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 Dynamical Systems Approaches to Collective Behaviour
We give a brief overview of some dynamical systems approaches, concepts and techniques that can be used to understand collective behaviour that spontaneously appears in coupled dynamical system models used for neuroscience modelling. We do not give a complete review of this area but try to highlight some of the approaches and how they interact; some examples of applications of these to neural systems are given in later chapters.
In the artificial neural network literature, a distinction is made between recurrent and feedforward networks; see for example [74]. A feedforward network is one that is coupled but contains no feedback loops—i.e. there are no directed loops, while a recurrent network does contain feedback loops. We note that the methodologies discussed in this section (including constraints from symmetries and groupoid structures) may be applied to networks regardless of whether they are feedforward or recurrent. In this review we will later mostly discuss examples that are recurrent, though there are many interesting and relevant questions for feedforward networks as these often appear as models for “input–output” processes in neural systems.
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 spatiotemporal 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 indegree of the node i is the number of incoming connections (i.e. \(d_{\mathrm{in}}(i)=\sum_{j} A_{ij}\)), while the outdegree 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 scalefree 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 pathconnected 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 pathconnected 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 pulsecoupled 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 weakcoupling approximation is very helpful because it allows one to use various types of perturbation theory to investigate the dynamics [43]. For coupling of limitcycle 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 weakcoupling 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 semianalytic 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 welldeveloped numerical tools such as DDEBIFTOOL [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 realworld networks that have grown (e.g. giving rise to treelike structures) are expected to be well approximated by models that have large symmetry groups [113].

Description: one can identify symmetries of networks and dynamic states to help classify and differentiate between them.

Bifurcation: there is a welldeveloped theory of bifurcation with symmetry to help understand the emergence of dynamically interesting (symmetry broken) states from higher symmetry states.

Stability: bifurcation with symmetry often gives predictions about possible bifurcation scenarios that includes information as regards stability.

Generic dynamics: symmetries and invariant subspaces can provide a powerful structure with which one can understand more complex attractors such as heteroclinic cycles.

Design: one can use symmetries to systematically build models and test hypotheses.
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.
Some permutation symmetry groups that have been considered as examples of symmetries of coupled oscillator networks
Name  Symbol  Comments 

Full permutation  \(S_{N}\)  
Undirected ring  \(\mathbb {D}_{N}\)  
Directed ring  \(\mathbb {Z}_{N}\)  
Polyhedral networks  Various  [121] 
Lattice networks  \(G_{1}\times G_{2}\)  \(G_{1}\) and \(G_{2}\) could be \(\mathbb {D}_{k}\) or \(\mathbb {Z}_{k}\) 
Hierarchical networks  \(G_{1} \wr G_{2}\)  \(G_{1}\) is the local symmetry, \(G_{2}\) the global symmetry, and ≀ is the wreath product [122] 
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
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 fixedpoint subspaces of the same dimension where essentially the same dynamics will occur.
The fixedpoint 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).
Theorem 3.1
(Theorem 3.4 in [116])

\(H/K\) is cyclic,

K is an isotropy subgroup,

\(\dim \operatorname {Fix}(K)\geq2\),

H fixes a connected component of \(\operatorname {Fix}(K)/L_{K}\), where \(L_{K}\) is defined as above.
One way of saying this is that the only possible spatiotemporal 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 spatiotemporal 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 welldeveloped 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
 (a)
Identification of the marginally unstable modes (the directions that are losing stability: for equilibria, this corresponds to the eigenspace of the Jacobian where the eigenvalues have zero real part).
 (b)
Reduction to a centre manifold parametrised by the marginally unstable modes (generically this is one or twodimensional when only one parameter is varied).
 (c)
Study of the dynamics of the normal form for the bifurcation under generic assumptions on the normal form coefficients.
 (a′):

Identification of the marginally unstable modes (as discussed in Sect. 3.9, symmetry means there can generally be several of these that will become unstable at the same time).
 (b′):

Reduction to a centre manifold parametrised by the marginally unstable modes (these are preserved by the action of the symmetries and may be of dimension greater than two even for oneparameter bifurcations).
 (c′):

Study of the dynamics of the normal form for the symmetric bifurcation under generic assumptions on the normal form coefficients (the symmetries mean that some coefficients may be zero, some are constrained to be equal while others may be forced to satisfy nontrivial and sometimes obscure algebraic relationships).
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 saddletype invariant set, and switches between different behaviours.
In higherdimensional 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.
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 socalled 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 structurepreserving 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 twocell 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 Coupled LimitCycle Oscillators
If the coupling between two or more limitcycle oscillators is relatively large, it can affect not only the phases but also the amplitudes, and a general theory of strongly interacting oscillators is likely to be no more or less complicated than a general theory of nonlinear systems. However, the theory of weak coupling is relatively well developed (see Sect. 5 and Sect. 6) and specific progress for strong coupling can sometimes be made for special choices of neuron model. Examples where one can do this include IF (see Sect. 4.3), piecewise linear models such as McKean [146], caricatures of FHN and ML [147], and singularly perturbed relaxation oscillators with linear [148] or fast threshold modulation coupling [149].
For linear coupling of planar oscillators, much is known about the general case [150, 151]. If the linear coupling is proportional to the difference between two state variables this is referred to as “diffusive”, and otherwise it is called “direct”. The difference between the two cases is most strongly manifest when considering the mechanism of oscillator death (see Sect. 3.4). The diffusive case is more natural in a neuroscience context as it can be used to model electrical gapjunction coupling (which depends on voltagedifferences). The existence of synchronous states in networks of identical units is inherited from the properties of the underlying single neuron model since in this case coupling vanishes, though the stability of this solution will depend upon the pattern of gapjunction connectivity.
Gap junctions are primarily believed to promote synchrony, though this is not always the case and they can also lead to robust stable asynchronous states [152], as well as “bursting” generated by cyclic transitions between coherent and incoherent network states [153]. For work on gap junctions and their role in determining network dynamics see for example [147, 154–158].
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 nonsmooth 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 PulseCoupled 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 pulsecoupled 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 pulsecoupled oscillators. See Mirollo and Strogatz [164] for the generalisation of this proof to populations of size N.
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 \(F(\phi)\equiv\phih(\phi)\). Now \(F(\delta)=\delta1<0\) 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 \(F^{\prime}(\phi)=1h^{\prime}(\phi)>2\).
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 inphase 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 pulsecoupled 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 suprathreshold 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 nonattracting 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 timecontinuous 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 spiketime event driven synaptic models, described in Sect. 2.5, analysis at the network level is hard for a general conductancebased model (given the usual expectation that the single neuron model will be highdimensional and nonlinear), though far more tractable for LIF networks, especially when the focus is on phaselocked 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
The results above, albeit valid for strong coupling, are only valid for LIF networks. To obtain more general results for networks of limitcycle oscillators it is useful to consider a reduction to phase models.
5 Reduction of LimitCycle Oscillators to Phase–Amplitude and Phase Models
Comparison of the three conventions for a phase variable that we use in this review. The ϕ is used for IF models
Symbol  Phase space  Uncoupled equation  Period  Advantages  Disadvantages 

ϑ  \(\mathbb {R}/T\mathbb {Z}\), [0,T)  \(\frac {\mathrm{d}}{\mathrm{d}t}\vartheta= 1\)  T  Simplicity of uncoupled equation, interpretation of ϑ as “time”  Phase space depends on parameters and initial conditions 
θ  \(\mathbb {R}/2\pi \mathbb {Z}\), [0,2π)  \(\frac{\mathrm{d}}{\mathrm{d}t}\theta= \omega\)  \(\frac{2\pi}{\omega}\)  Phase space fixed, good for heterogeneous oscillators  Equation needs scaling 
ϕ  \(\mathbb {R}/\mathbb {Z}\), [0,1)  \(\frac {\mathrm{d}}{\mathrm{d}t}\phi= \frac{1}{\Delta}\)  Δ  Phase space fixed, good for heterogeneous oscillators  Equation needs scaling 
We now review some techniques of reduction which can be employed to study the dynamics of (17) when \(\epsilon\neq0\) so that the perturbations may take the dynamics away from the limit cycle. In doing so we will reduce for example to an ODE for \(\vartheta(t)\) taken modulo T. Clearly any solution of an ODE must be continuous in t and typically \(\vartheta(t)\) will be unbounded in t growing at a rate that corresponds to the frequency of the oscillator. Strictly speaking, the coordinate we are referring to in this case is on the lift of the circle \(\mathbb {T}\) to a covering space \(\mathbb {R}\), and for any phase \(\vartheta\in[0,T)\) there are infinitely many lifts to \(\mathbb {R}\) given by \(\vartheta+ kT\) for \(k\in \mathbb {Z}\). However, in common with most literature in this area we will not make a notational difference between whether the phase is understood on the unit cell e.g. \(\theta \in [0,2\pi)\) or on the lift, e.g. \(\theta\in \mathbb {R}\) modulo 2π.
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 planepolar 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 continuationbased algorithm introduced by Osinga and Moehlis [180], the geometric approach of Guillamon and Huguet to find highorder approximations to isochrons in planar systems [181], quadratic and higherorder 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.
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 higherdimensional 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: ShearInduced 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 nonattracting 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.
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.
Shearinduced 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 nonconstant 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 shearinduced 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, shearinduced 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 shearinduced 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 shearinduced 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 phaseonly 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 highdimensional 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, modelocking 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 phaselocking 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
6 Weakly Coupled Phase Oscillator Networks
The theory of weakly coupled oscillators [5, 213] is now a standard tool of dynamical systems theory and has been used by many authors to study oscillatory neural networks; see for example [213–217]. The book by Hoppensteadt and Izhikevich provides a very comprehensive review of this framework [43], which can also be adapted to study networks of relaxation oscillators (in some singular limit) [146, 218].
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 bidirectional 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 bidirectionally 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 weakcoupling theory for phaselocked 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 phaselocking 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 selfconsistently in Eq. (13).
6.2.1 Synchrony
If the synchronous solution exists then the Jacobian is given by \(\epsilon H'(0)\mathcal{L}\) where \(\mathcal{L}\) is the graphLaplacian with components \(\mathcal{L}_{ij}=\delta_{ij}\sum_{k} w_{ik}w_{ij}\). We note that \(\mathcal{L}\) has one zero eigenvalue, with eigenvector \((1,1,\ldots,1,1)\). Hence if all the other eigenvalues of \(\mathcal{L}\) lie on one side of the imaginary axis then stability is solely determined by the sign of \(\epsilon H'(0)\). This would be the case for a weighted connectivity matrix with all positive entries since the graphLaplacian in this instance would be positive semidefinite. For example, for global coupling we have \(\mathcal{L}_{ij} = \delta_{ij}N^{1}\), and the (\(N1\) degenerate) eigenvalue is +1. Hence the synchronous solution will be stable provided \(\lambda= \epsilon H'(0)<0\).
6.2.2 Asynchrony
6.2.3 Clusters for Globally Coupled Phase Oscillators
Theorem 6.1
(Theorem 3.1 in [118])
It is a nontrivial problem to discover which of these subspaces contain periodic solutions. Note that the inphase case corresponds to \(\ell =m=1\), \(k_{1}=N\) while splay phase corresponds to \(\ell=k_{1}=1\), \(m=N\). The stability of several classes of these solutions can be computed in terms of properties of \(H(\varphi)\); see for example Sect. 6.2.1 and Sect. 6.2.2 and for other classes of solution [118, 120, 228].
6.2.4 Generic Loss of Synchrony in Globally Coupled Identical Phase Oscillators

Pathfollowing 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 twocluster states \(S_{k} \times S_{Nk}\).

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.
6.3 Phase Waves
The phasereduction 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
Constant speed waves in a chain of identical oscillators can be generated by considering phaselocked 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 \(\Delta\omega_{1} +W_{+} H(\varphi) =0\), \(\Delta\omega_{N1} W_{}H(\varphi) =0 \) and \(\Delta\omega_{i} =0\), for \(i=2,\ldots, N2\). 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 \(\Delta \omega_{N1}=0\) 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
6.3.1 Phase Turbulence
7 Heteroclinic Attractors
In addition to dynamically simple periodic attractors with varying degrees of clustering, the emergent dynamics of coupled phase oscillator systems such as (32) can be remarkably complex even in the case of global coupling and similar effects can appear in a wide range of coupled systems. In the case of globally coupled phase oscillators, the dynamical complexity depends only on the phase interaction function H and the number of oscillators N. Chaotic dynamics [243] can appear in four or more globally coupled phase oscillators for phase interaction functions of sufficient complexity. We focus now on attractors that are robust and apparently prevalent in many such systems: robust heteroclinic attractors.
In a neuroscience context such attractors have been investigated under several related names, including slow switching [231, 244–246] where the system evolves towards an attractor that displays slow switching between cluster states where the switching is on a time scale determined by the noise, heteroclinic networks [137, 232, 247] or winnerless competition [248–250]. The effects can be seen in “microscale” models of small numbers of neurons or in “macroscale” models of cognitive function. In all these cases there are a number of possible patterns of activity that compete with each other but such that each pattern is unstable to some perturbations that take it to another pattern—this can be contrasted to winnertakesall competition where there is attraction to an asymptotically stable pattern.
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 twocluster state that, surprisingly, is unstable. This is a paradox because only a lowdimensional 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.
Robust heteroclinic attractors also appear in a range of coupled phase oscillator models where the coupling is not global (alltoall) 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 pulsecoupled 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 closelyrelated 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 longterm 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 Stochastic Oscillator Models
Noise is well known to play a constructive role in the neural encoding of natural stimuli, leading to increased reliability or regularity of neuronal firing in single neurons [265, 266] and across populations [267]. From a mathematical perspective it is natural to consider how noise may affect the reduction to a phase oscillator description. Naively one may simply consider the addition of noise to a deterministic phase oscillator model to generate a stochastic differential equation. Indeed models of this type have been studied extensively at the network level to understand noiseinduced first and secondorder phase transitions, and new phenomenon such as noiseinduced synchrony [268–270] or asynchrony [271], and noiseinduced turbulence [272]. We refer the reader to the review by Lindner [273] for a comprehensive discussion. More recently Schwabedal and Pikovsky have extended the foundations of deterministic phase descriptions to irregular, noisy oscillators (based on the constancy of the mean first return times) [274], Ly and Ermentrout [275] and Nakao et al. [276] have built analytical techniques for studying weak noise forcing, and Moehlis has developed techniques to understand the effect of white noise on the period of an oscillator [277].
At the network level (global coupling) a classic paper examining the role of external noise in IF populations, using a phase description, is that of Kuramoto [278], who analysed the onset of collective oscillations. Without recourse to a phase reduction it is well to mention that Medvedev has been pioneering a phase–amplitude approach to studying the effects of noise on the synchronisation of coupled stochastic limitcycle oscillators [194, 279], and that Newhall et al. have developed a Fokker–Planck approach to understanding cascadeinduced synchrony in stochastically driven IF networks with pulsatile coupling and Poisson spiketrain external drive [280]. More recent work on pairwise synchrony in network of heterogeneous coupled noisy phase oscillators receiving correlated and independent noise can be found in [281]. However, note that even in the absence of synaptic coupling, two or more neural oscillators may become synchronised by virtue of the statistical correlations in their noisy input streams [282–284].
8.1 Phase Reduction of a Planar System with StateDependent Gaussian White Noise
8.2 Phase Reduction for Noise with Temporal Correlation
9 LowDimensional Macroscopic Dynamics and Chimera States
The selforganisation of large networks of coupled neurons into macroscopic coherent states, such as observed in phase locking, has inspired a search for equivalent lowdimensional dynamical descriptions. However, the mathematical step from microscopic to macroscopic dynamics has proved elusive in all but a few special cases. For example, neural mass models of the type described in Sect. 2.6 only track mean activity levels and not the higherorder correlations of an underlying spiking model. Only in the thermodynamic limit of a large number of neurons firing asynchronously (producing null correlations) are such rate models expected to provide a reduction of the microscopic dynamics. Even here the link from spike to rate is often phenomenological rather than rigorous. Unfortunately only in some rare instances has it been possible to analyse spiking networks directly (usually under some restrictive assumption such as global coupling) as in the spikedensity approach [289], which makes heavy use of the numerical solution of coupled PDEs. Recently however, exact results for globally pulsecoupled oscillators described by the Winfree model [219] have been obtained by Pazó and Montbrió [290], making use of the Ott–Antonsen (OA) ansatz. The OA anstaz was originally used to find solutions on a reduced invariant manifold of the Kuramoto model [278], and essentially assumes that the distribution of phases has a simple unimodal shape, capable of describing synchronous (peaked) and asynchronous (flat) distributions, though is not capable of describing clustered states (multipeak phase distributions), and see below for a more detailed discussion. The major difference between the Winfree and Kuramoto phase oscillator models is that the former has interactions described by a phase product structure and the latter a phasedifference structure.
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 meanfield 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 gapjunction coupling has also recently been achieved by Laing [294], using the OA ansatz, showing how this can destroy certain spatiotemporal patterns, such as localised “bumps”, and create others, such as travelling waves and spatiotemporal chaos. The OA ansatz has also proved remarkably useful in understanding nontrivial solutions such as chimera states (where a subpopulation 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].
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 Rneighbour 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 twodimensional 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 chimeralike 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 singleharmonic phase interaction function used in (62), (63).
More recent work includes investigations of chimeras (or chimeralike 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 pulsecoupled LIF networks [314], and hypothesised to underly coordinated oscillations in unihemispheric slowwave sleep, whereby one brain hemisphere appears to be inactive while the other remains active [315].
10 Applications
We briefly review a few examples where mathematical frameworks are being applied to neural modelling questions. These cover functional and structural connectivity in neuroimaging, central pattern generators (CPGs) and perceptual rivalry. There are many other applications we do not review, for example deep brain stimulation protocols [316] or modelling of epileptic seizures where network structures play a key role [71].
10.1 Functional and Structural Connectivity in Neuroimaging
10.2 Central Pattern Generators
CPGs are (real or notional) neural subsystems that are implicated in the generation of spatiotemporal 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 realtime 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.
One can also view CPGs as a window into more fundamental problems of how small groups of neurons coordinate to produce a range of spatiotemporal 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 threecell “motif” networks of bursters and classify a range of emergent spatiotemporal 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 spatiotemporal 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 wellstudied 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].
11 Discussion
As with any review we have had to leave out many topics that will be of interest to the reader. In particular we have confined ourselves to “cell” and “systemlevel” dynamics rather that “subcellular” behaviour of neurons. We briefly mention some other active areas of mathematical research relevant to the science of rhythmic neural networks. Perhaps the most obvious area that we have not covered in any depth is that of single unit (cell or population) forcing, which itself is a rather natural starting point for gaining insights into network behaviour and how best to develop mathematical tools for understanding response [342, 343]. For a general perspective on modelocked responses to periodic forcing see [344] and [76], and for the role of spatially correlated input in generating oscillations in feedforward neuronal networks see [345]. Other interesting recent work includes uncovering some surprising nonlinear dynamical properties of feedforward networks [346, 347].
For a more recent discussion of the importance of modelocking in auditory neuroscience see [348, 349] and in motor systems, see [350]. However, it is well to note that not much is known about nonlinear systems with three or more interacting frequencies [351], as opposed to periodically forced systems where the notions of Farey tree and the devil’s staircase have proven especially useful. We have also painted the notion of synchrony with a broad mathematical brush, and not discussed more subtle notions of envelope locking that may arise between coupled bursting neurons (where the within burst patterns may desynchronise) [352]. This is especially relevant to studies of synchronised bursting [353] and the emergence of chaotic phenomena [354]. Indeed, we have said very little about coupling between systems that are chaotic, such as described in [355], the emergence of chaos in networks [356, 357] or chaos in symmetric networks [243].
The issue of chaos is also relevant to notions of reliability, where one is interested in the stability of spike trains against fluctuations. This has often been discussed in relation to stochastic oscillator forcing rather than those arising deterministically in a highdimensional setting [267, 358–360]. Of course, given the sparsity of firing in cortex means that it may not even be appropriate to treat neurons as oscillators. However, some of the ideas developed for oscillators can be extended to excitable systems, as described in [361–363]. As well as this it is important to point out that neurons are not point processors, and have an extensive dendritic tree, which can also contribute significantly to emergent rhythms as described in [241, 364], as well as couple strongly to glial cells. Although the latter do not fire spikes, they do show oscillations of membrane potential [365]. At the macroscopic level it is also important to acknowledge that the amplitude of different brain waves can also be significantly affected by neuromodulation [366], say, through release of norepinephrine, serotonin and acetylcholine, and the latter is also thought to be able to modulate the PRC of a single neuron [367].
This review has focussed mainly on the embedding of weakly coupled oscillator theory within a slightly wider framework. This is useful in setting out some of the neuroscience driven challenges for the mathematical community in establishing inroads into a more general theory of coupled oscillators. Heterogeneity is one issue that we have mainly sidestepped, and we recall that the weakly coupled oscillator approach requires frequencies of individual oscillators to be close. This can have a strong effect on emergent network dynamics [368], and it is highly likely that even a theory with heterogeneous phase response curves [369] will have little bearing on real networks. The equationfree coarsegraining approach may have merit in this regard, though is a numerically intensive technique [370].
We suggest a good project for the future is to develop a theory of strongly coupled heterogeneous networks based upon the phase–amplitude coordinate system described in Sect. 5.2, with the challenge to develop a reduced network description in terms of a set of phase–amplitude interaction functions, and an emphasis on understanding the new and generic phenomena associated with nontrivial amplitude dynamics (such as clustered phase–amplitude chaos and multiple attractors). To achieve this one might further tap into recent ideas for classifying emergent dynamics based upon the group of structural symmetries of the network. This can be computed as the group of automorphisms for the graph describing the network. For many realworld networks, this can be decomposed into direct and wreath products of symmetric groups [113]. This would allow for the use of tools from computational group theory [371] and open up a way to classify the generic forms of behaviour that a given network may exhibit using the techniques of equivariant bifurcation theory.
Indeed, the human brain consists of the order of 10^{11} neurons, but of the order of 100–1000 types http://neuromorpho.org meaning there is a very high replication of cells that are only different by their location and exact morphology.
In this section we assume little about the dynamics of the nodes—they may be “chaotic oscillators”.
Glossary: some of the abbreviations used within the text
 DDE:

Delay differential equation
 IF:

Integrate and fire (model for neural oscillator)
 iPRC:

Infinitesimal phase response curve
 ISI:

Interspike interval
 FHN:

FitzHugh–Nagumo equation (model for neural oscillator)
 HH:

Hodgkin–Huxley equation (model for neural oscillator)
 LIF:

Leaky integrate and fire (model for neural oscillator)
 ML:

Morris–Lecar equation (model for neural oscillator)
 MSF:

Master stability function
 ODE:

Ordinary differential equation
 PDE:

Partial differential equation
 PRC:

Phase response curve
 QIF:

Quadratic integrate and fire (model for neural oscillator)
 SDE:

Stochastic differential equation
 SHC:

Stable heteroclinic channel
 SNIC:

Saddle node on an invariant circle (bifurcation)
 WLC:

Winnerless competition
Declarations
Acknowledgements
We would like to thank many people for commenting on draft versions of the manuscript, in particular Chris Bick, Áine Byrne, Kurtis Gibson, Diego Pázo, Mason Porter and Kyle Wedgwood. We thank Kyle Wedgwood for providing Fig. 11. SC was supported by the European Commission through the FP7 Marie Curie Initial Training Network 289146, NETT: Neural Engineering Transformative Technologies.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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