This will run from 20th to 22nd April 2016, Harrison building room 203 University of Exeter, Exeter.

### Registration

To register your interest, please fill out this form**.**

### Aim/Scope

The aim of this research workshop is to discuss the state of the art with regard to the effect of generalised coupling (or phase interaction) functions and the emergent patterns of synchrony in coupled oscillator networks, with a particular emphasis to connections in mathematical neuroscience. There will be a number of talks and opportunity for contributions and collaborative discussions during the workshop.

The workshop is partly sponsored by the EU Marie Curie project GECO and partly by the EPSRC “Centre for Predictive Modelling in Healthcare“. Further details will be published here in due course. To register your interest, please fill out this form and/or contact the organizers: Chris Bick, Peter Ashwin, Kyle Wedgwood with the assistance of Sarah Warren.

### Speakers include

- R Borisyuk (Plymouth)
- O Burylko (Kiev)
- A Daffertshofer (VU Amsterdam)
- M Field (IC London)
- G Huguet (Barcelona)
- Y Maistrenko (Kiev/TU Berlin)
- E Martens (Copenhagen)
- G Medvedev (Drexel)
- O Omelchenko (WIAS Berlin)
- M Porter (Oxford)
- T Stankovski (Lancaster)
- M Wolfrum (WIAS Berlin)

**Schedule**

**Wednesday 20th April 2016 (all in Harrison 203)**

*H203 is available for informal discussions until 12.00*

*12:00-13:00 Lunch*

13:00-14:00 Mason Porter:*Multilayer Networks*

14:00-15:00 Georgi Medvedev:*Synchronization of coupled systems: Relating network structure to dynamics*

*15:00-15:30 Break*

15:30-16:00 Yuri Maistrenko:*Solitary states*

16:00-16:30 Oleksandr Burylko:*Coexistence of Hamiltonian-like and dissipative dynamics in chains of coupled phase oscillators with skew-symmetric coupling*

*Evening: Special meal (invited speakers only): Meet Exeter Central railway station at 17:45***Thursday 21st April 2016****(in Harrison 203 except where noted)**

09:30-10:30 Andreas Daffertshofer:*Network-network interactions in systems of phase oscillators – implications for neuroscience*

*10:30-11:00 Break***(in Harrison 209)**

11:00-11:30 Erik Martens:*Chimera states in two populations with heterogeneous phase-lag*

11:30-12:00 Matthias Wolfrum:*Synchronization Transitions in Systems of Coupled Phase Oscillators*

12:00-12:30 Oleh Omel’chenko:*Creative control for chimera states*

*12:30-14:00 Lunch/Posters/Demonstration of software***(in Harrison 209)**

14:00-15:00 Tomislav Stankovski:*Universality of Coupling Functions: describing the mechanisms of general anaesthesia*

15:00-15:30 Gemma Huguet:*Phase and Amplitude response functions: mathematical tools for phase control in transient-states of neuronal oscillators*

15:30-16:00 Chris Bick:*Isotropy of Angular Frequencies and Chaotic Weak Chimeras with Broken Symmetry*

*16:00-16:30 Break***(in Harrison 209)**

*16:30- Collaboration time*

*Evening: informal arrangements, e.g. meal in Exeter***Friday 22nd April 2016****(all in Harrison 203)**

09:30-10:30 Roman Borisyuk:*Mathematical and computational modelling of the Xenopus tadpole spinal cord: Biologically realistic models of connectivity and functionality*

*10:30-11:00 Break*

11:00-11:30 Kyle Wedgwood:*Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis*

11:30-12:30 Mike Field:*Models for power and micro grids*

*12:30- Finish/Informal arrangements for lunch/Collaboration time*

### Travel Information

For general advice on how to get to the University of Exeter, see the University maps and directions pages.

**By plane:**

*Fly to Exeter (EXT)*: Onward travel from Exeter Airport is available by bus or taxi. The bus is cheap (a few pounds), but does not run late. The taxi costs about £25. There are direct flights between Exeter many British and European airports, including London City Airport, Schipol (Amsterdam), and Paris CDG; in particular the latter two are options to connect to other flights.

*Fly to Bristol (BRS):*Bristol airport is the next closest airport (*total travel time approx. 1.5 hrs*). Follow the signs for the Bristol Flyer bus and purchase a ticket to Bristol Temple Meads Rail Station (journey time is 20-30 minutes). From Bristol Temple Meads take a train to Exeter St David’s train station (journey time approximately one hour).

*Fly to London Heathrow (LHR)*:*From Heathrow by bus (approx. 3.5hrs, nonstop)*, you can take the National Express Bus which runs fairly frequently to Exeter Bus and Coach station right in the city centre and is inexpensive compared to trains.*From Heathrow by rail (approx. 3 hrs, 1 stop)*, you can take the Railair bus service from Heathrow to Reading train station and then take a train from there to Exeter St. David’s (book a through ticket at Heathrow Bus Station). Alternatively, you can take either the Heathrow Express or Heathrow Connect (slower but much cheaper) to London Paddington train station and then take the train from Paddington to Exeter St. David’s; yet another option is to take a bus from Heathrow to Woking and take the train from there to Exeter Central.

*Fly to London Gatwick (LGW)*:*From Gatwick by rail*, you can travel by train either via Reading to Exeter St. Davids (*approx. 3.5 hrs, 1 stop*) or via Clapham Junction (cheaper,*approx. 4 hrs*) to Exeter Central train station. There are National Express Bus services as well.

**By train: **

- The closest train station to the university is
**Exeter St. Davids**. The closest train station to ABode is**Exeter Central**. Rail prices and timetables (including for onward travel from airports other than Exeter) can be found at National Rail Enquiries. Generally,**Advance**or**Return**tickets come with significant discounts.

**By car: **

- Pay and Display parking on campus is incredibly limited. Visitors may park in Car Park C and 15 marked bays in Car Park A. Their locations are marked on the Streatham Campus map. Please note that you must arrive early to find a space. Parking costs £6 per day.

### Venue

The workshop will take place in the Harrison Building, room 203. Please see the Streatham campus map. Both Exeter St. Davids train station and ABode are close to campus. You can either walk to campus (a healthy 20min walk up the hill) or catch a bus as indicated below.

**From Exeter St. Davids: **

- By bus: Take the H bus, which leaves from just outside the station. Buses run every 15 minutes and costs £1.
- Walking: Follow the directions as indicated here.

**From ABode: **

- By bus: Take the D or H bus, which leave from the High Street, just behind the hotel (away from the Cathedral). Buses leave approximately every 15 minutes. The D bus costs £1 and the H bus costs £1.70.
- Walking: Follow the directions as indicated here.

### Abstracts

**Christian Bick:**

**Isotropy of Angular Frequencies and Chaotic Weak Chimeras with Broken Symmetry**

Symmetrically coupled systems of phase oscillator can give rise to localized dynamics, commonly known as chimera states. We phrase frequency synchronization in terms of symmetries of the system which allows to generalize the definition of a weak chimera–recently introduced by Ashwin and Burylko to rigorously characterize chimera states in finite dimensional systems of phase oscillators—to more general settings. While symmetries of solutions translate into symmetries of frequencies, we give a result that there are weak chimeras that have symmetries in the frequencies that are not present in the solutions.

**Roman Borisyuk:
Mathematical and computational modelling of the Xenopus tadpole spinal cord: Biologically realistic models of connectivity and functionality**

In close collaboration with neurobiologists from the University of Bristol (Alan Roberts and Steve Soffe) we have developed a new computational method to define synaptic connectivity (in the form of a “connectome”) between neurons in the tadpole spinal cord. The connectome is generated by a “developmental” process where the growing axons intersect dendrites and create connections. The resulting network has around 1,500 neurons with around 100,000 connections, and its statistical properties are similar to experimental measurements. We study the properties of the connectome using graph theory methods and find some similarities with the C. Elegans network, in terms of how close to a small world network it is. We have developed a functional network of spiking (Hodgkin-Huxley) neurons and used the generated connectome to produce a pattern of neural activity. Remarkably, the generated activity is very stable and corresponds with the typical pattern seen in vivo during fictive swimming (anti-phase oscillations between left and right sides of the body). Mathematical study of a simplified functional model shows that there is another limit cycle corresponding to synchrony (in-phase oscillations of two body sides), which can be stable under some conditions but has a small basin of attraction in comparison with that of swimming. These results are in a good agreement with experimental study of swimming and synchrony patterns in the tadpole spinal cord. (with Robert Merrison-Hort, Andrea Ferrario)

**Oleksandr Burylko:**

**Coexistence of Hamiltonian-like and dissipative dynamics in chains of coupled phase oscillators with skew-symmetric coupling**

We consider chains of coupled phase oscillators with anisotropic coupling and periodic boundary conditions. When the coupling is skew-symmetric, i.e. when the anisotropy is balanced in a specific way, the system shows robustly a coexistence of Hamiltonian-like and dissipative regions in the phase space. We relate this phenomenon to the time-reversibility property of the system. The geometry of low-dimensional examples up to five oscillators in described in details. In particular, we show that the boundary between the dissipative and Hamiltonian-like regions consists of families of heteroclinic connections. For larger chains, some sufficient conditions are provided, and in the limit of N → ∞ oscillators, amplitude equations are derived formally, which have the form of the Schrodinger equation for the skew-symmetric coupling case. (With Alexander Mielke (WIAS Berlin), Matthias Wolfrum (WIAS Berlin), Serhiy Yanchuk (TU Berlin))

**Andreas Daffertshofer:**

**Network-network interactions in systems of phase oscillators – implications for neuroscience**

Functional brain networks are a popular topic of research. We will discuss how spiking neurons can be combined to neural masses. Connecting different neural masses may yield large-scale synchronization patterns that can be described by systems phase oscillators. These steps will be explicated for Wilson-Cowan and Freeman neural mass models which give rise to Kuromoto-like networks. If two or more of such networks are again coupled to one another, one may ask to what extend their behavior agrees with the dynamics of a single network of oscillators with natural frequency drawn from a bi- or multimodal frequency distribution. This will be answered in detail for two coupled, symmetric (sub)populations with unimodal frequency distributions. It can be proven that the resulting synchronization dynamics resembles that of a single population with bimodally distributed frequencies. However, a generalization to networks consisting of multiple (sub)populations vis-à-vis networks with multimodal frequency distributions appears difficult if not impossible. Finally, a system-identification approach will be sketched that allows for pinpointing the dynamical structure of such synchronization patterns in encephalographic signals.

**Mike Field: **

**Models for power and micro grids**

Motivated by recent work of Dorfler & Bullo, we discuss models for power grids, microgrids, frequency dependent loads and droop controllers with a particular focus on two problems (1) The insertion/removal of a microgrid from the main power grid; (2) the analysis of transmission line or generator breakdown in a power grid. In both cases, we are influenced by ideas from asynchronous networks and synchronized dynamics.

**Gemma Huguet:
Phase and Amplitude response functions: mathematical tools for phase control in transient-states of neuronal oscillators**

The phase response curve (PRC) is a powerful tool to study the effect of a perturbation on the phase of an oscillator, assuming that all the dynamics can be explained by the phase variable. However, factors like the rate of convergence to the oscillator, strong forcing or high stimulation frequency may invalidate the above assumption and raise the question of how is the phase variation away from an attractor.

I will present a numerical method to perform the effective computation of the phase advancement when we stimulate an oscillator which has not reached yet the asymptotic state (a limit cycle) using the concept of isochrons. To do so, we first perform a careful study of the theoretical grounds (the parameterization method for invariant manifolds), which allow us to describe the isochronous sections of the limit cycle. From it, we can control changes in phase and amplitude variables. I will show some examples of the computations we have carried out for some well-known biological models and its possible implications for neural communcation.

**Yuri Maistrenko:**

**Solitary states**

See: Maistrenko_Abstract_Exeter2016.

**Erik A. Martens:**

**Chimera states in two populations with heterogeneous phase-lag**

Networks of coupled oscillators are pervasive in natural and engineered systems. These systems are known to exhibit a variety of dynamical behaviors including uniform synchronous oscillation and chimera states, patterns with coexisting coherent and incoherent domains. We study the simplest network of coupled phase-oscillators exhibiting chimera states — two populations with disparate intra- and inter-population coupling strengths — and explore the effects of heterogeneous phase-lags between the two populations. Such heterogeneity arises naturally in various settings, for example as an approximation to transmission delays. Breaking the phase-lag symmetry results in a variety of states with uniform and non-uniform synchronization, including states where both populations remain desynchronized. This is joint work with Mark Panaggio (Northwestern University) and Christian Bick (University of Exeter).

**Georgi Medvedev:**

**Synchronization of coupled systems: Relating network structure to dynamics**

We will discuss dynamics in two representative models of dynamical systems on graphs: coupled chaotic maps and coupled excitable systems driven by white noise. For both models, we analyze synchronization and explain what structural features of the network favor synchronization. For the second model, we also describe other dynamical regimes such as spontaneous oscillations and formation of clusters. The role of network topology in shaping each of these patterns will be explained.

**Oleh Omel’chenko:
Creative control for chimera states**

In this talk, we give an overview of the new types of chimera states, which can be found in systems of nonlocally coupled phase oscillators in the presence of a proportional control involving global order parameter. Among others, we discuss properties of chimera states close to the coherence and present some analytical results motivating our control scheme. This is joint work with M. Wolfrum and J. Sieber.

**Mason Porter:
Multilayer Networks**

One of the most active areas of network science, with an explosion of publications during the last few years, is the study of “multilayer networks,” in which heterogeneous types of entities can be connected via multiple types of ties that change in time. Multilayer networks can include multiple subsystems and “layers” of connectivity, and it is important to take multilayer features into account to try to improve our understanding of complex systems. In this talk, I’ll give an introduction to multilayer networks. I’ll indicate some applications and discuss some dynamical processes on multilayer networks. I’ll also comment briefly on synchronization on multilayer networks.

**Tomislav Stankovski:
Universality of Coupling Functions: describing the mechanisms of general anaesthesia**

See: Stankovski_Abstract_Exeter2016.

**Kyle Wedgwood:**

**Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis**

**Matthias Wolfrum:
Synchronization Transitions in Systems of Coupled Phase Oscillators
**We investigate the transition to synchrony in systems of coupled Kuramoto-Sakaguchi phase oscillators. In globally coupled systems with certain unimodal frequency distributions, there appear unusual types of synchronization transitions , where with increasing coupling strength synchrony can decay, incoherence can regain stability, or multistability between partially synchronized states and/or the incoherent state can appear. We present a detailed bifurcation analysis for these phenomena, which is based on the Ott Antonsen reduction, and show that they can appear already for arbitrarily small values of the phase lag parameter in the interaction function.

### Posters

The poster session will take place in Harrison 209 on Thursday, April 21, 2016. If you are presenting a poster, you can put it up in the morning and please do remove it before the end of the day.

Peter Ashwin:

**Equivariant Hopf bifurcation and non-pairwise coupled phase oscillators**

Sebastian Eydam:

**Mode-locking in systems of phase oscillators with higher harmonic coupling**

Robert Merrison:

**Software demonstration: Fireflies**

### Sponsorship

This workshop is partly sponsored by the EU Marie Curie project GECO and partly by the EPSRC “Centre for Predictive Modelling in Healthcare“.

For more information, contact the organizers: Chris Bick, Peter Ashwin, Kyle Wedgwood with the assistance of Sarah Warren.