This event will take place on **April 9-10, 2018** at the University of Exeter (Streatham Campus) in Exeter.

### Registration

Please **register here by March 31, 2018**. There is no registration fee.

### Aim/Scope

Many real-world systems—from biological to social systems—can be modelled as dynamical networks of interacting units. Thus, advancing the understanding of the dynamics of such networks, both theoretically and with respect to applications, will help to tackle challenges in a range of fields. To this end, the proposed workshop (which celebrates the appointments of Dr Christian Bick and Dr Kyle Wedgwood as new lecturers in Mathematics at the University of Exeter) will bring leading researchers together to discuss the latest developments in network dynamics. This joint workshop will specifically bridge mathematical advances (Dr Bick’s research) and applications of mathematics in the sciences and biology (Dr Wedgwood’s research).

**We gratefully acknowledge support from the London Mathematical Society.**

*This event will be followed by the workshop on Brain Networks and Neurological Disorders: from Theory to Clinic. Please register separately to attend.*

### Speakers include

- R Baker (Oxford)
- S Johnson (Birmingham)
- A Maltsev (Queen Mary)
- G Pruessner (Imperial)
- R Sanchez Garcia (Southampton)
- K Wedgwood (Exeter)
- C Bick (Exeter)

**Prospective Schedule**

**Monday April 9, 2018 (LSI Seminar Room A/B)**

*12:00-13:00 Arrival, registration, lunch*

13:00-13:40 A Maltsev: Intracellular calcium signalling and the Ising model

13:45-14:25 K Wedgwood: Interface approaches in spiking neural networks

*14:30-15:00 Coffee break*

~~15:00-15:40 R Baker: Topology-dependent density optima for efficient simultaneous network exploration~~

15:00-15:40 M Goodfellow:

15:45-16:25 S Johnson: The trophic structure of directed networks: What we can learn from ecosystems

*16:30-16:45 Break*

16:45-17:25 R Sanchez Garcia: Exploiting symmetries in network analysis

*Evening: Workshop dinner (limited slots available, please indicated interest in registration form)***Tuesday April 10, 2018****(Laver LT6)**

09:15-09:55 C Bick: Frequency synchrony and chaos in small oscillator networks

10:00-10:40 G Pruessner: A Field theoretic method for processes on graphs

10:45-10:55 Closing

*11:00-11:30 Coffee break*

11:30-**Workshop Brain Networks and Neurological Disorders: from Theory to Clinic**at the Living Systems Institute

### Abstracts

**Marc Goodfellow**

**TBA**

**S Johnson**

**The trophic structure of directed networks: What we can learn from ecosystems**

Rainforests, coral reefs and other very large ecosystems seem to be the most stable in nature, but this has long been regarded as mathematically paradoxical. More generally, the relationship between structure and dynamics in complex systems is the subject of much debate. I will discuss how ‘trophic coherence’, a recently identified property of food webs and other biological networks, is key to understanding many dynamical and structural features of complex systems. In particular, it allows networks to become more stable with increasing size and complexity, determines whether a given system will be in a regime of high or of negligible feedback, and influences spreading processes such as epidemics or cascades of neural activity. See also: https://en.wikipedia.org/wiki/Trophic_coherence

**A Maltsev**

**Intracellular calcium signalling and the Ising model**

Intracellular Ca signals represent a universal mechanism of cell function. Messages carried by Ca are local, rapid, and powerful enough to be delivered over the thermal noise. A higher signal to noise ratio is achieved by a cooperative action of Ca release channels arranged in clusters (release units) containing a few to several hundred release channels. The channels synchronize their openings via Ca-induced-Ca-release, generating high-amplitude local Ca signals known as puffs in neurons and sparks in muscle cells. Despite positive feedback nature of the activation, Ca signals are strictly confined in time and space by an unexplained termination mechanism. We construct an exact mapping of such molecular clusters to an Ising model and demonstrate that the collective transition of release channels from an open to a closed state is identical to the phase transition associated with the reversal of magnetic field. This is joint work with Prof. Stern’s laboratory at the National Institutes of Health.

**G Pruessner**

**A Field theoretic method for processes on graphs**

Field theoretic methods have been used regularly to analyse reaction diffusion processes on regular lattices and in the continuum. Under certain prerequisites, these methods can also be employed on arbitrary graphs. I will discuss these conditions and the methodological limitations using the example of the number of distinct sites visited by a branching random walk on regular lattices, fractals and certain graphs.

**R Sanchez Garcia**

**Exploiting symmetries in network analysis**

Network models of real-world complex systems have been extremely successful at revealing structural and dynamical properties of these systems. One property of interest is the presence of structural redundancies, which manifest themselves as symmetries in a network model. It has been shown that real-world networks possess a large number of symmetries, and that this has important consequences for network structural, spectral and dynamical properties, such as cluster synchronisation. Crucially, network symmetries are inherited by any measure or metric on the network, and any derived matrix such as the graph Laplacian.

In this talk, I will present a complete theory for the study of symmetry in empirical networks and their effects on arbitrary network measures, and show how this can be exploited in practice in a number of ways, from redundancy compression, to computational reduction. We also uncover the spectral signatures of symmetry for an arbitrary network measure, predicting and explaining most of the discrete part of the spectrum of a network measure such as the graph Laplacian. We show that computing network symmetries and motifs is very efficient in practice, testing real-world examples up to several million nodes. Our theoretical framework generalise, and helps understand, other network symmetry results in the literature.

**K Wedgwood**

**Interface approaches in spiking neural networks**

Persistent localised activity in neural networks has been associated with, amongst others, working memory, somatosensory perception and the formation of grid cell firing patterns used in spatial navigation. There is thus a need to understand how non-locally coupled networks can robustly establish such coherent firing activity. Pioneering work on neural field models (see Amari, Cowan, Ermentrout) provided a key framework for understanding these patterns at the tissue level. In particular, through approximations based on an all-or-nothing response, the activity across the network can be reduced to one which considers only the boundaries (or interfaces) of the localised activity. Moreover, linear stability properties of these solutions can be assessed by considering perturbations only at these interfaces.

There is a growing desire in the mathematical neuroscience community (and biology as a whole) to understand how neural activity at the level of single cells feeds into these tissue-level patterns. Indeed, small changes to intrinsic firing patterns, as observed in certain dementias, can have dramatic effects on the network activity. In this talk, I will show how interface approaches can be used to understand the formation of localised activity patterns in spiking neural networks. In this approach, knowledge of the single cell dynamics feed directly in to the macroscopic network description through approximation by an analog of a spiking neural field model.

**C Bick**

**Frequency synchrony and chaos in small oscillator networks**

Network interactions allow identical oscillators to generate dynamics with distinct average frequencies. Nonchaotic dynamics with distinct frequencies arise in small networks of phase oscillators that are organized into two populations with disparate coupling strengths. We analyze how these solutions arise and bifurcate, both in theory and in experiment. We then turn to two population networks of identical Kuramoto oscillators with a more generic coupling scheme where both coupling strengths and phase-lags between populations are distinct. We give numerical evidence that there are chaotic attractors on which the oscillators evolve with distinct frequencies. Hence, complicated dynamics with localized frequency synchrony are expected even in the simplest description of oscillator networks.

### 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 (fast but expensive) 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 the city center 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 Living Systems Institute (Monday) and Laver (Tuesday) buildings. Please see the Streatham campus map. Exeter St. Davids train station is 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 the City Center: **

- By bus: Take the D or H bus, which leave from the High Street. 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.

### Sponsorship

We gratefully acknowledge support from the London Mathematical Society.

For more information, contact the organizers Chris Bick or Kyle Wedgwood.