EPSRC Reference: 
EP/P007031/1 
Title: 
Threshold networks 
Principal Investigator: 
Coombes, Professor S 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Mathematical Sciences 
Organisation: 
University of Nottingham 
Scheme: 
Standard Research 
Starts: 
02 January 2017 
Ends: 
01 April 2020 
Value (£): 
341,073

EPSRC Research Topic Classifications: 
Mathematical Analysis 
Nonlinear Systems Mathematics 

EPSRC Industrial Sector Classifications: 

Related Grants: 

Panel History: 

Summary on Grant Application Form 
Networks are ubiquitous, and we experience them in our daily lives. Every phone call we make is routed through a telecommunications network, every product we buy has come through a complex manufacturing, supply and transport network. Our families, friends and colleagues form our social networks, and when we see a doctor, the conditions we discuss usually originate from interacting biomolecular networks. An important property of networks is their structure, which provides information about the connectivity amongst the elements that form the network. In recent years it has become apparent that in addition to the network structure, the dynamics of the network elements is vital for our understanding of the emergent behaviour of realworld networks. In this proposal we will develop a novel mathematical framework that will allow us to analyse a large variety of dynamical networks across numerous disciplines.
Our theory will be based on two concepts. Firstly, we will approximate the complex (nonlinear) behaviour of network elements in the real world by an appropriately chosen set of simpler (piecewise linear) models. This approach has proven extremely helpful in elucidating behaviours in engineered systems. Secondly, we will assume that network elements communicate with each other through events. For instance, in the telecommunications example above, the event corresponds to the situation where placing a phone call triggers activity in network routers.
While these two concepts allow us to describe numerous networks in a unifying language, existing mathematical techniques cannot be used to analyse them. This shortcoming mainly results from the fact that current mathematical approaches assume what is known as smooth dynamics. In contrast, we will deal with nonsmooth systems. A main part of the proposal is to develop ideas from general nonsmooth dynamical systems and make them fit for investigating eventdriven dynamical networks. A common assumption in the study of networks is that elements communicate with each other through weak signals. This assumption is often made for mathematical convenience. We will go beyond weak signals and develop an approach for strong signals, which more closely resembles what is observed in real world networks.
Once the mathematical techniques are in place we will showcase the versatility of this new approach by studying three different applications. Firstly, we will investigate in more detail how a heartbeat is generated. What we perceive as a single heartbeat is the orchestrated action of millions of connected muscle cells. Within each muscle cell numerous biochemically coupled networks shape the single cell behaviour. A better understanding of the cardiac muscle network is crucial for treating and even preventing cardiac arrhythmias such as atrial fibrillation. As a second application we will study patterns of activity in the brain. Here, specialised neural cells form a complex network, and they communicate with each other through chemical and electrical pulses. We will be primarily interested in activity patterns when neurons suddenly stop communicating or become hyperactive, as these dynamics are often observed in medical conditions such as epilepsy. Thirdly, we will apply our technique to information spread on social networks. A special emphasis will be on when such information travels extremely quickly amongst a large number of people (socalled viral activity). By achieving a better understanding of how ideas, behaviours or styles become viral, and how that depends on network structure and characteristics of the network elements, we can develop means for intervention strategies (e.g. quelling London riots quickly) and for the spread of desirable behaviours (e.g. a healthy lifestyle amidst the obesity epidemic).

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Organisation Website: 
http://www.nottingham.ac.uk 