# Day 4: Thursday 1. Oct.

## Dynamics on Networks, Attractors, Feedback and Bifurcations

The generic class of processes describing biological (and most other) phenomena are non-linear. In general much less strong mathematical results are availability when compared to theorems based on linearity and conservation principles. Nevertheless mathematics has produced in the last decades a rich literature on attractors, bifurcation theory and feedback. Bifurcation theory is currently combined with the concept of networks and network architecture. More refined methods can be obtained if the given non-linear system defined on the network obeys symmetry principles. For biology non-linear processes defined on networks are the essential modelling technique to understand genetic regulation, metabolic pathways, or neuronal activity.

**10:00am - 10:30am** Welcome Tea & Coffee, Mathematics Common Room.

**10:30am - 11:00am** Overview to workshop day, Markus Kirkilionis and Ian Stewart

**11:00am - 12:30am**

Josef Hofbauer (Wien) - ** Replicator Dynamics, **45min

Peter Ashwin (Exeter) - * Heteroclinic Switching - examples and challenges*, 45min

**12:30am - 1:30pm** Lunch Break, Mathematics Common Room.

**1:30pm - 3:00pm**

Mike Field (Houston) - * Dynamics, Structure & Equivalence of Coupled Dynamical Systems*, 45min

Jonathan Cave (Warwick) - * Networked economy or economic network - old ideas in new bottles*, 45min

**3:00pm - 3:30pm** Tea Break, Mathematics Common Room.

**3:30pm - 4:15
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Edmond Rolls (Warwick)**Memory, Vision, Attention, Decision-Making, and their disorders: linking cellular and subcellular properties to global behaviour with attractor network, 45min**** **** **

Abstracts:

**Replicator Dynamics**

by Josef Hofbauer

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**Heteroclinic switching - examples and challenges**

by Peter Ashwin

**Abstract:** We will discuss examples of heteroclinic switching that occurs robustly in networks of coupled dynamical systems, where the only attracting state may consists of a network of unstable cluster states and connecting invariant manifolds. These attractors have a number of interesting properties, including being embeddings of discrete-state computational systems that may coherently and reliably switch between different states depending on inputs. In addition to discussing some examples, we briefly outline some challenges both in terms of theory and application to population behaviour in neuroscience, including G. Orosz, P. Ashwin and S.B. Townley, Learning of spatiotemporal codes in a coupled oscillator system. IEEE Transactions in Neural Networks, vol 20, 1135-1147 (2009).

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**Dynamics, Structure & Equivalence of Coupled Dynamical Systems**

by Mike Field

**Abstract:** We start by describing some of the motivation for an approach to

coupled dynamical systems that is inspired by linear systems theory

and transfer function methodology. The set-up is broad enough to

encompass continuous dynamical systems (ODEs) with

general phase spaces, discrete dynamical systems and hybrid systems.

We illustrate the combinatorial and constructive aspects with the

example of 'inflation' and give some simple examples of how robust

heteroclinic cycles can occur in small asymmetric networks with low

dimensional dyanmics.

The remainder of the talk will be concerned with

dynamical equivalence of coupled dynamical systems and the concepts of

input and output equivalence. We present some of the main results which

include a simple characterization of dynamical equivalence for continuous

dynamical systems (and many classes of discrete dynamical systems)

together with explicit algorithms for realizing the equivalence.

Parts of the work reported on are joint with Manuela Aguiar and Ana Dias (Porto),

Peter Ashwin (Exeter) and Nikita Agarwal (Houston).

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**Networked economy or economic network - old ideas in new bottles**

by Jonathan Cave

**Abstract:** Economics has since its inception implicitly recognised the importance of relationships among entities with choices to make. But much economic theory has suspended this in favour of tractability e.g. by treating people as individuals rather than recognising the networked nature of identity, neglecting the structure of connections in favour of theories based on group membership and assuming that 'relational' or 'social' capital can be analysed as an aggregate. Applications have been more welcoming, especially in relation to industries based on measurable networks (roads, electricity distribution, telecommunications, citation and co-authorship, etc.). These applications have also followed mainstream economics in differentiating performance (flows through networks, including learning and contagion) from capability (network structures). More recently, game theorists have begun to investigate the influence of networks on strategic behaviour (e.g. local games) and to treat network structures themselves as objects of strategic choice (e.g. goods with network externalities). These developments draw attention, in particular, to the need to surmount the different tractability assumptions made in the most common physical and mathematical network models: networks as collections of binary links, linkages without much in the way of 'colour' (direction, strength, state dependence, duration, subjectivity) and models of network formation based on random rather then intentional behaviour. This creates a potentially fertile middle ground, in particular when qualitatively different networks (e.g. of individuals, firms, institutions and ideas) overlap and interact, when the new tools make possible the analysis of network goods occupying the middle ground between public and private goods, and when the complexity of networks and imperfect information interact to determine both the scope for rational decisions within networks and the consequences for robustness of capability and volatility of performance. The talk will summarise some of these aspects and discuss a couple of extensions under current development.

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**Memory, Vision, Attention, Decision-Making, and their disorders: linking cellular and subcellular properties to global behaviour with attractor networks**

by Edmund T Rolls, Oxford Centre for Computational Neuroscience, Oxford, UK (http://www.oxcns.org, Edmund.Rolls@oxcns.org).

Cortical attractor networks provide a basis for understanding memory, attention, and decision-making. Analysis of systems found in the brain using mean-field approaches and integrate-and-fire neuronal network simulations shows how stochastic noise generated by the probabilistic firing times of neurons influences the operation of these processes, often to advantage. These approaches allow subcellular effects such as changes in synaptic receptor-activated ion channel conductances and neuron-level events such as spiking on the global operation of the whole network and its functions in behaviour to be predicted and analyzed. This is leading to the use of dynamically realistic models incorporating stochastic noise to understand memory, attention, and decision-making, and some of their disorders. This new understanding in turn has implications for treatments for these disorders.

Rolls,E.T. and Deco,G. (2010) The Noisy Brain: Stochastic Dynamics as a Principle of Brain Function. Oxford University Press: Oxford.

Deco,G. and Rolls,E.T. (2006) A neurophysiological model of decision-making and Weber's law. European Journal of Neuroscience 24: 901-916.

Loh,M., Rolls,E.T. and Deco,G. (2007) A dynamical systems hypothesis of schizophrenia. PLoS Computational Biology 3 (11): e228.

Rolls,E.T. (2005) Emotion Explained. Oxford University Press: Oxford.

Rolls,E.T. (2008) Memory, Attention, and Decision-Making: A Unifying Computational Neuroscience Approach. Oxford University Press: Oxford.

Rolls,E.T. and Deco,G. (2002) Computational Neuroscience of Vision. Oxford University Press: Oxford.

Rolls,E.T. and Stringer,S.M. (2006) Invariant visual object recognition: a model, with lighting invariance. Journal of Physiology - Paris 100: 43-62.

Deco,G., Rolls, E.T. and Romo,R. (2009). Stochastic dynamics as a principle of brain function. Progress in Neurobiology 88: 1-16.

Rolls,E.T., Loh,M., Deco,G. and Winterer,G. (2008) Computational models of schizophrenia and dopamine modulation in the prefrontal cortex. Nature Reviews Neuroscience 9: 696-709.

Rolls,E.T., Loh,M. and Deco,G. (2008) An attractor hypothesis of obsessive-compulsive disorder. European Journal of Neuroscience 28: 782-793.

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