Workshop on Nonlinear Neurodynamics

University of Exeter

18th-19th September 2008

There will be a two-day workshop on Nonlinear Neurodynamics at the University of Exeter, Harrison Building, on 18th and 19th September. This workshop will aim to look at recent developments and problems in the application of nonlinear dynamics methods to problems in neuroscience. Agreed speakers include: Mikhail Rabinovich (San Diego), Piotr  Suffcyznski (Warsaw), Martin Krupa (New Mexico/Nijmegen), Netta Cohen (Leeds), David Chik (Plymouth), Svitlana Popovych (Cologne), Yulia Timofeeva (Warwick), Stuart Townley (Exeter) and there will be a poster session on Thursday.

It will start mid-day on 18th September and finish after lunch on 19th September. The tentative schedule is as follows:
The meeting will start late morning of 18th September and finish afternoon of 19th September. Those needing overnight accommodation can find information below; we intend to organize a meal at a local restaurant on the Thursday evening.

For more information or to register your interest, please contact Peter Ashwin or Abul Al-Azad. This workshop will be partially supported by the Mathematical Neuroscience Network of the EPSRC and a British Council/DAAD cooperation grant between Exeter, Plymouth and Cologne; participation is registration-free for members of these networks and includes lunch/tea and coffee; others will be charged a nomial registration fee. Exeter has some good accommodation facilities for visitors. Below is a list of bed and breakfast accommodation close to the University and city centre.


David Chik (University of Plymouth)

Constructive effects of noise in neural systems

Our brain is immersed in a noisy environment. This talk provides an introduction of how noise may sometimes become beneficial, which is a counter-intuitive phenomenon. We shall glance through five interesting examples: how noise can help detecting weak signals (so-called stochastic resonance); how noise can induce coherent firing of a neuron; how noise can improve the degree of coherence of activities of a neural population; how a common noise can induce coherence between uncoupled oscillators; and how noise can accelerate the synchronization process. Underlying mechanisms and implications will be discussed briefly.

Netta Cohen (University of Leeds)

Groovy worms: from swimming to crawling in the nematode C. elegans

Martin Krupa (Donders Institute for Brain, Cognition and Behaviour, Radboud Universiteit Nijmegen)

Gamma rhythm in an excitatory-inhibitory network with excitatory neurons of Hodgkin-Huxley type

Piotr  Suffcyznski (University of Warsaw)

Risk assessment of epileptic transitions

In a study of electroencephalographic (EEG) signals recorded in patients with mesial temporal lobe epilepsy (MTLE) undergoing invasive pre-surgical evaluation, we found that a specific feature of the neural system’s response to periodic stimulation, the so called relative “phase clustering index” (rPCI) can be related to the instantaneous risk of the transition to an epileptic seizure. To better understand the significance of this index in terms of the system’s dynamical and physiological properties we performed a simulation study using a computational model of seizure generation in a hippocampal network. Our study indicates that a stimulation - based paradigm can (1) reveal more information about the dynamics of the brain with respect to the transition to seizures than the pure observation of on-going activity does and (2) may reconstruct the physiological changes that precede a transition to a seizure. These findings give support to the development and application of active paradigms with the aim of predicting the occurrence of a transition to an epileptic seizure.

Svitlana Popovych (University of Cologne)

Spike timing-dependent plasticity in coupled phase oscillators

We present a generalized Kuramoto model of coupled phase oscillators with spike timing-dependent plasticity (STDP) given by symmetric and asymmetric plasticity functions. We found a coexistence of synchronized and desynchronized states in large domains in parameter space. We investigate bifurcation mechanisms of the transitions between plasticity-induced synchronous and desynchronous dynamical regimes.

Mikhail Rabinovich (University of California, San Diego)

Reproducible transient dynamics in a heirarchical "brain"

Yulia Timofeeva (University of Warwick)

The subthreshold dynamics in the Spike-Diffuse-Spike framework

Dendrites that are involved in receiving and integrating thousands of synaptic inputs from other neurons are equipped with voltage-gated ion channels. Dendritic spines that stud dendrites of many neurons are often loci for such channels. The recently proposed spike-diffuse-spike model has been shown to provide a reasonable caricature of spiny dendrites  with suprathreshold dynamics of active channels. However, it has been shown that ionic channels present in the cell membrane can also sustain subthreshold (resonant-like) responses. Here I will introduce a mathematical model of a dendritic tree based upon a generalisation of the spike-diffuse-spike model where the active spines are assumed to be distributed along a resonant dendritic structure. I will consider both continuous and discrete limits for spine distribution and show how solitary and periodic travelling waves can be studied in such a system.

Stuart Townley (University of Exeter)

Lyapunov-based adaptive learning for coupled oscillator systems

The purpose of this talk is to illustrate the use of Lyapunov-based adaptive control techniques in synchronization and learning for coupled oscillator systems. We consider the following specific problem. We take two systems of coupled oscillators: A Teaching System has fixed, but unknown, parameters or "weights". The Teaching System dynamics are assumed to follow a heteroclinic cycle between clustered states;  a second Learning System has variable weights. These two systems are coupled and the weights of the learning system are tuned via error dynamics. Our aim is to force the learning system to (i) synchronize with the teaching system and (ii) learn the weights of the teaching system. By using a Lyapunov-based analysis and borrowing ideas from adaptive control we show how to choose the coupling functions and tuning laws so as to satisfy these two aims.


Mathieu Desroches, Hinke Osinga, Bernd Krauskopf (University of Bristol)

Computing slow manifolds and canards in the self-coupled FitzHugh-Nagumo system

We study the so-called 'self-coupled FitzHugh-Nagumo system'. It has been reported that the firing rate of the neuron in this system slows down dramatically when the self-coupling is activated. This can be explained by slow-fast dynamics in the silent-phase system, with the presence of canard orbits that organize this region of the phase space and force trajectories to complete a certain number of small amplitude oscillations before firing, hence preventing them from firing successive action potentials. We have an accurate method that provides numerical evidence of this phenomenon by computing attracting and repelling slow  manifolds together with associated canard solutions.

Abul Kalam al Azad, Peter Ashwin (University of Exeter)

On the spike and burst synchronization of neuronal elliptic bursters

Neurons such as thalamic relay and reticular is, primary afferent in the brain stem circuits, and in many other areas of the brain show rhythmic pattern referred to as elliptic bursting, which involves recurrent alternation between active phases of large amplitude oscillation and silent phases of small oscillation. In this study we analyzed the topological normal form elliptic burster characterized by a codimension-2 Bautin bifurcation, first studied by Izhikevich (2004). The dynamics of the bursting involves a subcritical Andronov-Hopf bifurcation at the onset of repetitive spiking while the quiescent state occurs via fold limit cycle bifurcation. We studied synchronization behaviour of two and more elliptic Bautin bursters in both linear and nonlinear coupling scheme. In the previous study by Izhikevich, bursting synchrony is found to be prevalent behaviour among coupled bursting cells, while spike synchronization is hard to achieve. Our study involves the latter aspect of the coupled bursters. Using a modified form of the model and subjecting the coupled system under special slow manifold constraint, we were able to observe many novel synchrony patterns involving spikes in the burst dynamics. Using suitable control mechanism involving system parameters and higher order terms of the governing system we observed in-phase, anti-phase, chaotic and semi-synchronous dynamics among the spikes of the coupled bursters. This dynamical aspect of the spike synchrony may be a key mechanism for emergence of cluster or partial synchronization in the neuronal population, which is a hallmark synchrony in biological systems. 

Carlos Trenado and Daniel J. Strauss (University of Saarland, Homburg Germany)

Study of neural population phase synchronization and its connection to psychological  behavior: the case of attention and habituation

The functional behavior of the brain is encoded in spatio-temporal structures that have been  mathematically represented by a non-linear dynamics and spatial interconnection. It is generally accepted that the information of such pattern formation can be extracted from the dynamics of macroscopic quantities such as the EEG and MEG that may result from the synchronized post-synaptic activity of ensembles of neurons. In such respect, the seminal work of Freeman propelled the use of neural-mass models as an alternative to study the dynamics of such ensembles. In this paper, we make use of a neural large-scale model and a novel stochastic approach so as to study the phase dynamics of neural population responses rejected in evoked potentials during states of focal and non-focal human attention and habituation. Our simulated results are compared to experimental evoked potentials so that our hypothesis can be validated. It is concluded that our approach enforces experimental and theoretical results regarding the degree of neural synchronization as concomitant of attention and habituation processes.

Ozkan Karabacak and Peter Ashwin (University of Exeter)

Heteroclinic ratchets for a system of coupled oscillator systems

We investigate an unusual but robust phenomenon that appears for a system of four coupled oscillators. The system admits an unusual heteroclinic attractor, which we call heteroclinic ratchet, and this includes phase slips in one direction similar to a mechanical ratchet. The heteroclinic ratchet responds to a specific detuning d between a certain pair of the oscillators by a breaking of phase locking for an arbitrary d greater than 0 but not for d less than 0. Similarly, arbitrarily small noise destroys synchronization of oscillators as a result of the presence of a heteroclinic ratchet.

17th September 2008, Peter Ashwin