10:30 Arrival and coffee

11:00 Prof. Stephen Coombes (University of Nottingham)

**Title**: Modelling Brain Waves

**Abstract**: In this talk I will explore the way in which synaptically coupled neural networks may generate and maintain travelling waves of activity. Although these models are inherently non-local, a combination of mathematical approaches (predominantly drawn from non-smooth dynamical systems) means that we are now in a position to address fundamental questions about the effects of intrinsic ionic currents, synaptic processing, and anatomical connectivity on travelling waves in neural tissue. I will present a number of examples from both one and two dimensions, focusing on the contributions of axonal delays, adaptation, refractoriness, and slow hyper-polarisation activated currents, to brain waves seen in the cortex, thalamus, and hippocampus.

12:00 Áine Byrne (University of Nottingham)

**Title**: Next generation neural mass modelling

**Abstract**: Electromagnetic recordings of the brain that show transitions from high amplitude to low amplitude signals are likely caused by an underlying changes in the synchrony of neuronal population firing patterns. A classic example is the event-related oscillatory phenomenon known as post-movement beta-rebound (PMBR), where a sharp increase in EEG or MEG power is seen at beta frequency following movement termination. A related phenomenon is movement related beta decrease (MRBD), whereby beta rhythms are suppressed during movement.

Traditionally neural mass models have been used to model large-scale brain dynamics, however they fail to account for the degree of synchronisation within a population. This could be tracked within a large-scale model of synaptically interacting conductance based neurons, though at the expense of analytical tractability. Thus it is of interest to seek levels of description that provide a bridge between microscopic single neuron dynamics and coarse grained neural mass models, while preserving some notion of within-population coherence.

I will present a parsimonious model for the dynamics of synchrony within a synaptically coupled spiking network that can replicate a human MEG power spectrogram showing the evolution from MRBD to PMBR. Importantly the high-dimensional spiking model has an exact mean field description that allows considerable insight into the cause of beta-rebound. Interestingly the reduced model takes the form of a generalised neural mass model where the standard sigmoidal firing rate has been replaced by a derived quantity that is a function of the Kuramoto order parameter for synchrony.

12:30 Lunch and poster presentations by Mayte Bonilla Quintana, Joshua Davis and Aytul Gökçe (provisional list)

14:00 **Dr. Yulia Timofeeva** (University of Warwick)

**Title**: Mechanisms of synaptic vesicle exocytosis in small central synapses

**Abstract**: Increases in concentration of free Ca2+ ions in presynaptic boutons of neuronal cells trigger the vesicular release of neurotransmitters. Changes in Ca2+ concentration are primarily due to Ca2+ influx through voltage-gated calcium channels located at the plasma membrane and activated during an action potential or spontaneously. In this talk I will discuss the role of voltage-gated calcium channels in action potential-evoked and in spontaneous miniature neurotransmitter release and demonstrate how modelling can compliment experimental work and provide explanations for observations that are incompletely understood otherwise. I will also cover the effects of different buffers on Ca2+-dependent machinery of synaptic transmission and discuss a way of estimating Ca2+ kinetics from a single set of experimental conditions when a priori information about endogenous Ca2+ buffering is limited.

15:00 **Dr. Wilten Nicola** (Imperial College London)

**Title**: Mean-Field Analysis of Networks of Integrate-and-Fire Neurons with Spike-Timing Dependent Plasticity

**Abstract**: One of the most basic properties of spiking neuronal networks is the ability to adapt to new environments and circumstances and retain any new information learned. The adaptability of a neuronal network comes from its intrinsic ability to alter the connectivity between the individual neurons through spike-timing dependent plasticity. Unfortunately, analyzing the behavior of an individual weight in a large, recurrently coupled network is typically intractable.

In order to overcome this intractability, we have derived a closed mean-field system of equations for a network of integrate-and-fire neurons with spike-timing dependent plasticity that governs the dynamics of critical moments in the network, such as the mean synaptic weight. This system is derived through a separation of time scales assumption applied to the Fokker-Planck system for the network of neurons in the large network limit. The resulting equations are low dimensional and analytically tractable for the behavior of the network. The mean-field system for the network of neurons is amenable to bifurcation analysis. The mean-field system can also be extended to obtain higher order moments that quantitatively and qualitatively predict network behavior.

15:30 **Dr. Jennifer Creaser** (University of Exeter)

**Title**: Modelling noise-induced escape problems in networks

**Abstract**: Mathematical models of excitable cells, such as neurones, are often characterised by different dynamic regimes, such as alternating excited and rest states. The transient dynamics responsible for the transition between dynamic states are often discounted or overlooked. However, study of transient dynamics could provide important insights into, for example, the evolution of epileptic seizures or the initiation of tremors associated with Parkinson’s disease. We consider small networks of coupled noisy dynamic nodes. Each dynamic node has two stable states, a stable equilibrium representing the rest state, and a stable limit system representing the excited state. The addition of noise means that stochastic fluctuations on each node drive the system to switch between rest and excited states. The time taken for the system to switch between rest and excited states is called the the escape-time. We are interested in investigating the effects of network structure, timescale separation and noise on the escape-time problem.

16:00 Tea

16:30 Elif Köksal-Ersöz (INRIA Paris)

**Title**: Canard-Mediated (De)Synchronization in Coupled Phantom Bursters

**Abstract**: In this work, we study canard-mediated transitions in mutually coupled phantom bursters. We extend a multiple-timescale model which provides a sequence of dynamic events, i.e., transition from a frequency modulated relaxation cycle to a quasi-steady-state and resumption of the relaxation regime through small amplitude oscillations. Folded singularities and associated canard solutions have a particular impact on the dynamics of the original system, which consists of two feedforward coupled FitzHugh--Nagumo oscillators, where the slow subsystem (regulator) controls the periodic behavior of the fast subsystem (secretor). We first investigate the variability in the dynamics depending on the canard mechanism that occurs near the folded singularities of the four-dimensional secretor-regulator configuration. Then, we introduce a second secretor and focus on the slow-fast transitions in the presence of a linear coupling between the secretors. In particular, we explore the impact of the relationship between the canard structures and the coupling on patterns of synchronization and desynchronization of the collective dynamics of the resulting six-dimensional system. We identify two different sources of desynchronization induced by canards, near a folded-saddle singularity and a folded-node singularity, respectively.

17:00 Dr. Kyle Wedgwood (University of Exeter)

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

**Abstract**: At the single cell level, neurons typically exhibit an all-or-nothing response, dependent on the summation of input currents they receive from the rest of the network. Due to far reaching processes, neurons can form connections with distant parts of the network, allowing for rapid communication across long distances.

Certain neural systems show computation through patterned activity; persis- tent localised activity, in the form of bumps has been linked to working memory, whilst the propagation of activity in the form of waves has been associated with binocular rivalry tasks. The assumption of infinitely slow synapses allows for the replacement of firing patterns with firing rates, resulting in a neural field model that is amenable to perturbative analysis. This description of the network averages out fluctuations in both space and time ignoring these small scale effects. Our aim is to perform analysis on a network that retains these small scale effects, but whose large scale effects can be predicted in an analogous way to neural field models.

We present analysis of a network of minimal three-state neurons whose transitions are probabilistic. By taking appropriate limits, we demonstrate the existence and compute stability of spatiotemporally patterned activity across the network. We then go on to show how coarse-grained analysis can be used to construct bifurcation diagrams for the network when these limits are relaxed and show how these can be used to reduce the complexity of the dynamics.

17:30 Close