Applied Dynamical Systems and
Mathematical Neuroscience Seminar
Fall 2007- Spring 2008
Questions or comments? Please e-mail Igor Belykh
Atlanta Computational Neuroscience Conference
as a part of our regular seminar.
DATE: Friday, April 8
LOCATION: Loudermilk Center, GSU
Round table seminar (special guest: Nancy Kopell, Boston University)
DATE: Friday, April 7
TIME: 2:30 PM- 6:30 PM
LOCATION: COE 796
DATE: Friday, April 7
TIME: 4:30 PM
LOCATION: COE 796
Speaker: Gennady Cymbalyuk (Department of Physics and Astronomy)
Title: "Co-existence of bursting with other regimes ”
DATE: Friday, April 7
TIME: 5:00 PM
LOCATION: COE 796
Speaker: Igor Belykh (Math and Stat)
Title: "Fast threshold modulation and synchronization of excitatory bursting neurons"
DATE: Friday, March 7
TIME: 5:30 PM
LOCATION: COE 796
Speaker: Mukesh Dhamala (Physics and Astronomy)
Title: "Estimating information flow in dynamic networks with nonparametric granger causality"
DATE: Friday, April 4
TIME: 1:00 PM
LOCATION: COE 796
Speaker: Tatiana Malaschenko (Physics and Astronomy)
Title: "Co-existence of silent and oscillatory regimes of a single neuron's activity"
DATE: Friday, March 28
TIME: 1:00 PM
LOCATION: COE 796
Speaker: Andrey Shilnikov (Math and Stat)
Title: "Polyrhythmic synchronization in network motifs"
DATE: Friday, March 21
TIME: 1:00 PM
LOCATION: COE 796
Speaker: Paul Channell (Math and Stat)
Title: "Variability of bursting patterns in a neuronal model"
DATE: Friday, March 14
TIME: 1:00 PM
LOCATION: COE 796
Speaker: Mukesh Dhamala (Physics and Astronomy)
DATE: Friday, February 1
TIME: 1:15 PM
LOCATION: COE 796
Speaker: Andrey Shilnikov (Math and Stat)
Title: "Methods of the qualitative theory for the Hindmarsh-Rose model."
Abstract. Homoclinic bifurcations of both equilibria and periodic orbits are argued to be critical for understanding the dynamics of the Hindmarsh-Rose model in particular, as well as in some square-wave bursting models of neurons of the Hodgkin-Huxley type. They explain very well various transitions between the tonic spiking and bursting oscillations in the model. We present the approach that allows for constructing Poincar´e return mapping via the averaging technique. We show that a modified model can exhibit the blue sky bifurcation, as well as, a bistability of the coexisting tonic spiking and bursting activities. A new technique for localizing the slow motion manifold and periodic orbits on it is also presented.
References:
Shilnikov A. L. and Kolomiets M.L., Methods of the qualitative theory for the Hindmarsh-Rose model: a case study. SIADS 2008, submitted.
DATE: Friday, November 30
TIME: 2:00 PM
LOCATION: COE 796
Speaker:
Dr. Sergey Dashkovskiy (Department of Mathematics and Informatics, University of Bremen, Germany).
Title: "Small-gain theorems for networks of ISS systems".
Abstract.
We will consider a number of systems which are interconnected in a network. Each system is assumed to be input-to-state stable (ISS). In general such an interconnection is not ISS. We are looking for stability conditions for such networks. In case of feedback interconnection of two ISS systems such condition of a small-gain type were derived by Jiang et al. in 1994. A construction of an ISS-Lyapunov function for this case was performed in Jiang et al. 1996. In my presentation I will show a stability condition for a general interconnection of many systems and a construction of an ISS-Lyapunov function for a network. As we will see our condition is a natural generalization of the results mentioned above. We will consider some interpretations of this generalized small-gain condition and discuss on approaches how to check this condition numerically.
DATE: Friday, November 9
TIME: 2:00 PM
LOCATION: COE 796
Speaker: Sajiya Jalil, University of Toronto
Title: "Role of synaptic plasticity in the generation of complex patterns and ability to learn by the brain".
Abstract.
Short-term synaptic plasticity contributes significantly to the function of synapses. Similarly, long-term synaptic plasticity is thought to be at the heart of complex patterns such as learning and memory. Consequently, studies of synaptic weight evolution over different time scales are thriving areas of research in neuroscience. Specifically following two studies will be discussed:
• Novel bursting pattern emerging from model inhibitory networks with synaptic depression (research collaboration with Professor Frances Skinner)
• Computational consequences of experimentally derived STDP based learning rule (research collaboration with Professor Thomas Trappenberg) .
DATE: Friday, August 24
TIME: 2:00 PM
LOCATION: COE 796
Speaker: Dr. Oleksandr Burilko
Institute of Mathematics, Kiev, Ukraine/Institute of Medicine, Julich Research Center, Julich, Germany
Title: "Bifurcation to heteroclinic cycles and sensitivity in three
and four coupled phase oscillators".
Abstract.
We study the bifurcation and dynamical behaviour of the system of N globally coupled identical phase oscillators introduced by Hansel, Mato and Meunier, in the cases N=3 and N=4. This model has been found to exhibit robust `slow switching' oscillations that are caused by the presence of robust heteroclinic attractors. We consider bifurcations that occur in a system of identical oscillators on varying parameters in the coupling function. These bifurcations preserve the permutation symmetry of the system. We then investigate implications of these bifurcations for the sensitivity to detuning (i.e. the size of the smallest perturbations that give rise to loss of frequency locking). For N=3 we find three types of heteroclinic bifurcation that are codimension-one with symmetry. On varying two parameters in the coupling function we find three curves giving (a) an S3-transcritical homoclinic bifurcation, (b) a saddle-node/heteroclinic bifurcation and (c) a Z3-heteroclinic bifurcation. We also identify several global bifurcations with symmetry that organize the bifurcation diagram; these are codimension-two with symmetry. For N=4 oscillators we determine many (but not all) codimension-one bifurcations with symmetry, including those that lead to a robust heteroclinic cycle. A robust heteroclinic cycle is stable in an open region of parameter space and unstable in another open region. Furthermore, we verify that there is a subregion where the heteroclinic cycle is the only attractor of the system, while for other parts of the phase plane it can coexist with stable limit cycles. We finish with a discussion of bifurcations that appear for this coupling function and general N, as well as for more general coupling functions.