58 research outputs found
Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations
In this paper we perform the parameter-dependent center manifold reduction
near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf
bifurcations in delay differential equations (DDEs). This allows us to
initialize the continuation of codimension one equilibria and cycle
bifurcations emanating from these codimension two bifurcation points. The
normal form coefficients are derived in the functional analytic perturbation
framework for dual semigroups (sun-star calculus) using a normalization
technique based on the Fredholm alternative. The obtained expressions give
explicit formulas which have been implemented in the freely available numerical
software package DDE-BifTool. While our theoretical results are proven to apply
more generally, the software implementation and examples focus on DDEs with
finitely many discrete delays. Together with the continuation capabilities of
DDE-BifTool, this provides a powerful tool to study the dynamics near
equilibria of such DDEs. The effectiveness is demonstrated on various models
Tutorial of numerical continuation and bifurcation theory for systems and synthetic biology
Mathematical modelling allows us to concisely describe fundamental principles
in biology. Analysis of models can help to both explain known phenomena, and
predict the existence of new, unseen behaviours. Model analysis is often a
complex task, such that we have little choice but to approach the problem with
computational methods. Numerical continuation is a computational method for
analysing the dynamics of nonlinear models by algorithmically detecting
bifurcations. Here we aim to promote the use of numerical continuation tools by
providing an introduction to nonlinear dynamics and numerical bifurcation
analysis. Many numerical continuation packages are available, covering a wide
range of system classes; a review of these packages is provided, to help both
new and experienced practitioners in choosing the appropriate software tools
for their needs.Comment: 14 pages, 2 figures, 2 table
Parameter-sweeping techniques for temporal dynamics of neuronal systems: case study of Hindmarsh-Rose model
Background: Development of effective and plausible numerical tools is an imperative task for thorough studies of nonlinear dynamics in life science applications.
Results: We have developed a complementary suite of computational tools for twoparameter screening of dynamics in neuronal models. We test a ‘brute-force’ effectiveness of neuroscience plausible techniques specifically tailored for the examination of temporal characteristics, such duty cycle of bursting, interspike interval, spike number deviation in the phenomenological Hindmarsh-Rose model of a bursting neuron and compare the results obtained by calculus-based tools for evaluations of an entire spectrum of Lyapunov exponents broadly employed in studies of nonlinear systems.
Conclusions: We have found that the results obtained either way agree exceptionally well, and can identify and differentiate between various fine structures of complex dynamics and underlying global bifurcations in this exemplary model. Our future planes are to enhance the applicability of this computational suite for understanding of polyrhythmic bursting patterns and their functional transformations in small networks
Nonlinear synchrony dynamics of neuronal bursters
We study the appearance of a novel phenomenon for coupled identical bursters:
synchronized bursts where there are changes of spike synchrony within each burst.
The examples we study are for normal form elliptic bursters where there is a periodic
slow passage through a Bautin (codimension two degenerate Andronov-Hopf)
bifurcation. This burster has a subcritical Andronov-Hopf bifurcation at the onset
of repetitive spiking while the end of burst occurs via a fold limit cycle bifurcation.
We study synchronization behavior of two Bautin-type elliptic bursters for
a linear direct coupling scheme as well as demonstrating its presence in an approximation
of gap-junction and synaptic coupling. We also find similar behaviour
in system consisted of three and four Bautin-type elliptic bursters. We note that
higher order terms in the normal form that do not affect the behavior of a single
burster can be responsible for changes in synchrony pattern; more precisely, we
find within-burst synchrony changes associated with a turning point in the spontaneous
spiking frequency (frequency transition). We also find multiple synchrony
changes in similar system by incorporating multiple frequency transitions. To explain
the phenomenon we considered a burst-synchronized constrained model and
a bifurcation analysis of the this reduced model shows the existence of the observed
within-burst synchrony states.
Within-burst synchrony change is also found in the system of mutually delaycoupled
two Bautin-type elliptic bursters with a constant delay. The similar phenomenon
is shown to exist in the mutually-coupled conductance-based Morris-Lecar
neuronal system with an additional slow variable generating elliptic bursting.
We also find within-burst synchrony change in linearly coupled FitzHugh-Rinzel
2
3
elliptic bursting system where the synchrony change occurs via a period doubling
bifurcation. A bifurcation analysis of a burst-synchronized constrained system
identifies the periodic doubling bifurcation in this case.
We show emergence of spontaneous burst synchrony cluster in the system of
three Hindmarsh-Rose square-wave bursters with nonlinear coupling. The system
is found to change between the available cluster states depending on the stimulus.
Lyapunov exponents of the burst synchrony states are computed from the
corresponding variational system to probe the stability of the states. Numerical
simulation also shows existence of burst synchrony cluster in the larger network of
such system.Exeter Research Scholarship
Mathematical modelling and brain dynamical networks
In this thesis, we study the dynamics of the Hindmarsh-Rose (HR) model which studies the spike-bursting behaviour of the membrane potential of a single neuron. We study the stability of the HR system and compute its Lyapunov exponents (LEs). We consider coupled general sections of the HR system to create an undirected brain dynamical network (BDN) of Nn neurons. Then, we study the concepts of upper bound of mutual information rate (MIR) and synchronisation measure and their dependence on the values of electrical and chemical couplings. We analyse the dynamics of neurons in various regions of parameter space plots for two elementary examples of 3 neurons with two different
types of electrical and chemical couplings. We plot the upper bound Ic and the order parameter rho (the measure of synchronisation) and the two largest Lyapunov exponents LE1 and LE2 versus the chemical coupling gn and electrical coupling gl. We show that, even for small number of neurons, the dynamics of the system depends on the number of neurons and the type of coupling strength between them. Finally, we evolve a network of Hindmarsh-Rose neurons by increasing the entropy of the system. In particular, we choose the Kolmogorov-Sinai entropy: HKS (Pesin identity) as the evolution rule. First, we compute the HKS for a network of 4 HR neurons connected simultaneously by two undirected electrical and two undirected chemical links. We get different entropies with the use of different values for both the chemical and electrical couplings. If the entropy of the system is positive, the dynamics of the system is chaotic and if it is close to zero, the trajectory of the system converges to one of the fixed points and loses energy.
Then, we evolve a network of 6 clusters of 10 neurons each. Neurons in each cluster are connected only by electrical links and their connections form small-world networks. The six clusters connect to each other only by chemical links. We compare between the combined effect of chemical and electrical couplings with the two concepts, the information
flow capacity Ic and HKS in evolving the BDNs and show results that the brain networks might evolve based on the principle of the maximisation of their entropies
- …