27 research outputs found
Complex oscillations in the delayed Fitzhugh-Nagumo equation
Motivated by the dynamics of neuronal responses, we analyze the dynamics of
the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system
provides a canonical example of a canard explosion for sufficiently small
delays. Beyond this regime, delays significantly enrich the dynamics, leading
to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a
delay-induced subcritical Bogdanov-Takens instability arising at the fold
points of the S-shaped critical manifold. Underlying the transition from
canard-induced to delay-induced dynamics is an abrupt switch in the nature of
the Hopf bifurcation
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
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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
Computational bifurcation analysis to find dynamic transitions of the corticotroph model
The corticotroph model is a 7th order nonlinear differential equation
system derived for representing the action potential dynamics of corticotrophs; one
of the endocrine cells that are responsible for stress regulation. Here we use
numerical continuation methods to perform bifurcation analysis since controlling
bifurcations in the hormonal dynamics may bring some new insights in the
treatment of stress-related disorders. We study the bifurcation structure of the
system as a function of the BK-channel dynamic parameters and all maximal
conductances. We identify the regions of bistability and bifurcations that shape the
transitions between resting, bursting, and spiking behaviors, and which lead to the
appearance of bursting which is directly connected to stress regulation.
Furthermore, we find that there are two routes to bursting, one is the experimentally
observed BK-channel dynamics and the other is Ca2+ channel conductance only.
Finally, we discuss how some of the described bifurcations affect the dynamic behavior
and can be tested experimentally.No sponso
Shared inputs, entrainment, and desynchrony in elliptic bursters: from slow passage to discontinuous circle maps
What input signals will lead to synchrony vs. desynchrony in a group of
biological oscillators? This question connects with both classical dynamical
systems analyses of entrainment and phase locking and with emerging studies of
stimulation patterns for controlling neural network activity. Here, we focus on
the response of a population of uncoupled, elliptically bursting neurons to a
common pulsatile input. We extend a phase reduction from the literature to
capture inputs of varied strength, leading to a circle map with discontinuities
of various orders. In a combined analytical and numerical approach, we apply
our results to both a normal form model for elliptic bursting and to a
biophysically-based neuron model from the basal ganglia. We find that,
depending on the period and amplitude of inputs, the response can either appear
chaotic (with provably positive Lyaponov exponent for the associated circle
maps), or periodic with a broad range of phase-locked periods. Throughout, we
discuss the critical underlying mechanisms, including slow-passage effects
through Hopf bifurcation, the role and origin of discontinuities, and the
impact of noiseComment: 17 figures, 40 page
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
Nonlinear Dynamics, Synchronisation and Chaos in Coupled FHN Cardiac and Neural Cells
Physiological systems are amongst the most challenging systems to investigate from a
mathematically based approach. The eld of mathematical biology is a relatively recent
one when compared to physics. In this thesis I present an introduction to the physiological
aspects needed to gain access to both cardiac and neural systems for a researcher trained
in a mathematically based discipline. By using techniques from nonlinear dynamical
systems theory I show a number of results that have implications for both neural and
cardiac cells. Examining a reduced model of an excitable biological oscillator I show how
rich the dynamical behaviour of such systems can be when coupled together. Quantifying
the dynamics of coupled cells in terms of synchronisation measures is treated at length.
Most notably it is shown that for cells that themselves cannot admit chaotic solutions,
communication between cells be it through electrical coupling or synaptic like coupling,
can lead to the emergence of chaotic behaviour. I also show that in the presence of
emergent chaos one nds great variability in intervals of activity between the constituent
cells. This implies that chaos in both cardiac and neural systems can be a direct result
of interactions between the constituent cells rather than intrinsic to the cells themselves.
Furthermore the ubiquity of chaotic solutions in the coupled systems may be a means of
information production and signaling in neural systems
The Interplay of Intrinsic Dynamics and Coupling in Spatially Distributed Neuronal Networks
We explore three coupled networks. Each is an example of a network whose spatially coupled behavior is dratically different than the behavior of the uncoupled system. 1. An evolution equation such that the intrinsic dynamics of the system are those near a degenerate Hopf bifurcation is explored. The coupled system is bistable and solutions such as waves and persistent localized activity are found. 2. A trapping mechanism that causes long interspike intervals in a network of Hodgkin Huxley neurons coupled with excitatory synaptic coupling is unveiled. This trapping mechanism is formed through the interaction of the time scales present intrinsically and the time scale of the synaptic decay. 3. We construct a model to create the spatial patterns reported by subjects in an experiment when their eyes were stimulated electrically. Phase locked oscillators are used to create boundaries representing phosphenes. Asymmetric coupling causes the lines to move, as in the experiment. Stable stationary solutions and waves are found in a reduced model of evolution/ convolution type