119 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
Excitable neurons, firing threshold manifolds and canards
We investigate firing threshold manifolds in a mathematical model of an excitable neuron. The model analyzed investigates the phenomenon of post-inhibitory rebound spiking due to propofol anesthesia and is adapted from McCarthy et al. (SIAM J. Appl. Dyn. Syst. 11(4):1674-1697, 2012). Propofol modulates the decay time-scale of an inhibitory GABAa synaptic current. Interestingly, this system gives rise to rebound spiking within a specific range of propofol doses. Using techniques from geometric singular perturbation theory, we identify geometric structures, known as canards of folded saddle-type, which form the firing threshold manifolds. We find that the position and orientation of the canard separatrix is propofol dependent. Thus, the speeds of relevant slow synaptic processes are encoded within this geometric structure. We show that this behavior cannot be understood using a static, inhibitory current step protocol, which can provide a single threshold for rebound spiking but cannot explain the observed cessation of spiking for higher propofol doses. We then compare the analyses of dynamic and static synaptic inhibition, showing how the firing threshold manifolds of each relate, and why a current step approach is unable to fully capture the behavior of this model
Mixed-Mode Oscillations in a Stochastic, Piecewise-Linear System
We analyze a piecewise-linear FitzHugh-Nagumo model. The system exhibits a
canard near which both small amplitude and large amplitude periodic orbits
exist. The addition of small noise induces mixed-mode oscillations (MMOs) in
the vicinity of the canard point. We determine the effect of each model
parameter on the stochastically driven MMOs. In particular we show that any
parameter variation (such as a modification of the piecewise-linear function in
the model) that leaves the ratio of noise amplitude to time-scale separation
unchanged typically has little effect on the width of the interval of the
primary bifurcation parameter over which MMOs occur. In that sense, the MMOs
are robust. Furthermore we show that the piecewise-linear model exhibits MMOs
more readily than the classical FitzHugh-Nagumo model for which a cubic
polynomial is the only nonlinearity. By studying a piecewise-linear model we
are able to explain results using analytical expressions and compare these with
numerical investigations.Comment: 25 pages, 10 figure
Weakly coupled two slow- two fast systems, folded node and mixed mode oscillationsM
We study Mixed Mode Oscillations (MMOs) in systems of two weakly coupled
slow/fast oscillators. We focus on the existence and properties of a folded
singularity called FSN II that allows the emergence of MMOs in the presence of
a suitable global return mechanism. As FSN II corresponds to a transcritical
bifurcation for a desingularized reduced system, we prove that, under certain
non-degeneracy conditions, such a transcritical bifurcation exists. We then
apply this result to the case of two coupled systems of FitzHugh- Nagumo type.
This leads to a non trivial condition on the coupling that enables the
existence of MMOs
Hunting French Ducks in a Noisy Environment
We consider the effect of Gaussian white noise on fast-slow dynamical systems
with one fast and two slow variables, containing a folded-node singularity. In
the absence of noise, these systems are known to display mixed-mode
oscillations, consisting of alternating large- and small-amplitude
oscillations. We quantify the effect of noise and obtain critical noise
intensities above which the small-amplitude oscillations become hidden by
fluctuations. Furthermore we prove that the noise can cause sample paths to
jump away from so-called canard solutions with high probability before
deterministic orbits do. This early-jump mechanism can drastically influence
the local and global dynamics of the system by changing the mixed-mode
patterns.Comment: 60 pages, 9 figure
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