1,862 research outputs found
Qualitative Models of Neural Activity and the Carleman Embedding Technique.
The two variable Fitzhugh Nagumo model behaves qualitatively like the four variable Hodgkin-Huxley space clamped system and is more mathematically tractable than the Hodgkin Huxley model, thus allowing the action potential and other properties of the Hodgkin Huxley system to be more readily be visualized. In this thesis, it is shown that the Carleman Embedding Technique can be applied to both the Fitzhugh Nagumo model and to Van der Pol\u27s model of nonlinear oscillation, which are both finite nonlinear systems of differential equations. The Carleman technique can thus be used to obtain approximate solutions of the Fitzhugh Nagumo model and to study neural activity such as excitability
An organizing center in a planar model of neuronal excitability
The paper studies the excitability properties of a generalized
FitzHugh-Nagumo model. The model differs from the purely competitive
FitzHugh-Nagumo model in that it accounts for the effect of cooperative gating
variables such as activation of calcium currents. Excitability is explored by
unfolding a pitchfork bifurcation that is shown to organize five different
types of excitability. In addition to the three classical types of neuronal
excitability, two novel types are described and distinctly associated to the
presence of cooperative variables
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
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