The two-variable FitzHugh-Nagumo model for space-clamped...

The two-variable FitzHugh-Nagumo model for space-clamped nerve axon firing with an external applied current $I_{a}$ is $$ \frac{d v}{d t}=v(a-v)(v-1)-w+I_{a} . \quad \frac{d w}{d t}=b v-\gamma w $$ where $0<a<1$ and $b, \gamma$ and $I_{e}$ are positive constants. Here $v$ is directly related to the transmembrane potential and $w$ is the variable which represents the effects of the various chemical ion-generated potentials. Determine the local maximum and minimum for the $v$ null cline in terms of $a$ and $I_{u}$ and hence give the corresponding piecewise linear approximate form. Show that there is a confined set for the model system. Using the piecewise linear model, determine the conditions on the parameters such that the positive steady state is stable but excitable. Find the conditions on the parameters, and the relevant window $\left(I_{1}, I_{2}\right)$ of applied currents, for the positive steady state to be linearly unstable and hence for a limit cycle solution to exist. For a fixed set of parameters $a, b$ and $\gamma$, find the period of the small amplitude limit cycle when $I_{a}$ is just greater than the bifurcation value I 1 . |Use the Hopf bifurcation result near bifurcation.I.

The two-variable FitzHugh-Nagumo model for space-clamped nerve axon firing with an external applied current $I_{a}$ is
$$
\frac{d v}{d t}=v(a-v)(v-1)-w+I_{a} . \quad \frac{d w}{d t}=b v-\gamma w
$$
where $0<a<1$ and $b, \gamma$ and $I_{e}$ are positive constants. Here $v$ is directly related to the transmembrane potential and $w$ is the variable which represents the effects of the various chemical ion-generated potentials.

Determine the local maximum and minimum for the $v$ null cline in terms of $a$ and $I_{u}$ and hence give the corresponding piecewise linear approximate form.
Show that there is a confined set for the model system. Using the piecewise linear model, determine the conditions on the parameters such that the positive steady state is stable but excitable. Find the conditions on the parameters, and the relevant window $\left(I_{1}, I_{2}\right)$ of applied currents, for the positive steady state to be linearly unstable and hence for a limit cycle solution to exist. For a fixed set of parameters $a, b$ and $\gamma$, find the period of the small amplitude limit cycle when $I_{a}$ is just greater than the bifurcation value I 1 . |Use the Hopf bifurcation result near bifurcation.I.

Key Concepts

Limit Cycle Oscillations and Excitability

Limit cycle oscillations are closed trajectories in phase space that represent sustained periodic behavior. Their existence is particularly significant in excitable systems, where a perturbation can lead to a cycle of activity even if the steady state is stable. The analysis involves determining parameter regimes where the positive steady state becomes unstable, leading to the creation of a limit cycle. This interplay between stability, excitability, and recurrent oscillations is central to understanding how the system responds to external stimuli.

Hopf Bifurcation

A Hopf bifurcation occurs in a dynamical system when a pair of complex-conjugate eigenvalues of the linearized system crosses the imaginary axis as a parameter is varied. This bifurcation typically signals the onset of a small amplitude limit cycle bifurcating from a fixed point. In the analysis of the FitzHugh-Nagumo model, the Hopf bifurcation is key to understanding how oscillatory behavior emerges from a change in the stability of the steady state.

Stability Analysis of Fixed Points

Stability analysis examines whether small perturbations around a steady state decay or grow over time. In the context of the FitzHugh-Nagumo model, this involves linearizing the system around fixed points and determining the eigenvalues of the Jacobian matrix. Such an analysis helps to determine conditions under which the steady state is both stable and excitable, and it provides insight into the mechanisms leading to oscillatory behavior.

Confined Set in Phase Space

A confined set, or invariant set, in phase space is a bounded region where the trajectories of the dynamical system remain once they enter. Establishing the existence of such a confined set is important as it guarantees that the system's behavior remains bounded over time. This concept is pivotal in demonstrating that the dynamics are contained within a finite region, which is a common requirement for further stability analysis.

Nullclines

Nullclines are the sets of points in the phase space of a dynamical system where the derivative of one of the variables is zero. They are essential tools in phase plane analysis because their intersections determine the fixed points of the system. Analyzing nullclines helps in understanding the geometry of the system's trajectories, and in this context, they are used to locate the local extrema and ascertain the conditions for excitability and oscillatory behavior.

Piecewise Linear Approximation

Piecewise linear approximation is a method to simplify a nonlinear model by approximating its nonlinear functions with linear segments. This approach helps in analyzing complex systems by providing analytical tractability while maintaining the key qualitative features of the original nonlinear dynamics. In the study of excitable systems, such approximations facilitate the derivation of conditions for stability and the onset of oscillatory behavior.

Local Extrema Analysis

Local extrema analysis involves finding the maximum and minimum points of a curve, in this case, the v nullcline. Determining these extrema is crucial as they define the boundaries of different dynamical regimes and contribute to understanding the threshold behavior in excitable systems. This analysis contributes to creating a piecewise linear representation that captures the essential dynamics near these critical points.

FitzHugh-Nagumo Model

The FitzHugh-Nagumo model is a simplification of the Hodgkin–Huxley model that captures the essential features of excitability in neurons. It is a two-variable dynamical system where one variable represents the excitation (or voltage) and the other represents a recovery or inhibition mechanism. This model is used to study the generation of action potentials and to explore phenomena such as excitability, threshold behavior, and oscillations in nerve cells.

Please give Ace some feedback