Morris-Lecar Phase Plane

Preset Hopf · type IISaddle-node · type ICustom parameters

Damped oscillationCross HopfSaddle & thresholdCross saddle-node

Near boundary Reset Copy link

Applied current IµA/cm² Recovery rate φ

0.04 ms⁻¹

Phase plane

Zoom to equilibriumClear paths

V̇ = 0 ṅ = 0 FlowWˢ Wᵘ Eigenvectors

Voltage versus potassium activation: nullclines, equilibria, trajectories and saddle manifolds. Numerical values are available beside the plot.

Click to launch a trajectory. Select a numbered equilibrium below.Straddle Wˢ

Jacobian at equilibrium

δẋ = Jδx

$${\lambda }_{1,2}=\frac{\tau }{2}\pm \frac{\sqrt{{\tau }^2-4D}}{2}$$

Eigenvalues in the complex plane; horizontal real part controls stability, vertical imaginary part controls oscillation. Numerical values are available beside the plot.

What does the linearization predict?

Full modelJδx

Perturbation

1 mV

Direction

0 °

Duration 150 ms400 ms1000 ms Apply perturbation

Local perturbation in scaled coordinates, comparing nonlinear and linear trajectories and eigenvector directions. Numerical values are available beside the plot.

Voltage over time, comparing the full nonlinear model to the Jacobian prediction. Numerical values are available beside the plot.

Playt

80 ms

Fit local view

Follow the equilibria as current changes

StableUnstableSaddle

All equilibrium voltage branches against applied current, with Hopf crossings and saddle-node folds. Numerical values are available beside the plot.

Real and imaginary eigenvalue components against applied current for the selected equilibrium branch. Numerical values are available beside the plot.

Click the voltage branch to select a point; drag horizontally to vary current. Solid and dashed branches show equilibria, not limit-cycle amplitudes.

Read the same geometry in the trace–determinant plane

Trace determinant plane with saddle, node, focus and Hopf boundaries. Numerical values are available beside the plot.

D < 0 → saddle. One eigenvalue is negative, the other positive. Wˢ approaches the saddle; Wᵘ leaves it.

D = 0 → a zero eigenvalue. At a generic saddle-node, two equilibria meet and disappear. The flow slows along the direction whose eigenvalue tends to zero.

τ = 0, D > 0 → imaginary eigenvalues. At a generic Hopf crossing, a focus changes stability while ω stays nonzero. Nonlinear terms determine which periodic orbits exist and their stability.

τ² = 4D → real ↔ complex. Crossing this parabola changes a node into a focus. It is not itself a loss of stability.

Equations, parameters & numerical method

C V̇ = I − g_Cam_∞(V)(V − E_Ca) − g_Kn(V − E_K) − g_L(V − E_L)

ṅ = q(V)[n_∞(V) − n],   q(V) = φ cosh[(V − V₃)/(2V₄)]

m_∞(V) = ½[1 + tanh((V − V₁)/V₂)]
n_∞(V) = ½[1 + tanh((V − V₃)/V₄)]

At equilibrium n* = n_∞(V*). Thus J₂₁ = q n′_∞: the term q′(n_∞ − n) vanishes here. J₁₂ has units mV/ms and J₂₁ has units (mV·ms)⁻¹; both eigenvalues have units ms⁻¹.

Analytic Jacobian; equilibrium continuation parameterized by voltage, with turning-point brackets scanned every 0.25 mV and bisection at folds and trace crossings. Degenerate boundaries are not certified. Nonlinear paths use RK4, Δt = 0.05 ms. Linear paths use exp(Jt). Manifolds are finite orbit approximations seeded ±10⁻⁵ along the saddle eigenvectors in scaled coordinates; Wˢ is computed backward in time, with arrows showing forward time. Paths stop when they leave the integration bounds.

A saddle-node is a local statement. A SNIC additionally requires a closed global return; the Jacobian alone cannot establish it. At a Hopf crossing, linearization cannot distinguish a nonlinear center, a weak focus, or the stability of an emerging cycle.

Model and bifurcation reference: Bard Ermentrout’s XPP tutorial. Hopf preset uses g_Ca = 4.4; type I uses g_Ca = 4, V₃ = 12, V₄ = 17.4, φ = 0.0667. Critical currents are recomputed for the displayed parameters.