Phase-plane layers

Layers use the current applied current. The trajectory can include earlier current values.

V nullcline · dV/dt = 0 w nullcline · dw/dt = 0 EquilibriaTrajectory tail 100 msTail duration (ms)

Shows the last 0–500 ms of model time in the phase plane. Set 0 to show only the current state. Independent of playback speed and the trace window.

Phase-plane view

Equilibria: ● stable (filled circle), ○ unstable (open circle), ◇ saddle (open diamond). These classify local linear stability.

Direction field

Arrows show direction in plot coordinates; their lengths do not represent speed.

Trace windowPlayback speed

1× = 1 ms of model time per ms of playback. Slow playback preserves the integration time step.

Cellular biophysics · NSCI 351

Morris–Lecar Simulator

0 ms

Phase plane Click to set V and w (0–1) · nullclines show zero derivatives

━ Trajectory / ● current state━ dV/dt = 0━ dw/dt = 0

Membrane voltage V (mV)

Potassium activation w

Applied current I (µA/cm²)

Running. Applied-current changes preserve state and history.

Try this in class

Choose Type I below onset, then slow firing. Use the near-rest phase view to find the bottleneck where the nullclines nearly touch.
For Type I, increase I from 40 to 45 µA/cm². The firing rate rises from a very low value.
Compare Type II at I = 88 and 90 µA/cm². Repetitive firing starts at a finite frequency; the near-onset orbit already has a substantial period-to-period rate.
Choose Type II depolarization block at I = 250 µA/cm². After the transient, compare the stable depolarized equilibrium with repetitive firing at I = 100 µA/cm².
Pause and click either side of a nullcline. Predict the initial direction. A nullcline intersection is an equilibrium; its stability matters.
Choose Type I below onset (I = 39 µA/cm²), then enable the saddle manifolds. The stable manifold is the local separatrix: nearby trajectories on opposite sides can have different excursions. The unstable branches leave the saddle in forward time.

Model & parameters

The reduced Morris–Lecar model describes membrane voltage $V$ (mV) and potassium activation $w$ (dimensionless, 0–1). Calcium activation is assumed to equilibrate instantaneously: $m = m_{\infty} (V)$ . An applied current density $I_{app}$ (µA/cm²) drives the membrane.

Membrane and recovery equations

C \frac{d V}{d t} = I_{app} - g_{Ca} m_{\infty} (V) (V - E_{Ca}) - g_{K} w (V - E_{K}) - g_{L} (V - E_{L})

\frac{d w}{d t} = \frac{w_{\infty} (V) - w}{τ_{w} (V)}

The three ionic terms are calcium, potassium, and leak current densities (µA/cm²). Each is a conductance density multiplied by its driving voltage. The potassium gate relaxes toward its voltage-dependent steady state.

Steady-state activation

m_{\infty} (V) = \frac{1 + \tanh (\frac{V - V_{1}}{V_{2}})}{2}

w_{\infty} (V) = \frac{1 + \tanh (\frac{V - V_{3}}{V_{4}})}{2}

The potassium relaxation time is $τ_{w} (V) = \frac{1}{φ \cosh (\frac{V - V_{3}}{2 V_{4}})}$ (ms). The parameters V₁ and V₃ set half-activation voltages; V₂ and V₄ set activation slopes.

Parameter sets used by the starting examples

These are the preset values; sliders modify the active simulation. Type I examples illustrate gradual frequency onset near a saddle-node on an invariant circle (SNIC). Type II examples illustrate finite-frequency onset in a Hopf regime, with initial-condition dependence near onset.

Symbol	Meaning	Type I	Type II	Units
C	Membrane capacitance	20	20	µF/cm²
g_Ca	Calcium conductance	4	4.4	mS/cm²
g_K	Potassium conductance	8	8	mS/cm²
g_L	Leak conductance	2	2	mS/cm²
E_Ca	Calcium reversal potential	120	120	mV
E_K	Potassium reversal potential	−84	−84	mV
E_L	Leak reversal potential	−60	−60	mV
V₁	Calcium half-activation voltage	−1.2	−1.2	mV
V₂	Calcium activation slope	18	18	mV
V₃	Potassium half-activation voltage	12	2	mV
V₄	Potassium activation slope	17.4	30	mV
φ	Potassium rate factor	0.067	0.04	ms⁻¹

Applied current density: Type I examples use 39, 40, or 45 µA/cm²; Type II examples use 0, 88, 90, 100, or 250 µA/cm². The 250 µA/cm² example demonstrates depolarization block.

Numerical simulation

Integration uses fourth-order Runge–Kutta with $Δ t = 0.05$ ms. Default initial conditions are $V (0) = - 60$ mV and $w (0) = w_{\infty} (- 60)$ . Clicking the phase plane chooses a new initial state. Playback speed changes the rate of model time, while the integration step remains fixed.

Changing applied current preserves the evolving state and recorded history; changing another model parameter restarts the trajectory. Trace windows span 50–10,000 ms; CSV export contains the visible time window. Nullclines and equilibrium markers use the current parameter values.

See Chapter 14 of Cellular Biophysics and Modeling and the references below for the model’s derivation and phase-plane interpretation.