Morris-Lecar Model

The ML model uses millivolt units and real ionic currents (Ca²⁺, K⁺, leak) — all axis labels are in mV, ms, mA/cm². The Phase Plane tab shows the V-nullcline (N-shaped, from the Boltzmann Ca²⁺ activation) and w-nullcline (sigmoidal K⁺ activation), which are more complex than FHN's simple cubic+line. The Type I/II toggle changes V_3 and φ, switching the bifurcation type — observe how the fixed point position on the V-nullcline shifts and the trajectory shape changes. The Conductance tab shows g_Ca(t) and g_K(t) as functions of V in real time.

Ca²⁺ channels in the Morris-Lecar model (barnacle muscle Ca²⁺ channels, and similar to HVA Ca²⁺ channels in neurons) activate very rapidly — time constant τ_m < 1 ms at physiological voltages. K⁺ channels activate on a timescale of 5–50 ms (τ_w). The timescale separation is τ_w/τ_Ca ≫ 1, justifying the quasi-static approximation m ≈ m∞(V). This is identical to HH's observation that τ_m ≪ τ_n — Hodgkin and Huxley also treated m as quasi-static in some analyses. The quasi-static approximation reduces a 3D system (V,m,w) to 2D (V,w), enabling full phase-plane analysis.

ML is the standard model for: (1) Cortical excitability classification — theorists use ML (rather than HH) to derive the Rinzel-Ermentrout Type I/II classification because of its analytical tractability; (2) Cardiac myocyte modelling — early models of cardiac Ca²⁺ channels used ML-type equations (later extended to Luo-Rudy); (3) Smooth muscle — Ca²⁺-dependent contractions in arterial smooth muscle use ML-type models; (4) Endocrine cells — pancreatic beta cell bursting (Chay-Keizer model) is a direct extension of ML; (5) Bursting pacemakers — thalamic and hippocampal cells use ML with additional currents. ML is preferred over FHN for these because it uses biophysically interpretable parameters.

V_3 sets the midpoint of the K⁺ activation curve w∞(V). Shifting V_3 moves the w-nullcline horizontally in the V-w phase plane. When V_3 is small (e.g., +2 mV for Type I): the w-nullcline intersects the left (stable) branch of the V-nullcline — fixed point is on the stable branch, SNIC bifurcation possible. When V_3 is larger (+12 mV for Type II): the w-nullcline shifts right, its intersection moves to the middle (unstable) branch — the fixed point destabilises via Hopf. A 10 mV shift in the midpoint of one Boltzmann function is enough to change the mathematical class of the entire neuron's excitability. Biologically, this small shift corresponds to different K⁺ channel subtypes or different neuromodulatory states.

φ (phi) is a temperature scaling factor for the K⁺ activation kinetics: dw/dt = φ × (w∞−w)/τ_w. Larger φ → faster K⁺ recovery → shorter APs, higher maximum firing rate, faster spike-frequency adaptation. The original Morris-Lecar model used φ = 0.04 (barnacle muscle at 5°C). For Type II parameters: φ = 0.23 (faster, mammalian-like). φ affects the Hopf bifurcation onset: increasing φ increases the minimum frequency at onset (ω_Hopf = φ × Im(eigenvalue)/τ_w). In the simulation, dragging φ from 0.04 to 0.23 with Type II parameters shows the transition from near-Type I (slow recovery, very low onset frequency) to clear Type II (fast recovery, 20+ Hz minimum firing rate).

The theta model (Ermentrout & Kopell, 1986) is the exact normal form of the SNIC bifurcation: dθ/dt = 1 − cos θ + I(1 + cos θ). Near the SNIC bifurcation in the Morris-Lecar Type I model, the dynamics on the invariant circle (just above I_SNIC) are exactly described by the theta model. The relationship: V ≈ tan(θ/2), so the ML action potential corresponds to a single rotation of θ from −π to +π. The ML model is thus the 2D biophysical embedding of the 1D theta model: away from the bifurcation, ML adds the w variable providing recovery; near the bifurcation, the w dynamics are enslaved and ML reduces to theta. This connection enables exact analytical results for ML synchronisation properties using theta model analysis.

Three key reasons: (1) Only 2 variables — V and w can be drawn as a 2D phase portrait; HH needs 4D. (2) Clearly separated timescales — Ca²⁺ is instantaneous (m≈m∞), K⁺ is slow (w), making the fast-slow decomposition transparent. (3) Tunable excitability type — by changing V_3 and φ, you can demonstrate Type I vs Type II within the same model framework, which is impossible in HH without changing the channel type. For HH, all phase plane analysis requires projecting the 4D system onto 2D subspaces, losing intuition. ML provides all the qualitative richness of HH in a form where every feature of the trajectory can be understood geometrically.