🧠 Tier: Standard Undergraduate / Graduate · Dynamical Systems · Phase Plane
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§ 01
Interactive Simulation
State var 1
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State var 2
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Input I
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Firing rate
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Spike count
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Mode
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Time
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Playback
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Parameters
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Input I—
T sim (ms)500
Overlays
§ 02
The Idea, Step by Step
▸ From a wind-map to a spiking neuron
Picture a hiking map where every single spot has a tiny arrow showing which way the wind would push you. Drop a marble anywhere and it drifts along the arrows, tracing a path. A few rare spots have no arrow at all — stand there and nothing pushes you, so you stay put. Those no-push spots are the fixed points, and that whole field of arrows is the phase plane of a neuron. Reading the map tells you almost everything the neuron will do, without ever solving a hard equation.
A neuron model juggles two quantities at once: its voltage $V$ and a slower "recovery" variable $w$. Instead of drawing each against time, we plot $w$ against $V$ — that is the phase plane. Two guide-curves organise the whole picture. The V-nullcline is every place where the voltage is momentarily frozen, $dV/dt = 0$; the w-nullcline is where recovery is frozen, $dw/dt = 0$. Where the two curves cross, both are frozen, so the cell sits still — that crossing is the resting state. For the FitzHugh–Nagumo neuron the V-nullcline is the N-shaped cubic $w = V - V^3/3 + I$ and the w-nullcline is the straight line $w = (V+a)/b$. With $a=0.7$, $b=0.8$ and input $I=0$ they meet near $V^* \approx -1.2$ — the cell's resting voltage.
Is that rest point stable? Zoom in close and the curved equations look linear, captured by the Jacobian matrix $J$. Read off two numbers from it: the trace and the determinant. When $\mathrm{Tr}(J) < 0$ and $\mathrm{Det}(J) > 0$, every small nudge fades away and the cell returns to rest (a stable node or spiral); when $\mathrm{Det}(J) < 0$ you have a saddle, which is the true geometric meaning of a firing threshold. Now turn up the input current $I$: the cubic slides upward, the crossing climbs onto the unstable middle branch, the rest point loses stability, and a limit cycle is born — the neuron fires again and again. The sliders map straight onto this story: Parameter 1 and Parameter 2 reshape the nullclines, Input I slides the cubic up and down, and T sim sets how long you watch the trajectory unfold.
TRY THIS — 1
Open the Phase Plane tab and switch on Nullclines. Watch the yellow dot sit exactly where the cyan V-nullcline and the magenta w-nullcline cross — that intersection is the fixed point.
TRY THIS — 2
Raise Input I slowly and watch the fixed point creep up the cubic. At a critical value the trajectory stops settling and breaks into a closed loop — you have just crossed the bifurcation into repetitive spiking.
TRY THIS — 3
Set Input I back to a low value and give the neuron different starting nudges. Small ones fall straight back to rest; a big enough one loops once around the cubic before returning — that single loop is one action potential, and it shows why threshold is a curve in the plane, not one fixed voltage.
§ 03
Equation Derivation
▸ Phase Plane Analysis — Core Definitions
Phase plane analysis is the geometric study of 2D dynamical systems. For a neuron model dx/dt = f(x,y), dy/dt = g(x,y), the phase plane reveals all qualitative dynamics: fixed points, limit cycles, separatrices, and trajectories.
$$\text{V-nullcline: } \frac{dV}{dt} = 0 \;\Rightarrow\; \text{curve in (V,w) space where V is momentarily constant}$$
$$\text{w-nullcline: } \frac{dw}{dt} = 0 \;\Rightarrow\; \text{curve where w is momentarily constant}$$
$$\text{Fixed points: intersections of nullclines} \;\Rightarrow\; f(V^*,w^*) = 0 \;\text{AND}\; g(V^*,w^*) = 0$$
Stability of fixed point (V*, w*) from Jacobian J = ∂(f,g)/∂(V,w) evaluated at (V*,w*):
At any point (V,w) in the phase plane, the vector (dV/dt, dw/dt) = (f(V,w), g(V,w)) gives the instantaneous direction of motion. The collection of all such vectors is the direction field (or vector field). Trajectories are integral curves of this field — lines everywhere tangent to the vector. Nullclines divide the plane into regions with definite signs of dV/dt and dw/dt, allowing qualitative prediction of all trajectories without solving the ODE explicitly.
STEP 2 — Nullcline Geometry for FHN
V-nullcline of FHN: w = V − V³/3 + I (cubic, N-shaped). The three branches have specific roles: left branch (V < −√1 = −1): dV/dt < 0 above, > 0 below → stable. Middle branch (−1 < V < +1): dV/dt > 0 above, < 0 below → unstable. Right branch (V > 1): stable again. The w-nullcline (w = (V+a)/b) intersects the V-nullcline at the fixed point(s). The stability is determined by which branch the intersection occurs on.
STEP 3 — Separatrix and Threshold
When a fixed point is a saddle, it has two invariant manifolds: the stable manifold (separatrix) along which trajectories approach the saddle, and the unstable manifold along which they depart. In excitable systems, the stable manifold of the (unstable) threshold saddle point divides the phase plane into sub-threshold (trajectories return to rest) and suprathreshold (trajectories make the AP loop) regions. This is the geometric definition of "threshold" — it is not a voltage but a curve (separatrix) in 2D phase space.
STEP 4 — Limit Cycle Existence (Poincaré-Bendixson)
If a trajectory remains in a bounded region and there are no fixed points in that region, it must approach a limit cycle (Poincaré-Bendixson theorem). For FHN: construct a trapping region (a large rectangle that all trajectories enter and none leave). Show there is an unstable fixed point inside. Then by Poincaré-Bendixson, a limit cycle exists — the AP waveform. The limit cycle disappears via bifurcation (Hopf or SNIC) as I decreases below threshold.
STEP 5 — Stable and Unstable Manifolds
The stable manifold of a saddle point is computed by integrating backward in time from the saddle point along its stable eigenvectors. The unstable manifold is computed by integrating forward. These manifolds act as "walls" in phase space that trajectories cannot cross (by uniqueness of solutions). In excitable neurons, the threshold curve is the stable manifold of the saddle — trajectories starting slightly above it go around the AP loop; slightly below, they return to rest.
🔬 SimulationWhat does the Phase Plane tab show — how do I read it?▼
The Phase Plane shows 6 elements: (1) Cyan V-nullcline — dV/dt=0 curve; trajectory moves horizontally when it crosses this; (2) Magenta w-nullcline — dw/dt=0 curve; trajectory moves vertically here; (3) Yellow dot — current fixed point (intersection of nullclines); (4) Green trajectory — the path (V(t), w(t)) traced over time; (5) Arrow direction — current direction of motion; (6) Direction field — small arrows showing the flow everywhere. Direction of motion in each quadrant: above V-nullcline → V decreasing (←); below → V increasing (→); above w-nullcline → w decreasing (↓); below → w increasing (↑). The trajectory always follows the vector sum of these forces.
Key takeaway: To read a phase plane: V-nullcline = where horizontal motion changes direction; w-nullcline = where vertical motion changes direction. Fixed point = intersection. Trajectory always follows the direction field.
🧠 ConceptualWhat is the geometric meaning of 'threshold' in the phase plane?▼
In a 2D neuron model, 'threshold' is NOT a single voltage value — it is the stable manifold of the saddle point. This is a 1D curve in 2D phase space that divides the basin of attraction of the rest state from the basin of the AP loop. Stimuli that push the state just below the separatrix produce subthreshold responses (trajectory returns to rest via the left branch of the V-nullcline). Stimuli just above produce full APs (trajectory orbits the right branch). This means threshold depends on the initial conditions in both V AND w — a neuron in the relative refractory period (high w) has a higher effective threshold voltage than a fully rested neuron.
Key takeaway: Threshold in 2D is the stable manifold of the saddle point — a curve, not a voltage. The effective threshold voltage depends on w (recovery state), which is why the refractory period raises threshold.
🌍 AppliedHow does phase plane analysis apply to designing stimulation protocols for neurons?▼
Phase plane analysis directly guides cochlear implant, retinal prosthetic, and DBS stimulation design. By computing the separatrix (threshold curve) in the V-w phase plane of the target neuron model, engineers can design current pulses that reliably cross the threshold with minimum charge injection. The 'strength-duration curve' for electrical stimulation (threshold current vs pulse width) is derived directly from the phase plane geometry: short pulses must have large amplitude to overcome the fast V-nullcline dynamics; long pulses can be weaker because they drive V slowly past the saddle point's stable manifold. Knowing the exact separatrix also allows 'cathodic-first biphasic' pulse design that minimises charge while maximising threshold crossing probability.
Key takeaway: Phase plane separatrix analysis gives the minimum stimulation charge to trigger a spike — critical for cochlear implant, retinal prosthetic, and DBS design. Optimal pulse shapes are derived from the geometry of the threshold curve.
💡 Non-ObviousCan a system with only one fixed point oscillate — how does the Poincaré-Bendixson theorem guarantee a limit cycle?▼
Yes — a single unstable fixed point in a bounded region guarantees a limit cycle (Poincaré-Bendixson theorem, 1886/1901). The conditions: (1) the system is 2D; (2) the fixed point is unstable (Tr(J) > 0); (3) there exists a trapping region — a bounded region with no trajectory leaving it. For FHN above I_Hopf: construct the rectangle {−2.5 ≤ V ≤ 2.5, −2 ≤ w ≤ 2} and verify all vector field arrows on the boundary point inward. The unstable fixed point is inside. By P-B, a limit cycle must exist between the fixed point and the boundary. This is the mathematical guarantee of repetitive firing — no explicit construction of the limit cycle is needed.
Key takeaway: Poincaré-Bendixson: unstable fixed point + bounded trapping region → limit cycle guaranteed. This is the mathematical proof that repetitive firing must occur when the rest state is destabilised — no other attractor exists in 2D.
📐 ComputationalHow do you numerically draw nullclines for an arbitrary 2D neuron model?▼
To draw the V-nullcline: for each V in a grid, solve f(V,w) = 0 for w. If f is linear in w: w = −A(V)/B(V) analytically. For FHN: w = V − V³/3 + I. For HH projected to 2D: substitute m = m∞(V), h = h∞(V) and solve the cubic in V. To draw the w-nullcline: solve g(V,w) = 0 for w — usually straightforward (w = w∞(V) for most models). Plot both curves. For Morris-Lecar, both nullclines require numerical root-finding per V value. Direction field: at each grid point (V_i, w_j), compute (f(V_i,w_j), g(V_i,w_j)), normalise, and draw an arrow. For the trajectory: integrate the ODE with RK4 starting from initial conditions.
Key takeaway: V-nullcline: set f=0, solve for w at each V. w-nullcline: set g=0, solve for w. Direction field: normalise (f,g) at each grid point. Trajectory: RK4 integration from initial conditions. All can be computed without any special software.
🎓 DeepWhat is an isocline and how does it relate to nullclines?▼
An isocline is a curve along which dV/dt = c for some constant c (not necessarily 0). Nullclines are the special case c = 0 isoclines. The family of all isoclines (for all c) forms a coordinate system in the phase plane — every isocline of dV/dt = c is a curve where trajectories cross it with the same horizontal velocity c. Graphically, trajectories cross the V-nullcline (c=0) horizontally; they cross other isoclines at slopes determined by c/g(V,w). Isoclines are used in 'isocline method' for sketching phase portraits by hand: draw several isoclines and use them as guides for the trajectory direction. For neurons, the V=0 isocline (w = V−V³/3+I) and the w=0 isocline (w=(V+a)/b) are the nullclines; the V=1 isocline shows where dV/dt=1 mV/ms etc.
Key takeaway: Isoclines are level curves of dV/dt = constant. Nullclines = zero-isoclines. The full family of isoclines gives a coordinate system for the phase plane, useful for sketching trajectories by hand without numerical integration.
§ 04 Best Resources
Izhikevich — Dynamical Systems in Neuroscience, MIT Press, 2007. Free: dynamicalsystems.org
Gerstner et al. — Neuronal Dynamics, Cambridge, 2014. neuronaldynamics.epfl.ch
Strogatz — Nonlinear Dynamics and Chaos, Westview, 2015
Ermentrout & Terman — Mathematical Foundations of Neuroscience, Springer, 2010
§ 05
Misconceptions & Common Errors
Sub-block A — Conceptual Misconceptions
❌"A stable fixed point means the system does not oscillate and cannot produce action potentials."
✅A stable fixed point means only that small perturbations decay to rest. Large perturbations (above threshold) can still produce full AP excursions before the trajectory returns to rest — this is excitability, not oscillation. The system is excitable, not oscillating. A stable fixed point coexisting with a large limit cycle (in the subcritical Hopf regime) also allows firing above a bistability threshold. Additionally, a stable fixed point on the outer branch of the V-nullcline (Type I, SNIC geometry) will produce APs in response to suprathreshold stimuli even though it is stable.
❌Drawing nullclines by plotting where V is large or small — confusing the nullcline with the trajectory maximum/minimum.
✅The V-nullcline is the set of (V,w) where dV/dt = 0 — it is a curve in phase space where the VELOCITY in the V-direction is zero, NOT where V takes extreme values. The maximum of the trajectory in V occurs when dV/dt = 0, so the trajectory maximum DOES cross the V-nullcline — but the V-nullcline is not the trajectory itself. The trajectory maximum occurs at the point where the trajectory crosses the V-nullcline. Confusing these gives incorrect phase portraits where the nullcline is drawn as the envelope of trajectories rather than the zero-velocity curve.
🔍 Why: Very common error in hand-drawn phase portraits: students connect the trajectory peaks/troughs instead of the zero-velocity locus.
§ 05 References
Izhikevich — Dynamical Systems in Neuroscience, MIT Press, 2007
Gerstner et al. — Neuronal Dynamics, Cambridge, 2014. neuronaldynamics.epfl.ch
Ermentrout & Terman — Mathematical Foundations of Neuroscience, Springer, 2010
Strogatz — Nonlinear Dynamics and Chaos, Westview, 2015