Phase Plane Analysis

Seeing a neuron's entire future in two dimensions

"What if you could see a neuron's entire future — not just its voltage trace, but every possible trajectory it could ever take? That's what a phase portrait shows you. And once you can read one, you can predict whether a neuron will rest, spike, oscillate, or burst — just by looking at two curves."

We'll build this intuition piece by piece using the FitzHugh-Nagumo model — a two-variable reduction of Hodgkin-Huxley that preserves its essential dynamics while being simple enough to visualize completely.

Here $v$ is a fast voltage-like variable and $w$ is a slow recovery variable. The parameters $a = 0.7$, $b = 0.8$, $\varepsilon = 0.08$ shape the dynamics. $I$ is an external current we can inject.

1. State Space — Where Is the Neuron?

Instead of plotting voltage against time, we plot voltage ($v$) against the recovery variable ($w$). Each point in this plane is a complete description of the neuron's state at one instant. There is no time axis — just "where the neuron is right now."

Try it: Drag the dot around. You're placing the neuron in a different initial condition. Right now this plane is empty — we can't tell where the neuron would go from any given state. That changes next.

2. The Vector Field — Which Way Does the State Move?

At every point in state space, the differential equations define a direction — like a current in a river. The state is carried along by this flow. The arrows below show the direction and speed of this current everywhere simultaneously.

Try it: Drag the dot and watch the large arrow showing the exact $(dv/dt,\; dw/dt)$ vector. Where does the flow converge? Where does it diverge? Is there a point where the arrow shrinks to nothing?

3. Trajectories — Following the Flow

Now release a "particle" into the current and watch where it goes. Each click launches a trajectory from that initial condition, integrated forward using the differential equations. The moving dot traces where the neuron's state travels through time.

Try it: Click at different starting points — corners, edges, center. Do all trajectories end up in the same place? How fast do they get there? Try starting from the top-left versus the bottom-right.

4. Nullclines — The Map of the Territory

The vector field is rich but hard to read at a glance. We need landmarks. The two most important curves in any phase portrait are the nullclines — the places where one of the two variables is momentarily frozen.

4a. The v-nullcline

Setting $dv/dt = 0$ gives us $w = v - v^3/3 + I$. This cubic curve divides the plane into regions where voltage is increasing (right of the curve) and decreasing (left of it).

Try it: Click above the cyan curve, then below it. The horizontal component of the trajectory reverses across the nullcline. The curve is a boundary between "voltage rising" and "voltage falling."

4b. Adding the w-nullcline

Setting $dw/dt = 0$ gives us $w = (v + a)/b$. This straight line divides the plane into regions where recovery is increasing (below the line) and decreasing (above it).

Where both nullclines cross, both variables are frozen — $dv/dt = 0$ and $dw/dt = 0$ simultaneously. That's a fixed point: an equilibrium where the neuron sits forever unless perturbed. The green circle marks it.

Try it: Launch trajectories from all four quadrants defined by the nullclines. In each quadrant, the flow has a characteristic direction — up-right, up-left, down-left, down-right. The nullclines carve the plane into regions of qualitatively different flow.

5. The Current Slider — Watching a Bifurcation

So far the neuron rests. What makes it fire? In the FitzHugh-Nagumo model, injected current $I$ shifts the v-nullcline vertically. As it shifts, it drags the fixed point along the w-nullcline. At some critical current, something qualitative changes.

Try it: Slowly drag the current slider from 0 toward 0.5. Watch the cyan nullcline rise, pulling the fixed point up the gold line. At the critical current, the green dot turns red — the fixed point became unstable. Now click to launch a trajectory. Instead of converging to rest, it spirals outward into a limit cycle. The neuron fires repetitively.

This qualitative change — from stable rest to repetitive spiking — is a Hopf bifurcation. It happens when the eigenvalues of the linearized system cross the imaginary axis. You just watched it happen geometrically: the fixed point slid past the "knee" of the cubic, and the balance of forces reversed.

6. Two Views — Phase Portrait and Voltage Trace

The loop in phase space is the repetitive spike train in time. Here we show both views simultaneously: the phase portrait on the left and the voltage trace $v(t)$ on the right. A synced cursor connects the two representations.

Try it: Click "Relaxation osc." — notice the sharp, spike-like waveform and the trajectory hugging the cubic nullcline. Now click "Sinusoidal" — the oscillation becomes smooth and round. The parameter $\varepsilon$ controls time-scale separation: small $\varepsilon$ means the recovery variable is much slower than voltage, producing sharp action-potential-like spikes.