Bifurcation Theory & Neural Excitability

How qualitative changes in dynamics create two fundamentally different kinds of neurons

1. One Dimension — Birth and Death of Fixed Points

Before we touch neurons, let's build intuition with the simplest possible dynamical system. One variable, $x$, evolving according to:

$$\frac{dx}{dt} = f(x) = x^2 + \mu$$

The parameter $\mu$ controls the shape of $f(x)$. Fixed points — where $dx/dt = 0$ — are the places where $f(x)$ crosses zero. The sign of $f(x)$ tells you which way $x$ flows.

At $\mu = 0$, two fixed points appear out of thin air — one stable (attracting, shown as a filled dot), one unstable (repelling, shown as an open dot). This is a saddle-node bifurcation, and it's how many neurons transition from resting to spiking. The stable fixed point is the resting membrane potential; the unstable one is the firing threshold. When they collide and annihilate, the neuron has to spike because there's nowhere to rest.

2. The Bifurcation Diagram

Instead of looking at one value of $\mu$ at a time, we can build a picture of all the fixed points for all parameter values at once. This is the bifurcation diagram.

The solid curve is a stable rest state. The dashed curve is the threshold. At the bifurcation point ($\mu = 0$), they collide and vanish — the neuron has to spike because there's nowhere stable to rest.

3. Two Types of Neuron — The Fork in the Road

The saddle-node bifurcation isn't the only way a neuron can start spiking. There's a fundamentally different mechanism — the Hopf bifurcation — and it creates a completely different kind of neuron. Let's see both side by side.

The theta neuron starts firing at zero frequency — arbitrarily slow spikes appear right at threshold, and the rate climbs continuously as $\sqrt{I}$. The FitzHugh-Nagumo model starts firing at a finite minimum frequency — it jumps from silence to rhythmic spiking. Two different bifurcations, two different kinds of neurons.

4. How to Tell Them Apart

These aren't just mathematical curiosities — they predict measurable electrophysiology.

Impedance Profile

Another signature: inject oscillating current at different frequencies and measure the voltage response amplitude. Integrators respond most to slow input. Resonators have a peak.

5. The Hopf Bifurcation — Watching Eigenvalues Cross

Why does the Hopf bifurcation create oscillations? The answer lives in the eigenvalues of the Jacobian matrix at the fixed point. Two complex conjugate eigenvalues $\lambda = \alpha \pm i\beta$ encode two things: the real part $\alpha$ is the growth/decay rate, and the imaginary part $\beta$ is the oscillation frequency.

When $\alpha < 0$: perturbations decay — stable spiral. When $\alpha > 0$: perturbations grow — unstable spiral, and a limit cycle catches the trajectory. The bifurcation happens at $\alpha = 0$, exactly when the eigenvalues cross the imaginary axis.

The entire story of the Hopf bifurcation is here: eigenvalues crossing the imaginary axis. Negative real part → stable. Positive real part → unstable. The imaginary part determines the oscillation frequency. This is why resonators have a preferred frequency — it's baked into the imaginary part of their eigenvalues.

6. Which Type Are Hippocampal Neurons?

The type of bifurcation isn't just mathematics — it predicts measurable properties. Different hippocampal cell types sit on different sides of this dynamical divide.

PV interneurons as resonators selectively lock to gamma oscillations. Pyramidal cells as integrators perform graded rate coding. This dynamical diversity is interneuron diversity, expressed in the language of bifurcations.