Three ideas that explain how neurons compute
Every neuron in your brain is a dynamical system — its voltage rises, falls, and sometimes explodes into a spike. To understand why, you need just three mathematical ideas: how things change (derivatives), how change compounds (exponentials), and how to read the DNA of a system (eigenvalues).
Before you can model a neuron, you need to read a voltage trace. Not just the voltage — the speed at which it changes. That speed is the derivative, \frac{dV}{dt}.
Drag the dot along the curve below. Watch the tangent line — it shows the direction and steepness of change at that instant.
When \frac{dV}{dt} > 0, the neuron is depolarizing — voltage climbs. When \frac{dV}{dt} < 0, it repolarizes. Notice the relationship between shape and speed.
The aha: \frac{dV}{dt} = 0 at peaks and troughs — the voltage is momentarily frozen. The steepest change happens at the zero crossings, where the curve is most vertical. Increase the frequency (decrease T) and the derivative grows — faster oscillations require faster voltage changes.
If you inject current into a neuron and then stop, what happens? The voltage doesn't snap back instantly. It decays — quickly at first, then slower and slower, following an exponential curve.
This curve is the solution to \frac{dV}{dt} = -\frac{V - V_{eq}}{\tau}. The membrane time constant \tau controls how fast the neuron forgets a perturbation. After one \tau, 63% of the way back. After 3\tau, 95%. After 5\tau, effectively home.
Why this matters: A PV basket cell (τ ≈ 8 ms) forgets in ~40 ms — it tracks fast input faithfully. A pyramidal cell (τ ≈ 25 ms) integrates over ~125 ms — it smooths and accumulates. The time constant is a design parameter that shapes computation.
So far, we have one equation and one variable — voltage on a line. A real neuron has at least two: voltage and a recovery variable (like a slow potassium current). This changes everything.
Left: In 1D, the system can only flow left or right along a line. The green dot is the equilibrium — slide the current I to shift it. Right: In 2D, drag the state point around the V–w plane. The arrow shows where the system wants to go — notice how the direction depends on both coordinates.
In 1D, the system can only converge or diverge. In 2D, it can spiral, oscillate, overshoot. This is where neurons become interesting — and where eigenvalues become essential.
A 2D linear system near equilibrium is controlled by a 2×2 matrix. That matrix stretches, rotates, and flips the state space. Eigenvalues tell you how.
Set the matrix entries below, then press Apply. Watch how the arrow gets stretched and rotated. The eigenvalues are the stretch factors along special directions.
With the default diagonal matrix, the horizontal component stretches by 2 and the vertical component shrinks by 0.5. Try setting off-diagonal entries to see rotation and shearing.
Every 2×2 matrix has a trace (sum of diagonal) and determinant. These two numbers alone classify all possible behaviors near an equilibrium. Drag the point below across the plane and watch the phase portrait change.
The trace tells you whether the system grows or decays overall. The determinant tells you whether trajectories stay in one direction or spiral. Together, they classify every possible linear behavior near an equilibrium — including a neuron's subthreshold dynamics.
The parabola \text{tr}^2 = 4\,\text{det} is the boundary between monotone and oscillatory behavior. Below it: spirals. Above it: straight-line convergence or divergence. Neurons that resonate (like many cortical interneurons) live below the parabola. Neurons that integrate (like some pyramidal cells near rest) live above it.
These three tools — derivatives for reading instantaneous change, exponentials for understanding how perturbations decay, and eigenvalues for classifying behavior near equilibria — are everything you need to analyze a neuron as a dynamical system. The derivative tells you what's happening now. The exponential tells you how fast the system forgets. The eigenvalues tell you what kind of behavior to expect.
Next, we put them together and read full phase portraits — nullclines, fixed points, limit cycles, and the geometry of spiking.
→ Phase Plane Explorer (coming next)