The Eigenvalue Value (in Neuroscience)

An interactive walk-through of Christodoulou & Vogels (2022) for NEU200C. Each demo below mirrors a figure from the paper — drag the eigenvalues, sliders, and initial conditions to build intuition.

Setup. A linear neuronal network evolves as

$$\tau \frac{d\mathbf{x}(t)}{dt} = -\mathbf{x}(t) + \mathbf{W}\mathbf{x}(t) + \mathbf{I}(t).$$

We absorb the leak term, set $\tau=1$, and study $\dot{\mathbf{x}} = \mathbf{M}\mathbf{x}$ with $\mathbf{M} = \mathbf{W} - \mathbf{I}$. The eigenvalues of $\mathbf{M}$ tell us almost everything about how the network behaves — and the eigenvectors fill in what they hide.

Fig 2 — Toy 2-neuron systems

Drag the two eigenvalues on the complex plane. The matrix $\mathbf{W}$ on the right, the trajectories $x_1(t), x_2(t)$ in the bottom-left, and the phase-plane vector field on the bottom-right all update live. Switch between real and complex eigenvalue modes, and use the preset buttons (A–D) to jump to the four panels in the paper.

Fig 3 — What the spectrum tells us

Three lessons from the eigenspectrum alone (no eigenvector geometry yet): stability, timescales, and dominant directions.

Fig 4 — What the spectrum hides

Same eigenvalues, very different transient behavior. Crank the eigenvector skew slider and watch the response norm $\|\mathbf{x}(t)\|$ spike before decaying — this is non-normality and transient amplification, invisible to the spectrum alone.

Eigenvalues fixed at −1, −2

eigenvector skew θ = 0.00 rad

W (becomes non-normal)

-1.00	0.00
0.00	-2.00

W

‖off-diagonal of Schur T‖ = 0.000

‖x(t)‖ over time

As θ grows the norm spikes before decaying.

Schur form T (upper triangular)

1.00	-8.00
2.00	-3.00

W (Fig 2C example)

→

-1.00	-3.46
3.46	-1.00

T = U* W U

Diagonal entries are the eigenvalues; the off-diagonal block carries the feed-forward coupling that creates transient amplification.

ε-pseudospectra (pre-rendered)

A · low non-normalityB · medium non-normalityC · high non-normality

Color rings show ε-pseudospectrum bands (ε ∈ {0.01, 0.1, 1, 5, 20, ∞}). Wider rings around the eigenvalues indicate higher non-normality.

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Fig 5 — Random matrix laws

When the entries of $\mathbf{W}$ are drawn from a distribution, the spectrum has a predictable shape. Switch between IID Gaussian (Girko’s circular law), symmetric (Wigner’s semicircle), correlated (elliptic), and E/I networks (Dale’s law).

Bonus — Amplification via dominant eigenvector (Bondanelli & Ostojic 2020)

The classic “non-normality” picture says that to get transient amplification you need the eigenvectors of $\mathbf{J}$ to be non-orthogonal. Bondanelli & Ostojic sharpened this: the precise criterion is on the symmetric part of the connectivity,

and you get amplified trajectories if and only if its largest eigenvalue exceeds 1, i.e., $\lambda_{\max}(\mathbf{J}_S) > 1$. This can hold even when $\mathbf{J}$ itself is stable (all eigenvalues of $\mathbf{J}$ have negative real part). The optimal input direction $\mathbf{r}_0$ that achieves the maximum amplification is precisely the eigenvector of $\mathbf{J}_S$ associated with $\lambda_{\max}(\mathbf{J}_S)$.

In the demo below: drag the eigenvalues of $\mathbf{J}$ on the left, watch the spectrum of $\mathbf{J}_S$ on the right (always real). When the indicator says “Amplifying”, click Snap r₀ to top eigenvector of J_S to maximize the transient.

Reference: Christodoulou, G., & Vogels, T. P. (2022). The Eigenvalue Value (in Neuroscience).