The missing bridge

Your existing tutorials already contain the pieces. Mean field turns a stochastic population into a tractable macroscopic state. Criticality asks how that state changes near a phase transition. Information geometry asks how rapidly the entire predicted probability distribution changes as model parameters move. The Widom line sits between the last two ideas.

Start with the thing we can actually count

Imagine pausing the network at time $t$. Let $A(t)$ be the number of neurons that are active at that instant and let $N$ be the total number of neurons. We turn that count into a fraction:

The Greek letter $\rho$ (“rho”) is only a name. It is not a new physical object. If 200 of 1,000 neurons are active, then $\rho=200/1000=0.2$. Using a fraction rather than a raw count lets networks of different sizes be compared on the same 0-to-1 scale.

Physicists call $\rho$ an order parameter. In normal language: it is the dashboard number we watch to tell what the whole system is doing. Near zero means mostly silent; a sustained value above zero means ongoing population activity.

Now introduce the two knobs—because they do different jobs

The first knob is $\kappa=w_0/w_{0c}$. It tells us how strong recurrent excitation is relative to the model's critical E–I balance. Turning $\kappa$ changes whether activity tends to die out or recruit more activity. This is the main tuning knob we sweep from subcritical to supercritical.

The second knob is $h$, the external background push. In the code it represents spontaneous activation or external drive. It acts directly on the thing we are measuring by nudging inactive neurons toward activity. Physicists call this a field; here, “small direct push” is the useful translation.

A response curve is just an input–output curve

Pick one value of $\kappa$. Set the background push $h$, let the mean-field system settle, and record its settled activity. Increase $h$ slightly, let it settle again, and record the new activity. Repeating that gives an ordinary input–output curve:

The input is the push $h$. The output is the activity level after settling, $\rho^*$. The star does not mean multiplication. It is mathematical shorthand for “the fixed, stationary, or settled value of $\rho$.”

We usually do not care about the entire curve at once. We care about its slope at the operating point. A steep slope means a tiny extra push causes a large activity change; a shallow slope means the same push barely matters. That slope is what “susceptibility” means here:

“Local” means local on the input–output curve: the slope at the current $h$, not the average slope across every possible input. It does not mean that only a spatially local cluster of neurons responds.

The project also measures fluctuations

There is a second way to ask whether the system is excitable: do not change $h$ at all; watch the spontaneous ups and downs of $\rho(t)$. Large variance means the network's activity is wandering over a broad range even under the same conditions:

Where correlation length enters

In ordinary language, correlation length asks: how far through the network does one neuron's change remain statistically related to others? The symbol $\xi$ names that typical reach. A large $\xi$ means a nudge recruits a large coordinated patch, which helps explain why population susceptibility becomes large.

Formally, if $G(r)$ measures activity correlation at separation $r$, equilibrium intuition gives $\chi\propto\int G(r)\,d^dr$. In your all-to-all aggregate model there is no literal spatial distance, so relaxation time, covariance modes, and population response are the practical analogues; one should not claim a measured spatial $\xi$ without adding topology.

Where the toy equation—and its response—come from

We want the smallest equation that tells a sensible story about population activity. Read each term as something that raises or lowers the active fraction:

This equation is a normal form—plainly, a stripped-down model that keeps only the ingredients needed near the transition. It is motivated by the fuller mean-field Wilson-Cowan drift, “activation of available neurons minus recovery.” Expanding that fuller drift near low activity produces linear growth, nonlinear saturation, and drive terms of this shape. The toy is therefore justified as a local teaching model, not claimed to be the project's exact simulator.

Derive the stationary state: set “still changing” to zero

The dot in $\dot\rho$ means “rate of change of activity.” A stationary state is a value where the gains and losses balance, so activity is no longer drifting upward or downward. Therefore set $\dot\rho=0$:

That is an ordinary quadratic equation in $\rho^*$. The quadratic formula gives two roots. The minus root is negative, so it cannot represent a fraction of active neurons. Keeping the nonnegative root gives:

Derive susceptibility: measure the slope of the settled activity

Now ask the concrete question: at fixed $\kappa$, how much does that settled activity change if $h$ increases by a tiny amount? Define $\chi_{\mathrm{resp}}=\partial\rho^*/\partial h$ and differentiate the balance equation itself:

So neither formula was dropped from the sky. The stationary-state function comes from balancing the toy dynamics. The response susceptibility comes from taking the slope of that stationary solution with respect to the direct push.

Why the critical point becomes a rounded peak

At $h=0$, exactly zero activity can become absorbing, and the stable solution changes sharply at $\kappa_c=1$. At $h>0$, the external push prevents perfect silence, so the sharp transition is smoothed into a crossover. The infinite mathematical sensitivity disappears, but a finite maximum remains near the old threshold.

A Widom line is a ridge, built one scan at a time

Choose one drive $h_j$, scan coupling, and find the maximum of a declared susceptibility. Repeat for several drives. The maximizers $\kappa_w(h_j)$ form a discrete ridge; interpolation makes it look like a line.

The line locates quasi-criticality; the cross-section measures robustness

The Widom line answers where is the response maximal? A robustness window answers how far can parameters be mistuned while response stays high? Those are different estimands. At each fixed $(N,h)$, the connected full width at half maximum is:

Information geometry turns the parameter plane into a statistical manifold

So far, each point $\theta=(\kappa,h)$ produced a mean and a susceptibility. Information geometry replaces that point with an entire probability model $p(y\mid\theta)$ for an observation $y$—perhaps stationary activity, block summaries, or complete paths.

The equilibrium identity that creates the famous connection

In a Gibbs family with field $h$ and extensive order parameter $M$, write $p(x\mid h)\propto e^{\beta hM(x)}$. Then the score is centered magnetization:

This is why Fisher information often peaks or diverges near equilibrium critical points: nearby parameter values become highly distinguishable. But your nonequilibrium stationary process does not inherit this identity for free.

Map the idea onto the actual Wilson-Cowan study

Normalize coupling by each architecture's analytic zero-drive critical coordinate so the scan compares like with like:

One cell, one residence-time estimator

What the repository's calibration has already taught us

A clean claim ladder

1. Descriptive: show Mono and CoTx susceptibility profiles with stationarity and topology flags.
2. Integrated robustness: compare matched-seed areas on the frozen common $\kappa$ domain.
3. Width robustness: only after both nearest crossings are estimable, compare connected FWHM.
4. Finite-size statement: model how peak height, location, width, or area changes with $N$.
5. Thermodynamic claim: require a scaling law; a wider finite-size region is not automatically an extended critical phase.

The whole story, compressed

Continue through the tutorial sequence

Sources and project records

1. R. V. Williams-García, M. Moore, J. M. Beggs, and G. Ortiz (2014), “Quasicritical brain dynamics on a nonequilibrium Widom line.”
2. A. de Candia, A. Sarracino, I. Apicella, and L. de Arcangelis (2021), “Critical behaviour of the stochastic Wilson-Cowan model.”
3. S. Amari and H. Nagaoka (2000), Methods of Information Geometry.
4. Project methods: docs/SUSCEPTIBILITY_WIDOM_PROTOCOL.md, the 2026-08-12 pilot decision, LNA validation decision, and 2026-08-13 frozen integrated protocol.