Measuring Criticality Robustness
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How wide is the critical region, and does co-transmission make it more robust?
Based on de Candia, Sarracino, Apicella & de Arcangelis (2021). Critical behaviour of the stochastic Wilson-Cowan model. PLoS Comput Biol 17(8): e1008884.
A central finding in computational neuroscience is that neural networks can exhibit critical behaviour — a state at the boundary between ordered and disordered dynamics, characterized by scale-free avalanche distributions with power-law exponents τS = 3/2 and τT = 2 (the mean-field branching process universality class).
In the stochastic Wilson-Cowan model, criticality occurs at a specific value of the control parameter w0 = wE − wI, which expresses the relative balance between excitatory and inhibitory synaptic strength. De Candia et al. (2021) showed that this critical value is:
But this raises a fundamental question: if criticality requires exactly the right parameter value, how could a biological system ever achieve it? This is the fine-tuning problem.
In a finite neural network, criticality is not a single point but a region in parameter space. How wide is this region? Does it shrink as the network grows? And critically: do co-transmitting networks have a wider critical region than mono-transmitting ones?
This tutorial walks through our methodology for answering these questions using critical window analysis — a principled approach that leverages the theoretically derived critical point to measure the width of the “approximately critical” parameter region as a function of system size.
In the standard (mono-transmission) Wilson-Cowan model with symmetric weights (wEE = wIE = wE, wII = wEI = wI) and equal population sizes (NE = NI = N), the fixed-point equation for the average activity Σ0 is:
For h = 0, linearizing tanh around Σ0 = 0 yields the critical condition w0 · β = α, giving:
When co-transmitting (C) neurons are introduced, they contribute to both excitatory and inhibitory synaptic input. With wc_ratio = 0.5, each C neuron excites with weight wE and inhibits with weight 0.5 · wI. The net contribution of the C population shifts the effective w0 upward, reducing the external w0 needed for criticality:
Both critical points are independent of population proportions in the mean-field limit, because the weight matrix is symmetric across target populations. The proportion of co-transmitting neurons (prop_C) only affects finite-size noise structure, not the location of the critical point. This is a key theoretical result.
| Parameter | Value | Meaning |
|---|---|---|
| α | 0.1 ms¹ | Deactivation (recovery) rate |
| β | 1.0 ms¹ | Activation rate scale |
| ws = wE + wI | 13.8 | Total synaptic weight (fixed) |
| Symmetric weights | wEE=wIE, wII=wEI | All-to-all equal within type |
| wc_ratio | 0.5 | C neurons inhibit at half the I weight |
| h | 10−6 | External input (near-zero) |
At infinite system size, the critical point is a single value w0c. At finite N, there exists a region of w0 values around w0c where the system exhibits approximately critical behaviour. We call this the critical window.
At each simulation point, we fit three critical exponents from the avalanche data and compute a combined error measuring deviation from the theoretical mean-field branching process values:
This gives a single number between 0 (perfect criticality) and 1 (completely non-critical). We define threshold levels:
Strict. Requires exponents very close to theoretical. Often too noisy for finite simulations.
Primary metric. Robust to stochastic noise while still discriminating critical from non-critical.
Permissive. Useful for seeing the broadest region of approximate criticality.
The window width is computed as the range of w0 values where error < threshold, expressed relative to w0c:
A value of 100% means the entire tested range (w0c ± 50%) passes the threshold. As N increases, this should shrink toward 0%, converging to the true critical point.
Our robustness sweep is a systematic computational experiment. Here is the precise algorithmic process:
In the theory of critical phenomena, finite systems exhibit a critical region whose width scales as a power law with system size:
where ν̄ is the correlation-length exponent. Taking logarithms:
On a log-log plot, this is a straight line with slope −1/ν̄.
The critical window shrinks faster with N. The system is less robust — it requires more precise tuning of w0 at large N.
The critical window shrinks slowly with N. The system is more robust — a wider range of w0 produces critical behaviour even at large N.
A high R² (say > 0.8) means we can confidently extract a scaling exponent. A low R² (< 0.5) means the slope estimate is unreliable. More N values, more runs, and larger Nmax all help improve R².
With only 6 N values from 20k to 2M, the first 3-4 points all had window ≈ 100% (ceiling), leaving only 2-3 points in the shrinking regime to define the slope. The “thorough” mode adds intermediate N values (50k, 350k, 750k, 1.5M) to fill the transition region and improve fit quality.
A striking feature of the robustness data is that the critical window narrows almost entirely from one side — the supercritical (high w0) side. The subcritical side remains surprisingly “flat,” with errors staying around 6-9% even far from w0c.
The absorbing state Σ0 = 0 is stable. The system returns to silence, but noise from the external input h drives intermittent avalanches. These noise-driven avalanches mimic critical scaling at finite N because the branching dynamics during each burst are approximately critical.
A stable nonzero fixed point Σ0 > 0 appears. At large N, the system enters sustained activity with few or no avalanches. The transition is unambiguous: avalanche counts drop to near zero.
De Candia et al. describe this precisely:
“The occurrence of the avalanche activity […] is indeed related to the presence of a critical point. If the system is moved away from the critical point, this kind of behaviour persists as long as the size of the system is small enough and disappears for larger sizes. More precisely, the system size must be smaller than the squared coefficient of variation of the firing rate.”
— de Candia et al. (2021), PLoS Comput Biol
The critical window width is effectively determined by the supercritical boundary. On the subcritical side, our error metric is insensitive to deviations at these system sizes. This means the “robustness” we measure is specifically the robustness to excess excitatory drive — how far above w0c the system can tolerate before entering sustained, non-avalanche-like activity.
The Wilson-Cowan model exhibits a phase transition with an absorbing state (Σ = 0, all neurons quiescent). This is characteristic of the directed percolation universality class, though the mean-field (fully-connected) version falls into the mean-field branching process class. The asymmetry we observe is a hallmark of absorbing-state transitions: below the critical point, the system can only be activated by external perturbations, creating finite-size avalanche behaviour that mimics criticality.
The Error Profile plot shows error vs. relative distance from w0c, with one curve per N value. As N increases:
The log-log plot of window width vs. N is the central deliverable. Each network type produces a line; the slope tells us the scaling exponent, and the vertical offset tells us the overall window size.
| Feature | What to look for | What it means |
|---|---|---|
| Slope | Negative value | Window shrinks with N (expected for genuine criticality) |
| Magnitude of slope | Closer to 0 = more robust | How fast the critical region narrows |
| R² | Close to 1 | Power-law scaling holds well |
| Vertical offset | Higher = wider window | Absolute robustness at a given N |
| CoTx vs Mono | CoTx above Mono | Co-transmission is more robust |
Plotting τS, τT, and γ against relative distance at the largest N shows how the exponents deviate from theoretical values as we move away from w0c. At the theoretical critical point, we expect values close to 1.5, 2.0, and 2.0 respectively.
From our initial sweep (6 N values, 15 distances, 3 runs), the 10% threshold window at N = 2,000,000:
Window: 69.7%
Slope: −0.064
Window: 80.1%
Slope: −0.030
Window: 81.7%
Slope: −0.033
Our results show that co-transmitting networks retain a wider critical window at large N, with the window narrowing roughly 2x more slowly (slope −0.030 vs −0.064 for mono).
The mechanism is intuitive: co-transmitting neurons release both excitatory and inhibitory neurotransmitters. When excitatory drive increases (w0 moves above w0c), the C population's inhibitory component partially counteracts the increase, providing a natural buffering against runaway excitation.
If neural systems operate near criticality for computational advantages (maximizing dynamic range, information transmission, sensitivity to perturbations), then co-transmission provides a built-in homeostatic mechanism that helps maintain criticality without requiring precise fine-tuning of synaptic weights.
The idea that neural criticality requires some form of self-organization or robustness mechanism has been widely explored: