A model-free gauge for criticality in neural activity

What the work claims

This is a methods and theory paper, authored by Yuewei Du, Alberto Liardi, Hardik Rajpal, and Henrik Jeldtoft Jensen of Imperial College London, spanning physics, computing, the Centre for Complexity Science, and the I-X Centre for AI in Science.¹ The target problem is real and long-standing: the criticality hypothesis in neuroscience, the idea that neural systems operate at the boundary between ordered and disordered dynamics, remains debated in large part because measuring distance from criticality is methodologically fragile. Avalanche power-law fits depend sensitively on thresholding and time binning, and apparent power laws can arise without true criticality. Branching-parameter estimators assume an uncorrelated branching process, which contradicts the long-range correlations that define criticality. The temporal renormalization group approach is principled but requires fitting an explicit dynamical model, so it inherits the model's limitations.

The proposed gauge is the Fisher Information Metric, an information-geometric measure of how sensitively a system's probability distribution responds to changes in a parameter, interpretable as a generalized susceptibility that peaks at the critical point. The authors validate it on three models of increasing biological complexity: a prototypical branching process, a spiking neural network, and a whole-brain model. In every case, FIM computed with respect to the model's true control parameter peaks at the known critical point, with peak height growing and peak position converging to criticality as system size increases. The practically important step is the second one: for real systems the control parameter is unknown, so they instead compute FIM of the observed branching ratio, the ratio of activity between consecutive time points. That quantity also peaks, at a branching ratio of 1, and the peak becomes highest and narrowest precisely when the system is closest to criticality. Peak height and width therefore give a continuous, model-free readout of proximity to criticality.

How it works

The logic runs in two stages. First, establish that FIM locates criticality when you know the control parameter. In the branching process (critical mean offspring of 1, systems up to 1,000 generations), in the spiking network (critical largest connectivity eigenvalue of 1, across neuron counts N and recording lengths T), and in the whole-brain model (critical global coupling g), the Fisher information of the control parameter shows a pronounced peak at the known critical point, sharpening with system size in exactly the way finite-size scaling theory predicts for a susceptibility. In the whole-brain model the signature survives across anatomical parcellations examined in the supplement.

Second, replace the unknowable control parameter with an observable. The observed branching ratio sigma(t) is simply the ratio of total activity at one time step to the previous one. Two results make it usable. Its residence time (how long the system spends near sigma equals 1) behaves non-trivially: in the two neural network models it dips to a minimum at the critical point while the temporal standard deviation of activity peaks, reflecting the broad fluctuations of the critical regime; the naive idea that hovering near sigma equals 1 means being critical is directly refuted, because the FIM peak sits at sigma equals 1 across all values of the true control parameter. What changes with distance from criticality is the peak's geometry: near criticality it is high and narrow, and it progressively broadens and flattens as the system moves away. The width is quantified as the second central moment of the Fisher information distribution, giving two numbers, height and width, that together encode how close the system runs to its critical point.

The authors are explicit about a conceptual distinction that falls out of the analysis: the control parameter sets the global distance from criticality (a slow, structural property, changing only with connectivity), while the observed branching ratio tracks the instantaneous dynamical state (a fast, fluctuating property on a fixed network). A useful criticality gauge has to read the fast variable while its interpretation is calibrated by the slow one, and that separation of timescales is what the height-and-width pair provides.

Where a skeptic should push

The single most load-bearing assumption is that a signature demonstrated in three simulations will survive contact with real recordings, and on this point the paper is honest to a fault: there are none. Every result, from the branching process to the whole-brain model, comes from systems where the critical point is known by construction and stationarity is guaranteed. The authors list empirical extension as future work and name the obstacles themselves. Non-parametric FIM estimation is data-intensive in both space and time: you need fine-grained resolution and long stationary windows to estimate the underlying probability distributions reliably, and living cultures drift, mature, and sometimes die on exactly those timescales.

Second, the estimator carries its own knobs. Binarization threshold, the adaptive bin width (chosen so each bin holds at least a minimum number of samples), and density-estimator hyperparameters are all free parameters, and while the authors report robustness across a broad threshold range in the supplement, the finite-size scaling analysis that would nail down convergence was left out as too numerically demanding. Third, keep the physics frame in view: this is a susceptibility peak, which means the readout is intrinsically resolution-hungry near the very point of greatest interest, and a finite dish will always smear it. None of this is fatal; all of it bounds how strong a claim one should hang on the method before it has been benchmarked on at least one real dataset with a ground-truth manipulation of the control parameter.

Keeping organoid cultures parked at criticality

Organoid intelligence has borrowed the criticality hypothesis as an engineering target: if computation in living tissue is best near the order-disorder boundary, then the field needs cultures maintained there, and homeostatic plasticity is usually nominated as the mechanism that does the maintaining. What the field has lacked is a gauge. Experiments report avalanche statistics after the fact, debate binning choices, and cannot say whether today's culture is closer to criticality than last week's. A branching-ratio FIM is close to ideal as an online gauge for this purpose: it needs only the total activity time course, which any multi-electrode array or calcium-imaging rig already produces; it requires no knowledge of the effective connectivity (which in a dish is unknown, uncontrolled, and developmentally drifting); and it returns a continuous readout, height and width of the sigma-equals-1 peak, rather than a binary power-law verdict.

The non-obvious implication concerns how you would close the loop. The paper's global-versus-instantaneous distinction maps almost perfectly onto the control structure an organoid bioreactor actually has. The slow variable, the control parameter, is what development and long-term plasticity set; you cannot dial it in real time, but you can influence it over days through activity, neuromodulation, or network size. The fast variable, the instantaneous branching ratio, is what the tissue is doing right now. A controller built on this gauge would act on two timescales: use slow interventions (passaging density, stimulation regimens, pharmacology) to keep the FIM peak narrow, and use fast readouts of sigma to detect when the culture has wandered off its operating point between measurements. That is a concrete, buildable control architecture, and it turns homeostatic plasticity from a hypothesized internal mechanism into an externally enforceable setpoint with a measurable error signal.

The threat is history repeating with better mathematics. Neuronal avalanche analysis generated a decade of contested claims partly because power-law fitting admitted too many degrees of freedom, and the field should be wary of a new criticality estimator arriving with simulation-only validation and several binning hyperparameters. An FIM readout that drifts because a culture is non-stationary could be misread as the tissue leaving criticality, when the estimator is the thing that moved; and a smeared finite-size peak in a 100,000-neuron culture could be mistaken for subcriticality. The opportunity, if the method survives a real-dataset benchmark, is a standard gauge that lets different labs compare cultures and interventions on one scale, which organoid intelligence currently cannot do. The bottom line for the field: the doctrine says stay near criticality; this paper supplies the first instrument designed for exactly that job, but it is still an instrument on the bench, not yet in the dish.

The bottom line

Established: FIM computed with respect to the true control parameter peaks at the known critical point in all three models, with the expected finite-size sharpening; FIM of the observed branching ratio peaks at sigma equals 1 for every operating regime, and its height and width track distance from criticality continuously and without model assumptions; and the naive reading of the branching ratio as a direct criticality meter is refuted by the residence-time analysis. Not established: any of this on experimental recordings, convergence of the estimator under finite data in non-stationary systems, or robustness beyond the examined parameter ranges. The claim is confirmed if the gauge, computed from a recording with an independently manipulated control parameter, tracks the manipulated distance to criticality; it is broken if peak geometry proves sensitive to stationarity violations that real cultures cannot avoid. For organoid intelligence the significance is that criticality maintenance finally has a candidate instrument; the next experiment that matters is calibrating it against a dish whose operating point is deliberately moved.