Criticality is easy to claim and hard to measure
Neural criticality is the hypothesis that neuronal systems sit at a boundary between order and disorder, where activity neither dies out nor explodes. The usual signature is avalanches: bursts of activity whose sizes and durations follow power laws. The paper's motivating problem is that the exponents of those power laws are hard to estimate empirically, and their interpretation often depends on the model assumed.
The authors also point to a second route, estimating the branching parameter from ratios of activity at consecutive time points. That route presumes neural activity is well described by an uncorrelated branching process, which conflicts with the long-range correlations that are themselves a hallmark of criticality. A third approach, temporal renormalization, requires fitting an explicit dynamical model such as an autoregressive process. The gap the paper targets is therefore specific: a measure that stays tied to the underlying dynamics without requiring the true control parameter.
What the Fisher Information Metric actually computes
The FIM is described in the paper as a measure of generalized susceptibility. Operationally, it quantifies how sharply a probability distribution of some observable responds when a parameter is changed. Near a critical point, susceptibility grows, so a peak in FIM marks a region of high sensitivity.
The paper's key move is to compute FIM twice for each system. First it uses the model's theoretical control parameter, the quantity that directly tunes the system toward or away from criticality. Then it uses the observed branching ratio, defined as the ratio of activity at two consecutive time points, which can be measured without knowing the control parameter. The second calculation is the one meant to emulate real recordings. Because the underlying distributions are not analytically tractable, all FIM values are estimated non-parametrically from simulated data, with bootstrap resampling used to reconstruct the distributions.
The observed branching ratio alone does not mark criticality
Before presenting FIM results, the authors test whether the observed branching ratio itself can serve as a proximity measure. They examine its residence time, the fraction of time the ratio spends in a small interval around a given value, as a function of each model's control parameter.
In the two neural systems, the residence time shows a minimum at the critical point rather than a peak, while the temporal standard deviation of activity peaks there. The branching process behaves differently again, showing a monotonic trend, which the authors relate to difficulties in handling realizations that never terminate. The conclusion is that in none of the three cases does the observed branching ratio peak around a value of one. Proximity to criticality cannot be read directly from the statistics of the observed ratio, which is what motivates computing its FIM instead.
FIM of the control parameter peaks at the critical point
Across all three models, FIM computed with respect to the theoretical control parameter shows a pronounced peak at the known critical point. In the Galton-Watson branching process, the control parameter is the true average branching ratio, and the FIM peak sits at a value of one; its height grows with the maximum number of generations while its position converges to one.
In the spiking network, the control parameter is the largest eigenvalue of the connectivity matrix, with a phase transition at one. The FIM peak appears there, and its height increases with both spatial system size and recording length, consistent with susceptibility diverging in the infinite-system limit. In the whole-brain model, FIM of the global coupling peaks near a coupling of about 1.1 for the 1000-region parcellation. Rescaling the coupling by the largest eigenvalue of the structural connectivity collapses the peaks from three parcellation sizes onto a common location, and peak height grows with the number of regions.
FIM of the branching ratio sharpens near criticality
The practically important result is the second calculation. When FIM is computed with respect to the observed branching ratio, it peaks at a ratio of one across the explored range, from subcritical to supercritical regimes. What changes with distance from criticality is the shape of that peak: it is highest and narrowest near the critical point and broadens as the system moves away.
The paper quantifies peak width as the second central moment of the FIM distribution, using normalized weights. In the branching process, the broadening is asymmetric around the critical point, which the authors plausibly attribute to the asymmetry between subcritical processes, where generation size shrinks exponentially, and supercritical ones, where it grows and may never terminate. In the spiking network, the supercritical regime retains a peak that is largely invariant to the control parameter, which the authors plausibly link to inhibitory feedback bounding average activity. Finite-size analysis at the critical point shows the peak growing and narrowing as the network size increases.
A real recording example and what it does not settle
The paper applies the branching-ratio FIM to real neuronal spike recordings in awake and non-REM sleep states. Peaks at a ratio of one appear in both states, consistent with criticality. However, peak height and width are not significantly distinguishable between the two states, so this example supports the presence of a peak but does not separate the states by proximity to criticality.
This is the clearest boundary on the method's practical resolution. The simulated results establish that peak height and width track distance from criticality when the underlying model is known and the control parameter can be varied systematically. The real-data example shows the peak is detectable, but not that the shape measures are sensitive enough to rank two brain states against each other.
Read peak height and width together, not the peak alone
The paper's practical claim is that FIM peak width and height give continuous, model-free readouts of proximity to criticality without knowledge of the true control parameter. The peak position alone is less informative, because a peak at a branching ratio of one appears across the whole explored range, including away from criticality.
The discriminating information is in the shape. A high, narrow peak indicates the system is maximally sensitive to deviations of the ratio from one, which the simulations associate with the critical region. A lower, broader peak indicates lost sensitivity. The authors' interpretation is that FIM of an activity-dependent observable acts simultaneously as a detector of criticality and as a quantitative measure of proximity to it, and they present this as promising for empirical recordings such as multi-electrode arrays, calcium imaging, or fMRI time series where control parameters are inaccessible.
The priority for a reader comparing systems is therefore not whether a peak exists at a ratio of one, but how sharp it is. The paper's own real-data comparison is the caution: two states can both show the peak without their widths and heights being distinguishable, so the shape readout should be treated as a continuous indicator whose resolving power still has to be demonstrated in the setting where it is applied.
Sources used in this answer
Fisher Information Metric as a model-free measure of proximity to criticality in neural systems
FIM of the observed branching ratio peaks at a ratio of one and becomes higher and narrower near criticality across a branching process, a spiking network, and a whole-brain model, providing a model-free readout that does not require the true control parameter.