Power-law and Frequency Scaling

Power-law and frequency scaling in electrophysiological data
Self-organized critical (SOC) states are found for many complex systems in nature, from earthquakes to avalanches. Such systems are characterized by scale invariance, which is usually identified as a power-law distribution of variables such as event duration or the waiting time between events. 1/f noise is usually considered as a footprint of such systems. 1/f frequency scaling is interesting, because it betrays long-lasting correlations in the system, similar to the behavior near critical points. SOC states were found in neuronal cultures, but the presence of such critical states in the awake brain remains controversial.

We started to investigate the presence of SOC states in cerebral cortex in vivo, with Claude Bedard and Helmut Kröger [1]. Global variables such as the electroencephalogram, were previously reported to display 1/f frequency scaling, which could be the sign of SOC states in the awake brain. Indeed, by analyzing simultaneous recordings of global and neuronal activities, we confirmed the 1/f scaling of global variables (such as LFPs) for selected frequency bands. However, by analyzing neuronal activities, we did not find the typical power-law scaling of SOC states (“avalanche analysis”), which suggests that neuronal activity does not stem from critical states. The 1/f scaling of LFPs can be explained by a model which does not rely on critical states, but is rather due to a filtering process from extracellular space. This latter hypothesis was recently shown to be plausible [2]. The predictions of this model are testable experimentally, and are presently under investigation (see also Section 1.3).

In a subsequent study with Jonathan Touboul, we investigated whether the LFP signal can show evidence for SOC states [3], as found by other authors in awake monkeys. By using the same techniques, we could show that indeed, the statistics of negative LFP peaks (which are related to increase of firing), can show power-law scaling which could be taken as evidence for SOC. However, we did the same analysis for positive LFP peaks, which are unrelated to firing activity, and found the same results. Moreover, shuffled peaks also demonstrated apparent power-law scaling, suggesting that power-law scaling may be a generic property of thresholded stochastic processes. We next showed that, indeed, spurious power-law scaling can appear from stochastic processes without the presence of underlying self-organized criticality. However, this power-law is only apparent in logarithmic representations, but does not resist to more severe analysis such as the Kolmogorov-Smirnoff test. We conclude that logarithmic representations can lead to spurious power-law scaling induced by the stochastic nature of the phenomenon, and should be demonstrated by more stringent statistical tests (see details in [3]).

Another approach to power-law relations (in collaboration with Sami El Boustani, Olivier Marre and Yves Fregnac at the ICN) consists of extracting correlations from the power spectral density (PSD) of the Vm recorded in vivo. Vm recordings with different stimulus types (from moving gratings to natural images) revealed a different frequency scaling as a function of the stimulus. To explain this, we are developing a model where the “effective connectivity” leads to special correlations in the Vm activity, which may appear as different slopes in the Vm PSD. This approach could lead to methods to estimate the effective connectivity from the Vm, and is still under study. We also plan to test this method experimentally in dynamic-clamp (in collaboration with Sebastien Behuret and Thierry Bal at the ICN), where the amount of correlation will be controlled within synthetic inputs, and related to the PSD of the Vm obtained experimentally (see details in [4]).

To check if other models than SOC can account for unit activity, we examined Ising type models to predict the occurrence of population patterns in distributed spiking activity [5]. Using a maximum entropy principle with a Markovian assumption, we elaborated a model that accounts for both spatial and temporal pairwise correlations among neurons. This model was tested on model data as well as on experimental data, and it was shown that this approach correctly predicts the occurrence probabilities of spatio-temporal patterns of spikes, significantly better than Ising type models only based on pairwise correlations. The same approach can be used to generate surrogates that reproduce the spatial and temporal correlations of a given data set.

These studies suggest that SOC does not seem to be a satisfactory model to explain the statistics of electrical brain activity, both at the level of the LFP and at the level of unit activity. Simple models based on pairwise correlations (such as the Ising model or its variants) seem to account for much of the statistics measured experimentally.

These conclusions were later confirmed by an avalanche analysis from cat, monkey and human cerebral cortex, in wake and sleep states [6]. This analysis made use of recently acquired unit and LFP recordings in humans where a spatiotemporal correlation analysis was performed [7] (see Section 4.4). The avalanche analysis of such data, as well as from similar high-density recordings in cat and monkey (from 96 to 160 electrodes), all pointed to the absence of power-law distributions, in both units, and LFP recordings [6]. We also found that correlations in neuronal activity stayed high across large distances, but only for interneurons [7]. In conclusion, these data and analyses suggest that the awake and sleeping brain display dynamical states more complex than SOC, and that highly-correlated inhibition seems to play a crucial role.

More recently, we discovered a fundamental cause for power-law relations in the brain, and more generally in natural systems [8]. We found that neural networks can display power-law statistics, but they are not associated to criticality. The power-law scaling survives the replacement of individual neurons by stochastic processes, thus demonstrating that it can be obtained without criticality. We also show that such power-law relations can be explained by Boltzmann’s molecular chaos, which is a universal property of many natural systems. These results provide a firm demonstration that power-law distributions constitute no proof for criticality in experimental systems, and that other criteria should be used. It also shows that systems made of a large number of weaky-correlated units will display power-law statistics, which applies to many natural systems.

[1] Bedard, C., Kröger, H. and Destexhe, A. Does the 1/f frequency-scaling of brain signals reflect self-organized critical states ? Physical Review Letters 97: 118102, 2006 (see abstract) [2] Bedard, C. and Destexhe, A. Macroscopic models of local field potentials the apparent 1/f noise in brain activity. Biophysical Journal 96: 2589-2603, 2009 (see abstract) [3] Touboul, J. and Destexhe, A. Can power-law scaling and neuronal avalanches arise from stochastic dynamics? PLoS-One 5:e8982, 2010 (see abstract) [4] El Boustani, S., Marre, O., Behuret, S., Baudot, P., Yger, P., Bal, T., Destexhe, A. and Frégnac, Y. Network-state modulation of power-law frequency-scaling in visual cortical neurons. PLoS Computational Biology 5:e1000519, 2009 (see abstract) [5] Marre, O., El Boustani, S., Frégnac, Y. and Destexhe, A. Prediction of spatio-temporal patterns of neural activity from pairwise correlations. Physical Review Letters 102: 138101, 2009 (see abstract) [6] Dehghani, N., Hatsopoulos, N.G., Haga, Z.D., Parker, R.A., Greger, B., Halgren, E., Cash, S.S., and Destexhe, A. Avalanche analysis from multi-electrode ensemble recordings in cat, monkey and human cerebral cortex during wakefulness and sleep. Frontiers Physiol. 3:302, 2012 (see abstract) [7] Peyrache, A., Dehghani, N., Eskandar, E.N., Madsen, J.R., Anderson, W.S., Donoghue, J.S., Hochberg, L.R., Halgren, E., Cash, S.S., and Destexhe, A. Spatiotemporal dynamics of neocortical excitation and inhibition during human sleep. Proc. Natl. Acad. Sci. USA 109: 1731-1736, 2012 (see abstract) [8] Touboul, J. and Destexhe, A. Power-law statistics and universal scaling in the absence of criticality. Phys. Rev. E 95:012413, 2017 (see abstract)

Alain Destexhe