Neuronal avalanches: Sandpiles of self-organized criticality or critical dynamics of brain waves?

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Front. Phys.

2023, Vol. 18

Issue (4) : 45301 https://doi.org/10.1007/s11467-023-1273-7

RESEARCH ARTICLE

Neuronal avalanches: Sandpiles of self-organized criticality or critical dynamics of brain waves?

Vitaly L. Galinsky¹(

), Lawrence R. Frank^1,²(

)

¹. Center for Scientific Computation in Imaging, University of California at San Diego, La Jolla, CA 92037-0854, USA
². Center for Functional MRI, University of California at San Diego, La Jolla, CA 92037-0677, USA

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Abstract

Analytical expressions for scaling of brain wave spectra derived from the general nonlinear wave Hamiltonian form show excellent agreement with experimental “neuronal avalanche” data. The theory of the weakly evanescent nonlinear brain wave dynamics [Phys. Rev. Research 2, 023061 (2020); J. Cognitive Neurosci. 32, 2178 (2020)] reveals the underlying collective processes hidden behind the phenomenological statistical description of the neuronal avalanches and connects together the whole range of brain activity states, from oscillatory wave-like modes, to neuronal avalanches, to incoherent spiking, showing that the neuronal avalanches are just the manifestation of the different nonlinear side of wave processes abundant in cortical tissue. In a more broad way these results show that a system of wave modes interacting through all possible combinations of the third order nonlinear terms described by a general wave Hamiltonian necessarily produces anharmonic wave modes with temporal and spatial scaling properties that follow scale free power laws. To the best of our knowledge this has never been reported in the physical literature and may be applicable to many physical systems that involve wave processes and not just to neuronal avalanches.

Keywords nonlinear waves critical exponent Hamiltonian system neuronal avalanches critical dynamics

Corresponding Author(s): Vitaly L. Galinsky,Lawrence R. Frank

Issue Date: 17 March 2023

Cite this article:

Vitaly L. Galinsky,Lawrence R. Frank. Neuronal avalanches: Sandpiles of self-organized criticality or critical dynamics of brain waves?[J]. Front. Phys. , 2023, 18(4): 45301.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1273-7
https://academic.hep.com.cn/fop/EN/Y2023/V18/I4/45301

Fig.1 (a) Comparison of the analytical expression (32) for the effective spiking frequency

ω s = 2 π / T s

(red) and the frequency estimated from numerical solution of (21) and (22) (blue) as a function of the criticality parameter

γ / γ c

. In the numerical solution only

γ

was varied and the remaining parameters were the same as parameters reported in Ref. [36]. (b) Spiking solutions for typical parameters producing temporal ((21) and (22), red) and spatial ((41) and (42), blue) spiking profiles where some signal of width

δ t s

or

δ l s

was detected and surrounded by quiet area with the total effective period

T s

or

L s

.

Fig.2 (a) The avalanche durations distribution for all wave modes compared with the ?2 exponent. (b) WETCOW modes randomly distributed and propagated on a 1000 by 1000 grid. Two examples of temporal signal snapshots with different values of signal threshold are shown (color pallet encodes the change of frequencies from the smallest (blue) to the largest (red). Localized regions of wave activity in the spatial domain are clearly evident.

Fig.3 (a) Analytical probability density spectra as a function of brain waves criticality parameter

S / S c

show excellent agreement with the experimental avalanche data [(b), from Refs. [12, 13]] reproducing not only the overall shape of the spectra with the ?3/2 power exponent at the initial scale free part of the spectra and the steep falling edge in the vicinity of the critical point, but also reproduce the fine details such as the small bump-like flattening of the spectra at the transition from ?3/2 leg to the steep falling edge that is clearly evident in experimental spectra.

Fig.4 Analytical probability density spectra multiplied by an

(S / S c) 3 / 2

as a function of brain waves criticality parameter

S / S c

plotted for several values of the phase shift

Φ

.

Fig.5 Examples of complete wave mode trajectory snapshots for two randomly selected parameters and initial conditions. The trajectories was randomly selected from an ensemble of 10⁶ WETCOW modes used for generation of probability distributions of Fig.6. The brain substrate includes separate regions for gray and white matter, such that the cortical region (gray matter) is semi-transparent and sub-cortical area is not transparent, instead, it is completely opaque. The trajectories are only visible if they are confined in the cortical tissue and would be obscured by white-gray matter surface. Therefore the figure clearly show that the wave trajectories mostly propagate through the cortical tissue.

Fig.6 Plots of spatial (a) and temporal (b) probability density spectra obtained by binning oscillatory signal of ensemble of 10⁶ WETCOW modes randomly distributed and propagated through cortical tissue. Two examples of temporal signal (dots or “spikes”) snapshots with different values of signal threshold are shown in (c) and (d) (color pallet encodes the change of frequencies from the smallest (blue) to the largest (red).

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