The criticality hypothesis that living systems will self-organize to operate near a critical point has been believed and discussed for quite some time[1]. The hypothesis was inspired by a number of research work, in particular, P. Bak and collaborators [2,3]. But most of the work is focused on dynamical criticality. Recently Bialek and collaborators, by using the maximum-entropy model for an equilibrium system, found criticality in a number of neural systems like the retinal ganglion cells of salamander [4], and the CA1 region of a mouse hippocampus [5].

The pairwise spin model (PSM) based on equilibrium maximum-entropy assumption is used by us to look at three systems. In the first system, calcium images of neural activity in the CA1 region of freely moving nice collected with a miniscope by our collaborators, the lab of Prof. Dong Chun Wu of China Medical University, were analyzed. Not only we found similar result as Bialek’s group, there is an interesting new result indicating that the resolution of single neuron is not essential for criticality [6]. This has motivated us to collaborate with several professors in NYCU to work on the fMRI signals for human subjects under visual stimuli and at resting. The experimental data of the whole brain is again best fitted with a critical state of the PSM. The vast different scale between fMRI for human brains and the much smaller region of CA1 region of mice seems to be irrelevant for the criticality has inspired us to examine a third system: the internal state of an AI or a convolutional neural network (CNN) while it is trained with the images of handwritten numerals 0-9 (the MNIST database). How the criticality emerged with training will be discussed along with many open questions.

References

[1] J.M. Beggs, The criticality hypothesis: how local cortical networks might optimize information processing, Phil. Trans. R. Soc. A 366 , 329-343 (2008).

[2] P. Bak, How nature works: the science of self-organized criticality. 1996, New York, NY: Copernicus.

[3] P. Bak, C. Tang, & K. Wiesenfeld, Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. 59, 381–384 (1987).

[4] G. Tkačik, T. Mora, O. Marre, D. Amodei, S.E. Palmer, M.J. Berry, W. Bialek, Thermodynamics and signatures of criticality in a network of neurons, Proc. Natl. Acad. Sci. 11508–11513, 112 (2015).

[5] L. Meshulam, J.L. Gauthier, C.D. Brody, D.W. Tank, W. Bialek, Coarse Graining, Fixed Points, and Scaling in a Large Population of Neurons, Phys. Rev. Lett. 123, 178103 (2019).

[6] Y. L. Chen, C. C. Chen, Y. Y. Mei, N. Zhou, D. Wu, and T. K. Lee, Ubiquitous proximity to a critical state for collective neural activity in the CA1 region of freely moving mice, Chinese J. Physics 77, 497-510 (2022).