Being critical of criticality in the brain

Abstract

Relatively recent work has reported that networks of neurons can produce avalanches of activity whose sizes follow a power law distribution. This suggests that these networks may be operating near a critical point, poised between a phase where activity rapidly dies out and a phase where activity is amplified over time. The hypothesis that the electrical activity of neural networks in the brain is critical is potentially important, as many simulations suggest that information processing functions would be optimized at the critical point. This hypothesis, however, is still controversial. Here we will explain the concept of criticality and review the substantial objections to the criticality hypothesis raised by skeptics. Points and counter points are presented in dialog form.

Keywords: Ising model; avalanche; criticality; multi-electrode array; network; scale-free; statistical physics.

Figure 1

A simple diagram of spins in the Ising model. (Left) At low temperature, nearest neighbor interactions dominate over thermal fluctuations. As a result, almost all the spins align in the same direction, producing a very ordered state. (Right) At high temperature, thermal fluctuations dominate over nearest neighbor interactions. As a result, the spins point in different directions, producing a very disordered state. (Center) At some critical temperature, nearest neighbor interactions and thermal fluctuations balance to produce a complex state.

Figure 2

Hypothetical positions of two spins as a function of time. (Top) At high temperature, the spin orientations fluctuate greatly, but independently of one another, producing a low dynamic correlation value. (Middle) At the critical temperature, the spin orientations fluctuate somewhat and the fluctuations are coordinated, producing a high dynamic correlation value. (Bottom) At low temperature, the spin orientations do not fluctuate very much, yielding a low dynamic correlation value.

Figure 3

Average dynamic correlation as a function of distance. At high and low temperatures, the average dynamic correlation between two lattice sites decreases rapidly toward 0 as the distance between the lattice sites is increased. At the critical temperature, the average dynamic correlation also decreases toward 0 as the distance is increased, but much more gradually.

Figure 4

Correlation length as a function of temperature for a simulation of the Ising Model. Near the critical temperature the correlation length rapidly approaches a maximum value. This sharp peak separates the ordered phase from the disordered phase and occurs at the phase transition point.

Figure 5

Hypothetical relationship between the average dynamic correlation between two lattice sites and the distance between those lattice sites at the critical temperature in a small simulation of the Ising model.

Figure 6

Probability distribution of neuronal avalanche size. (Black) Size measured using the total number of activated electrodes. (Teal) Size measured using total LFP amplitude measured at all electrodes participating in the avalanche (Beggs and Plenz, 2003).

Figure 7

Avalanche size distributions in local field potential data collected with a 60-channel microelectrode array from rat cortical slice networks. (A) Subcritical regime; excitatory antagonist (3 mM CNQX) applied. (B) Critical regime; normal network. (C) Supercritical regime; inhibitory antagonist (2 mM PTX) applied (Haldeman and Beggs, 2005).

Figure 8

Average avalanche shapes for avalanches of three distinct durations (Friedman et al., 2012).

Figure 9

Rescaled avalanche shapes from Figure 8 (Friedman et al., 2012).