Figure 1. The sandpile model of self-organized criticality from Figure 1 in the book “How Nature Works: the science of self-organized criticality
“…consider the scenario of a child at the beach letting sand trickle down to form a pile (Figure 1). In the beginning, the pile is flat, and the individual grains remain close to where they land. Their motion can be understood in terms of their physical properties. As the process continues, the pile becomes steeper, and there will be little sand slides. As time goes on, the sand slides become bigger and bigger. Eventually, some of the sand slides may even span all or most of the pile. At that point, the system is far out of balance, and its behavior can no longer be understood in terms of the behavior of the individual grains. The avalanches form a dynamic of their own, which can be understood only from a holistic description of the properties of the entire pile rather than from a reductionist description of individual grains: the sandpile is a complex system.”
Per Bak, “How Nature Works: the science of self-organized criticality
Living organisms operate at stable points far from equilibrium. Homeostasis, a process characteristic of living things, maintains the set point (or points) far from equilibrium. Even the simplest of life forms such as viruses exist far from equilibrium. The genetic code they carry is information that takes energy to get in formation and to maintain. Equilibrium, lack of information, and death seem synonymous. So it probably isn’t a surprise when we say that brain activity is a process that is carried out in states far from equilibrium.
That said, non-living things can exist at stable points far from equilibrium. The whole idea first came to my attention with the work of Ilya Prigogine, the 1977 Nobel Prize winner in Chemistry. Prigogine mixed chemicals together that, when energy was added to the mixture, self organized into intricate structures. These structures sucked in and used energy. He called them dissipative structures. Dissipative structures are self organized systems that exist at set points far from equilibrium. If energy input is cut off, the dissipative structure falls back into chaos and then randomness. To equilibrium.
Around 1987 the physicist Per Bak introduced the idea of criticality. Per Bak pointed out that a large amount of variability in our universe is the normal rather than exceptional condition and he defined systems with large variability as complex. He said these complex systems that exist far from equilibrium exhibit universal phenomena no matter if they are weather patterns, a biological organism, or a chemical reaction.
In my previous post Engineered Proteins Enable Watching High Resolution Electrical Activity Across the Brain, the research team that published the paper “Voltage Imaging of Waking Mouse Cortex Reveals Emergence of Critical Neuronal Dynamics” published December 10, 2014 in The Journal of Neuroscience was particularly interested in finding out if activity in the mammalian cortex adhere to the rules of critical dynamics.
Figure 2. Cascade size probability distributions approach power-law form during recovery from anesthesia. Figure 3 in the paper “Voltage Imaging of Waking Mouse Cortex Reveals Emergence of Critical Neuronal Dynamics” (published December 10, 2014 in The Journal of Neuroscience)
While recording electrical activity across mouse brains, the research team saw cascades of various sizes travel across the cerebral cortex (for a movie and more information see Engineered Proteins Enable Watching High Resolution Electrical Activity Across the Brain). Locally isolated and short-lived electrical cascades were observed more often than large cascades. In the anesthetized state, very large cascades of activity were seen that were not usually seen in other brain states.
The probability of observing a cascade of a particular size was used as a measure of cortical dynamics during discrete 20 minute time periods (see Figure 2 above). For example, each blue point in the Figure 2A left hand graph is the probability that a cascade of the size displayed on the x-axis (notice the x-axis is in logarithmic scale) occurs in the anesthetized mouse’s cortex. The blue points correspond to the time period represented by the blue bar (the first 20 minutes of the experiment) in the bar chart in Figure 2C. In the same way, each red point in the Figure 2A right hand graph represents the probability that a cascade of the size displayed on the x-axis occurs in the awake mouse’s cortex. The awake time period is represented by the red bar (the last 20 minutes of the experiment from 180 to 200 minutes) in the bar chart in Figure 2C.
These data show that the probability of observing a cascade of a particular size in the anesthetized mouse does not follow a power law relationship but in the awake animal it does. The lack of a power law relationship is consistent with the lack of criticality in an anesthetized brain and the the power law relationship measured in awake brains is consistent with brain dynamics adhering to the physical laws of self organizing criticality.
Criticality predicts a power law relationship in the brain. It also predicts that cortical dynamics should be independent of scale. In my next post we’ll look at how this research team addresses the question of scale independence. We’ll also consider how this work may provide us with useful insight to help us learn more about how our brains work.