The book “The cortex and the critical point: understanding the power of emergence” by John M. Beggs was recommended to me by my supervisor when I began asking too many questions about the critical brain hypothesis. Did I find my answers there? Some. Was it useful? Certainly. 

Criticality with respect to a certain phenomenon means that this phenomenon operates around a critical point. Critical point is a concept from thermodynamics which means a state of equilibrium. The idea that the brain operates at a critical point has been around for a while, and therefore, some body of literature has accumulated. The author of the book is a leading scholar in this field, and the book is a collection of his research and his knowledge. The criticality hypothesis can be formulated as follows: if neurons in a network operate near a critical point, the information processing in such a network will be optimal.

The brain criticality started from being a mathematical assumption and recently developed into a set of testable hypotheses. Let me outline the primary three and then explain them one by one to the best of my understanding (which is regrettably only partial):

1. The branching ratio should be equal to 1. 

An active network of neurons transmits signals (information) from one neuron (sender) to another (receiver). The branching ratio is a number of receiver neurons that one sender could activate (one sender neuron may be physically connected to many receiver neurons but activate not all of them). 

If the branching ratio is greater than 1, then the sender can activate more than one neuron downstream. In this case, signal will be amplified and eventually overwhelm the network. 

If the branching ratio is smaller than 1, then the sender would activate less than one neuron downstream. It means that activity will fade away soon. 

If the branching ratio is equal to 1, the sender neuron would transmit the signal to one receiver neuron. Under such a scenario, the activity will persist in the network without fading away and without overexciting it.

2. The network has the potential to exist in different phases (or states) and experience phase transitions (from one state to another).

The distinct phases are: inactive state, active state, and transition state. In inactive state, branching ratio is smaller than 1; in active state, branching ratio is larger than 1. The transition state is continuous, meaning that the network cannot ‘jump’ but instead smoothly transitions from active to inactive state and back. An example of the phase transition IRL is boiling water turning into steam. If the network operates in a transition state, it is critical, or it operates at a critical point. 

3. When in a transition state (at a critical point), network activity would follow power laws and many scales.

The power law is a certain type of relation between two variables that can be approximated by the function x-α. Essentially, it means that if one variable is increasing, another will experience an exponential decrease. Power laws are observed in many systems. Classical example: earthquakes. Stronger earthquakes occur much less frequently than minor ones, therefore, the strength of earthquakes would be inversely associated (according to the power law) with their frequency of occurrence. 

In a critical system, there should be several power laws occuring at several scales (or levels). In the case of the brain, several empirical measures of activity demonstrate power laws. For instance, firing in the network of neurons occurs in cascades (or avalanches), and avalanche size and duration are related via the power law. Moreover, the spectrum of EEG data (or frequency-to-power relation) demonstrates power law, as well as amplitude dynamics at a particular frequency, e.g., in alpha oscillations. 

These three prerequisites have been demonstrated in vitro and in vivo, in several species, and on several time scales, with one substantial correction. It turned out that neurons operate slightly below the critical point, in a slightly subcritical regime. The author terms it quasi-criticality. 

The last chapter is devoted to speculations about criticality and human brain. One claim there is that primary sensory areas perform “easy” tasks that don’t require being at the critical point, while association areas solve “complex” problems and hence operate closer to the critical point. Another claim is that layers II-III of the neocortex expanded the most through evolution, and experiments show that they operate closer to the critical point than, for instance, layer V. These speculations are intriguing but require rigorous testing.

The book is very well written. Complex mathematical concepts are mostly absent, but a general theoretical framework is described in much detail. The author balances on the phase transition between being critical of critics and being critical of himself. This honest and open way of conveying the story engaged and inspired me. However, the book left many of my questions unanswered. For instance, how do neurons regulate this collective behaviour? Although neurons in the network certainly self-organise, the single neuron doesn't have the information about excitation and inhibition in the whole network, does it? How can it integrate input in such a way to hold the system at a critical point? Also, I cannot say I agree with some speculations. The assumption that phylogenetically old regions (evolutionary old) of the brain operate in a less critical regime doesn’t converge, in my view, with the fact that the majority of epyleptical foci are in the temporal lobe, which is phylogenetically old. It seems that there neurons are more likely to tip into the supercritical regime. Although, maybe, for this, quasi-criticality people have the answer.

I would recommend this book to every experienced or novice reader, with or without interest in brain criticality. The book is truly enjoyable.

Memorable quote: “only few parameter combinations [in the model] lead to complex activity and these occur at the border between disorder and order”

April, 2024