&Bullet; physics 14, 28

A constant bombardment of stimuli drives the dynamics of the brain from a critical point to a “quasi-critical” state.

The dynamics of many different physical systems can be described by a single mathematical model. Systems that seem to have little in common – for example, water seeping through sand or cracks spreading through rocks – are all critical phenomena of the same “universality class” (see Common Ground in Avalanche-Like-Like Events). Some research has found that networks of neurons in the brain represent another critical system in this universality class, while others have revealed neural behaviors that deviate from criticality. Leandro Fosque of Indiana University Bloomington and colleagues are now showing that the brain may be “quasi-critical” in that it is driven away from the critical point by the sustained flood of external stimuli [1] . You find that this shift is not arbitrary; it happens in a way that maximizes the brain’s responsiveness to stimuli – a property that is central to the processing of information in the brain.

In the brain, criticality refers to the way in which neural activity is triggered by a stimulus, such as sensory input. This activity spreads out as a series of avalanches – periods of time

$T$

during the

$\mathrm{S.}$

Neurons fire in quick succession. Distributions of

$T$

and

$\mathrm{S.}$

Over time, follow power laws that indicate scale-free, critical behavior [2] . Cultivated in neural networksin vitro, the corresponding power law exponents

${\tau }_{T}$

and

${\tau }_{\mathrm{S.}}$

(which determined the temporal distribution of avalanches with

$T$

and

$\mathrm{S.}$

Values) are constant and compatible with a universality class that can be modeled by a “directed percolation” branching process (Fig. 1). In this model, each node represents a firing neuron, and the branches represent the activity that propagates from that neuron to one or more “offspring”. The dynamics of the network is critical when the branching parameter (the number of offspring per active neuron) crosses the boundary between two phases: on the one hand, the cascade is quickly dampened; on the other hand, the cascade is self-supporting.

The networks developed in the laboratory that have this universality class are small and relatively simple. But living brains are complex, and though

$T$

and

$\mathrm{S.}$

still show a power law distribution, experiments show that

${\tau }_{T}$

and

${\tau }_{\mathrm{S.}}$

differ depending on the type, test conditions, time and stimuli – a finding that fundamentally contradicts the fact that they occupy a single universality class [3] .

Fosque and colleagues explain the variation in these exponents by combining simulations and analysis of neural data from rodents. Their critical bifurcation model takes into account an important aspect of directed percolation that differs between neural cultures and living brains: the presence of an absorbent state transition. This transition defines the cessation of brain activity that occurs when all neurons happen to be inactive at the same time. Such a state – essential for an avalanche to end – is easy to achieve in a small, isolated network, but more difficult to achieve in a large, networked network.

The reason for this difference is that a cultured network is small and interactions between neurons are necessarily local. A piece of the in vivoIn contrast, the brain’s cortex receives about 50% of its input from remote areas [4] . Fosque and colleagues’ model takes these distant stimuli into account by allowing each neuron to be spontaneously activated with a probability

${p}_{\mathrm{S.}}$

. In the image of directed percolation, this means that avalanches can start spontaneously at any point in time instead of just letting a single neuron ignite an avalanche. These intermittent activations increase the duration and size of avalanches and result in smaller values ​​of

${\tau }_{T}$

and

${\tau }_{\mathrm{S.}}$

which means that the system is no longer part of the universality class. This spontaneous activation of avalanches also breaks the strict temporal scale invariance, as it implies a characteristic time scale of spontaneous activation

$~1/{p}_{\mathrm{so}}$

, and the system is no longer critical in the strict sense [5] .

Although in living brains the values ​​of

${\tau }_{T}$

and

${\tau }_{\mathrm{S.}}$

vary, they do not vary arbitrarily. Rather, as recently stated,

${\tau }_{T}$

and

${\tau }_{\mathrm{S.}}$

are closely linked by a simple linear relationship [6] (See New Evidence for Brain Criticality). Fosque and colleagues show that this linear relationship arises when the parameters of their model are tuned to “the point of maximum susceptibility”, which means that the neurons are most sensitive to stimuli. They call this state “quasi-critical”: as critical as possible in view of the disturbing, spontaneous activations.

The team’s strong conclusion that ramified models of brain activity can only be truly critical without external drive has far-reaching ramifications. For example, brain regions can work at different distances from the critical point, depending on their input and – possibly – the function they perform. The picture of Fosque and colleagues also agrees with recent findings that the cortex can actually be somewhat subcritical [7, 8] .

But why should brain dynamics even be near a critical point, considering that some of the brain’s desirable properties – that it is sensitive to external stimuli and that distant regions interact – push the system away from criticality? The answer might be that the brain’s critical dynamics are intertwined with its problem-solving skills. Solvable and unsolvable problems are separated by a sharp boundary, with the most difficult problems to solve right on the edge – a phase transition [9] . A clear confirmation or rejection of this hypothesis would be an important step towards understanding a central principle of biological computing.

The study raises other interesting questions. For example, how is the critical dynamics affected by inhibitory neurons that reduce the activity of their target? These cells make up about one fifth of all cells in the cortex and are critical in preventing out of control arousal [10] . Another question is whether there could be other forms of criticality than those captured by bifurcation models and directed percolation? One possibility that warrants investigation is that the brain is hovering between a hyperactive state on one side and an oscillating rather than silent state on the other [6] . Finally, how do the geometry and density of connections between neurons change critical exponents? Models normally assume random, closely connected networks, which leads to an average field behavior. Could non-midfield behavior in the brain play a role, possibly leading to different exponents? As research gradually reveals the fascinating link between statistical imbalance physics and brain dynamics, we can look forward to finding answers to these and other questions in the years to come.

## References

1. LJ Fosque et al., “Evidence of quasi-critical brain dynamics”, Phys. Rev. Lett.126, 098101 (2021).
2. JM Beggs and D. Plenz, “Neural Avalanches in Neocortical Circuits”, J. Neurosci.23, 11167 (2003).
3. N. Goldenfeld, Lectures on phase transitions and the renormalization group(CRC press, Boca Raton, 2019)[Amazon] [WorldCat] .
4. V. Braitenberg and A. Schüz, Anatomy of the Cortex: Statistics and Geometry(Springer-Verlag, Berlin, 1991)[Amazon] [WorldCat] .
5. Williams-García motorhome et al., “Quasi-critical brain dynamics on a non-equilibrium Widom line”, Phys. Rev. E90, 062714 (2014).
6. AJ Fontenele et al., “Criticality between cortical states”, Phys. Rev. Lett.122, 208101 (2019).
7. V. Priesemann et al., “Spike avalanches in vivo indicate a driven, slightly subcritical brain state”, Front. Syst. Neurosci.24 (2014).
8. J. Wilting and V. Priesemann, “Between perfectly critical and completely irregular: A reverberation model records and predicts the spread of cortical spikes”, Cereb. cortex29, 2759 (2019).
9. P. Käsemann et al., “Where the really tough problems are”, in Proceedings of the 12th International Joint Conference on Artificial Intelligence, IJCAI’91, vol. 1(Morgan Kaufmann Publishers Inc., San Francisco, 1991), p. 331[Amazon] [WorldCat] .
10. C. van Vreeswijk and H. Sompolinsky, “Chaos in neuronal networks with balance excitatory and inhibitory activity”, science274, 1724 (1996).

## About the author

Moritz Helias is group leader at Forschungszentrum Jülich and assistant professor for physics at RWTH Aachen University. His research interests lie in recurrent biological networks and neural information processing as well as in the development of statistical physics and field theory approaches that are applied to these areas.

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