scale-free

Scale-free free properties (and universality) are usually described by extremely simple models of basic interaction dynamics.

If it werent so, i.e. the specifics of each different layer would play a role in the dynamics, then it would likely not show the same behavior across different scales.
The model should not just be independent of scale, but also other specifics like species, type of neurotransmitter, cell types, … or even type of the substrate, …

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Scale-free avalanche properties.

Two avalanches of sizes $S_1,S_2$, have the probabilities $P(S)=k{S}^{-\tau }$ occuring, where $k$ is a constant and $\tau$ controls the avalanche size distribution.
The ratio between the two sizes is $a=\frac{S_2}{S_1}$. What is the ratio between their probabilities?
Since $S_2=a{S}_1$, we can wirte the ratio as $\frac{P(S_2)}{P(S_1)}=\frac{ka{S}_1^{-\tau }}{k{S}_1^{-\tau }}=a^{-\tau }$.
→ The ratio is preserved, regardless of avalanche size (10/100, 10k/100k) → scale-free
Another scale-free property for e.g. for neuronal avalanches is the number of neurons evolved in the avalanche over time. The “avalanche shape” at different scales is the same but scaled by a constant.
Such scale-free properties are indicative of criticality.

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(See also the power law note)