Scale Invariance: A Cautionary Tale Against Reductionism

How long is the coast of Britain?  It doesn’t matter how good your geography is — the answer depends on the size of your measuring stick.  The coast of Britain has twists and turns at all spatial scales, from kilometers to millimeters.  And the smaller the measuring stick used, the longer the measured length of the coastline.

Coast of Britain Knowing Neurons

How does the coast of Britain relate to the brain?  Consider the Blue Brain Project, an undertaking by the Brain and Mind Institute in Switzerland to simulate the mammalian brain neuron by neuron.  This project assumes that by simulating the smallest functional scales of the brain, we will understand the entire brain across all functional scales.  The problem?  Like the coast of Britain, the brain lacks an average or privileged scale.

Averages are often a convenient measure that allow us to condense large amounts of information into a single number.  If you’ve taken a statistics class, you’re familiar with the normal distribution: a bell-shaped curve showing the distribution of a variable, such as IQ, with a well defined average.  And yet, many phenomena in nature, such as the size of earthquakes and the flow of the Nile River, follow power law distributions.  A power law distribution is given by a function where one quantity varies according to another quantity raised to some exponent. Such distributions are often summarized by an 80-20 rule: 80% of one quantity (e.g., destruction) is accounted for by only 20% of the other quantity (e.g., earthquakes).  An interesting property of such distributions is that they have no clear average!

Distributions Knowing Neurons

In the brain, power laws are ubiquitous.  Recordings of electrical activity from the scalp and cortex show signal fluctuations of all sizes following a power law.  Similarly, the presence of brain waves of different frequencies is power law distributed such that no average frequency exists.  Other examples of power law distributed variables in the brain include neuronal spiking, phase resets, opening and closing of ion channels, neurotransmitter exocytosis, and the topology of brain networks.

Beautiful examples of scale invariance in nature and mathematics are seen in fractals, which have repeating patterns at all scales.

If power law distributions are so common in the brain, they must be telling us something about how it operates.  Why does the brain transcend bell-curve averages?  One possible explanation is that the brain lacks a privileged scale because its functioning cannot be reduced to component parts (i.e., neurons).  Rather, it is the complex interactions between parts which give rise to phenomena at all spatial and temporal scales. If this hypothesis is true, it does not bode well for the Blue Brain Project.  Like averages, reductionism is deeply ingrained in our scientific thinking.  Water is explained in terms of molecules, molecules in terms of atoms, etc.  If the brain is reducible to simpler parts, it should also exhibit a privileged scale of organization.

And yet, it does not.  A unifying mechanism for power law behavior in the brain and other systems is that of self-organized criticality (SOC).  According to this model, systems such as the brain operate on the brink of instability, exhibiting slow processes that build energy and fast processes that dissipate energy.  In such systems, small causes have effects of many sizes. Imagine you are at the beach building a sand pile.  As you add sand, the pile gets taller until its slope reaches a critical angle where it can barely support more sand.  Steadily adding more sand will result in avalanches ranging in size from a few grains to significant portions of the pile.  The avalanches are a scale invariant emergent property. Studying individual grains of sand tells you little about avalanches.


We do not know for certain, yet, if SOC can explain scale invariance in the brain.  What we do know is that the brain is more than the sum of its parts.  It’s nice to know neurons, but to truly understand the brain, we must know how neurons interact.



Bak, Per. How nature works: the science of self-organized criticality. Springer Science & Business Media, 2013.

Markram, Henry. “The blue brain project.” Nature Reviews Neuroscience 7.2 (2006): 153-160.

Ward, Lawrence M., and Priscilla E. Greenwood. “1/f noise.” Scholarpedia 2.12 (2007): 1537.

Frohlich J, Irimia A, Jeste S. Trajectory of Frequency Stability in Typical Development. Brain Imaging Behav. 2015 Mar;9(1):5-18. doi: 10.1007/s11682-014-9339-3.

Images made by Jooyeun Lee and adapted from wikimedia commons and

Joel Frohlich

Joel Frohlich is a postdoc studying consciousness in the lab of Martin Monti at UCLA. He is interested in using brain activity recorded with EEG to infer when a person is conscious. Joel earned his PhD from UCLA in 2018 studying EEG markers of neurodevelopmental disorders in the lab of Shafali Jeste. You can also check out Joel's blog Consciousness, Self-Organization, and Neuroscience on Psychology Today. For more about Joel's research and writing, please visit Joel's website at

6 thoughts on “Scale Invariance: A Cautionary Tale Against Reductionism

  • January 6, 2016 at 11:46 am

    A note from one of our readers:
    THANKS, a fine bit of Biophysics!
    & meshes well with Quantum Neurophysiology.
    All of your points are nowadays rather “obvious”, yet continue to be over-looked by the main-stream Human social-thought as well as specialists trained in the old paradigm.
    Francis Jeffrey,
    Monterey, CA, USA

    • January 6, 2016 at 12:42 pm

      Hi Francis,

      Thanks for your note! I anticipate that these concepts of scale-invariance and SOC will become increasingly relevant to physiologists and neuroscientists in coming years.


    • January 9, 2016 at 5:40 am

      Power law distributions would be observed in sillico everywhere even if neurons were modeled as binary input/output processor with a normally distributed probability of firing when receiving an input.

  • January 9, 2016 at 5:37 am

    Power law distributions would be observed in sillico everywhere even if neurons were modeled as binary input/output processor with a normally distributed probability of firing when receiving an input.

  • January 12, 2016 at 10:28 pm

    Bounded power law distributions do have an expectation. All the examples given are bounded by the plank constant at least and by other much larger bounds, such as the size of the skull. So there is an average.

  • January 22, 2016 at 9:18 pm

    But the project is precisely NOT studying individual neurons; it is studying/simulating the interaction of (many) neurons. Simulations are completely capable of handling emergent properties.

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