People like simplicity. Each decade, corporate logos grow progressively minimalistic, pop songs use ever simpler melodies, and visual art embraces simpler compositions, as Monet gives way to Picasso and Picasso gives way Rothko. This zeitgeist, summarized as “simplicity is the ultimate sophistication,” shapes our perceptions of physiology in interesting ways. The thumping of a beating heart is often celebrated as nature’s beautifully simple rhythm. Listening through a doctor’s stethoscope, one expects any deviation from perfect rhythmicity to be an omen of disease.
No one is surprised to learn that cardiac arrhythmia may result in cardiac arrest. But just the opposite is also true: a heartbeat which is too simple and too rhythmic may be a warning sign of congestive heart failure. The perfect, idealized rhythm we expect to hear through the stethoscope is often the heart’s swan song. Moreover, many heartbeats categorized as cardiac arrhythmia receive this label by nature of being too fast or too slow, not necessarily by being overly complex or aperiodic.
Perhaps the heart is an anomaly, you say. But moving from cardiology to neurology, epileptic seizures manifest in the brain as transitions from electrical activity which appears highly chaotic to highly regular and organized. Continuing to psychiatry, researchers using mathematics to quantify the complexity of speech patterns have found that individuals with autism have highly regular speech patterns as compared with healthy controls.
We all know that life is complex. Yet, the intimate relationship between complexity and health is surprising and counterintuitive. Even in complex systems, we expect a sort of orderliness which appears pathological. A previous Knowing Neurons piece, which described how the brain lacks a privileged level of organization, also introduced the concept of self-organized criticality (SOC). SOC is analogous to a sandpile that is built up until its steepness gives way to avalanches of all sizes. The sizes of avalanches follow a power law distribution, which cannot be predicted by studying individual grains of sand.
How does SOC explain the necessity of complexity in healthy physiological systems? The answer in a moment. But first — because the take home message of the previous Knowing Neurons piece was easy to miss — a quick dialogue with our readers is in order:
Thomas Pfeffer (Twitter handle @thmspfffr:) says, “Nice summary! However, scale-invariance [sic] isn’t everything as there *are* typical scales in the spectrum (eg alpha).”
Thanks, Thomas! The alpha rhythm does deviate from the power law distribution of the EEG spectrum. However, despite being discovered in 1929, the mechanism of the alpha rhythm remains unknown almost a century later. This is likely because studying single neurons tells us little about how they interact to generate alpha rhythms. There is no one scale which can be studied to tell us everything about the brain.
William Blamey on WordPress says, “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.“
William is technically correct, but his comment misses the point. Mathematical models are approximations. While the real number line stretches from negative to positive infinity, the physical world does have boundaries. However, the scales which are relevant for studying brain activity stretch from nanometers to centimeters, and everything in between must also be observed for a complete understanding of the brain.
Albert James Teddy on WordPress says, “Power law distributions would be observed in sillico [sic] everywhere even if neurons were modeled as binary input/output processor with a normally distributed probability of firing when receiving an input.”
Multiplying two uniform random distributions often gives a log-normal distribution, a distribution which is normal after log transforming the x-axis. This might occur in a brain simulation because neurons carry out multiplicative operations. Log-normal distributions are easy to mistake for power law distributions, but they are not the same. A true power law distribution is indicative of scale invariant behavior which is unlikely to be simulated without accounting for the nonlinear interactions or scale invariant connectivity between neurons.
At this point, the main point of the previous Knowing Neurons piece should be clear. But why is complexity healthy? And how does it arise? Let’s return to SOC. In the example of stacking sand at the beach, we add sand to a sandpile until the slope is too steep to support more sand. Avalanches ranging in size from a few grains to a large portion of the pile result from a slow process (adding sand) which builds energy and a fast process (the force of gravity overcoming the force of friction) which dissipates energy. This instability is formally known as criticality, and it is self-organized because the slow process brings the sandpile closer to this state and keeps it there. Other critical states, such as the boundary between the gas and liquid phases of a chemical substance, are not self-organized because the substance continues past the critical point without being held there by further changes in temperature and pressure.
So why is complexity healthy? It is likely that physiological systems are poised at the edge of criticality to allow for maximum flexibility. For this reason, a loss of criticality (and thus complex behavior) is pathological. As a consequence, complex behavior observed in the electrical activity of the brain or heart or may serve as a useful biomarker of disease risk or prognosis. This doesn’t mean that having a crazy, erratic heartbeat is healthy. A useful definition of complexity is a balance between opposing tendencies such as order and disorder or stability and instability. Using mathematical tools which quantify complexity, researchers have already identified potential biomarkers of psychiatric disorders such as autism and schizophrenia in EEG signals recorded non-invasively from the scalp.
While SOC is a strong hypothesis for explaining the role of scale invariant behavior in physiology, another possible source of such behavior is the architecture of brain networks. Imagine anatomical brain regions as pages on Facebook, with “friends” connected by white matter tracts. The number of connections each region has follows a power law distribution: 20% of the brain regions account for 80% of the connections. Like social networks, brain regions also form densely interconnected “cliques,” mini-networks embedded within the larger network. These cliques are often part of still larger cliques, a condition known as modularity. The same modularity occurs in social networks: your closest friends might be part of a larger, looser collection of friends, who in turn are part of an even larger community, such as a town or university.
Perhaps as a consequence of the fact that brain networks and social networks both follow similar scale invariant architecture, the number of page views for Knowing Neuron‘s previous scale invariance piece follows a power law distribution! As the piece is shared each day by readers on Twitter and Facebook, the number of daily views reflects the number of friends or followers of those readers. Most readers have a healthy but modest number of followers, yet a few readers have a very large number of followers, resulting in an enormous number of page views! Interestingly, the distribution of page views can also be explained by SOC. Readers share the piece to relieve “tension,” a process which builds further tension in other readers, keeping the system in a perpetual state of criticality. Days on which the page receives an enormous number of hits represent “avalanches” in the critical system. In reality, a combination of network science and SOC can probably explain both scale invariant behavior in the brain and on social media.
The science of complexity itself is complex and concerns the largest questions in physics, biology, sociology, economics, and other fields. While it’s unclear exactly which mechanisms give way to complex behavior in physiology, such behavior is important to understanding what goes awry in brain disorders and how we can identity biomarkers of such disorders. In an upcoming Knowing Neurons post, we’ll explore how interactions between simple parts give rise to complexity in cellular automata. The complexity debate may not be settled for years to come, nor is it closed to non-specialists. Give us your opinion in the comments below. Why is complexity necessary, and what model best explains the complexity of the brain?
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