Chaos & the Brain
A dissection of neuroscience and neurological conditions from the perspective of Chaos & Dynamical Systems. And no, we will not call 'Chaos a ladder' at any point.
A. Chaos as a Concept:
The concept of chaos is often ascertained to imply a state of confusion, or that of disorder. Chaos theory is the mathematical approach that aims to explain the state of such systems. These are dynamic systems with apparent random states of irregularities and disarray, in which the states are governed by deterministic laws. In other words, what appears to be random in various systems, biological or otherwise, may not necessarily be stochastic in nature. Furthermore, the underlying patterns that are embedded in these systems are highly sensitive to the perturbations in the initial state of that system. A fractional change in a given non-linear deterministic system can result in significant changes to subsequent states, highlighting a rather strong dependence on initial conditions.
Henri Poincaré summarizes this notion in his essay Science and Method writing:
“It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.”1
At the intersection of time and constant change, chaos emerges as a fundamental force. The phenomena of chaos ripple through every facet of our existence, touching upon domains as diverse as the capricious weather, fluctuating food prices, ever-shifting population numbers, the unpredictability of stock markets and the volatile swings of industrial averages.
The roots of our fascination with chaos can be traced back to determinism, a concept popularized by Laplace. Henri Poincaré, in his quest to fathom the evolution of physical systems, expanded on this notion. It was in the early 1960s that chaos theory experienced a renaissance, spearheaded by Edward Lorenz, a meteorology professor with a remarkable vision. Lorenz embarked on a journey to refine weather forecasting, speculating that mainframe computers could reduce errors in predicting atmospheric conditions and optimize the planning of satellite launches and military strategies.
In his pursuit, Lorenz embarked on an ambitious mission, constructing a series of equations to capture the essence of weather. These equations were translated into the language of special vacuum-tube-based computers of the time. It was here that serendipity took hold. As he scrutinized the output, Lorenz realized that seemingly insignificant rounding of decimal values had nudged the data into a subtly different direction. This minuscule deviation in initial conditions bore astonishing consequences, leading to vastly divergent forecasts from his expectations.
It was within this revelation that the "butterfly effect" fluttered into the realm of scientific discourse. This metaphorical concept illuminated the extreme sensitivity of complex systems to their initial states, echoing the notion that the delicate flapping of a butterfly's wings could trigger a tornado on the other side of the world.
Chaos, with its intrinsic unpredictability, heralds the absence of conventional order. It is here that chaos reveals its intriguing dual nature. While chaos can complicate the task of forecasting, a task of paramount importance in realms ranging from financial markets to political landscapes and biological ecosystems, it also beckons itself as an irresistible enigma to be unraveled. Deciphering the underlying patterns concealed within the turbulent embrace of chaos becomes an imperative for scientific inquiry.
Edward Lorenz, the architect of chaos theory, encapsulated the essence of his discovery by characterizing it as the study of seemingly random and unpredictable behaviors lurking within the embrace of deterministic laws. He vividly illustrated the notion of "deterministic chaos" through graphical representations, unveiling the intricate dance of nearby points in phase space that wove an ever-evolving narrative. As these points moved through space and time, they were influenced by their proximity, leading to a cascade of separations and reorganizations. In a delightful twist of irony, the outcome bore a resemblance to the delicate wings of a butterfly, a captivating symbol for the interconnectedness and unpredictability of the world.
Lorenz, in his ceaseless exploration of chaos, uncovered this intriguing phenomenon and decided to call them "strange attractors." These were intricate and multifaceted formations, each bearing a distinct signature in the behavior of chaotic systems2. Soon after, the realm of strange attractors expanded as mathematician Michael Hénon introduced the captivating “Hénon attractor”, while the ‘Poincaré–Bendixson theorem’ gracefully stipulated that these enigmatic structures could only manifest in three or more dimensions. Thus, chaos transcended the confines of mere measurement or numerical computation, revealing an unsettling truth: even the slightest deviations in the initial conditions could send a system's trajectory spiraling into a cacophony of diverse outcomes, rendering any long-term predictions futile3.
Yet, within this apparent chaos, determinism offered a paradoxical solace. It asserted that the future unfurled in the embrace of a unique revolution, meticulously choreographed by the interplay of initial conditions, with only a whisper of stochastic behavior. This union of determinism and chaos, though initially perplexing, has now been embraced as a ubiquitous force, permeating many facets of the natural world. Chaotic behavior, with its capricious dance, manifests in the rhythms of heartbeat irregularities, the meandering of fluid flows, and, of course, the tumultuous climate4. Similarly, chaos unfurls its unruly banner in artificial domains, such as the labyrinthine labyrinth of road traffic and the tempestuous seas of the stock market. In these complex realms, mathematical models and analytical tools like Poincaré maps and recurrence plots offer glimpses into the intricate choreography of chaos.
The pursuit of taming non-linear equations has spanned decades, with a symphony of scientific disciplines contributing valuable insights. From meteorologists unearthing the first strange attractors in their quest to grasp the essence of unpredictability to biologists employing the quadric map to decipher the dynamics of populations5, the chorus of contributors has grown. Engineers, applied mathematicians, and computer scientists have seized upon the enigma of non-linear dynamical systems to tackle complex problems within their respective domains. The allure of non-linear dynamics lies in its ability to orchestrate interdisciplinary harmonies, bringing together diverse perspectives from the realms of mathematics and physics.6
Although chaos theory sprang from the turbulent cauldron of weather patterns, its wings now cast a wide shadow, touching various disciplines and contexts. In the realm of robotics, it has fueled the construction of groundbreaking technologies, expanding the horizons of automation and artificial intelligence. In psychology, it serves as a guiding beacon, navigating the intricate landscape of cognitive analysis and the human psyche7. And in the recent chapters of public health, the dynamics of pandemics, epitomized by the relentless march of COVID-198, have become the subject of intricate investigation, offering insights that hold the potential to save lives and shape the course of humanity.
B. The Non-linear nature of our biological systems
The Central Nervous System (CNS), often hailed as the epicenter of all bodily functions and mental faculties, is the sanctuary of our existence. Within the convolutions of the brain, it not only orchestrates the symphony of life-sustaining processes and cognitive prowess but also bestows upon us the extraordinary gifts that make us distinctly human - our vast repertoire of cognitive capabilities. Among these remarkable traits, language, emotions, and the (still) enigmatic realm of mental illnesses stand as complex frontiers that defy the explanatory powers of traditional scientific paradigms. The more we know, the more we tend to realize we don’t know.
The conventional belief in regularity within natural processes has long been relegated to the annals of outdated ideas within the scientific community. Linear approaches, once the cornerstone of analysis, are now seen as insufficient in unraveling the intricate mysteries of the brain and other biological systems. Think about it: biological data, numerous variables, including the vast & often indecipherable patterns of neural activity, exhibit non-linear characteristics and temporal variability. It is within this paradigm that chaos theory emerges as a torchbearer, illuminating the path toward a deeper understanding of cognitive function and the convoluted processes that underlie dysfunction.
In contrast to conventional perspectives on the brain, contemporary researchers have embraced a paradigm shift, perceiving the brain as a dynamic networked system. A healthy brain thrives on forging connections and ensuring the precision of information transfer. In this state of harmonious operation, the myriad networks within the brain harmonize to support a spectrum of cognitive abilities, from the art of problem-solving to the nuances of executive function, attention, and the intricate dance of language. Even though the brain's spontaneous activity arises without the prodding of external forces, it must be dynamic, capable of adapting to the ever-changing external environment. Consequently, the human brain stands as a remarkable entity, continually reshaping its internal connections with precision and adaptability9.
Within the intricate matrix of neurons, each component assumes critical importance, with discernible functions in some instances. However, when we zoom out to view the brain as a whole, the contribution of each individual neuron may appear minuscule in contrast to the majestic symphony of operations conducted by its expansive networks. Research within this framework has focused on deciphering the dance of activity at both the microcosmic neuron level and the macrocosmic network level. Remarkably, the occurrence of chaotic phases, typically associated with seemingly random-like activity, is a characteristic often found within the expansive networks that govern the brain's operations.
C. The Mind & Chaos
The allure of non-linear systems theory lies in its interdisciplinary nature, providing researchers with a distinct advantage when unraveling the intricate connection between chaos and the enigmatic realm of the brain. Understanding the intricacies of this relationship necessitates the exploration of connections and interactions that span both macroscopic and microscopic levels of activity.
These include the intersectionality of cellular activity, behavioral patterns, neural assemblies, and various other phenomena generated by the complex interactions of nerve cells. To navigate this terrain, researchers require sophisticated tools and models that are finely attuned to handling biological time-series data, whether noisy or non-stationary.
In the quest to unveil chaos within the central nervous system (CNS), a pressing need emerges: the discernment of stochastic patterns from deterministic ones.10 It becomes paramount to distinguish whether the observed variability in neural activations stems from some underlying deterministic order or whether it's a manifestation of true randomness. Several studies point out, that variability is an essential ingredient for survival and successful behavior in all living systems, underscoring the need to separate the effects of noise from those arising from the complex interplay of multiple interacting non-linear elements.
For instance, some problems among others are formations of memories during alterations of mental states and nature of a barrier that divides mental states which often leads to the process called dissociation. This process is related to a formation of groups of neurons which often synchronize their firing patterns in a unique spatial manner.
When one looks at the relationship between the level of moving and oscillating mental processes and their neurophysiological substrate, it opens up questions about principles of organization of conscious experiences and how these experiences arise in the brain. Chaotic self-organization11 provides a unique theoretical and experimental tool for deeper understanding of dissociative phenomena and enables us to study how dissociative phenomena can be linked to ‘epileptiform discharges’ which are related to various forms of psychological and somatic manifestations. Organizing principles that constitute human consciousness and other mental phenomena from this point of view may be described by analysis and reconstruction of underlying dynamics of psychological or psychophysiological measures. Some analysis also lends credence to a possible role of chaotic transitions in the processing of dissociated memory. There are also supportive findings for a possible chaotic process related to dissociation found in patients with schizophrenia and depression. In fact, the self-organizing theory of dreaming is particularly important with respect to problem of memory formation and processing during dissociative states characteristic for dreams. Recent data supports the conceptual view of dynamic ordering factors and self-organization underlying psychological processes and brain physiology.
This tumultuous mind-and-brain relationship has captivated researchers over the past decade or so. In the quest for comprehension, studies continue to explore various trends in cognitive neuroscience and psychology, with the application of chaos theory and non-linear dynamics showing great promise. In this realm, captivating areas of inquiry involve deciphering the explanatory power of chaos theory concerning altered mental states and the transitions between mental states that lead to dissociation.
Dumb-founded? I was too. Let me break it down for you.
Wait, What does that even mean? Let me elaborate. First off, let's get a couple of essential concepts out of the way: [1] 'Self-organization' is a process in which a system spontaneously orders itself without any external intervention. This is often seen in complex systems, such as the human brain, where individual components interact in a way that produces an overall pattern that is greater than the sum of its parts. [2] 'Dissociation' is a mental process in which a person's thoughts, feelings, memories, or sense of identity become separated from each other. This can happen as a result of trauma, stress, or other mental health conditions. [3] 'Epileptiform discharges' are to put it crudely, abnormal electrical signals in the brain that (possess the ability to) cause seizures. What we are stating here is simply this: Chaos theory can help us to understand how memories are formed and how they change. For example, chaos theory suggests that memories are not formed in a linear fashion. Instead, they are formed through a complex process of self-organization. This means that memories are not simply encoded and stored in the brain, rather, they emerge from the interactions of many different neurons. In addition, we all know that our mental states are constantly changing. We may be happy one moment and sad the next. We may be focused one moment and distracted the next. This is where chaos theory comes in - it can help us to understand how mental states change and how they can transition from one state to another. It suggests that mental states are not governed by linear equations. Instead, they are governed by nonlinear equations that cause extreme changes even with deterministic patterns to them. Put simply, small changes in our mental state can lead to large and unpredictable changes in our behavior.
TLDR? Several studies of the brain & psychology points to the fact that chaos theory is a useful tool for understanding how the mind and brain work, particularly in the context of dissociation and other mental health disorders. Scholars argue that the chaotic behavior of the brain may play a role in how memories are formed and processed, and that this may be linked to epileptiform discharges. It is also stated that self-organizing theory of dreaming may be supported by chaos theory.
Continuing on the same thread on how neurons interact, in the early days of neuroscience, there was a prevailing belief that the cognitive foundation of human behavior could be traced to the individual neurons. According to Hubel & Wiesel, this "single unit approach" posited that behavior could be neatly explained by the activity of individual cells in response to stimuli. However, chaos theory challenges this view. While the neuron is undeniably the brain's most elementary unit, there is an inexplicable, dynamic interconnectedness of complex electromagnetic phenomenology, an array of neurotransmitters, and the ever-present influence of autonomous activities which cast a profound shadow over our neural activity. The brain doesn't function as a passive responder to stimuli; any theory assuming so would be overly simplistic. Instead, the brain functions as a chaotic system, demanding a holistic perspective that considers factors such as internal feedback. In this regard, chaos theory presents a more encompassing and realistic viewpoint.
The advent of neuroimaging technology has lent strong support to the existence of networks within the brain. These collective dynamics of the brain have been explored through the lens of mathematical physics, further reinforced by empirical studies. For instance, Haghighi and Markazi delved into the mechanisms behind seizure generation in epilepsy and discovered evidence pointing toward the involvement of non-linear processes. Electroencephalographic (EEG) recordings from the brain's cortex, analogous to electrocardiography data from the heart, have provided compelling evidence that brain activity possesses chaotic attributes and, to some extent, unpredictability. Intriguingly, strange attractors have made their appearance on phase-space diagrams of brain data. These fractal strange attractors within the brain undergo a fascinating reorganization during cognitive differentiation processes12. Hebbian learning principles suggest that neurons and their connections must be regularly activated to stay vital. The unexpected firing of previously inactive neurons might serve as a mechanism for sustaining brain health, underscoring the essential role of chaos in maintaining a healthy, functioning brain. Moreover, while the background noise in the brain maintains stability, its electrical activity seems to operate in a chaotic fashion. These chaotic responses and activities facilitate rapid state transitions, crucial for efficient information processing. Without these transitions, cognitive processes such as sensation and perception would plod along at an excruciatingly slow pace. Indeed, the human body, including the brain, mirrors the dynamic and intricate nature of nature itself, exhibiting fractal dimensions. It is, without a doubt, safe to assert that human beings are inherently creatures of chaos.
D. The Intersection of Neuroscience & Chaos
Future research on neural systems and other higher brain functions will most likely focus on combining traditional reductionist neuroscience with non-linear science. However, applying concepts and tools developed to describe noise-free and low dimensional mathematical models to biological systems such as the brain has not been easy The question of how neurons in the human brain assemble and give rise to a complex biological machine that outperforms even the most advanced computers continues to motivate research in this area.
The CNS is infamously complex. This complexity emerges from the interaction of different elements and variables resulting in a non-linear dynamical system. The intricate interplay makes it challenging to understand even the healthy brain’s functioning fully. Although significant advances have been made in the understanding of genetics and behavior of neural systems over the past decade, a plethora of questions remain unanswered.
Perhaps one of the most significant complexities is the structure and wiring of the human brain. Neurons, approximately 8.6 × 10^11 of them in the human brain, emerge from a combination of extracellular signals and transcription gradient factors acting on neocortical cells. These neurons connect with each other forming over one hundred trillion synapses. Furthermore, new evidence for the brain’s ability to produce new neurons adds to the already overwhelming complexity of the system.13
Developing in parallel with these complex connections are chaotic patterns. Using models such as the Huxley and Hodgkin model and the Hindmarsh and Rose model of bursting neurons, researchers attempt to determine the non-linear patterns in higher brain functions.14 Based on a radical hypothesis, the brain’s processing, perception, and storage capabilities may in fact be the end-result of “strange attractors” that Lorenz came across. Thus, modifications in the system can result in variation of cognitive outputs. Once again, the initial status is critical for the final product of a system. Regulation of excitatory and inhibitory activity in neural circuits is necessary for functional stability. In other words, optimal brain functioning requires a healthy balance between inhibition and excitation processes and failure to maintain this balance may results in various neuropsychiatric conditions.
E. Where Chaos & Disorder Reign Supreme?
An important question concerns the processes that characterize neurologic and psychiatric disorders. In this context, the application of chaos theory has been valuable for understanding a variety of pathologies.15 For example, principles of non-linear dynamics have been used to analyze and interpret EEG recordings in patient populations. Chaos theory has also opened up possibilities for studying the relationship between environmental and genetic factors in various pathologies.
Non-linear system research has revealed critical functions and characteristics of both physical and biological systems. Findings in particular have highlighted the appearance of random events across time, resulting in identification of mathematical elements of different systems. Some examples of this phenomenon include the variability of heartbeats, the coding sequences of DNA, and the flow of information across neurons. Identification of these patterns is critical to understanding both normal and pathological processes. With a solid understanding about complex functions and chaotic systems, it may be possible to differentiate between healthy and unhealthy levels of chaos in the brain.16
In understanding disorders of the brain, neurotransmitters have a critical role. When released, these chemical molecules have the critical task of carrying messages between neurons via synapses. Neurotransmitters, namely dopamine, serotonin, glutamate, gamma-aminobutyric acid (GABA), and acetylcholine, are associated with different disorders of the CNS.
These neurotransmitters are responsible for behavioral, psychological and cognitive patterns of activity. Because dopamine has been implicated in a vast number of diseases, it has been one of the most widely researched neurotransmitters. Dopamine receptors are divided into two main categories. This first category consists of D1 and D5 receptors. These receptors are responsible for the activation of adenylyl cyclase enzymes. The second category of dopamine receptors is the D2, D3, and D4 receptors. Their primary function is to inhibit the adenylyl cyclase enzymes.17 Given this complex interaction, it is not surprising that dopamine is associated with pathways linked to addiction disorders, psychosis, and bipolar disorder. Other conditions related to dopamine include Parkinson’s disease, restless leg syndrome, and attention-deficit hyperactivity disorder (ADHD).
Serotonin, another well-studied neurotransmitter, usually behaves as a multifunctional biochemical particle. This is demonstrated by its distinguishing role in behavioral and mood patterns. Serotonin imbalances are associated with epileptic seizures, migraine, and major depressive disorder just to name a few examples.18
GABA is defined as one of the primary inhibitory neurotransmitters. It is responsible for regulating excitement levels and muscle tones. Its receptors are usually associated with drugs that act as modulators. A surplus of GABA in the CNS is usually associated with anxiety reduction and anti-convulsion. Abnormally low levels of GABA are usually associated with anxiety disorders and convulsive disorders such as epilepsy.
Acetylcholine has a modulatory role in the CNS. It is present in the neuromuscular junction, the parasympathetic system, and the autonomic nervous system and has been associated with issues of learning, motivation, attention, arousal, and addiction.
Problems in the production and regulation of acetylcholine have also been linked to memory impairments, a hallmark of Alzheimer’s disease. Noradrenaline is a neurotransmitter in the catecholamine family. As such, it can be identified as both a hormone and a neurotransmitter. Noradrenaline is produced from different sources, including the sympathetic ganglia and is responsible for the mobilization of functions such as alertness, arousal, and attention. Based on the current evidence, noradrenaline plays a vital role in pathologies. Some of the health issues associated with this neurotransmitter include different psychiatric disorders and neuropathic pain.
The disorders mentioned above are associated with a malfunction in the production and/or regulation of different neurotransmitters. For instance, movement disorders such as Parkinson’s disease are attributed to issues with dopamine and serotonin production and epilepsy, can be attributed to dysregulation of dopamine and GABA production. In addition to neurochemical factors, atypical electrical activity observed in epilepsy has also been shown to be consistent with chaotic systems.
Needless to say, the interaction of neurotransmitters through various receptors is notably complex, even in the normal brain. This vulnerability however, is not specific to neurotransmitters or just to healthy brains. Disruption in the fluid dynamics of the brain or electrical activity can play a role in mental illnesses. Moreover, application of chaos theory and non-linear analyses have proven to be a valuable approach for understanding psychiatric conditions including psychosis, bipolar disorder, depression, and schizophrenia. In concordance with chaos theory, a small imbalance or oscillation in the brain can result in a system that functions unpredictably. Based on the evidence to date, it appears that this may very well be the case in neuropsychiatric conditions.
The Convergence Needed to Understand this?
The answer to this is Convergence. We need convergence - of multiple disciplines coalescing towards potential solutions. On their own, in their silos, physiology, neuroscience, physics, chaos theory and computational biology are all insufficient to tackle it.
An interdisciplinary approach including physics, mathematics, neuroscience, and related fields will be beneficial for furthering our understanding of chaotic mechanisms in biological systems. Research on both healthy and disordered populations is needed to delineate the nature and function of chaos in the brain and to develop new models of pathological disorders / neuropathology.
An Extended Hiatus Over - Personal Note
On a personal note for the Savant in Space newsletter / blog (whatever you wish to call it), it has been a tumultuous time for me. I managed to reach 13-something thousand subscribers before I lost a good chunk of them due to my own inactivity. I am glad for the support from all of you.
I know this has been a long-time pending and that this update has taken forever. It’s almost been upwards of almost two quarters (!?) since I posted on Savant in Space.
Things have gotten out of hand, priorities have been realigned and the main factor has been an internal reluctance from my own self regarding what I write and post.
More exciting writeups incoming, with a different, less-abstract focus as we juggle different interests than what we aligned with in the beginning. For an overview of what kind of facets we will be incorporating into the Substack going forward, check out ‘The Future’ section on my Substack.
Taken from James Gleick, Chaos: Making A New Science (1988)
S H Kellert, In the wake of chaos: Unpredictable order in dynamical systems, University of Chicago Press (1993)
Christian Oestreicher, A History of Chaos Theory
The New Scientist’s Guide to Chaos
Steven Strogatz, Nonlinear Dynamics & Chaos (2018)
On Chaotic Self-Organization: See Barton, Chaos, Self-organization & psychology
J Briggs, Fractals: The patterns of Chaos (1992)
Robert Sapolsky, Behave
Henri Korn & Phillippe Faure, Is there chaos in the brain? II. Experimental evidence and related models
Rae Fortunato Blackerby, Application of Chaos Theory to Psychological Models
On Dopamine Receptors & their functional understanding generally, see: Biochemistry, Dopamine Receptors
G B Richerson & G F Buchanan, The serotonin axis: Shared mechanisms in seizures, depression and SUDEP