Abstract
Converging from a number of disciplines, non-linear systems theory and in particular chaos theory offer new descriptive and prescriptive insights into physiological systems. This paper briefly reviews an approach to physiological systems from these perspectives and outlines how these concepts can be applied to the study of migraine. It suggests a wide range of potential applications including new approaches to classification, treatment and pathophysiological mechanisms. A hypothesis is developed that suggests that dysfunctional consequences can result from a mismatch between the complexity of the environment and the system that is seeking to regulate it and that the migraine phenomenon is caused by an incongruity between the complexity of mid brain sensory integration and cortical control networks. Chaos theory offers a new approach to the study of migraine that complements existing frameworks but may more accurately reflect underlying physiological mechanisms.
Keywords
Introduction
The predominant scientific paradigm views natural systems operating at a state of equilibrium with feedback eliminating environmental challenge. System variables change in a smoothly continuous fashion and unexplained variation is viewed as random behaviour that can be described by statistical methods. Individual knowledge of all the parts of a system add up to an understanding of the system as a whole. Converging from a number of disparate disciplines, the science of dynamic non-linear systems offers new insights into how natural systems operate that challenge this perspective. Over the last 20 years there has been an increasing suspicion that non-linear dynamics and the associated behaviour known as chaos may play an important role in the functioning of living systems and the concept of a dynamic disease has now been recognized in a wide range of clinical areas (1).
The aim of the paper is to illuminate the potential of this new approach to the study of physiological systems in health and disease, and in particular the area of migraine.
The paper is constructed in three parts. The first section provides a brief introduction to non-linearity and chaos theory. The second part outlines how these approaches can be applied to physiological systems. Finally, the implications for the study of migraine using chaos theory are explored from the perspectives of description, treatment and pathophysiology.
Non-linearity and chaos
A wide range of physical and biological systems demonstrate properties that emerge from a network of elements that interact predominately at a local level and which cannot always be explained using traditional scientific approaches. For example, the brain is a network of interconnecting neurons that are excitatory or inhibitory. How each element responds to the information it is presented with is determined by an activation rule that weighs and sums the inputs to determine whether there is an output. Activation rules are directed by discontinuous data and recursive feedback (i.e. the output of one interaction is fed back as the input of the next) that give rise to non-linear features.
The mathematical definition of non-linearity is beyond the scope of this text but in qualitative terms a number of important features can be recognized. For example:
Systems cannot be understood by a reduction into their component parts.
Rarely is there a simple relationship between cause and effect – small inputs can lead to large system changes, large inputs may have little impact.
No one element is in control or has an ‘overview’ of the system. Information about the system is encoded in a distributed manner.
System behaviour evolves from the interaction of elements at a local level without external direction or the presence of internal control. This property is known as emergence and gives systems the flexibility to adapt and self-organize in response to external challenge. Emergence is a pattern of system behaviour that could not have been predicted by an analysis of the component parts of that system.
Chaotic behaviour is an important feature of non-linear systems. Although non-linearity is a prerequisite for chaos, non-linear systems are not necessarily chaotic. Chaotic behaviour was suspected over 100 years ago but it has only been the availability of computational power that has enabled scientists to probe the complex interior of non-linear systems from a mathematical perspective in areas. The publication of James Gleick's Chaos: Making a New Science (2) alerted a wider audience to the importance of an area that has now found applications as widespread as the study of weather, organizations, biological systems and the behaviour of stock markets.
Chaotic systems have a number of important features:
Predictability or determinism. Chaos can be understood by comparing it with two other types of behaviour – randomness and periodicity. Chaos has characteristics of both behaviours. Although it looks random, it is predictable.
There is extreme sensitivity of behaviour to initial conditions. Small changes in a variable in the system at one point will make a very large difference in the behaviour of a system at some future point. This has been termed the ‘butterfly effect’. For example, the weather is a chaotic system and a butterfly flapping its wings in New York can be responsible for a hurricane in Tokyo. The Lyapunov exponent is a measure of the divergence of points that are initially very close and can be used to quantify chaotic systems. In practice, it is this extreme sensitivity to initial conditions that makes chaotic systems so unpredictable.
Fractal scaling. Chaotic systems demonstrate similar characteristics and different levels of scale or magnification. The fractal dimension is another approach to describing chaotic systems and is defined as the slope of the function relating the numbers of points or elements contained in a given ‘magnification’ to the magnification itself.
The presence of a chaotic attractor. Although chaotic behaviour appears random, when studied in a particular way, patterned features are discernible. One way of describing a dynamic system is geometrically, plotting its trajectory with time. If a system is described using n variables and each variable is allocated one dimension, the trajectory of each system element can be plotted with time in an n dimensional graph or phase space. In a chaotic system, the trajectory will never repeat itself but forms a unique pattern as it is attracted to a particular area of phase space – a chaotic attractor. The dimension of this attractor, gives an indication of the complexity of the system.
In summary, chaotic behaviour is a feature of non-linear systems that gives rise to a number of important characteristics that can be identified and quantified using mathematical techniques.
Chaos – the mother of physiology
Over the last decade, what was original thought to be random variation in physiological systems has been shown to be low-dimensional chaos that may play an important functional role in terms of efficiency and adaptability (3–5). Chaotic characteristics have been identified in variables such as blood glucose (6), heart rhythm (7) and brain electrical activity (8). In fact, chaos appears to be the healthy signature of physiology and leads to a radically different model from those based on homeostasis and central control.
Fractal scaling in physiological systems
Biological objects demonstrate statistically self-similar fractal patterns in both space and time and fractal scaling is a feature of a wide range of physiological systems (9). Complex fractal patterns can be produced with simple mathematical expressions allowing complex biological structures to be coded by relatively few genes. Fractal structures also allow for rapid and efficient transport and communication over spatially distributed systems such as the circulation or nervous system.
Fractal scaling can also be demonstrated over time by following the dynamics of a single physiological variable, but under pathological conditions fractal scaling is lost. For example, heart rate variability demonstrates fractal scaling under normal conditions but in disease states transforms into a periodic output such as ventricular fibrillation dominated by a single scale or into uncorrected randomness such as atrial fibrillation (3).
Quantifying the complexity of a physiological system
Although there are a number of approaches to quantifying chaotic systems, the dimension of the system's chaotic attractor (a marker of system complexity) is the more prevalent approach to quantifying physiological systems. This dimension fluctuates within a limited range depending on arousal. For example, there are changes in chaotic activity during sleep and particularly rapid eye movement (REM) sleep (10) and changes in the complexity of visual stimulus are reflected in corresponding changes in brain complexity (11, 12).
Physiological ageing is associated with a generalized loss of complexity in the dynamics of organ system function including the brain and cardiovascular system (13, 14). Dimensionality changes with brain maturation, increasing with age from the neonatal period and declining again with old age (15).
During pathological conditions, systems demonstrate lower dimensional chaos or even periodicity, e.g. cardiac arrhythmia (16), brain electrical activity during epileptic seizures (17) or nocturnal migraine (18). Measures of chaotic dimension can also offer an alternative approach to predicting pathological change based on time series data that focus on individuals rather than stochastic approaches that draw inferences from a number of subjects. For example, epileptic seizures can be anticipated using non-linear analysis (19).
In summary, identification and measurement of chaotic features may offer alternative descriptive and diagnostic perspectives that may be more aligned with underlying physiological mechanisms than traditional methods of inquiry.
Applying non-linear insights to migraine
Migraine is a multifactorial primary headache disorder (20). Imaging studies show disturbances at distributed sites and a wide range of medications act at different foci. Non-linear analysis may offer alternative insights in a number of areas.
Insights for description and prediction
Stochastic linear approaches to EEG analysis in migraine are limited (21) but measurements of chaotic dimension can offer alternative approaches to the investigation of migraine and its prediction. Although there are a number of studies in the field of epilepsy (19), to date the experimental base in this area in migraine is small and only two studies have been identified.
Latka (22) studied middle cerebral artery blood flow in humans using transcranial Doppler ultrasonography. It was found that in migraineurs, fractal properties of axial blood flow velocity were significantly reduced, reflecting a significant loss of short-term adaptability that was termed ‘fractal rigidity’. Strenge (18) found that EEG measures during spontaneous nocturnal migraine demonstrate a loss of dimensional complexity during non-REM sleep states in the migraine night, providing evidence of a global dimensional decrease that was related to cortical network changes during a migraine attack.
Although the continual monitoring of brain electrical activity in migraineurs may present practical difficulties in the experimental setting, other systems that are more readily accessible to investigation may demonstrate associated and relevant behaviour. Migraineurs demonstrate changes in a number of physiological variables between and approaching attacks (23) and particularly in the autonomic nervous system (24). Autonomic dysfunction measured through heart rate fluctuations has produced conflicting results when analysed using linear approaches, but non-linear analysis may offer a more promising line of enquiry. An alternative practical approach is the measurement of interictal psychological symptoms (25), and methods have been developed to detect and measure chaos in psychological variables that may offer more accessible avenues to the investigation of chaotic dynamics (26). However, a major drawback is the large number of data points required to demonstrate and measure chaotic behaviour.
Implications for classification
Current classification of headache disorders in based largely on clinical features rather than underlying mechanisms (27). Often headache types overlap and it has been suggested that primary headaches form a spectrum that may be united by a common mechanism (28). Classification based on the chaotic dynamics of time-dependent processes may offer new insights into taxonomy that more accurately reflect underlying mechanisms.
Insights for treatment
The current therapeutic research paradigm adopts a reductionist approach, assuming that a system can be understood by an analysis of its constituent parts. Adopting a non-linear perspective may offer alternative approaches to treatment and control.
It is possible to control chaotic behaviour in a desired manner using either exogenous or endogenous signals (29) and non-linear control techniques have been applied successfully in a wide range of physical systems including the brain and cardiovascular system (30, 31). Remarkably, such techniques are model independent, i.e. they require no a priori knowledge of the underlying mechanisms.
Control of chaotic behaviour in neural systems has potential therapeutic applications in migraine either using non-pharmaceutical interventions such as neuro-feedback techniques or pharmacological approaches. For example, drugs can alter the dimensionality of neural networks (32) and in vivo are more likely to act on the dynamics of a system rather than specific fixed points under equilibrium conditions. Exploring the impact of drugs on the non-linear dynamics of migraineurs may offer powerful models for pharmacological exploration and development.
Insights for pathophysiology
The dynamic interplay of positive and negative recursive feedback loops in the brain gives rise to chaotic characteristics that can be quantified by their complexity or the dimension of the attractor that characterizes the system's behaviour. Experimental evidence suggests that health is associated with complexity while disease is associated with complexity loss. The interacting regulatory processes operating over multiple time scales prime the organism for efficient and adaptive response. Errors in this mechanism are likely to lead to consequences that are recognized as pathological.
At a central level, this regulatory mechanism requires the integration and control of distributed cortical networks (33) and it has been suggested that synchronization of neuronal activity can serve to define functionally relevant relationships between spatially distributed neural subsystems (34). Synchronization phenomena are likely to play a major role in establishing the communication between different regions of the brain (35) and non-linear approaches can provide an alternative and possibly more relevant measure of this synchronization phenomenon (36, 37).
From a general systems perspective, it is the function of a regulator to reduce variety, so retaining stability in a system, despite the high level of variety outside it. Ashby's Law of Requisite Variety is a fundamental principle of control systems and states that the complexity and speed of regulatory response must match the complexity and speed of changes in environmental stimuli (38). My suggestion is that in migraineurs, there are deficiencies in the coherence of chaotic dimension or complexity of synchronized neural subsystems and in particular between the mid-brain and cortex. Why should this be?
Evidence suggests that the habituation response during stimulus repetition is abnormal between attacks due to inadequate control originating in both the brain stem (39) and cortex where there is a lack of cortical inhibition causing delayed habituation (40–42). Further, synchronization phenomena in EEGs recorded from migraine patients in the presence of repetitive visual stimuli using non-linear techniques show that migraine patients have overactive regulatory mechanisms which are prone to instability and render them more sensitive to environmental factors (43). The reason for these dynamic abnormalities are not known but magnetic resonance spectroscopy studies suggest that abnormalities in energy metabolism of brain mitochondria at a cellular level may be implicated (44).
In summary, my suggestion is that in health, interrelated neural systems operate within a narrow range of attractor dimensionality, i.e. their complexity is coherent. In migraineurs, the mid-brain networks that integrate sensory inputs cannot always accommodate the necessary dimensional range. Under certain conditions, the gap between the attractor dimensions of sensory integration and cortical control networks becomes too great, resulting in a loss of synchronization and a global transition to a significantly lower dimensional state which is a feature of dysfunctional physiological systems – in this context, the migraine phenomenon. This provokes a behavioural response that in turn reduces the level of sensory input, restoring the sensory integrating network to a compatible attractor dimension and resolution of the attack.
This model would explain why migraine is rare at the extremes of age when cortical dimensions are lower and there is less possibility of dimensional incoherence between neural subsystems.
How could this hypothesis be tested? Increasing the complexity of sensory integration by increasing the incongruity of sensory modalities should precipitate migraines in susceptible individuals. For example, mirror visual feedback techniques have been described to test the hypothesis that incongruence between motor output, proprioception and visual sensory input produces complex regional pain syndrome (45). Similar approaches could be used in migraineurs. If the hypothesis was confirmed, this approach could offer a potential therapeutic approach to resetting the dimensionality of sensory integration networks through a process of de-sensitization.
Conclusion
According to classical concepts of physiological control, healthy systems are self-regulated to reduce variability and maintain physiological constancy. However, contrary to the predictions of homeostasis, the output of a wide range of systems fluctuates in a complex manner that is underpinned by non-linear mechanisms and the low dimensional dynamics of chaos. Chaos provides new concepts and methods of analysis that help to understand the dynamics of neural networks in both health and disease that complement existing approaches and may lead to new investigative opportunities.
The insights of this paper are speculative and to date there is limited experimental evidence and application of non-linear techniques in the field of migraine. However, as chaotic features have been found in so many other physiological systems in both health and disease, it seems likely that this new approach will offer new and exciting opportunities for further study that complement existing approaches but with the potential to reflect more accurately underlying physiological mechanisms.