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doi:10.1111/j.1468-2982.2005.00934.x

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Original Article

Migraine – new perspectives from chaos theoryD Kernick

REVIEW

Migraine – new perspectives from chaos theory

D Kernick

St Thomas Health Centre, Exeter, UK

Kernick D. Migraine – new perspectives from chaos theory. Cephalalgia 2005;25:561–566. London. ISSN 0333-1024

Converging from a number of disciplines, non-linear systems theory and in par-ticular chaos theory offer new descriptive and prescriptive insights into physio-logical systems. This paper briefly reviews an approach to physiological systemsfrom these perspectives and outlines how these concepts can be applied to thestudy of migraine. It suggests a wide range of potential applications includingnew approaches to classification, treatment and pathophysiological mechanisms.A hypothesis is developed that suggests that dysfunctional consequences canresult from a mismatch between the complexity of the environment and thesystem that is seeking to regulate it and that the migraine phenomenon is causedby an incongruity between the complexity of mid brain sensory integration andcortical control networks. Chaos theory offers a new approach to the study ofmigraine that complements existing frameworks but may more accurately reflectunderlying physiological mechanisms.

�

Chaos, migraine, non-linear

David Kernick MD, FRCGP, DCH, DRCOG, DA, St Thomas Health Centre, Cowick Street, Exeter EX4 1HJ, UK. E-mail su1838@eclipse.co.uk Received 26 March 2004, accepted 15 November 2004

Introduction

The predominant scientific paradigm views naturalsystems operating at a state of equilibrium withfeedback eliminating environmental challenge. Sys-tem variables change in a smoothly continuous fash-ion and unexplained variation is viewed as randombehaviour that can be described by statistical meth-ods. Individual knowledge of all the parts of a sys-tem add up to an understanding of the system as awhole. Converging from a number of disparate dis-ciplines, the science of dynamic non-linear systemsoffers new insights into how natural systems operatethat challenge this perspective. Over the last 20 yearsthere has been an increasing suspicion that non-lin-ear dynamics and the associated behaviour knownas chaos may play an important role in the function-ing of living systems and the concept of a dynamicdisease has now been recognized in a wide range ofclinical areas (1).

The aim of the paper is to illuminate the potentialof this new approach to the study of physiological

systems in health and disease, and in particular thearea of migraine.

The paper is constructed in three parts. The firstsection provides a brief introduction to non-linearityand chaos theory. The second part outlines how theseapproaches can be applied to physiological systems.Finally, the implications for the study of migraineusing chaos theory are explored from the perspec-tives of description, treatment and pathophysiology.

Non-linearity and chaos

A wide range of physical and biological systemsdemonstrate properties that emerge from a networkof elements that interact predominately at a locallevel and which cannot always be explained usingtraditional scientific approaches. For example, thebrain is a network of interconnecting neurons thatare excitatory or inhibitory. How each elementresponds to the information it is presented with isdetermined by an activation rule that weighs andsums the inputs to determine whether there is an

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output. Activation rules are directed by discontinu-ous data and recursive feedback (i.e. the output ofone interaction is fed back as the input of the next)that give rise to non-linear features.

The mathematical definition of non-linearity isbeyond the scope of this text but in qualitative termsa number of important features can be recognized.For example:

•

Systems cannot be understood by a reduction intotheir component parts.

•

Rarely is there a simple relationship betweencause and effect – small inputs can lead to largesystem changes, large inputs may have littleimpact.

•

No one element is in control or has an ‘overview’of the system. Information about the system isencoded in a distributed manner.

•

System behaviour evolves from the interaction ofelements at a local level without external directionor the presence of internal control. This propertyis known as emergence and gives systems the flex-ibility to adapt and self-organize in response toexternal challenge. Emergence is a pattern of sys-tem behaviour that could not have been predictedby an analysis of the component parts of thatsystem.

Chaotic behaviour is an important feature of non-linear systems. Although non-linearity is a prereq-uisite for chaos, non-linear systems are notnecessarily chaotic. Chaotic behaviour was sus-pected over 100 years ago but it has only been theavailability of computational power that hasenabled scientists to probe the complex interior ofnon-linear systems from a mathematical perspec-tive in areas. The publication of James Gleick’s

Chaos: Making a New Science

(2) alerted a wideraudience to the importance of an area that has nowfound applications as widespread as the study ofweather, organizations, biological systems and thebehaviour of stock markets.

Chaotic systems have a number of importantfeatures:

•

Predictability or determinism. Chaos can beunderstood by comparing it with two other typesof behaviour – randomness and periodicity. Chaoshas characteristics of both behaviours. Although itlooks random, it is predictable.

•

There is extreme sensitivity of behaviour to initialconditions. Small changes in a variable in the sys-tem at one point will make a very large differencein the behaviour of a system at some future point.This has been termed the ‘butterfly effect’. Forexample, the weather is a chaotic system and a

butterfly flapping its wings in New York canbe responsible for a hurricane in Tokyo. TheLyapunov exponent is a measure of the diver-gence of points that are initially very close andcan be used to quantify chaotic systems. Inpractice, it is this extreme sensitivity to initialconditions that makes chaotic systems sounpredictable.

•

Fractal scaling. Chaotic systems demonstratesimilar characteristics and different levels of scaleor magnification. The fractal dimension isanother approach to describing chaotic systemsand is defined as the slope of the function relat-ing the numbers of points or elements containedin a given ‘magnification’ to the magnificationitself.

•

The presence of a chaotic attractor. Although cha-otic behaviour appears random, when studied ina particular way, patterned features are discern-ible. One way of describing a dynamic system isgeometrically, plotting its trajectory with time. If asystem is described using

n

variables and eachvariable is allocated one dimension, the trajectoryof each system element can be plotted with timein an

n

dimensional graph or phase space. In achaotic system, the trajectory will never repeatitself but forms a unique pattern as it is attractedto a particular area of phase space – a chaoticattractor. The dimension of this attractor, gives anindication of the complexity of the system.

In summary, chaotic behaviour is a feature of non-linear systems that gives rise to a number of impor-tant characteristics that can be identified andquantified using mathematical techniques.

Chaos – the mother of physiology

Over the last decade, what was original thought tobe random variation in physiological systems hasbeen shown to be low-dimensional chaos that mayplay an important functional role in terms of effi-ciency and adaptability (3–5). Chaotic characteristicshave been identified in variables such as blood glu-cose (6), heart rhythm (7) and brain electrical activity(8). In fact, chaos appears to be the healthy signatureof physiology and leads to a radically differentmodel from those based on homeostasis and centralcontrol.

Fractal scaling in physiological systems

Biological objects demonstrate statistically self-simi-lar fractal patterns in both space and time and fractal

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scaling is a feature of a wide range of physiologicalsystems (9). Complex fractal patterns can be pro-duced with simple mathematical expressions allow-ing complex biological structures to be coded byrelatively few genes. Fractal structures also allow forrapid and efficient transport and communicationover spatially distributed systems such as the circu-lation or nervous system.

Fractal scaling can also be demonstrated over timeby following the dynamics of a single physiologicalvariable, but under pathological conditions fractalscaling is lost. For example, heart rate variabilitydemonstrates fractal scaling under normal condi-tions but in disease states transforms into a periodicoutput such as ventricular fibrillation dominated bya single scale or into uncorrected randomness suchas atrial fibrillation (3).

Quantifying the complexity of a physiological system

Although there are a number of approaches toquantifying chaotic systems, the dimension of thesystem’s chaotic attractor (a marker of system com-plexity) is the more prevalent approach to quantify-ing physiological systems. This dimension fluctuateswithin a limited range depending on arousal. Forexample, there are changes in chaotic activity duringsleep and particularly rapid eye movement (REM)sleep (10) and changes in the complexity of visualstimulus are reflected in corresponding changes inbrain complexity (11, 12).

Physiological ageing is associated with a general-ized loss of complexity in the dynamics of organsystem function including the brain and cardiovas-cular system (13, 14). Dimensionality changes withbrain maturation, increasing with age from the neo-natal period and declining again with old age (15).

During pathological conditions, systems demon-strate lower dimensional chaos or even periodicity,e.g. cardiac arrhythmia (16), brain electrical activityduring epileptic seizures (17) or nocturnal migraine(18). Measures of chaotic dimension can also offeran alternative approach to predicting pathologicalchange based on time series data that focus on indi-viduals rather than stochastic approaches that drawinferences from a number of subjects. For example,epileptic seizures can be anticipated using non-linear analysis (19).

In summary, identification and measurement ofchaotic features may offer alternative descriptiveand diagnostic perspectives that may be morealigned with underlying physiological mechanismsthan traditional methods of inquiry.

Applying non-linear insights to migraine

Migraine is a multifactorial primary headache disor-der (20). Imaging studies show disturbances at dis-tributed sites and a wide range of medications act atdifferent foci. Non-linear analysis may offer alterna-tive insights in a number of areas.

Insights for description and prediction

Stochastic linear approaches to EEG analysis inmigraine are limited (21) but measurements of cha-otic dimension can offer alternative approaches tothe investigation of migraine and its prediction.Although there are a number of studies in the fieldof epilepsy (19), to date the experimental base in thisarea in migraine is small and only two studies havebeen identified.

Latka (22) studied middle cerebral artery bloodflow in humans using transcranial Doppler ultra-sonography. It was found that in migraineurs, frac-tal properties of axial blood flow velocity weresignificantly reduced, reflecting a significant loss ofshort-term adaptability that was termed ‘fractalrigidity’. Strenge (18) found that EEG measures dur-ing spontaneous nocturnal migraine demonstratea loss of dimensional complexity during non-REMsleep states in the migraine night, providing evi-dence of a global dimensional decrease that wasrelated to cortical network changes during amigraine attack.

Although the continual monitoring of brainelectrical activity in migraineurs may presentpractical difficulties in the experimental setting,other systems that are more readily accessible toinvestigation may demonstrate associated and rel-evant behaviour. Migraineurs demonstratechanges in a number of physiological variablesbetween and approaching attacks (23) and partic-ularly in the autonomic nervous system (24).Autonomic dysfunction measured through heartrate fluctuations has produced conflicting resultswhen analysed using linear approaches, but non-linear analysis may offer a more promising line ofenquiry. An alternative practical approach is themeasurement of interictal psychological symptoms(25), and methods have been developed to detectand measure chaos in psychological variables thatmay offer more accessible avenues to the investi-gation of chaotic dynamics (26). However, a majordrawback is the large number of data pointsrequired to demonstrate and measure chaoticbehaviour.

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Implications for classification

Current classification of headache disorders in basedlargely on clinical features rather than underlyingmechanisms (27). Often headache types overlap andit has been suggested that primary headaches forma spectrum that may be united by a common mech-anism (28). Classification based on the chaoticdynamics of time-dependent processes may offernew insights into taxonomy that more accuratelyreflect underlying mechanisms.

Insights for treatment

The current therapeutic research paradigm adopts areductionist approach, assuming that a system canbe understood by an analysis of its constituent parts.Adopting a non-linear perspective may offer alter-native approaches to treatment and control.

It is possible to control chaotic behaviour in adesired manner using either exogenous or endoge-nous signals (29) and non-linear control techniqueshave been applied successfully in a wide range ofphysical systems including the brain and cardiovas-cular system (30, 31). Remarkably, such techniquesare 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 migraineeither using non-pharmaceutical interventions suchas neuro-feedback techniques or pharmacologicalapproaches. For example, drugs can alter the dimen-sionality of neural networks (32) and

in vivo

aremore likely to act on the dynamics of a system ratherthan specific fixed points under equilibrium condi-tions. Exploring the impact of drugs on the non-linear dynamics of migraineurs may offer powerfulmodels for pharmacological exploration and devel-opment.

Insights for pathophysiology

The dynamic interplay of positive and negativerecursive feedback loops in the brain gives rise tochaotic characteristics that can be quantified by theircomplexity or the dimension of the attractor thatcharacterizes the system’s behaviour. Experimentalevidence suggests that health is associated withcomplexity while disease is associated with com-plexity loss. The interacting regulatory processesoperating over multiple time scales prime the organ-ism for efficient and adaptive response. Errors in thismechanism are likely to lead to consequences thatare recognized as pathological.

At a central level, this regulatory mechanismrequires the integration and control of distributedcortical networks (33) and it has been suggested thatsynchronization of neuronal activity can serve todefine functionally relevant relationships betweenspatially distributed neural subsystems (34). Syn-chronization phenomena are likely to play a majorrole in establishing the communication betweendifferent regions of the brain (35) and non-linearapproaches can provide an alternative and possiblymore relevant measure of this synchronization phe-nomenon (36, 37).

From a general systems perspective, it is the func-tion of a regulator to reduce variety, so retainingstability in a system, despite the high level of varietyoutside it. Ashby’s Law of Requisite Variety is a fun-damental principle of control systems and states thatthe complexity and speed of regulatory responsemust match the complexity and speed of changes inenvironmental stimuli (38). My suggestion is that inmigraineurs, there are deficiencies in the coherenceof chaotic dimension or complexity of synchronizedneural subsystems and in particular between themid-brain and cortex. Why should this be?

Evidence suggests that the habituation responseduring stimulus repetition is abnormal betweenattacks due to inadequate control originating in boththe brain stem (39) and cortex where there is a lackof cortical inhibition causing delayed habituation(40–42). Further, synchronization phenomena inEEGs recorded from migraine patients in the pres-ence of repetitive visual stimuli using non-lineartechniques show that migraine patients have over-active regulatory mechanisms which are prone toinstability and render them more sensitive to envi-ronmental factors (43). The reason for these dynamicabnormalities are not known but magnetic reso-nance spectroscopy studies suggest that abnormali-ties in energy metabolism of brain mitochondria ata cellular level may be implicated (44).

In summary, my suggestion is that in health, inter-related neural systems operate within a narrowrange of attractor dimensionality, i.e. their complex-ity is coherent. In migraineurs, the mid-brain net-works that integrate sensory inputs cannot alwaysaccommodate the necessary dimensional range.Under certain conditions, the gap between theattractor dimensions of sensory integration and cor-tical control networks becomes too great, resultingin a loss of synchronization and a global transitionto a significantly lower dimensional state which is afeature of dysfunctional physiological systems – inthis context, the migraine phenomenon. This pro-vokes a behavioural response that in turn reduces

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the level of sensory input, restoring the sensory inte-grating network to a compatible attractor dimensionand resolution of the attack.

This model would explain why migraine is rare atthe extremes of age when cortical dimensions arelower and there is less possibility of dimensionalincoherence between neural subsystems.

How could this hypothesis be tested? Increasingthe complexity of sensory integration by increasingthe incongruity of sensory modalities shouldprecipitate migraines in susceptible individuals. Forexample, mirror visual feedback techniques havebeen described to test the hypothesis that incongru-ence between motor output, proprioception andvisual sensory input produces complex regionalpain syndrome (45). Similar approaches could beused in migraineurs. If the hypothesis wasconfirmed, this approach could offer a potentialtherapeutic approach to resetting the dimensionalityof sensory integration networks through a process ofde-sensitization.

Conclusion

According to classical concepts of physiological con-trol, healthy systems are self-regulated to reducevariability and maintain physiological constancy.However, contrary to the predictions of homeostasis,the output of a wide range of systems fluctuates ina complex manner that is underpinned by non-linearmechanisms and the low dimensional dynamics ofchaos. Chaos provides new concepts and methods ofanalysis that help to understand the dynamicsof neural networks in both health and disease thatcomplement existing approaches and may lead tonew investigative opportunities.

The insights of this paper are speculative and todate there is limited experimental evidence andapplication of non-linear techniques in the field ofmigraine. However, as chaotic features have beenfound in so many other physiological systems inboth health and disease, it seems likely that this newapproach will offer new and exciting opportunitiesfor further study that complement existingapproaches but with the potential to reflect moreaccurately underlying physiological mechanisms.

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