Chaos Theory is as a relatively new branch of mathematics that deals with complex, nonlinear systems whose behavior is highly sensitive to slight changes in initial conditions. It deals with final data that can be severely altered, had small alterations been made at the beginning of the data collection. The systems that the theory describes are apparently disordered and unsystematic, but Chaos Theory is used to find the underlying order in the apparently random data.The first true experiments and studies conducted with Chaos Theory were performed by meteorologist Edward Lorenz. In 1960 Lorenz worked on the problem of weather prediction. He set up a computer with twelve separate equations in an attempt to model the weather and predict changes. His equations and computer did not successfully predict the weather itself, but they did theoretically predict what the weather might be (Chaos Theory: A Brief Introduction).

- Lorenz's experiment: the difference between the inputted starting values of these curves is only .000127. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)In an attempt to duplicate the sequence, but save time, Lorenz attempted the same equations, but started in the middle of the sequence as opposed to the beginning. He found that the end of the sequence had changed completely. Instead of the same pattern, he observed that the curve diverged as it approached the end of the sequence, as seen in the above figure. He attributed this divergence to a very small rounding difference in the initial value inputted. The difference between the two inputs was only 0.00127, but the final difference was much larger. This occurrence, commonly referred to as the butterfly effect, is common to Chaos Theory. It demonstrates a systems high sensitivity to initial conditions. Because of this, Lorenz determined it to be impossible to predict future weather accurately. Lorenzs failed experiments with weather prediction did, however, lead him to many other great contributions to what eventually became known as Chaos Theory. His work lead to the discovery of what is now known as the Lorenz attractor, a stronger understanding of bifurcation diagrams, fractals, the Mandelbrot set, and much more.Thanks to fractals and chaos, it is possible to accurately describe and understand various parts of the human body. Researchers have been working to find solutions to biological phenomenon and problems such as cancer, diabetes, influenza, cystic fibrosis, and much more for countless years now. Currently, most of the possible solutions and treatments to these hazardous diseases involve dangerous or controversial methods and testing. These include, but are not limited to, stem cell research and radiation treatment. With an understanding of Chaos Theory, it could be possible to find solutions to these diseases and problems in biology and medicine safely, utilizing only mathematical models and equations.Chaos Theory offers the possibility to find brief, yet organized patterns within chaotic systems such as the human body. Chaos Theory is still a relatively new area of study, and many of its properties and methods may change or become clearer in the future. There are many possibilities for advances in the field. In Biology, chaotic systems can be utilized to demonstrate rhythms of heartbeats, brain waves, walking strides, and even the effects of aging. Fractals can be employed to model the structures of nerve networks, lungs, circulatory systems, and strands of DNA. Chaos Theory has already been applied to resolve many problems in the world, such as the calculation of turbulent events in fluid dynamics and quantifying the pathways of molecules during Brownian motion. There are now attempts to utilize it in the field of Biology to help resolve problems such as the prediction of epileptic seizures. (Herbert, Donald E)Chaos Theory being applied to biology was first considered and investigated in the early 1970s. Researchers were attempting to use Chaos Theory to model population trends. After the Mandelbrot Set was discovered and papers on the subject were published, this application began to take off, and Chaos Theory was being utilized in many areas of biology. The term Dynamical Disease was coined by physicists at the time. This term describes any disease that is shown to exhibit chaotic behavior. There are now organizations dedicated to research into dynamical diseases with Chaos Theory and fractals throughout the field of Biology.Chaos Theory applied specifically to the brain is still in its early stages. There is still a great deal of work to be done on the topic. Research has, however, made many advances on the subject. Neuroscientists have been able to organize the brain strictly through the laws of chaos. Brain waves can be modeled and categorized easily into time series graphs, and areas of the brain can also be modeled and displayed using fractal geometry.The fundamental unit of the brain is called a neuron. This is the scientific term for a single brain cell within an organism. Neurons communicate with other neurons through electric impulses, known as potentials, and chemical emissions, known as neurotransmitters. Electrical input to one neuron comes from many other neurons, each having a specific and different amount of influence, or power on each other. When multiple neuronal networks are activated, firing off electrical impulses to each other, they produce a noticeable change in potential, or voltage, which can be captured and displayed graphically by a piece of equipment called an electroencephalograph, EEG for short. EEG recordings appear as up and down fluctuating, squirming lines across a time x-axis. These recordings are measured in hertz, cycles per second, and they are believed to exhibit chaotic behavior, and thus can be described through nonlinear dynamics. (DasMeet, Atin)-A typical EEG time series recording showing the apparentlyrandom and unpredictable fluctuation of potential (voltage), across a time x-axis.These time series graphs have helped organize and categorize brain waves into four different categories based on the frequency of their potentials: Beta Waves, Alpha Waves, Theta Waves, and Delta Waves. Each category of brain frequency is explained below in the following chart. (Exploring the Brain and Brain Waves) Electroencephalograph data are very important for many branches of neurosciences. Many experiments in cognitive science have shown that electroencephalograph and induced potentials are strongly correlated with specific cognitive tasks as seen in the image above. EEG data also serve to diagnose specific diseases of the mind, such as Epilepsy.Through various experiments involving EEG data, it can be shown that neuronal activity shown in the recordings exhibits many characteristics of chaos. Therefore, it is possible for one to conclude that the overall system which gives birth to the changes in potential, namely the brain, is in a chaotic state. Paul Rapp, a neuroscientist for the Medical College of Pennsylvania comments on the discovery: For the first time we are able to see changes in the geometry of EEG activity that occur as the result of human cognitive activity I expected to see something very boring that did not significantly change as the subject began to think. The moment these structures flooded onto the screen and began to rotate, I knew I was seeing something very extraordinary (Briggs). Dr. Freeman, a researcher of Chaos Theory in the brain, developed mathematical models for EEG signals generated by the brain systems in charge of the sense of smell in rabbits. These models exhibit very important dynamical features observed in the EEG recordings, including a noticeable transitional period from one odor to the next (Freeman).Although there is some debate on the topic, these researchers suggest that learning and recognition of aromas, as well as recollection of familiar aroma, can be explained through chaotic dynamics. In response to those who believe the brain to be un-chaotic, psychiatrist and dynamicist Arnold Mandell voices his frustration saying, More than fifty transmitters, thousands of cell types, complex electromagnetic phenomenology, and continuous instability based on autonomous activity on all levelsand still the brain is thought of as a chemical point-to-point switchboard (Gleick). Scientists researching brain activity are unsatisfied with anything less than a chaotic theory of the mind. Freeman concludes, We have found that brain function cannot be explained in terms of features of neurons taken individually or as part of a local network, nor is it adequately characterized as a passive reaction to stimuli (Freeman). The brain is a chaotic system, complexly related by internal feedback that must be analyzed as a whole. Small internal uncertainties are amplified over time, making long term predictions of brain activity impossible (Skarda and Freeman). The abnormal behavior in the brain has only been logically explained by non linear mathematics (Briggs). There is a strong theory that chaos is essential for the human body, specifically the brain. Chaos is needed to transmit information, solve problems, generate and learn new information, perform creative processes, adapt to new circumstances, access memories, create new abilities, and most importantly, the random firing of neurons keeps them healthy. If the human did not have a chaotic brain, it would be like a machine with a set amount of functions it could perform and access. Nothing new would be learned or thought about. Human beings have the ability to create because of chaotic systems in their bodies. The brains chaotic activity creates new solutions - an internal process critical to learning (Briggs). Chaos is essential to the health of the brain as well. Neurons must be exercised consistently to assure their proper function. The neurons will die if left unused for long periods of time. The random firing of inactive neurons provides a suitable process for maintaining neuron health.From the beginnings of the field of Pathology, which is the study and practice of diagnosing diseases, it has been thought that disorder caused disease in the body. Now physicians and scientists have begun to see chaos as health (Gleick). Arnold Mandell wonders, Is it possible that mathematical pathology, chaos, is health? And that mathematical health, which ispredictability and differentiabilityis disease? (Gleick). This theory is not only possible, but it also makes perfect sense and seems likely. A linear process, given a slight nudge, tends to remain off track. A nonlinear process, given the same nudge, will return to its starting point. For this reason, nonlinear systems can withstand small jolts and operate over a vast range of environmental conditions. Systems disordered in the right way are highly adaptable and balanced at a critical point, ready to react quickly to any change it might encounter in a given environment (Ward 146). From seizures to leukemia, disease is being recognized as an attack of order throughout the field of Pathology. The amount of chaos in an epileptics brain has been shown to decrease as neurons at the seizures focus begin to entice other neurons in their area to fire in sync with them. This process, known as dynamic entrainment, has been shown to start from hours to days before an attack. This is how it is believed to be possible to predict and prevent seizures. If an algorithm could be constructed to accurately and precisely detect dynamic entrainment within a reasonable amount of warning time, it can be used to predict seizures, which would create the possibility of preventing them as a whole.Epilepsy is a neurological condition, affecting the patients nervous system. It is referred to as a seizure disorder, and is usually diagnosed after a person has had at least two seizures that were not caused by some known medical condition such as alcohol withdrawal or extremely low blood sugar. Sometimes though, Epilepsy can be diagnosed after one seizure, if it is expected that the person has a condition that places him or her at a high risk for having another. As far as what causes someone to contract Epilepsy, the cause is not always known. Sometimes it is related to a brain injury or family tendency, but for the most part the cause is unknown. (What Is Epilepsy)This condition is defined as an abnormal excessive or synchronous neuronal activity (Epilepsy.com). There are systems in the brain that limit the spread of electrical activity, and during a seizure, these limits are broken down. This results in abnormal amounts of electrical discharges to occur and spread to whole groups of neighboring cells at once. This creates a storm of electrical activity in the brain, which is the seizure. (What Is Epilepsy)There are many different types of Epilepsy, which are determined based on the type of seizures the patient has, the time of day the patient has them, and the age at which the seizures begin. The following diagram illustrates the EEG recordings of normal adult brain waves, and then the brain waves that come with two of the many different types of possible seizure types.

-The figure demonstrates normal adult brain waves shown in EEG recordings, an adults EEG recordings during an Absence Seizure, and the adults EEG recordings during a Tonic-Clonic Seizure.Epilepsy and seizures affect 1 in every 100 United States adults. The disorder can cause many adverse effects on a persons life and can also lead to possible memory loss and potential injury or death. It is a condition that currently has no cure, only treatments that sometimes limit the amount of seizures a patient has. (What is Epilepsy) A possible way of predicting the seizures could prove very useful to many people around the world whose lives are affected every day by the disease.In 1988, chaos science began offering mathematical methods for seeing order in events that previously appeared random. In the 1990s, after analyzing seemingly random EEG data from a 10 day period, scientists found a transitional period that seemed to occur from minutes to hours before the seizure event had initiated. This transitional period was previously believed to occur only seconds before the event of a seizure. This discovery could yield the possibility of the development of a chip implemented into a patients head that could identify the pre-seizure transitional period early enough for the patient to prepare him or herself for a seizure, or take steps to prevent said seizure from occurring at all, such as looking away from flashing lights if the person had photosensitive Epilepsy and did not realize they were looking into a seizure causing light. The devise could also possibly deliver some kind of electrical impulse that could stop the synchronous firing of neurons and return the brain to its normal chaotic state. A similar device has been developed for patients with diabetes. (Chaos Theory Epilepsy)

- This is a diagram of a possible closed-loop intervention device described above. The long-term goal is to develop algorithms to predict epileptic seizures with high sensitivity and precision. Neuroscientist Chris Sackellares, along with bioengineer Leonidas Lesemidis, in the early 2000s, developed a technique that they claimed could identify seizures in the making ninety percent of the time with up to seventy-five minutes notice. They did this by using dense mathematics to calculate a Lyapunov exponent. A Lyapunov exponent is a quantity of a dynamical system that measures the rate of separation, or divergence of infinitesimally close trajectories. This allowed them to measure the amount of chaos in a patients brain to try and detect the pre-seizure transitional period through EEG recordings. They would look for the maximum Lyapunov exponent, because this would determine the notion of predictability and chaos in the brain. (Iasemidis and Sackellares)The method of creating algorithms for seizure prediction utilizes moving-window analysis of the time series EEG graphs. The algorithms would look at a certain window of EEG data and calculate the maximum Lyapunov exponent, and then repeat this process for the following window of data, as well as the next, and so on. The windows are usually ten to forty seconds worth of data each. The algorithm would look for a decrease in the maximum Lyapunov exponent as it continued calculating from window to window, and if it detected a decrease, it would predict an upcoming seizure. The ultimate goal of the algorithms is to be prospective. This means that the algorithm should be able to take in any amount and type of information at any time, and output whether or not a seizure will follow soon. (Swiderski)The following figures are examples of how some seizure prediction algorithms operate.

-The figure above shows continuous EEG recordings being analyzed by means of a moving-window analysis. The data covered in the orange window are transformed into a single value (the maximum Lyapunov exponent) in the time profile of multivariate characterizing measure. When the values graphed in the time profile cross a certain pre-defined threshold, the algorithm should issue a warning in the form of an alarm that alerts that a seizure should be occurring soon. This alarm can either be true or false, depending on whether a seizure does indeed follow it.

-This figure demonstrates true and false warnings from a seizure prediction algorithm. Whether an alarm is true or false is dependent on a certain prediction horizon, or warning time. This is the period of time after an alarm from which a seizure was expected to occur. If an alarm is followed by a seizure within the given warning time, the alarm is classified as a true positive. If there was no seizure occurring within the prediction horizon after an alarm, the alarm is classified as a false positive. The grey area represents examples of the time under a false warning where there is a waiting period for a seizure that did not happen. The green area represents the time under a correct warning, where a seizure did indeed occur. The time period shown on the x-axis could typically represent one day.

-This is another view of a seizure prediction algorithms results. It is the basic operation of a prediction method during an interictal (between seizure) and a preictal (before seizure) period. a.) An example of the EEG recording with seizures being represented by a full vertical line. b.) A time course of a feature extracted by a seizure prediction algorithm. The solid, horizontal line indicates the threshold for raising alarms. Alarm events and two consecutive time intervals characterize a prediction. c.)SPH is the seizure prediction horizon, SOP is the seizure occurrence period. The interictal phase had a false warning at about 12 minutes. During the preictal phase a warning for a seizure occurred at about 25 minutes, and a seizure occurred at about 45 minutes. The algorithm worked, but it also predicted a false positive.

The efficiency of an algorithm is based on its sensitivity and its specificity. Sensitivity is calculated as the number of seizures with at least one alarm divided by the total number of seizures that occurred. Specificity is calculated by the number of false predictions divided by the number of hours. Ideally, the algorithm should have a high sensitivity and a low specificity. To determine if the algorithm was efficient, the results are compared to those predicted from a random predictor.Results from such algorithms were highly optimistic in the late 1990s to early 2000s. Researchers such as Sackellares were displaying very efficient results with their algorithms. However, much of this data was deemed invalid and lacked reliability in the past five years. The focus on these early studies was limited to analyzing short and selected EEG recordings where it was known that seizures would occur. The research was only done on preictal phases of the EEG data and was entirely limited to this period. There was not enough research that included an evaluation of control recordings from a seizure free interval, therefore, specificity could not be calculated from this data. Also, the earlier found algorithm success could not be reproduced later, on randomly selected and larger sections of EEG data. Therefore, the data was not worth much and there was further debate on whether nonlinear measurements should be used to analyze EEG time series graphs, or whether this preictal phase will ever be accurately and precisely predicted.The search for an efficient algorithm for seizure prediction is still ongoing. There is currently a contest at the annual International Workshop on Seizure Prediction. This is a public competition where contestants download parts of continuous long-term EEG recordings from three different patients. The contestants then test their algorithms on this data, and submit them to be tested on the remaining parts of the data that they did not previously receive. All the contestants need to do to win is to outperform chance level and predict a percentage higher than the percentage of time under a false warning. No one has yet to create an algorithm that can do this and the competition continues to be open to the public.Techniques and algorithms previously considered suitable are not nearly as good as first thought. The next milestone in seizure prediction algorithms is to prove that an algorithm can be designed to run prospectively on an unselected, out of sample data at a higher precision and accuracy than a random predictor. Just as many of the topics covered in this course, this is just another example of researchers being so close, yet so far away from coming to a conclusion. Chaos Theory is still relatively new compared to nearly all other branches of math and science. It is exciting how much is being discovered that can be explained by it. It simply provides a new method of viewing math and data.While many potential applications of Chaos Theory to medicine have yet to be dreamed of, many are currently being researched. From early warning of heart attack by monitoring chaos levels in heart beats, to respirators that operate on chaotic patterns like healthy working ones, to the prediction of epidemics by the identification of the chaotic attractor in the spread of disease, and to artificial intelligence based on the chaotic model of the mind, the possible applications of Chaos Theory to medicine are endless and unpredictable. The human body is a complex dynamic system that operates by the laws of chaos. We are beings composed of chaos, living in a world surrounded by fractals and chaos. Once physicians and scientists gain more understanding of our chaotic state and the diseases caused by abnormal order, the potential for practical application is enormous. Chaos in us and our surroundings should not be ignored.

Bibliography

Briggs, John. Fractals: the Patterns of Chaos. Touchstone, Simon and Schuster Inc. New York, NY. 1992.

"Chaos Theory: A Brief Introduction | IMHO."IMHO | In My Humble Opinion. Web. 10 Dec. 2011. .

Chaos Theory Epilepsy, Leonidas D. Iasemidis & J. Chris Sackellares. The Neuroscientist 2:118-126, 1996

DasMeet, Atin. "Brain and Chaos: When Two Giants Meet.""Brain & Mind" Magazine - WWW Home Page. Web. 10 Dec. 2011. .

"Exploring the Brain and Brain Waves."Neurofeedback, Your Brain, and Your Health. Web. 10 Dec. 2011. .

Freeman, Walter J. Chaos in the CNS: Theory and Practice. Unpublished, prepared for Dahlem Workshop on Flexibility and Constraint in Behavioral Systems.

Gleick, James. Chaos: Making a New Science. Penguin Books, New York NY. 1987.

Herbert, Donald E., and Paul Croft.Chaos and the Changing Nature of Science and Medicine: a Introduction : Mobile, AL, April 1995. Woodbury, NY: AIP, 1996. Print.

Iasemidis, L. D., and J. C. Sackellares. " REVIEW : Chaos Theory and Epilepsy."The Neuroscientist2.2 (1996): 118-26. Print.

How brains make chaos in order to make sense of the world,Skarda C.A. & Freeman W.J, Behav Brain Sci 1987;10:161-195.

Milton, John.Epilepsy as a Dynamic Disease: 11 Tables. Berlin [u.a.: Springer, 2003. Print.Simulation of chaotic EEG patterns with a dynamic model of the olfactory system. Freeman W.J., Biol Cybern 1987; 56:139-150.

Skara, Christine A. and Freeman, Walter J. Chaos and the New Science of the Brain. Concepts in Neuroscience vol. 1, no. 2 p275. March 1990.

Stewart, Ian.Does God Play Dice?: the Mathematics of Chaos. Cambridge, MA: Blackwell, 1989. Print.

Swiderski B., Osowski S., Rysz A.: Lyapunov Exponent of EEG Signal for Epileptic Seizure Characterization. (2005) IEEE Conf. European Circuit Theory and Design Cork

Ward, Mark Beyond Chaos: The Underlying Theory Behind Life, the Universe and Everything. St. Martins Press, New York, NY. 2001.

"What Is Epilepsy? | Epilepsy.com."Epilepsy and Seizure Information for Patients and Health Professionals | Epilepsy.com. Web. 10 Dec. 2011. .