1. Chaos in the Brain Jaeseung Jeong, Ph.D Department of Bio and Brain Engineering, KAIST

2. Chaos 1. the formless shape of matter that is alleged to have existed before the Universe was given order. 2. complete confusion or disorder. 3. Physics; a state of disorder and irregularity that is an intermediate stage between highly ordered motion and entirely random motion. Nonlinear dynamics and Chaos : the tiniest change in the initial conditions produces a very different outcome, even when the governing equations are known exactly - neither predictable nor repeatable

3. (2) King Oscar II (1829 – 1907) offered a prize of 2500 crowns to anyone solve the n-body problem  stability of the Solar System Nonlinear dynamics and Chaos

4. N-body problem The classical n-body problem is that given the initial positions and velocities of a certain number (n) of objects that attract one another by gravity, one has to determine their configuration at any time in the future.. Nonlinear dynamics and Chaos

5. (3) Jules Henri Poincarè (1854 – 1912) Won the Oscar II’s contest,  not for solving the problem, but for showing that even the three-body problem was impossible to solve. (over 200 pages ) Nonlinear dynamics and Chaos “…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” - in a 1903, essay "Science and Method"

6. N-body problem This problem arose due to a deterministic way of thought, in which people thought they could predict into the future provided they are given sufficient information. However, this turned out to be false, as demonstrated by Chaos Theory. Nonlinear dynamics and Chaos

8. Systems behaving in this manner are now called “chaotic.” They are essentially nonlinear, indicating that initial errors in measurements do not remain constant, rather they grow and decay nonlinearly (usually exponentially) with time. Since prediction becomes impossible, these systems can appear to be irregular, but this randomness is only apparent because the origin of their irregularities is different: they are intrinsic, rather than due to external influences.

9. What is chaos? The meteorologist E. Lorenz He modeled atmospheric convection in terms of three differential equations and described their extreme sensitivity to the starting values used for their calculations. The meteorologist R May He showed that even simple systems (in this case interacting populations) could display very “complicated and disordered” behavior. D. Ruelle and F. Takens They related the still mysterious turbulence of fluids to chaos and were the first to use the name ‘strange attractors.’

10. Nonlinear dynamics and Chaos Lorenz attractor

11. Nonlinear dynamics and Chaos

12. The Logistic equation X n+1 =AX n (1-X n )

13. The Logistic equations

14. R

15. Laminar(regular) / Turbulent(chaotic) Turbulent of gas flows Nonlinear dynamics and Chaos

16. High flow rate : Laminar  Turbulent Department of BioSystems Nonlinear dynamics and Chaos

17. What is Chaos? M Feigenbaum He revealed patterns in chaotic behavior by showing how the quadratic map switches from one state to another via periodic doubling. TY Li and J Yorke They introduced the term ‘chaos’ during their analysis of the same map. A. Kolmogorov and YG Sinai They characterized the properties of chaos and its relations with probabilistic laws and information theory.

18. Taffy – pulling machine Nonlinear dynamics and Chaos

19. The strength of science It lies in its ability to trace causal relations and so to predict future events. Newtonian Physics Once the laws of gravity were known, it became possible to anticipate accurately eclipses thousand years in advance. Determinism is predictability The fate of a deterministic system is predictable This equivalence arose from a mathematical truth: Deterministic systems are specified by differential equations that make no reference to chance and follow a unique path.

20. Chaos systems Newtonian deterministic systems (Deterministic, Predictable) Probabilistic systems (Non-deterministic, Unpredictable) Chaotic systems (Deterministic, Unpredictable)

21. Dynamical system and State space A dynamical system is a model that determines the evolution of a system given only the initial state, which implies that these systems posses memory. The state space is a mathematical and abstract construct, with orthogonal coordinate directions representing each of the variables needed to specify the instantaneous stae of the system such as velocity and position Plotting the numerical values of all the variables at a given time provides a description of the state of the system at that time. Its dynamics or evolution is indicated by tracing a path, or trajectory, in that same space. A remarkable feature of the phase space is its ability to represent a complex behavior in a geometric and therefore comprehensible form (Faure and Korn, 2001).

22. Phase space and attractor

24. For any phenomena, they can all be modeled as a system governed by a consistent set of laws that determine the evolution over time, i.e. the dynamics of the systems.

25. Linear vs. Nonlinear Conservative vs. Dissipative Deterministic vs. Stochastic A dynamical system is linear if all the equations describing its dynamics are linear; otherwise it is nonlinear. In a linear system, there is a linear relation between causes and effects (small causes have small effects); in a nonlinear system this is not necessarily so: small causes may have large effects. A dynamical system is conservative if the important quantities of the system (energy, heat, voltage) are preserved over time; if they are not (for instance if energy is exchanged with the surroundings) the system is dissipative. Finally a dynamical system is deterministic if the equations of motion do not contain any noise terms and stochastic otherwise.

26. Attractors A crucial property of dissipative deterministic dynamical systems is that, if we observe the system for a sufficiently long time, the trajectory will converge to a subspace of the total state space. This subspace is a geometrical object which is called the attractor of the system. Four different types of Attractors: Point attractor: such a system will converge to a steady state after which no further changes occur. Limit cycle attractors are closed loops in the state space of the system: period dynamics. Torus attractors have a more complex ‘donut like’ shape, and correspond to quasi periodic dynamics: a superposition of different periodic dynamics with incommensurable frequencies (Faure and Korn, 2001; Stam 2005).

27. Stam, 2005

28. Chaotic attractors The chaotic (or strange) attractor is a very complex object with a so-called fractal geometry. The dynamics corresponding to a strange attractor is deterministic chaos. Chaotic dynamics can only be predicted for short time periods. A chaotic system, although its dynamics is confined to the attractor, never repeats the same state. What should have become clear from this description is that attractors are very important objects since they give us an image or a ‘picture’ of the systems dynamics; the more complex the attractor, the more complex the corresponding dynamics.

29. Three-dimensional Lorenz attractors

30. Characterization of the attractors I If we take an attractor and arbitrary planes which cuts the attractor into two pieces (Poincaré sections), the orbits which comprise the attractor cross the plane many times. If we plot the intersections of the orbits and the Poincaré sections, we can know the structure of the attractor.

31. Characterization of the attractors II The dimension of a geometric object is a measure of its spatial extensiveness. The dimension of an attractor can be thought of as a measure of the degrees of freedom or the ‘complexity’ of the dynamics. A point attractor has dimension zero, a limit cycle dimension one, a torus has an integer dimension corresponding to the number of superimposed periodic oscillations, and a strange attractor has a fractal dimension. A fractal dimension is a non integer number, for instance 2.16, which reflects the complex, fractal geometry of the strange attractor.

32. Fractal dimension of the Attractor

33. Characterization of the attractors III Lyapunov exponents can be considered ‘dynamic’ measures of attractor complexity. Lyapunov exponents indicate the exponential divergence (positive exponents) or convergence (negative exponents) of nearby trajectories on the attractor. A system has as many Lyapunov exponents as there are directions in state space.

34. Characterization of the attractors IV A chaotic system can be considered as a source of information: it makes prediction uncertain due to the sensitive dependence on initial conditions. Any imprecision in our knowledge of the state is magnified as time goes by. A measurement made at a later time provides additional information about the initial condition. Entropy is a thermodynamic quantity describing the amount of disorders in a system.

35. Control parameters and multistability Control parameters are those system properties that can influence the dynamics of the system and that are either held constant or assumed constant during the time the system is observed. Parameters should not be confused with variables, since variables are not held constant but are allowed to change. Multistability: For a fixed set of control parameters, a dynamical system may have more than one attractor. Each attractor occupies its own region in the state space of the system. Surrounding each attractor there is a region of state space called the basin of attraction of that attractor. If the initial state of the system falls within the basin of a certain attractor, the dynamics of the system will evolve to that attractor and stay there. Thus in a system with multi stability the basins will determine which attractor the system will end on.

36. The escape time plot gives the basin of attraction.

37. Bifurcations In a multistable system, the total of coexisting attractors and their basins can be said to form an ‘attractor landscape’ which is characteristic for a set of values of the control parameters. If the control parameters are changed this may result in a smooth deformation of the attractor landscape. However, for critical values of the control parameters the shape of the attractor landscape may change suddenly and dramatically. At such transitions, called bifurcations, old attractors may disappear and new attractors may appear (Faure and Korn, 2001; Stam 2005).

38. Bifurcations

39. This EEG time series shows the transition between interictal and ictal brain dynamics. The attractor corresponding to the inter ictal state is high dimensional and reflects a low level of synchronization in the underlying neuronal networks, whereas the attractor reconstructed from the ictal part on the right shows a clearly recognizable structure. (Stam, 2003)

40. Route to Chaos Period doubling As the parameter increases, the period doubles: period- doubling cascade, culminating into a behavior that becomes finally chaotic, i.e. apparently indistinguishable visually from a random process Intermittency A periodic signal is interrupted by random bursts occurring unpredictably but with increasing frequency as a parameter is modified. Quasiperiodicity A torus becomes a strange attractor.

41. R

42. Intermittency

43. Detecting chaos in experimental data Bottom-up approach We can apply nonlinear dynamical system methods to the dynamical equations, if we know the set of equations governing the basic systems variables. Top-down approach However, the starting point of any investigation in experiments is usually not a set of differential equations, but rather a set of observations. The way to get from the observations of a system with unknown properties to a better understanding of the dynamics of the underlying system is nonlinear time series analysis. Starting with the output of the system, and working back to the state space, attractors and their properties.

44. General strategy of nonlinear dynamical analysis Nonlinear time series analysis is a procedure that consists of three main steps: (i) reconstruction of the system’s dynamics in the state space using delay coordinates and embedding procedure. (ii) characterization of the reconstructed attractor using various nonlinear measures (iii) checking the validity (at least to a certain extent) of the procedure using the surrogate data methods.

45. Reconstruction of system dynamics [problem] our measurements usually do not have a one to one correspondence with the system variables we are interested in. For instance, the actual state space may be determined by ten variables of interest, while we have only two time series of measurements; each of these time series might then be due to some unknown mixing of the true system variables.

46. Delay coordinate and Embedding procedure With embedding, one time series are converted to a series or sequence of vectors in an m-dimensional embedding space. If the system from which the measurements were taken has an attractor, and if the embedding dimension m is sufficiently high, the series of reconstructed vectors constitute an ‘equivalent attractor’ (Whitney, 1936). Takens has proven that this equivalent attractor has the same dynamical properties (dimension, Lyapunov spectrum, entropy etc.) as the true attractor (Takens, 1981). We can obtain valuable information about the dynamics of the system, even if we don't have direct access to all the systems variables.

47. Takens has shown that, if we measure any single variable with sufficient accuracy for a long period of time, it is possible to reconstruct the underlying dynamic structure of the entire system from the behavior of that single variable using delay coordinates and the embedding procedure. Takens’ Embedding theorem (1981)

48. Time-delay embedding We start with a single time series of observations. From this we reconstruct the m-dimensional vectors by taking m consecutive values of the time series as the values for the m coordinates of the vector. By repeating this procedure for the next m values of the time series we obtain the series of vectors in the state space of the system. The connection between successive vectors defines the trajectory of the system. In practice, we do not use values of the time series of consecutive digitizing steps, but use values separated by a small ‘time delay’ d.

49. Stam, 2005

50. Parameter choice Time delay d: a pragmatic approach is to choose l equal to the time interval after which the autocorrelation function (or the mutual information) of the time series has dropped to 1/e of its initial value. Embedding dimension m: repeat the analysis (for instance, computation of the correlation dimension) for increasing values of m until the results no longer change; one assumes that is the point where m>2d (with d the true dimension of the attractor).

51. Spatial Embedding The m coordinates of the vectors are taken as the values of the m time series at a particular time; by repeating this for consecutive time points a series of vectors is obtained. The embedding dimension m is equal to the number of channels used to reconstruct the vectors. The spatial equivalent of the time delay d is the inter electrode distance. The advantage of spatial embedding is that it achieves a considerable data reduction, since the dynamics of the whole system is represented in a single state space. The disadvantage is that the spatial ‘delay’ cannot be chosen in an optimal way. Some groups advocated spatial embedding (Lachaux et al., 1997), whereas others suggested it may not be a valid embedding procedure (Pritchard et al., 1996, 1999; Pezard et al., 1999).

52. Nonlinear dynamical analysis attractor Jeong, 2002

53. How to quantify dynamical states of physiological systems Physiological system States Physiological Time series Embedding procedure (delay coordinates) 1-dimensional time series  multi-dimensional dynamical systems Attractor in phase space Dynamical measures (L1, D2) A deterministic (chaotic) system Topologically equivalent

54. C(r)  r D2 D2 algorithm Nonlinear measure: correlation dimension (D2)

55. Correlation integral attractor Scaling region

56. Nonlinear measures: The first positive Lyapunov exponent

57. Why determinism is important? Whether a time series is deterministic or not decides our approach to investigate the time series. Surrogate data method This method detects nonlinear determinism. Surrogate data are linear stochastic time series that have the same power spectra as the raw time series. They are randomized to destroy any deterministic nonlinear structure that may be present. Statistical differences of nonlinear measures between the raw data and their surrogate data imply the presence of nonlinear determinism in the original data.

58. Stam, 2005

59. Bursting as an information carrier of temporal spiking patterns of nigral dopamine neurons (a) Dopamine neurons in substantia nigra Substantia nigra, a region of the basal ganglia that is rich in dopamine-containing neurons, is thought to be etiologies of Parkinson’s disease, Schizophrenia, Tourette's syndrome etc.

60. Electrophysiology of DA neurons in substantia nigra Irregular and complex single spiking and bursting states in vivo The presence of nonlinear deterministic structure in ISI firing patterns (Hoffman et al. Biophysical J, 1995) Deterministic structure of ISI data produced by nigral DA neurons reflects interactions with forebrain structures (Hoffman et al. Synapse 2000)

61. No determinism of non-bursting DA neurons Histogram D2s of ISI data of DA neurons D2s of surrogate ISI data Embedding dim. vs. D2

62. Nonlinear determinism of bursting DA neurons D2s of ISI data of DA neuronsD2s of ISI surrogate data HistogramEmbedding dim. vs. D2

63. The source of nonlinear determinism in ISI firing patterns of DA neurons Materials (a)Estimation of correlation dimension (b)Surrogate data method (c)Burst separation method Methods (a) Non-bursting neurons (3/7) (b) Bursting neurons (4/7) (ISI <80ms, 160ms) 7 Male Sprague-Dawley rats anesthetized with chloral hydrate Original ISI Burst time series Single spike time series

64. Nonlinear determinism of burst time series D2s of ISI burst time seriesD2s of its surrogate ISI time series

65. No determinism of single spike time series D2s of ISI single spike time seriesD2s of its surrogate ISI time series

66. Nonlinear determinism of inter-burst interval data D2s of IBI data D2s of surrogate IBI data

67. Suprachiasmatic nucleus(SCN) Computational modeling of single neurons and small neuronal circuits

68. SCN neurons exhibit the circadian rhythm in their mean firing rates.

69. Spontaneous Spiking activity of SCN neurons SCN neurons exhibit irregular spontaneous firing patterns, accompanied by intermittent bursts, and thus generate complex ISI patterns, although the average SFR seems to maintain a circadian rhythm.

70. Temporal Dynamics Underlying Spiking Patterns of the Rat Suprachiasmatic Nucleus in vitro. I. Nonlinear Dynamical Analysis (Jeong et al., 2005) Among 173 neurons, 16 neurons were found to exhibit deterministic ISI patterns of spikes.

71. Temporal Dynamics Underlying Spiking Patterns of the Rat Suprachiasmatic Nucleus in vitro. II. Fractal stochastic Analysis. (Kim et al., 2003) 1/f A

72. Temporal Dynamics Underlying Spiking Patterns of the Rat Suprachiasmatic Nucleus in vitro. II. Fractal stochastic Modeling A B A CD

73. Nonlinear analysis of the sleep EEG In many of these studies it was suggested that sleep EEG reflects low-dimensional chaotic dynamics (Cerf et al., 1996, Fell et al., 1993, Kobayashi et al., 1999, Kobayashi et al., 2001, Niestroj et al., 1995, Pradhan et al., 1995, Pradhan and Sadasivan, 1996, Röschke, 1992, Röschke and Aldenhoff, 1991 and Röschke et al., 1993).Cerf et al., 1996Fell et al., 1993Kobayashi et al., 1999Kobayashi et al., 2001Niestroj et al., 1995Pradhan et al., 1995Pradhan and Sadasivan, 1996Röschke, 1992Röschke and Aldenhoff, 1991Röschke et al., 1993 The general pattern that emerges from these studies is that deeper sleep stages are almost always associated with a ‘lower complexity’ as exemplified by lower dimensions and lower values for the largest Lyapunov exponent. This type of finding has suggested the possible usefulness of nonlinear EEG analysis to obtain automatic hypnograms.

74. Nonlinear analysis of the sleep EEG An analysis of an all night sleep recording found an evidence for weak nonlinear structure but not low-dimensional chaos (Achermann et al., 1994 and Achermann et al., 1994; Fell et al. (1996a) using the nonlinear cross prediction (NLCP) to search for nonlinear structure in sleep EEGs of adults and infants.Achermann et al., 1994 Fell et al. (1996a) sleep EEG of young infants showed nonlinear structure mostly during quiet sleep (Ferri et al., 2003).Ferri et al., 2003 The nonlinear measures were better in discriminating between stages I and II, whereas the spectral measures were superior in separating stage II and slow wave sleep: Nonlinear structure may be most outspoken in stage II. nonlinear and asymmetric coupling during slow wave sleep in infants (Pereda et al., 2003).Pereda et al., 2003

75. Nonlinear EEG analysis of Coma and anesthesia Matousek et al. (1995) studied the correlation dimension (based upon a spatial embedding) in a small group of 14 healthy subjects aged from 1.5 to 61 years. They found an increase of the dimension during drowsiness as compared to the awake state. The usefulness of nonlinear EEG analysis as a tool to monitor anesthetic depth was suggested (Watt and Hameroff, 1988). The correlation dimension correlated with the estimated level of sevoflurane in the brain (Widman et al., 2000; Van den Broek, 2003).

76. A dynamical brain disorder Spatial synchronization of brain electrical / magnetic activity Brain fails to function as a multi-task multi-processing machine Hallmarks of epilepsy: - Interictal spikes - Epileptic seizures (detectable from electroencephalograms – EEGs) Epilepsy as a dynamical disorder

77. EEGs in Epileptic seizures there is now fairly strong evidence that seizures reflect strongly nonliner brain dynamics (Andrzejak et al., 2001b, Casdagli et al., 1997, Ferri et al., 2001, Pijn et al., 1991, Pijn et al., 1997 and Van der Heyden et al., 1996).Andrzejak et al., 2001bCasdagli et al., 1997Ferri et al., 2001Pijn et al., 1991Pijn et al., 1997Van der Heyden et al., 1996 Epileptic seizures are also characterized by nonlinear interdependencies between EEG channels. Other studies have investigated the nature of interictal brain dynamics in patients with epilepsy. In intracranial recordings, the epileptogenic area is characterized by a loss of complexity as determined with a modified correlation dimension (Lehnertz and Elger, 1995).Lehnertz and Elger, 1995 A time dependent Lyapunov exponent calculated from interictal MEG recordings could also be used to localize the epileptic focus (Kowalik et al., 2001)

78. The importance of seizure prediction The importance of seizure prediction can easily be appreciated: if a reliable and robust measure can indicate an oncoming seizure twenty or more minutes before it actually starts, the patient can be warned and appropriate treatment can be installed. Ultimately a closed loop system involving the patient, a seizure prediction device and automatic administration of drugs could be envisaged (Peters et al., 2001).Peters et al., 2001

79. In 1998, within a few months time, two papers were published that, in restrospect, can be said to have started the field of seizure prediction. The first paper showed that the dimensional complexity loss L, previously used by the same authors to identify epileptogenic areas in interictal recordings, dropped to lower levels up to 20 min before the actual start of the seizure (Elger and Lehnertz, 1998 and Lehnertz and Elger, 1998).Elger and Lehnertz, 1998Lehnertz and Elger, 1998 The second paper was published in Nature Medicine by a French group and showed that intracranially recorded seizures could be anticipated 2–6 minutes in 17 out of 19 cases (Martinerie et al., 1998).Martinerie et al., 1998 Schiff spoke about ‘forecasting brainstorms’ in an editorial comment on this paper (Schiff, 1998).Schiff, 1998 Controversial about seizure prediction I

80. One possible answer for why seizures occur is that: Seizures have to occur to reset (recover) some abnormal connections among different areas in the brain. Seizures serve as a dynamical resetting mechanism.

81. It was shown that seizure prediction was also possible with surface EEG recordings (Le van Quyen et al., 2001b). This was a significant observation, since the first two studies both involved high quality intracranial recordings.Le van Quyen et al., 2001b Next, it was shown that seizure anticipation also worked for extra temporal seizures (Navarro et al., 2002).Navarro et al., 2002 This early phase was characterized by great enthusiasm and a hope for clinical applications (Lehnertz et al., 2000).Lehnertz et al., 2000 Controversial about seizure prediction I

82. Changes in D 2 of epileptic patients

83. Le Van Quyen M et al., Nonlinear interdependencies of EEG signals in human intracranially recorded temporal lobe seizures. Brain Res. 792(1):24-40 (1998).

84. EEG characteristics of seizures Traditional view: i nterictal  ictal  postictal Seizures ’ occurrences are random Random occurrence of interictal spikes The transition from an interictal to ictal state is very abrupt (seconds) Ictal activity may spread from the epileptogenic focus to other normal brain areas after seizure ’ s onset

85. Emerging view: interictal  preictal  ictal  postictal Seizures or spikes are NOT random events Existence of a preictal state The transition from the interictal to preictal to ictal state is progressive (minutes to hours) Preictal and ictal spatio-temporal entrainment of the epileptogenic focus with normal brain sites Seizures reset: postictal disentrainment of the epileptogenic focus from normal brain sites. EEG characteristics of seizures

87. Aschenbrenner-Scheibe et al. (2003). These authors showed that with an acceptable false positive rate the sensitivity of the method was not very high.Aschenbrenner-Scheibe et al. (2003) The results of Martinerie et al. were also critically re-examined. McSharry et al. suggested that the measure used by Martinerie et al. was sensitive to signal amplitudes and that the good results might also have been obtained with a linear method (McSharry et al., 2003).McSharry et al., 2003 Another group attempted to replicate the results of Le van Quyen et al. in predicting seizures from surface EEG recordings (De Clercq et al., 2003). These authors could not replicate the results in their own group.De Clercq et al., 2003 Controversial about seizure prediction II

88. Online real-time seizure prediction

89. General features of EEGs in AD The hallmark of EEG abnormalities in AD patients is slowing of the rhythms and a decrease in coherence among different brain regions: A major promising candidate is the cholinergic deficit. AD is thought to be a syndrome of neocortical disconnection, in which profound cognitive losses arise from the disrupted structural and functional integrity of long cortico-cortical tracts

90. Correlation dimension analysis of the EEG in AD patients The D2 reflects the number of independent variables that are necessary to describe the dynamics of the system, and is considered to be a reflection of the complexity of the cortical dynamics underlying EEG recordings. Thus, reduced D2 values of the EEG in AD patients indicate that brains injured by AD exhibit a decrease in the complexity of brain electrical activity (Woyshville and Calabrese (1994) Besthorn et al., 1995 and Jeong et al., 1998, Stam et al., 1995 and Yagyu et al., 1 997).

91. Non-linear dynamical analysis of the EEG in Alzheimer's disease with optimal embedding dimension. Jeong et al. (1998) Electroencephalogr Clin Neurophysiol AD patients have significantly lower nonlinear complex measures than those for age-approximated healthy controls, suggesting that brains afflicted by Alzheimer's disease show less chaotic behaviors than those of normal healthy brains. EEG dynamics in patients with Alzheimer’s disease

92. Pathophysiological implications of the decreased EEG complexity in AD A decrease in dynamic complexity of the EEG in AD patients might arise from neuronal death, deficiency of neurotransmitters like acetylcholine, and/or loss of connectivity of local neuronal networks. The reduction of the dimensionality in AD is possibly an expression of the inactivation of previously active networks. Also, a loss of dynamical brain responsivity to stimuli might be responsible for the decrease in the EEG complexity of AD patients. AD patients do not have D2 differences between in eyes-open and eyes-close conditions, whereas normal subjects have prominently increased eyes-open D2 values compared with eyes-closed D2 values, suggesting a loss of dynamical brain responsivity to external stimuli in AD patients.

93. Nonlinear measures as a diagnostic indicator of AD Pritchard et al (1994) assessed the classification accuracy of the EEG using nonlinear measures and a neural-net classification procedure in addition to linear methods. The combination of linear and nonlinear analyses improves the classification accuracy of the AD/control status of subjects up to 92%. Besthorn et al. (1997) reported that the D2 correctly classified AD and normal subjects with an accuracy of 70%. Good correlations are found between nonlinear measures and the severity of the disease, a slowing of EEG rhythms, and neuropsychological performance. Furthermore, the global entropy can quantify EEG changes induced by drugs, suggesting a possibility that nonlinear measures is capable of quantifying the effect of drugs on the course of the disease.

94. Nonlinear dynamical analysis of the EEG in patients with Alzheimer's disease and vascular dementia. Jeong et al., J Clin Neurophysiol (2001) VaD patients have relatively increased values of nonlinear measures compared with AD patients, and have an uneven distribution of D2 values over the regions than AD patients and healthy subjects.

95. Controversial on EEG complexity in Schizophrenia The majority of these studies focused upon the question whether schizophrenia is characterized by a loss of dynamical complexity or rather by an abnormal increase of complexity, reflecting a ‘loosening of neural networks’. Many and especially more recent studies have found a lower complexity in terms of a lower correlation dimension or lower Lyapunov exponent (Jeong et al., 1998, Kim et al., 2000, Kotini and Anninos, 2002, Lee et al., 2001 and Rockstroh et al., 1997).Kim et al., 2000Kotini and Anninos, 2002Rockstroh et al., 1997 However, increases in dimension and Lyapunov exponent have also been reported in the older studies (Elbert et al., 1992, Koukkou et al., 1993 and Saito et al., 1998).Elbert et al., 1992 Koukkou et al., 1993Saito et al., 1998

96. Decreased complexity of cortical dynamics in Schizophrenic patients Jaeseung Jeong, Dai-Jin Kim, Jeong-Ho Chae, Soo Yong Kim, et al. Nonlinear analysis of the EEG of Schizophrenics with optimal embedding dimension. Medical Engineering and Physics (1998). Dai-Jin Kim, Jaeseung Jeong, Jeong-Ho Chae, et al. The estimation of the first positive Lyapunov exponent of the EEG in patients with Schizophrenia. Psychiatry Research (2000). Jeong-Ho Chae, Jaeseung Jeong, Dai-Jin Kim, et al. The effect of antipsychotic medications on nonlinear dynamics of the EEG in schizophrenic patients. Clinical Neurophysiology (2003)

97. Jaeseung Jeong, Dai-Jin Kim, Soo Yong Kim, Jeong-Ho Chae, et al. Effect of total sleep deprivation on the dimensional complexity of the waking EEG. Sleep (2001) Jeong-Ho Chae, Jaeseung Jeong, Bradley S. Peterson, Dai-Jin Kim, Seung-Hyun Jin, et al. Dimensional complexity of the EEG in patients with Posttraumatic stress disorder. Psychiatry Research: neuroimaging (2003) Dai-jin Kim, Jaeseung Jeong, Kook Jin Ahn, Kwang-Soo Kim, Jeong-Ho Chae, et al. Complexity change in the EEG in alcohol dependents during alcohol cue exposure. Alcoholism: Clinical & Experimental Research (2003) Dai-Jin Kim, Won Kim, Su-Jung Yoon, Yong-Ku Kim, Jaeseung Jeong, Effects of alcohol hangover on cytokine production in healthy subjects. Alcohol (2003)

98. Nonlinear dynamics of the EEG during photic and auditory stimulation Jaeseung Jeong, Moo Kwang Joung and Soo Yong Kim. Quantification of emotion by nonlinear analysis of the chaotic dynamics of EEGs during perception of 1/f music. Biological Cybernetics (1998) Seung Hyun Jin, Jaeseung Jeong, Dong-Gyu Jeong, Dai-Jin Kim et al. Nonlinear dynamics of the EEG separated by Independent Component Analysis after sound and light stimulation. Biological Cybernetics (2002) Jaeseung Jeong, Sangbaek Han, Bradley S. Peterson, and Soo Yong Kim, "The effect of photic and auditory stimulation on nonlinear dynamics of the human electroencephalogram," Clinical Neurophysiology (in press)

99. Information flow during Tic suppression of Tourette’s syndrome patients T4T4 T3T3 T4T3 Resting EEG: normal vs. TS TS: Normal vs. Tic suppression

100. The effect of alcohol on the EEG complexity measured by Approximate entropy

102. Perspectives For the last thirty years, progress in the field of nonlinear dynamics has increased our understanding of complex systems dy namics. This framework can become a valuable tool in scientific fields such as neuroscience and psychiatry where objects possess natural time dependency (i.e. dynamical properties) and non-linear characteristics. Relative estimates of nonlinear measures can reliably characterize different states of normal and pathologic brain function. Nonlinear dynamical analysis provides valuable information for developing mathematical models of the systems.

103. Multimodal approach for psychiatric disorders For example, the combination of EEG and PET variables results in approximately 90% of overall correct classification with a specificity of 100% (Jelic et al., 1999).Jelic et al., 1999 EEG and MRI measurements of the hippocampus obtain the highest scores of abnormalities in patients with probable AD ( Jonkman, 1997).Jonkman, 1997 Furthermore, CT- and MRI-based measurements of hippocampal atrophy provide a useful early marker of AD ( Scheltens, 1999).Scheltens, 1999 These neuroimaging techniques can offer not only supplementary information for diagnosis of AD, but also an opportunity to explore structural, functional, and biochemical changes in the brain leading to new insights into the pathogenesis of AD.

104. Coronal MRI at the level of the hippocampi showing no significant atrophy, but FDG-PET SPM indicates posterior cingulate hypometabolism Coronal MRI slices perpendicular to the long axis of the hippocampus showing a smaller hippocampus in an MCI patient. (Nestor et al., 2004)

105. Measures of nonlinear interdependency The brain can be conceived as a complex network of coupled and interacting subsystems. Higher brain functions depend upon effective processing and integration of information in this network. This raises the question how functional interactions between different brain areas take place, and how such interactions may be changed in different types of pathology.

106. J Jeong, JC Gore, BS Peterson. Mutual information analysis of the EEG in patients with Alzheimer's disease. Clin Neurophysiol (2001) Mutual information of the EEG The MI between measurement x i generated from system X and measurement y j generated from system Y is the amount of information that measurement x i provides about y j.

107. Recent MI studies on the EEG Schlogl A, Neuper C, Pfurtscheller G. Estimating the mutual information of an EEG-based Brain-Computer Interface. Biomed Tech. 2002;47(1-2):3-8. Na et al., EEG in schizophrenic patients: mutual information analysis. Clin Neurophysiol. 2002;113(12):1954-60. Huang L, Yu P, Ju F, Cheng J. Prediction of response to incision using the mutual information of electroencephalograms during anaesthesia. Med Eng Phys. 2003;25(4):321-7.

108. Phase synchronization in chaotic systems Coupled chaotic oscillators can display phase synchronization even when their amplitudes remain uncorrelated (Rosenblum et al., 1996). Phase synchronization is characterized by a non uniform distribution of the phase difference between two time series. It may be more suitable to track nonstationary and nonlinear dynamics.

109. Phase synchronization ‘Synchronization of chaos refers to a process, wherein two (or many) systems (either equivalent or nonequivalent) adjust a given property of their motion to a common behavior due to a coupling or to a forcing (periodical or noisy)’ (Boccaletti et al., 2002).

110. Nonlinear coupling among cortical areas

111. Phase synchronization and interdependence Jansen et al., Phase synchronization of the ongoing EEG and auditory EP generation. Clin Neurophysiol. 2003;114(1):79-85. Le Van Quyen et al., Nonlinear interdependencies of EEG signals in human intracranially recorded temporal lobe seizures. Brain Res. (1998) Breakspear and Terry. Detection and description of non-linear interdependence in normal multichannel human EEG data. Clin Neurophysiol (2002) Definition of synchronization: two or many subsystems sharing specific common frequencies Broader notion: two or many subsystems adjust some of their time- varying properties to a common behavior due to coupling or common external forcing

112. Generalized Synchronization Generalized synchronization exists between two interacting systems if the state of the response system Y is a function of the state of the driver system X: Y=F(X). Cross prediction is the extent to which prediction of X is improved by knowledge about Y, which allows the detection of driver and response systems. The nonlinear interdependence is not a pure measure of coupling but is also affected by the complexity or degrees of freedom of the interacting systems

113. Decreased EEG synchronization in MCI Koenig et al. (2005) Stam et al. (2003)