Non-linear forecasting measurements of multichannel EEG dynamics

Time Irreversibility of Resting-State Activity in the Healthy Brain and Pathology

Characterizing brain activity at rest is of paramount importance to our understanding both of general principles of brain functioning and of the way brain dynamics is affected in the presence of neurological or psychiatric pathologies. We measured the time-reversal symmetry of spontaneous electroencephalographic brain activity recorded from three groups of patients and their respective control group under two experimental conditions (eyes open and closed). We evaluated differences in time irreversibility in terms of possible underlying physical generating mechanisms. The results showed that resting brain activity is generically time-irreversible at sufficiently long time scales, and that brain pathology is generally associated with a reduction in time-asymmetry, albeit with pathology-specific patterns. The significance of these results and their possible dynamical etiology are discussed. Some implications of the differential modulation of time asymmetry by pathology and experimental condition are examined.

Seizure prediction and its applications

Epilepsy is characterized by intermittent, paroxysmal, hypersynchronous electrical activity that may remain localized and/or spread and severely disrupt the brain's normal multitask and multiprocessing function. Epileptic seizures are the hallmarks of such activity. The ability to issue warnings in real time of impending seizures may lead to novel diagnostic tools and treatments for epilepsy. Applications may range from a warning to the patient to avert seizure-associated injuries, to automatic timely administration of an appropriate stimulus. Seizure prediction could become an integral part of the treatment of epilepsy through neuromodulation, especially in the new generation of closed-loop seizure control systems.

The nonlinear theory of schizophrenia

OBJECTIVE

Nonlinear properties exist within the brain across a hierarchy of scales and within a variety of critical neural processes. Only a few studies of brain activity in schizophrenia, however, have used nonlinear methods. This review paper evaluates the contribution of the nonlinear sciences towards understanding schizophrenia.

METHOD

Applications of nonlinear methods to the study of schizophrenia symptoms and to healthy and schizophrenia functional neuroscience data are reviewed. The main flaws of nonlinear algorithms and recent methods to correct these are also appraised.

RESULTS

Initial research methods utilized in the study of nonlinearity in schizophrenia have fundamental methodological limitations. In the last decade, many of these problems have been addressed, facilitating future progress. Research incorporating these improvements has been applied to normal electroencephalogram (EEG) data and to the symptoms of schizophrenia, but not systematically to brain imaging data collected from patients with schizophrenia.

CONCLUSION

There is strong statistical evidence for weak nonlinearity in normal EEG and in the fluctuations of the symptoms of schizophrenia. However, the contribution of nonlinear processes to brain dysfunction in schizophrenia is yet to be properly established or accurately quantified. Despite this, recent methodological advances suggest that a 'nonlinear theory' of schizophrenia may be helpful in understanding this disorder.

Epileptic seizure prediction and control

Epileptic seizures are manifestations of epilepsy, a serious brain dynamical disorder second only to strokes. Of the world's approximately 50 million people with epilepsy, fully 1/3 have seizures that are not controlled by anti-convulsant medication. The field of seizure prediction, in which engineering technologies are used to decode brain signals and search for precursors of impending epileptic seizures, holds great promise to elucidate the dynamical mechanisms underlying the disorder, as well as to enable implantable devices to intervene in time to treat epilepsy. There is currently an explosion of interest in this field in academic centers and medical industry with clinical trials underway to test potential prediction and intervention methodology and devices for Food and Drug Administration (FDA) approval. This invited paper presents an overview of the application of signal processing methodologies based upon the theory of nonlinear dynamics to the problem of seizure prediction. Broader application of these developments to a variety of systems requiring monitoring, forecasting and control is a natural outgrowth of this field.

The algorithmic complexity of multichannel EEGs is sensitive to changes in behavior

Symbolic measures of complexity provide a quantitative characterization of the sequential structure of symbol sequences. Promising results from the application of these methods to the analysis of electroencephalographic (EEG) and event-related brain potential (ERP) activity have been reported. Symbolic measures used thus far have two limitations, however. First, because the value of complexity increases with the length of the message, it is difficult to compare signals of different epoch lengths. Second, these symbolic measures do not generalize easily to the multichannel case. We address these issues in studies in which both single and multichannel EEGs were analyzed using measures of signal complexity and algorithmic redundancy, the latter being defined as a sequence-sensitive generalization of Shannon's redundancy. Using a binary partition of EEG activity about the median, redundancy was shown to be insensitive to the size of the data set while being sensitive to changes in the subject's behavioral state (eyes open vs. eyes closed). The covariance complexity, calculated from the singular value spectrum of a multichannel signal, was also found to be sensitive to changes in behavioral state. Statistical separations between the eyes open and eyes closed conditions were found to decrease following removal of the 8- to 12-Hz content in the EEG, but still remained statistically significant. Use of symbolic measures in multivariate signal classification is described.

[Comparison of spectral analysis and non-linear analysis of EEG in patients with cognitive decline]

Twenty-five elder subjects were classified in two groups according to the MMS score and the cognitive evoked potentials. Normal subjects (n = 15) had mean MMS = 27.6 and mean P3 amplitude = 7.1 uV), while patients with cognitive decline (n = 10) had respective values of 18 (MMS) and 3.3 uV (P3). Spectral analysis and non-linear analysis of EEG (recurrence plots of dynamical systems) were performed and both showed statistically significant differences between groups for all the parameters analysed. Subjects' classification with discriminant analysis was slightly better using the non-linear parameters. The recurrence plot method applied to EEGs, gave similar results as the dimension of correlation (D2) calculation, and was in favour of a more constraint and less complex dynamics of brain activity associated with cognitive decline.

Relationship of nonlinear analysis, MRI and SPECT in the lateralization of temporal lobe epilepsy

OBJECTIVES

The purpose of this study was to investigate the correlation of lateralization by nonlinear analysis, magnetic resonance imaging (MRI) and interictal single-photon emission computed tomography (SPECT) in patients with temporal lobe epilepsy.

METHODS

Twenty-three patients (7 males, 16 females) were examined by MRI, interictal SPECT and EEG. Nonlinear dynamic properties of neuronal networks were estimated by calculating correlation dimensions on interictal EEG signals and corresponding surrogate data. Lateralization was detected based on the criteria introduced in this study. Concordance rates of the results among the three methods were compared.

RESULTS

Epileptogenic foci were shown in the temporal areas in 21 patients using the nonlinear method (8 left, 2 right, 11 both), while 20 patients showed abnormalities in temporal lobes on MR images (13 left, 5 right, 2 both). Low cerebral blood flows of the temporal lobes were detected in all patients (11 left, 8 right, 4 both). Completely concordant lateralization was observed in 8 patients (35%) for the nonlinear method and MRI, in 9 patients (39%) for the nonlinear method and SPECT, and in 10 patients (43%) for MRI and SPECT. There were no significant differences among the concordance rates for these different methods.

CONCLUSIONS

Our results revealed that correlation dimension is useful for differentiating dynamic properties of neuronal networks in the interictal state, and can provide informative data for localizing epileptogenic foci in epileptic patients. Therefore, the present nonlinear method is recommended for use with patients during presurgical evaluation.

Detection and description of non-linear interdependence in normal multichannel human EEG data

OBJECTIVES

This study examines human scalp electroencephalographic (EEG) data for evidence of non-linear interdependence between posterior channels. The spectral and phase properties of those epochs of EEG exhibiting non-linear interdependence are studied.

METHODS

Scalp EEG data was collected from 40 healthy subjects. A technique for the detection of non-linear interdependence was applied to 2.048 s segments of posterior bipolar electrode data. Amplitude-adjusted phase-randomized surrogate data was used to statistically determine which EEG epochs exhibited non-linear interdependence.

RESULTS

Statistically significant evidence of non-linear interactions were evident in 2.9% (eyes open) to 4.8% (eyes closed) of the epochs. In the eyes-open recordings, these epochs exhibited a peak in the spectral and cross-spectral density functions at about 10 Hz. Two types of EEG epochs are evident in the eyes-closed recordings; one type exhibits a peak in the spectral density and cross-spectrum at 8 Hz. The other type has increased spectral and cross-spectral power across faster frequencies. Epochs identified as exhibiting non-linear interdependence display a tendency towards phase interdependencies across and between a broad range of frequencies.

CONCLUSIONS

Non-linear interdependence is detectable in a small number of multichannel EEG epochs, and makes a contribution to the alpha rhythm. Non-linear interdependence produces spatially distributed activity that exhibits phase synchronization between oscillations present at different frequencies. The possible physiological significance of these findings are discussed with reference to the dynamical properties of neural systems and the role of synchronous activity in the neocortex.

Nonlinear analysis of EEG after repetitive transcranial magnetic stimulation

The purpose of this study was to investigate whether repetitive transcranial magnetic stimulation (rTMS) can change nonlinear dynamic properties of the cerebral cortex. Two rTMS trains (10 Hz, 3 seconds, 100% of motor threshold) were administered to the left frontal area in healthy subjects. EEG signals were collected at 14 electrode sites before and after stimulation, and were filtered digitally into delta, theta, alpha, beta, and gamma bands. Basing on an improved algorithm introduced in the authors' recent study, dimension estimates were calculated on these signals as well as on the corresponding surrogate data. Sham treatment was designed into this study. The data showed that EEG signals obviously exhibited lower dimension estimates than the surrogate data, whereas the theta and alpha rhythms presented the lowest values among the frequency components. rTMS increased the dimension estimates of EEG signals during the first 2 minutes. Similar findings were also obtained on the delta and beta components. This study revealed that EEG signals in the normal state can be described by a nonlinear dynamic process. This process can be affected temporarily by rTMS. Neuronal networks revealed by EEG signals show more complicated properties after stimulation.

Nonlinear analysis of orthostatic posture in patients with vertigo or balance disorders

The stabilometry signals involve irregular and unpredictable components. In order to identify the hidden dynamics that underlie the multi-link networks consisted of the multiple sensory systems, motor components and central integration, we applied a nonlinear analysis to these signals. We evaluated the postural control differences between eyes open and closed by means of the dynamical closeness between two states, known as similarity index, for the patients with vestibular disorders. We were able to demonstrate that some patients (eight of 21) showed a difference between the conventional and nonlinear measures. Especially, the similarity index tended to reflect the clinical course of the vestibular compensation and the findings in the patients with benign paroxysmal positional vertigo (BPPV) demonstrated that its vestibular function may include various pathologies besides canalithiasis. These results suggest that nonlinear analysis can elucidate the complex postural control networks and this procedure may also be able to provide the new findings of the stabilometry examinations.

Epileptic event forewarning from scalp EEG

The authors present a model-independent approach to quantify changes in the dynamics underlying nonlinear time-serial data. From time-windowed datasets, the authors construct discrete distribution functions on the phase space. Condition change between base case and test case distribution functions is assessed by dissimilarity measures via L1 distance and chi2 statistic. The discriminating power of these measures is first tested on noiseless data from the Lorenz and Bondarenko models, and is then applied to detecting dynamic change in multichannel clinical scalp EEG data. The authors compare the dissimilarity measures with the traditional nonlinear measures used in the analysis of chaotic systems. They also assess the potential usefulness of the new measures for robust, accurate, and timely forewarning of epileptic events.

Investigation of non-linear properties of multichannel EEG in the early stages of Parkinson's disease

OBJECTIVES

Modifications of brain activity in the early stages of Parkinson's disease (PD) are difficult to detect using electroencephalography (EEG) signals and are often biased by L-DOPA treatment. We compare here the performances of both linear and non-linear methods in differentiating EEG of L-DOPA naive PD patients from that of control subjects.

METHODS

Resting multichannel EEG (20 electrodes, 30 s epochs) of 9 patients with PD in Hoehn and Yahr stages 1-2 (4 women, 5 men, mean age 54.3 years, range 48-63 years) were compared with those of 9 control subjects (7 women, two men, mean age 51.3 years, range 43-61 years). The following measurements were computed: theta-, alpha- and beta-band relative powers constituted the linear indices; localized entropy, slope asymmetry and number of non-linear EEG segments constituted the non-linear indices.

RESULTS

In the case of linear quantification, only a decrease in the beta-band was observed for patients. Significant non-linear structures were observed in our EEG data. Non-linear quantifiers demonstrate an increase in entropy and in the number of non-linear EEG segments for the patients.

CONCLUSIONS

Changes in EEG dynamics observed here in L-DOPA naive PD patients may represent early signs of cortical dysfunction produced by subcortical dopamine depletion.

Correlation dimension of the human electroencephalogram corresponds with cognitive load

This study aimed at assessing the effects of cognitive activity and mental task load on the correlation dimension of the human electroencephalogram (EEG). Three experimental conditions were created: a baseline condition and two cognitive task conditions, a calculation task and a time estimation task. The calculation task was supposed to induce a higher mental load than the time estimation task, which is regarded as a less complex one. This was verified by a subjective rating scale. All conditions differed significantly in subjective estimated task load. The correlation dimension appeared to be higher in both task conditions compared to the baseline condition. A comparison of the two tasks indicated that the difference in correlation dimension between calculation and time estimation was also significant, with the highest value for calculation. It is concluded that cognitive and mental activity is associated with a higher correlation dimension in the EEG. This implies that the correlation dimension is a sensitive parameter in the analysis of electrical brain activity.

Nonlinear EEG analysis in early Alzheimer's disease

Nonlinear EEG analysis attempts to characterize the dynamics of neural networks in the brain. Abnormalities in nonlinear EEG measures have been found repeatedly in Alzheimer's disease (AD). The present study was undertaken to investigate whether these abnormalities could already be found in the early stage of AD. In a representative sample of 49 community-dwelling elderly, Alzheimer's disease was diagnosed in 7 subjects. Correlation dimension (D2) and nonlinear prediction were measured at 16 electrodes and in two different activational states. Also, 10 surrogate data sets were generated for each EEG epoch in order to investigate the presence of nonlinear dynamics. Differences between nonlinear statistics derived from original and from surrogate data sets were expressed as Z-scores. We found lower D2 and higher predictability in the demented subjects compared to the normal subjects. The results obtained with the Z-scores pointed to changed nonlinear dynamics in frontal and temporal areas in demented subjects. However, the major differences between demented and healthy subjects are not due to nonlinearity. From this it appears that linear dynamics change first in the course of AD, followed by changes in nonlinear dynamics.

Chaos theories and therapeutic commonalities among depression, Parkinson's disease, and cardiac arrhythmias

This report reviews and compares all therapies that have shown efficacy in depression and Parkinson's disease, although some are not in current use and others are at the experimental stage. They include pharmacological modification of neurotransmitter pathways, electroconvulsive therapy (ECT), sleep deprivation, psychosurgery, electrical stimulation through cerebral electrodes, and transcranial magnetic stimulation. Stemming from a pathophysiological model that stresses the brain as an open, complex, and nonlinear system, all therapies have been attributed a common mechanism of action. This report suggests that the therapeutic isomorphism is related to their ability to help the CNS deactivate cortical-subcortical circuits that are dysfunctionally coupled. These circuits are self-organized among neurons of their informational subsystem (rapid conduction) and modulating subsystem (slow conduction). Finally, this report extends the analysis and comparison of these therapies to some cardiac arrhythmias.

Entropy maps characterize drug effects on brain dynamics in Alzheimer's disease

Non-linear quantifiers of brain electrical dynamics (entropy maps computed from the degradation of temporal forecasting of EEG signals) were studied in relation to drug treatment of Alzheimer's disease. A placebo condition was compared to three drug doses (50, 100 and 200 mg). A significant general effect of the drug was found when compared to placebo and specific contrasts between placebo and each of the three drug doses only reveal a significant entropy increase for the highest dose. These effects were localized bilaterally in fronto-temporal areas and support changes in the dynamics of the cerebral structures involved in memory processes.

Adaptive regulation of the nonlinear dynamics of electrical activity of the brain

A programmable system was used to provide contingent reinforcement of EEG cycles corresponding to a selected criterion in a dynamic regime. Use of automated reinforcing stimulation of emotionally positive zones of the hypothalamus led to a significant increase in the number of cycles with the characteristics specified by the dynamic regime within the dominant EEG frequency bands. This effect lasted for some time after withdrawal of reinforcing stimulation, and then died down gradually. These changes in the EEG activity structure did not occur in conditions of nonassociated hypothalamic stimulation. Pseudoreinforced background EEG cycles showed complex nonlinear dynamics with competitive interactions between processes in which the large dimensionality of the attractor was difficult to interpret because of indeterminacy in the trends of the dominant process. In contingent hypothalamic stimulation, the form of the correlation integral changed towards a predominance of a single nonlinear process determining all the activity recorded. In fact, a single dominant nonlinear process was formed, which became responsible for the entire dynamics of the system with concordance of its internal structure.

[Contribution of non-linear mathematics (chaos theory) to EEG analysis]

Since 1963, nonlinear dynamics or "chaos theory" were widely used in various area of physics. The first application to the analysis of the human electroencephalogram (EEG) was performed in 1985. A tutorial revue of some concepts of nonlinear dynamics is presented with the various results obtained since 1985. The dimensional complexity of the EEG (DC) seems a good descriptor of the "desynchronisation" or decorrelation of the EEG: DC is high with eyes open, during a mental task and also during a tonic epileptic discharge. DC is low with eyes closed, during slow wave sleep, and during a clonic epileptic discharge. The measure of DC could allow a more objective comparison between various states of electrical cerebral activity.

Application of non-linear filters based on the median filter to experimental and simulated multiunit neural recordings

Two non-linear, high-pass filters based on the median filter are proposed and tested as substitutes for linear filtering in applications involving multiunit neural recordings. The first, the median-based high-pass (MH) filter, operates by subtracting the output from the input of the median filter; it is aimed at preserving the shape of the impulses. The second, the negative median-based high-pass (NMH) filter, sets at zero the positive values in the output of the MH filter; it is aimed at transforming the impulses into monophasic waves placed on a flat baseline. When applied to experimental recordings and to a template action potential, the two median-based filters clearly outperformed two corresponding procedures based on a linear filter (moving-average filter). They did not produce appreciable distortions of the impulses, whereas their two counterparts induced or enlarged lateral lobes, as is the rule for linear high-pass filters. The recording display was much improved and impulse identification was made easier. When the two filters were applied to simulated recordings and the mean output was estimated by averaging and cross-correlation, a certain degree of performance deterioration was assessed in conditions of sustained activity and/or noise, with a resulting growing similarity to the mean output of the two corresponding, moving-average-based filters.

Working-memory load and dimensional complexity of the EEG

This investigation shows that a working-memory load induced by a memory scanning task has an effect on nonlinear descriptors of the EEG dynamics. The effect was locally specific above the fronto-temporal (right) cortex and it may be described as a reduction in the dimensional complexity of cortical brain activity. The meaning of the effects seems to differ from that of EEG spectral power, which varied with time during the experiment and not with changes in the working-memory load conditions. Behaviorally controlled over- and underload led to increased variance of the dimensional complexity, thus indicating that dimensional complexity correlates more closely with actual performance than with more general descriptions of brain states. Habitual response speed had an effect at the parietal lead, thus indicating that fast responders reduced their dimensional complexity as the task demand increased. In contrast, the slower responders showed no such definite trend.

On the validity of estimating EEG correlation dimension from a spatial embedding

We demonstrate by using simulations that spatial embedding of single-variable time series data does not reliably reconstruct state-space dynamics. Instead, correlation dimension estimated from spatially embedded data is largely a measure of linear cross-correlation in the data set. For actual electroencephalographic (EEG) data, we demonstrate a high negative correlation between spatial correlation dimension and the average amount of lag-zero cross-correlation between "nearest-neighbor" embedding channels (the greater the cross-correlation, the lower the dimension). We also show that the essential results obtained from spatially embedding EEG data are also obtained when one spatially embeds across a set of highly cross-correlated stochastic (second-order autoregressive) processes. Although, with appropriate surrogate data, correlation dimension estimated from spatially embedded data detects nonlinearity, its use is not recommended because correlation dimension estimated from temporally embedded data both reconstructs state-space dynamics and, with appropriate surrogate data, detects nonlinearity as well.

Depression as a dynamical disease

Mathematical models are helpful in the understanding of diseases through the use of dynamical indicators. A previous study has shown that brain activity can be characterized by a decrease of dynamical complexity in depressive subjects. The present paper confirms and extends these conclusions through the use of recent methodological advances: first episode and recurrent patients strongly differ in their dynamical response to therapeutic interventions. These results emphasize the need for clinical follow-ups to avoid recurrence and the necessity of specific therapeutic intervention in the case of recurrent patients.

The analysis of the EEG

The quantitative analysis of the electroencephalogram (EEG) relies heavily on methods of time series analysis. A quantitative approach seems indispensable for research (be it clinical or basic neurophysical research), but it can also be a useful information for purely clinical purposes. Apart from the ongoing spontaneous EEG, evoked potentials (EPs) also play an important role. They can be elicited by simple sensory stimuli or more complex stimuli. Their analysis requires methods which are different from those for the spontaneous EEG. Those methods operate usually in the time domain and offer many challenging problems to statisticians. Methods for analysing the spontaneous EEG usually work in the frequency domain in terms of spectra and coherences. Biomedical engineers who take care of the equipment are usually also trained in time series analysis. Thus, they have contributed much more to methodological progress for analysing EEGs and EPs, compared with statisticians. However, the availability of a sample of subjects, and the associated problems in modelling followed by an inferential analysis could make a larger influence from the statistical side quite profitable. This paper tries to give an overview of a fascinating area. In doing so we treat more extensively problems with some statistical appeal. This leads inevitably to some overlap with our own work.

Re-examination of the evidence for low-dimensional, nonlinear structure in the human electroencephalogram

We have re-examined single channel EEG data obtained from normal human subjects. In the original analysis, calculation of the correlation dimension with the Grassberger-Procaccia algorithm produced results consistent with and interpretation of low-dimensional behavior. The re-examination suggests that the previous indication of low-dimensional structure was an artifact of autocorrelation in the oversampled signal. Calculations with a variant of the Grassberger-Procaccia algorithm modified to be less sensitive to autocorrelations, and comparison with linear gaussian surrogate data, indicate that these data may be more appropriately modeled by linearly filtered noise. Discriminant analysis further indicates that the correlation dimension is a poor discriminator for distinguishing between EEGs recorded at rest and during periods of cognitive activity. It remains possible that the application of other nonlinear measures or the examination of multichannel EEG data may resolve structures not found in these calculations.