Bronzino J.D.. Berbari. E.J.. Johnson P.L.. Smith. W.M."Bioelectronics and Instruments The electrical Engineering Handbook Ed. Richard C. dorf Boca Raton CRC Press llc. 2000
Bronzino, J.D., Berbari, E.J., Johnson, P.L., Smith, W.M. “Bioelectronics and Instruments” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
15 Bioelectronics and Joseph D. Bronzino finity College/Biomedical Allience Instruments for Central Connecticut(BOACON) Edward J. Berbari Purdue University 115.1 The Electroencephalogram Philip L. Johnson Recording Techniques Frequency Analysis of the EEG University of Alabama at Nonlinear Analysis of the EEG. Topographic Mapping 115.2 The Electrocardiograph William M. smith Physiology. Instrumentation.Conclusions University of Alabama at 115.3 Pacemakers/Implantable Defibrillators Pacemakers. Implantable Cardioverter Defibrillators 115.1 The Electroencephalogram Joseph d. bronzino Electroencephalograms(EEGs)are recordings of the minute (generally less than 300 uV) electrical potential produced by the brain. Since 1924, when Hans Berger reported the measurements of rhythmic electrical activit the neuronal bases for specific behaviors and has offered great promise to reveal correlations between patha a on the human scalp, it has been suggested that these patterns of bioelectrical origin may provide clues regard logical processes and the electrical activity of specific regions of the brain. Over the years, EEG analyses have been conducted primarily in clinical settings, to detect gross organic pathologies and the epilepsies, and in research facilities to quantify the central effect of new pharmacological agents. As a result of these efforts, cortical EEG patterns have been shown to be modified by a wide variety of variables including biochemical, metabolic, circulatory, hormonal, neuroelectric, and behavioral factors. In the past, interpretation of the EEG was limited to visual inspection by a trained electroencephalographer capable of distinguishing normal activity from localized or generalized abnormalities of particular types from relatively long EEG records. This approach has left clinicians and researchers alike lost in a sea of EEG paper records. Computer technology has permitted the application of a host of methods to quantify EEG changes. With this mind, this section provides an introduction to some of the basic concepts underlying the generation of the EEG, a review of the basic approaches used in quantifying alterations in the EEG, and some insights regarding quantitative electrophysiology techniques The Language of the Brain The mass of brain tissue is composed of bundles of nerve cells(neurons) which constitute the fundamental uilding blocks of the nervous system. Figure 115. 1 is a schematic drawing of just such a cell. It consists of the cell body (or soma), the receptor zone(or dendrites), and the axon, which carries electrical signals from the soma to target sites such as muscles, glands, or other neurons. Numbering approx imately 20 billion in each human being, these tiny cells come in a variety of sizes and shapes. Although neurons are anatomically distinct units having no physical continuity between their processes, the axon ends on the as a spherical enlargement at the end of the axon to whis synapse. Under the microscope this often stands out soma and the dendrites of other cells in what is called a various names have been given, for example, boutons, c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 115 Bioelectronics and Instruments 115.1 The Electroencephalogram The Language of the Brain • Historical Perspective • EEG Recording Techniques • Frequency Analysis of the EEG • Nonlinear Analysis of the EEG • Topographic Mapping 115.2 The Electrocardiograph Physiology • Instrumentation • Conclusions 115.3 Pacemakers/Implantable Defibrillators Pacemakers • Implantable Cardioverter Defibrillators 115.1 The Electroencephalogram Joseph D. Bronzino Electroencephalograms (EEGs) are recordings of the minute (generally less than 300 µV) electrical potentials produced by the brain. Since 1924, when Hans Berger reported the measurements of rhythmic electrical activity on the human scalp, it has been suggested that these patterns of bioelectrical origin may provide clues regarding the neuronal bases for specific behaviors and has offered great promise to reveal correlations between pathological processes and the electrical activity of specific regions of the brain. Over the years, EEG analyses have been conducted primarily in clinical settings, to detect gross organic pathologies and the epilepsies, and in research facilities to quantify the central effect of new pharmacological agents. As a result of these efforts, cortical EEG patterns have been shown to be modified by a wide variety of variables including biochemical, metabolic, circulatory, hormonal, neuroelectric, and behavioral factors. In the past, interpretation of the EEG was limited to visual inspection by a trained electroencephalographer capable of distinguishing normal activity from localized or generalized abnormalities of particular types from relatively long EEG records. This approach has left clinicians and researchers alike lost in a sea of EEG paper records. Computer technology has permitted the application of a host of methods to quantify EEG changes. With this in mind, this section provides an introduction to some of the basic concepts underlying the generation of the EEG, a review of the basic approaches used in quantifying alterations in the EEG, and some insights regarding quantitative electrophysiology techniques. The Language of the Brain The mass of brain tissue is composed of bundles of nerve cells (neurons) which constitute the fundamental building blocks of the nervous system. Figure 115.1 is a schematic drawing of just such a cell. It consists of three major components: the cell body (or soma), the receptor zone (or dendrites), and the axon, which carries electrical signals from the soma to target sites such as muscles, glands, or other neurons. Numbering approximately 20 billion in each human being, these tiny cells come in a variety of sizes and shapes. Although neurons are anatomically distinct units having no physical continuity between their processes, the axon ends on the soma and the dendrites of other cells in what is called a synapse. Under the microscope this often stands out as a spherical enlargement at the end of the axon to which various names have been given, for example, boutons, Joseph D. Bronzino Trinity College/Biomedical Allience for Central Connecticut (BOACON) Edward J. Berbari Purdue University Philip L. Johnson University of Alabama at Birmingham William M. Smith University of Alabama at Birmingham
end-plate, or synaptic terminals. This ending does not actually make physical contact with the soma or dendrite but is separated by a narrow cleft(gap)of approximately Axonal Terminals cleft. Each of these synaptic endings contains a large num ber of submicroscopic spherical structures(synaptic vesi in be detected only under an electron Dendrite microscope. These synaptic vesicles, in turn, are essentially is released into the synaptic cleft on excitation. Axon When an individual neuron is excited, an electrical sig- nal is transmitted along its axon to many tiny branching, diverging fibers near its far end. These axonal terminals Axonal nd as synapse on a large number of other neurons. when an electrical pulse arrives at the synapse, it triggers the FIGURE 115.1 Basic structure of the neuron lease of a tiny amount of transmitter substance which crosses the synaptic cleft thereby altering the membrane potential of the receiving neuron. If the change is above a certain threshold value, the neuron is activated and generates an action potential of its own which propagated along its axon, and the process is repeated. Neurons are involved in every conceivable action taken by the body, whether it is to control its own internal nvironment or to respond to changes in the external world. As a result, they are responsible for such essential ons as: Accepting and converting sensory information into a form that can be processed within the nervous system by other neurons. Processing and analyzing this information so that an"integrated portrait"of the incoming data can be obtained Translating the final outcome or decision"of this analysis process into appropriate electrical or chemical form needed to stimulate glands or activate muscles. Evolution has played a role in the development of these unique neurons and in the arrangement and development of interconnections between nerve cells in the various parts of the brain. Since the brain is a most complex organ, it contains numerous regions designed for specific tasks. One might, in fact, consider it to be a collection of organs arranged together to act in the harmony of activity we recognize as the individuals state of consciousness or as life itself. Over the years, anatomists and physiologists have identified and named most pathways(tracts), most groups of neurons(nuclei), and most of the major parts of the human brain. Such attention to detail is certainly not necessary here. It will serve our purpose to simply provide a broad overview of the organization of the brain and speak of three general regions: the brainstem, cerebellum, and the cerebral cortex. The brainstem, or old brain, is really an extension and elaboration of the spinal chord. This section of the brain evolved first and is the location of all the centers that control the regulatory systems, such as respiration, necessary for physical survival of the organism. In addition, all sensory pathways find their way into the brainstem, thereby permitting the integration of complex input patterns to take place within its domain Above the brainstem is a spherical mass of neuronal tissue called the cerebellum. This remarkable structure is a complex monitor and modifier of body movements. The cerebellum does not initiate movements, but only modifies motor control activate er areas. Cerebellar operation is not only dependent on evolutionary development but relies heavily on actual use and patterns of learned motor behavior acquired throughout life. It is for this reason that the movements of a gymnast are smooth and seemingly effortless The most conspicuous part of all in the human brain is the cerebral cortex. Compared to most mammals, it is so large in man that it becomes a covering that surrounds and hides most of the other regions of the brain. Wrinkled and folded, the cerebral tissue is literally pressed into the limited space allocated to it. although it has been possible to ascertain that certain cortical areas such as visual cortex, the sensory projection area, and the motor strip are associated with specific functions, the overall operation of this complex structure is still c2000 by CRC Press LLC
© 2000 by CRC Press LLC end-plate, or synaptic terminals. This ending does not actually make physical contact with the soma or dendrite but is separated by a narrow cleft (gap) of approximately 100 to 200 Å (10–9 m) wide. This is known as the synaptic cleft. Each of these synaptic endings contains a large number of submicroscopic spherical structures (synaptic vesicles) that can be detected only under an electron microscope. These synaptic vesicles, in turn, are essentially “chemical carriers” containing transmitter substance that is released into the synaptic cleft on excitation. When an individual neuron is excited, an electrical signal is transmitted along its axon to many tiny branching, diverging fibers near its far end. These axonal terminals end as synapse on a large number of other neurons. When an electrical pulse arrives at the synapse, it triggers the release of a tiny amount of transmitter substance which crosses the synaptic cleft thereby altering the membrane potential of the receiving neuron. If the change is above a certain threshold value, the neuron is activated and generates an action potential of its own which is propagated along its axon, and the process is repeated. Neurons are involved in every conceivable action taken by the body, whether it is to control its own internal environment or to respond to changes in the external world. As a result, they are responsible for such essential functions as: • Accepting and converting sensory information into a form that can be processed within the nervous system by other neurons. • Processing and analyzing this information so that an “integrated portrait” of the incoming data can be obtained. • Translating the final outcome or “decision” of this analysis process into appropriate electrical or chemical form needed to stimulate glands or activate muscles. Evolution has played a role in the development of these unique neurons and in the arrangement and development of interconnections between nerve cells in the various parts of the brain. Since the brain is a most complex organ, it contains numerous regions designed for specific tasks. One might, in fact, consider it to be a collection of organs arranged together to act in the harmony of activity we recognize as the individual’s state of consciousness or as life itself. Over the years, anatomists and physiologists have identified and named most pathways (tracts), most groups of neurons (nuclei), and most of the major parts of the human brain. Such attention to detail is certainly not necessary here. It will serve our purpose to simply provide a broad overview of the organization of the brain and speak of three general regions: the brainstem, cerebellum, and the cerebral cortex. The brainstem, or old brain, is really an extension and elaboration of the spinal chord. This section of the brain evolved first and is the location of all the centers that control the regulatory systems, such as respiration, necessary for physical survival of the organism. In addition, all sensory pathways find their way into the brainstem, thereby permitting the integration of complex input patterns to take place within its domain. Above the brainstem is a spherical mass of neuronal tissue called the cerebellum. This remarkable structure is a complex monitor and modifier of body movements. The cerebellum does not initiate movements, but only modifies motor control activated in other areas. Cerebellar operation is not only dependent on evolutionary development, but relies heavily on actual use and patterns of learned motor behavior acquired throughout life. It is for this reason that the movements of a gymnast are smooth and seemingly effortless. The most conspicuous part of all in the human brain is the cerebral cortex. Compared to most mammals, it is so large in man that it becomes a covering that surrounds and hides most of the other regions of the brain. Wrinkled and folded, the cerebral tissue is literally pressed into the limited space allocated to it. Although it has been possible to ascertain that certain cortical areas such as visual cortex, the sensory projection area, and the motor strip are associated with specific functions, the overall operation of this complex structure is still FIGURE 115.1 Basic structure of the neuron
Parietal Lobe Frontal Occipital Temporal Lobe Cerebellum Brainstem FIGURE 115.2 Major divisions of the cerebral cortex. not completely understood. However, for the sake of convenience, it has been arbitrarily divided( based primarily on anatomical considerations) into the following areas: frontal lobe, parietal lobe, temporal lobe, and occipital lobe( Fig. 115.2). Each of these segments of the cortex, which is the source of intellectual and imaginative capacities, includes millions of neurons and a host of interconnections. It is generally agreed that brain function is based on the organization of the activity of large numbers of neurons into coherent patterns. Since the primary mode of activity of these nerve cells is clectrical in nature, it is not surprising that a composite of this activity can be detected in the form of electrical signals Of extreme interest, then, are the actual oscillations, rhythms, and patterns seen in the cryptic flow of electrical energy oming from the brain itself, i. e, in the EEG Historical Perspective In 1875, Caton published the initial account of the recording of the spontaneous electrical activity of the brain from the cerebral cortex of an experimental animal. The amplitude of these electrical oscillations was so lot that is, on the order of microvolts, that Catons discovery is all the more amazing because it was made 50 years before suitable electronic amplifiers became available. In 1924, Hans Berger, of the University of Jena in Austria, arried out the first human EEG recordings using electrical metal strips pasted to the scalps of his subjects as electrodes and a sensitive galvanometer as the recording instrument. Berger was able to measure the irregula relatively small electrical potentials (i.e. 50 to 100 uV) coming from the brain. By studying the successive positions of the moving element of the galvanometer recorded on a continuous roll of paper, he was able to observe the resultant patterns in these brain waves as they varied with time. From 1924 to 1938, Berger laid the foundation for many of the present applications of electroencephalography. He was the first to use the word electroencephalogram in describing these brain potentials in man. Berger noted that these brain waves were although these brain waves were slow(i. e, exhibited a synchronized patter of high amplitude and low frequency, <3 Hz)in sleep and states of depressed function, they were faster(i.e, exhibited a desynchronized pattern of low amplitude and high frequency, 15-25 Hz)during waking behavior. He suggested, quite correctly, that the brain's activity changed in a consistent and recognizable fashion when the general status of the subject changed as from relaxation to alertness. Berger also concluded that these brain waves could be greatly affected by certain pathological conditions after noting the marked increase in the amplitude of these brain waves brought about by convulsive seizures. However, in spite of the insights provided by these studies, Berger's original paper published in 1929 did not excite much attention. In essence, the efforts of this most remarkable pioneer were largely ignored until similar investigations were carried out and verified by British investigators It was not until 1934 when Adrian and Matthews published their classic paper verifying Berger's findings that the reality of human brain waves was accepted and EEG studies were put on a firmly established basis One of their primary contributions was the identification of certain rhythms in the EEG, regular oscillations at approximately 10-12 Hz in the occipital lobes of the cerebral cortex. They found that this alpha rhythm in the EEG would disappear when the brain displayed any type of attention or alertness or focused on objects in ne visual field. The physiological basis for these results, the"arousing influence "of external stimuli on the e 2000 by CRC Press LLC
© 2000 by CRC Press LLC not completely understood.However, for the sake of convenience, it has been arbitrarily divided (based primarily on anatomical considerations) into the following areas: frontal lobe, parietal lobe, temporal lobe, and occipital lobe (Fig. 115.2). Each of these segments of the cortex, which is the source of intellectual and imaginative capacities, includes millions of neurons and a host of interconnections. It is generally agreed that brain function is based on the organization of the activity of large numbers of neurons into coherent patterns. Since the primary mode of activity of these nerve cells is electrical in nature, it is not surprising that a composite of this activity can be detected in the form of electrical signals. Of extreme interest, then, are the actual oscillations, rhythms, and patterns seen in the cryptic flow of electrical energy coming from the brain itself, i.e., in the EEG. Historical Perspective In 1875, Caton published the initial account of the recording of the spontaneous electrical activity of the brain from the cerebral cortex of an experimental animal. The amplitude of these electrical oscillations was so low, that is, on the order of microvolts, that Caton’s discovery is all the more amazing because it was made 50 years before suitable electronic amplifiers became available. In 1924, Hans Berger, of the University of Jena in Austria, carried out the first human EEG recordings using electrical metal strips pasted to the scalps of his subjects as electrodes and a sensitive galvanometer as the recording instrument. Berger was able to measure the irregular, relatively small electrical potentials (i.e., 50 to 100 mV) coming from the brain. By studying the successive positions of the moving element of the galvanometer recorded on a continuous roll of paper, he was able to observe the resultant patterns in these brain waves as they varied with time. From 1924 to 1938, Berger laid the foundation for many of the present applications of electroencephalography. He was the first to use the word electroencephalogram in describing these brain potentials in man. Berger noted that these brain waves were not entirely random, but instead displayed certain periodicities and regularities. For example, he observed that although these brain waves were slow (i.e., exhibited a synchronized patter of high amplitude and low frequency, <3 Hz) in sleep and states of depressed function, they were faster (i.e., exhibited a desynchronized pattern of low amplitude and high frequency, 15–25 Hz) during waking behavior. He suggested, quite correctly, that the brain’s activity changed in a consistent and recognizable fashion when the general status of the subject changed, as from relaxation to alertness. Berger also concluded that these brain waves could be greatly affected by certain pathological conditions after noting the marked increase in the amplitude of these brain waves brought about by convulsive seizures. However, in spite of the insights provided by these studies, Berger’s original paper published in 1929 did not excite much attention. In essence, the efforts of this most remarkable pioneer were largely ignored until similar investigations were carried out and verified by British investigators. It was not until 1934 when Adrian and Matthews published their classic paper verifying Berger’s findings that the reality of human brain waves was accepted and EEG studies were put on a firmly established basis. One of their primary contributions was the identification of certain rhythms in the EEG, regular oscillations at approximately 10–12 Hz in the occipital lobes of the cerebral cortex. They found that this alpha rhythm in the EEG would disappear when the brain displayed any type of attention or alertness or focused on objects in the visual field. The physiological basis for these results, the “arousing influence” of external stimuli on the FIGURE 115.2 Major divisions of the cerebral cortex
cortex,was not formulated until 1949 when Moruzzi and Magoun demonstrated the existence of widely spread pathways through the central reticular core of the brainstem capable of exerting a diffuse activating influence on the cerebral cortex. This reticular activating system has been called the brain s response selector because it alerts the cortex to focus on certain incoming information while ignoring other. It is for this reason that a sleeping mother will immediately be awakened by her crying baby or the smell of smoke, and yet ignore the traffic outside her window or the television still playing in the next room. An in-depth discussion of these early studies is beyond the scope of this presentation; however, for the interested reader an excellent historical review of this early era in brain research has been recorded in a fascinating text by Brazier [1968 EEG Recording TechI eques Scalp recordings of spontaneous neuronal activity in the brain, identified as the EeG, allow measurement of potential changes over time between a signal electrode and a reference electrode[ Kondraski, 1986]. Compared to other biopotentials, such as the electrocardiogram, the EEG is extremely difficult for an untrained observer to interpret. As might be expected, partially as a result of the spatial mapping of functions onto different regions of the brain, correspondingly different waveforms are visible, depending on electrode placement. Recognizing that some standardizati vas necessary for comparison of research as well as clinical EEG records, the International Federation in Electroencephalography and Clinical Neurophysiology adopted the 10-20 electrode placement system, [Jasper, 1958]. Additional electrodes to monitor extracerebral contaminants of the EEG such as eye movement, EKG, and muscle activity are essential. The acquisition of EEG for quantitative analysis should also require the ability to view the EEG during collection on a polygraph or high-resolution video display Since amplification, filtering, and digitization determine the frequency characteristics of the EEG and the source of potential artifacts, the acquisition parameters must be chosen with an understanding of their effects on signal acquisition and subsequent analysis. Amplification, for example, increases the amplitude range(volts) of the analog-to-digital (A/D)converter. The resolution of the A/D converter is determined by the smallest plitude of steps that can be sampled. This is calculated by dividing the voltage range of the A/D converter by 2 to the power of the number of bits of the A/D converter. For example, an A/D converter with a range of +5 V with 12-bit resolution can resolve samples as small as #2. 4 mV. Appropriate matching of amplification and A/D converter sensitivity permits resolution of the smallest signal while preventing clipping of the largest ignal amplitudes. The bandwidth of the filters and the rate of digitization determine the frequency components of interest that are passed, while other frequencies outside the band of interest that may represent potential artifacts, such as aliasing, are rejected. A filter's characteristics are determined by the rate of the amplitude decrease at the bandwidths upper and lower edges. Proper digital representation of the analog signal depends on the rate of data sampling, which is governed by the Nyquist theorem that states that data sampling should be at least twice the highest frequency of interest. In addition to the information available from spontaneous electrical activity of the EEG, the brains electrical response to sensory stimulation can contribute data as to the status of cortical and subcortical regions activate by sensory input. Due to the relatively small amplitude of a stimulus-evoked potential as compared to the spontaneous EEG potentials, the technique of signal averaging is used to enhance the stimulus-evoked respo Stimulus averaging takes advantage of the fact that the brains electrical response is time-locked to the onset of the stimulus and the nonevoked background potentials are randomly distributed in time. Consequently, the average of multiple stimulus responses will result in the enhancement of the time-locked activity, while the averaged random background activity will approach zero. The result is an evoked response that consists of a lumber of discrete and replicable peaks that occur, depending upon the stimulus and the recording parameters, at predicted latencies from the onset of stimulation. The spatial localization of maximum peak amplitudes has been associated with cortical generators in y sensory cortex. Instrumentation required for EEG recordings can be simple or elaborate[Kondraski, 1986].(Note: Although the discussion presented in this section is for a single-channel system it can be extended to simultaneous multichannel recordings simply by multiplying the hardware by the number of channels required In cases that do not require true simultaneous recordings, special electrode selector panels can minimize hardware require- ments)Any EEG system consists of electrodes, amplifiers(with appropriate filters)and a recording device c2000 by CRC Press LLC
© 2000 by CRC Press LLC cortex, was not formulated until 1949 when Moruzzi and Magoun demonstrated the existence of widely spread pathways through the central reticular core of the brainstem capable of exerting a diffuse activating influence on the cerebral cortex. This reticular activating system has been called the brain’s response selector because it alerts the cortex to focus on certain incoming information while ignoring other. It is for this reason that a sleeping mother will immediately be awakened by her crying baby or the smell of smoke, and yet ignore the traffic outside her window or the television still playing in the next room. An in-depth discussion of these early studies is beyond the scope of this presentation; however, for the interested reader an excellent historical review of this early era in brain research has been recorded in a fascinating text by Brazier [1968]. EEG Recording Techniques Scalp recordings of spontaneous neuronal activity in the brain, identified as the EEG, allow measurement of potential changes over time between a signal electrode and a reference electrode [Kondraski, 1986]. Compared to other biopotentials, such as the electrocardiogram, the EEG is extremely difficult for an untrained observer to interpret. As might be expected, partially as a result of the spatial mapping of functions onto different regions of the brain, correspondingly different waveforms are visible, depending on electrode placement. Recognizing that some standardization was necessary for comparison of research as well as clinical EEG records, the International Federation in Electroencephalography and Clinical Neurophysiology adopted the 10–20 electrode placement system, [Jasper, 1958]. Additional electrodes to monitor extracerebral contaminants of the EEG such as eye movement, EKG, and muscle activity are essential. The acquisition of EEG for quantitative analysis should also require the ability to view the EEG during collection on a polygraph or high-resolution video display. Since amplification, filtering, and digitization determine the frequency characteristics of the EEG and the source of potential artifacts, the acquisition parameters must be chosen with an understanding of their effects on signal acquisition and subsequent analysis. Amplification, for example, increases the amplitude range (volts) of the analog-to-digital (A/D) converter. The resolution of the A/D converter is determined by the smallest amplitude of steps that can be sampled. This is calculated by dividing the voltage range of the A/D converter by 2 to the power of the number of bits of the A/D converter. For example, an A/D converter with a range of ±5 V with 12-bit resolution can resolve samples as small as ±2.4 mV. Appropriate matching of amplification and A/D converter sensitivity permits resolution of the smallest signal while preventing clipping of the largest signal amplitudes. The bandwidth of the filters and the rate of digitization determine the frequency components of interest that are passed, while other frequencies outside the band of interest that may represent potential artifacts, such as aliasing, are rejected. A filter’s characteristics are determined by the rate of the amplitude decrease at the bandwidth’s upper and lower edges. Proper digital representation of the analog signal depends on the rate of data sampling, which is governed by the Nyquist theorem that states that data sampling should be at least twice the highest frequency of interest. In addition to the information available from spontaneous electrical activity of the EEG, the brain’s electrical response to sensory stimulation can contribute data as to the status of cortical and subcortical regions activated by sensory input. Due to the relatively small amplitude of a stimulus-evoked potential as compared to the spontaneous EEG potentials, the technique of signal averaging is used to enhance the stimulus-evoked response. Stimulus averaging takes advantage of the fact that the brain’s electrical response is time-locked to the onset of the stimulus and the nonevoked background potentials are randomly distributed in time. Consequently, the average of multiple stimulus responses will result in the enhancement of the time-locked activity, while the averaged random background activity will approach zero. The result is an evoked response that consists of a number of discrete and replicable peaks that occur, depending upon the stimulus and the recording parameters, at predicted latencies from the onset of stimulation. The spatial localization of maximum peak amplitudes has been associated with cortical generators in primary sensory cortex. Instrumentation required for EEG recordings can be simple or elaborate [Kondraski, 1986]. (Note: Although the discussion presented in this section is for a single-channel system it can be extended to simultaneous multichannel recordings simply by multiplying the hardware by the number of channels required. In cases that do not require true simultaneous recordings, special electrode selector panels can minimize hardware requirements.) Any EEG system consists of electrodes, amplifiers (with appropriate filters) and a recording device
Commonly used scalp electrodes consist of Ag-AgCl disks, 1 to 3 mm in diameter, with a very flexible long lead that can be plugged into an amplifier. Although it is desirable to obtain a low-impedance contact at the lectrode ski interface(less than 10 kQ2), this objective is confounded by hair and the difficulty of mechanically stabilizing the electrodes Conductive electrode paste helps obtain low impedance and keep the electrodes in place A type of cement( collodion) is used to fix small patches of gauze over electrodes for mechanical stability, and leads are usually taped to the subject to provide some strain relief. Slight abrasion of the skin is sometimes sed to obtain better electrode impedances, but this can cause irritation and sometimes infection(as well as pain in sensitive subjects). For long-term recordings, as in seizure monitoring, electrodes present major problems. Needle electrodes, which must be inserted into the tissue between the surface of the scalp and skull, are sometimes useful. However, ne danger of infection increases significantly. Electrodes with self-contained miniature amplifiers are somewhat more tolerant because they provide a low-impedance source to interconnecting leads, but they are expensive Despite numerous attempts to simplify the electrode application process and to guarantee long-term stability none has been widely accepted Instruments are available for measuring impedance between electrode pairs. The procedure is recommended trongly as good practice, since high impedance leads to distortions that may be difficult to separate from actual EEG signals. In fact, electrode impedance monitors are built into some commercial devices for recording EEGs Standard dc ohmmeters should not be used, since they apply a polarizing current that causes build-up of noisy electrode potential at the skin-electrode interface. Commercial devices apply a known-amplitude sinusoidal voltage(typically 1 kHz) to an electrode pair circuit and measure root mean square(rms)current, which directly related to the magnitude of the impedance From carefully applied electrodes, signal amplitudes of I to 10 uV can be obtained Considerable amplification (gain = 106)is required to bring these levels up to an acceptable level for input to recording devices. Because of long electrode leads and the common electrically noisy environment where recordings take place, differential amplifiers with inherently high input impedance and high common mode rejection ratios are essential for high In some facilities, special electrically shielded rooms minimize environmental electrical noise, particularly 60-Hz alternating current(ac) line noise. Since much of the information of interest in the EEG lies in the frequency bands less than 40 Hz, low-pass filters in the amplifier can be switched into attenuate 60-Hz noise sharply. For attenuating ac noise when the low-pass cutoff is greater than 60 Hz, many EEG amplifiers have notch filters that attenuate only frequencies in a narrow band centered around 60 Hz. Since important signal infor mation may also be attenuated, notch filtering should be used as a last resort; one should try to identify and eliminate the source of interference instead In trying to identify 60-Hz sources to eliminate or minimize their effect, it is sometimes useful to use a lummy source, such as a fixed 100-kQ2 resistor attached to the electrodes. An amplifier output represents only contributions from interfering sources. If noise can be reduced to an acceptable level (at least by a factor of 10 less than EEG signals) under this condition, one is likely to obtain uncontaminated EEG records Different types of recording instruments obtain a temporary or permanent record of the eeg. The most ommon recording device is a pen or chart recorder(usually multichannel) that is an integral part of most commercially available EEG instruments. The bandwidth of clinical EEGs is relatively low (less than 40 Hz) and therefore within the frequency response capabilities of these devices. Recordings are on a long sheet of continuous paper(from a folded stack), fed past the moving pen at one of several selectable constant speed The paper speed translates into distance per unit time or cycles per unit time, to allow EEG interpreters to identify different frequency components or patterns within the EEG. Paper speed is selected according to the monitoring situation at hand: slow speeds(10 mm/s)for observing the spiking characteristically associat with seizures and faster speeds(up to 120 mm/s )for the presence of individual frequency bands in the EEG In addition to (or instead of)a pen recorder, the EEg may be recorded on a multichannel frequency modulated(FM) analog tape recorder. During such recordings, a visual output device such as an oscilloscope or video display is necessary to allow visual monitoring of signals, so that corrective action(reapplying the electrodes and so on) can take place immediately if necessary. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC Commonly used scalp electrodes consist of Ag-AgCl disks, 1 to 3 mm in diameter, with a very flexible long lead that can be plugged into an amplifier. Although it is desirable to obtain a low-impedance contact at the electrode ski interface (less than 10 kW), this objective is confounded by hair and the difficulty of mechanically stabilizing the electrodes. Conductive electrode paste helps obtain low impedance and keep the electrodes in place. A type of cement (collodion) is used to fix small patches of gauze over electrodes for mechanical stability, and leads are usually taped to the subject to provide some strain relief. Slight abrasion of the skin is sometimes used to obtain better electrode impedances, but this can cause irritation and sometimes infection (as well as pain in sensitive subjects). For long-term recordings, as in seizure monitoring, electrodes present major problems. Needle electrodes, which must be inserted into the tissue between the surface of the scalp and skull, are sometimes useful. However, the danger of infection increases significantly. Electrodes with self-contained miniature amplifiers are somewhat more tolerant because they provide a low-impedance source to interconnecting leads, but they are expensive. Despite numerous attempts to simplify the electrode application process and to guarantee long-term stability, none has been widely accepted. Instruments are available for measuring impedance between electrode pairs. The procedure is recommended strongly as good practice, since high impedance leads to distortions that may be difficult to separate from actual EEG signals. In fact, electrode impedance monitors are built into some commercial devices for recording EEGs. Standard dc ohmmeters should not be used, since they apply a polarizing current that causes build-up of noisy electrode potential at the skin-electrode interface. Commercial devices apply a known-amplitude sinusoidal voltage (typically 1 kHz) to an electrode pair circuit and measure root mean square (rms) current, which is directly related to the magnitude of the impedance. From carefully applied electrodes, signal amplitudes of 1 to 10 mV can be obtained. Considerable amplification (gain = 106 ) is required to bring these levels up to an acceptable level for input to recording devices. Because of long electrode leads and the common electrically noisy environment where recordings take place, differential amplifiers with inherently high input impedance and high common mode rejection ratios are essential for highquality EEG recordings. In some facilities, special electrically shielded rooms minimize environmental electrical noise, particularly 60-Hz alternating current (ac) line noise. Since much of the information of interest in the EEG lies in the frequency bands less than 40 Hz, low-pass filters in the amplifier can be switched into attenuate 60-Hz noise sharply. For attenuating ac noise when the low-pass cutoff is greater than 60 Hz, many EEG amplifiers have notch filters that attenuate only frequencies in a narrow band centered around 60 Hz. Since important signal information may also be attenuated, notch filtering should be used as a last resort; one should try to identify and eliminate the source of interference instead. In trying to identify 60-Hz sources to eliminate or minimize their effect, it is sometimes useful to use a dummy source, such as a fixed 100-kW resistor attached to the electrodes. An amplifier output represents only contributions from interfering sources. If noise can be reduced to an acceptable level (at least by a factor of 10 less than EEG signals) under this condition, one is likely to obtain uncontaminated EEG records. Different types of recording instruments obtain a temporary or permanent record of the EEG. The most common recording device is a pen or chart recorder (usually multichannel) that is an integral part of most commercially available EEG instruments. The bandwidth of clinical EEGs is relatively low (less than 40 Hz) and therefore within the frequency response capabilities of these devices. Recordings are on a long sheet of continuous paper (from a folded stack), fed past the moving pen at one of several selectable constant speeds. The paper speed translates into distance per unit time or cycles per unit time, to allow EEG interpreters to identify different frequency components or patterns within the EEG. Paper speed is selected according to the monitoring situation at hand: slow speeds (10 mm/s) for observing the spiking characteristically associated with seizures and faster speeds (up to 120 mm/s) for the presence of individual frequency bands in the EEG. In addition to (or instead of) a pen recorder, the EEG may be recorded on a multichannel frequency modulated (FM) analog tape recorder. During such recordings, a visual output device such as an oscilloscope or video display is necessary to allow visual monitoring of signals, so that corrective action (reapplying the electrodes and so on) can take place immediately if necessary
Sophisticated FM cassette recording and playback systems allow clinicians to review long EEG recordings over a greatly reduced time, compared to that required to flip through stacks of paper or observe recordings as they occur in real time. Such systems take advantage of time compensation schemes, whereby a signal recorded at one speed(speed of the tape moving past the recording head of the cassette drive)is played back at a different faster speed. The ratio of playback to recording speed is known, so the appropriate correction factor can be applied to played-back data to generate a properly scaled video display. A standard ratio of 60 1 is often used. Thus, a trained clinician can review each minute of real-time EEG in 1 s. The display appears to be scrolled at a high rate horizontally across the display screen. Features of these instruments allow the clinician to freeze segment of EEG on the display and to slow down or accelerate tape speed from the standard playback as needed. A time mark channel is usually displayed as one of the traces as a convenient reference(vertical tick "mark displayed at periodic intervals across the screen) Computers can also be recording devices, digitizing( converting to digital form) one or several amplified EEG channels at a fixed rate. In such sampled data systems, each channel is repeatedly sampled at a fixed time interval(sample interval)and this sample is converted into a binary number representation by an A/D converter The A/D converter is interfaced to a computer system so that each sample can be saved in the computer nemory. A set of such samples, acquired at a sufficient sampling rate(at least two times the highest frequency component in the sampled signal), is sufficient to represent all the information in the waveform. To ensure that the signal is band-limited, a low-pass filter with a cutoff frequency equal to the highest frequency of interest is used. Since physically realizable filters do not have the ideal characteristics, the sampling rate is usually greater than two times the filters cutoff frequency. Furthermore, once converted to a digital format, digital filtering es can On-line computer recordings are only practical for short-term recordings or for situations in which the EEG immediately processed. This limitation is primarily due rage requirements. For example, a typical sampling rate of 128 Hz yields 128 new samples per second that require storage. For an 8-channel recording, ,024 samples are acquired per second. A 10-minute recording period yields 614, 400 data points. Assuming 8- bit resolution per sample, over 0.5 megabyte(MB)of storage is required to save the 10-minute recording Processing can consist of compression for more efficient storage(with associated loss of total information content), as in data record or epoch averaging associated with evoked responses, or feature extraction and subsequent pattern recognition, as in automated spike detection in seizure monitoring. Frequency analysis of the EEG In general, the EEG contains information regarding changes in the electrical pot of the brain obtained om a given set of recording electrodes. These data include the characteristic waveform with its variation in amplitude, frequency, phase, etc and the occurrence of brief electrical patterns, such as spindles. Any analysis procedure cannot simultaneously provide information regarding all of these variables. Consequently, the selec tion of any analytic technique will emphasize changes in one particular variable at the expense of the others This observation is extremely important if one is to properly interpret the results obtained by any analytic chnique. In this chapter, special attention is given to frequency analysis of the EEG In early attempts to correlate the EEG with behavior, analog frequency analyzers were used to examine single channels of EEG data. Although disappointing, these initial efforts did introduce the utilization of frequency analysis to study gross brain wave activity. Although, power spectral analysis, i.e., the magnitude square of Fourier transform, provides a quantitative measure of the frequency distribution of the EEG, it does so as mentioned above, at the expense of other details in the EEg such as the amplitude distribution, as well as the presence of specific patterns in the EEG The first systematic application of power spectral analysis by general-purpose computers was reported in 1963 by Walter; however, it was not until the introduction of the fast Fourier transform(FFT) by Cooley and Tukey in the early 1970s that machine computation of the EEG became commonplace. Although an individual FFT is ordinarily calculated for a short section of EEG data(e.g, from 1 to 8 s epoch), such segmentation of a signal with subsequent averaging over individual modified periodograms has been shown to provide a consistent estimator of the power spectrum, and an extension of this technique, the compressed spectral array, has been particularly useful for computing EEG spectra over long periods of time. A detailed review of the c2000 by CRC Press LLC
© 2000 by CRC Press LLC Sophisticated FM cassette recording and playback systems allow clinicians to review long EEG recordings over a greatly reduced time, compared to that required to flip through stacks of paper or observe recordings as they occur in real time. Such systems take advantage of time compensation schemes, whereby a signal recorded at one speed (speed of the tape moving past the recording head of the cassette drive) is played back at a different, faster speed. The ratio of playback to recording speed is known, so the appropriate correction factor can be applied to played-back data to generate a properly scaled video display. A standard ratio of 60:1 is often used. Thus, a trained clinician can review each minute of real-time EEG in 1 s. The display appears to be scrolled at a high rate horizontally across the display screen. Features of these instruments allow the clinician to freeze a segment of EEG on the display and to slow down or accelerate tape speed from the standard playback as needed. A time mark channel is usually displayed as one of the traces as a convenient reference (vertical “tick” mark displayed at periodic intervals across the screen). Computers can also be recording devices, digitizing (converting to digital form) one or several amplified EEG channels at a fixed rate. In such sampled data systems, each channel is repeatedly sampled at a fixed time interval (sample interval) and this sample is converted into a binary number representation by an A/D converter. The A/D converter is interfaced to a computer system so that each sample can be saved in the computer’s memory. A set of such samples, acquired at a sufficient sampling rate (at least two times the highest frequency component in the sampled signal), is sufficient to represent all the information in the waveform. To ensure that the signal is band-limited, a low-pass filter with a cutoff frequency equal to the highest frequency of interest is used. Since physically realizable filters do not have the ideal characteristics, the sampling rate is usually greater than two times the filter’s cutoff frequency. Furthermore, once converted to a digital format, digital filtering techniques can be used. On-line computer recordings are only practical for short-term recordings or for situations in which the EEG is immediately processed. This limitation is primarily due to storage requirements. For example, a typical sampling rate of 128 Hz yields 128 new samples per second that require storage. For an 8-channel recording, 1,024 samples are acquired per second. A 10-minute recording period yields 614,400 data points. Assuming 8- bit resolution per sample, over 0.5 megabyte (MB) of storage is required to save the 10-minute recording. Processing can consist of compression for more efficient storage (with associated loss of total information content), as in data record or epoch averaging associated with evoked responses, or feature extraction and subsequent pattern recognition, as in automated spike detection in seizure monitoring. Frequency Analysis of the EEG In general, the EEG contains information regarding changes in the electrical potential of the brain obtained from a given set of recording electrodes. These data include the characteristic waveform with its variation in amplitude, frequency, phase, etc. and the occurrence of brief electrical patterns, such as spindles. Any analysis procedure cannot simultaneously provide information regarding all of these variables. Consequently, the selection of any analytic technique will emphasize changes in one particular variable at the expense of the others. This observation is extremely important if one is to properly interpret the results obtained by any analytic technique. In this chapter, special attention is given to frequency analysis of the EEG. In early attempts to correlate the EEG with behavior, analog frequency analyzers were used to examine single channels of EEG data. Although disappointing, these initial efforts did introduce the utilization of frequency analysis to study gross brain wave activity. Although, power spectral analysis, i.e., the magnitude square of Fourier transform, provides a quantitative measure of the frequency distribution of the EEG, it does so as mentioned above, at the expense of other details in the EEG such as the amplitude distribution, as well as the presence of specific patterns in the EEG. The first systematic application of power spectral analysis by general-purpose computers was reported in 1963 by Walter; however, it was not until the introduction of the fast Fourier transform (FFT) by Cooley and Tukey in the early 1970s that machine computation of the EEG became commonplace. Although an individual FFT is ordinarily calculated for a short section of EEG data (e.g., from 1 to 8 s epoch), such segmentation of a signal with subsequent averaging over individual modified periodograms has been shown to provide a consistent estimator of the power spectrum, and an extension of this technique, the compressed spectral array, has been particularly useful for computing EEG spectra over long periods of time. A detailed review of the
development and use of various methods to analyze the EEG is provided by Givens and Redmond [1987 Figure 115.3 provides an overview of the computational processes involved in 28 performing spectral analysis of the EEG, i.e, including computation of auto and cross spectra [Bronzino, 1984. It is to be noted that the power spectrum is the utocorrellogram, i.e. the correlation of the signal with itself. As a result, the power CALCULATE spectrum provides only magnitude information in the frequency domain; it does RAW SPECTRA not provide any data regarding phase. The power spectrum is computed by: P(=Re2X(]+Im2[X(0) (1151) CALCULATE POWER SPECTRAL DENSITY where X() is the Fourier transform of the EEG. Power spectral analysis not only provides a summary of the EEG in a convenient graphic form, but also facilitates statistical analysis of EEG changes which may not ALCULATE be evident on simple inspection of the records. In addition to absolute power CROSS SPECTRA derived directly from the power spectrum, other measures calculated from absolute power have been demonstrated to be of value in quantifying various aspects of the <EG. Relative power expresses the percent contribution of each frequency band to SMOOTH DATA total power and is calculated by dividing the power within a band by the total power across all bands Relative power has the benefit of reducing the intersubject variance associated with absolute power that arises from intersubject differences in skull and scalp conductance. The disadvantage of relative power is that an increase in one frequency band will be reflected in the calculation by a decrease in CALCULATE COHERENCE other bands; for example, it has been reported that directional shifts between high and low frequencies are associated with changes in cerebral blood flow and metab olism. Power ratios between low(0-7 Hz) and high(10-20 Hz) frequency bands CALCULATE PHASE SHIFT have been demonstrated to be an accurate estimator of changes in cerebral activity during these metabolic changes. Although the power spectrum quantifies activity at each electrode, other vari- FIGURE 115.3 Block dia. ables derivable from FFT offer a measure of the relationship between activity gram of measures determined recorded at distinct electrode sites. Coherence( which is a complex number), cal- from spectral analysis. culated from the cross-spectrum analysis of two signals, is similar to cross-corre- lation in the time domain. The magnitude squared coherence(MSC) values range from 1 to 0, indicating maximum or no synchrony, respectively, and are independent of power. The temporal relationship between two signals is expressed by phase, which is a measure of the lag between two signals for common frequency components or bands. Phase is expressed in units of degrees, 0o indicating no time lag between signals or 180 if the signals are of opposite polarity. Phase can also be transformed into the time domain, giving a measure of the time difference between two frequencies Cross spectrum is computed by Cross spectrum=X(r where X(, Y) are Fourier transforms and* indicates complex conjugates and coherence is calculated by Coherence Cross spectrum (115.3) PX()-PY) Since coherence is a complex number, the phase is simply the angle associated with the polar expression of that number. MSC and phase represent measures that can be employed to investigate the cortical interactions of cerebral activity. For example, short(intracortical)and long( cortico-cortical) pathways have been proposed e 2000 by CRC Press LLC
© 2000 by CRC Press LLC development and use of various methods to analyze the EEG is provided by Givens and Redmond [1987]. Figure 115.3 provides an overview of the computational processes involved in performing spectral analysis of the EEG, i.e., including computation of auto and cross spectra [Bronzino, 1984]. It is to be noted that the power spectrum is the autocorrellogram, i.e., the correlation of the signal with itself.As a result, the power spectrum provides only magnitude information in the frequency domain; it does not provide any data regarding phase. The power spectrum is computed by: P(f) = Re2[X(f)] + Im2[X(f)] (115.1) where X(f) is the Fourier transform of the EEG. Power spectral analysis not only provides a summary of the EEG in a convenient graphic form, but also facilitates statistical analysis of EEG changes which may not be evident on simple inspection of the records. In addition to absolute power derived directly from the power spectrum, other measures calculated from absolute power have been demonstrated to be of value in quantifying various aspects of the EEG. Relative power expresses the percent contribution of each frequency band to the total power and is calculated by dividing the power within a band by the total power across all bands. Relative power has the benefit of reducing the intersubject variance associated with absolute power that arises from intersubject differences in skull and scalp conductance. The disadvantage of relative power is that an increase in one frequency band will be reflected in the calculation by a decrease in other bands; for example, it has been reported that directional shifts between high and low frequencies are associated with changes in cerebral blood flow and metabolism. Power ratios between low (0–7 Hz) and high (10–20 Hz) frequency bands have been demonstrated to be an accurate estimator of changes in cerebral activity during these metabolic changes. Although the power spectrum quantifies activity at each electrode, other variables derivable from FFT offer a measure of the relationship between activity recorded at distinct electrode sites. Coherence (which is a complex number), calculated from the cross-spectrum analysis of two signals, is similar to cross-correlation in the time domain. The magnitude squared coherence (MSC) values range from 1 to 0, indicating maximum or no synchrony, respectively, and are independent of power. The temporal relationship between two signals is expressed by phase, which is a measure of the lag between two signals for common frequency components or bands. Phase is expressed in units of degrees, 0° indicating no time lag between signals or 180° if the signals are of opposite polarity. Phase can also be transformed into the time domain, giving a measure of the time difference between two frequencies. Cross spectrum is computed by: Cross spectrum = X(f) Y*(f) (115.2) where X(f), Y(f) are Fourier transforms and * indicates complex conjugates and coherence is calculated by (115.3) Since coherence is a complex number, the phase is simply the angle associated with the polar expression of that number. MSC and phase represent measures that can be employed to investigate the cortical interactions of cerebral activity. For example, short (intracortical) and long (cortico-cortical) pathways have been proposed Coherence = Cross spectrum PX( )f - PY(f ) FIGURE 115.3 Block diagram of measures determined from spectral analysis
as the anatomic substrate underlying the spatial frequency and patterns of coherence. Therefore, discrete cortical regions linked by such fiber systems should demonstrate a relatively high degree of synchrony, whereas the time lag between signals, as represented by phase, quantifies the extent to which one signal leads another Nonlinear Analysis of the EEG As mentioned earlier, the EEG has been studied extensively using signal-processing schemes, most of which are based on the assumption that the Eeg is a linear, gaussian process. Although linear analysis schemes are computationally efficient and useful, they only utilize information retained in the autocorrelation function(ie the second-order cumulant). Additional information stored in higher-order cumulants is therefore ignored by linear analysis of the EEG. Thus, while the power spectrum provides the energy distribution of a stationary process in the frequency domain, it cannot distinguish nonlinearly coupled frequencies from spontaneously generated signals with the same resonance condition [Nikias and Raghvveer, 1987] There is evidence showing that the amplitude distribution of the EEg often deviates from gaussian behavior. It has been reported, for example, that the EEG of humans involved in the performance of mental arithmetic ask exhibits significant nongaussian behavior. In addition, the degree of deviation from gaussian behavior of the EEG has been shown to depend to the behavioral state, with the state of slow-wave sleep showing less gaussian behavior than quiet waking, which is less gaussian than rapid eye movement(REM)sleep [Ning and Bronzino, 1989, b]. Nonlinear signal-processing algorithms such as bispectral analysis are therefore necessary to address nongaussian and nonlinear behavior of the EEG in order to better describe it in the frequency domain. But what exactly is the bispectrum? For a zero-mean, stationary process (X(K)l, the bispectrum, by definition, is the Fourier transform of its third-order cumulant(TOC)sequence B(a,02)=∑∑(mn)m 朋三一n三一 The TOC sequence C(m, n)) is defined as the expected value of the triple product C(m, m)=x(k)(k+ m x(k+ n) (115.5) If process X(k)is purely gaussian, then its third-order cumulant C(m, n)is zero for each(m, n), and consequently, Fourier transform, the bispectrum, B(O,, o, )is also zero. This property makes the estimated bispectrum an immediate measure describing the degree of deviation from gaussian behavior. In our studies [Ning and Bronzino, 1989, b], the sum of magnitude of the estimated bispectrum was used as a measure to describe the EEG's deviation from gaussian behavior, that is, D=∑, (o @ 2 Using bispectral analysis, the existence of significant quadratic phase coupling(QPC) in the hippocampa EEG obtained during REM sleep in the adult rat was demonstrated [Ning and Bronzino, 1989a, b, 1990]. The result of this nonlinear coupling is the appearance, in the frequency spectrum, of a small peck centered at approximately 13 to 14 Hz(beta range) that reflects the summation of the two theta frequency(i.e, in the 6- 7-Hz range)waves. Conventional power spectral (linear)approaches are incapable of distinguishing the fact that this peak results from the interaction of these two generators and is not intrinsic to either. To examine the phase relationship between nonlinear signals collected at different sites, the cross-bispectrum is also a useful tool. For example, given three zero-mean, stationary processes"x(n)j=1, 2, 31, there are two nventional methods for determining the cross-bispectral relationship, direct and indirect. Both methods first divide these three processes into M segments of shorter but equal length. The direct method computes the c2000 by CRC Press LLC
© 2000 by CRC Press LLC as the anatomic substrate underlying the spatial frequency and patterns of coherence. Therefore, discrete cortical regions linked by such fiber systems should demonstrate a relatively high degree of synchrony, whereas the time lag between signals, as represented by phase, quantifies the extent to which one signal leads another. Nonlinear Analysis of the EEG As mentioned earlier, the EEG has been studied extensively using signal-processing schemes, most of which are based on the assumption that the EEG is a linear, gaussian process. Although linear analysis schemes are computationally efficient and useful, they only utilize information retained in the autocorrelation function (i.e., the second-order cumulant). Additional information stored in higher-order cumulants is therefore ignored by linear analysis of the EEG. Thus, while the power spectrum provides the energy distribution of a stationary process in the frequency domain, it cannot distinguish nonlinearly coupled frequencies from spontaneously generated signals with the same resonance condition [Nikias and Raghvveer, 1987]. There is evidence showing that the amplitude distribution of the EEG often deviates from gaussian behavior. It has been reported, for example, that the EEG of humans involved in the performance of mental arithmetic task exhibits significant nongaussian behavior. In addition, the degree of deviation from gaussian behavior of the EEG has been shown to depend to the behavioral state, with the state of slow-wave sleep showing less gaussian behavior than quiet waking, which is less gaussian than rapid eye movement (REM) sleep [Ning and Bronzino, 1989a,b]. Nonlinear signal-processing algorithms such as bispectral analysis are therefore necessary to address nongaussian and nonlinear behavior of the EEG in order to better describe it in the frequency domain. But what exactly is the bispectrum? For a zero-mean, stationary process {X(k)}, the bispectrum, by definition, is the Fourier transform of its third-order cumulant (TOC) sequence: (115.4) The TOC sequence {C(m, n)} is defined as the expected value of the triple product (115.5) If process X(k) is purely gaussian, then its third-order cumulant C(m, n) is zero for each (m, n), and consequently, its Fourier transform, the bispectrum, B(w1, w2) is also zero. This property makes the estimated bispectrum an immediate measure describing the degree of deviation from gaussian behavior. In our studies [Ning and Bronzino, 1989a,b], the sum of magnitude of the estimated bispectrum was used as a measure to describe the EEG’s deviation from gaussian behavior, that is, (115.6) Using bispectral analysis, the existence of significant quadratic phase coupling (QPC) in the hippocampal EEG obtained during REM sleep in the adult rat was demonstrated [Ning and Bronzino, 1989a,b, 1990]. The result of this nonlinear coupling is the appearance, in the frequency spectrum, of a small peck centered at approximately 13 to 14 Hz (beta range) that reflects the summation of the two theta frequency (i.e., in the 6- to 7-Hz range) waves. Conventional power spectral (linear) approaches are incapable of distinguishing the fact that this peak results from the interaction of these two generators and is not intrinsic to either. To examine the phase relationship between nonlinear signals collected at different sites, the cross-bispectrum is also a useful tool. For example, given three zero-mean, stationary processes ”xj (n)j = 1, 2, 3}, there are two conventional methods for determining the cross-bispectral relationship, direct and indirect. Both methods first divide these three processes into M segments of shorter but equal length. The direct method computes the B Cm ne j wm wn n a m a w w aa 1 2 1 2 ( , , ) = ( ) - + ( ) = -= - ÂÂ Cm n E XkXk mXk n ( , ) = { } ( ) ( + ) ( + ) D B = ( ) ( ) Â w w w w 1 2 1 2
Fourier transform of each segment for all three processes and then estimates the cross-bispectrum by taking the average of triple products of Fourier coefficients over M segments, that is ∑x(o)xo)x"on+2) (1157) x12 3 where X (o) is the Fourier transform of the mth segment of x(n)l, and*indicates the complex conjugate. The indirect method computes the third-order cross-cumulant sequence for all segments Cm(=∑增小)(m+A)r(+ (1158) where t is the admissible set for argument n. The cross-cumulant sequences of all segments will be averaged to give a resultant estimate k,1) (1159) xrx The cross-bispectrum is then estimated by taking the Fourier transform of the third-order cross-cumulant (,1) (115.10) Since the variance of the estimated cross-bispectrum is inversely proportional to the length of each segment omputation of the cross-bispectrum for processes of finite data length requires careful consideration of both the length of individual segments and the total number of segments to be used. The cross-bispectrum can be applied to determine the level of cross-QPC occurring between x,(n) and Ix2(n)) and its effects on x(n). For example, a peak at Bx, x2*(O1,o2) suggests that the energy component O2 of [,(n) is generated due to the QPC betw of x(n)). In theory, the absence of QPC will generate a flat cross-bispectrum. However, due to the finite data length encountered in practice, peaks may appear in the cross-bispectrum at locations where there is no significant cross-QPC. To avoid improper interpretation, the cross-bicoherence index, which indicates the significance level of cross-QPC, can be computed as follows: (115.11) P(,P(OP(o, +02 where Pr (o) is the power spectrum of process x(n)). The theoretical value of the bicoherence index ranges In situations where the interest is the presence of QPC and its effects on ix( n)l, the cross-bispectrul equations can be modified by replacing Ix(n)) and x(n)) with (x(n)) and (x,(n)I with n(n)b, that is, Bn(o,0)=∑xo)y"(o) X"(o1+0 (115.12) e 2000 by CRC Press LLC
© 2000 by CRC Press LLC Fourier transform of each segment for all three processes and then estimates the cross-bispectrum by taking the average of triple products of Fourier coefficients over M segments, that is, (115.7) where Xj m(w) is the Fourier transform of the mth segment of {xj (n)}, and * indicates the complex conjugate. The indirect method computes the third-order cross-cumulant sequence for all segments: (115.8) where t is the admissible set for argument n. The cross-cumulant sequences of all segments will be averaged to give a resultant estimate: (115.9) The cross-bispectrum is then estimated by taking the Fourier transform of the third-order cross-cumulant sequence: (115.10) Since the variance of the estimated cross-bispectrum is inversely proportional to the length of each segment, computation of the cross-bispectrum for processes of finite data length requires careful consideration of both the length of individual segments and the total number of segments to be used. The cross-bispectrum can be applied to determine the level of cross-QPC occurring between {x1(n)} and {x2(n)} and its effects on {x3(n)}. For example, a peak at Bx1x2x3(w1, w2) suggests that the energy component at frequency w1 + w2 of {x3(n)} is generated due to the QPC between frequency w1 of {x1(n)} and frequency w2 of {x2(n)}. In theory, the absence of QPC will generate a flat cross-bispectrum. However, due to the finite data length encountered in practice, peaks may appear in the cross-bispectrum at locations where there is no significant cross-QPC. To avoid improper interpretation, the cross-bicoherence index, which indicates the significance level of cross-QPC, can be computed as follows: (115.11) where Pxj(w) is the power spectrum of process {xj (n)}. The theoretical value of the bicoherence index ranges between 0 and 1, i.e., from nonsignificant to highly significant. In situations where the interest is the presence of QPC and its effects on {x(n)}, the cross-bispectrum equations can be modified by replacing {x1(n)} and {x3(n)} with {x(n)} and {x2(n)} with {y(n)}, that is, (115.12) B M xxx XXX m m M m m 123 12 1 1 12 23 1 2 1 ww w w w w , * ( ) = ( ) ( ) ( + ) = Â C k l x nx n kx n l xxx m m n m m 123 12 3 ( , ) = Â ( ) ( + ) ( + ) et C kl M C kl xxx xxx m m M 123 123 1 1 ( , , ) = ( ) = Â B C kl xxx xxx jk l lk 123 123 1 2 w w 1 2 w w a a a a ( , , ) = ( ) - + ( ) = -= - ÂÂ bic B PP P xxx xxx xx x 123 123 12 3 1 2 1 2 1 2 12 w w w w w w ww , , ( ) = ( ) ( ) ( ) ( + ) B M xyz XY X m m M m m ww w w w w 12 1 1 2 12 1 , * ( ) = ( ) ( ) ( + ) = Â
