THE FRACTAL TIME BEHAVIOR OF SPONTANEOUS PERINATAL BEHAVIORS ASSOCIATED WITH REM SLEEP: A POSSIBLE ONTOGENETIC ADAPTATION AND SOURCE OF PLASTICITY UNDERLYING THE EMERGENCE OF BEHAVIORAL NEOPHENOTYPES
by
Carl M. Anderson
A Dissertation Submitted to the Faculty of
The College of Science
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
Florida Atlantic University
Boca Raton, Florida
December 1995
Copyright by Carl M. Anderson 1995
This dissertation was prepared under the direction of the candidate's dissertation advisor, Dr. Leslie M. Terry, Department of Psychology and has been approved by the members of his supervisory committee. It was submitted to the faculty of the College of Science and was accepted in partial fulfillment of the requirements for the degree of doctor of philosophy.
SUPERVISORY COMMITTEE:
___________________________________
Dr. Leslie M. Terry
Chairperson of dissertation committee
___________________________________
Dr. Arnold J. Mandell
___________________________________
Dr. Ingrid B. Johanson
_______________________________ ___________________________________
Chairperson, Department of Psychology Dr. Larry S. Liebovitch
_______________________________ ___________________________________
Dean, College of Science Dr. Timothy J. Iverson
_______________________________ ___________________________________
Dean of Graduate Studies and Research Date
Acknowledgements
The work and concepts expressed in this dissertation owe a great debt in large part to hundreds of people. I attempt here to spontaneously list and describe then in the order in which they occur to me. To Mary, my wife, I owe more than I say for all of her love, cheer, motivation, intellectual stimulation and unsurpassed editorial guidance, without you nothing would exist. To Corinna, my daughter, I owe the inspiration of her laughter and the joy of her spontaneity. I also thank my parents, Corrine, Gene and my brother Arthur Anderson for their love, support and uncritical acceptance through all the long years. I thank my in-laws, Margaret and Carl Pfeiffer, Jim Kolodny and Susan Thompson for all of their support and understanding throughout my endless education.
I would also like to acknowledge a huge debt to Dr. Leslie M. Terry, my longtime friend, chairperson and coadvisor who took me under her wing at the 87 Neuroscience in Dallas, to my first ISDP meeting, to dinner after my poster and has been unselfish in her support ever since. Thanks for letting me go my own way. To Dr. Clinton "Clint" Kilts, I owe the excellent mentorship and friendship and encouragement to proceed with my scientific career. To my coadvisor Dr. Arnold Mandell, for letting me play 1/f my way in your band and really listening. Arnold, thanks for all of your hard work over the years, particularly "Toward a neuropharmacology of habituation: a vertical integration" which inspired me to have that long talk with you and Karen at Santa Fe and to come to FAU and work with you. Thanks also for creating the "Bagels and cream cheese seminar in mathematical neuroscience and psychology" by which the early intuitions embodied in my 1992 talk "1/f noise in rapid eye movement sleep a developmental hypothesis" caught the flame that burns brightly to this day. Many thanks also to Dr. Karen Selz for her skill with cartoons and Lévy fits. Also I would like to thank all those in the Department of Psychology and the Center for Complex Systems for your help all these years. Among them, Bill McLean for your courage and help with the activity monitor, Tom and Kieko Holroyd and Colin Brown for my hours of friendship and tutoring. Also Pam Case, Gene Wallenstein, John Buchanan, Paul Treffner, Sue and Betty for your friendship. Special thanks to Dr. Larry Liebovitch for teaching me Hurst analysis and helping with the programming. Also to Alison Tannenbaum, thanks for your help in securing the Grass 7P511 and the Metaphane. Thanks also to Mingzhou Ding and Steve Bressler for the excellent courses and conversation. Thanks again to Dr. Ted Hall who put me on course to Dr. Ingrid Johanson's office and FAU. Thanks to Ingrid for her instruction and the generous use of the lab and materials. Thanks also to Dr. Tim Iverson for many helpful discussions about kids and how this work may benefit them. Thank you Jane Roberts and Rob Butts for your efforts and the idea that spontaneous activity informs conscious action. Last but not least, thanks to Dr. Bill Smotherman for his friendly comments and the Binghamton and Cornell groups for having the foresight to collect the data that comprises a sizable part of this dissertation, and generosity in allowing my access to it.
ABSTRACT
Rapid Eye Movement (REM) sleep in adult and neonatal mammals is characterized by episodes of high variability and bursting in brainstem sites associated with spontaneous tonic and phasic behavioral events such as REMs, nuchal inactivity and twitches of the body. REM sleep is the principal behavioral state during fetal and neonatal life and as has been demonstrated by various REM deprivaiton proceedures to be indispensable during this period and to lead to long lasting behavioral defects in adult life. The guiding hypothesis throughout this dissertation is that the variability of REM-associated nuchal atonia episodes and other spontanous motor events reflects the fractal time patterns of a global fetal REM sleep state over mutiple timescales serving as a transient behavioral ontogenetic adaptation to changing developmental environments. Further, spontanous activity over many levels of
organization, including phasic REM motor activity during ontogeny, could play a fundamental role in the development of appetitive behavioral processes (e.g., searching and orienting) and other forms of neuroplasticity (e.g., learning and dynamic regulation of receptor fields and maps). The nature of this variability was investigated by measuring the durations of nuchal atonia over extended periods in fetal sheep and neonatal rats, species which are in a REM sleep-like state > 50% of the time. Hurst's rescaled range analysis, which affords comparisons between natural time series with short- and long-term correlated fluctuations, indicated that variability in both species over short time scales is statistically similar to longer time scales (i.e., is fractal in time) and remarkably stable over the developmental periods examined. Spontaneous nuchal events in both species were also found to be described by convolutionally stable self-similar Lévy distributions, suggesting that activity associated with fetal REM sleep could provide a stable, scale invariant source of correlated stimulation, facilitating integration of new neural changes into developing motor and cortical networks over gestation. These fractal time descriptions of spontaneous prenatal behaviors have implications for conceptualizing the evolutionary mechanisms underlying heterochrony (shifting self-affine relationships between the timing of gene expression and behavioral activity) and the plasticity essential to the genesis of behavioral neophenotypes.
To Mary, Corinna And The World As Dream
Table of Contents
I. INTRODUCTION..........................................................................1
FRACTAL TIME AND SPONTANEOUS PRENATAL BEHAVIOR.....4
REM SLEEP IN FETUSES AND NEONATES AS AN
ONTOGENETIC ADAPTATION..............................................16
II. EXPERIMENT 1: ANALYSIS OF NUCHAL ATONIA SEQUENCES IN FETAL SHEEP: AN EXAMPLE OF THE FRACTAL TIME STRUCTURE OF A SPONTANEOUS PRENATAL BEHAVIOR.........................................21
SPECIFIC METHODS..........................................................22
RESULTS.........................................................................29
DISCUSSION....................................................................52
III. EXPERIMENT 2: ANALYSIS OF NUCHAL INACTIVITY SEQUENCES IN NEONATAL RATS: AN EXAMPLE OF THE FRACTAL TIME STRUCTURE OF A SPONTANEOUS POSTNATAL BEHAVIOR.................................60
SPECIFIC METHODS..........................................................73
RESULTS.........................................................................75
DISCUSSION...................................................................108
IV. GENERAL DISCUSSION: ORDER AND RANDOMESS IN BIOLOGICAL SYSTEMS AND SPONTANEOUS NEONATAL BEHAVIORS.................114
THE ORIGIN AND FUNCTION OF FRACTAL TIME PATTERNS ASSOCIATED WITH FETAL REM SLEEP: A TRANSIENT BEHAVIORAL ONTOGENETIC ADAPTATION TO CHANGING ENVIRONMENTS?.......117
FRACTAL TIME, FETAL REM SLEEP, HETEROCHRONY AND THE GENESIS OF BEHAVIORAL NEOPHENOTYPES................................129
THE FRACTAL TIME NATURE OF SPONTANEOUS PRENATAL BEHAVIORS: PARALLELS WITH CRITICAL POINT FLUCTUATIONS AND OTHER COMPLEX SELF-ORGANIZING SYSTEMS............................133
THE FRACTAL TIME ORIGINS OF SPONTANEOUS APPETITIVE BEHAVIORS: REM SLEEP, NEURAL PLASTICITY AND THE ORIENTING RESPONSE...............................................................................137
THE RETICULAR FORMATION, REM SLEEP, ORIENTING AND DEVELOPMENT: A GUIDING IMAGE.............................................139
THE UTILITY OF FRACTAL DESCRIPTIONS OF SPONTANEOUS PRENATAL BEHAVIOR ..............................................................165
V. CONCLUSIONS.........................................................................167
VI. REFERENCES...........................................................................170
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INTRODUCTION
Intermittent spontaneous prenatal behavior has long been recognized as an integral component of fetal development (Coghill, 1929a; 1929b; Corner, 1977; 1978; 1990; 1994; Hamburger, 1963; 1973; Hall and Oppenheim, 1987; Preyer, 1885; Robinson and Smotherman, 1988) and has been proposed as a transient ontogenetic adaptation to the in ovo or in utero environment (Oppenheim, 1981; 1984). Wihelm Preyer, in his monumental 1885 work Specielle Physiologie des Embryo. Untersuchunge üher die Lebensersc heinungen vor der Geburt , "Special physiology of the Embryo: Investigations of the Phenomena of Life before Birth," called these spontaneous movements "impulsive movements" because of the distinctiveness of their uncoordinated, aimless, seemingly nonadaptive nature (Gottlieb, 1973). The underlying motivation behind much of Preyer's developmental work was the problem of "psychogenesis" or the genesis of mind. Through his motor primacy theory, Preyer proposed a central role for these impulsive, autogenous movements in physiogenesis and ultimately in psychogenesis.
George Coghill, who greatly admired Preyer as a behavioral embryologist, translated and republished portions of Specielle Physiologie des Embryo, and in a similar fashion sought to understand the nature of spontaneity and its relation to the development of more complex behaviors. In a philosophical dialogue with the famous neuroanatomist C. Judson Herrick (Herrick, 1948), Coghill refers to the "integration and spontaneity that can be recognized throughout both the organic and the inorganic realms (p.222)" as the source of mind or psychogenesis in Preyer's terms. Coghill further clarifies the roots of his Gestalt view of development of embryonic motility in Amblystome as the "progressive
expansion of a perfectly integrated total pattern and the individuation within it....[of] degrees of discreteness (Coghill, 1929, p. 38)." In a further quote from this dialogue: "This thing that we agree upon is that integrative experience, as you call it, and mentation, as I call it, is a total-pattern type of activity common to all organisms and that there has been progressive expansion and individuation of this pattern in evolutionary history and in personal development (Herrick, 1948, p. 216)."
In this dissertation I will attempt to reanimate the germs of these ideas with the concepts of fractal geometry by demonstrating that the same distinctive spontaneous perinatal activities perceived by Preyer and Coghill, which lack a unique measurement scale in time or space in which to fully capture their true nature, can be described as fundamental biological processes intrinsically associated with adult REM sleep. It is my contention, and the subtext of this dissertation, that the true wellspring of creative spontaneity and cognitive flexibility that serves as the foundation of consciousness has its source in the fractal time nature of recurrent phasic REM processes over a lifetime. And furthermore, that the rich creativity, diversity, adaptability and impulsivity of vertebrate forms result from what Coghill describes as "....a perfectly integrated total [fractal] pattern " of interactions over time and space that is first manifested in spontaneous perinatal behaviors.
Endogenously generated spontaneous activity, as opposed to evoked activity, is common to all vertebrate embryos, larvae and fetuses and it is associated with emergent behavioral and neurobiological processes from which adult REM originates (Blumberg and Lucas, 1994b; in press; Corner, 1978; 1990; Mirmiran, 1986) although true autonomous activity is only observed in fish and bird embryos (Hamburger, 1973). This spontaneous activity, whether movements (Coghill, 1929a; 1929b; Corner, 1977; Dawes, 1988; Oppenheim, 1982), neuronal activity (Corner and Crain, 1972; Corner, 1994; Ripley and Provine, 1972; Sharma, Provine, Hamburger, Sandel, 1970) or state changes (Prechtl, 1990) is often described as a sequence of independent random events lacking the coordination found in adult species-typical motor patterns (Bekoff and Lau, 1980; Blumberg and Lucas, 1994b; Corner, 1978; Gottlieb, 1976; Hamburger, 1973; Narayanan et al., 1970; Oppenheim, 1981; Windle, 1944; Robinson and Smotherman, 1988). Implicitly, species-typical adult behavior is conceptualized as originating from "...the activation of a neuronal circuitry that to a large extent has its origins in a basic neurobiological substrate laid down in the embryo and whose functional manifestation is spontaneous prenatal behavior...; Hall and Oppenheim, 1987, p.96." If this theoretical assumption is correct, how can this ontogenetic organizing capability of apparently uncorrelated random prenatal behavior be identified and characterized? Can the biological process giving rise to spontaneous prenatal behavior be a truly independent random process if spontaneous prenatal behavior arises over many scales of space and time in the fetus and ultimately results in global behavioral patterns over a range of space and time scales? Why do fetuses spend most of their time and energy in a state similar to adult REM sleep (Nijhuis, Martin and Prechtl, 1984; Richardson, 1991; Roffwarg, Muzio and Dement, 1966) if it only results in the spontaneous generation of independent random sequences of behavior? This dissertation attempts a comparative approach to investigating the time structure of spontaneous prenatal behavior and its inferred role in development. By demonstrating the recurrence of invariant temporal patterns in spontaneous prenatal behavior of the sheep fetus or the postnatal rat, I am suggesting that "impulsive" spontaneous behaviors need not be conceptualized as independent random processes. Perhaps the ontogenetic organizing capability of spontaneous prenatal behavior resides in a subtle attribute of its time structure.
FRACTAL TIME AND SPONTANEOUS PRENATAL BEHAVIOR
Many apparently random processes in nature have been demonstrated to have fractal order in space and time (Mandelbrot, 1982; 1983). If spontaneous prenatal behavior is not an independent random process but instead is correlated over many time scales (i.e, is fractal in time), then the intra-organism correlations provided by spontaneous prenatal behavior may illustrate a previously unrecognized form of non-cyclic, non-mode-locked coherence in time-dependent developmental systems (Mandell and Shlesinger, 1990; Smotherman, Seltz and Mandell, in press) which could represent a transient behavioral ontogenetic adaptation to the fetal environment and a substrate for the generation of novel behavioral pheotypes.
What is fractal time? This statistical concept is perhaps best described within the context of determining characteristic times from a series of observables such as sums of sequences of one spontaneous perinatal behavior, such as nuchal atonia durations, investigated at length in this work. Nuchal atonia or the loss of nuchal muscular tone in the primary anti-gravity muscles of the neck is a key indication of the onset of REM sleep in many animal species. If is an independent random process with a finite mean and variance, , the serves as one characteristic time of the nuchal atonia episode sequence. Another time, t, is derived from the product of two nuchal atonia
episodes computed over increasing numbers of intervals along the sequence, the autocorrelation function, . yields a characteristic time of decay of the amplitude of the joint products over the nuchal atonia episode sequence; an independent random sequence would proceed exponentially , where k is a constant. If the nuchal atonia sequences lack well defined central moments, say , then we would be without as a characteristic time, but instead, would have some probability distribution of times with a "stretched exponential" tail, or a "power law" tail, (see Schroeder, 1991 for a discussion of power laws and fractals). The decay of autocorrelations, , in this sequence without a well defined characteristic time is slower than exponential, often approximated by in which b also has a fractional value, 0 < b < 1.
Processes of this sort without a characteristic scale of time , or event correlation time, , have times of all orders which can often be described by distributions and correlation functions with fractional exponents. A fractal measure provides a single quantity to describe behavioral fluctuations over many scales of space or time. For example, if a process is fractal in time then the correlations among or similarity in the variations of a measured property of the process at one sampling rate (e.g., fluctuations of fetal behavior at one second) is statistically similar to the correlation or similarity in the variations of a measured property at a lower sampling rate (e.g., fluctuations of fetal behavior over an hour). This repetition of "self-likeness" in behavioral variation can be described in several ways. In geometric terms, if variation at different scales of measurement occur along one dimension, such as time, it is called "self-similar ". If the similarity dimensions of fluctuations are greater than one, such as time and amplitude, they are termed "self-affine". In this circumstance, the time series is invariant under a transformation that scales different coordinates by different amounts (for more detailed descriptions of these terms, see Barton & La Pointe, 1995a &b; Bassingthwaighte, Liebovitch and West, 1994; Fan, Neogi, and Yashima 1991; Feder, 1988; 1991; Hastings and Sugihara, 1993; Nonnenmacher, Losa and Weibel 1994; Peitgen, Jürgens and Saupe, 1992; Peters, 1991; and references therein). Naturally occurring self-affine processes (i.e., fluctuations in atomic clocks and sand hour glasses, wobbling of the earth on its axis, current fluctuations in electrical components, etc.) are called fractal or scaling noises (see Jensen, Todoeschuck, Crossley and Gregotski, 1991; West and Shlesinger, 1989; Voss, 1988a &b; 1992). Although, fractal time behavior was originally observed and defined for physical systems (Mandelbrot, 1982; 1983; see also Montroll and Shlesinger, 1984; Shlesinger, 1987; 1988), fractal time fluctuations in biological systems can be characterized by the same methods used to describe non-biological fractal time fluctuations. These "statistical fractals" may quantify the similarity of intermittent behavior across several different time scales, an example of which is the clustering within clusters of some fetal behaviors (Szeto, Dwyer, Cheng and Decena, 1990; Szeto et al., 1992).
Fluctuations of fetal behavior often appear as clusters in time, possibly self-affine, observed in spontaneous motility in the chick (Ripley & Provine, 1972) and other species (Corner, 1978) as well as in fetal breathing (Szeto et al., 1992) and nuchal electromyographic EMG recordings (see Figure 1), both markers of fetal REM sleep in sheep. This intermittent clustering in time among episodes of fetal behavior over minutes, hours or days can be distinguished from randomly ordered sequences by Hurst's rescaled range analysis (the Range of the sum of deviations from a local mean divided by the
Figure 1. On E123 of gestation, erratic intermittent bursting is apparent in nuchal activity with intervening periods of atonia. Nuchal EMG activity along the abscissa (2 volt range normalized to 250 standard units), plotted over 100 minutes on the ordinate (top trace). Rescaling the amplitude, abscissa, and time, the ordinate, over a 600 second subset of the original series (middle trace) reveals smaller clusters within larger clusters of both muscle discharges and atonia. Rescaling both dimensions again over 65 seconds (bottom trace) demonstrates similar qualitative "self-affinity" which is bounded from below by the sampling rate
Standard deviation from the local mean; R/S) developed by the British hydrologist Harold E. Hurst and popularized by Mandelbrot and Wallis (1968, 1969a-d and 1995).
Hurst needed to determine from historical records if the yearly flows of the Nile were random or clustered from year to year, for the construction of the Aswan dam (Bassingthwaighte et al. 1994, Bassingthwaighte and Raymond, 1994; Feder, 1988; 1991; Hurst, Black and Simaika, 1965). If high and low Nile flows were random over successive years (i.e., did not cluster in sequences of high or low years), then the reservoir size estimate could be based on the average of the recorded flows. On the other hand, if years of large Nile flows were not independent, but clustered or demonstrating serial dependency over successive years, then the "memory", or carry over between a series of wet years, could create a situation where reservoir capacity would have to be larger than the estimate based on the mean. Hurst examined 800 years of Nile flows recorded at the Roda gauge and determined they were not random but tended to cluster in runs of high or low years, supporting the need for a larger reservoir. He also examined 837 records of other natural phenomena (e.g., annual river levels, rainfall, temperature and pressure records, tree rings, varves, sunspot activity) finding non-random positive correlations in most of them (Hurst et al., 1965; see also Bassingthwaighte and Raymond, 1994; Mandelbrot and Wallis, 1995; and Peters, 1991 for Hurst analysis of capital markets).
Hurst's R/S analysis thus provides a measure of sequential dependency in the increments of records of natural fluctuations over different time scales of observation, . Mandelbrot and Wallis (1968 and 1969a-d) and Mandelbrot and Van Ness (1968) generalized and extended Hurst's R/S analysis to a class of Gaussian Brownian motions called fractional Brownian motions (fBm), which need not conform to the condition of serial independence of Brownian motion. The dependency between increments in fBm is quantified by H (called the Hurst exponent by Mandelbrot in honor of Hurst's work) which is calculated from the slope of the log-log relationship between the ratio R/S and the length of the record examined; R/S varies as .
To distinguish between statistically independent and dependent increments in a discrete time record, the difference in magnitude between two consecutive increments (e.g., nuchal atonia durations) DM = M(t2) - M(t1) is related to their difference in time = t2 - t1 by the scaling exponent H, or DM = (Dt)H (see eq. 9-23 in Feder 1988). This can be determined simply for the time record by plotting the ratio R/S versus increasing in a log-log plot and calculating its slope. For Brownian motion, fBm with H = 0.5 indicates that there is no dependency between steps (generated by flipping an unbiased coin and accumulating steps to the left or right, or recording points along the trajectory of a pollen grain in water). The sum of the independent steps or increments leads to a rescaled range of variation that scales with the square root of the number of steps or units of time : R/S . Due to the absence of positive or negative correlations between steps, the scaling exponent H equals 0.5 (Feller. 1951). Fractional Brownian motions with a scaling exponent H 0.5 indicate the presence of serial dependencies in the time series. When H > 0.5, positive correlations or dependency exists between the cumulative increments. Mandelbrot (1983) termed this "persistence" because increases (or decreases) are more likely to be followed by increases (or decreases) when observed at any time scale. In contrast, some biological processes such as closed loop control of upright stance or heart beat intervals have fluctuations with H < 0.5, called "antipersistence" because increases (or decreases) are more likely to be followed by decreases (or increases) when observed at any time scale (Collins and De Luca, 1993, 1995; Peng et al., 1993). The cumulative variation in a Gaussian distribution of increments in classical Brownian motion is proportional to the time between increments, which in the case of H = 0.5, classical Brownian motion, reduces to . H, by relating the variance-normalized cumulative deviation from a Gaussian zero-mean process over increasing time windows, provides a statistically scale invariant measure of correlation or dependency between increments over different time periods. In addition, H can be related to the coefficient of correlation between successive increments by the formula [see Bassingthwaighte et al., 1994 or Hastings and Sugihara, 1993 for a discussion of the relationship between H and the Pearson product moment correlation and coefficient of correlation] or to the slope, , of the spectral density function or power spectrum by the formula (Bassingthwaighte et al., 1994; Mandelbrot and Van Ness, 1968; Tatom, 1995; Voss, 1988a&b, 1989). Hurst's key discovery that many apparently irregular natural records could not be treated as the sums of independent random processes has still not had a great impact on the field of statistics (Bassingthwaighte and Raymond, 1994). This may in part be due to the non-convergence of the moments (see Hays, 1963 for a good treatment of statistical moments) in these fractal time series (non-finite mean and/or variance) and the absence of non-finite variance distribution-dependent tests of significance.
The distributions of times, , in a fractal time process with non-finite variance and long tails nonetheless manifest invariants with respect to indices on their identically distributed sums. The accumulated increments distribute like the probabilities of any single one of its motions. These Lèvy distributions without finite variance (and sometimes without a convergent mean as well) are in this way invariant across summation; this defining property is called "convolutional stability" (Lèvy, 1937; Gnedenko and Kolmogorov, 1968; Montroll and Badger, 1974; Takayasu, 1989). The Lèvy distribution is typically represented as a complex valued, exponential distribution function with four parameters indicating, respectively, location, symmetry, global scale, and rate of convergence of the tail (Mandelbrot 1982; 1983; 1993, Montroll and Shlesinger, 1984; Peters, 1991; Takayasu, 1984; 1987; 1989). Disregarding the location and symmetry and letting , the Lèvy distribution can be represented more simply as in which controls the relative size and the rate of convergence of the tail of across the range of values of t. In a Gaussian process with finite variance, ; if , the variance is nonconvergent but the mean, , can be computed; is the well known Cauchy distribution. If , the process is without a finite mean and will require the use of the median of interquartile indicators to locate the center of the distribution. Gaussian distributions are thus special stable Lévy distributions, with characteristic exponents a = 2 which by the Central Limit Theorem (Winer, Brown and Michels, 1991) converge to a finite mean and variance with sufficent sample size (Takayasu, 1989). For experimentalists who are accustomed to ignoring outliers in their data, the most remarkable feature of stable Lévy distributions is that the longer the period of observation, the greater the value for an outlier that might be observed. This signature, common to many fractal time processes, is the antithesis of the Central Limit Theorem governing normal stable Gaussian processes; that is, as more data points are accumulated, the variance and, in some cases, the mean are divergent. Can spontaneous fetal behavior such as nuchal atonia, a marker of fetal REM sleep, be described as a fractal time process?
I examined episodes of nuchal atonia, strongly associated with REM sleep in fetal sheep, with Hurst's rescaled range analysis to test whether this spontaneous fetal behavior would qualify as an independent random process. Developmentally, during embryonic days (E) 95-107, eye movements, nuchal muscle activity, breathing movements, and desynchronized low voltage fast (DLVF) electrocorticographic activity (ECoG) are almost continuous. Nuchal atonia first appears with differentiation of ECoG into synchronized high voltage slow (SHVS) waves and an increased mean amplitude of DLVF ECoG beginning near E120. At this time tonic nuchal activity starts to break up into periods of atonia. From E107 until E120 nuchal activity is actually increased during eye movement and breathing episodes. Following E120, long contractions of the nuchal muscles only accompany SHVS ECoG and are less active or absent during low-voltage activity (Clewlow et al., 1983). This reorganization of tonic nuchal activity into alternating periods of atonia and activity appears analogous in a general way to the emergence of spontaneous electrophysiological responses in early chick muscle primordium (Landmesser and Morris, 1975) or coordinated sequences of muscle contractions seen during later spontaneous motility in the maturing embryo (Bekoff, 1981). In addition, from about E120 on, spontaneous changes in fetal movements (e.g., rapid irregular fetal breathing, mouth and tongue, diaphragmatic, and isolated body twitches and generalized bursts of limb movements, changes in tracheal and arterial pressure) are correlated with most of the criteria of REM sleep in adult sheep (e.g., rapid eye movements, DLVF, increased cerebral blood flow; nuchal atonia; myoclonic twitches; Dawes, Fox, Leduc, Liggins and Richards, 1972; Parks, 1991; see figure 23-4, Rigatto, 1989; Szeto and Hinman, 1985). The sheep fetus, like other mammalian fetuses, is in DLVF sleep with atonia 40-60% of the time, proportionately more than time spent in any other behavioral state (Dawes et al., 1972; Rigatto, Moore, and Cates, 1986; Szeto and Hinman, 1985). Also, micturition, myoclonic twitches complex bursts of body movements and increased variability in heart rate and blood pressure fluctuations appear more likely to accompany periods of DLVF and atonia than SHVS ECoG (Robertson, 1987; Ruckebusch, 1971; Ruckebusch, Gaujouz and Eghbali, 1977; Wlodek, Thorburn and Harding, 1989). In accordance with these observations, in the third trimester of both sheep and humans, fetal REM has been suggested to provide a global organizing framework for perinatal behavioral states (Hofer, 1988; Kohyama, Shimohira and Iwakawa, 1994). The occurrence of nuchal atonia episodes, therefore, would appear to provide a primary index of this global fetal REM state.
Nuchal atonia in the late gestation sheep fetus probably originates from activity in behavioral state dependent neuronal populations in the reticular formation (RF), a brainstem structure which plays an important role in other phasic REM sleep phenomena in adults (Chase and Morales, 1990; Glenn, 1985). It therefore provides a convenient external marker of multiple state-dependent processes involving the adult RF, such as changes in single unit activity (Glenn and Dement, 1982; Peterson, Pitts and Fukushima, 1978) which are strongly associated with REM sleep. Although the neuroanatomical substrates responsible for phasic REM processes such as nuchal atonia are largely uninvestigated in the RF of fetal sheep, Peterson et al. (1978) observed that the probable origins of nuchal atonia in the adult cat involved large lateral inhibitory reticulospinal fibers originating in nucleus reticularis ventralis that project caudally as far as neck motor neurons, producing strong IPSPs in nuchal motor neurons when electrically stimulated. Atonia episodes result from a cluster of spontaneous phasic-event-related hyperpolarizing potentials invading motor neurons from these reticulospinal fibers and reducing the tonus of the nuchal muscle contraction (Glenn and Dement, 1982). Nuchal atonia is also a sensitive measure of the disruption of ECoG sleep patterns by infusions of drugs such as methadone or cocaine (Burchfield, Graham, Abrams and Gerhardt, 1990; Derks et al., 1993; Szeto, 1983; Szeto and Hinman , 1983), as well as, other dopamine and noradrenergic agonists or antagonists (Bamford, Dawes and Ward 1986 and Bamford, Dawes, Denny and Ward, 1986). These drugs are associated with the induction of a prolonged state of arousal in the sheep fetus. The presence of nuchal tone accompanied by DLVF ECoG (Ioffe, Janson, Russell and Chernick, 1980) is routinely used to distinguish this "arousal-like" state from REM sleep (Clewlow et al., 1983; Szeto, 1985). Therefore, transitions in nuchal EMG activity indicative of REM sleep are possibly representative of the state dependent reorganization of firing patterns in multiple sub-regions and neurotransmitter specific sub-populations in the RF (Hobson and Steriade, 1986; Sakai, 1985).
The interrelationships among the above mentioned fetal REM markers and nuchal atonia are often characterized as inconstant (Dawes et al., 1973; Parks, 1991), suggesting these fluctuations might be more completely described as fractal patterns in time. As I will describe spontaneous sequences of atonia episodes, over one embryonic day, have a clustered fractal structure in time that is different from random independent sequences. Statistical analysis of nuchal atonia episodes encompassing the observed gestation period demonstrated that longer episodes were observed with increased sample size resulting in nonconvergent cumulative means and standard deviations, the signature of fractal processes (Bassingthwaighte et al., 1994). Also, consistent with a developmental role for this clustered fractal time structure of atonia sequences, and inconsistent with other findings of Poisson random process for spontaneous prenatal behavior (Narayanan et al., 1970), distributions of atonia episodes were found to be well approximated by stable Lévy distributions over the gestational period examined, suggesting these may be invariant properties indicative of more general developmental processes.
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REM SLEEP IN FETUSES AND NEONATES AS AN ONTOGENETIC ADAPTATION
The use of the term ontogenetic adaptation attempts to bring an ecologically valid perspective to interpretations of morphogenic and behavioral changes undergone during different developmental stages of an organism (e.g., embryonic, fetal, larval, postnatal and juvenile stages; Oppenheim, 1981, 1984). In essence the adaptive aspect of ontogenetic adaptation refers to those neurobehavioral and neurophysiological developmental patterns, processes and traits that enhance an individual's chances of perpetuating its genes. Although, as Oppenheim points out: "...there is always some danger in assuming that all traits of an organism are both adaptive and the direct result of natural selection...[and] it is not possible in most cases to test experimentally for the adaptiveness of a trait. In almost any discussion of adaptations one is forced...to rely on logical arguments [just so stories], inference and the consensus of fellow scientists as to whether or not a trait is adaptive (1984; p. 18)."
The core problem of developmental biology is the study of the temporal cause and effect relationship between transient antecedent structures (e.g., pharyngeal arches or gills, the pronephros, the egg-tooth) or behaviors (e.g., motility patterns, suckling, play) and more mature forms of those structures (e.g., adult facial features) and behaviors (crawling, walking, sleep-wake cycles, feeding, social interactions). Oppenheim's central purpose in introducing the concept of ontogenetic adaptation was to propose that viewing developmental change as a series of genetically controlled transient adaptations rather than a simple cumulative progression of developmental innovations provides a useful explanation
of the origins of age specific behaviors and structures in terms of meeting the unique ecological demands stemming from the organism's morphological and physiological immaturity (Oppenheim, 1984; Hall and Oppenheim, 1987). As Hofer (1988) points out: "The concept of ontogenetic adaptation can be useful to organize and understand many of the special features of the intrauterine period, as well as some of the features of subsequent development" ( p.6). In particular, in light of the fact that the occurrences of behavioral ontogenetic adaptations grossly characterize the developmental stages of an organism, they can also be used to assess the viability of the developing nervous system.
Why is REM sleep considered an ontogenetic adaptation? The dominance of REM sleep in fetal and infant behavioral states during prenatal and early postnatal life was first noted by Roffwarg and co-workers in the mid 1960's. Roffwarg et al. (1966) proposed that developmental neurobiologists "give consideration to the possibility that REM sleep plays a role in stimulating structural maturation and maintenance within the central nervous system" (p.617). REM sleep is the most metabolically active state of the brain in the fetus (Abrams, Hutchison, Jay, Sokoloff, & Kennedy, 1988; Richardson, 1991) and if it did not represent an adaptive innovation to the fetal (whether egg, pouch or uterus) or postnatal environment (nest or nursing niche), it would likely have been abandoned early in evolution for metabolic reasons; instead, it is ubiquitous in vertebrates and has analogous forms in higher invertebrates (Karmanova, 1982). An underlying experimental-theoretical foundation for understanding the how or why of this highly energetic REM state with its associated behavioral manifestations and the role it plays during ontogeny is almost completely lacking.
Corner (1990) has reviewed the connection between ultradian motility patterns in the embryos of vertebrates and invertebrates, their brainstem origins and the ontogeny of sleep states. He suggests a robust connection between the ontogenetic coalescence of body twitches, REM's, breathing moments, EEG, etc. in the developing animal and the maturation of the brainstem. The importance of REM is also supported by the observation that deprivation of sleep, especially REM sleep, in developing animals can have detrimental effects. Electrolytic lesions of pathways from the reticular core to the lateral geniculate nucleus (LGN) in kittens, which isolate the LGN from REM associated patterns of neuronal activity, were found to impair the structural and physiological maturation of cells in the LGN (Davenne and Adrien, 1985) Also, selective REM deprivation in kittens by non-invasive awakening produced anatomical changes in the LGN similar to eyelid occlusion (Shaffery et al., 1993). Rat pups deprived of REM sleep over the first three postnatal weeks with clonidine or alpha-methyl dopa, when tested as adults, showed a constellation of behavioral abnormalities such as hyperactivity, hyperanxiety, attentional distractability, sleep disturbances, reduced sexual performance and reduced cerebral cortical size (Mirmiran, 1986).
Hofer (1988) has referred to REM sleep as an ontogenetic adaptation "whereby early neural function acts as a vital ingredient in the developmental plan of the organism. Presumably, as the infant grows older, its interaction with the environment increasingly takes the place of this state specific internal stimulation. From its peak in the late fetal period, REM sleep proportion declines rapidly during the first postnatal year [in the human infant], possibly representing the decline of its role in neural development " (p. 16). This decline of the increased proportion of REM over the developmental period is analogous to the disappearance of other ontogenetic adaptations (i.e., shedding the fetal membrane and placenta at birth, the loss of sucking or crawling behaviors with the development of independent ingestion and walking).
If the REM sleep associated activity patterns represent an ontogenetic adaptation, then genetically homogeneous individuals would be expected to show similar patterns. Using a variety of methods for describing the temporal patterns of sleep in adults, several investigators have observed that general sleep structure as well as REM associated events exhibit a genetic component. Using Markov analysis, Zung and Wilson (1967) found almost complete concordance between all night EEG and REM patterns in adult monozygotic twins but not in dizygotic twins. Using standard methods, Webb and Campbell (1983) found significant correlations between the structural measures of sleep (e.g., onset latencies, awakening measures, stage changes and rapid eye movement amounts) in adult monozygotic but not in dizygotic twins. Using different standard methods, Hori (1986), in agreement with Webb and Campbell, found that the number of body movements associated with REM sleep were significantly correlated between adult monozygotic twins. However, Hori also found more similarities between structural measures of sleep in dizygotic twins than Webb and Campbell (1983). In a recent non EEG observational study of one neonatal pair of conjoined twins, both were found to have almost identical REM epochs per sleep bout and identical lengths of intervals between bouts, but temporal patterns of the occurrence of bouts were strongly individual (Sackett and Korner, 1993). Thus, these studies argue that the number of REM related activity patterns may have some genetic component, but that the temporal patterns of occurrence of these events is a characteristic of the individual. It is possible however, that these characteristic activity patterns show long run correlations in time, not detected in previous studies, that are invariant among members of a species and between species. Support for this conclusion comes from the work of Thoman, Zeidner and Denenberg (1981) who found commonalities among analog recordings of state-related motility patterns from rats, rabbits and humans scored by different observers, suggesting cross-species invariance in state-related motiltiy patterns.
Addressing the patterns of REM-associated activity, as well as motility during awake states, Thoman et al. (1986) have pioneered the interpretation of characteristic breathing and body movements signifying rest-activity cycles or "patterns of state behaviors" as an indicator of neurobehavioral stability in infants. Known associations between neurological defects and patterns of state behaviors furnish further evidence for these interpretations. For example, Tanguay, Ornitz, Forsythe and Ritvo (1976) found that eye movements (EM) in normal children did not become organized into bursts until 40 weeks gestational age; thereafter changes in the clustering of the bursts of EM were correlated with developmental age. From 2 to 24 weeks postnatal, as total REM decreases, the number of EM's remain constant resulting in an increase in the mean number of EMs/sec of REM. Between 3 months and 5 years of age, the major organizational change in the patterns of EMs is the increasing tendency of bursts of EMs to cluster, with more and shorter EMs packed into bursts within bursts. However, autistic children show substantially less clustering of EMs. In fact, no significant differences between burst structure in 2-5 year old autistics and younger (<18 month) normal children could be found. Thoman and colleagues have also found significant correlates between the stability of activity patterns and different developmental disorders (e.g., low stability scores were associated with major developmental disorders such as aplastic anemia, infantile seizures, hypsarrhythmia, SIDS and hyperactivity). The only criticism of the work of Thoman or predecessors such as Tanguay is not with the phenomena, because it has been so well supported in the literature, but with the use of standard methods of analysis (e.g. calculation of statistical moments such as the mean and variance under the possibly incorrect assumption of convergence) for biological processes that may not have convergent moments. Therefore, other methods of analysis sensitive to the temporal strucure of these activity patterns that do not rely on these standard statistical assumptions may provide new insights into their developmental roles.
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EXPERIMENT 1: ANALYSIS OF NUCHAL ATONIAL SEQUENCES IN FETAL SHEEP: AN EXAMPLE OF THE FRACTAL TIME STRUCTURE OF A SPONTANEOUS PRENATAL BEHAVIOR
In the first of two studies, I examined spontaneous sequences of nuchal atonia associated with REM sleep behavior in late gestation fetal sheep (E121-133) with Hurst's rescaled range analysis, which affords comparisons between natural time series with short- and long-term correlated fluctuations (e.g., annual patterns of riverflow, rainfall, tree rings). This analysis indicats that nuchal atonia sequences over short time scales are statistically similar to longer time scales (i.e., is fractal in time). Hurst exponents (H) are distributed around 0.70 (H in an independent random process = 0.5) suggesting an invariant measure among animals and across age. Longer nuchal atonia durations are observed as more likely to be followed by longer durations, and shorter nuchal atonia events by shorter durations in a pattern called persistence. Several properties of nuchal atonia episodes are inconsistent with a Poisson process: 1) mean variance; 2) H 0.5; 3) probabilities are in the family of non-finite moment Lèvy stable distributions with a characteristic (tail) exponent of a < 2.0 ( a 1.8). Nuchal atonia has temporal coherence with other markers of REM sleep such as eye movements and electrocorticogram and breathing pattern changes consistent with its use as a marker of this sleep state. Based on the experimental findings detailed below I conclude that the fractal time patterns of nuchal atonia sequences appear to reflect the temporal structure of the global fetal REM sleep state over mutiple timescales which serves as a transient behavioral ontogenetic adaptation to changing developmental environments.
SPECIFIC METHODS
Subjects. Five time-mated pregnant ewes (Ovis aries, Rambouillet Columbia stock) obtained from the Cornell University breeding facility provided fetuses for this study. Ewes were acclimatized to the laboratory for one week prior to surgery. Ewes were housed in individual metabolism stalls, which permitted a range of movements and body postures but prohibited turning around so as not to damage catheters and electronic leads.
Preparation of Pregnant Ewes and Fetal Subjects. All surgeries were performed at E112-114 using ketamine (5-10mg/kg) and glycopyrollate preanesthetic and tracheal halothane which also anesthetizes the fetus. An antibiotic treatment of 1g ampicillin sodium (Polycilin-N; Bristol Laboratories, Syracuse, NY) was administered prophylactically, and 1g chloramphenicol sodium succinat (LyphoMed Inc., Melrose Park, Il) was administered i.v. to the ewe at surgery. Midline laparotomy and incision through the uterine wall provided access to the fetus. Chronic catheters were placed in the external jugular of mother and fetus to monitor blood gases as a means of determining preparation viability. EMG stitch-electrodes were placed on nuchal and geniohyoid muscles to enable recording of electro-myographic activity. Fetal electrocortical activity (ECoG) was monitored through gold electrodes placed on the dura overlying the parietal lobes to left and right of midline (see Robinson et al., 1995 and [Gluckman and Parsons, 1984], for more extensive details of the surgical procedures).
Data Acquisition and Analysis. Raw measurements from fetal EMG and ECoG leads were recorded in a computerized data acquisition system. Specifically, 50-200 mV
signals were amplified up to 2 volt full scale deflections and bandpass filtered (3-30 Hz for ECoG or 80 to 300 Hz for EMG signals), full wave rectified and low pass filtered at 10 Hz to create the envelope of signals. Fetal EMG and ECoG were then resampled with a 12 bit A/D converter at 32Hz and averaged over 1-sec intervals. Nuchal tone and ECoG measurements were normalized to 0 to 250 amplitude units and recorded continuously from E121 to 133. A 200 Hz EMG recoding made prior to low pass filtering was examined.
Two partitions of nuchal EMG were examined to distinguish baseline noise artifacts and to determine an unbiased estimate of atonia episode duration. The first partition (P1) was constructed by observing the range of variation of baseline EMG amplitudes across 13 days (typically 3-7 out of a possible 250 amplitude units) and recording the durations of EMG amplitude threshold range (e.g., values < 7 would count as atonia; events of amplitude > 8 would not). The second more stringent partition (P2) was contingent on the total absence of nuchal tone and was applied by observing the lowest EMG amplitude across the record and counting the duration in seconds of atonia.
Statistical Properties of Nuchal Atonia Episodes. Changes in mean nuchal atonia episode length and number over E121-E133 were assessed by 2-way ( 2 Partions ¥ 13 Fetal Days) Analysis of Variance (ANOVA ), with Fetal Day as the repeated measure. Simple main effects were analyzed by 1-way (Fetal Day) ANOVAs. Orthogonal linear trend analysis was used to further examine developmental trends for the number and average mean nuchal atonia episode length and number over the period of observation.
To examine the statistical properties of the Moments (e.g., cumulative mean and variance) a 3-way (2 Moments ¥ 2 Partions ¥ 13 Fetal Days) ANOVA was calculated, with Fetal Day as the repeated measure, for data sets ranging from 1 to 13 days in length. In addition, the cumulative means and standard deviations for each partition were tested individually for statistical convergence over development by 1-way ( 13 Fetal Days) ANOVA, with Fetal Day as the repeated measure. The criterion for type 1 errors was set on all comparisons at 0.05 and the Huhyn-Feldt procedure was used to assess violations of circularity (Winer, Brown and Michels, 1991).
Hurst's Rescaled Range Analysis. The Hurst exponents were estimated by the following segmentation procedure which is visually illustrated in Figure 2: (1) the mean over the total available record was calculated. (2) A lag () or window up to and including the total record length (i.e., 2048, 1024, 512, 256,128, 64, 32, 16, 8, 4) was chosen. (3) The local accumulated differences for nonoverlapping segments of length equal to were determined by subtracting episode length from the mean of the total record. (4) The local range (R), standard deviation (S) and ratio R/S determined for each was then calculated. (5) Local R/S ratios were then averaged. (6) The procedure was repeated for all window sizes or 's. (7) The logarithms of the average R/S values are then plotted against the logarithms of the fraction of points in the time window (e.g., for a 4 increment window out of 4096 increments the fraction would be 4/4096 or 0.0097). The linear regression of this line results in the slope which is equal to the Hurst exponent (H).
Two methods of validating Hurst analysis were used. The first involved randomizing the data set by the following procedure to destroy the correlations present in the data (Collins and De Luca, 1994 and 1995; Peters, 1991; Rapp, 1994). To accomplish this, a set of random numbers was generated for each file analyzed that ranged from zero to the size of the data set N and was stored in a parallel array. The random numbers were sorted in ascending order by size. In this way the random number corresponding to a particular N was used as an index to reorder the values of a original nuchal atonia data file. The reordered nuchal atonia data file or first order surrogate data set (Rapp, 1994) was then resubjected to Hurst analysis. Hurst and randomized exponent estimates were then
Figure 2. Diagrammatic representation of the processes of rescaled range analysis. The top line represents the entire analysis period (4096 sequential nuchal atonia episodes) where the first estimate of R/S is obtained by dividing the range by the standard deviation. Subsequent lines represent subsets of the original set which have undergone the same procedure to obtain multiple estimates that are then averaged. Average estimates of R/S are then plotted on a double logarithmic plot, and the slope determined by linear regression.
compared by a one factor (2 exponent estimates) ANOVA. The second method involved testing the Hurst algorithm on artificial data sets of fractional Brownian motion as generated by a spectral synthesis method with known values of H as described by Feder (1987) and Peters (1991).
Developmental changes in estimated Hurst exponents for original and randomized data sets were compared by 2-way (2 Partions ¥ 13 Fetal days) ANOVA, with Fetal day treated as a repeated measure. Developmental trends were tested by orthogonal linear trend analysis of average H values over the period of observation.
Estimating the Lévy Tail Exponent. I approximated the normalized density distributions of atonia episodes from each animal for E121-123 and 131-133 (i.e., for two 72 hour recording periods relatively early and late in fetal development) with parametrically fitted algebraic Lévy curves of the form , where C is a constant (Mantegna, 1991). I used a Simplex, direct search minimization of quadratic (i.e., sum of least squares) error (O'Neill, 1971; Griffiths and Hill, 1985) with larger initial step sizes (i.e., tolerances) designed to avoid local least squares minima while fitting and in combination. This is an exhaustive, "brute force" fitting procedure with relatively large but bounded regions of the parameter space (i.e., ). As noted below, the results of fitting to the density distributions of the nuchal atonia durations yielded which is consistent with processes termed anomalously diffusive in statistical mechanics; . Whereas the range of the summed increments of a subset of Lévy processes composed of Gaussian (or Poisson) independent random events with individual increments distributed such that grows with time in a process called normal diffusion, Lévy processes with individual events distributed such that are called superdiffusive (Shlesinger, 1987) and are consistent with the observed absence of finite (see results).
Nonlinear estimation is an art form. The pitfalls are many; parameter interdependencies and local minima among them. Here I have used variable, larger tolerances (i.e, 10-5 to 10-2) to test, and therefore avoid most local minima. The Simplex method of nonlinear estimation is robust in the presence of many discontinuities, as opposed to other methods (e.g., Quasi-Newton). Similarly, Simplex does not require that the model is differentiable in the parameter region of interest . The Simplex method is also immune to round off error because it recomputes estimates at each iteration.
The asymptotic standard errors were computed by central differencing finite estimation of the Hessian matrix after the parameter estimates were completed. Using this "final" matrix, rather than an iteratively updated one, I avoided the tolerance and iteration history dependence of some asymptotic standard error estimates.
RESULTS
Nuchal EMG records. Recordings of fetal sheep nuchal muscle potentials as normalized mVs over time appear at first glace to have little structure. However, after rescaling in both the dimensions of amplitude and time (see Figure 1, p. 7) 60 sec periods appeared qualitatively similar to those of 600 seconds which in turn resembled those over 60 minutes in a relation called self-affinity. In addition, this fractal time structure of nuchal atonia sampled at 1Hz appears to extend to nuchal muscle potentials recorded at 200Hz (see Figure 3, p. 30). The scaling structure of the irregular patterns of intermittent bursts of nuchal muscle activity and their absence appeared to have psychobiological significance: they corresponded roughly with other indicators of REM sleep including low voltage faster activity in the ECoG, eye muscle activity in the EOG, muscle activity in the geniohyoid
group, and changes in fetal breathing patterns recorded as tracheal pressure (see Figure 4, p.32 ).
Developmental Changes in Statistical Properties of Nuchal Atonia Episodes. Overall the results of the analysis suggest two main findings: As the fetus matures, there is a reduction in the number of nuchal atonia episodes (Figure 5a) and an increase in the mean length of nuchal atonia periods (Figure 5b). Without exception, application of the stringent partition P2 resulted in larger numbers of atonia events with shorter durations than with the less stringent partition P1 (Figure 5a and 5b). This may be due to small baseline fluctuations in EMG amplitudes which resulted in fragmentation of longer periods of atonia. Two 2-way ANOVAs ( 2 Partitions ¥ 13 Fetal days) comparing Partitions for
Figure 4. Digitized analog recordings from a representative sheep fetus on day E133 of gestation portray nuchal atonia episodes as a low or nearly flat baseline in nuchal EMG trace. Fetal REM sleep is characterized by lower nuchal muscle tone and increased geniohyoid EMG and eye movement EOG potentials corresponding roughly in time with lower voltage faster frequency patterns in the electocorticogram record, ECoG. Periods of decreased geniohyoid EMG and eye movement EOG potentials overlap with increased nuchal tone and higher voltage lower frequency ECOG.
Figure 5. Developmental changes over days E121-E133 in mean number of atonia episodes (a) and mean length of atonia episodes (b) over a day for the lower amplitude, more sensitive partition (P1) and higher amplitude, less sensitive partition (P2) in all animals. Error bars indicate the standard error of the mean for each day.
mean nuchal atonia episode length and for nuchal atonia number revealed significant main effects of Partition, F(1,4) = 2.502, p<.05, e = 1, and F(1,4) = 8.954, p<.01, e = 1, respectively. A significant main effect of Fetal Day was also present for atonia event number F(12,48) = 9.285, p<.01, e = 0.223, but not for mean length F(12,48) = 1.664, p>.05, e = 0.213. This suggests there is a strong developmental trend for decreasing nuchal atonia episode number over gestation which was evident with further 1-way tests of Partition and Day. The number of atonia events was found to significantly decrease with Fetal Day for P1, [F(4,12) = 10.460, p<.0001, e = 1] and for P2 [ F(4, 12) = 4.879, p<.05, e = 0.222], (p<.0001 before correcting for violation of the assumption of circularity) over the period of observation. In agreement with reports showing a decline in fetal DLVF ECoG with development (Dawes et al., 1983; Clewlow et al., 1983; and Szeto, 1985), trend analysis indicated linear developmental trends for decreases in the number of atonia events with P1 F(1,48) = 113.755, p<.0001,e = 1 and for P2 F(1,48) = 49.119, p<.001,e = 0.222, which accounted for 90.62% and 83.89% of the variance while departures from linearity accounted for 9.38% and 16.11%, respectively, of the remaining variance but did not reach criterion.
Although, the main effect of Fetal Day on mean nuchal atonia episode length was not significant, as a result of the large variability of atonia episodes, further analysis by 1 way (Fetal days) ANOVA resulted in an apparently significant effect of Fetal Day on P1 [F(4, 12) = 19.755, p<.05]. However, after corrections for a departure from the assumptions of circularity, this was found not to be significant , p >0.05, e = .624. Analysis of P2 was also non-significant [F(4, 12) = 19.755, p >.05 ,e = .118]. A priori trend analysis, justified by an apparent trend in the data (Figure 5b) indicated a linear increase in episode length with developmental age in mean nuchal atonia episode length for both P1[ F(1,64) = 12.831, p<.005,e = .624] and P2 [ F(1,64) = 43.117, p<.01,e = .118] which accounted for 54.15% and 68.42% of the variance, while departures from linearity did not reach criterion.
To test whether sequences of nuchal atonia are statistically similar to a Poisson process (e.g., for a Poisson process the mean equals the variance) in addition to the effects of Partition and Day, the cumulative Moments (e.g., mean and varience of nuchal atonia events summed progressively over 13-days) were analyzed by a 3-way ANOVA (2 Moments ¥ 2 Partitions ¥ 13 Fetal Days) with Fetal Day as the repeated measure. This
test detected a significant main effect of cumulative Moments [F(1,4) = 43.043, p<.0028, e = 1]. None of the 2 or 3-way interactions, however, proved to be significant. The nature of the effect of Moments is evident in Figure 6a and b where the magnitude of difference between the cumulative mean and variance for both Partitions over 24 hrs and development is seen to be quite large. Specifically, the magnitude of the difference is greater for P2 than P1. The difference between the cumulative mean and variance for P1 (M = 14.104 vs. V = 355.357 seconds) was significant [F(1,4) = 47.159, p<.005, e = 1] by 2-way repeated measures ANOVA (2 Moments ¥ 13 Fetal days). For P2, however, although the cumulative variance (V= 585.666 seconds) was larger than the cumulative mean (M = 8.539 seconds) and significantly different [F(1,4) = 12.069, p<.05, e = 1], the significance level was smaller. The extreme difference between the cumulative mean and variance argue against the possibility that nuchal atonia is a Poisson process.
To investigate whether the cumulative means and variance demonstrated any
Figure 6. (a)The cumulative mean duration and variance derived for both partitions over increasing numbers of nuchal atonia episodes for the five subjects during 24hr and (b) over increasing days of development E121-133, both of which fail to demonstrate the expected convergence of the variance in general and the mean in particular of a Poisson process.
statistical convergence with development both partitions were tested by a priori trend analysis following a 1-way ANOVA (Moment). None of the main effects of the 1-way ANOVAs were significant after corrections for departures from the assumptions of circularity. However, the cumulative mean and variance of P1 significantly increased with development [F(1,64) = 20.178, p<.05, e = .168] and [F(1,64) = 11.208, p<.005,e = .204], which accounted for 42.037% and 23.351% respectively, of the varience while other trend components were not significant.
To summarize the findings thus far, significant developmental decreases in the nuchal atonia episode number across all subjects were observed. This is compatible with previous reports of developmental declines in fetal REM sleep (Dawes et al., 1983; Clewlow et al.,1983; and Szeto, 1985). The cumulative mean duration and variance of atonia episodes were significantly different, highly variable and non-convergent in terms of the central limit theorem because the mean and variance consistently increased with sample size. The high variability also contributed to the finding of a non-significant effect of days on mean atonia length, although a priori trend analysis indicates a highly significant developmental increase in the length of atonia periods. These data are consistent with the hypothesis that REM sleep periods, when indexed by nuchal atonia, tend to consolidate into longer clusters with increasing age.
Results of Hurst Analysis of Nuchal Atonia Sequences. Hurst analysis, quantified by H, measures long-run statistical dependency or correlations within a timeseries (Bassingthwaighte et al., 1994). Hurst exponents greater than 0.5, which are significantly different than Hurst exponents from randomized surrogate data sets, suggest that the assumption of statistical independence between events can be rejected. Hurst exponents calculated for P1 ranged from 0.553 to 0.779 and from 0.620 to 0.843 for P2 during the period of observation (Figure 7), suggesting the presence of long-run dependency between Figure 7. The least squares fits of the mean of the Hurst exponents illustrate the changes in persistence (defined by H > 0.5) for the original sequences of nuchal atonia episodes for each partition. The R represents the Hurst values (H 0.5) computed in the same way following randomization of the sequences data for partition 1 and 2 over development, reflecting the expected loss of long range correlations following shuffling of the sequences.
nuchal atonia episodes over 24 hours. A histogram of the distribution of Hurst exponents for both P1 and P2, original and randomized is displayed in Figure 8a and 8b. A 2-way ANOVA (2 Partions ¥ 2 Fetal days) was used to assess the effects of Partition and Fetal Day on estimated Hurst exponents. This analysis indicated the significant main effect of Partition F(1,4) = 11.169, p>0.05,e = 1 and Day F(12,48) = 4.377, p>0.0001,e =1. Over all days examined, Hurst exponent estimates for P2 were higher than those for P1 (Figure 7). In addition, a substantial decreasing linear trend of Hurst exponent estimates with Day was evident regardless of Partition F(1,129) = 47.533, p<.0001,e =1.
Hurst exponents calculated from randomized surrogate data sets of P1 and P2 ranged from 0.430 to 0.587 and 0.408 to 0.590, respectively. Hurst exponent estimates from the original data sets were found to be significantly different, independent of Partition, from estimates derived from randomized surrogate data sets by 2-factor ANOVA (2 Hurst Estimates ¥ 2 Partitions) P1[ F(1, 4) = 572.221, p<.0001, e = 1] and P2 [F(1, 4) = 723.525, p<.0001, e = 1] confirming the existence of long-run dependency. A priori trend analysis also indicated strong linear trends for decreases for both P1 [F(1,64) = 12.831, p<.005, e = .624] and P2 [F(1,64) = 43.117, p<.01, e = .118] which accounted for 73.13% and 67.16% of the variance while trend components were not significant.
In summery, the magnitudes of Hurst exponent estimates were dependent on the Partition. Hurst exponent estimates from original timeseries were significantly different from Hurst exponent estimates from randomized surrogate data for both partitions and all days. In keeping with the strong developmental decline in nuchal atonia episode number, a significantly decreasing linear trend in Hurst exponent estimate magnitudes with day was observed. This decline is not unexpected due to the developmental decrease in nuchal
Figure 8. The distribution composed of all the 24 hour values of Hurst exponent (H) estimates for (a) original and (b) randomized sets for both P1 and P2 partitions over the period of gestation examined. The largest probability mass of the Hurst exponents for the original sequence was in the neighborhood of H 0.70. H 0.5 for the randomized set.
atonia event numbers and the effects of smaller sample sizes on Hurst estimates (see Bassingthwaighte and Raymond, 1994). Therefore, higher resolution measurements of
nuchal atonia events are needed to counter the developmental effect of the declining number of nuchal atonia events before the existence of a developmental trend in the Hurst exponents can be confirmed.
The probability density distributions describing nuchal atonia. To test the hypothesis that nuchal atonia episodes might not be described by normal Gaussian distributions, the goodness of fit to other Lévy distribution curves, as indexed by the derived characteristic exponents a and g, was examined. Fitting the probability distribution yielded estimates of a ranging from 1.8224 to 1.8300 and g's from 3.0062 to 3.0745 across animals and days. The single parameter Simplex estimates converged quickly and had small asymptotic standard error estimates (all A.S.E. < 0.008). No consistent changes in a or g were found between animals or within each animal across gestational days. Also, the finding that the normalized probability density distributions of atonia episodes calculated for the two partitions were generally similar in form over development (see Figure 9) is consistent with both the unbiased nature of the estimation of atonia events by the partitioning procedure and the finding of invariance in the characteristic exponents with maturation.
Figure 9. The log-linear plots of the probability density distributions for nuchal atonia episode durations on representative days of a representative animal demonstrate the highest densities at the lower bounds of the resolution of its sampling rate in animal A, as in a Poisson process. In addition, they display long tails which failed to converge over sample length or days. This distribution remained invariant over the 13 days of observation for all fetuses despite a shift in the average length of nuchal atonia episodes.
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DISCUSSION
Fractal description is useful in developmental psychobiology because it unifies experimental accounts of diverse and apparently independent random events of behavior by representing relations among these patterns over many scales of space or time. The erratic cluster-within-clusters appearance of nuchal EMG activity in Figure 1 would seem to portend large within-group variances and concomitant difficulty discerning developmental trends using traditional methods of analysis. However, when this EMG activity is viewed at different timescales, a new symmetry of similar erratic behavior independent of timescale
is evident. Many neurobiological and behavioral processes appear, at first glance, similarly erratic over time. For example, ion channel currents (Liebovitch and Tóth, 1990), interspike intervals (Gerstein and Mandelbrot, 1964; Selz and Mandell, 1992; Teich, 1989), animal search behavior (Coughlin, Strickler, and Sanderson, 1992; Cole, 1991; Fourcassié, Coughlin and Traniello, 1992) plant and animal population dynamics (Hassell, Comins and May, 1991; Tilman and Wedin, 1991) and even taxonomic diversity (Burlando, 1990;1993) all have this self-affine signature of clusters within clusters. The analysis of the intervals of atonia of Figure 1, or nuchal atonia associated with REM sleep, revealed surprising statistical properties, such as cumulative moments that are non convergent with larger samples (i.e., violate the law of large numbers). Nuchal atonia is therefore neither Gaussian nor Poissonian, but highly ordered in that statistical interdependence extends between these intervals of atonia over many time scales. Another implication of this novel descriptor "fractal time" is that there is no natural time scale for the measurement of spontaneous prenatal behavioral patterns. This perspective on
the measurement of biological processes should be considered when examining complex patterns of behavior. Conversely, the measurement of a spontaneous prenatal behavior pattern, such as a nuchal atonia episode sequence or perhaps spontaneous motility, then depends on the time scale of measurement. This intrinsic variability of developmental fractal processes with time scale may underlie some of the difficulties in defining fetal behavioral states.
The implications of fractal time descriptions for defining fetal states. The definition of REM sleep and other states in the fetus is problematical because it is defined in relation to a prototypical adult state, with implicit assumptions that may be inappropiate when applied to the changing behavioral state of fetal REM sleep. In defining REM sleep in the fetus, characteristic markers used in the adult definition have either not matured or have greater variability in the fetus. In terms of REM sleep, the similarities with the adult state are many: DLVF ECoG (Clewlow et al, 1982); increased brain metabolism (Richardson, 1991), nuchal atonia (Ioffe et al, 1980; Szeto and Hinman, 1983), rapid eye and irregular fetal breathing movements (Dawes et al, 1972; Ruckebusch, 1972) -all classical indicators of REM in adult sleep physiology. However, the extent to which REM sleep markers correlate with other state-markers is much more variable in fetal or neonatal animals (Briem, 1986; Ruckebusch, 1972; Parks, 1991). Thus efforts to unambiguously delineate the behavioral states of sleep and wakefulness in fetal sheep or any fetus are beset from the start by the conceptual bias introduced by backwards generalization from adult states to fetal states. Further complications mark any attempt to define the in utero state of alert wakefulness in studies of fetuses where behavioral or drug manipulations have disturbed the fetus and its environment. In these instances of fetal disruption, DLVF becomes disassociated from nuchal atonia, and this is considered indicative of a state of arousal which has been reported in utero (Clewlow et al, 1983; Szeto and Hinman, 1983). Parks (1991) points out the dangers of circular arguments in using descriptions of fetal activity to test the validity of criteria for behavioral states and concludes that fetal "states" identical to coma, sleep or wakefulness do not exist in utero and are unique to adults.
My observation of increased variation with additional nuchal atonia episodes, a hallmark of fractal processes, suggests that difficulties of delineating the behavioral states of sleep and wakefulness in fetal sheep may result from the presence of fractal processes. The pattern of fetal behavioral states, if fractal in form, would not have a single natural time scale, and measurement of changes in these states would depend on the timescale of measurement or the size of time window used to classify an electrocorticogram tracing (Dement and Mitler, 1974). In support of this viewpoint, recent analysis of heart rate variability in the human fetus has demonstrated fractal structure in time (Gough, 1992; 1993; Shono et al.,1991), suggesting that the absence of a single natural time scale for making measurements may underlie the difficulties associated with diagnosing fetal distress and may partially explain the parallel increase in use of cesarean surgery with the availability of fetal monitoring (McCutcheon-Rosegg and Rosegg, 1984; Szeto, Personal Communication).
Nuchal atonia as a marker of REM sleep in fetal sheep. I have reviewed behavioral and neurophysiological studies supporting nuchal atonia as a correlate of REM sleep in adult species. The findings of the present study also support nuchal atonia as a marker of a REM sleep-like state in the sheep fetus. I observed a significant developmental pattern of decline in the number of nuchal atonia episodes across all animals and partitions with maturation, analogous to the ontological consolidation of markers of REM sleep in other species (Jouvet-Mounier, Astic and Lacote 1970; Roffwarg et al., 1966). I also observed a significant difference in the number of atonia episodes depending on the method of partitioning EMG activity, with P2 showing significantly more events that P1. I believe this difference is due in part to the sensitivity of P2, the most rigorous partition, to low amplitude digitization noise, resulting in the interruption of an ongoing absence of EMG activity and an associated generation of multiple atonia periods from single events. Also, for both partitions,the possibility exists that in some cases atonia periods were terminated prematurely by fetal movement artifacts from other channels, although this seems unlikely due to time lags observed between limb EMG and nuchal or ECoG recordings at higher resolution (data not shown). If, however, movement artifacts are present, my bias is that these occurrences represent in all likelihood concurrent phasic REM phenomena that could be confounded with nuchal atonia measurements without significantly affecting or invalidating the conclusions concerning the fractal nature of fetal REM associated processes. In fact, I would hypothese that other fetal behaviors (e.g. leg, trunk, and eye movements; breathing, heart beats) could be described as fractal time processes with similar Hurst exponents.
How does the presence of fractal correlations and non-Gaussian distributions influence the validity of statistical procedures, such as repeated measures ANOVA which are used to assess developmental changes in nuchal atonia? This is a fundamental problem that is at present unresolved. Fractal biological processes can run counter to the basic statistical assumptions involved in estimating stable population means or variances from stable sample means or variances because these processes do not show convergence with increasing sample size (Bassingthwaighte et al. 1994). Nuchal atonia distributions are stable over the course of development as was demonstrated by the invariance of the characteristic exponents and the form of the normalized probability distribution. This suggests that these processes are at least stationary over long periods of development. These confounds of non-convergent moments would prove more severe for non-repeated measures comparisons. Non-repeated measures ANOVA procedures allow comparisons between groups of uncorrelated means under the assumption that the means are normally distributed and have homogeneity of variance. Repeated measures ANOVA, however, by definition, compare correlated means and attempt to correct the F-test and probability levels for violations of assumptions by approximations such as the Hunyh-Feldt epsilon (Winer et al., 1991). The large partition dependent variations in both length and number of within subject atonia episodes probably contributed to the observed violations of the assumption of circularity. The observed violations of circularity are assumed to be another indication, along with the non-convergence of the cumulative moments, of the dissimilarity of fractal sequences of nuchal atonia episodes from normal Gaussian processes.
Given the above considerations, I conclude that nuchal atonia episodes investigated in this study are indicative of REM sleep for the following reasons: 1) At least one other REM associated state marker (low fast ECoG, EOG activity, tracheal pressure changes, movements of mouth or limbs) was usually observed to be coincident with nuchal atonia events, although this was not rigorously quantified for all days and animals; 2) the percent decline in total nuchal atonia developmentally, about 65% for E121 to 50% for E133, calculated by totaling atonia episode lengths over 24 hours, is similar to other reports of total REM time for sheep fetuses in this gestational range (Dawes et al., 1983; Clewlow et al., 1983; and Szeto, 1985); 3) A significantly linear trend of increasing nuchal atonia mean length with development suggests the possible consolidation of short nuchal atonia episodes into longer duration episodes, a trend reported for the clustering of rapid eye moments in human infants (Petre-Quadens, De Lee and Remy, 1971; Tanquay, Ornitz, Forsythe and Ritvo, 1976).
Non-convergence of Non-Poissonian Nuchal Atonia Episode Sequences. As I have reported, one key descriptive characteristic of nuchal atonia episode sequences is their deviation from normal probability distributions and the central limit theorem assumptions. This may seem abnormal given the pervasive use of Gaussian distributions. However, when viewed from the perspective of all Lévy distributions, the properties of normal Gaussian distributions, such as convergence of the mean and variance with increased sample size, are seen as unusual in a space of distributions where non-convergence of one or both moments is the norm (Takayasu, 1989). For one partition of nuchal atonia events, P1, I found the cumulative mean and variance were significantly non-convergent. I also report that, although nuchal atonia might appear, by the shape of the probability distribution, to be a Poisson distributed process (see Figure 9), the cumulative mean and standard deviation are significantly different for both P1 and P2. Therefore, nuchal atonia episodes are not consistent with a Poisson random process for the following reasons: 1) The cumulative mean is significantly different from the cumulative standard deviation; 2) The long tails of atonia distributions contribute to increased variance which does not converge with a larger "n", unlike the "law of large numbers" for Gaussian processes; 3) The Hurst exponent is > 0.5, suggesting long range correlations or dependency in the increments which violate assumptions of statistical independence.
Spontaneous Prenatal Behaviors as Non-Poisson Fractal Time Processes. Spontaneous prenatal behaviors such as spontaneous motility have been described previously and conceptualized as stochastic processes. Narayanan, Fox and Hamburger (1971), for example, reported that episodes of spontaneous motility in the E17-20 rat fetus could not be statistically distinguished from a Poisson random process. Recently, Blumberg and Lucas (1994a; 1994b) reported that limb twitching in normal and spinal transacted 5- and 8-day old rat pups also displayed Poisson random behavior as indicated by log-survivor analysis. The results of these experiments appear to differ from my findings that nuchal atonia events are not Poisson distributed.
One possibility is that traditional data collection methods are biased in that they are not sensitive to infrequent events (Fagen & Young, 1987) because an extended observation period or higher sampling of a fractal process would inevitably result in the appearence of outliers and higher variances. For example, Narayanan et al. recorded the length of periods of spontaneous unevoked activity and atonia in and ex utero for 15 minutes only. In their analysis of the distribution of spontaneous activity, the 15 minute observation time was divided into 45 periods of 20 seconds during which the number of movements was recorded. This method effectively high-pass-filters behavioral observations, restricting them to one "characteristic" timescale, 20 seconds, which effectively eliminates long periods of movement and restricts short movements to the sampling rate. Examining all relevant timescales is critical to obtaining a complete overview of temporal patterns of behavior. For example, in the analysis of behavioral repertoire size, the complexity of a species behavior repertoire increases with the sample size and length of the period of observation, thus necessitating very long records to adequately estimate the full range of behaviors (Fagen, 1987). When Fagen (1987) analyzed different distributions of complex behavior repertoires for optimum curve fits, the lognormal distribution, a prototypical fractal distribution (Montroll and Shlesinger, 1982; 1984; West and Shlesinger, 1990) was found to provide the best fit.
Validating the results found for fetal sheep, analysis of surface EMG recording of spontaneous motility from 2- and 5-day old rat pups collected at high sampling rates (300Hz) over long sampling times (one hour) reveal non-Poisson distributions and Hurst exponents similar to the sleep nuchal atonia (See Experiment 2). Similar to findings reported here for the clustering of nuchal atonia events are descriptions of clustering in the spontaneous discharge of single pyramidal track neurons during REM sleep in adult monkeys that have been shown to differ from a Poisson distribution because of an excess of short and long interspike intervals (Evarts, 1964; 1967). Evarts was able to demonstrate the non-random nature and behavioral state dependency of these discharge patterns by using an extensive range of counters and recording continuously from subjects over long periods. Consequently, spontaneous prenatal behaviors should be investigated with longer observation times or finer sampling rates along with the proper statistical tools (Bassingthwaighte et al., 1994; Rapp, 1994) such as Hurst analysis, before conclusions about the random nature of these developmental processes can be accurately drawn.
Hurst Analysis of Fractal Time Nuchal Atonia Sequences. The presence of Hurst exponents of greater than 0.5 for nuchal atonia episode sequences for all data partitions, days and subjects, indicating correlations between REM processes over many timescales, is an important finding of this study. Due to the necessity of examining 24 hr records (defined as one fetal day), I was limited to three orders of magnitude of nuchal atonia episodes in my Hurst analysis. Recent work by Berge et al. (1994) concerning Hurst analysis of particle flow through a single pore argues for five or more orders of magnitude to rule out crossover effects; these can lead to a spread in different Hurst estimates for the same subject or experiment situation resulting in a distribution of estimates centering on an effective H @ 0.75 (see figure 4.19 in Bassingthwaighte et al., 1994). I found, similarly, a distribution of Hurst estimates centering on H @ 0.70. Further investigation of these possible crossover effects, as reported by Berge et al., over time periods longer than 24hr in this data may prove valuable in elucidating the developmental emergence of circadian rhythmicity in the fetus (Robertson, 1988) which current results do not address. Another difficulty with Hurst analysis reported here is the developmental decline in number of nuchal atonia events from E121 to E133. Hurst analysis of short correlated sequences results, spuriously in H @ 0.5 (Bassingthwaighte and Raymond, 1994). A possible solution to the problem of nuchal atonia episode number influences on Hurst estimates would be to observe nuchal EMG at higher sampling rates (e.g., average over 500, 250 or 100 msecs). Anmuth, Goldberg and Mayer (1994) report fractal scaling relationships between EMG structure and muscle activation in humans with high sampling rates. Analysis of interspike intervals from single motor units may also be informative (Rapp, et al. 1993). Any of these procedures would enable high resolution examination by Hurst analysis for possible periodic ultradian processes which I did not observe in the current analysis. In addition, all physical and biological fractals have an effective range of timescales or spacescales over which they manifest scaling behavior (Bassingthwaighte et al. 1994). Addressing this issue, analysis of 200 Hz recordings of nuchal activity from other sheep fetuses (illustrated in Figure 3, p.31) indicate that the scaling relationships reported here for 24 hours are present with similar H values over time periods of 5 seconds (Anderson et al., unpublished). In order to determine if patterns of nuchal activity as indexed by the Hurst exponent and distrubutional charateristics would be similar in different species a series of observations of nuchal inactivity sequences in infant rats were carried out and are reported in the next section.
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EXPERIMENT 2: ANALYSIS OF NUCHAL INACTIVITY SEQUENCES IN NEONATAL RATS: AN EXAMPLE OF THE FRACTAL TIME STRUCTURE OF A SPONTANEOUS POSTNATAL BEHAVIOR
Neonatal rats display spontaneous movements or twitches of the limbs, the trunk, the head and tail indicative of a state of REM sleep (Blumberg and Lucas, in press; Gramsbergen, Schwartzen and Prechtal, 1970; Jouvet-Mounier, Astic and Lacote,1970; Tamásy, Korányi and Lissák, 1980; Van Someren et al. 1990). These movements are associated with fluctuations of muscle tone which reflect central processes associated with the induction of muscle inactivity and REM sleep. Over the first postnatal week, one key indicator of REM DLVF ECoG is difficult to record reliably due to masking by EMG artifacts. In fact, the other classical indicators of REM, loss of nuchal muscle tone and paroxysmal musclar twitches accompanied by rapid conjugate eye movements (observable on PN 5-6), are the only reliable indicators of this state in infant rats (Jouvet-Mounier et al., 1970).
In this experiment, I investigated the temporal structure of spontaneous fluctuations of nuchal muscle tone in the form of inactivity sequences in infant rats on postnatal (PN) days 2, 4, 6, 8, and 10 with Hurst's rescaled range analysis, a statistical technique that distinguishes between natural time series with correlated and uncorrelated fluctuations (Mandelbrot and Wallis, 1995). I chose the term nuchal inactivity rather than nuchal atonia to distinguish the recording technique used here from that used to measure nuchal atonia in fetal sheep (e.g., surface EMG as opposed to intramuscular wire EMG). My analysis indicates that nuchal inactivity sequences are fractal patterns in time and that
over short time scales are statistically similar to those over longer time scales as in the nuchal atonia sequences found in fetal sheep. Hurst exponents (H) are distributed around 0.77 for the 0 hr deprivation condition and 0.80 for the 2 hr deprivation condition over all ages (H for an independent random process would = 0.5). The mean H exponent is within 0.07 units of the H values for fetal sheep, strongly suggesting that the H exponent is measuring a process associated with neonatal REM sleep across age and species. Further supporting the finding that the fractal nature of these patterns of nuchal inactivity sequences are inconsistent with a Poisson process are the following observations: 1) the mean does not equal the variance; 2) H 0.5; and 3) probabilities are in the family of non-finite moment Lèvy stable distributions with a characteristic (tail) exponent of a < 2.0 ( a 1.8). The early postnatal rat has been proposed to bear similarities to the third trimester fetus, suggesting it may spend a similar amount of time in a REM state similar to that of the fetal sheep. Based on the experimental findings detailed below, I conclude that these fractal time patterns of nuchal inactivity sequences represent an extension of the temporal structure of the global fetal REM sleep state into the postnatal period. This extension supports the notion that these properties may serve as a transient behavioral ontogenetic adaptation to changing developmental environments. In addition, the change in the clustered patterns of nuchal inactivity (i.e., shift to more persistent and prolonged bouts of clusters) with 2 hours of maternal deprivation as indicated by the increase in H (increase in presistance) and decrease in the Lèvy tail exponent (increase in cluster size), further support the usefulness of fractal time measures as a window into heterochromatic mechanisms underlying the plasticity of developing organisms to changing environmental factors.
SPECIFIC METHODS
Subjects. Forty male and forty female infant rats from 26 litters were tested (25 of which are reported here) from the age groups postnatal day (P) 2,4,6,8 and 10. The average body weights of the animals used are represented in Figure 10. All infant rats were born to Sprague-Dawley females (Charles River CD strain) in the animal colony at Florida Atlantic University. Water and Purina Lab Chow (#5012) were provided ad libitum. All animals were housed under temperature- and humidity-controlled conditions (21-23oC and 40-70% relative humidity) on a 14:10 hour light:dark cycle (lights on at 0700 hrs). Infant rats were raised in litters culled to 14 rats within 2 days after birth (day of birth = Day 0). Larger litters were used to ensure that littermates remaining after removals would not grow excessively large. Due to the necessity of collecting 2-hour-long time series recordings, only 2-3 infant rats/litter were examined per age.
Preparation of Subjects. Infant rats were quickly collected from the dam to minimizing disturbance to the the subject and remaining littermates. The subjects were transported in a small covered container (to attenuate thermal shock) filled with home cage bedding and then quickly weighed on an electronic balance before beginning electrode placement. Electrode placement and design was informed by Loeb and Gans (1986). Small amounts of Nuprep (D.O. Weaver & Co., Aurora Co), a cleaning abrasive, were applied to the neck and dorsal trunk to remove oils and lightly abrade the skin to improve electrode conductance. Three small electrodes (fabricated by cutting 4 mm [approx.] section from the tip of a conical microcentrifuge tube, inserting coiled 1mm bare silver wire through the
Figure 10. Bar chart of average body weights and standard errors of subjects used in these experiments illustrates the normal developmental changes in body weight. Doubling of body weight over this postnatal period is evident from linear trend analysis indicating a strong linear increase in weight with Postnatal Day F(1,16) = 158.457, p<.0001, which accounted for 99.46% of the variance.
tube wall and filling with electrode paste) were attached to the sites by super-glue. Two electrodes were applied bilaterally to the neck region approximately 5mm lateral to the midline, and the ground electrode was applied to the base of the tail. Anatomical guides were used to define nuchal muscle in the infant by extrapolating from the adult rat (Greene, 1963, p. 35; Hebel and Stromberg, 1976, p. 19; and Wingerd 1988, p. 19).
Soon after this procedure (the entire process takes less than 5min), the subjects were placed in a jacketed chamber with the temperature and relative humidity maintained close to thermoneutrality (between 34.5-35.8oC and 50%-64%, respectively) for testing (Satinoff, 1991). Subjects responded well to the procedure and appeared to enter REM sleep quickly after being placed in the testing chamber, as evidenced by the rapid appearance of twitches (Szymusiak, Stinoff, Schallert and Whishaw, 1980). Unlike the fetal sheep described in the last section, neonatal rats when removed from the dam are subject to maternal deprivation effects (see Anderson, 1991; Hofer, 1976). Maternal deprivation is therefore a treatment condition in these experiments.
Data Acquisition and Analysis. A Grass model 7P511L high performance AC amplifier in combination with a Grass model 7HIP5G high impedance probe was used to record neck surface EMG and movement artifacts generating an analog -5 to 5mV signal proportional in amplitude to the µV range biological source signal. Additional recordings of movements detected with a electrostatic motion sensitive device were also made. In this case movement patterns were detected and converted into analog 0-1V signals proportional in amplitude to the movements by a capacitive proximity sensor monitored by a radio-frequency circuit using a phase-locked-loop in the 1 megHz frequency range. The voltage output from one or both amplifiers was sent to a MacADIOS II jr 12-bit A/D (equipment, GW Instruments, Inc., Somerville, MA) sampled at 300 Hz and processed offline.
Nuchal inactivity sequences were detected by first full wave rectifing the 300 Hz
Figure 11. A segment of raw unrectified nuchal EMG (-5 to 5 mV) showing intermittent bursts of nuchal activity from 6194 to 6218 seconds. At 6218 seconds a large sustained burst of nuchal muscle contraction ensues. The regions of intermittent bursting of nuchal EMG associated with behavioral twitching would result in the detection of long episodes of inactivity. The sustained burst of EMG associated with behaviors such as orienting head movements would result in many short episodes of inactivity.
EMG signals. An example of the raw unrectified nuchal EMG can be found in Figure 11.
Two partitions of the resulting signal were then examined to distinguish baseline noise
artifacts and to determine an unbiased estimate of the duration of inactivity episodes. The first partition (P1) was constructed by observing the range of variation of baseline EMG amplitudes (typically 0 to 0.1 mVs after rectifying) and then recording the durations of EMG amplitude threshold range (e.g., values < 0.1 mV would count as inactivity; events of amplitude > 0.2 would not). The second partition (P2) examined the range of 0.1 to 0.5
mVs. Other partitions, 0.5 to 1.0 and 1.0 to 2.0 mVs, were also examined but are not reported here. These partitions were well above the background noise levels ( < 0.01 mVs) in most cases. In addition, two periods of the recording, the first 5 and last 5 minutes of the 2 hour recording session were compared to determine the effects of maternal deprivation on nuchal inactivity episodes. Representative histograms generated from these partitions are illustrted in Figure 12. In anesthetized subjects (Methoxyflurane[Metafane Pittman Moore, Washington Crossing, NJ]), complete loss of muscle tonus was observed and no segmentation of nuchal inactivity occurred (see Figures 15a and b). This manipulation constitutes a control for non-specific noise resulting from the instrumentation as well as heart beat artifacts.
Statistical Properties of Nuchal Inactivity Episodes. Changes in mean nuchal inactivity episode length and number over P2-10 were assessed by 3-way ( 2 Partitions ¥ 5 Postnatal Days ¥ 2 Deprivation conditions) Analysis of Variance (ANOVA ). Orthogonal linear trend analysis was used to further examine developmental trends for the number and average mean nuchal inactivity episode length over the period of observation.
To examine the statistical properties of the Moments (i.e., cumulative mean and variance) a 2-way (2 Moments ¥ 2 Partitions ) ANOVA was calculated for data sets 5
Figure 12. Representative histograms of inactivity episodes generated from partition P1 [0.1- 0.2 mV] in (a), P2, [0.1- 0.5 mV] in (b) and a partition at [0.5- 1.0 mV] in (c) are illustrated for a 10-day-old subject after 2 hrs of deprivation. The log of episode number per bin size is plotted on the ordinate and the length of inactivity episodes in time units (300/sec) is represented on the abscissa. Notice how the number of short inactivity episodes decreases with increasing partition threshold and the number of long inactivity episodes increases with partition threshold.
minutes in length. The criterion for type 1 errors was set on all comparisons at p = 0.05 and the Huhyn-Feldt procedure was used to assess violations of circularity when repeated measures were used (Winer, Brown and Michels, 1991).
Hurst's Rescaled Range Analysis. The Hurst exponents were estimated by the segmentation procedure detailed for fetal sheep (p.24). Validation of Hurst analysis was accomplished by the randomization procedure described for fetal sheep (p.24). Developmental changes in estimated Hurst exponents for original and randomized data sets were compared by 2-way (2 Partitions ¥ 13 Fetal days) ANOVA, with fetal day treated as a repeated measure. Developmental trends were tested by orthogonal linear trend analysis of average H values over the period of observation.
Estimating the Lévy Tail Exponent. I approximated the normalized density distributions of inactivity episodes for all animals for each age P 2 -10 with parametrically fitted algebraic Lévy curves as described for fetal sheep (p. 27). The asymptotic standard errors were computed as was described for fetal sheep. For neonatal rats, these matrices were singular (i.e., not all positive with nonzero roots), and the standard errors of the multparameter estimates were not computable. A common solution to this problem is to set one of the model parameters constant and refit the simpler model. Because my primary concern was the parameter and because the convergent values varied little across the 10 files (5 days ¥ 2 partitions), I set = 3.1, its mean value across all files. The single parameter Simplex estimates converged quickly and as was described for fetal sheep had small asymptotic standard error estimates (all A.S.E. < 0.008).
RESULTS
Nuchal EMG records. Recordings made of neonatal rat nuchal muscle potentials appear remarkable similar to intramuscular potentials in fetal sheep (see Figure 13, compare with Figure 1 p. 8). Self-affinity is evident when rescaling in both the dimensions of amplitude and time so that 20 second periods appear qualitatively similar to those of 2 seconds which in turn resemble those over the time scale of minutes. This fractal time structure of nuchal muscle potentials sampled at 300 Hz in the neonatal rat appears to be very similar to nuchal muscle potentials recorded at 200 Hz in fetal sheep (see Figure 3, p. 31). As in examination of the fetal sheep, the scaling structure of the irregular patterns of intermittent bursts of nuchal muscle activity and inactivity appeared to also have psychobiological significance: they corresponded roughly with other indicators of REM sleep such as muscle twitches, here detected by changes in the electrical capacitance of the recording chamber (see Figure 14).
Statistical Properties of Nuchal Inactivity Episodes. Overall the results of the analysis suggest two main findings: As the neonate matures, there is: 1) a reduction in the number of nuchal inactivity episodes (Figure 16a) and 2) an increase in the mean length of nuchal inactivity periods (Figure 16b). This is concordant with the findings for nuchal atonia defined for 1Hz data over 24 hours in fetal sheep reported in experiment 1. This further supports the observation of the fractal nature of nuchal EMG which implies that behavioral activity, whether recorded at 200 Hz or 1Hz, displays the same time structure.
Figure 13. Erratic intermittent bursting is apparent in nuchal activity with intervening periods of inactivity in a rectified recording from a 2-day old neonatal rat. Nuchal EMG activity along the abscissa (rectified, 10 mvolt / division), plotted over 20 seconds (6000 points at 300 Hz) on the ordinate (top trace). Rescaling the amplitude, abscissa, and time, the ordinate, over a 2 second subset of the original series (middle trace), reveals smaller clusters within larger clusters of both muscle discharges and inactivity. Rescaling both dimensions again over .67 seconds (bottom trace) demonstrates similar qualitative "self-affinity" which is bounded from below by the sampling rate.
Figure 14. The top plot illustrates changes in electrical capacitance of the recording chamber (in mVolts on the ordinate) due to a burst of twitching behavior in a 3-day-old infant rat (in msecs on abscissa). The bottom plot corresponds to the EMG of nuchal muscle (in mVolts on the ordinate) during the time (same scale as above) that twitches were observed in the burst of the top plot indicative of the state of REM sleep. The EMG recording was made before a high impedance probe was available, resulting in the high level of background noise.
Figure 15. This figure illustrates the control condition of anaesthesia. The subject, a 10-day-old pup, was administered Metafane shortly before being placed in the testing chamber. After a complete loss of muscle tonus was observed the electrode leads were connected and the EMG record in Figure 15a was recorded. Figure 15b was recorded 120 seconds after 15a and represents partial recovery from the Metafane. One long period of nuchal inactivity resulted without any interruptions. This manipulation constitutes a control for non-specific noise resulting from the instrumentation as well as heart beat artifacts and suggests that data presented only represents nuchal and movement events
Figure 16. Developmental changes over P2-10 in (a) mean number of nuchal inactivity episodes and (b) mean length of episodes over the non-deprived 5min recording period for the lower amplitude, more sensitive partition (P1) and higher amplitude, less sensitive partition (P2) in all animals. Error bars indicate the standard error of the mean for each 5min period.
Figure 17. Developmental changes over P2-10 in (a) mean number of nuchal inactivity episodes and (b) mean length of nuchal inactivity episodes over the deprived 5 min recording period for the lower amplitude, more sensitive partition (P1) and higher amplitude, less sensitive partition (P2) in all animals. Error bars indicate the standard error of the mean for each 5min period.
Two hours of maternal deprivation result in: 1) a reduction in the number of nuchal inactivity episodes at all ages (Figure 17a) compared with no deprivation and 2) a larger increase in the mean length of nuchal inactivity periods at all ages (Figure 17b) suggestive of larger clusterings of inactivity events. Without exception, application of the lower partition P1 resulted in larger numbers of inactivity events with shorter durations than inactivity events in P2 (Figures 16a & b and 17a & b). This may be the result of noise in the EMG recordings which resulted in fragmentation of longer periods of inactivity. Two 3-way ANOVAs ( 2 Partitions ¥ 5 Postnatal days ¥ 2 Deprivation conditions) comparing Partitions for mean nuchal inactivity episode length and for nuchal inactivity number revealed significant main effects of Partition, F(1, 80) = 20.013, p<.0001, and F(1, 80) = 63.329, p<.0001, respectively. A significant main effect of Postnatal Day was also present for inactivity number F(4, 80) = 4.847, p<.005, and for mean length F(4, 80) = 4.735, p<.005. This was suggestive of a strong developmental trend for decreasing nuchal inactivity episode number over gestation which was evident with further trend analysis of Postnatal Day. In agreement with reports showing a decline in postnatal markers of REM sleep (such as DLVF ECoG) with development (Gramsbergen et al., 1970; Jouvet-Mounier et al., 1970; Tamásy et al., 1980; Van Someren et al., 1990), trend analysis indicated linear developmental trends for decreases in the number of nuchal inactivity events with Postnatal Day F(1,16) = 14.475, p<.0005, and mean nuchal inactivity episode length with Postnatal Day F(1,16) = 14.479, p<.0005, which accounted for 74.66% and 76.45% of the variance. Departures from linearity accounted for 25.34% and 23.55%, respectively, of the remaining variance but did not reach criterion. In addition, a significant main effect of Deprivation condition was also present for nuchal inactivity number[ F(1, 80) = 9.416, p<.005] and for nuchal mean length [F(1, 80) = 7.630, p<.01] suggesting that the amount of time spent away from the dam could be a critical influence on nuchal inactivity episode patterns. Finially, a 2-way interaction of Partitions of nuchal inactivity number with Deprivation conditions proved to be significant: F(1, 80) = 4.811, p<.05 . A test of the effect of partitioning on nuchal inactivity number was found to be significantly different by orthogonal contrast [F(1, 96) = 9.416, p<.0029] under the condition of deprivation but not under non-deprived conditions.
To test whether sequences of nuchal inactivity are statistically similar to a Poisson process (e.g., for a Poisson process the mean equals the variance) and to test the effects of Partition and Postnatal Day, the cumulative Moments (i.e., mean and variance of nuchal inactivity events summed progressively over the 5 minute recording window) were analyzed by a 3-way ANOVA (2 Moments ¥ 2 Partitions ¥ 5 Postnatal Days). This test detected a significant main effect of cumulative Moments [F(1, 180) = 5.104, p<.05] and Partitions [F(1, 180) = 3.987, p<.05]. None of the 2 or 3-way interactions, however, proved to be significant. The nature of the effect of Moments is evident in Figure 18a and b where the magnitude of difference between the cumulative mean and variance for both Partitions over development is seen to be quite large for both the non-deprived (Figure 18a) and deprived conditions (Figure 18b). Also, the magnitude of the difference between the cumulative mean and variance is greater for P2 than P1. The large differences between the cumulative mean and variance strongly argue against the possibility that nuchal inactivity is a Poisson process and confirms the findings of large differences between the cumulative mean and variance in fetal sheep.
Summarizing the findings so far, significant developmental decreases in the nuchal inactivity episode number were observed over the developmental period examined. This is compatible with previous reports of developmental declines in REM sleep with development in the rat (Gramsbergen et al., 1970; Jouvet-Mounier et al., 1970; Tamásy et al., 1980; Van Someren et al., 1990). The second partition (.1-.5 mV) demonstrated the
Figure 18 The cumulative mean duration and variance (a) derived for both partitions over increasing numbers of nuchal inactivity episodes for the five subjects during the non-deprived period and (b) during the deprived period. Both of these conditions fail to demonstrate the expected convergence of the variance in general and the mean in particular of a Poisson process and are completely concordat with similar findings in fetal sheep (Figures 6 a&b, p. 39).
largest change with development. In the non-deprived condition, nuchal inactivity episode number went from 7500 events to 3500 events (Figure 16a), and in the deprived condition from 7500 to 2200 events (Figure 17a). In addition, the change in mean length of nuchal inactivity was also larger for P2, where it tripled from 5 to 15 points (16.7 to 50 msecs) in the non-deprived condition compared with 2.5 to 27 points (8.3 to 90 msecs) for the deprived condition. The cumulative mean duration and variance of inactivity episodes were significantly different from each other and highly variable, as indicated by the large standard errors, and possibly non-convergent in terms of the central limit theorem although longer sequences of nuchal inactivity episodes (>5min) should be examined to see if the mean and/or variance converge with sample size. These data are consistent with the hypothesis that REM periods, when indexed by nuchal inactivity periods, lack a charateristic timescale like other fractal time processes and tend to consolidate into longer clusters with increasing age.
Results of Hurst Analysis of Nuchal inactivity Sequences. Hurst exponents calculated for different partitions of the nondeprived condition ranged from 0.687 to 0.856 for P1 and from 0.648 to 0.846 for P2 during the period of observation (Figures 19a and b), suggesting the presence of long-run dependency between nuchal inactivity episodes over 5 minutes. For the deprived condition, Hurst exponents ranged from 0.741 to 0.867 for P1 and from 0.701 to 0.866 for P2. Histograms of the distribution of Hurst exponents for both P1 and P2, original and randomized, are displayed in Figures 20a and 20b for the non-deprived condition and in Figures 21a and 21b for the deprived condition.
A 3-way repeated measures ANOVA (2 Partitions ¥ 2 Deprivation Conditions ¥ 2 Hurst Estimates [original and randomized]) was used to assess the effects of Partition and Deprivation Conditions on estimated Hurst exponents. This analysis indicated the significant main effect of Deprivation Condition F(1,24) = 5.692, p>0.05,e = 1 and Hurst Figure 19. The least squares fits of the mean of the Hurst exponents for the non deprived (a) and deprived conditions (b) illustrate the changes in persistence (defined by H > 0.5) for the original sequences of nuchal inactivity episodes for each partition. The R represents the Hurst values (H 0.5) computed in the same way following randomization of the sequences data for partition 1 and 2 over development, reflecting the expected loss of long range correlations. Notice the similarity with Hurst values from fetal sheep (Figure 8, p. 43). Notice also that H values for the deprived condition (b) are higher than the non-deprived (a)
Figure 20. The distribution composed of all the 5 min values of Hurst exponent (H) estimates for (20a) original non-deprived and (20b) randomized sets for non-deprived for both P1 and P2 partitions over the postnatal period examined. The largest probability mass of the Hurst exponents for the original non-deprived sequence was in the neighborhood of H 0.78 for deprived. H0.55 for the randomized set.
Figure 21. The distribution composed of all the 5 min values of Hurst exponent (H) estimates for (a) original deprived and (b) randomized sets for deprived for both P1 and P2 partitions over the postnatal period examined. The largest probability mass of the Hurst exponents for the original non deprived sequence was in the neighborhood of H 0.80 for deprived subjects which was significantly different from the non-deprived estimates. H 0.55 for both randomized partitions.
Estimates F(1,24) = 1502.990, p>0.0001,e =1. In addition, a 2-way interaction of Deprivation Condition and Hurst Estimates was also significant F(1,24) = 6.653, p>0.05,e = 1, indicating that deprivation increased the magnitude of the normal Hurst exponents suggesting an increase in behavioral clustering. A test of the effect of Deprivation on the magnitude of the normal Hurst exponent was found to be significantly different by orthogonal contrast F(1, 196) = 5.385, p<.0029, under the condition of deprivation but not under non-deprived condition (see Figure 19b, p. 97).
Hurst exponents calculated from randomized surrogate data sets of P1 and P2 ranged from 0.483 to 0.586 and 0.438 to 0.573, respectively, for the non-deprived condition and from 0.458 to 0.584 and 0.474 to 0.585, respectively, for the deprived condition. Hurst exponent estimates from the original data sets were found to be significantly different from estimates derived from randomized surrogate data sets, independent of Partition, confirming the existence of long-run dependency.
In summary, the magnitudes of Hurst exponent estimates were dependent on the deprivation condition, with 2hrs of deprivation resulting in larger estimates. As a demonstration of the statistical validity of these estimates, Hurst exponents derived the from original time series were significantly different from Hurst exponent estimates from randomized surrogate data for both partitions and deprivation conditions.
The probability density distributions describing nuchal inactivity episodes. To test the hypothesis that nuchal inactivity episodes, as with fetal sheep nuchal atonia episodes, might not be described by normal Gaussian distributions, the goodness of fit to other Lévy distribution curves, was examined, by the derived characteristic exponents a at fixed g. Fitting the probability distribution yielded estimates of a for each postnatal day examined. Nuchal inactivity intervals from Partion 2 were examined. The values of a for non- deprived animals were 1.5839 for postnatal day 2; 1.5076 for postnatal day 4; 1.6125 for postnatal day 6; 1.5749 for postnatal day 8 ; and 1.6050 for postnatal day 10 at g = 3.0062. The values of a for deprived animals were 1.8750 for postnatal day 2; 1.8656 for postnatal day 4; 1.8916 for postnatal day 6; 1.8809 for postnatal day 8 ; and 1.8823 for postnatal day 10 at g = 3.0062. Non-deprived animals collapsed across age were found to be significantly different from deprived animals by 1-way repeated measures ANOVA (Deprivation Conditions), F(1, 4) = 415.028, p>0.0001, e = 1, suggesting that deprivation changes the nature of the underlying probability density distributions by shifting them toward vaules of a where both the first and second moment are non-convergent. The non deprived a value is remarkably similar to that found for fetal sheep (p. 49). In addition, the normalized probability density distributions of nuchal inactvity episodes calculated for the two partitions were found to be remarkably similar in form over species and development (compare Figure 22 with Figure 9, p. 51) which is consistent with both the unbiased nature of the estimation of inactivity events by the partitioning procedure and the finding of invariance in the characteristic exponents with maturation.
Figure 22. The log-linear plots of the probability density distributions for nuchal inactivity episode durations in representative 5 min periods for partition P1 and P2 post-deprivation demonstrate the highest densities at the lower bounds of sampling rate resolution in a 2- and 10-day-old subject, as in a Poisson process. In addition, they display long tails which failed to converge over the sample period. This distribution remained invariant over the 9 days of observation for subjects despite a shift in the average length of nuchal inactivity episodes, concordant with the findings in the sheep fetus (compare with Figure 9, p. 50).
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DISCUSSION
The erratic cluster-within-clusters appearance of nuchal EMG activity over the 100 minute period illustrated in Figure 1 from the E123 sheep fetus is barely distinguish-able from the intermittent bursting behavior of nuchal activity recorded from the 2-day-old neonatal rat over 20 seconds in Figure 13. This observation is striking in that it suggests that developing biological systems are richly endowed with multiple levels of order extending over a spectrum of timescales and that the study of the fractal temporal structure of organisms is as important as the study of spatial organization in understanding the roles of behavioral processes such as REM sleep in adaptations throughout development. Another surprising conclusion from this comparison is that nuchal activity recorded over 100 minutes in one species is similar to nuchal activity recorded over 20 seconds in another species. In fact, all of the findings derived from 24 hr data in sheep fetuses are almost identical to the findings derived from infant rats over a 5 minute period. This observed self-similarity affords several important implications: 1) the electro-physiological behavior of the nuchal muscle is statistically similar over 4 orders of magnitude in the sheep and probably in the neonatal rat; 2) similarly, REM related behavior is statistically correlated over many timescales; 3) different species, possibly including humans, share this invariant time structure. The principal conclusion from both experiments is that normal statistical analysis of this data (e.g., not investigating order over many different scales of time) would most likely not be sensitive to this developmental invariant. The analysis of the intervals of inactivity of Figure 13 also revealed surprising statistical properties, such as
non-convergent cumulative moments, not found in normally distributed data. Nuchal inactivity in the rat, like atonia in the fetal sheep, is therefore neither Gaussian nor Poissonian, but is highly ordered in that statistical interdependence extends between these intervals of inactivity over many time scales.
Nuchal inactivity as a marker of REM sleep in neonatal rats. I have reviewed behavioral and neurophysiological studies supporting nuchal atonia as a correlate of REM sleep in adult species and fetal sheep. The findings of Experiment 2 also support nuchal atonia as a marker of a REM sleep-like state in the neonatal rat, which more closely resembles the third trimester fetus than other precocious rodents such as the hamster. I observed a significant developmental pattern of decline in the number of nuchal inactivity episodes and increase in mean length across all animals and partitions with maturation, analogous to the ontological consolidation of markers of REM sleep in other species (Jouvet-Mounier et al., 1970; Roffwarg et al., I also observed a significant difference in the number of inactivity episodes depending on the method of partitioning EMG activity, with the more sensitive (lower absolute threshold) P1 showing significantly more events that P2. I believe this difference is due in part to the sensitivity of more sensitive partition to low amplitude noise events in the EMG not observed in the control experiment (see Figure 15a), resulting in the interruption of an ongoing inactivity event and resulting in the generation of multiple inactivity periods from single events. Also, for both partitions, inactivity periods could have possibily been terminated prematurely by startle and twitching movement artifacts (see Figure 14). I believe these movement artifacts are present, and that these occurrences represent concurrent phasic REM phenomena with a similar fractal structure that could be confounded with nuchal inactivity measurements without significantly affecting or invalidating the conclusions concerning the fractal nature of neonatal REM sleep just as described for the fetal sheep observations.
Nuchal inactivity distributions also appear stable over the course of development in the postnatal rat as was demonstrated by the invariance of the characteristic exponents and the form of the normalized probability distribution. This suggests that these concurrent phasic REM phenomena are at least stationary over long periods of development. In addition, the relatively large number of nuchal events observed in this study may be due in part to the normal polyneuronal innervation of skeletal muscle that has been observed in the newborn rat (Brown, Jansen and Van Essen, 1976). Brown et al. found a rapid drop in the precentage of soleus muscle fibres innervated by more than one motor neuron axon after day 10, suggesting activity dependent synaptic elimination resulting in higher background activity. Brown et al. also observed much greater fluctuations in maximal end plate potentials from days 2 to 10 in muscle fibers which may be correlated with the greater quantity of twitching observed during this period (Jouvet-Mounier et al., 1970).
From a different perspective, Blumber and Lucas (in press), suggest that the "simple random firing mechanism" or "desynchronized" twitching during this period is critical in uniquely identifying different "cartels" (as defined by Colman and Lichtman, 1993) of motor neurons actively competing with each other to "monopolize" innervation of muscle fibers. I will detail my objections to the notion of random mechanisms in a later section; here, however, I propose that a view of activity patterns as having order imparted by fractal time processes does provide a richer perspective from which to view these complex interactions occurring within and among the components of cytoplasm, membranes, neuromuscular junctions, muscle fibers, motor neuron "cartels", muscles, muscle groups, etc. In fact, just the possibility of common fractal patterns in the activity of both muscles and the motor neurons that innervate them would be more suggestive of vertical and horizontal co-interactions and cooperation than competition and monopolization.
Given the above considerations, I conclude that nuchal atonia episodes investigated