Fractal properties of sympathetic nerve discharge

Nonlinear analysis of neuronal firing modulated by sinusoidal stimulation at axons in rat hippocampus

Introduction

Electrical stimulation of the brain has shown promising prospects in treating various brain diseases. Although biphasic pulse stimulation remains the predominant clinical approach, there has been increasing interest in exploring alternative stimulation waveforms, such as sinusoidal stimulation, to improve the effectiveness of brain stimulation and to expand its application to a wider range of brain disorders. Despite this growing attention, the effects of sinusoidal stimulation on neurons, especially on their nonlinear firing characteristics, remains unclear.

Methods

To address the question, 50 Hz sinusoidal stimulation was applied on Schaffer collaterals of the rat hippocampal CA1 region in vivo. Single unit activity of both pyramidal cells and interneurons in the downstream CA1 region was recorded and analyzed. Two fractal indexes, namely the Fano factor and Hurst exponent, were used to evaluate changes in the long-range correlations, a manifestation of nonlinear dynamics, in spike sequences of neuronal firing.

Results

The results demonstrate that sinusoidal electrical stimulation increased the firing rates of both pyramidal cells and interneurons, as well as altered their firing to stimulation-related patterns. Importantly, the sinusoidal stimulation increased, rather than decreased the scaling exponents of both Fano factor and Hurst exponent, indicating an increase in the long-range correlations of both pyramidal cells and interneurons.

Discussion

The results firstly reported that periodic sinusoidal stimulation without long-range correlations can increase the long-range correlations of neurons in the downstream post-synaptic area. These results provide new nonlinear mechanisms of brain sinusoidal stimulation and facilitate the development of new stimulation modes.

Dynamics of neuronal firing modulated by high-frequency electrical pulse stimulations at axons in rat hippocampus

Objective. The development of electrical pulse stimulations in brain, including deep brain stimulation, is promising for treating various brain diseases. However, the mechanisms of brain stimulations are not yet fully understood. Previous studies have shown that the commonly used high-frequency stimulation (HFS) can increase the firing of neurons and modulate the pattern of neuronal firing. Because the generation of neuronal firing in brain is a nonlinear process, investigating the characteristics of nonlinear dynamics induced by HFS could be helpful to reveal more mechanisms of brain stimulations. The aim of present study is to investigate the fractal properties in the neuronal firing generated by HFS.Approach. HFS pulse sequences with a constant frequency 100 Hz were applied in the afferent fiber tracts of rat hippocampal CA1 region. Unit spikes of both the pyramidal cells and the interneurons in the downstream area of stimulations were recorded. Two fractal indexes-the Fano factor and Hurst exponent were calculated to evaluate the changes of long-range temporal correlations (LRTCs), a typical characteristic of fractal process, in spike sequences of neuronal firing.Mainresults. Neuronal firing at both baseline and during HFS exhibited LRTCs over multiple time scales. In addition, the LRTCs significantly increased during HFS, which was confirmed by simulation data of both randomly shuffled sequences and surrogate sequences.Conclusion. The purely periodic stimulation of HFS pulses, a non-fractal process without LRTCs, can increase rather than decrease the LRTCs in neuronal firing.Significance. The finding provides new nonlinear mechanisms of brain stimulation and suggests that LRTCs could be a new biomarker to evaluate the nonlinear effects of HFS.

Rhythmic firing of neurons in the medulla of conscious freely behaving rats: rhythmic coupling with baroreceptor input

Recent investigations emphasized the importance of neural control of cardiovascular adjustments in complex behaviors, including stress, exercise, arousal, sleep-wake states, and different tasks. Baroreceptor feedback is an essential component of this system acting on different time scales from maintaining stable levels of cardiovascular parameters on the long-term to rapid alterations according to behavior. The baroreceptor input is essentially rhythmic, reflecting periodic fluctuations in arterial blood pressure. Cardiac rhythm is a prominent feature of the autonomic control system, present on different levels, including neuron activity in central circuits. The mechanism of rhythmic entrainment of neuron firing by the baroreceptor input was studied in great detail under anesthesia, but recordings of sympathetic-related neuron firing in freely moving animals remain extremely scarce. In this study, we recorded multiple single neuron activity in the reticular formation of the medulla in freely moving rats during natural behavior. Neurons firing in synchrony with the cardiac rhythm were detected in each experiment (n = 4). In agreement with prior observations in anesthetized cats, we found that neurons in this area exhibited high neuron-to-neuron variability and temporal flexibility in their coupling to cardiac rhythm in freely moving rats, as well. This included firing in bursts at multiples of cardiac cycles, but not directly coupled to the heartbeat, supporting the concept of baroreceptor input entraining intrinsic neural oscillations rather than imposing a rhythm of solely external origin on these networks. It may also point to a mechanism of maintaining the basic characteristics of sympathetic neuron activity, i.e., burst discharge and cardiac-related rhythmicity, on the background of behavior-related adjustments in their firing rate.

The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus

This is the third essay advocating the use the (non-integer) fractional calculus (FC) to capture the dynamics of complex networks in the twilight of the Newtonian era. Herein, the focus is on drawing a distinction between networks described by monfractal time series extensively discussed in the prequels and how they differ in function from multifractal time series, using physiological phenomena as exemplars. In prequel II, the network effect was introduced to explain how the collective dynamics of a complex network can transform a many-body non-linear dynamical system modeled using the integer calculus (IC) into a single-body fractional stochastic rate equation. Note that these essays are about biomedical phenomena that have historically been improperly modeled using the IC and how fractional calculus (FC) models better explain experimental results. This essay presents the biomedical entailment of the FC, but it is not a mathematical discussion in the sense that we are not concerned with the formal infrastucture, which is cited, but we are concerned with what that infrastructure entails. For example, the health of a physiologic network is characterized by the width of the multifractal spectrum associated with its time series, and which becomes narrower with the onset of certain pathologies. Physiologic time series that have explicitly related pathology to a narrowing of multifractal time series include but are not limited to heart rate variability (HRV), stride rate variability (SRV) and breath rate variability (BRV). The efficiency of the transfer of information due to the interaction between two such complex networks is determined by their relative spectral width, with information being transferred from the network with the broader to that with the narrower width. A fractional-order differential equation, whose order is random, is shown to generate a multifractal time series, thereby providing a FC model of the information exchange between complex networks. This equivalence between random fractional derivatives and multifractality has not received the recognition in the bioapplications literature we believe it warrants.

Hidden Signals-The History and Methods of Heart Rate Variability

The understanding of heart rate variability (HRV) has increased parallel with the development of modern physiology. Discovered probably first in 1847 by Ludwig, clinical applications evolved in the second part of the twentieth century. Today HRV is mostly used in cardiology and research settings. In general, HRV can be measured over shorter (e.g., 5-10 min) or longer (12 or 24 h) periods. Since 1996, most measurements and calculations are made according to the standard of the Task Force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology. As the first step, the series of times between successive R-peaks in the ECG are in milliseconds. It is crucial, however, to identify and remove extrasystoles and artifacts according to standard protocols. The series of QRS distances between successive heartbeats can be analyzed with simple or more sophisticated algorithms, beginning with standard deviation (SDNN) or by the square root of the mean of the sum of squares of differences between adjacent normal RR (rMSSD). Short-term HRV is frequently analyzed with the help of a non-parametric fast Fourier transformation quantifying the different frequency bands during the measurement period. In the last decades, various non-linear algorithms have been presented, such as different entropy and fractal measures or wavelet analysis. Although most of them have a strong theoretical foundation, their clinical relevance is still debated.

Elements of the cellular metabolic structure

A large number of studies have demonstrated the existence of metabolic covalent modifications in different molecular structures, which are able to store biochemical information that is not encoded by DNA. Some of these covalent mark patterns can be transmitted across generations (epigenetic changes). Recently, the emergence of Hopfield-like attractor dynamics has been observed in self-organized enzymatic networks, which have the capacity to store functional catalytic patterns that can be correctly recovered by specific input stimuli. Hopfield-like metabolic dynamics are stable and can be maintained as a long-term biochemical memory. In addition, specific molecular information can be transferred from the functional dynamics of the metabolic networks to the enzymatic activity involved in covalent post-translational modulation, so that determined functional memory can be embedded in multiple stable molecular marks. The metabolic dynamics governed by Hopfield-type attractors (functional processes), as well as the enzymatic covalent modifications of specific molecules (structural dynamic processes) seem to represent the two stages of the dynamical memory of cellular metabolism (metabolic memory). Epigenetic processes appear to be the structural manifestation of this cellular metabolic memory. Here, a new framework for molecular information storage in the cell is presented, which is characterized by two functionally and molecularly interrelated systems: a dynamic, flexible and adaptive system (metabolic memory) and an essentially conservative system (genetic memory). The molecular information of both systems seems to coordinate the physiological development of the whole cell.

On the dynamics of the adenylate energy system: homeorhesis vs homeostasis

Biochemical energy is the fundamental element that maintains both the adequate turnover of the biomolecular structures and the functional metabolic viability of unicellular organisms. The levels of ATP, ADP and AMP reflect roughly the energetic status of the cell, and a precise ratio relating them was proposed by Atkinson as the adenylate energy charge (AEC). Under growth-phase conditions, cells maintain the AEC within narrow physiological values, despite extremely large fluctuations in the adenine nucleotides concentration. Intensive experimental studies have shown that these AEC values are preserved in a wide variety of organisms, both eukaryotes and prokaryotes. Here, to understand some of the functional elements involved in the cellular energy status, we present a computational model conformed by some key essential parts of the adenylate energy system. Specifically, we have considered (I) the main synthesis process of ATP from ADP, (II) the main catalyzed phosphotransfer reaction for interconversion of ATP, ADP and AMP, (III) the enzymatic hydrolysis of ATP yielding ADP, and (IV) the enzymatic hydrolysis of ATP providing AMP. This leads to a dynamic metabolic model (with the form of a delayed differential system) in which the enzymatic rate equations and all the physiological kinetic parameters have been explicitly considered and experimentally tested in vitro. Our central hypothesis is that cells are characterized by changing energy dynamics (homeorhesis). The results show that the AEC presents stable transitions between steady states and periodic oscillations and, in agreement with experimental data these oscillations range within the narrow AEC window. Furthermore, the model shows sustained oscillations in the Gibbs free energy and in the total nucleotide pool. The present study provides a step forward towards the understanding of the fundamental principles and quantitative laws governing the adenylate energy system, which is a fundamental element for unveiling the dynamics of cellular life.

Multifractal analysis of information processing in hippocampal neural ensembles during working memory under Δ⁹-tetrahydrocannabinol administration

BACKGROUND

Multifractal analysis quantifies the time-scale-invariant properties in data by describing the structure of variability over time. By applying this analysis to hippocampal interspike interval sequences recorded during performance of a working memory task, a measure of long-range temporal correlations and multifractal dynamics can reveal single neuron correlates of information processing.

NEW METHOD

Wavelet leaders-based multifractal analysis (WLMA) was applied to hippocampal interspike intervals recorded during a working memory task. WLMA can be used to identify neurons likely to exhibit information processing relevant to operation of brain-computer interfaces and nonlinear neuronal models.

RESULTS

Neurons involved in memory processing ("Functional Cell Types" or FCTs) showed a greater degree of multifractal firing properties than neurons without task-relevant firing characteristics. In addition, previously unidentified FCTs were revealed because multifractal analysis suggested further functional classification. The cannabinoid type-1 receptor (CB1R) partial agonist, tetrahydrocannabinol (THC), selectively reduced multifractal dynamics in FCT neurons compared to non-FCT neurons.

COMPARISON WITH EXISTING METHODS

WLMA is an objective tool for quantifying the memory-correlated complexity represented by FCTs that reveals additional information compared to classification of FCTs using traditional z-scores to identify neuronal correlates of behavioral events.

CONCLUSION

z-Score-based FCT classification provides limited information about the dynamical range of neuronal activity characterized by WLMA. Increased complexity, as measured with multifractal analysis, may be a marker of functional involvement in memory processing. The level of multifractal attributes can be used to differentially emphasize neural signals to improve computational models and algorithms underlying brain-computer interfaces.

Attractor metabolic networks

BACKGROUND

The experimental observations and numerical studies with dissipative metabolic networks have shown that cellular enzymatic activity self-organizes spontaneously leading to the emergence of a Systemic Metabolic Structure in the cell, characterized by a set of different enzymatic reactions always locked into active states (metabolic core) while the rest of the catalytic processes are only intermittently active. This global metabolic structure was verified for Escherichia coli, Helicobacter pylori and Saccharomyces cerevisiae, and it seems to be a common key feature to all cellular organisms. In concordance with these observations, the cell can be considered a complex metabolic network which mainly integrates a large ensemble of self-organized multienzymatic complexes interconnected by substrate fluxes and regulatory signals, where multiple autonomous oscillatory and quasi-stationary catalytic patterns simultaneously emerge. The network adjusts the internal metabolic activities to the external change by means of flux plasticity and structural plasticity.

METHODOLOGY/PRINCIPAL FINDINGS

In order to research the systemic mechanisms involved in the regulation of the cellular enzymatic activity we have studied different catalytic activities of a dissipative metabolic network under different external stimuli. The emergent biochemical data have been analysed using statistical mechanic tools, studying some macroscopic properties such as the global information and the energy of the system. We have also obtained an equivalent Hopfield network using a Boltzmann machine. Our main result shows that the dissipative metabolic network can behave as an attractor metabolic network.

CONCLUSIONS/SIGNIFICANCE

We have found that the systemic enzymatic activities are governed by attractors with capacity to store functional metabolic patterns which can be correctly recovered from specific input stimuli. The network attractors regulate the catalytic patterns, modify the efficiency in the connection between the multienzymatic complexes, and stably retain these modifications. Here for the first time, we have introduced the general concept of attractor metabolic network, in which this dynamic behavior is observed.

Fractal physiology and the fractional calculus: a perspective

This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a fractal statistical process. Control of physiologic complexity is one of the goals of medicine, in particular, understanding and controlling physiological networks in order to ensure their proper operation. We emphasize the difference between homeostatic and allometric control mechanisms. Homeostatic control has a negative feedback character, which is both local and rapid. Allometric control, on the other hand, is a relatively new concept that takes into account long-time memory, correlations that are inverse power law in time, as well as long-range interactions in complex phenomena as manifest by inverse power-law distributions in the network variable. We hypothesize that allometric control maintains the fractal character of erratic physiologic time series to enhance the robustness of physiological networks. Moreover, allometric control can often be described using the fractional calculus to capture the dynamics of complex physiologic networks.

Variation in acoustic signalling traits exhibits footprints of sexual selection

Phenotypic variation is ubiquitous in nature and a precondition for adaptive evolution. However, theory predicts that the extent of phenotypic variation should decrease with increasing strength of selection on a trait. Comparative analyses of trait variability have repeatedly used this expectation to infer the type or strength of selection. Yet, the suggested influence of selection on trait variability has rarely been tested empirically. In the present study, I compare estimates of sexual selection strength and trait variability from published data. I constricted the analysis to acoustic courtship traits in amphibians and insects with known variability and corresponding results of female binary choice experiments on these traits. Trait variability and strength of sexual selection were significantly correlated, and both were correlated with signal duration. Because traits under stronger selection had lower variation even after the effect of signal duration was eliminated, I conclude that traces of the strength of selection can be observed with respect to variation of acoustic signaling traits in insects and amphibians. The analysis also shows that traits under stabilizing selection have significantly lower phenotypic variability than traits under directional selection.

Quantification of clustering in joint interspike interval scattergrams of spike trains

Joint interval scattergrams are usually employed in determining serial correlations between events of spike trains. However, any inherent structures in such scattergrams that are often seen in experimental records are not quantifiable by serial correlation coefficients. Here, we develop a method to quantify clustered structures in any two-dimensional scattergram of pairs of interspike intervals. The method gives a cluster coefficient as well as clustering density function that could be used to quantify clustering in scattergrams obtained from first- or higher-order interval return maps of single spike trains, or interspike interval pairs drawn from simultaneously recorded spike trains. The method is illustrated using numerical spike trains as well as in vitro pairwise recordings of rat striatal tonically active neurons.

Global self-regulation of the cellular metabolic structure

BACKGROUND

Different studies have shown that cellular enzymatic activities are able to self-organize spontaneously, forming a metabolic core of reactive processes that remain active under different growth conditions while the rest of the molecular catalytic reactions exhibit structural plasticity. This global cellular metabolic structure appears to be an intrinsic characteristic common to all cellular organisms. Recent work performed with dissipative metabolic networks has shown that the fundamental element for the spontaneous emergence of this global self-organized enzymatic structure could be the number of catalytic elements in the metabolic networks.

METHODOLOGY/PRINCIPAL FINDINGS

In order to investigate the factors that may affect the catalytic dynamics under a global metabolic structure characterized by the presence of metabolic cores we have studied different transitions in catalytic patterns belonging to a dissipative metabolic network. The data were analyzed using non-linear dynamics tools: power spectra, reconstructed attractors, long-term correlations, maximum Lyapunov exponent and Approximate Entropy; and we have found the emergence of self-regulation phenomena during the transitions in the metabolic activities.

CONCLUSIONS/SIGNIFICANCE

The analysis has also shown that the chaotic numerical series analyzed correspond to the fractional Brownian motion and they exhibit long-term correlations and low Approximate Entropy indicating a high level of predictability and information during the self-regulation of the metabolic transitions. The results illustrate some aspects of the mechanisms behind the emergence of the metabolic self-regulation processes, which may constitute an important property of the global structure of the cellular metabolism.

Global self-organization of the cellular metabolic structure

BACKGROUND

Over many years, it has been assumed that enzymes work either in an isolated way, or organized in small catalytic groups. Several studies performed using "metabolic networks models" are helping to understand the degree of functional complexity that characterizes enzymatic dynamic systems. In a previous work, we used "dissipative metabolic networks" (DMNs) to show that enzymes can present a self-organized global functional structure, in which several sets of enzymes are always in an active state, whereas the rest of molecular catalytic sets exhibit dynamics of on-off changing states. We suggested that this kind of global metabolic dynamics might be a genuine and universal functional configuration of the cellular metabolic structure, common to all living cells. Later, a different group has shown experimentally that this kind of functional structure does, indeed, exist in several microorganisms.

METHODOLOGY/PRINCIPAL FINDINGS

Here we have analyzed around 2.500.000 different DMNs in order to investigate the underlying mechanism of this dynamic global configuration. The numerical analyses that we have performed show that this global configuration is an emergent property inherent to the cellular metabolic dynamics. Concretely, we have found that the existence of a high number of enzymatic subsystems belonging to the DMNs is the fundamental element for the spontaneous emergence of a functional reactive structure characterized by a metabolic core formed by several sets of enzymes always in an active state.

CONCLUSIONS/SIGNIFICANCE

This self-organized dynamic structure seems to be an intrinsic characteristic of metabolism, common to all living cellular organisms. To better understand cellular functionality, it will be crucial to structurally characterize these enzymatic self-organized global structures.

Long-range correlation of renal sympathetic nerve activity in both conscious and anesthetized rats

In this study we employed both detrended fluctuation analysis (DFA) and multiscale entropy (MSE) measurements to compare the long-range temporal correlation (LRTC) of multifibre renal sympathetic nerve activity (RSNA) between conscious and anesthetized Wistar rats. It was found that both methods showed the obvious LRTC properties in conscious state. Moreover, the scaling exponent of the RSNA in conscious rats was significantly higher than that in anesthetized rats. The results of MSE analysis showed that the entropy values, derived from the conscious group, increased on small time scales and then stabilized to a relatively constant value whereas the entropy measure, derived from anesthetized animals, almost monotonically decreased. This suggests that the fractal properties of underlying dynamics of the system have been reduced by anesthesia. The results demonstrate that apparently random fluctuations in multifibre RSNA are dictated by a complex deterministic process that imparts "long-term" memory to the dynamic system. However, this memory is significantly weakened by anesthesia.

Nonlinear analysis of discharge patterns in monkey basal ganglia

Spontaneous discharge of basal ganglia neurons is often analyzed with time- or frequency-domain methods. However, it has been shown that sequences of inter-spike interval series are not fully described by such linear procedures. We therefore carried out a characterization of the nonlinear features of spontaneous discharge of neurons in the primate basal ganglia. We studied the spontaneous activity of neurons in the subthalamic nucleus (22 cells), as well as neurons in the external and internal pallidal segments (53 and 39 cells, respectively), recorded with standard extracellular recording methods in two awake Rhesus monkeys. As a measure of the statistical irregularity of neuronal discharge, we compared the approximate entropy of inter-spike interval sequences with that of shuffled representations of the same data. In all three basal ganglia structures, approximately 95% of the original data showed lower approximate entropy values than the shuffled data, suggesting a temporal organization in the original sequence. Fano factor analysis confirmed the presence of a temporal organization of inter-spike interval sequences, and indicated the presence of self-similarity in the great majority of them. In addition, Hurst exponent analysis showed that the inter-spike interval series are persistent. Hurst exponents often differ between short and long scaling ranges. Subsequent principal component analyses allowed us to identify three distinct patterns of the temporal evolution of inter-spike interval sequences in the phase space. These types were found in varying distributions in all three nuclei. Our analyses demonstrate that the discharge of most neurons in the basal ganglia of awake monkeys has nonlinear features that may be important for information coding in the basal ganglia.

Fractal Noises and Motions in Time Series of Presympathetic and Sympathetic Neural Activities

We used Allan factor analysis to classify time series of the discharges of single presympathetic neurons in the cat medullary lateral tegmental field (LTF) and rostral ventrolateral medulla (RVLM) and of the postganglionic vertebral sympathetic nerve. These time series fell into two classes of fractal-based point processes characterized by statistically self-similar behavior reflecting long-range correlations among data points. Classification of a time series as either a fractional Gaussian noise (fGn)–or fractional Brownian motion (fBm)–based point process depended on the scaling exponent, α, of the power law in the Allan factor curve. fGn is defined as 0 < α < 1 and fBm as 1 < α < 3. The process responsible for the fractal spike trains of 11 of 12 classifiable LTF neurons with sympathetic nerve-related activity was fGn. In contrast, the process responsible for the fractal spike trains of eight of nine classifiable RVLM presympathetic neurons was fBm. The time series of simultaneously recorded vertebral sympathetic nerve discharge and the arterial pulse also were fBm-based signals. Because a fBm signal is the cumulative sum of the elements comprising the corresponding fGn signal, these results show smoothing of fractal time series in a feedforward direction from medullary presympathetic neurons to postganglionic sympathetic neurons. This may involve integration by RVLM neurons of their LTF inputs or independent fractal processes acting at different levels of the network controlling sympathetic nerve discharge. Whether feedforward smoothing of fractal signals is a feature in other neural systems is open to investigation.

Long-range correlations in rabbit brain neural activity

We have analyzed the presence of persistence properties in rabbit brain electrical signals by means of non-equilibrium statistical physics tools. To measure long-memory properties of these experimental signals, we have first determined whether the data are fractional Gaussian noise (fGn) or fractional Brownian motion (fBm) by calculating the slope of the power spectral density plot of the series. The results show that the series correspond to fBm. Then, the data were studied by means of the bridge detrended scaled windowed variance analysis, detecting long-term correlation. Three different types of experimental signals have been studied: neural basal activity without stimulation, the response induced by a single flash light stimulus and the average of the activity evoked by 200 flash light stimulations. Analysis of the series revealed the existence of persistent behavior in all cases. Moreover, the results also exhibited an increasing correlation in the level of long-term memory from recordings without stimulation, to one sweep recording or 200 sweeps averaged recordings. Thus, brain neural electrical activity is affected not only by its most recent states, but also by previous states much more distant in the past.

Fractal fluctuations in human respiration

The present study was designed to characterize respiratory fluctuations in awake, healthy adult humans under resting conditions. For this purpose, we recorded respiratory movements with a strain-gauge pneumograph in 20 subjects. We then used Allan factor, Fano factor, and dispersional analysis to test whether the fluctuations in the number of breaths, respiratory period, and breath amplitude were fractal (i.e., time-scale-invariant) or random in occurrence. Specifically, we measured the slopes of the power laws in the Allan factor, Fano factor, and dispersional analysis curves for original time series and compared these with the slopes of the curves for surrogates (randomized data sets). In addition, the Hurst exponent was calculated from the slope of the power law in the Allan factor curve to determine whether the long-range correlations among the fluctuations in breath number were positively or negatively correlated. The results can be summarized as follows. Fluctuations in all three parameters were fractal in nine subjects. There were four subjects in whom only the fluctuations in number of breaths and breath amplitude were fractal, three subjects in whom only the fluctuations in number of breaths were fractal, and two subjects in whom only fluctuations in breath number and respiratory period were fractal. Time-scale-invariant behavior was absent in the two remaining subjects. The results indicate that, in most cases, apparently random fluctuations in respiratory pattern are, in fact, correlated over more than one time scale. Moreover, the data suggest that fractal fluctuations in breath number, respiratory period, and breath amplitude are controlled by separate processes.

Fractal properties of human muscle sympathetic nerve activity

Muscle sympathetic nerve activity (MSNA) in resting humans is characterized by cardiac-related bursts of variable amplitude that occur sporadically or in clusters. The present study was designed to characterize the fluctuations in the number of MSNA bursts, interburst interval, and burst amplitude recorded from the peroneal nerve of 15 awake, healthy human subjects. For this purpose, we used the Allan and Fano factor analysis and dispersional analysis to test whether the fluctuations were time-scale invariant (i.e., fractal) or random in occurrence. Specifically, we measured the slopes of the power laws in the Allan factor, Fano factor, and dispersional analysis curves. In addition, the Hurst exponent was calculated from the slope of the power law in the Allan factor curve. Whether the original time series contained fractal fluctuations was decided on the basis of a comparison of the values of these parameters with those for surrogate data blocks. The results can be summarized as follows. Fluctuations in the number of MSNA bursts and interburst interval were fractal in each of the subjects, and fluctuations in burst amplitude were fractal in four of the subjects. We also found that fluctuations in the number of heartbeats and heart period (R-R interval) were fractal in each of the subjects. These results demonstrate for the first time that apparently random fluctuations in human MSNA are, in fact, dictated by a time-scale-invariant process that imparts “long-term memory” to the sequence of cardiac-related bursts. Whether sympathetic outflow to the heart also is fractal and contributes to the fractal component of heart rate variability remains an open question.

Fractal activity generated independently by medullary sympathetic premotor and preganglionic sympathetic neurons

In anesthetized cats with cervical spinal cord transection, Fano factor analysis was used to test for time-scale invariant (fractal) fluctuations in spike counts of single preganglionic cervical sympathetic neurons (PSNs) and putative sympathetic premotor neurons located in the rostral ventrolateral medulla (RVLM) and caudal medullary raphe. The medullary neurons exhibited cardiac-related activity, and their axons projected to the spinal cord, as demonstrated by antidromic activation. The variance-to-mean spike count ratio (Fano factor) was plotted as a function of the window size used to count spikes. The Fano factor curves for seven PSNs, eight RVLM neurons, and eight raphe neurons contained a power law relationship extending over more than one time scale. In these cases, random shuffling of the interspike intervals in the original time series eliminated the power law relationship. Thus the power law relationships can be attributed to long-range correlations among interspike intervals rather than simply to the distribution of the intervals that is not changed by shuffling the data. It is concluded that PSNs and sympathetic premotor neurons in the medulla can independently generate fractal firing patterns.