Relevant beat-to-beat measures of local electrical responses during complex cardiac rhythms are interpreted as successive iterates of a low dimensional mapping. That simplified view is supported by previously reported experimental and numerical work. In that approximate theory, low dimensional dynamics (not restricted to chaos) also can be perturbed and controlled, much in the same way as in the Ott et al method for controlling chaos in nonlinear dynamical systems. In the problem at hand, which involves nonlinear waves and spatial degrees of freedom, the task is much more complicated and the phenomena less well understood. Recordings from an in vitro model of ventricular fibrillation are analyzed searching for deterministic recurrences in the local period of activation.

We investigate the dynamics of a pair of electrically coupled pacemaking sino-atrial node cells based on a physiologically detailed model. Each cell has distinguished oscillation properties. It is found that at low, yet still physiologically reasonable coupling conductance values, complex dynamics including chaos can arise. Occurrence of these complex dynamics in coupled pacemaker cells may provide an explanation for the origin of certain cardiac arrhythmias.

The point process formed by the sequence of human heartbeats exhibits long-duration power- law correlation. We obtain the normalized coincidence rate g⁽²)((tau) ) of the underlying point process and demonstrate that the correlation is stronger for patients with normal hearts than those with heart failure. This is consistent with the greater rate fluctuations observed in the normal heart. A number of statistical measures are used to establish the existence and reveal the form of the correlation, including rescaled range analysis, pulse- number distribution, Fano-factor time curve (FFC), and power spectral density. The normalized coincidence rate is obtained from the FFC. The long-duration, power-law correlation observed in the sequence of heartbeats is similar to that observed at a number of neurophysiological loci in a variety of species. We also obtain the box-counting estimate of the attractor's fractal dimension from a phase-space reconstruction and analysis of the trajectory of the number of heartbeat events. This approach reveals that the heartbeats of normal patients exhibit an attractor of higher dimension than those of heart-failure patients.

This note discusses how information contained in a neural message is transmitted depending on two schemes of encoding: stochastic or deterministic. For the first case, it is shown that the rate of information loss is minimized for a range of signal to noise ratios entering the channel with noise and signal amplitude of the same order of magnitude. In contrast, at the deterministic limit, (i.e., signal amplitude very large compared with the noise) the rate of information loss increases; approximately by a power law of the distance traveled by the message. The exponent depends linearly on the time constant of the function relating speed of propagation vs period of excitation of the axon.

Various anxiety states have been linked with disorders of the autonomic nervous system. These autonomic disorders may be revealed by analysis of physiological time series such as the heart rate interbeat interval series. The present paper reports a general model of biological system functioning and related assessment indices based on recent nonlinear dynamical systems approaches. In particular, two experimental studies are reported which suggest the utility of heart rate nonlinear dynamics in the assessment of anxiety.

The application of wavelet transforms (WT) to experimental data from the nervous system has been hindered by the lack of a straightforward method to handle noise. A noise reduction technique, developed recently for use in wavelet cluster analysis in cosmology and astronomy, is here adapted for electroencephalographic (EEG) time-series data. Noise is filtered using control surrogate data sets generated from randomized aspects of the original time-series. In this study, WT were applied to EEG data from human patients undergoing brain mapping with implanted subdural electrodes for the localization of epileptic seizure foci. EEG data in 1D were analyzed from individual electrodes, and 2D data from electrode grids. These techniques are a powerful means to identify epileptic spikes in such data, and offer a method to identity the onset and spatial extent of epileptic seizure foci. The method is readily applied to the detection of structure in stationary and non-stationary time-series from a variety of physical systems.

We discuss issues involved in estimating the dimension of a fractal point process. We first define the term fractal point process and provide some examples of experimental phenomena for which they serve as suitable models. We then develop mathematical formulations of fractal point processes, and present two methods for dimension estimation and an analysis of their performance. Finally, we compare this analysis with results from both simulated and natural examples of fractal point processes.

A network of chaos elements has been presented as an information processor where each element consists of two oscillators and it acts as a neuron by making use of the synchronized state of the two oscillators. The model is considered as a dynamical model of the brain, and brain dynamics is metaphorically analyzed with the use of the model. The time sequences of Hopfield's energy, which are generated by the network when it solves a traveling salesman problem, are investigated with the use of a fluctuation spectrum theory. The change of the energy reflects the active motion of neurons, and we consider that the time sequence corresponds to a brain wave. If the control parameters of the neuron are chosen properly, the model can efficiently find the solution where a low intermittent `brain wave' is observed. On the other hand, the model will have epileptic fits if a certain control parameter takes a small value.

Interspike interval patterns of brain stem neurons that project directly or indirectly to much of the neocortex interactively influence electroencephalographically-defined states of consciousness and modulate patterns of temporal-spatial coherence, `binding,' in cortical field potential oscillations. Neurochemical classes of brain stem neurons manifest discriminable dynamical characteristics apart from the statistics of their firing rates. These sequences of interspike intervals are not well described by either harmonic functions or the Poisson statistics of renewal processes. We cast these patterns within the context of information bearing processes by using moment partitions and symbolic dynamics. We describe the expanding behavior of model and real brain stem neurons in relationship to states of resonance (the presence of complex singularities in the power spectrum with amplitudes related to the persistence of unstable fixed points in the nonexponential decay of correlations), synchronization (how closely the measure of maximal entropy comes to equaling the Sinai- Ruelle-Bowen area measure), and lexical redundancy (as repetitions of symbol subsequences).

Studies on how chaos theory may be applied to neural disorders is a very challenging theoretical problem. But, to determine the applications of chaos theory cellular functions, it is best to study the genesis of chaos and its characteristics using a minimal model of cellular excitability. In this paper we present two neuronal models which gives rise to interesting types of bursting and chaos. The first model is based on the model of Chay, in which the bursting of neuronal cells is caused by voltage- and time-dependent inactivation of calcium channels. The second model is based on Chay's work in which the bursting is caused by the conformational transformation of the calcium channels that is induced by binding of Ca²⁺ ion to the receptor site. With these two models, we elucidate how the periodic states and chaos can be evolved when the properties of two types of inward current change. Our bifurcation diagram reveals new types of bifurcations and chaos which were not seen in the other non-linear dynamic models. The predicted chaos from the models closely resembles that observed experimentally in neuronal cells. An implication of our finding is that chaos theory may be used to understand and improve the treatment of certain irregular activities in the brain.

Power spectral scaling and correlation properties of physical and biological dynamical systems are useful in system characterization and in giving insight into their mechanisms. Since, heart rate has been found to vary with respect to time in a very complicated manner, analysis of this variation using power spectral scaling and correlation techniques can give insight into the various physiologic systems which are involved in heart rate control. 1/f power spectrum, one of the most ubiquitous types of power spectra found in nature, has previously been found to be characteristic of normal cardiac interbeat interval time series for frequencies less than 2 X 10^-2 Hz. This frequency domain corresponds to relatively long-term interbeat interval variation. The scaling properties of short-term heart rate variability (related to short-term heart rate control by the baroreceptor reflex), on the other hand, have not as yet been examined analytically. To accomplish this now, we analyzed the scaling properties of the power spectra of cardiac interbeat interval time series of five minute durations in 10 normal individuals and in 10 patients with heart failure. By studying the scaling and correlation properties of the power spectra of short-term interbeat interval time series we may gain more insight into the non-linear characteristics of baroreceptor reflex heart rate regulation.

Blood oxygen saturation level dynamics measured during sleep simultaneously with pulse rate data by means of standard pulseoxymeter with a 1 second time resolution are analyzed in a number of obstructive sleep apnoea (OSA) and control cases by nonlinear dynamics methods. Minima of the blood oxygen saturation level and maxima of the pulse rate were analyzed and used to construct three types of return maps. On a global time scale, in OSA cases, a long term relaxation type oscillation was found. A group of OSA cases may be defined for which the characteristic times of such oscillations seem to be a feature specific for the individual. On a local timescale--when the data is viewed through a time window of 20 - 45 minutes-- characteristic, complex shapes of the return maps have been identified. For controls these shapes universal and can be found in different controls. For OSA cases there is only a similarity between the shapes of the maps which, upon comparison, appear distorted and rescaled.

Adding random noise to a weak periodic signal can enhance the flow of information through certain nonlinear physical systems, via a process known as stochastic resonance (SR). We have used crayfish mechanoreceptor cells to investigate the possibility that SR can be induced in neurophysiological systems. Various signal-to-noise ratio (SNR) measurements were derived from the action potentials (spikes) of single receptor cells stimulated with weak periodic signals. Spike noise was controlled by one of two methods: (1) adding external noise to the stimulus, or (2) altering internal noise sources by changing the temperature of the cell. In external noise experiments, an optimal noise level can be identified at which the SNR is maximized. In internal noise experiments, although the SNR increases with increasing noise, no SNR maximum has been observed. These results demonstrate that SR can be induced in single neurons, and suggest that neuronal systems may also be capable of exploiting SR.

A new proposal to the study of large-scale neural networks is reported. It is based on the use of similar graphs to the Feynman diagrams. A first general theory is presented and some interpretations are given. A propagator, based on the Green's function of the neuron, is the basis of the method. Application to a simple case is reported.

Human subjects performed a set of mouse-driven computer tasks that required them to `remove dots from the screen' under fast and slow dot return (redraw) conditions. The binary partition of the 1D phase space generated a shift map, (sigma) , on two symbols. Assuming this sequence to be the symbolic dynamic product of a parametrized, symmetric tent map, the parity sequence associated with (sigma) was used in an inverse autoexpansion to recover the tent height parameter, (mu) , with 1 <EQ (mu) <EQ 2. We obtained a single convergent (mu) value for most subject sequences. In the case of T(mu) (X), the topological entropy is equal to the height of the tent, h_T equals (mu) , and is monotonic increasing with respect to (mu) . We, therefore, may order the subjects (with convergent (mu) 's) in terms of the orbital complexity made available to them by their associated (mu) equals h_T values. The distribution of (mu) 's in the sample could not be distinguished from a normal distribution. Linear multiple regression models (with higher order terms) were built, significantly relating (mu) _f (fast redraw), (mu) _s (slow redraw), and (mu) _d equals ((mu) _f - (mu) _s) to established quantitative personality variables. The subject-map equivalence used here also allows the extension of sample lengths (unavailable in the experimental setting) in self-initiated tasks with widely differing N, while preserving topological properties of the subjects' performance.

The time course of the velocity of vibration was measured in individual sensory cells (outer hair cells and Hensen's cells from the third and fourth turns of a guinea-pig temporal-bone preparation) for a variety of applied acoustic stimuli. The methods of preparation and interferometric measurement have been described earlier. To observe the cell's response for a continuous and large range of acoustic intensities, an amplitude modulated tone with fixed modulation characteristics and a wide range of carrier frequencies f_c was used. The peak acoustic intensities of the applied tones were in excess of 90 dB:re .0002 dynes/cm², which is at the high intensity end of the hearing range of most mammals. A spectrogram was used to analyze the velocity response of the cell. This is a 3D representation that exhibits the time evolution of the amplitudes of all of the spectral components in the response.

It is shown that the pupil latency can be estimated from pupil cycling measurements when the pupil light reflex is clamped with piecewise constant negative feedback. The solution of the mathematical model previously shown to describe these oscillations is utilized to develop a variety of strategies to estimate latency and to evaluate the effects of noise on these estimates. The results demonstrate that the pupil latency shows considerable variation.

Recurrence plots were used to evaluate pupil dynamics of subjects with narcolepsy. Preliminary data indicate that this nonlinear method of analyses may be more useful in revealing underlying deterministic differences than traditional methods like FFT and counting statistics.

We analyze the Bonhoeffer-van der Pol equations in a parameter range where no `overt' limit cycle exists, rather the dominating feature in phase space is a focus. Exciting the system by an external pulse, its response depends upon this pulse's size. For small pulses, a quick return to the focus occurs. For large pulses, extending beyond the separatrix, the orbits traverse along a `hidden' structure. This structure initially resembles a temporary limit cycle and then spirals into the focus. The response of the system to single excitations of different sizes at different points of the `hidden' structure is used to understand its response to a train of pulses of different periods. Thus, e.g. the boundaries of the phase-locking regions are easily calculated and the explanation of the appearance of `below threshold' responses for the pulse-train case becomes straightforward.

In this paper we attempt to give an affirmative answer to the title question. In addition, we present results showing that nonlinear oscillators forced at two incommensurate frequencies can exhibit a variety of novel dynamical phenomena, including two-frequency quasi- periodicity, three-frequency quasi-periodicity, chaos and in particular, strange nonchaotic attractors.

In this work we investigate the effect of direct and alternating current on spatial and spatiotemporal patterns in two ionic Brusselator models. While the electrical field strength is assumed to be spatially homogeneous in one model, the other model takes local variations of the electrical field caused by the different mobility of ions into account. An externally applied electrical field distorts stationary Turing patterns and induces oscillatory behavior. If bulk oscillations in a spatially distributed system are perturbed by alternating current, deterministic chaos may be found.

We have investigated the dynamics of electrical wave propagation in a ring of excitable cardiac tissue modeled by the Beeler-Reuter membrane equations. By adiabatically eliminating the fast ionic channel variables, we reduce this electrophysiological partial differential equation (PDE) model to a single-front free-boundary problem in which the dynamics is solely determined by the `slow' variables. This free-boundary problem is further reduced to a 3D coupled map by using the intensity of wavefront velocity. Stability analysis of this discrete map shows the existence of a period doubling bifurcation at a certain wave period. The critical period, below which there occurs an alternation in action potential duration, depends sensitively on the dynamics of the calcium channel. These results are in good agreement with finding from direct PDE simulations. Our work supports the hypothesis that electrical alternans result in spiral breakup, which might be one possible mechanism for the transition from ventricular tachycardia to fibrillation.

The peroxidase-oxidase system is the only in vitro single-enzyme reaction which has been shown to oscillate chaotically. Difficulties in reproducing literature results have led to careful attempts to specify conditions for reproducing experimental parameters. Progress in specifying reaction conditions is reported. Parameters which require further elucidation before control can be adequately achieved are also specified.

A model is derived for a `simple' motor control task. An artificial delay is introduced in the experiments to assess the dynamic influence it may have on normal and/or pathological conditions. The model takes the form of a delay-differential equation containing two time delays, associated with two (proprioceptive and visual) negative feedback loops. A linear stability analysis reveals a rich structure in the parameter values destabilizing the equilibrium. A nonlinear analysis, by a reduction on a center manifold when two Hopf bifurcations interact, reveals the existence of stable and unstable 2D tori. These results are contrasted with systems involving a single feedback loop, and a single time delay.

Sequential response organization is assessed experimentally based on the degree of association between behavioral elements in a simple binary choice task. The thermodynamic formalism for dynamical systems provides a quantitative framework to determine fluctuations of response organization. The order parameter approach proposed by Haken is used to extract organizational principles that determine the association of consecutive behavioral elements. The results indicate that (1) the fluctuations of local dynamical entropies are significantly larger in humans compared to the randomized data sets; (2) the switching and duration order parameter determine critically the sequential organization of response elements in the binary choice task paradigm. Thus. sequential response organization is characterized by the predictable repetition of similar behavioral actions that are initiated frequently or predictable repetitive switching between different behavioral actions. In contrast, unpredictable sequences of behavioral actions emerge from infrequently initiated and moderate switching between response elements.

A number of factors influence the chaotic dynamics of heart function. Genetics, age, sex, disease, the environment, experience, and of course the mind, play roles in influencing cardiovascular dynamics. The mind is of particular interest because it is an emergent phenomenon of the body admittedly seated and co-occurrent in the brain. The brain serves as the body's controller, and commands the heart through complex multipathway feedback loops. Structures deep within the brain, the hypothalamus and other centers in the brainstem, modulate heart function, partially as a result of afferent input from the body but also a result of higher mental processes. What can chaos in the body, i.e., the nonlinear dynamics of the heart, tell of the mind? This paper presents a brief overview of the spectral structure of heart rate activity followed by a summary of experimental results based on phase space analysis of data from semi-structured interviews. This paper then describes preliminary quantification of cardiovascular dynamics during different stressor conditions in an effort to apply more quantitative methods to clinical data.

In this paper we show the effect that several desired signals have on the performance of a neural network dynamic classifier for transient detection. We compare performances of the same neural network trained with the conventional 1/0 desired signal, a prediction framework and a desired signal composed of noise during the background. This last choice is the one that works best. We show that in terms of statistical decision theory this choice of desired signal should work as well as the optimal a posteriori detector. We provide an explanation why the noise during the background works for transient detection. Finally we comment on the implications of this choice of desired signal for learning in biological networks.

In this paper we extend recent work concerning the emergence of macroscopic patterns in a thalamo-cortical network. A small number of thalamic `cells' were used to drive a network of cortical cells structured on a square lattice consisting of 2-torus connectivity. It was previously shown that the network can exhibit irregular and periodic spatial behavior with the latter including both standing and traveling waves. A systematic analysis of this network across a wide range of driving frequencies revealed several properties of its spatiotemporal behavior which resemble those of the 8 - 12 Hz cortical alpha rhythm. We extend these results by examining the time-dependent amplitude of the spatial modes of this network and derive a system of ODEs to characterize their dynamics. Finally, we investigate the relationship between changes at the microscopic (cellular) level of this system and their effect on these macroscopic parameters.

We observed previously that hydraulic pressure in rat renal proximal tubules changes from the periodic oscillations found in normal animals to random-appearing fluctuations during the development of hypertension. It was suggested that the random-appearing fluctuations seen in the hypertensive rats were due to chaotic dynamics in the systems regulating tubular pressure and flow. To test this hypothesis we have now produced surrogate data sets from the originals by randomizing the phase of the power spectra. We have applied both correlation dimension estimation and forecasting error as discriminating statistics. The results show that the original experimental time series can be distinguished from linear and static nonlinear correlated noise, which confirms that the nonlinear behavior is due to the intrinsic dynamics of the system. This finding suggests that there is a low dimensional chaotic attractor that governs renal hemodynamics in the hypertensive stage. To our knowledge this represents the first rigorous demonstration of a transition to chaotic dynamics in an integrated physiological control system occurring in association with a pathological condition.

Nonlinear dynamical systems with feedback (dynamic) noise exhibit solutions which visit a large number of individual trajectories. Such systems will occasionally visit trajectories which converge to stable fixed points (or limit cycles), and will remain in these stable regimes until ejected by noise. Such systems exhibit ensemble behavior which is quite different from that of the noise-free case. We examine the effects of feedback noise in the Hodgkin-Huxley system for both constant and sinusoidally modulated applied current. We demonstrate that increasing noise levels causes the system to occasionally visit an otherwise rarely visited stable region of the manifold. Further increasing the noise level eventually obscures this behavior. We explore the potential for generating chaotic behavior via dynamic noise in this system under nominally periodic conditions. Ramifications for the analysis of systems undergoing transitions to chaos are discussed.

A method of controlling chaotic to laminar flows in the Lorenz equations using fixed points dictated by minimizing the Lyapunov functional was proposed by Singer, Wang and Bau. Using different fixed points, we find that the solutions in a chaotic regime can also be periodic. Since the lasers equations are isomorphic to the Lorenz equations, we use this new method to control chaos when the laser is operated over the pump threshold. Furthermore, by solving the laser equations with an occasional proportional feedback mechanism, we recover the essential lasers controlling features experimentally discovered by Roy, Murphy, Jr., Maier, Gills and Hunt. This method of control chaos is now extended to various medical and biological systems.