Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator

Membrane potential responses of a Hodgkin-Huxley oscillator to an externally-applied sinusoidal current were numerically calculated with relation to bifurcation parameters of the amplitude and the frequency of the stimulating current. The Hodgkin-Huxley oscillator, or the Hodgkin-Huxley axon in the state of self-sustained oscillation of action potentials, was realized by immersing the axon in calcium-deficient sea water. The forced oscillations were analysed by the stroboscopic plots and/or the Lorenz plots. The results show that the periodically forced Hodgkin-Huxley oscillator exhibits not only periodic motions (harmonic or sub-harmonic synchronization) but also non-periodic motions (quasi-periodic or chaotic oscillation), that the motions were determined by the amplitude and the frequency of the stimulating current, and that the characteristic motions obtained in the present study were in reasonable agreement with those of our previous results, found experimentally in squid giant axons. Also, two kinds of routes to the chaotic oscillations were found; successive period-doubling bifurcations and formation of the intermittently chaotic oscillation from sub-harmonic synchronization.     

Zum Mechanismus der biologischen 24-Stunden-Periodik

Summary The level (=arithmetic average of all instantaneous values)of a self-sustained oscillation in general influences all properties of the oscillation, including period, amplitude and shape of the oscillation, and the rate of exchange of energy between the oscillator and its environment. Only when the non-linear damping factor does not depend on the instantaneous value of the oscillating function, but only on the amplitude of the oscillation, are the other properties independent of the average level. The differential equations describing self-sustained oscillations cannot be solved exactly, but methods of approximation are applicable. Numerical solutions to several different forms of the equations will be discussed.In the simplest case (van der Pol equation) all properties of the self-sustained oscillation (e.g. period, amplitude) are extreme when the level is zero. The oscillation continues only within a given range of levels (oscillating range); outside this range, the oscillation damps out. In other modifications of the equation, the oscillating function cannot assume a zero value. In all cases, the extent to which the average level influences the different properties depends on the factor , which describes the position of the oscillation within the range between harmonic and relaxation types of oscillation.In the elementary van der Pol equation, the correlation between level and frequency changes sign within the oscillating range; that is, the circadian rule, demanding an always positive correlation between level and frequency, cannot be fulfilled. Only with an additional non-linearity in the energy of recoil does the correlation remain unchanged in sign throughout the oscillating range. A stability condition demands a positive sign for this non-linearity, and hence, for the correlation (fulfilling the circadian rule); if the sign is negative (violating the circadian rule), the oscillation becomes unstable. With an additional term of the third order, the oscillation acquires a two-peaked shape typical of many circadian oscillations.A simple differential equation describing all general properties of the circadian periodicity must fulfil these conditions: the oscillation must be self-sustained and limited to positive values; and the energy of recoil must be non-linear with a positive coefficient to obtain the appropriate correlation between level and frequency. In the equations here developed the environment directly influences only one parameter of the oscillation, i.e. the level. In addition to the circadian periodicity, the differential equations here examined describe the behavior of several other biological oscillations.

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Die benutzten mathematischen Begriffe folgen — soweit dort angeführt — den Benennungen des DIN-Blattes 1311; im Anhang I sind die wichtigsten Begriffe noch einmal zusammenfassend definiert.     

Frequency encoding in renal blood flow regulation

With a model of renal blood flow regulation, we examined consequences of tubuloglomerular feedback (TGF) coupling to the myogenic mechanism via voltage-gated Ca channels. The model reproduces the characteristic oscillations of the two mechanisms and predicts frequency and amplitude modulation of the myogenic oscillation by TGF. Analysis by wavelet transforms of single-nephron blood flow confirms that both amplitude and frequency of the myogenic oscillation are modulated by TGF. We developed a double-wavelet transform technique to estimate modulation frequency. Median value of the ratio of modulation frequency to TGF frequency in measurements from 10 rats was 0.95 for amplitude modulation and 0.97 for frequency modulation, a result consistent with TGF as the modulating signal. The simulation predicted that the modulation was regular, while the experimental data showed much greater variability from one TGF cycle to the next. We used a blood pressure signal recorded by telemetry from a conscious rat as the input to the model. Blood pressure fluctuations induced variability in the modulation records similar to those found in the nephron blood flow results. Frequency and amplitude modulation can provide robust communication between TGF and the myogenic mechanism.     

Modeling the emergence of circadian rhythms in a clock neuron network

Circadian rhythms in pacemaker cells persist for weeks in constant darkness, while in other types of cells the molecular oscillations that underlie circadian rhythms damp rapidly under the same conditions. Although much progress has been made in understanding the biochemical and cellular basis of circadian rhythms, the mechanisms leading to damped or self-sustained oscillations remain largely unknown. There exist many mathematical models that reproduce the circadian rhythms in the case of a single cell of the Drosophila fly. However, not much is known about the mechanisms leading to coherent circadian oscillation in clock neuron networks. In this work we have implemented a model for a network of interacting clock neurons to describe the emergence (or damping) of circadian rhythms in Drosophila fly, in the absence of zeitgebers. Our model consists of an array of pacemakers that interact through the modulation of some parameters by a network feedback. The individual pacemakers are described by a well-known biochemical model for circadian oscillation, to which we have added degradation of PER protein by light and multiplicative noise. The network feedback is the PER protein level averaged over the whole network. In particular, we have investigated the effect of modulation of the parameters associated with (i) the control of net entrance of PER into the nucleus and (ii) the non-photic degradation of PER. Our results indicate that the modulation of PER entrance into the nucleus allows the synchronization of clock neurons, leading to coherent circadian oscillations under constant dark condition. On the other hand, the modulation of non-photic degradation cannot reset the phases of individual clocks subjected to intrinsic biochemical noise.     

Asymmetry in the self-sustained oscillation ofPhysarum plasmodial strands

Summary The characteristics of the self-sustained oscillation in the plasmodial strand ofPhysarum polycephalum have been investigated in one steady and two transient conditions. An isolatedPhysarum strand changes its length periodically when it is suspended. In the behaviour of the self-sustained oscillation under the conditions, we provide the first demonstration that the changes in the periods of the oscillation can be ascribed to effects on the shortening phase, while the lengthening periods are almost unaffected. This result means that the asymmetric self-sustained oscillation of thePhysarum strand is composed of an active contracting process, presumably due to actin filaments and myosin-like molecules in the strand, and a passive lengthening process which is merely an extension of the strand under a load.     

Transfer characteristics between EMG activity and muscle tension under isometric conditions in man.

The relationship between motor unit activity and a voluntarily produced, sinusoidally modulated isometric tension was evaluated as a function of the modulation frequency. These date are reported in terms of the gain and phase difference of the motor activity (input) and tension (output) relationship, the gain being the logarithmic ratio of the amplitudes of the output and input sinusoids. It was found that an increase in the modulation amplitude of the motor unit activity was required to produce the same amount of modulation of the output tension as the modulation frequency was increased. For example, the modulation amplitude of the activity is about twice as much at 2 Hz as at 0.25 Hz and about 4 times as much at 5 Hz. It was also found that the maximum tension which could be produced voluntarily during brief jerks at 5 Hz was the same as the maximum sustained tension which could be attained. This latter finding emphasizes the importance of recruitment and especially synchronization of motor unit activity to the gradation of output tension.     

Concerning the amplitude modulation of the human EEG

Amplitude-modulated processes can be formally presented as a product of two or more sinusoids. This makes it possible to study them by means of analysis of multiplicative phenomena using the Fast Fourier Transform (FFT). To assess the contribution of amplitude EEG modulation to the dynamic of electrical activity of the human brain, the results of the FFT of simulated signals obtained by multiplication of oscillatory processes with different parameters were compared with the results of the FFT of a single EEG recording from a subject at rest. We studied the temporal dynamics of spectral components calculated with different spectral resolution under similar conditions for real and simulated signals. An attempt was made to analyze and interpret the amplitude-modulated EEG processes using the additive properties of the FTT. It was shown that processes of amplitude modulation are present in electrical brain activity and determine the synchronism of changes in time in the majority of frequency components of the EEG spectrum. The presence of the amplitude modulation in bioelectrical processes is of a fundamental nature, since it is a direct reflection of the control, synchronization, regulation, and intersystem interaction in the nervous and other body systems. The study of this modulation gives a clue to the mechanisms of these processes.     

The parallelisms in of sound signal of domestic sheep and Northern fur seals

The parallelisms in communicative behavior of domestic sheep and Northern fur seals within a herd are accompanied by parallelisms in parameters of sound signal, the calling scream. This signal ensures ties between babies and their mothers at a long distance. The basis of parallelisms is formed by amplitude modulation at two levels: the one being a direct amplitude modulation of the carrier frequency and the other--modulation of the carrier frequency oscillation. Parallelisms in the signal oscillatory process result in corresponding parallelisms in the structure of its frequency spectrum.     

A model for self-sustained potential oscillation of lipid bilayer membranes induced by the gel-liquid crystal phase transitions.

To clarify the mechanism of self-sustained oscillation of the electric potential between the two solutions divided by a lipid bilayer membrane, a microscopic model of the membrane system is presented. It is assumed, on the basis of the observed results (Yoshikawa, K., T. Omachi, T. Ishii, Y. Kuroda, and K. liyama. 1985. Biochem. Biophys. Res. Commun. 133:740-744; Ishii, T., Y. Kuroda, T. Omochi, and K. Yoshikawa. 1986. Langmuir. 2:319-321; Toko, K., N. Nagashima, S. liyama, K. Yamafuji, and T. Kunitake. Chem. Lett. 1986:1375-1378), that the gel-liquid crystal phase transition of the membrane drives the potential oscillation. It is studied, by using the model, how and under what condition the repetitive phase transition may occur and induce the potential oscillation. The transitions are driven by the repetitive adsorption and desorption of proton by the membrane surface, actions that are induced the periodic reversal of the direction of protonic current. The essential conditions for the periodic reversal are (a) at least one kind of cations such as Na+ or K+ are included in the system except for proton, and the variation of their permeability across the membrane due to the phase transition is noticeably larger than that of proton permeability; and (b) the phase transition has a hysteresis. When these conditions are fulfilled, the self-sustained potential oscillation may be brought about by adjusting temperature, pH, and the cation concentration in the solutions on both sides of the membrane. Application of electric current across the membrane also induces or modifies the potential oscillation. Periodic, quasiperiodic, and chaotic oscillations appear especially, depending on the value of frequency of the applied alternating current.     

Comparative analysis of EEG correlation synchronism and EEG amplitude relationships in all-night sleep

We used a new methodological approach to the evaluation of EEG synchronization based on correlation between amplitude modulation processes (EEG envelopes). We revealed: left-hemispheric dominance and dominance of frontal over occipital regions characteristic of all sleep stages; differences in synchronization in frequency bands and their patterns characteristic of a specific sleep stage; stage-dependent differences in inter-hemispheric synchrony and patterns of their changes from the frontal to occipital regions; and stage-dependent topographical distributions of high synchronization foci with respect to frequency domains. Analysis of amplitude topography also revealed left-hemispheric dominance and many significant differences in activity distribution patterns over parasagittal chains of electrodes (meridians) depending on sleep stages and frequency domains. The combination of EEG synchrony estimates with the amplitude spectral estimates made it possible to perform a reliable discriminant recognition of five sleep stages with errors in the range of 3-20%.     

A contribution to the problem of the concept “biological clock” (part I)

Summary In the first parts of this study on the concept of the biological clock it has been investigated how it is used in the field of biorhythmology. The analysis of the contents of the concept is preceded by a survey of the current research in this field.There are two general hypotheses with respect to the ultimate origin of rhythmic phenomena: the Endogenous Timer Hypothesis and the Exogenous Timer Hypothesis. Within the Endogenous Timer Hypothesis two contrasting viewpoints with respect to the structure of the biological clock can be distinguished, which have been called: the Discrete Entity view, and the Organizational view.It has been shown that in the application of the general clock-idea upon the organism a logical distortion is present. The organism is conceived tobe a clock and tohave a clock simultaneously.In anticipation of the analysis of the explanatory value of the concept of the biological clock in part II, it has been put forward that in a number of cases time-measurement can be considered as a superfluous (redundant) element in the explanation of the phenomena involved.Finally it has been demonstrated that in the case of the Exogenous Timer Hypothesis the concept of the biological clock is not relevant. The Exogenous Timer Hypothesis has been compared with the Endogenous Timer Hypothesis with respect to the criterion of the principle of parsimony, orOccam's razor.List of rhythmological terms autonomous system not under the influence of a periodic source of energy,i.e. selfsustained and free oscillations - non-autonomous system under the influence of a periodic source of energy,i.e. forced oscillations whether in active or passive systems - period time after which a definite phase of the oscillation reoccurs - frequency reciprocal of period - phase instantaneous state of an oscillation within a period, represented by the value of the variable and all its time derivatives - phase angle value of the abscissa corresponding to a point of the curve (phase) given either in radians, in degrees or in other fractions of the whole period. It can be given in units of time, if the length of the period is stated - synchronization state in which two or more oscillations have the same frequency due to mutual or unilateral influences - entrainment coupling of a self-sustained oscillation to a Zeitgeber (forcing oscillation) with the result that either both oscillations have the same frequency (synchronization) or that the frequencies are integral multiples (frequency-demultiplication): possible only within limited ranges of frequencies - Zeitgeber that forcing oscillation which entrains a biological rhythm - free-running rhythms selfsustained oscillations under constant conditions - response-curve indicates, in biology, how the amount and the sign of a phase shift, induced by a single stimulus, depends on the phase in which the stimulus is applied     

The fundamental organization of cardiac mitochondria as a network of coupled oscillators

Mitochondria can behave as individual oscillators whose dynamics may obey collective, network properties. We have shown that cardiomyocytes exhibit high-amplitude, self-sustained, and synchronous oscillations of bioenergetic parameters when the mitochondrial network is stressed to a critical state. Computational studies suggested that additional low-amplitude, high-frequency oscillations were also possible. Herein, employing power spectral analysis, we show that the temporal behavior of mitochondrial membrane potential (DeltaPsi(m)) in cardiomyocytes under physiological conditions is oscillatory and characterized by a broad frequency distribution that obeys a homogeneous power law (1/f(beta)) with a spectral exponent, beta = 1.74. Additionally, relative dispersional analysis shows that mitochondrial oscillatory dynamics exhibits long-term memory, characterized by an inverse power law that scales with a fractal dimension (D(f)) of 1.008, distinct from random behavior (D(f) = 1.5), over at least three orders of magnitude. Analysis of a computational model of the mitochondrial oscillator suggests that the mechanistic origin of the power law behavior is based on the inverse dependence of amplitude versus frequency of oscillation related to the balance between reactive oxygen species production and scavenging. The results demonstrate that cardiac mitochondria behave as a network of coupled oscillators under both physiological and pathophysiological conditions.     

Frequency modulation of large oscillatory neural networks

Dynamical systems which generate periodic signals are of interest as models of biological central pattern generators and in a number of robotic applications. A basic functionality that is required in both biological modelling and robotics is frequency modulation. This leads to the question of whether there are generic mechanisms to control the frequency of neural oscillators. Here we describe why this objective is of a different nature, and more difficult to achieve, than modulating other oscillation characteristics (like amplitude, offset, signal shape). We propose a generic way to solve this task which makes use of a simple linear controller. It rests on the insight that there is a bidirectional dependency between the frequency of an oscillation and geometric properties of the neural oscillator’s phase portrait. By controlling the geometry of the neural state orbits, it is possible to control the frequency on the condition that the state space can be shaped such that it can be pushed easily to any frequency.     

Caffeine-induced modulation of network oscillation in a molluscan olfactory center

Calcium release from intracellular stores has various actions in neurons, but its effects on network oscillation have not been well understood. The olfactory center (procerebrum, PC) of the terrestrial slug Limax valentianus shows a regular oscillation in the local field potential (LFP). Here we report that caffeine, which is an agonist for ryanodine receptors and triggers calcium release from intra-cellular stores, has strong modulatory effects on the PC. In isolated PC neurons, caffeine enhanced the cytoplasmic calcium concentration, and this was blocked by ryanodine. Caffeine elevated the frequency and amplitude of the LFP oscillation, which was also blocked by ryanodine. The time lag between the frequency and amplitude effects suggests distinct mechanisms for the modulation of these two parameters. These results suggest that calcium release from intracellular stores through ryanodine receptors activates network activity in the PC.     

Metabolic flux and cell cycle analysis indicating new mechanism underlying process oscillation in continuous ethanol fermentation with Saccharomyces cerevisiae under VHG conditions

Process oscillation characterized by long oscillation period and large oscillation amplitude was observed in continuous ethanol fermentation with Saccharomyces cerevisiae under very high gravity conditions. Metabolic flux analysis was applied to the fermentation system, and the results indicated that carbon flux distributions at the metabolic notes oscillated, correspondingly, and the root reason for the process oscillation was the intracellular metabolism of yeast cells. Cell cycle analysis with the flow cytometry showed that no cell-cycle-dependent synchronization of the daughter and mother cells occurred within the duration of the oscillation, and thus different mechanism existed compared with the oscillation observed in the continuous culture of Saccharomyces cerevisiae and triggered by the synchronization of the daughter and mother cells under specific conditions. Furthermore, the overall metabolic activity of the yeast cells was examined, which was found not exactly out of phase but lag behind ethanol concentration that accumulated within the fermentation system and its inhibition on the yeast cells as well, which supported the mechanistic speculation for the process oscillation: the lag response of yeast cells to ethanol inhibition.     

Effect of local anesthetics on the electrical characteristics of an excitable model membrane composed of dioleyl phosphate II. Dynamic response

The effects of local anesthetics (tetracaine, procaine and lidocaine) on self-sustained electrical oscillations were studied for a lipid membrane comprising dioleyl phosphate (DOPH). This model membrane exhibits oscillation of the membrane potential in a manner similar to that of nerve membranes, i.e., repetitive firing, in the presence of an ion-concentration gradient, on the application of d.c. electric current. Relatively weak anesthetics such as procaine and lidocaine increased the frequency of self-sustained oscillation, and finally induced aperiodic, rapid oscillation. The strong anesthetic tetracaine inhibited oscillation.     

Load dependence of simulated central tremor

Previous studies have argued that tremors of central versus peripheral origin can be distinguished based on their load dependence: the frequency of peripheral tremor decreases when a weight is added to the tremulous limb, while the frequency of central tremors remains unchanged. The present study scrutinizes the latter statement. We simulated central tremor using a simple network of coupled neural oscillators, which receives proprioceptive feedback from the motor periphery. The network produced a self-sustained, stable oscillation. When the gain of proprioceptive feedback was high, oscillation frequency decreased in the presence of an inertial load. When the gain was low, the oscillation frequency was load independent. We conclude that load dependence is not an exclusive property of peripheral tremors but may be found in tremors of central origin as well. Therefore, the load test is not sufficient to reject a central tremor origin. Received: 1 October 1998 / Accepted in revised form: 28 January 1999     

A model for modulation of neuronal synchronization by D4 dopamine receptor-mediated phospholipid methylation

We describe a new molecular mechanism of dopamine-induced membrane protein modulation that can tune neuronal oscillation frequency to attention-related gamma rhythm. This mechanism is based on the unique ability of D4 dopamine receptors (D4R) to carry out phospholipid methylation (PLM) that may affect the kinetics of ion channels. We show that by deceasing the inertia of the delayed rectifier potassium channel, a transition to 40 Hz oscillations can be achieved. Decreased potassium channel inertia shortens spike duration and decreases the interspike interval via its influence on the calcium-dependent potassium current. This mechanism leads to a transition to attention-related gamma oscillations in a pyramidal cell-interneuron network. The higher frequency and better synchronization is observed with PLM affecting pyramidal neurons only, and recurrent excitation between pyramidal neurons is important for synchronization. Thus dopamine-stimulated methylation of membrane phospholipids may be an important mechanism for modulating firing activity, while impaired methylation can contribute to disorders of attention. Action Editor: Upinder Bhalla     

Detection of gaps in sinusoids by frog auditory nerve fibers: importance in AM coding

Physiological studies were carried out in the frog (Rana pipiens pipiens) eighth nerve to determine: (i) whether the modulation rate or the silent gap was the salient feature that set the upper limit of time-locking to pulsed amplitude-modulated (PAM) stimuli, (ii) the gap detection capacity of individual eighth nerve fibers. Time-locked responses of 79 eighth nerve fibers to PAM stimuli (at the fiber's characteristic frequency) showed that the synchronization coefficient was a low-pass function of the modulation rate. In response to PAM stimuli having different pulse durations, a fiber gave rise to non-overlapping modulation transfer functions. The upper cut-off frequency of time locking was higher when tonepulses in PAM stimuli had shorter duration. The fact that the cut-off frequency was different for the different PAM series suggested that the AM rate was neither the sole, nor the main, determinant for the decay in time-locking at high AM rates. Gap detection capacity was determined for 69 eighth nerve fibers by assessing fiber's spiking activities to paired tone-pulses during an OFF-window and an ON-window. It was found that the minimum detectable gap of eighth nerve fibers ranged from 0.5 to 10 ms with an average of 1.23–2.16 ms depending on the duration of paired tone pulses. For each fiber, the minimum detectable gap was longer when the duration of tone pulses comprising the twin-pulse stimuli was more than four times longer. When the synchronization coefficient was plotted against the silent gap between tones pulses in the PAM stimuli, the gap response functions of a fiber as derived from multiple PAM series were equivalent to gap response functions deriving from twin-pulse series suggesting that it was the silent gap which primarily determined the upper limit of time-locking to PAM stimuli.Abbreviations MTF modulation transfer function - PAM pulse amplitude modulated - SAM sinusoidally amplitude modulated - SC synchronization coefficient - TW time window     

Nephron blood flow dynamics measured by laser speckle contrast imaging

Tubuloglomerular feedback (TGF) has an important role in autoregulation of renal blood flow and glomerular filtration rate (GFR). Because of the characteristics of signal transmission in the feedback loop, the TGF undergoes self-sustained oscillations in single-nephron blood flow, GFR, and tubular pressure and flow. Nephrons interact by exchanging electrical signals conducted electrotonically through cells of the vascular wall, leading to synchronization of the TGF-mediated oscillations. Experimental studies of these interactions have been limited to observations on two or at most three nephrons simultaneously. The interacting nephron fields are likely to be more extensive. We have turned to laser speckle contrast imaging to measure the blood flow dynamics of 50-100 nephrons simultaneously on the renal surface of anesthetized rats. We report the application of this method and describe analytic techniques for extracting the desired data and for examining them for evidence of nephron synchronization. Synchronized TGF oscillations were detected in pairs or triplets of nephrons. The amplitude and the frequency of the oscillations changed with time, as did the patterns of synchronization. Synchronization may take place among nephrons not immediately adjacent on the surface of the kidney.