Phase resetting and phase singularity of an insect circannual oscillator

In circadian rhythms, the shape of the phase response curves (PRCs) depends on the strength of the resetting stimulus. Weak stimuli produce Type 1 PRCs with small phase shifts and a continuous transition between phase delays and advances, whereas strong stimuli produce Type 0 PRCs with large phase shifts and a distinct break point at the transition between delays and advances. A stimulus of an intermediate strength applied close to the break point in a Type 0 PRC sometimes produces arrhythmicity. A PRC for the circannual rhythm was obtained in pupation of the varied carpet beetle, Anthrenus verbasci, by superimposing a 4-week long-day pulse (a series of long days for 4 weeks) over constant short days. The shape of this PRC closely resembles that of the Type 0 PRC. The present study shows that the PRC to 2-week long-day pulses was Type 1, and that a 4-week long-day pulse administered close to the PRC’s break point induced arrhythmicity in pupation. It is, therefore, suggested that circadian and circannual oscillators share the same mode in phase resetting to the stimuli.     

Pulse coupled oscillators and the phase resetting curve

Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators. In these oscillators, the interactions take the form of brief pulses such that the effect of one input dies out before the next is received. A phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. PRCs can be used to predict phase locking in networks of pulse coupled oscillators. In some studies of pulse coupled oscillators, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is generated using a perturbation that approximates the input received in the real biological network. In general, this approach requires that circuit architecture and a specific firing pattern be assumed. This allows the construction of discrete maps from one event to the next. The fixed points of these maps correspond to periodic firing modes and are easier to locate and analyze for stability compared to locating and analyzing periodic modes in the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks.     

Control of multistability in ring circuits of oscillators

The essential dynamics of some biological central pattern generators (CPGs) can be captured by a model consisting of N neurons connected in a ring. These circuits, like many oscillatory nonlinear circuits of sufficient complexity, are capable of multistability, that is, of generating different firing patterns distinguished by the phasic relationships between the firing in each circuit element (neuron). Moreover, a shift in firing pattern can be induced by a transient perturbation. A systematic approach, based on phase-response curve (PRC) theory, was used to determine the optimum timing for perturbations that induce a shift in the firing pattern. The first step was to visualize the solution space of the ring circuit, including the attractive basins for each stable firing pattern; this was possible using the relative phase of N−1 oscillators, with respect to an arbitrarily selected reference oscillator, as coordinate axes. The trajectories in this phase space were determined using an iterative mapping based only on the PRCs of the uncoupled component oscillators; this algorithm was called a circuit emulator. For an accurate mapping of the attractive basin of each pattern exhibited by the ring circuit, the emulator had to take into account the effect of a perturbation or input on the timing of two bursts following the onset of the perturbation, rather than just one. The visualization of the attractive basins for rings of two, three, and four oscillators enabled the accurate prediction of the amounts of phase resetting applied to up to N−1 oscillators within a cycle that would induce a transition from any pattern to any another pattern. Finally, the timing and synaptic characterization of an input called the switch signal was adjusted to produce the desired amount of phase resetting. Received: 29 May 1998 / Accepted in revised form: 18 September 1998     

The root–soil system of Norway spruce subjected to turning moment: resistance as a function of rotation

The reactions of trees to wind, rockfall, and snow and debris flow depend largely on how strong and deformable their anchorage in the soil is. Here, the resistive turning moment M of the root–soil system as a function of the rotation ϕ at the stem base plays the major role. M(ϕ) describes the behavior of the root–soil system when subject to rotational moment, with the maximum M(ϕ) indicating the anchorage strength M _a of the tree. We assessed M(ϕ) of 66 Norway spruce (Picea abies L. Karst) by pulling them over with a winch. These 45- to 170-year-old trees grew at sites of low and high elevation, with a diameter at breast height DBH = 14–69 cm and a height H = 9–42 m. M(ϕ) displayed a strong nonlinear behavior. M _a was reached at a lower ϕ for large trees than for small trees. Thus overhanging tree weight contributed less to M _a for the large trees. Overturning also occurred at a lower ϕ for the large trees. These observations show that the rotational ductility of the root–soil system is higher for small trees. M _a could be described by four monovariate linear regression equations of tree weight, stem weight, stem volume and DBH ² ·H (0.80 < R ² < 0.95), and ϕ at M _a, ϕ _a, by a power law of DBH²·H (R ² = 0.85). We found significantly higher M _a for the low-elevation spruces than for the high-elevation spruces, which were more shallowly anchored, but no significant difference in ϕ _a. The 66 curves of M(ϕ), normalized (_n) by M _a in M-direction and by ϕ _a in ϕ-direction, yielded one characteristic average curve: . Using and the predictions of M _a and ϕ _a, it is shown that M(ϕ) and the curves associated with M(ϕ) can be predicted with a relative standard error ≤25%. The parameterization of M(ϕ) by tree size and weight is novel and provides useful information for predicting with finite-element computer models how trees will react to natural hazards.     

Studies in mathematical biosociology of imitative behavior: I. Effects of income distribution

In connection with previous studies (Rashevsky,Mathematical Biology of Social Behavior, chap. xii), a situation is investigated in which the two mutually exclusive possible behaviors of a society consist of the desire to keep the present socioeconomical situation and the desire to change it inany way. The psychophysiological tendency ϕ towards either of the behaviors is considered to be proportional to the difference between the actual incomei of the individual and his needsi′. Assuming that the distribution functionN ₁(i′) of the needs is a given characteristic of the population, it is shown that the distribution functionN(ϕ) of ϕ in the society can be derived fromN ₁(i′) and from the distributionN ₂(i) of the incomesi. A particular case is worked out as an example. Conditions of stability of a socioeconomic structure are studied in their dependence on the income distribution.     

Cellularly-driven differences in network synchronization propensity are differentially modulated by firing frequency

Spatiotemporal pattern formation in neuronal networks depends on the interplay between cellular and network synchronization properties. The neuronal phase response curve (PRC) is an experimentally obtainable measure that characterizes the cellular response to small perturbations, and can serve as an indicator of cellular propensity for synchronization. Two broad classes of PRCs have been identified for neurons: Type I, in which small excitatory perturbations induce only advances in firing, and Type II, in which small excitatory perturbations can induce both advances and delays in firing. Interestingly, neuronal PRCs are usually attenuated with increased spiking frequency, and Type II PRCs typically exhibit a greater attenuation of the phase delay region than of the phase advance region. We found that this phenomenon arises from an interplay between the time constants of active ionic currents and the interspike interval. As a result, excitatory networks consisting of neurons with Type I PRCs responded very differently to frequency modulation compared to excitatory networks composed of neurons with Type II PRCs. Specifically, increased frequency induced a sharp decrease in synchrony of networks of Type II neurons, while frequency increases only minimally affected synchrony in networks of Type I neurons. These results are demonstrated in networks in which both types of neurons were modeled generically with the Morris-Lecar model, as well as in networks consisting of Hodgkin-Huxley-based model cortical pyramidal cells in which simulated effects of acetylcholine changed PRC type. These results are robust to different network structures, synaptic strengths and modes of driving neuronal activity, and they indicate that Type I and Type II excitatory networks may display two distinct modes of processing information.     

Cell Division Cycles and Orcadian Oscillators in Euglena

The algal flagellate Euglena grown photoautotrophically in L:D 3:3 displays a circadian rhythm of cell division. Oscillatory models for cell cycle (CDC) control (particularly those of the limit cycle variety) include the property of phase perturbation, or resetting. This prediction has been tested in synchronous cultures in which the free-running rhythm has been scanned by 3-hr light signals. A strong (Type 0) phase response curve (PRC), yielding both advances and delays as great as 15 hr, has been derived. A second prediction of the limit cycle model is that there exists a pulse of a critical intensity, which, if given at one specific phase of the rhythm (the singularity point), should result in a phaseless, motionless state in which the rhythmicity disappears. Such a point has been found in Euglena in the late subjective night for light pulses having an intensity ranging from 40 to 700 Ix. Finally, circadian oscillators typically display temperature-compensated period lengths within the physiological range of steady-state temperatures, although the length of the CDC is commonly thought to be highly temperature dependent. We have found that over a range of at least 10°C, the period of the division rhythm is only slightly affected, exhibiting a Q₁₀ of about 1.05-1.20. These observations, therefore, collectively implicate a circadian oscillator in the control of the CDC.     

EPR properties of mixed-valent μ-oxo and μ-hydroxo dinuclear iron complexes produced by radiolytic reduction at 77 K

 Radiolytic reduction at 77 K of oxo-/hydroxo-bridged dinuclear iron(III) complexes in frozen solutions forms kinetically stabilized, mixed-valent species in high yields that model the mixed-valent sites of non-heme, diiron proteins. The mixed-valent species trapped at 77 K retain ligation geometry similar to the initial diferric clusters. The shapes of the mixed-valent EPR signals depend strongly on the bridging ligands. Spectra of the Fe(II)OFe(III) species reveal an S=1/2 ground state with small g-anisotropy as characterized by the uniaxial component (g _z –g _av /2<0.03) observable at temperatures as high as ∼100 K. In contrast, hydroxo-bridged mixed-valent species are characterized by large g-anisotropy (g _z –g _av /2>0.03) and are observable only below 30 K. Annealing at higher temperatures causes structural relaxation and changes in the EPR characteristics. EPR spectral properties allow the oxo- and hydroxo-bridged, mixed-valent diiron centers to be distinguished from each other and can help characterize the structure of mixed-valent centers in proteins. Received: 27 June 1998 / Accepted: 25 February 1999     

Comparing Membrane Simulations to Scattering Experiments: Introducing the SIMtoEXP Software

SIMtoEXP is a software package designed to facilitate the comparison of biomembrane simulations with experimental X-ray and neutron scattering data. It has the following features: (1) Accepts number density profiles from simulations in a standard but flexible format. (2) Calculates the electron density ε(z) and neutron scattering length density ν(z) profiles along the z direction (i.e., normal to the membrane) and their respective Fourier transforms (i.e., F _ε [q _z ] and F _ν [q _z ]). The resultant four functions are then displayed graphically. (3) Accepts experimental F _ε (q _z ) and F _ν (q _z ) data for graphical comparison with simulations. (4) Allows for lipids and other large molecules to be parsed into component groups by the user and calculates the component volumes following Petrache et al. (Biophys J 72:2237–2242, 1997). The software then calculates and displays the contributions of each component group as volume probability profiles, ρ(z), as well as the contributions of each component to ε(z) and ν(z).     

Phase and Period Responses of the Two-Peak Circadian Rhythm of Trigonoscelis gigas Reitter (Coleoptera: Tenebrionidae) for 6-hr Light Pulses

Circadian rhythm of locomotor activity of the desert beetle T.gigas usually has two narrow peaks: morning (M) and evening (E). While entrained with diurnal (T_z = 24 hr) full or skeleton photoperiods, the M peak is precedes light, while the E peak coincides with light. In a variety of natural and laboratory conditions both peaks tend to maintain a stable mutual phase relationship, about 12 hr apart. The phase responses of the M and E peaks were studied using 6-hr, 30 lx green LED-light pulses applied around ct3, ?t12 and ct18. The PRC for the E peak, plotted versus ct0 (extrapolated moment of light-on) as abscissa, had the same position, as the PRC for the M peak. Both PRCs were asymmetric, but in an opposite way: for the M peak the area of phase advances was bigger, than the area of phase delays, while for the E peak, vice versa. The transient PRCs on day 1, 2 etc. did not differ from the steady state PRC, i.e, the phase response was accomplished virtually in one cycle. Period changes were almost all positive (period became longer after a light pulse). The only "dead zone" in the period response curve (decrease of Dt down to zero) was around subjective evening - early night. Here again, the M peak appeared more "eager" to phase advances than the E peak. Our data support the hypothesis that M and E peaks are controlled by putative separate oscillators. These oscillators seem to have different properties, tend to phase shift to a different extent, and are extremely strongly mutually coupled with phases locked at approximately 180°. The asymmetry of properties of the M and E oscillators has a clear adaptive significance.     

Limitations of perturbative techniques in the analysis of rhythms and oscillations

Perturbation theory is an important tool in the analysis of oscillators and their response to external stimuli. It is predicated on the assumption that the perturbations in question are “sufficiently weak”, an assumption that is not always valid when perturbative methods are applied. In this paper, we identify a number of concrete dynamical scenarios in which a standard perturbative technique, based on the infinitesimal phase response curve (PRC), is shown to give different predictions than the full model. Shear-induced chaos, i.e., chaotic behavior that results from the amplification of small perturbations by underlying shear, is missed entirely by the PRC. We show also that the presence of “sticky” phase–space structures tend to cause perturbative techniques to overestimate the frequencies and regularity of the oscillations. The phenomena we describe can all be observed in a simple 2D neuron model, which we choose for illustration as the PRC is widely used in mathematical neuroscience.     

Numerical design of RN<Stack><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript><Superscript>ν</Superscript></Stack> symmetry-based RF pulse schemes for recoupling and decoupling of nuclear spin interactions at high MAS frequencies

An approach for the efficient implementation of RN _n ^ν symmetry-based pulse schemes that are often employed for recoupling and decoupling of nuclear spin interactions in biological solid state NMR investigations is demonstrated at high magic-angle spinning frequencies. RF pulse sequences belonging to the RN _n ^ν symmetry involve the repeated application of the pulse sandwich {R _ϕ R _−ϕ}, corresponding to a propagator U _RF = exp(−i4ϕI _z), where ϕ = πν/N and R is typically a pulse that rotates the nuclear spins through 180° about the x-axis. In this study, broadband, phase-modulated 180° pulses of constant amplitude were employed as the initial ‘R’ element and the phase-modulation profile of this ‘R’ element was numerically optimised for generating RN _n ^ν symmetry-based pulse schemes with satisfactory magnetisation transfer characteristics. At representative MAS frequencies, RF pulse sequences were implemented for achieving ¹³C–¹³C double-quantum dipolar recoupling and through bond scalar coupling mediated chemical shift correlation and evaluated via numerical simulations and experimental measurements. The results from these investigations are presented here. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Kinetic processes initiated by a nanosecond high-current discharge in hot air

Results from numerical investigations of kinetic processes initiated by a pulsed nanosecond discharge in hot (T ₀ ≥ 1000 K) air at atmospheric pressure are presented. The calculated results on the dynamics of the electron density, the population of the N₂(B³Π _g ) and N₂(C³Π _u ) states, and the atomic oxygen density in the axial discharge region agree with experiment. The method for determining the gas temperature by measuring the rotational structure of the transitions N₂(C³Π _u , ν) → N₂(B³Π _g , ν′) of the 2⁺ nitrogen system is analyzed. It is shown that, in relatively weak reduced electric fields typical of secondary discharge pulses, the electron impact excitation of the N₂(C³Π _u ) state from the ground state N₂(X¹Σ _g ⁺) can be accompanied by its additional step population from the N₂(B³Π _g ), N₂(a′Σ _u ⁻), and other electronic states. This effect substantially influences the rotational distribution of nitrogen molecules in the N₂(C³Π _u , ν) state; moreover, the temperature determined from this distribution can be substantially higher than the true gas temperature.     

Fault tolerance in the cardiac ganglion of the lobster

Canavier et al. (1997) used phase response curves (PRCs) of individual oscillators to characterize the possible modes of phase-locked entrainment of an N-oscillator ring network. We extend this work by developing a mathematical criterion to determine the local stability of such a mode based on the PRCs. Our method does not assume symmetry; neither the oscillators nor their connections need be identical. To use these techniques for predicting modes and determining their stability, one need only determine the PRC of each oscillator in the ring either experimentally or from a computational model. We show that network stability cannot be determined by simply testing the ability of each oscillator to entrain the next. Stability depends on the number of neurons in the ring, the type of mode, and the slope of each PRC at the point of entrainment of the respective neuron. We also describe simple criteria which are either necessary or sufficient for stability and examine the implications of these results. Received: 2 April 1998 / Accepted in revised form: 2 July 1998     

The influence of phosphate concentration on the kinetics of uptake by maize roots

In an experiment with native maize roots depending on different phosphorus concentration in the external solution (0.001 … 50 mM P), the multiphasic character of the kinetics of phosphate uptake has been stated. The single phases are characterized by the different values of K_m and V_max. In the wide range of concentrations the isotherm of the phosphate uptake has five evident phases. The character of kinetics for the uptake of phosphate is analogical to the kinetics of the enzymatic reactions described by the Michaelis-Menten equation. On the other hand the linear dependence for the inactivated root was determined,i.e. the uptake of phosphate versus different phosphorus concentration in the external solution. The graphic representation of the logarithmic values for the phosphorus taken up versus the different phosphorus concentration in the external solution gives the biphasic course including concentration less than 1.0 mM P and more than 1.0 mM P. Within the framework of the concentration range the following values of V_max, K_m and ϕ_in were calculated under the conditions if the concentration of phosphorus is less than 1.0mMP: V_max = 1.705 μmol P × g^-1h^-1, K_m = 0.057 mM P and ϕ_in = 0.83,i.e. if the concentration of phosphorus is more than 1.0mM P: V_max = 40 μmol P × g^-1 h^-1, K_m = 16.66 mM and ϕ_in = 20. According to these results, the phosphate concentration in the external solution influences the activity of the transport mechanisms concerning their conformative changes which discretely change their working regime of membrane transport. This is also demonstrated in the change of values V_max, K_m and ϕ_in.     

Phase response characteristics of model neurons determine which patterns are expressed in a ring circuit model of gait generation

In order to assess the relative contributions to pattern-generation of the intrinsic properties of individual neurons and of their connectivity, we examined a ring circuit composed of four complex physiologically based oscillators. This circuit produced patterns that correspond to several quadrupedal gaits, including the walk, the bound, and the gallop. An analysis using the phase response curve (PRC) of an uncoupled oscillator accurately predicted all modes exhibited by this circuit and their phasic relationships – with the caveat that in certain parameter ranges, bistability in the individual oscillators added nongait patterns that were not amenable to PRC analysis, but further enriched the pattern-generating repertoire of the circuit. The key insights in the PRC analysis were that in a gait pattern, since all oscillators are entrained at the same frequency, the phase advance or delay caused by the action of each oscillator on its postsynaptic oscillator must be the same, and the sum of the normalized phase differences around the ring must equal to an integer. As suggested by several previous studies, our analysis showed that the capacity to exhibit a large number of patterns is inherent in the ring circuit configuration. In addition, our analysis revealed that the shape of the PRC for the individual oscillators determines which of the theoretically possible modes can be generated using these oscillators as circuit elements. PRCs that have a complex shape enable a circuit to produce a wider variety of patterns, and since complex neurons tend to have complex PRCs, enriching the repertoire of patterns exhibited by a circuit may be the function of some intrinsic neuronal complexity. Our analysis showed that gait transitions, or more generally, pattern transitions, in a ring circuit do not require rewiring the circuit or any changes in the strength of the connections. Instead, transitions can be achieved by using a control parameter, such as stimulus intensity, to sculpt the PRC so that it has the appropriate shape for the desired pattern(s). A transition can then be achieved simply by changing the value of the control parameter so that the first pattern either ceases to exist or loses stability, while a second pattern either comes into existence or gains stability. Our analysis illustrates the predictive value of PRCs in circuit analysis and can be extended to provide a design method for pattern-generating circuits. Received: 20 November 1996 / Accepted: 29 July 1997     

Biological oscillators can be stopped—Topological study of a phase response curve

Many biological oscillators are stable against noise and perturbation (e.g. circadian rhythms, biochemical oscillators, pacemaker neurons, bursting neurons and neural networks with periodic outputs). The experiment of phase shifts resulting from discrete perturbation of stable biological rhythms was developed by Perkel and coworkers (Perkel et al., 1964). By these methods, they could get important insights into the entrainment behaviors of biological rhythms. Phase response curves, which are measured in these experiments, can be classified into two types. The one is the curve with one mapping degree (Type 1), and the other is that with zero mapping degree (Type 0) (Winfree, 1970). We define the phase response curve mathematically, and explain the difference between these two types by the homotopy theory. Moreover, we prove that, if a Type 0 curve is obtained at a certain magnitude of perturbation, there exists at least one lower magnitude for which the phase response curve cannot be measured. Some applications of these theoretical results are presented.     

Impact of adaptation currents on synchronization of coupled exponential integrate-and-fire neurons

The ability of spiking neurons to synchronize their activity in a network depends on the response behavior of these neurons as quantified by the phase response curve (PRC) and on coupling properties. The PRC characterizes the effects of transient inputs on spike timing and can be measured experimentally. Here we use the adaptive exponential integrate-and-fire (aEIF) neuron model to determine how subthreshold and spike-triggered slow adaptation currents shape the PRC. Based on that, we predict how synchrony and phase locked states of coupled neurons change in presence of synaptic delays and unequal coupling strengths. We find that increased subthreshold adaptation currents cause a transition of the PRC from only phase advances to phase advances and delays in response to excitatory perturbations. Increased spike-triggered adaptation currents on the other hand predominantly skew the PRC to the right. Both adaptation induced changes of the PRC are modulated by spike frequency, being more prominent at lower frequencies. Applying phase reduction theory, we show that subthreshold adaptation stabilizes synchrony for pairs of coupled excitatory neurons, while spike-triggered adaptation causes locking with a small phase difference, as long as synaptic heterogeneities are negligible. For inhibitory pairs synchrony is stable and robust against conduction delays, and adaptation can mediate bistability of in-phase and anti-phase locking. We further demonstrate that stable synchrony and bistable in/anti-phase locking of pairs carry over to synchronization and clustering of larger networks. The effects of adaptation in aEIF neurons on PRCs and network dynamics qualitatively reflect those of biophysical adaptation currents in detailed Hodgkin-Huxley-based neurons, which underscores the utility of the aEIF model for investigating the dynamical behavior of networks. Our results suggest neuronal spike frequency adaptation as a mechanism synchronizing low frequency oscillations in local excitatory networks, but indicate that inhibition rather than excitation generates coherent rhythms at higher frequencies.     

Mirror imaging phase response curves obtained for the circadian rhythm of a bat with single steps of light and darkness∗

Abstract

The circadian rhythm in the flight activity of a tropical microchiropteran bat Taphozous melanopogon responds at all phases with delay phase shifts to single light‐on steps (DD/LL transfers). The circadian rhythm responds at all phases with advance phase shifts to single light‐off steps (LL/DD transfers). Phase shifts were measured from the delays or advances of the onsets of flight activity on days following DD/LL and LL/DD transfers relative to the temporal course of the onsets of activity in controls. The magnitude of the phase shifts was a function of the phases in which the transfers were made. The On‐PRC and Off‐PRC plotted from such data are mirror‐images in their time‐course and wave‐form.

The phase shifts of the circadian rhythm in either direction were accompanied by changes in period (for the duration of our recordings after die transfer). The period lengthened following a delay shift and it shortened following an advance shift. The phase shifts are abrupt and discernible in the first cycle after perturbation. There are no transients.     

Analysis of clustered firing patterns in synaptically coupled networks of oscillators

Oscillators in networks may display a variety of activity patterns. This paper presents a geometric singular perturbation analysis of clustering, or alternate firing of synchronized subgroups, among synaptically coupled oscillators. We consider oscillators in two types of networks: mutually coupled, with all-to-all inhibitory connections, and globally inhibitory, with one excitatory and one inhibitory population of oscillators, each of arbitrary size. Our analysis yields existence and stability conditions for clustered states, along with formulas for the periods of such firing patterns. By using two different approaches, we derive complementary conditions, the first set stated in terms of time lengths determined by intrinsic and synaptic properties of the oscillators and their coupling and the second set stated in terms of model parameters and phase space structures directly linked to parameters. These results suggest how biological components may interact to produce the spindle sleep rhythm in thalamocortical networks. Received: 9 September 1999 / Revised version: 7 July 2000 / Published online: 24 November 2000