Emergence of adaptability to time delay in bipedal locomotion

Based on neurophysiological evidence, theoretical studies have shown that locomotion is generated by mutual entrainment between the oscillatory activities of central pattern generators (CPGs) and body motion. However, it has also been shown that the time delay in the sensorimotor loop can destabilize mutual entrainment and result in the failure to walk. In this study, a new mechanism called flexible-phase locking is proposed to overcome the time delay. It is realized by employing the Bonhoeffer–Van der Pol formalism – well known as a physiologically faithful neuronal model – for neurons in the CPG. The formalism states that neurons modulate their phase according to the delay so that mutual entrainment is stabilized. Flexible-phase locking derives from the phase dynamics related to an asymptotically stable limit cycle of the neuron. The effectiveness of the mechanism is verified by computer simulations of a bipedal locomotion model.     

Mathematical models of cochlear nucleus onset neurons: II. model with dynamic spike-blocking state

Onset (On) neurons in the cochlear nucleus (CN), characterized by their prominent response to the onset followed by little or no response to the steady-state of sustained stimuli, have a remarkable ability to entrain (firing 1 spike per cycle of a periodic stimulus) to low-frequency tones up to 1000 Hz. In this article, we present a point-neuron model with independent, excitatory auditory-nerve (AN) inputs that accounts for the ability of On neurons to both produce onset responses for high-frequency tone bursts and entrain to a wide range of low-frequency tones. With a fixed-duration spike-blocking state after a spike (an absolute refractory period), the model produces entrainment to a broad range of low-frequency tones and an On response with short interspike intervals (chopping) for high-frequency tone bursts. To produce On response patterns with no chopping, we introduce a novel, more complex, active membrane model in which the spike-blocking state is maintained until the instantaneous membrane voltage falls below a transition voltage. During the sustained depolarization for a high-frequency tone burst, the new model does not chop because it enters a spike-blocking state after the first spike and fails to leave this state until the membrane voltage returns toward rest at the end of the stimulus. The model entrains to low-frequency tones because the membrane voltage falls below the transition voltage on every cycle when the AN inputs are phase-locked. With the complex membrane model, On response patterns having moderate steady-state activity for high-frequency tone bursts (On-L) are distinguished from those having no steady-state activity (On-I) by requiring fewer AN inputs. Voltage-gated ion channels found in On-responding neurons of the CN may underlie the hypothesized dynamic spike-blocking state. These results provide a mechanistic rationale for distinguishing between the different physiological classes of CN On neurons.     

Non-parametric photic entrainment of Djungarian hamsters with different rhythmic phenotypes

To investigate the role of non-parametric light effects in entrainment, Djungarian hamsters of two different circadian phenotypes were exposed to skeleton photoperiods, or to light pulses at different circadian times, to compile phase response curves (PRCs). Wild-type (WT) hamsters show daily rhythms of locomotor activity in accord with the ambient light/dark conditions, with activity onset and offset strongly coupled to light-off and light-on, respectively. Hamsters of the delayed activity onset (DAO) phenotype, in contrast, progressively delay their activity onset, whereas activity offset remains coupled to light-on. The present study was performed to better understand the underlying mechanisms of this phenomenon. Hamsters of DAO and WT phenotypes were kept first under standard housing conditions with a 14:10 h light–dark cycle, and then exposed to skeleton photoperiods (one or two 15-min light pulses of 100 lx at the times of the former light–dark and/or dark–light transitions). In a second experiment, hamsters of both phenotypes were transferred to constant darkness and allowed to free-run until the lengths of the active (α) and resting (ρ) periods were equal (α:ρ = 1). At this point, animals were then exposed to light pulses (100 lx, 15 min) at different circadian times (CTs). Phase and period changes were estimated separately for activity onset and offset. When exposed to skeleton-photoperiods with one or two light pulses, the daily activity patterns of DAO and WT hamsters were similar to those obtained under conditions of a complete 14:10 h light–dark cycle. However, in the case of giving only one light pulse at the time of the former light–dark transition, animals temporarily free-ran until activity offset coincided with the light pulse. These results show that photic entrainment of the circadian activity rhythm is attained primarily via non-parametric mechanisms, with the “morning” light pulse being the essential cue. In the second experiment, typical photic PRCs were obtained with phase delays in the first half of the subjective night, phase advances in the second half, and a dead zone during the subjective day. ANOVA indicated no significant differences between WT and DAO animals despite a significantly longer free-running period (tau) in DAO hamsters. Considering the phase shifts induced around CT0 and the different period lengths, it was possible to model the entrainment patterns of both phenotypes. It was shown that light-induced phase shifts of activity offset were sufficient to compensate for the long tau in WT and DAO hamsters, thus enabling a stable entrainment of their activity offsets to be achieved. With respect to activity onsets, phase shifts were sufficient only in WT animals; in DAO hamsters, activity onset showed increasing delays. The results of the present paper clearly demonstrate that, under laboratory conditions, the non-parametric component of light and dark leads to circadian entrainment in Djungarian hamsters. However, a stable entrainment of activity onset can be achieved only if the free-running period does not exceed a certain value. With longer tau values, hamsters reveal a DAO phenotype. Under field conditions, therefore, non-photic cues/zeitgebers must obviously be involved to enable a proper circadian entrainment.     

The circadian locomotor rhythm of Hemideina thoracica (Orthoptera; Stenopelmatidae): the circadian clock as a population of interacting oscillators

ABSTRACT. The behaviour of the circadian locomotor rhythm of the New Zealand weta, Hemideina thoracica (White), supports the model that the underlying pacemaker consists of a population of weakly coupled oscillators. Certain patterns of locomotor activity, previously demonstrated almost exclusively in vertebrates, are presented here as evidence for the above hypothesis. They include after-effects of various pre-treatments, rhythm-splitting and spontaneous changes in the rhythm. After-effects, which describe the unstable behaviour of free-running circadian rhythms following particular experimental perturbations, have been observed in Hemideina following single light pulses, constant dim light, and laboratory and natural entrainment. Period changes occurred in the activity rhythm after single light pulses of 8-h and 12-h duration (25 lx). Constant dim light (0.1 lx) increased the free-running period (τ) of the activity rhythm, but the after-effect of constant dim light was either an increase or a decrease in τ. After-effects upon both τ and the active phase length of the activity rhythm were found following non-24-h light entrainment cycles with 8-h and 12-h light phases of 25 lx. Qualitative measurements of these after-effects upon τ are presented which reveal a relationship between both the direction and amount of change in τ, and the difference between entrainment cycle length (T) and pre-entrainment free-running period. The after-effect of natural entrainment was an initial short-period free-run (τ < 24h) lasting 5–10 days, generally followed by a rapid period lengthening to τ= 25–26 h. Support for the population model was provided by spontaneous dampening, recovery, and period changes of the rhythm, together with the disruption of the active phase following critical light perturbations, and rhythm-splitting. These Hemideina results are compared with the simulations of the Coupled Stochastic System of Enright (1980).     

Entrainment of the circadian system of the house sparrow: A population of oscillators in pinealectomized birds

Summary Removal of the pineal gland modifies the entrainment behavior of house sparrows. Abnormal entrainment occurs in pinealectomized sparrows exposed to skeleton photoperiods (light cycles composed of 2 pulses of light per 24-h cycle). This abnormal entrainment depends upon the state of the locomotor activity (rhythmic or arrhythmic) before exposure to the light cycle, and upon the interval between the 2 pulses of light which constitute the skeleton photoperiod. The conditions that produce abnormal entrainment in pinealectomized birds are strongly correlated with those that produce 2 stable phases of entrainment to skeleton photoperiods in normal birds (bistability phenomenon). These results suggest that after pinealectomy, there remains a population of oscillators whose combined output is reflected in the locomotor activity of individual sparrows.Abbreviations LD 12 12 light-dark cycle with 12 h of light and 12 h of dark per 24-h cycle - CT circadian time     

Non‐Parametric Entrainment by Natural Twilight in the Microchiropteran Bat,Hipposideros Speoris Inside a Cave

The study aimed to determine the influence of repeated natural dawn and dusk twilight pulses in entraining the circadian flight activity rhythm of the microchiropteran bat, Hipposideros speoris, free‐running in constant darkness in a natural cave. The bats were exposed to repeated dawn or dusk twilight pulses at eight circadian phases. All bats exposed to dawn twilight pulses were entrained by advancing transients, and the stable entrainment was reached when the onset of activity occurred about 12 h before the lights‐on of the pulses, irrespective of the initial phase at which the bats were exposed to twilight. All bats exposed to dusk twilight pulses, however, were entrained by delaying transients, and the stable entrainment was reached when the onset of activity occurred about 1.6 h after the lights‐on of the pulses. The entrainment caused by dawn and dusk twilight pulses is discussed in the context of the postulated two photoreceptors: the short wavelength sensitive (S) photoreceptors mediating entrainment via dusk twilight, and the medium wavelength sensitive (M) photoreceptors mediating entrainment via dawn twilight.     

Sensory Feedback Mechanism Underlying Entrainment of Central Pattern Generator to Mechanical Resonance

Rhythmic body motions observed in animal locomotion are known to be controlled by neuronal circuits called central pattern generators (CPGs). It appears that CPGs are energy efficient controllers that cooperate with biomechanical and environmental constraints through sensory feedback. In particular, the CPGs tend to induce rhythmic motion of the body at a natural frequency, i.e., the CPGs are entrained to a mechanical resonance by sensory feedback. The objective of this paper is to uncover the mechanism of entrainment resulting from the dynamic interaction of the CPG and mechanical system. We first develop multiple CPG models for the reciprocal inhibition oscillator (RIO) and examine through numerical experiments whether they can be entrained to a simple pendulum. This comparative study identifies the neuronal properties essential for the entrainment. We then analyze the simplest model that captures the essential dynamics via the method of harmonic balance. It is shown that robust entrainment results from a strong, positive-feedback coupling of a lightly damped mechanical system and the RIO consisting of neurons with the complete adaptation property     

Mutual entrainment of two pacemaker cells. A study with an electronic parallel conductance model

Repetitive firing of pacemaker cells was simulated with the use of an electronic Hodgkin-Huxley-like membrane model having a capacitance, a sodium-, a potassium- and a leakage conductance connected in parallel. The frequency of firing of the model cells was controlled by the leakage conductance. Two such model cells of sometimes widely different intrinsic frequencies were coupled through a coupling resistance R_c. Mutual synchronization was allowed to occur by decreasing R_c in small steps from high to low values.The lower R_c, the more the ratio of the mean frequencies of both cells approaches unity. Around certain R_c values stable entrainment occurs during which the mean frequencies are related as simple integral values m: n (e.g. 2:1, 3:2, 4:3, 1:1 for decreasing R_c). Within the range of a fixed m: n entrainment the phase difference between the oscillations of the two cells declines with R_c. When R_c is close to the range ofm: n entrainment, “almost entrainment” occurs in which m: n entrainment is periodic. But also further away from the range of m: n entrainment periodic variations in action potential interval occur.The model cells were adjusted in such a way that a depolarizing current pulse can only advance the first occurring and all subsequent action potentials. This property is used to explain effects of R_c on the mean frequency of the two cells both within and without the R_c ranges of entrainment. Furthermore, it is illustrated how in certain cases the synchronized frequency at R_c = 0 ω follows from simple electrical considerations of the properties of the uncoupled cells.Several of the entrainment phenomena observed are well known in the fields of electronic relaxation oscillators, clinical electrocardiography and circadian rhythms, though with different terminologies. The present model results exemplify the general nature of these phenomena and clarify entrainment phenomena, recently observed in coupling experiments with pacemaker cells from embryonic heart tissue.     

Development of hamster circadian rhythms: Role of the maternal suprachiasmatic nucleus

Summary During development, the circadian rhythms of rodents become entrained to rhythmicity of the mother. Rhythms in behavior and in neuroendocrine function are regulated by a circadian pacemaker thought to be located within the suprachiasmatic nucleus (SCN) of the hypothalamus. Evidence indicates that this pacemaker begins to function and to be entrained by maternal rhythms before birth. Although the maternal rhythms which mediate prenatal entrainment of the fetal circadian pacemaker have not been identified, it is likely that they are regulated by the maternal SCN.The role of the maternal SCN in entrainment of the offspring was examined in Syrian hamsters (Mesocricetus auratus) by measuring the activity/rest rhythms of pups. Using the synchrony among the rhythms of pups within a litter as an indication that the pups had been entrained, the effect on entrainment of ablating the maternal SCN was determined. Lesions of the maternal SCN which were performed early in gestation (day 7) and which destroyed at least 75% of the SCN were found to disrupt the normal within litter synchrony among pups, indicating interference with the normal mechanism of entrainment.The effect of lesions on day 7 of gestation could mean that the maternal SCN is important for entrainment of the pups before birth, after birth, or during both of these times. To determine if the maternal SCN is specifically important for prenatal entrainment, lesions were performed two days before birth on day 14 of gestation. Lesions of the maternal SCN on day 14 were not as disruptive as were lesions on day 7. This suggests that the maternal SCN is important between days 7 and 14 of gestation and that the synchrony normally observed at weaning is already established, in part, on or before day 14 of gestation. This further suggests that an entrainable circadian pacemaker is present in the fetus only two weeks after fertilization.Abbreviations SCN suprachiasmatic nucleus - L:D light:dark - LL constant light - r mean vector length - 2DG 2-deoxyglucose - NAT N-acetyltransferase     

A pacemaker cell pair model based on the phase response curve

A pacemaker cell pair model and the dynamic interaction between the two pacemaker cells is described in this paper. It is an extension of our single pacemaker cell model, in which we studied its response to repetitive external depolarization stimulations. This model is a simple model based on the two most important functional properties of the cardiac pacemaker cells: its intrinsic pacemaker cycle length, which is an `internal' parameter of the cell, and the phase response curve (PRC), which is an `overall collective' function. The PRC contains all the `information' about the possible interactions of the pacemaker cell with the outside world (interaction with surrounding cells, external stimulus, etc.). First, we examined the properties and solutions of 1:1 synchronization between two pacemaker cells. We found that in order to achieve synchronization between two pacemaker cells, there should be limitations on the PRC parameters, which depend on the cells intrinsic cycle lengths. Next, we investigated the 2:1 entrainment state between two interacting pacemaker cells. We found that there is not necessarily a unique solution for this state as there was for the 1:1 state. Finally, we ran our computer model to investigate the properties of more complex patterns of entrainment between two pacemaker cells. As a result of our analytical study, we unveil two new important parameters, which are fully defined as a function of the PRC parameters: (1) the `accelerator factor' which describes the tendency of a pair of interacting pacemaker cells to synchronize at a common cycle length, which is closer to the faster cycle of the pair; (2) the `degree of coupling', which describes the range of the 1:1 synchronization and the `strength' of the interaction between a pair of interacting pacemaker cells. Those two interaction parameters arise as helpful `tools' for the understanding of synchronization and mutual entrainment mechanisms between pacemaker cells. Therefore, this study establishes the PRC as an important determinant and a useful approach for the understanding of the dynamic interaction of pacemaker cells among themselves and with the outside world. Received: 12 May 1997 / Accepted in revised form: 22 April 1998     

Astronomical cycles

Abstract

Talorchestia quoyana, a sand beach amphipod, shows a rhythm of locomotor activity controlled by a circadian clock and an inhibitory circatidal clock. This article reports on an investigation of the entrainment of the circadian dock to skeleton photoperiods. Four important mathematical models for circadian rhythms are examined with respect to the results of the entrainment experiments and to predictions from the phase response curve for Talorchestia. Significant differences between the models are described, and properties of circadian rhythms not accounted for by present models are outlined.     

PHOTIC ENTRAINMENT OF CIRCADIAN ACTIVITY PATTERNS IN THE TROPICAL LABRID FISH HALICHOERES CHRYSUS

Yellow wrasses (Halichoeres chrysus) show clear daily activity patterns. The fish hide in the substrate at (subjective) night, during the distinct rest phase. Initial entrainment in a 12h:12h light-dark (12:12 LD) cycle (mean period 24.02h, SD 0.27h, n = 16 was followed by a free run (mean period 24.42h, SD 1.33h) after transition into constant dim light conditions. Light pulses of a comparable intensity as used in the light part of the LD cycles did not result in significant phase shifts of the free-running rhythm in constant darkness. Application of much brighter 3h light pulses resulted in a phase-response curve (PRC) for a fish species, with pronounced phase advances during late subjective night. The PRCs differed from those mainly obtained in other vertebrate taxa by the absence of significant phase delays in the early subjective night. At that circadian phase, significant tonic effects of the light pulses caused a shortening of the circadian period length. Entrainment to skeleton photoperiods of 1:11 LD was observed in five of six wrasses exposed, also after a 3h phase advance of this LD cycle. Subsequently, a 1:11.25 LD cycle resulted in entrainment in four of the six fish. It is suggested that the expression of the circadian system in fish can be interpreted as a functional response to a weak natural zeitgeber, as present in the marine environment. This response allows photic entrainment as described here in the yellow wrasse. (Chronobiology International, 17(5), 613–622, 2000)     

Late emergence chronotypes of fruit flies Drosophila melanogaster exhibit higher accuracy of entrainment

Inter-individual variation in phase-of-entrainment (chronotype) is widely observed in many species, but the underlying mechanisms and its consequences remain largely unexplored. In light of considerable limitations of previous studies proposing that the late chronotypes exhibit weakly stable rhythms, we employed outbred Drosophila populations exhibiting early and late emergence chronotypes to re-visit such associations. Contrary to previous reports, we observed that the late chronotypes consistently exhibit higher stability in emergence and activity–rest rhythms as compared to the early chronotypes, both under laboratory and semi-natural conditions, which is not associated with higher precision of circadian clocks, thus demonstrating the existence of genetic correlations between accuracy of entrainment and chronotype. Our results, along with the previously reported clock property differences between the early and the late emergence chronotypes highlights a possible complex interplay of clock period, phase response curve and accuracy in determining phase-of-entrainment.     

Physio-chemical mathematical theory for transitions in cellular metabolism: An aspect of carcinogenesis

It is pointed out that the non-monotonic character of the chemical reaction rate expressions, together with the relative magnitude of the diffusivity constants, is likely to engender a multiplicity of locally stable steady-state solutions to the system of reaction-diffusion equations for the concentration distributions of molecular species through the volume of a living cell. A transition in cellular metabolism, i.e., the dynamical evolution from an initial locally stable steady-state solution for the concentration distributions to another distinct locally stable steady-state solution, can be induced by an etiologic agent which modifies the rate expressions significantly during an interval of time. Global inequality analysis is employed to derive a condition on the modified rate expressions that is sufficient to guarantee the occurrence of such a transition in cellular metabolism. The possibility of a transition induced by a chemical carcinogen is investigated by applying the latter sufficient condition, and it is found that the statistical frequency of carcinogenesis should depend essentially on the magnitude of the grouping (T ^{2 − α} D ^α) for a total doseD of carcinogen administered to an animal at a uniform rate (D/T) over a time interval of durationT, where α is a certain positive number less than 1. This theoretical result is shown to be supported by the available experimental evidence.     

A mechanistic model of the adaptation of phytoplankton photosynthesis

The mechanistic model of the phytoplankton photosynthesis-light intensity relationship by Eilers and Peeters (1988.Ecol. Modelling 42, 199–215) is investigated mathematically. The model is based on the physiological idealization of transition probabilities between states of the photosynthetic factories,PSF. The model was found to have under constant light condition a globally stable unique positive equilibrium, while under periodically varying light (e.g. daily periodicity) there exists a unique globally asymptotically stable periodic solution. Based on this, the adaptation to a change of light intensity is defined as a process by which the state ofPSF converges to an equilibrium. Assuming that phytoplankton regulates its photosynthetic production rate with a certain strategy which maximizes production, two such possible strategies were examined. Both the instantaneous and the integral maximal photosynthetic production were shown to have the same result. With realistic qualitative assumptions of the shape of the dependence of the four model parameters on the light intensity to which phytoplankton is adapted, the numerical values of parameters under both constant and periodically varying conditions are determined by applying Pontryagin's maximum principle.     

Conformational study of trinucleoside tetraphosphate d(pCpGpCp): Transition of right-handed form to left-handed form

Using the semiempirical potential functions, conformational energies of the model compounds DMP^?, d(pCp), d(pGp), and d(pCpGpCp) are calculated, and the B → Z transition is discussed along the pseudorotational path of the sugar ring. For dimethylmonophosphate anion, DMP^?, the energy contour map is presented and the stabilities of the phosphodiester conformations discussed. For the sugar ring without the base attached, the minimum energies for each sugar-puckering form are calculated along the pseudorotational path. The energy barrier of the interconversion between the C(3′)-endo form and the C(2′)-endo form is calculated to be about 2.0 kcal/mol. From the conformational energy calculations of the interconversions of mononucleoside diphosphates, d(pCp) and d(pGp), between the C(2′)-endo conformer and the C(3′)-endo conformer, the purine sugar segment is known to be more convertible than the pyrimidine sugar segment. The results also support the finding that the pseudorotational transition occurred with the O(1′)-endo form more easily than with the O(1′)-exo form. Based on the results of conformational studies of DMP^?, d(pCp), and d(pGp), a topological transition of the handedness of the model compound, d(pCpGpCp), is studied. The left-handed Z-form is found to be less stable by about 8.5 kcal/mol than is the right-handed B-form. The energy barrier of the Z → B transition is calculated to be about 17.4 kcal/mol. The contributions of the electrostatic and nonbonded energies to the energy barrier are discussed in connection with the conformation changes of the model compound, d(pCpGpCp).     

Territorial occupancy dynamics in a forest raptor community

A Markovian modeling approach was used to explore territorial interactions among three forest raptors coexisting in a forested natural area in southeast Spain: the booted eagle (Hieraaetus pennatus), the common buzzard (Buteo buteo) and the northern goshawk (Accipiter gentilis). Using field data collected over a period of 12 years, 11 annual transition matrices were built, considering four occupancy states for each territory. The model describes transitional processes (colonization, abandonment, replacement and persistence), permits temporal variations in the transition matrix to be tested, and simulates territorial occupation for a few subsequent years. Parameters for the species and community dynamics were described in terms of turnover times and damping ratio. A perturbation analysis was performed to simulate the effects of changes in the transition probabilities on the stable state distribution. Our results indicate the existence of a stable community, largely dominated by the booted eagles, and described by a time-invariant transition matrix. Despite the stability observed, the territorial system is highly dynamic, with frequent abandonment and colonization events, although interspecific territorial interactions (the replacement of one species by another) are uncommon. Consequently, the three species appear to follow relatively independent occupancy dynamics. Simulation of potential management actions showed that substantial increases in the number of territories occupied by the less common species (goshawk and buzzard) can only be attained if relatively large increases in their reoccupation and colonization rates are considered.