Simulation of atrial activity by a phase response curve based model of a two-dimensional pacemaker cells array: the transition from a normal activation pattern to atrial fibrillation

In this paper, we present an original model of the atria, based on our hypothesis that atrial cells have features of pacemaker cells, characterized by their normally longer intrinsic cycle lengths and different type of connection (stronger) than the, sino-atrial (SA) node pacemaker cells. The atrium is simulated by a two-dimensional array of pacemaker cells (25 × 25), composed of a region of SA node pacemaker cells (11 × 11) surrounded by atrial pacemaker cells. All pacemakers cells are characterized by only the most relevant functional properties, those which play the most direct role in the determination of the cardiac rate and in the mechanism of arrhythmias. These properties are: the intrinsic cycle length, τ, an `internal' feature of each pacemaker cell, and the phase-response curve (PRC), an `overall collective' function. The PRC embodies the interactions of each pacemaker cell with its neighboring cells, and thus represents the type of connection (strong, weak, etc.) of the pacemaker cell with its surroundings. In our model, the SA node region differs from the atrial region by cycle length distribution and PRCs. We studied the spatial interaction between SA node pacemaker cells and atrial pacemaker cells as a function of the regional variation of cells properties and as a function of the “electrical” coupling between cells (the PRC), in the SA node region, in the atrial region, and in a border zone between them. We investigated the influence of those parameters on the activation pattern, on the conduction time of the array, and on a pseudo-ECG signal. This study demonstrates that by representing the atrial cells as a population of `pacemaker-like' cells, similar to the SA node pacemaker cells, but differing markedly in their cycle lengths and cell-to-cell interaction (PRC), we can create a global picture of the atrial system by applying a simple physical-mathematical model. This approach enables us to explore physiological phenomena related to the genesis and maintenance of atrial activity. It also reveals the conditions which predispose to atrial arrhythmias and conduction disturbances (e.g. tachycardia, pacemaker shift, re-entry, fibrillation). In particular, it yields insight into the mechanism of transition from normal atrial activity to the disordered state of atrial fibrillation. Therefore, this study suggests a new way of looking at the development of cardiac arrhythmias of atrial origin. Received: 8 September 1997 / Accepted in revised form: 6 October 1998     

Synchronization of two coupled pacemaker cells based on the phase response curve

In this paper, the synchronization of a pair of pacemaker cells as Sino-Atrial (SA) and Atrio-Ventricullar (AV) nodes have been studied and a new approach for synchronization, based on the concept of Phase Response Curve (PRC), has been proposed. The paper starts with presenting the necessary and sufficient conditions for synchronization in terms of the PRC parameters. Such conditions are time dependent and thus, the paper proceeds with deriving some sufficient conditions, which are not time dependent. The time-delay between the firing time of SA node and when it reaches the AV node is also considered. When the conditions for spontaneous synchronization are not valid, the synchronization is achieved by applying pulses to the AV or the SA nodes or to both of the nodes, depending on the accessibility. The subject has been investigated and sufficient conditions were achieved for all three cases. In each case, the dynamical equations of coupled pacemakers have been determined and the stability analyses of delay dynamical equations between discharges of two pacemakers were performed. The number of excitation pulses and the time intervals for applying them to accessible pacemaker(s) were obtained and eventually some numerical examples were simulated to approve the accuracy of the theoretical results and conditions.     

Breath-by-breath interactions between inspiratory and expiratory duration in occlusive sleep apnea

We examined interactions between inspiratory duration (TI), expiratory duration (TE), and inspiratory (esophageal) pressure (Pes) generation in seven subjects with confirmed occlusive sleep apnea. Breath-by-breath values of TI, TE, and Pes were identified by digital computer during 21 260-s epochs of repetitive occlusive apnea during non-rapid-eye-movement sleep. The control theory of interacting nonlinear oscillators was used to categorize the interaction between TI and TE for each epoch as either 1) synchronization, the strongest possible interaction between biological oscillators; 2) relative entrainment, a moderate interaction between oscillators; or 3) relative coordination, a weak interaction. The latter two interactions were characterized by systemic oscillations in the moving cross-correlation between TI and TE. The relationship between TI and Pes was analyzed in a similar fashion. Significant oscillations were present in all three parameters (P less than 0.0001 for each). We observed significant negative correlations between TI and TE and between TI and Pes (P less than 0.001 for each) when all breaths for all epochs were pooled. In no epoch was there a significant positive correlation between TI and TE or Pes. All three interactions were observed between TI and TE: five epochs of synchronization, nine of relative entrainment, and seven of relative coordination. In contrast, 19 of 21 epochs exhibited synchronization between TI and Pes, with 2 epochs of relative entrainment. The relative frequency of TI vs. Pes synchronization was significantly greater than TI vs. TE synchronization (P less than 0.005).(ABSTRACT TRUNCATED AT 250 WORDS)     

Nonphotic entrainment in a diurnal mammal, the European ground squirrel (Spermophilus citellus).

Entrainment by nonphotic, activity-inducing stimuli has been investigated in detail in nocturnal rodents, but little is known about nonphotic entrainment in diurnal animals. Comparative studies would offer the opportunity to distinguish between two possibilities. (1) If nonphotic phase shifts depend on the phase of the activity cycle, the phase response curve (PRC) should be about 180 degrees out of phase in nocturnal and diurnal mammals. (2) If nonphotic phase shifts depend on the phase of the pacemaker, the two PRCs should be in phase. We used the diurnal European ground squirrel (Spermophilus citellus) in a nonphotic entrainment experiment to distinguish between the two possibilities. Ten European ground squirrels were kept under dim red light (<1 lux) and 20 +/- 1 degrees C. During the entrainment phase of the experiment, the animals were confined every 23.5 h (T) to a running wheel for 3 h. The circadian rhythms of 6 squirrels entrained, 2 continued to free run, and 2 possibly entrained but displayed arrhythmicity during the experiment. In a second experiment, a photic pulse was used in a similar protocol. Five out of 9 squirrels entrained, 1 did not entrain, and 3 yielded ambiguous results. During stable entrainment, the phase-advancing nonphotic pulses coincided with the end of the subjective day, while phase-advancing light pulses coincided with the start of the subjective day: mean psi(nonphotic) = 11.4 h; mean psi(photic) = 0.9 h (psi defined as the difference between the onset of activity and the start of the pulse). The data for nonphotic entrainment correspond well with those from similar experiments with nocturnal Syrian hamsters where psi(nonphotic) varied from 8.09 to 11.34 h. This indicates that the circadian phase response to a nonphotic activity-inducing stimulus depends on the phase of the pacemaker rather than on the phase of the activity cycle.     

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.     

A phase response curve for circannual rhythm in the varied carpet beetle <Emphasis Type="Italic">Anthrenus verbasci</Emphasis>

We know that entrainment, a stable phase relationship with an environmental cycle, must be established for a biological clock to function properly. Phase response curves (PRCs), which are plots of phase shifts that result as a function of the phase of a stimulus, have been created to examine the mode of entrainment. In circadian rhythms, single-light pulse PRCs have been obtained by giving a light pulse to various phases of a free-running rhythm under continuous darkness. This successfully explains the entrainment to light-dark cycles. Some organisms show circannual rhythms. In some of these, changes in photoperiod entrain the circannual rhythms. However, no single-pulse PRCs have been created. Here we show the PRC to a long-day pulse superimposed for 4 weeks over constant short days in the circannual pupation rhythm in the varied carpet beetle Anthrenus verbasci. Because the shape of that PRC closely resembles that of the Type 0 PRC with large phase shifts in circadian rhythms, we suggest that an oscillator having a common feature in the phase response with the circadian clock, produces a circannual rhythm.     

Functional characterization of LIKE HETEROCHROMATIN PROTEIN 1 in the moss Physcomitrella patens: its conserved protein interactions in land plants

In flowering plants, LIKE HETEROCHROMATIN PROTEIN 1 (LHP1)/TERMINAL FLOWER 2 (TFL2) is known to interact with polycomb group (PcG) and non‐PcG proteins and control developmental programs. LHP1/TFL2 is an ancient protein and has been characterized in the early‐divergent plant Physcomitrella patens. However, interacting partners of PpLHP1 other than the chromomethylase PpCMT have not been identified to date. Also, while functional polycomb repressive complex 2 (PRC2) is known to exist in P. patens, there is no experimental evidence to support the existence of PRC1‐like complexes in these mosses. In this study, using protein?protein interaction methods, transient expression assays and targeted gene knockout strategy, we report the conserved properties of LHP1/TFL2 using the Physcomitrella system. We show that a PRC1‐like core complex comprising of PpLHP1 and the putative PRC1 Really Interesting New Gene (RING)‐finger proteins can form in vivo. Also, the interaction between PpRING and the PRC2 subunit PpCLF further sheds light on the possible existence of combinatorial interactions between the Polycomb Repressive Complex (PRC) in early land plants. Based on the interaction between PpLHP1 and putative hnRNP PpLIF2‐like in planta, we propose that the link between PpLHP1 regulation and RNA metabolic processes was established early in plants. The conserved subnuclear distribution pattern of PpLHP1 in moss protonema further provides insight into the manner in which LHP1/TFL2 are sequestered in the nucleoplasm in discrete foci. The PpLHP1 loss‐of‐function plants generated in this study share some of the pleiotropic defects with multiple aberrations reported in lhp1/tfl2. Taken together, this work documents an active role for PpLHP1 in epigenetic regulatory network in P. patens.     

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     

Modeling of the influence of gap-junction coupling on synchronization of central and peripheral sinoatrial node cells

Mathematical modeling of the electric activity of pairs of true (central) and latent (peripheral) sinoatrial pacemaker cells coupled through gap junctions revealed that attenuation of coupling conductance increases the beat phase shift; below a critical conductance there is no synchronization. The phase difference also depends on the type of interacting cells, being maximal for a “central”-“peripheral” cell pair.     

Probabilistic modeling of rosette formation

Rosetting, or forming a cell aggregate between a single target nucleated cell and a number of red blood cells (RBCs), is a simple assay for cell adhesion mediated by specific receptor-ligand interaction. For example, rosette formation between sheep RBC and human lymphocytes has been used to differentiate T cells from B cells. Rosetting assay is commonly used to determine the interaction of Fc gamma-receptors (FcgammaR) expressed on inflammatory cells and IgG coated on RBCs. Despite its wide use in measuring cell adhesion, the biophysical parameters of rosette formation have not been well characterized. Here we developed a probabilistic model to describe the distribution of rosette sizes, which is Poissonian. The average rosette size is predicted to be proportional to the apparent two-dimensional binding affinity of the interacting receptor-ligand pair and their site densities. The model has been supported by experiments of rosettes mediated by four molecular interactions: FcgammaRIII interacting with IgG, T cell receptor and coreceptor CD8 interacting with antigen peptide presented by major histocompatibility molecule, P-selectin interacting with P-selectin glycoprotein ligand 1 (PSGL-1), and L-selectin interacting with PSGL-1. The latter two are structurally similar and are different from the former two. Fitting the model to data enabled us to evaluate the apparent effective two-dimensional binding affinity of the interacting molecular pairs: 7.19x10(-5) microm4 for FcgammaRIII-IgG interaction, 4.66x10(-3) microm4 for P-selectin-PSGL-1 interaction, and 0.94x10(-3) microm4 for L-selectin-PSGL-1 interaction. These results elucidate the biophysical mechanism of rosette formation and enable it to become a semiquantitative assay that relates the rosette size to the effective affinity for receptor-ligand binding.     

Altitudinal Variation In Phase Response Curves For The Himalayan Strains Of Drosophila Helvetica

In previous research, it was determined that the altitude of origin altered the parameters of photic entrainment and free‐running rhythmicity of adult locomotor activity of the high‐altitude Himalayan (haH) strain (Hemkund‐Sahib, 4121 m above sea level) of Drosophila helvetica compared to the low‐altitude Himalayan (laH) strain (Birahi, 1132 m above sea level) of the same species. The present study investigated whether the altitude of origin also affects the parameters of the light pulse phase response curve (PRC) of the adult locomotor activity rhythm of the haH strain. Light pulse PRCs were determined for both strains against the background of constant darkness. Although both were “weak” or type 1 PRCs, the PRC for the haH strain differed from that of the laH strain in three basic parameters. The PRC for the haH strain was of low amplitude, had a protracted dead zone, and showed a ratio of the advance to delay region (A/D>1), while the PRC of the laH strain was characterized by high amplitude, absence of dead zone, and a A/D ratio<1. The asymmetric PRCs of these strains might explain the process of photic entrainment to 24 h light‐dark cycles, as the long period of the free‐running rhythm (τ) of the haH strain is complemented with a larger advance portion of its PRC (A/D>1), whereas the short τ of the laH strain is matched with a larger delay portion of its PRC (A/D<1). Prolonged dead zone and low amplitude in the PRC of the haH strain imply that the photic sensitivity of this strain has been drastically diminished as an adaptation to environmental conditions at the altitude of its origin. While adults of this strain begin activity in very bright light in the forenoon due to non‐permissible low temperature in the morning, the converse is true for the laH strain.     

Circadian clock controlling gut-purge rhythm of the saturniidSamia cynthia ricini: its characterization and entrainment mechanism

Summary Experiments using various light-dark (LD) conditions demonstrated that an endogenous circadian clock controls gut-purge timing in the saturniid mothSamia cynthia ricini. A phase-response curve (PRC) based on the application of brief (15 min) light pulses is used to characterize the underlying pacemaking oscillation. The entrainment of the pacemaker to various LD cycles is interpreted in terms of this PRC. The effect of light immediately preceding gut purge was analyzed to account for the deviation of the actual gut-purge rhythm from the prediction made by considering only the action of the oscillation. Lack of precision in gut-purge timing in LD cycles with a very short scotophase has been explained by the failure of the oscillation in these conditions to attain the specific phase-point at which the clock information dictating gut-purge timing is released.Abbreviations AZT arbitrary Zeitgeber time - CT circadian time - PRC phase response curve     

Probing the circadian pacemaker of a mouse using two light pulses

In two separate sets of experiments, the phases of the locomotor activity rhythm of the nocturnal field mouse Mus booduga were probed using two light pulses (LPs). In the first set of experiments, the circadian pacemaker underlying the locomotor activity rhythm was perturbed at circadian time 14 (CT 14) using a resetting light pulse LP1 of 1000 lux intensity and 15 min duration. The phases of the resetting pacemaker were then probed at all even CTs between CT 16 and CT 14 using a PRC probing light pulse LP2 of equal strength. The "LP2 PRC" thus obtained was then compared with the single light pulse PRC in terms of the area under delay (D) and advance (A) zones of the PRCs. The time course and waveform of the two LP PRCs suggest that the LP2 PRC resembled the single LP PRC, displaced by 2 h toward the right. The LP1 PRC had smaller D compared to the single LP PRC (p = 0.007), whereas both the PRCs had A of equal magnitude (p = 0.23). This suggests that the pacemaker phase shifts rapidly after LP perturbations. In the second set of experiments, the LP1 was administered at CT 14. The phase of the pacemaker was then perturbed on day 1 (next cycle after LP1) either 2 h after activity onset (at ca. CT 14 of the transient cycle) or 8 h after activity onset (at ca. CT 20 of the transient cycle) using an LP2 of equal strength. It was observed that the steady-state phase shifts evoked by positioning an LP2, 2 h after activity onset, were positively correlated with the phase shifts observed on day 1. The steady-state phase shifts observed, when the LP2 was positioned, 8 h after activity onset, were negatively correlated with the phase shifts observed on day 1. These results suggest that the transient cycles do not mirror the state of the pacemaker oscillator.     

Robust synchronization of coupled circadian and cell cycle oscillators in single mammalian cells

Circadian cycles and cell cycles are two fundamental periodic processes with a period in the range of 1 day. Consequently, coupling between such cycles can lead to synchronization. Here, we estimated the mutual interactions between the two oscillators by time‐lapse imaging of single mammalian NIH3T3 fibroblasts during several days. The analysis of thousands of circadian cycles in dividing cells clearly indicated that both oscillators tick in a 1:1 mode‐locked state, with cell divisions occurring tightly 5 h before the peak in circadian Rev‐Erbα‐YFP reporter expression. In principle, such synchrony may be caused by either unidirectional or bidirectional coupling. While gating of cell division by the circadian cycle has been most studied, our data combined with stochastic modeling unambiguously show that the reverse coupling is predominant in NIH3T3 cells. Moreover, temperature, genetic, and pharmacological perturbations showed that the two interacting cellular oscillators adopt a synchronized state that is highly robust over a wide range of parameters. These findings have implications for circadian function in proliferative tissues, including epidermis, immune cells, and cancer.     

The temporal ‘structure’ and function of the insect photoperiodic clock: a tribute to Colin S. Pittendrigh

In 1936, Erwin Bünning suggested that photoperiodic time measurement was a function of the circadian system. Colin Pittendrigh became an ardent supporter of Bünning's hypothesis, drawing parallels between photoperiodism and his own group's investigations of adult eclosion rhythmicity in the fruit fly Drosophila pseudoobscura. They developed several more modern versions of Bünning's general hypothesis based on the entrainment of circadian oscillations to the light cycle, including ‘external coincidence’, which is a derivation of Bünning's original model, and ‘internal coincidence’, which relied upon seasonal changes in the mutual phase relationship of oscillators within a multi‐oscillator circadian system. This review considers the experimental evidence for the central role of the circadian system in photoperiodic timing and, in some species, for both external and internal coincidence. Pittendrigh, however, pursued the idea of internal coincidence further with his analysis of the pacemaker–slave organization of eclosion rhythmicity in D. pseudoobscura and proposed a similar theoretical model for photoperiodism comprising a group of slave oscillators driven by a light‐sensitive pacemaker. In this model, the phase relationships of the slaves to the pacemaker were affected by (i) the relative periods of the pacemaker and slave(s); (ii) the strength(s) of the coupling between the two; and (iii) the dampening coefficients of the various slaves. Manipulation of these variables showed that the slaves adopted different internal phase relationships (both to each other and to the pacemaker) under the influence of changes in daily photophase, the period of the Zeitgeber and phase shifts of the entraining light cycle.     

Synchronization and stochastic resonance of the small-world neural network based on the CPG

According to biological knowledge, the central nervous system controls the central pattern generator (CPG) to drive the locomotion. The brain is a complex system consisting of different functions and different interconnections. The topological properties of the brain display features of small-world network. The synchronization and stochastic resonance have important roles in neural information transmission and processing. In order to study the synchronization and stochastic resonance of the brain based on the CPG, we establish the model which shows the relationship between the small-world neural network (SWNN) and the CPG. We analyze the synchronization of the SWNN when the amplitude and frequency of the CPG are changed and the effects on the CPG when the SWNN’s parameters are changed. And we also study the stochastic resonance on the SWNN. The main findings include: (1) When the CPG is added into the SWNN, there exists parameters space of the CPG and the SWNN, which can make the synchronization of the SWNN optimum. (2) There exists an optimal noise level at which the resonance factor Q gets its peak value. And the correlation between the pacemaker frequency and the dynamical response of the network is resonantly dependent on the noise intensity. The results could have important implications for biological processes which are about interaction between the neural network and the CPG.     

Dynamics of discrete entrainment of the circadian rhythm in the rat pineal N-acetyltransferase activity during transient cycles

To elucidate entrainment of a pacemaker controlling the N-acetyltransferase (NAT) rhythm in the rat pineal gland, we studied the phase response curves (PRCs) of this rhythm. We exposed 50- to 60-day-old male Wistar rats maintained in a light-dark cycle (LD 12:12) to a 1-min light pulse at different times before midnight or at various times throughout the whole night. We then released them into constant darkness and studied the morning NAT decline during the night when rats were pulsed before midnight, as well as the evening NAT rise and the morning decline after 4 days following the pulses. The PRC for the first NAT decline and the PRCs for the NAT rise and decline after 4 days were compared with published transient PRCs (Illnerová and Van?cek, 1982b), in order to obtain a complete picture of the dynamics of the NAT rhythm entrainment during the transient cycles. Phase delays in the NAT rise due to a pulse before midnight were complete (i.e., identical to those of day 4) on day 1. Phase delays in the NAT decline were almost complete on day 1, while incomplete phase delays were observed on day 0. Phase advances in the NAT rise and decline due to a pulse past midnight had different dynamics: Advances in the decline were complete on day 1, while advances in the rise were absent on day 1 and much smaller than in the decline on day 4. The results are discussed in terms of a two-component (E-M) pacemaker controlling the NAT rhythm. The NAT rise may reflect the phase of the E-component, while the decline reflects the M-component. Phase delays of the E-component are accomplished within one cycle, and so are phase advances of the M-component. However, although delays of E already result in delays of M one cycle after the pulse, it takes several transient cycles before advances of M begin to induce advances of E.