Analysis of models for crustacean stretch receptors

 The mechanisms underlying the diverse responses to step current stimuli of models [Edman et al. (1987) J Physiol (Lond) 384: 649–669] of lobster slowly adapting stretch receptor organs (SAO) and fast-adapting stretch receptor organs (FAO) are analyzed. In response to a step current, the models display three distinct types of firing reflecting the level of adaptation to the stimulation. Low-amplitude currents evoke transient firing containing one to several action potentials before the system stabilizes to a resting state. Conversely, high-amplitude stimulations induce a high frequency transient burst that can last several seconds before the model returns to its quiescent state. In the SAO model, the transition between the two regimes is characterized by a sustained pacemaker firing at an intermediate stimulation amplitude. The FAO model does not exhibit such a maintained firing; rather, the duration of the transient firing increases at first with the stimulus intensity, goes through a maximum and then decreases at larger intensities. Both models comprise seven variables representing the membrane potential, the sodium fast activation, fast inactivation, slow inactivation, the potassium fast activation, slow inactivation gating variables, and the intra cellular sodium concentration. To elucidate the mechanisms of the firing adaptations, the seven-variable model for the lobster stretch receptor neuron is first reduced to a three-dimensional system by regrouping variables with similar time scales. More precisely, we substituted the membrane potential V for the sodium fast activation equivalent potential V _m , the potassium fast inactivation V _n for the sodium fast inactivation V _h , and the sodium slow inactivation V _l for the potassium slow inactivation V _r . Comparison of the responses of the reduced models to those of the original models revealed that the main behaviors of the system were preserved in the reduction process. We classified the different types of responses of the reduced SAO and FAO models to constant current stimulation. We analyzed the transient and stationary responses of the reduced models by constructing bifurcation diagrams representing the qualitatively distinct dynamics of the models and the transitions between them. These revealed that (1) the transient firings prior to reaching the stationary state can be accounted for by the sodium slow inactivation evolving more slowly than the other two variables, so that the changes during the transient firings reflect the bifurcations that the two-dimensional system undergoes when the sodium slow inactivation, considered as a parameter, is varied; and (2) the stationary behaviors of the models are captured by the standard bifurcations of a two-dimensional system formed by the membrane potential and the potassium fast inactivation. We found that each type of firing and the transitions between them is due to the interplay between essentially three variables: two fast ones accounting for the action potential generation and the post-discharge refractoriness, and a third slow one representing the adaptation. Received: 28 February 2000 / Accepted in revised form: 4 October 2000     

Inhibition modifies the effects of slow calcium-activated potassium channels on epileptiform activity in a neuronal network model

Generation of epileptiform activity typically results from a change in the balance between network excitation and inhibition. Experimental evidence indicates that alterations of either synaptic activity or intrinsic membrane properties can produce increased network excitation. The slow Ca²⁺-activated K⁺ currents (sI_AHP) are important modulators of neuronal firing rate and excitability and have important established and potential roles in epileptogenesis. While the effects of changes in sI_AHP on individual neuronal excitability are readily studied and well established, the effects of such changes on network behavior are less well known. The experiments here utilize a defined small network model of multicompartment pyramidal cells and an inhibitory interneuron to study the effects of changes in sI_AHP on network behavior. The benefits of this model system include the ability to observe activity in all cells in a network and the effects of interactions of multiple simultaneous influences. In the model with no inhibitory interneuron, increasing sI_AHP results in progressively decreasing burst activity. Adding an inhibitory interneuron changes the observed effects; at modest inhibitory strengths, increasing sI_AHP in all network neurons actually results in increased network bursting (except at very high values). The duration of the burst activity is influenced by the length of delay in a feedback loop, with longer loops resulting in more prolonged bursting. These observations illustrate that the study of potential antiepileptogenic membrane effects must be extended to realistic networks. Network inhibition can dramatically alter the observations seen in pure excitatory networks.     

Mixed mode oscillations as a mechanism for pseudo-plateau bursting

We combine bifurcation analysis with the theory of canard-induced mixed mode oscillations to investigate the dynamics of a novel form of bursting. This bursting oscillation, which arises from a model of the electrical activity of a pituitary cell, is characterized by small impulses or spikes riding on top of an elevated voltage plateau. Oscillations with these characteristics have been called “pseudo-plateau bursting”. Unlike standard bursting, the subsystem of fast variables does not possess a stable branch of periodic spiking solutions, and in the case studied here the standard fast/slow analysis provides little information about the underlying dynamics. We demonstrate that the bursting is actually a canard-induced mixed mode oscillation, and use canard theory to characterize the dynamics of the oscillation. We also use bifurcation analysis of the full system of equations to extend the results of the singular analysis to the physiological regime. This demonstrates that the combination of these two analysis techniques can be a powerful tool for understanding the pseudo-plateau bursting oscillations that arise in electrically excitable pituitary cells and isolated pancreatic β-cells.     

Multiple timescale mixed bursting dynamics in a respiratory neuron model

Experimental results in rodent medullary slices containing the pre-Bötzinger complex (pre-BötC) have identified multiple bursting mechanisms based on persistent sodium current (I _NaP) and intracellular Ca²⁺. The classic two-timescale approach to the analysis of pre-BötC bursting treats the inactivation of I _NaP, the calcium concentration, as well as the Ca²⁺-dependent inactivation of IP ₃ as slow variables and considers other evolving quantities as fast variables. Based on its time course, however, it appears that a novel mixed bursting (MB) solution, observed both in recordings and in model pre-BötC neurons, involves at least three timescales. In this work, we consider a single-compartment model of a pre-BötC inspiratory neuron that can exhibit both I _NaP and Ca²⁺ oscillations and has the ability to produce MB solutions. We use methods of dynamical systems theory, such as phase plane analysis, fast-slow decomposition, and bifurcation analysis, to better understand the mechanisms underlying the MB solution pattern. Rather surprisingly, we discover that a third timescale is not actually required to generate mixed bursting solutions. Through our analysis of timescales, we also elucidate how the pre-BötC neuron model can be tuned to improve the robustness of the MB solution.     

Analysis of an autonomous phase model for neuronal parabolic bursting

An understanding of the nonlinear dynamics of bursting is fundamental in unraveling structure-function relations in nerve and secretory tissue. Bursting is characterized by alternations between phases of rapid spiking and slowly varying potential. A simple phase model is developed to study endogenous parabolic bursting, a class of burst activity observed experimentally in excitable membrane. The phase model is motivated by Rinzel and Lee's dissection of a model for neuronal parabolic bursting (J. Math. Biol. 25, 653–675 (1987)). Rapid spiking is represented canonically by a one-variable phase equation that is coupled bi-directionally to a two-variable slow system. The model is analyzed in the slow-variable phase plane, using quasi steady-state assumptions and formal averaging. We derive a reduced system to explore where the full model exhibits bursting, steady-states, continuous and modulated spiking. The relative speed of activation and inactivation of the slow variables strongly influences the burst pattern as well as other dynamics. We find conditions of the bistability of solutions between continuous spiking and bursting. Although the phase model is simple, we demonstrate that it captures many dynamical features of more complex biophysical models.This research was partially supported by NSF-JOINT RESEARCH grant 8803573, grant from CONCYT and DGAPA(UNAM) Mexico for H. Carrillo, and for the S. M. Baer NSF DMS-9107538     

The role of a transient potassium current in a bursting neuron model

Presented here is a biophysical cell model which can exhibit low-frequency repetitive activity and bursting behavior. The model is developed from previous models (Av-Ron et al. 1991, 1993) for excitability, oscillations and bursting. A stepwise development of the present model shows the contribution of a transient potassium current (I _A ) to the overall dynamics. By changing a limited set of model parameters one can describe different firing patterns; oscillations with frequencies ranging from 2–200 Hz and a wide range of bursting behaviors in terms of the durations of bursting and quiescence, peak firing frequency and rate of change of the firing frequency.     

A multivariate population density model of the dLGN/PGN relay

Using a population density approach we study the dynamics of two interacting collections of integrate-and-fire-or-burst (IFB) neurons representing thalamocortical (TC) cells from the dorsal lateral geniculate nucleus (dLGN) and thalamic reticular (RE) cells from the perigeniculate nucleus (PGN). Each population of neurons is described by a multivariate probability density function that satisfies a conservation equation with appropriately defined probability fluxes and boundary conditions. The state variables of each neuron are the membrane potential and the inactivation gating variable of the low-threshold Ca²⁺ current I_T. The synaptic coupling of the populations and external excitatory drive are modeled by instantaneous jumps in the membrane potential of postsynaptic neurons. The population density model is validated by comparing its response to time-varying retinal input to Monte Carlo simulations of the corresponding IFB network composed of 100 to 1000 cells per population. In the absence of retinal input, the population density model exhibits rhythmic bursting similar to the 7 to 14 Hz oscillations associated with slow wave sleep that require feedback inhibition from RE to TC cells. When the TC and RE cell potassium leakage conductances are adjusted to represent cholingergic neuromodulation and arousal of the network, rhythmic bursting of the probability density model may either persists or be eliminated depending on the number of excitatory (TC to RE) or inhibitory (RE to TC) connections made by each presynaptic cell. When the probability density model is stimulated with constant retinal input (10–100 spikes/sec), a wide range of responses are observed depending on cellular parameters and network connectivity. These include asynchronous burst and tonic spikes, sleep spindle-like rhythmic bursting, and oscillations in population firing rate that are distinguishable from sleep spindles due to their amplitude, frequency, or the presence of tonic spikes. In this context of dLGN/PGN network modeling, we find the population density approach using 2,500 mesh points and resolving membrane voltage to 0.7 mV is over 30 times more efficient than 1000-cell Monte Carlo simulations. Action Editor: David Golomb     

The potassium A-current,low firing rates and rebound excitation in Hodgkin-Huxley models

It is widely believed, following the work of Connor and Stevens (1971,J. Physiol. Lond. 214, 31–53) that the ability to fire action potentials over a wide frequency range, especially down to very low rates, is due to the transient, potassium A-current (I _A). Using a reduction of the classical Hodgkin-Huxley model, we study the effects ofI _A on steady firing rate, especially in the near-threshold regime for the onset of firing. A minimum firing rate of zero corresponds to a homoclinic bifurcation of periodic solutions at a critical level of stimulating current. It requires that the membrane's steady-state current-voltage relation be N-shaped rather than monotonic. For experimentally based genericI _A parameters, the model does not fire at arbitrarily low rates, although it can for the more atypicalI _A parameters given by Connor and Stevens for the crab axon. When theI _A inactivation rate is slow, we find that the transient potassium current can mediate more complex firing patterns, such as periodic bursting in some parameter regimes. The number of spikes per burst increases asg _A decreases and as inactivation rate decreases. We also study howI _A affects properties of transient voltage responses, such as threshold and firing latency for anodal break excitation. We provide mathematical explanations for several of these dynamic behaviors using bifurcation theory and averaging methods.     

A simulation model of Bouteloua gracilis biomass dynamics on the North American shortgrass prairie

Summary A grassland primary producer model for simulating intraseasonal biomass dynamics as a function of temperature, moisture, light, and nitrogen was developed for Bouteloua gracilis (H.B.K.) Lag., the dominant C₄ grass of the North American shortgrass prairie. Plant state variables included young and mature leaves, crowns, and roots from three depth categories while simulated processes included spring regrowth, photosynthesis, respiration, photosynthate allocation, death, and litterfall. Sensitivity analyses revealed the model was most sensitive to changes in photosynthesis and photosynthate allocation and least sensitive to changes in initial values of state variables, leaf dark respiration rates, and rate of spring regrowth.An abiotic submodel driven by observed weather data was used in conjunction with the primary producer model to simulate plant biomass dynamics under a variety of conditions including untreated controls (C), nitrogen fertilization (F), irrigation (I), and irrigation plus fertilization (IF). Model predictions of life shoot biomass (B _s) and annual aboveground net primary production (NPP _A) followed the same trends as field measurements with B _sand NPP _Aof IF>I>F>C. Failure of the model to accurately predict measured declines in peak B _sand NPP _Aafter several years of irrigation may have been caused by failure to account for growth lags following water stress, inadequate simulation of interspecific competition, or failure to simulate response to some mineral nutrients which had become limiting after several years of this treatment. A simulated annual carbon budget for plants in the four treatments suggests that from 61% (IF) to 80% (C) of the net carbon fixed above ground is ultimately translocated and utilized below ground.     

A unified model for two modes of bursting in GnRH neurons

Gonadotropin-releasing hormone (GnRH) neurons exhibit at least two intrinsic modes of action potential burst firing, referred to as parabolic and irregular bursting. Parabolic bursting is characterized by a slow wave in membrane potential that can underlie periodic clusters of action potentials with increased interspike interval at the beginning and at the end of each cluster. Irregular bursting is characterized by clusters of action potentials that are separated by varying durations of interburst intervals and a relatively stable baseline potential. Based on recent studies of isolated ionic currents, a stochastic Hodgkin-Huxley (HH)-like model for the GnRH neuron is developed to reproduce each mode of burst firing with an appropriate set of conductances. Model outcomes for bursting are in agreement with the experimental recordings in terms of interburst interval, interspike interval, active phase duration, and other quantitative properties specific to each mode of bursting. The model also shows similar outcomes in membrane potential to those seen experimentally when tetrodotoxin (TTX) is used to block action potentials during bursting, and when estradiol transitions cells exhibiting slow oscillations to irregular bursting mode in vitro. Based on the parameter values used to reproduce each mode of bursting, the model suggests that GnRH neurons can switch between the two through changes in the maximum conductance of certain ionic currents, notably the slow inward Ca²⁺ current I _s, and the Ca²⁺ -activated K⁺ current I _KCa. Bifurcation analysis of the model shows that both modes of bursting are similar from a dynamical systems perspective despite differences in burst characteristics.     

A mathematical model of adult GnRH neurons in mouse brain and its bifurcation analysis

GnRH neurons are hypothalamic neurons that secrete gonadotropin-releasing hormone (GnRH) which stimulates the release of gonadotropins, one of the crucial hormones for sexual development, fertility and maturation. A mathematical model was built to help elucidate the mechanisms underlying electrical bursting and synchronous [Ca²⁺] transients in GnRH neurons (Lee et al., 2010). The model predicted that bursting in GnRH neurons (at least of the short-bursting type) requires the existence of a [Ca²⁺]-dependent slow after-hyperpolarisation current (sI_AHP-UCL), and this predicted current was found experimentally. GnRH behaviour under a wide range of conditions (inhibition of Na⁺ channels, IP₃ receptors, [Ca²⁺]-dependent K⁺ channels, or Ca²⁺ pumps, or in the presence of zero extracellular [Ca²⁺]) is successfully reproduced by the model. In this paper, a simplified version of the previous model, with the same qualitative behaviour, is constructed and studied using timescale separation techniques and bifurcation analysis.     

Complex nonlinear dynamics of the Hodgkin-Huxley equations induced by time scale changes

 The Hodgkin–Huxley equations with a slight modification are investigated, in which the inactivation process (h) of sodium channels or the activation process of potassium channels (n) is slowed down. We show that the equations produce a variety of action potential waveforms ranging from a plateau potential, such as in heart muscle cells, to chaotic bursting firings. When h is slowed down – differently from the case of n variable being slow – chaotic bursting oscillations are observed for a wide range of parameter values although both variables cause a decrease in the membrane potential. The underlying nonlinear dynamics of various action potentials are analyzed using bifurcation theory and a so-called slow–fast decomposition analysis. It is shown that a simple topological property of the equilibrium curves of slow and fast subsystems is essential to the production of chaotic oscillations, and this is the cause of the large difference in global firing characteristics between the h-slow and n-slow cases. Received: 9 August 2000 / Accepted in revised form: 10 January 2001     

Modelling the dynamic physical protection of soil organic carbon: Insights into carbon predictions and explanation of the priming effect

The role and significance of physically protected soil organic carbon (SOC) in regulating SOC dynamics remains unclear. Here, we developed a simple theoretical model (DP model) considering dynamic physical protection to simulate the dynamics of protected (C_p) and unprotected SOC (C_u), and compared the modelling results with a conventional two‐pool (fast vs. slow) model considering chemical recalcitrance. The two models were first constrained using extensive SOC data collected from soils with and without fresh carbon (C) inputs under incubation conditions, and then applied to project SOC dynamics and explore mechanisms underpinning the priming effect (PE). Overall, both models explained more than 99% of the variances in observed SOC dynamics. The DP model predicted that C_p accounted for the majority of total SOC. As decomposition proceeds, the proportion of C_p reached >90% and kept relatively constant. Although the similar performance of the two models in simulating observed total SOC dynamics, their predictions of future SOC dynamics were divergent, challenging the predictions of widely used pool‐based models. The DP model also suggested alternative mechanisms underpinning the priming of SOC decomposition by fresh C inputs. The two‐pool model suggested that the PE was caused by the stimulated decomposition rates, especially for the slow recalcitrant pool, while the DP model suggested that the PE might be the combined consequence of stimulated C_u decomposition, the liberation of C_p to decomposition and the inhibition of the protection of unprotected SOC. The model‐data integration provided a new explanation for the PE, highlighting the importance of liberation of initially physically protected SOC to decomposition by new C inputs. Our model‐data integration demonstrated the importance of simulating physical protection processes for reliable SOC predictions, and provided new insights into mechanistic understanding of the priming effect.     

Intrinsic bursting activity in the pre-Bötzinger Complex: Role of persistent sodium and potassium currents

Computational models of single pacemaker neuron and neural population in the pre-Bötzinger Complex (pBC) were developed based on the previous models by Butera et al. (1999a,b). Our modeling study focused on the conditions that could define endogenous bursting vs. tonic activity in single pacemaker neurons and population bursting vs. asynchronous firing in populations of pacemaker neurons. We show that both bursting activity in single pacemaker neurons and population bursting activity may be released or suppressed depending on the expression of persistent sodium (I_NaP) and delayed-rectifier potassium (I_K) currents. Specifically, a transition from asynchronous firing to population bursting could be induced by a reduction of I_K via a direct suppression of the potassium conductance or through an elevation of extracellular potassium concentration. Similar population bursting activity could be triggered by an augmentation of I_NaP. These findings are discussed in the context of the possible role of population bursting activity in the pBC in the respiratory rhythm generation in vivo vs. in vitro and during normal breathing in vivo vs. gasping.     

Cl− Currents Activated by Extracellular Nucleotides in Human Bronchial Cells

The perforated-patch technique was used to study the response of human bronchial cells to extracellular nucleotides. ATP or UTP (100 μm) elicited a complex response consisting of a large transient membrane current increase followed by a relatively small sustained level. These two phases were characterized by different current kinetics. Throughout the transient phase (2–3 min) the membrane current (I _p ) displayed slow activation and deactivation kinetics at depolarizing and hyperpolarizing potentials respectively. At steady-state (I _s ) the relaxation at hyperpolarizing potential disappeared whereas at positive membrane potentials the current became slightly deactivating. The I _s amplitude was dependent on the extracellular Ca²⁺ concentration, being completely inhibited in Ca²⁺-free medium. Cell pre-incubation with the membrane-permeable chelating agent BAPTA/AM prevented completely the response to nucleotides, thus suggesting that both I _p and I _s were dependent on intracellular Ca²⁺. The presence of a hypertonic medium during nucleotide stimulation abolished I _s leaving I _p unchanged. On the contrary, niflumic acid, a blocker of Ca²⁺-activated Cl⁻ channels, prevented completely I _p without reducing significantly I _s . 1,9-dideoxyforskolin fully inhibited I _s but also reduced I _p . Replacement of extracellular Cl⁻ with aspartate demonstrated that the currents activated by nucleotides were Cl⁻ selective. I _p resulted five times more Cl⁻ selective than I _s with respect to aspartate. Taken together, our results indicate that ATP and UTP activate two types of Cl⁻ currents through a Ca²⁺-dependent mechanism. Received: 15 August 1996/Revised: 6 December 1996     

Calmodulin dissociation regulates Myo5 recruitment and function at endocytic sites

Myosins‐I are conserved proteins that bear an N‐terminal motor head followed by a Tail Homology 1 (TH1) lipid‐binding domain. Some myosins‐I have an additional C‐terminal extension (C_ext) that promotes Arp2/3 complex‐dependent actin polymerization. The head and the tail are separated by a neck that binds calmodulin or calmodulin‐related light chains. Myosins‐I are known to participate in actin‐dependent membrane remodelling. However, the molecular mechanisms controlling their recruitment and their biochemical activities in vivo are far from being understood. In this study, we provided evidence suggesting the existence of an inhibitory interaction between the TH1 domain of the yeast myosin‐I Myo5 and its C_ext. The TH1 domain prevented binding of the Myo5 C_ext to the yeast WIP homologue Vrp1, Myo5 C_ext‐induced actin polymerization and recruitment of the Myo5 C_ext to endocytic sites. Our data also indicated that calmodulin dissociation from Myo5 weakened the interaction between the neck and TH1 domains and the C_ext. Concomitantly, calmodulin dissociation triggered Myo5 binding to Vrp1, extended the myosin‐I lifespan at endocytic sites and activated Myo5‐induced actin polymerization.     

The induction of inorganic carbon transport and external carbonic anhydrase in Chlamydomonas reinhardtii is regulated by external CO2 concentration

Induction of the carbon concentrating mechanism (CCM) has been investigated during the acclimation of 5% CO₂‐grown Chlamydomonas reinhardtii 2137 mt + cells to well‐defined dissolved inorganic carbon (C_i) limited conditions. The CCM components investigated were active HCO₃^? transport, active CO₂ transport and extracellular carbonic anhydrase (CA_ext) activity. The CA_ext activity increased 10‐fold within 6 h of acclimation to 0·035% CO₂ and there was a further slight increase over the next 18 h. The CA_ext activity also increased substantially after an 8 h lag period during acclimation to air in darkness. Active CO₂ and HCO₃^? uptake by C. reinhardtii cells were induced within 2 h of acclimation to air, but active CO₂ transport was induced prior to active HCO₃^? transport. Similar results were obtained during acclimation to air in darkness. The critical C_i concentrations effecting the induction of active C_i transport and CA_ext activity were determined by allowing cells to acclimate to various inflow CO₂ concentrations in the range 0·035–0·84% at constant pH. The total C_i concentration eliciting the induction and repression of active C_i transport was higher during acclimation at pH 7·5 than at pH 5·5, but the external CO₂ concentration was the same at both pHs of acclimation. The concentration of external CO₂ required for the full induction and repression of C_i transport and CA_ext activity were 10 and 100 μM , respectively. The induction of CA_ext and active C_i transport are not correlated temporally, but are regulated by the same critical CO₂ concentration in the medium.     

Multiple modes of a conditional neural oscillator

We present a model for a conditional bursting neuron consisting of five conductances: Hodgkin-Huxley type time- and voltage-dependent Na⁺ and K⁺ conductances, a calcium activated voltage-dependent K⁺ conductance, a calcium-inhibited time- and voltage-dependent Ca⁺⁺ conductance, and a leakage Cl(_– conductance. With an initial set of parameters (versionS), the model shows a hyperpolarized steady-state membrane potential at which the neuron is silent. Increasingg _Na and decreasingg _Cl, whereg _i , is the maximal conductance for speciesi, produces bursts of action potentials (BursterN). Alternatively, an increase ing _Ca produces a different bursting state (BursterC). The two bursting states differ in the periods and amplitudes of their bursting pacemaker potentials. They show different steady-stateI–V curves under simulated voltage-clamp conditions; in simulations that mimic a steady-stateI–V curve taken under experimental conditions only BursterN shows a negative slope resistance region. ModelC continues to burst in the presence of TTX, while bursting in ModelN is suppressed in TTX. Hybrid models show a smooth transition between the two states.     

Spike-Frequency Adaptation of a Generalized Leaky Integrate-and-Fire Model Neuron

Although spike-frequency adaptation is a commonly observed property of neurons, its functional implications are still poorly understood. In this work, using a leaky integrate-and-fire neural model that includes a Ca²⁺-activated K⁺ current (I _AHP), we develop a quantitative theory of adaptation temporal dynamics and compare our results with recent in vivo intracellular recordings from pyramidal cells in the cat visual cortex. Experimentally testable relations between the degree and the time constant of spike-frequency adaptation are predicted. We also contrast the I _AHP model with an alternative adaptation model based on a dynamical firing threshold. Possible roles of adaptation in temporal computation are explored, as a a time-delayed neuronal self-inhibition mechanism. Our results include the following: (1) given the same firing rate, the variability of interspike intervals (ISIs) is either reduced or enhanced by adaptation, depending on whether the I _AHP dynamics is fast or slow compared with the mean ISI in the output spike train; (2) when the inputs are Poisson-distributed (uncorrelated), adaptation generates temporal anticorrelation between ISIs, we suggest that measurement of this negative correlation provides a probe to assess the strength of I _AHP in vivo; (3) the forward masking effect produced by the slow dynamics of I _AHP is nonlinear and effective at selecting the strongest input among competing sources of input signals.