Electrical bistability in a neuron model with monostable dendritic and axosomatic membranes

In a two-compartment mathematical model, we studied the reason for and conditions of manifestation of electrical bistability in a neuron composed of monostable parts. One compartment of the model simulated the dendrites; their membrane was monostable at high depolarization and characterized by an N-shaped steady current-voltage (I–V) characteristic endowed by inward synaptic current through voltage-dependent channels sensitive to N-methyl-D-aspartate (NMDA). Another compartment simulated the axosomatic region with a positively sloped linearizedI–V characteristic of the membrane monostable at the resting membrane potential. For the whole cell, bistability was obvious at a subcritical intensity of NMDA activation; the reason was the current directed from the more depolarized dendritic region into the somatic region, and the necessary condition was that the above somatopetal core current must exceed the net inward transmembrane current (the latter was the sum of the inward synaptic and outward passive extrasynaptic currents) of the dendritic compartment. This relation essentially depended on the size of the dendrites. Neirofiziologiya/Neurophysiology, Vol. 32, No. 2, pp. 98–101, March–April, 2000.     

A compartmental model of carbon allocation in the vegetative barley plant

Abstract. The allocation of carbon in a vegetative barley plant is described as an open, three-compartment model; the three compartments are soluble (which exchanges material with the environment), storage, and structure (both of which exchange material with the soluble compartment). The model shows a good fit with data on ¹⁴C kinetics following ¹⁴CO₂ feeding and some of its assumptions, properties and implications are discussed.     

A model of interorganelle monoclonal antibody transport and secretion in mouse hybridoma cells

A three compartment model (ER --> Golgi --> extracellular medium) is used here to describe the interorganelle transport and final secretion of an IgG(2a) monoclonal antibody (MAb) in 9.2.27 murine hybridoma cells. Model simulations of pulse-chase and continuous labeling experiments are used to gain a better understanding of the kinetics of MAb interorganelle traffic. Simulation results for the continuous labeling case compare well with experimental data obtained during continuous labeling of 9.2.27 hybridoma cells. Incorporation of this compartmental transport model into our previously developed model of MAb synthesis and assembly can provide a useful tool for analyzing the dynamics and regulation of the complete antibody secretory pathway under different growth and/or nutritional conditions.     

Predicting abundances of plants and pollinators using a simple compartmental mutualistic model

Key gaps to be filled in population and community ecology are predicting the strength of species interactions and linking pattern with process to understand species coexistence and their relative abundances. In the case of mutualistic webs, like plant–pollinator networks, advances in understanding species abundances are currently limited, mainly owing to the lack of methodological tools to deal with the intrinsic complexity of mutualisms. Here, we propose an aggregation method leading to a simple compartmental mutualistic population model that captures both qualitatively and quantitatively the size-segregated populations observed in a Mediterranean community of nectar-producing plant species and nectar-searching animal species. We analyse the issue of optimal aggregation level and its connection with the trade-off between realism and overparametrization. We show that aggregation of both plants and pollinators into five size classes or compartments leads to a robust model with only two tunable parameters. Moreover, if, in each compartment, (i) the interaction coefficients fulfil the condition of weak mutualism and (ii) the mutualism is facultative for at least one party of the compartment, then the interactions between different compartments are sufficient to guarantee global stability of the equilibrium population.     

Synchronization of pacemaker cell firing by weak ELF fields: Simulation by a circuit model

Entrainment of output action potentials from repetitively firing pacemaker cells, brought about by regularly spaced excitatory or inhibitory postsynaptic inputs, is a well-known phenomenon. Synchronization of neural firing patterns by extremely low frequency (ELF) external electric fields has also been observed. Whereas current densities of ≈10 A-m⁻² are required for direct excitation of otherwise quiescent neural tissue, much lower peak current densities (≈10⁻² A-m²) have been reported to entrain spontaneously firing molluscan pacemaker cells. We have developed a neural spike generator circuit model that simulates repetitive spike generation by a space clamped patch (area ≈ 10⁻⁷ m²) of excitable membrane subjected to depolarizing current. Picoampere (pA) range variation of DC depolarizing current causes a corresponding smooth variation of neural spike frequency, producing a physiologically realistic stimulus-response (S-R) characteristic. When lower pA range 60 Hz AC current is superposed upon the DC depolarizing current, smooth variation of the S-R characteristic is distorted by subharmonic locking of the spike generator at 30, 20, 15, 12, 10 Hz, and higher order subharmonic frequencies. Although the additional superposition of a physiologically realistic level of “white” current noise, covering the bandwidth 4–200 Hz, suffices to obscure higher order subharmonic locking, locking at 30, 20, and 15 Hz is still clearly evident in the presence of noise. Subharmonic locking is observed at a root mean square AC simulated tissue current density of ≈10⁻⁵ A-m⁻². Bioelectromagnetics 19:92–97, 1998. © 1998 Wiley-Liss, Inc.     

Stochastic simulations of a multi-group compartmental model for Johne's disease on US dairy herds with test-based culling intervention

Infection elimination may be an important goal of control programs. Only in stochastic infection models can true infection elimination be observed as a fadeout. The phenomena of fadeout and variable prevalence are important in understanding the transmission dynamics of infectious diseases and these phenomena are essential to evaluate the effectiveness of control measures. To investigate the stochastic dynamics of Mycobacterium avium subsp. paratuberculosis (MAP) infection on US dairy herds with test-based culling intervention, we developed a multi-group stochastic compartmental model (a continuous time Markov chain model) with both horizontal and vertical transmission. The stochastic model predicted fadeout and within-herd prevalence to have a large variance. Although test-based culling intervention generally decreased prevalence over time, it took longer than desired by producers to eliminate the endemic MAP infection from a herd. Uncertainty analysis showed that, using annual culture test and culling of only high shedders or culling of both low and high shedders with a 12-month delay in culling of low shedders, MAP infection persisted in many herds beyond 20 years. While using semi-annual culture test and culling of low and high shedders with a 6-month delay in culling of low shedders, MAP infection in many herds would be extinct within 20 years. Sensitivity analysis of the cumulative density function of fadeout suggested that combining test-based culling intervention and reduction of transmission rates through improved management between susceptible calves and shedding animals may be more effective than either alone in eliminating endemic MAP infection. We also discussed the effects of other factors such as herd size, heifer replacement, and adult cow infection on the probability of fadeout.     

A model for the firing pattern of a paced nerve cell

When a nerve cell is paced by another cell or by external stimuli, its firing pattern is generally not regular. In a given stimulus and frequency interval the cell responds only on a fraction of the stimuli, and this fraction is not necessarily a nice simple fraction as 2, ;, etc. but may be any fraction less than one, why the firing pattern becomes accordingly irregular.The paper demonstrates the rules for these irregularities and supplies a mathematical tool to analyse the firing pattern. In the analysis only a cell with the so-called impulse dependent type of adaptation is considered, but similar analyses may be done with other cell types.The analysis is restricted to a case where a resting cell is stimulated by repetitively applied stimuli of constant strength and repetition frequency. Cases with more than one input, spontaneously firing cells, and transient phenomena are not included in the analysis.     

A novel mechanism for irregular firing of a neuron in response to periodic stimulation: irregularity in the absence of noise

Irregular firing of action potentials (AP's) is a characteristic feature of neurons in the brain. The variability has been attributed to noise from various sources. This study illustrates an alternative mechanism, namely, deterministic irregularity within a model of ionic conductances. Specifically, a model based on modern measurements of the Na⁺ and K⁺ current components from the squid giant axon fires irregularly in response to a continuous train of near-threshold current pulses. The interspike interval histogram from these simulations is multi-modal, a result which in other systems has been attributed to stochastic resonance. Moreover, the simulations exhibited short burst of spikes followed by relatively long quiescent periods, a result suggestive of patterned input to the model even though the input consisted of a train of regularly spaced current pulses. The variability of firing is attributable to variations in AP parameters, in particular AP amplitude. The action potential for squid giant axons is not all-or-none. Rather, it is fundamentally a continuous function of stimulus amplitude. That is, the membrane lacks a threshold. Variation in AP amplitude, and to a lesser extent, AP duration, can produce variations in the time to a subsequent AP, which represents a paradigm shift for understanding irregular neuronal firing. The emphasis is not as much on events prior to an AP as it is on the AP's themselves.     

Relating ion channel expression, bifurcation structure, and diverse firing patterns in a model of an identified motor neuron

Neurons show diverse firing patterns. Even neurons belonging to a single chemical or morphological class, or the same identified neuron, can display different types of electrical activity. For example, motor neuron MN5, which innervates a flight muscle of adult Drosophila, can show distinct firing patterns under the same recording conditions. We developed a two-dimensional biophysical model and show that a core complement of just two voltage-gated channels is sufficient to generate firing pattern diversity. We propose Shab and DmNa _v to be two candidate genes that could encode these core currents, and find that changes in Shab channel expression in the model can reproduce activity resembling the main firing patterns observed in MN5 recordings. We use bifurcation analysis to describe the different transitions between rest and spiking states that result from variations in Shab channel expression, exposing a connection between ion channel expression, bifurcation structure, and firing patterns in models of membrane potential dynamics.     

A compartmental model for the study of diurnal rhythms in cell proliferation

A mathematical model taking into account the observed diurnal variations in cell kinetics is presented. Its principle is to divide each phase of the cell cycle into a definite number of compartments and to assume time-dependent probabilities of transition from one compartment to the following; general properties of the model are derived.The particular case where the only time-dependent transition probabilities are those corresponding to the G₁ phase is studied. A characterization of the joint percentages of S and M cells variations is given. The application of the model to interpretation of published experimental data obtained in hamster cheek pouch epithelium is given.     

A targeted extracellular approach for recording long-term firing patterns of excitable cells: a practical guide

Excitable cells in many endocrine and neuronal systems display rhythms with periodicities on the order of many minutes. To observe firing patterns that represent the output of these rhythms requires a recording technique that can monitor electrophysiological activity for several hours without affecting cell behavior. A targeted extracellular approach (also known as loose-patch) accomplishes this objective. Because low resistance seals (<20 MΩ) do not influence the cell membrane and because the normal intracellular milieu is maintained, this approach is the least invasive method for monitoring the endogenous electrical activity of single cells. In this report, we detail our use of this technique to record the firing patterns of gonadotropin-releasing hormone (GnRH) neurons in brain slices continuously for several hours. Published: February 17, 2003 This publication makes use, with permission, of data and methodologies published in Nunemaker CS, DeFazio RA, Moenter SM. Estradiol-sensitive afferents modulate long-term episodic firing patterns of GnRH neurons.Endocrinology 2002; 143:2284–2292, Copyright 2002 by The Endocrine Society.     

A model for the origin of life

Summary A simple statistical model is constructed, describing the transition from disorder to order in a population of mutually catalytic molecules undergoing random mutations. The consequences of the model are calculated, and its possible relevance to the problem of the origin of life is discussed. The main conclusion of the analysis is that the model allows populations of several thousand molecular units to make the transition from disorder to order with reasonable probability.     

Analysis of a minimal model for p53 oscillations

Oscillatory behaviours in genetic networks are important examples for studying the principles underlying the dynamics of cellular regulation. Recently the team of Alon has reported a surprisingly rich oscillatory response of the p53 tumor suppressor to irradiation stress et al. [Lahav, G., Rosenfeld, N., Sigal, A., Geva-Zatorsky, N., Levine, A.J., Elowitz, M.B., Alon, U., 2004. Dynamics of the p53-Mdm2 feedback loop in individual cells. Nat. Genet. 36 (2), 147-150; Geva-Zatorsky, N., Rosenfeld, N., Itzkovitz, S., Milo, R., Sigal, A., Dekel, E., Yarnitzky, T., Liron, Y., Polak, P., Lahav, G., Alon, U., 2006. Oscillations and variability in the p53 system. Mol. Syst. Biol. 2, 2006.0033]. Several models for this system have been proposed by different groups, based essentially on negative feedback loops. In this paper we investigate in detail oscillations and stability in a deterministic time delayed differential model of the core circuit for p53 expression. This model is representative of a class of modelling approaches of this system, based on a minimal set of well-established biomolecular regulations. Depending on the protein degradation rates we show the existence of bifurcations between a stable steady state and oscillations both in presence and absence of stress.     

Heartbeat control in the medicinal leech: A model system for understanding the origin,coordination, and modulation of rhythmic motor patterns

We have analyzed in detail the neuronal network that generates heartbeat in the leech. Reciprocally inhibitory pairs of heart interneurons form oscillators that pace the heartbeat rhythm. Other heart interneurons coordinate these oscillators. These coordinating interneurons, along with the oscillator interneurons, form an eight-cell timing oscillator network for heartbeat. Still other interneurons, along with the oscillator interneurons, inhibit heart motor neurons, sculpting their activity into rhythmic bursts. Critical switch interneurons interface between the oscillator interneurons and the other premotor interneurons to produce two alternating coordination states of the motor neurons. The periods of the oscillator interneurons are modulated by endogenous RFamide neuropeptides. We have explored the ionic currents and graded and spike-mediated synaptic transmission that promote oscillation in the oscillator interneurons and have incorporated these data into a conductance-based computer model. This model has been of considerable predictive value and has led to new insights into how reciprocally inhibitory neurons produce oscillation. We are now in a strong position to expand this model upward, to encompass the entire heartbeat network, horizontally, to elucidate the mechanisms of FMRFamide modulation, and downward, to incorporate cellular morphology. By studying the mechanisms of motor pattern formation in the leech, using modeling studies in conjunction with parallel physiological experiments, we can contribute to a deeper understanding of how rhythmic motor acts are generated, coordinated, modulated, and reconfigured at the level of networks, cells, ionic currents, and synapses. © 1995 John Wiley & Sons, Inc.     

A model for translocation inheritance in man,segregation patterns for a single centric-fusion translocation

A probability model is developed for transmission of a centric-fusion translocation from one generation to the next, giving probability distributions for gametes, zygotes, and liveborn children of different types. The biological assumptions of the model and different alternatives to them are thoroughly discussed. Models of this type form an important part of a theory of population genetics for inherited structural chromosome rearrangements. The treatment is devoted to human populations only. With reference to two segregation analyses carried out by means of the model, the types of insights in the biological mechanisms obtainable by using such a model are demonstrated.     

用原图-对偶图法设计内部间隔最小的自然保护区

生境破碎是导致生物多样性损失的重要原因之一,在设计自然保护区时设法减少生境破碎是提高保护区有效性的重要方法.由于经济资源或地理因素制约不可能把连续的大片土地都划为保护区时,设计一个由相互分离的几部分组成的保护区是更为现实的做法.选择地块组成内部间隔最小的保护区是减少破碎化的一个重要途径,但结合空间特征的保护区地块选择模...     

A minimal model for decoding of time-limited Ca2+ oscillations

Calcium oscillations regulate several cellular processes by activating particular proteins. Most theoretical studies focused on the idealized situation of infinitely long oscillations. Here we analyze information transfer by time-limited calcium spike trains. We show that proteins can be selectively activated in a resonance-like manner by time-limited spike trains of different frequencies, while infinitely long oscillations do not show this resonance phenomenon. We found that proteins are activated more specifically by shorter oscillatory signals with narrower spikes.     

Dissection of a model for neuronal parabolic bursting

We have obtained new insight into the mechanisms for bursting in a class of theoretical models. We study Plant's model [24] for Aplysia R-15 to illustrate our view of these so-called parabolic bursters, which are characterized by low spike frequency at the beginning and end of a burst. By identifying and analyzing the fast and slow processes we show how they interact mutually to generate spike activity and the slow wave which underlies the burst pattern. Our treatment is essentially the first step of a singular perturbation approach presented from a geometrical viewpoint and carried out numerically with AUTO [12]. We determine the solution sets (steady state and oscillatory) of the fast subsystem with the slow variables treated as parameters. These solutions form the slow manifold over which the slow dynamics then define a burst trajectory. During the silent phase of a burst, the solution trajectory lies approximately on the steady state branch of the slow manifold and during the active phase of spiking, the trajectory sweeps through the oscillation branch. The parabolic nature of bursting arises from the (degenerate) homoclinic transition between the oscillatory branch and the steady state branch. We show that, for some parameter values, the trajectory remains strictly on the steady state branch (to produce a resting steady state or a pure slow wave without spike activity) or strictly in the oscillatory branch (continuous spike activity without silent phases). Plant's model has two slow variables: a calcium conductance and the intracellular free calcium concentration, which activates a potassium conductance. We also show how bursting arises from an alternative mechanism in which calcium inactivates the calcium current and the potassium conductance is insensitive to calcium. These and other biophysical interpretations are discussed.