Heuristics for the Hodgkin–Huxley system

Hodgkin and Huxley (HH) discovered that voltages control ionic currents in nerve membranes. This led them to describe electrical activity in a neuronal membrane patch in terms of an electronic circuit whose characteristics were determined using empirical data. Due to the complexity of this model, a variety of heuristics, including relaxation oscillator circuits and integrate-and-fire models, have been used to investigate activity in neurons, and these simpler models have been successful in suggesting experiments and explaining observations. Connections between most of the simpler models had not been made clear until recently. Shown here are connections between these heuristics and the full HH model. In particular, we study a new model (Type III circuit): It includes the van der Pol-based models; it can be approximated by a simple integrate-and-fire model; and it creates voltages and currents that correspond, respectively, to the h and V components of the HH system.     

Mathematical models and simulations of bacterial growth and chemotaxis in a diffusion gradient chamber

The diffusion gradient chamber (DGC) is a novel device developed to study the response of chemotactic bacteria to combinations of nutrients and attractants [7]. Its purpose is to characterize genetic variants that occur in many biological experiments. In this paper, a mathematical model which describes the spatial distribution of a bacterial population within the DGC is developed. Mathematical analysis of the model concerning positivity and boundedness of the solutions are given. An ADI (Alternating Direction Implicit) method is constructed for finding numerical solutions of the model and carrying out computer simulations. The numerical results of the model successfully reproduced the patterns that were observed in the experiments using the DGC. Received: 3 June 1997 / Revised version: 15 August 2000 / Published online: 20 December 2000     

On plasmid incompatibility

In this paper is presented a brief review of the current state of information on plasmid incompatibility followed by a detailed mathematical model dealing with incompatibility between autonomous homogenic plasmids and based on the assumption that the intracellular plasmid copy pool is randomized with respect to assortment during cell division. Two cases are considered: one in which each plasmid copy replicates once in each generation of cell growth (regular replication) and one in which plasmids are chosen at random for replication from a common pool, irrespective of their replication history (random replication). In both cases, it is assumed that the partition of plasmid copies to daughter cells at cell division is regular—existing plasmid copies are divided equally among the two daughter cells (equipartition). In the case of regular replication coupled with equipartition, it is shown that the survival of heteroplasmid cells (cells containing at least one copy of each of two incompatible plasmids) during exponential growth in a nonselective medium is given by H = H₀[1 − 1/(2N − 1)]ⁿ, where H₀ and H are the numbers of heteroplasmid cells after 0 and n generations of growth, respectively, and N is the plasmid copy number in newborn cells. In the second case, (random replication-equipartition), it is shown that the survival of the heteroplasmid population during exponential growth under nonselective conditions is given by H = H₀[(N − 1)(2N + 1)/(2N − 1)(N + 1)ⁿ. Sample calculations are presented to show that segregation is more rapid in the latter than in the former case. Finally, some of the plasmid-linked genetic determinants that might be expected to affect the expression of incompatibility between nonisogenic plasmids are briefly considered. These determinants include recognition specificity for replication origins, recognition specificity, specific activity of copy number control systems, and recognition specificity of partition systems.     

Phase locking of biological clocks

Radial isochron clocks (RICs) and their response to external signals and coupling with other RICs are studied. RICs are derived as phase approximations to self-sustained oscillators. Their response to single impulses (phase resetting) and to repetitive impulses is determined. This response may be harmonic or chaotic. Finally, the effect of coupling between clocks is studied. Simple coupling is shown to exhibit rhythm splitting like that observed in fish and small mammals. New phase locking results for general weakly coupled RIC systems are also derived.Supported in part by the National Science Foundation under grants MCS-80-15359 (FCH) and MCS-79-02505 (JPK)     

Inhibition of factor X, factor V and prothrombin activation by the bis(lactobionic acid amide) LW10082.

The minimum concentrations of heparin, dermatan sulfate, hirudin, and D-Phe-Pro-ArgCH2Cl required to delay the onset of prothrombin activation in contact-activated plasma also prolong the lag phases associated with both factor X and factor V activation. Heparin and dermatan sulfate prolong the lag phases associated with the activation of the three proteins by catalyzing the inhibition of endogenously generated thrombin. Thrombin usually activates factor V and factor VIII during coagulation. The smallest fragment of heparin able to catalyze thrombin inhibition by antithrombin III is an octadecasaccharide with high affinity for antithrombin III. In contrast, a dermatan sulfate hexasaccharide with high affinity for heparin cofactor II can catalyze thrombin inhibition by heparin cofactor II. A highly sulfated bis(lactobionic acid amide), LW10082 (Mr 2288), which catalyzes thrombin inhibition by heparin cofactor II and has both antithrombotic and anticoagulant activities, has been synthesized. In this study, we determined how the minimum concentration of LW10082 required to delay the onset of intrinsic prothrombin activation achieved this effect. We demonstrate that, like heparin and dermatan sulfate, LW10082 delays the onset of intrinsic prothrombin activation by prolonging the lag phase associated with both factor X and factor V activation. In addition, LW10082 is approximately 25% as effective as heparin and 10 times as effective as dermatan sulfate in its ability to delay the onset of prothrombin activation. The strong anticoagulant action of LW10082 is consistent with previous reports which show that the degree of sulfation is an important parameter for the catalytic effectiveness of sulfated polysaccharides on thrombin inhibition.     

A new approach for heparin standardization: combination of scanning UV spectroscopy, nuclear magnetic resonance and principal component analysis

The year 2007 was marked by widespread adverse clinical responses to heparin use, leading to a global recall of potentially affected heparin batches in 2008. Several analytical methods have since been developed to detect impurities in heparin preparations; however, many are costly and dependent on instrumentation with only limited accessibility. A method based on a simple UV-scanning assay, combined with principal component analysis (PCA), was developed to detect impurities, such as glycosaminoglycans, other complex polysaccharides and aromatic compounds, in heparin preparations. Results were confirmed by NMR spectroscopy. This approach provides an additional, sensitive tool to determine heparin purity and safety, even when NMR spectroscopy failed, requiring only standard laboratory equipment and computing facilities.     

Response of a solutivory chain to a nutrient pulse

A microbial community model is proposed that accounts for byproducts of one strain being nutrients for another and for cells passing in and out of states of torpor. It is shown that such models can sustain the propagation of a nutrient pulse as observed, for example, in methanogenesis.This work was supported in part by NSF Grant DMS9206677 and in part by NSF STC Center in Microbial Ecology, Michigan State University.     

Oscillatory models of the hippocampus: A study of spatio-temporal patterns of neural activity

Spatial patterns of theta-rhythm activity in oscillatory models of the hippocampus are studied here using canonical models for both Hodgkin's class-1 and class-2 excitable neuronal systems. Dynamics of these models are studied in both the frequency domain, to determine phase-locking patterns, and in the time domain, to determine the amplitude responses resulting from phase-locking patterns. Computer simulations presented here demonstrate that phase deviations (timings) between inputs from the medial septum and the entorhinal cortex can create spatial patterns of theta-rhythm phase-locking. In this way, we show that the timing of inputs (not only their frequencies alone) can encode specific patterns of theta-rhythm activity. This study suggests new experiments to determine temporal and spatial synchronization. Received: 31 July 1998 /Accepted in revised form: 20 April 1999     

A mathematical analysis of small mammal populations

Populations of Microtus montanus, the montane vole, have been extensively studied. It is known that their reproductive activity is closely linked to the availability of the chemicals in growing plants. We use a mathematical model here to study how the length of the vegetative season and the natural reproduction rhythm of voles are involved in the long term dynamics of the population numbers. In particular, we use data obtained from Timpie Springs, Utah, and from Jackson Hole, Wyoming, to formulate a model. The novelty of this model is its use of littering curves that highlight the temporally discrete nature of vole reproduction. The model shows how the timing of the vegetative season can influence vole population sizes.This work was supported in part by NSF Grant No. MCS-82-08986 and DMS85-14000 (FCH) and by the University of Utah where Dr. Murphy was a visitor