Dominant ionic mechanisms explored in spiking and bursting using local low-dimensional reductions of a biophysically realistic model neuron

The large number of variables involved in many biophysical models can conceal potentially simple dynamical mechanisms governing the properties of its solutions and the transitions between them as parameters are varied. To address this issue, we extend a novel model reduction method, based on “scales of dominance,” to multi-compartment models. We use this method to systematically reduce the dimension of a two-compartment conductance-based model of a crustacean pyloric dilator (PD) neuron that exhibits distinct modes of oscillation—tonic spiking, intermediate bursting and strong bursting. We divide trajectories into intervals dominated by a smaller number of variables, resulting in a locally reduced hybrid model whose dimension varies between two and six in different temporal regimes. The reduced model exhibits the same modes of oscillation as the 16 dimensional model over a comparable parameter range, and requires fewer ad hoc simplifications than a more traditional reduction to a single, globally valid model. The hybrid model highlights low-dimensional organizing structure in the dynamics of the PD neuron, and the dependence of its oscillations on parameters such as the maximal conductances of calcium currents. Our technique could be used to build hybrid low-dimensional models from any large multi-compartment conductance-based model in order to analyze the interactions between different modes of activity.     

The dynamic structure underlying subthreshold oscillatory activity and the onset of spikes in a model of medial entorhinal cortex stellate cells

Medial entorhinal cortex layer II stellate cells display subthreshold oscillations (STOs). We study a single compartment biophysical model of such cells which qualitatively reproduces these STOs. We argue that in the subthreshold interval (STI) the seven-dimensional model can be reduced to a three-dimensional system of equations with well differentiated times scales. Using dynamical systems arguments we provide a mechanism for generations of STOs. This mechanism is based on the “canard structure,” in which relevant trajectories stay close to repelling manifolds for a significant interval of time. We also show that the transition from subthreshold oscillatory activity to spiking (“canard explosion”) is controlled in the STI by the same structure. A similar mechanism is invoked to explain why noise increases the robustness of the STO regime. Taking advantage of the reduction of the dimensionality of the full stellate cell system, we propose a nonlinear artificially spiking (NAS) model in which the STI reduced system is supplemented with a threshold for spiking and a reset voltage. We show that the synchronization properties in networks made up of the NAS cells are similar to those of networks using the full stellate cell models. In memory of Angel A. Alonso     

Theoretical Assessment of Public Health Impact of Imperfect Prophylactic HIV-1 Vaccines with Therapeutic Benefits

This paper presents a number of deterministic models for theoretically assessing the potential impact of an imperfect prophylactic HIV-1 vaccine that has five biological modes of action, namely “take,” “degree,” “duration,” “infectiousness,” and “progression,” and can lead to increased risky behavior. The models, which are of the form of systems of nonlinear differential equations, are constructed via a progressive refinement of a basic model to incorporate more realistic features of HIV pathogenesis and epidemiology such as staged progression, differential infectivity, and HIV transmission by AIDS patients. The models are analyzed to gain insights into the qualitative features of the associated equilibria. This allows the determination of important epidemiological thresholds such as the basic reproduction numbers and a measure for vaccine impact or efficacy. The key findings of the study include the following (i) if the vaccinated reproduction number is greater than unity, each of the models considered has a locally unstable disease-free equilibrium and a unique endemic equilibrium; (ii) owing to the vaccine-induced backward bifurcation in these models, the classical epidemiological requirement of vaccinated reproduction number being less than unity does not guarantee disease elimination in these models; (iii) an imperfect vaccine will reduce HIV prevalence and mortality if the reproduction number for a wholly vaccinated population is less than the corresponding reproduction number in the absence of vaccination; (iv) the expressions for the vaccine characteristics of the refined models take the same general structure as those of the basic model.     

Neural field model of binocular rivalry waves

We present a neural field model of binocular rivalry waves in visual cortex. For each eye we consider a one-dimensional network of neurons that respond maximally to a particular feature of the corresponding image such as the orientation of a grating stimulus. Recurrent connections within each one-dimensional network are assumed to be excitatory, whereas connections between the two networks are inhibitory (cross-inhibition). Slow adaptation is incorporated into the model by taking the network connections to exhibit synaptic depression. We derive an analytical expression for the speed of a binocular rivalry wave as a function of various neurophysiological parameters, and show how properties of the wave are consistent with the wave-like propagation of perceptual dominance observed in recent psychophysical experiments. In addition to providing an analytical framework for studying binocular rivalry waves, we show how neural field methods provide insights into the mechanisms underlying the generation of the waves. In particular, we highlight the important role of slow adaptation in providing a “symmetry breaking mechanism” that allows waves to propagate.     

A mathematical model of decision-making neural circuits controlling behavioral model selection

Mathematical models of mutually interacting reverberatory neural circuits exhibit decision-making properties that can account for behavioral mode selection in terms of identifiable neural subsystems and measurable variables, without the need for invoking concepts such as “drive,” “motivation,” or “instinct”. The models can account for an animal's ability to persist in a given behavioral mode under appropriate stimulus conditions and to rapidly switch modes in response to significant stimulus changes via the properties of stable steady states, thresholds for steady-state transitions, and the lack of intrinsic oscillations.     

Two-point heterogeneous connections in a continuum neural field model

We examine a novel heterogeneous connection scheme in a 1D continuum neural field model. Multiple two-point connections are added to a local connection function in order to model the “patchy” connections seen in, for example visual cortex. We use a numerical approach to solve the equations, choosing the locations of the two-point connections stochastically. We observe self-sustained persistent fluctuations of activity which can be classified into two types (one of which is similar to that seen in network models of discrete excitable neurons, the other being particular to this model). We study the effect of parameters such as system size and the range, number and strength of connections, on the probability that a particular realisation of the connections is able to exhibit persistent fluctuations.     

Source-sink landscape theory and its ecological significance

Exploring the relationships between landscape pattern and ecological processes is the key topic of landscape ecology, for which, a large number of indices as well as landscape pattern analysis model were developed. However, one problem faced by landscape ecologists is that it is hard to link the landscape indices with a specific ecological process. Linking landscape pattern and ecological processes has become a challenge for landscape ecologists. “Source” and “sink” are common concepts used in air pollution research, by which the movement direction and pattern of different pollutants in air can be clearly identified. In fact, for any ecological process, the research can be considered as a balance between the source and the sink in space. Thus, the concepts of “source” and “sink” could be implemented to the research of landscape pattern and ecological processes. In this paper, a theory of sourcesink landscape was proposed, which include: (1) In the research of landscape pattern and ecological process, all landscape types can be divided into two groups, “source” landscape and “sink” landscape. “Source” landscape contributes positively to the ecological process, while “sink” landscape is unhelpful to the ecological process. (2) Both landscapes are recognized with regard to the specific ecological process. “Source” landscape in a target ecological process may change into a “sink” landscape as in another ecological process. Therefore, the ecological process should be determined before “source” or “sink” landscape were defined. (3) The key point to distinguish “source” landscape from “sink” landscape is to quantify the effect of landscape on ecological process. The positive effect is made by “source” landscape, and the negative effect by “sink” landscape. (4) For the same ecological process, the contribution of “source” landscapes may vary, and it is the same to the “sink” landscapes. It is required to determine the weight of each landscape type on ecological processes. (5) The sourcesink principle can be applied to non-point source pollution control, biologic diversity protection, urban heat island effect mitigation, etc. However, the landscape evaluation models need to be calibrated respectively, because different ecological processes correspond with different source-sink landscapes and evaluation models for the different study areas. This theory is helpful to further study landscape pattern and ecological process, and offers a basis for new landscape index design. __________ Translated from Acta Ecologica Sinica, 2006, 26(5): 1444–1449 [译自: 生态学报]     

On the spread of information with time and distance

The transmission of some information or behavior pattern is treated as a flow of “particles” which execute random motions over a population of individuals and which may multiply or disappear. Equations are derived for the number density of these “particles” and from this is calculated the number of individuals through which the “particles” have passed. The results are applied to a number of situations such as 1) uniform spatial distribution with multiplication factor decreasing with time because of loss of interest or confusion of the information, 2) multiplication factor constant but the rate of spreal decreasing with multiple hearings, 3) one-dimensional region with a small starting region with or without an absorbing barrier 4) two-dimensional region with absorbing barrier, 5) continous sources of information within a small region in one dimension, 6) uniform spatial distribution in which individuals do not respond to more than one hearing.     

Hybrid bounded error observers for uncertain bioreactor models

In this paper, we build bounded error observers for a common class of partially known bioreactor models. The main idea is to construct hybrid bounded observers “between” high gain observer, which has an adjustable convergence rate but requires perfect knowledge of the model, and asymptotic observer which is very robust towards uncertainty but has a fixed convergence rate. An hybrid bounded error observer which reconstructs the two state variables is constructed considering two steps: first step is similar to a high gain observer meaning that fast convergence rate but error depending on the knowledge of the model are obtained; second step is a switch to an observer similar to the asymptotic one meaning that fixed convergence rate towards an error as small as desired is obtained. Thus, a better convergence rate of estimated variables than the classical asymptotic observer is obtained.     

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.     

Integrate-and-fire models of insolation-driven entrainment of broadcast spawning in corals

The circa-annual cycle of gametogenesis produces mature gametes at the spawning “season” for successful mass spawning of broadcast corals. We develop a bioenergetic integrate-and-fire model that reveals how annual insolation rhythms can entrain the gametogenetic cycles in tropical hermatypic corals to the appropriate spawning season, since photosynthate is their primary source of energy. In the presence of short-term fluctuations in the energy input, a feedback regulatory mechanism is likely required to achieve coherence of spawning times to within one lunar cycle, in order for subsequent signals such as lunar and diurnal light cycles to unambiguously determine the “correct” night of spawning. The feedback mechanism can also provide robustness against population heterogeneity that may arise due to genetic and environmental effects. We solve the integrate-and-fire bioenergetic model numerically using the Fokker–Planck equation and use analytical tools such as rotation number to study entrainment.     

Two-scale modelling of Saponaria bellidifolia Sm. (Caryophyllaceae) abundance on limestone outcrops from its northern range periphery (Southeastern Carpathians)

We modelled the effect of habitat heterogeneity on the abundance of the submediterranean Saponaria bellidifolia, a red list species in Romania. The study was designed at two scales: 100 and 0.5 m². At larger scale, generalized additive models and canonical correspondence analysis were used to model the density of ramets, whereas at microscale, binomial logistic regression was employed to model the species’ occurrence. S. bellidifolia abundance responded sensitively to habitat type (classified as “grassy”, “rocky” and “scree”), rather than to microclimatic variables. At both scales, habitat type was the best predictor of ramet abundance, followed by slope and vegetation cover. At 0.5 m², soil depth was also a good predictor of species occurrence. The data revealed that screes are the most suitable habitats for hosting relatively large populations of this rare species, because of occasional natural disturbances and presumably lower interspecific competition.     

An indicator quantitatively comparing two treatment effect sizes on responder and non-responder groups – exponential of estimated interaction parameter

 Sometimes a specific treatment is effective in one subgroup but not in another. An indicator allowing quantitative comparison of treatment effect in two subgroups would be useful in clinical medicine. We have developed such an indicator. It is obtained by calculations using Cox’s proportional hazard or logistic model with therapy, subgroup, and confounding explanatory variables. The parameter of the interaction between therapy and subgroup can be estimated and tested statistically. The exponential value of the interaction parameter is what we tentatively call the “hazard ratio ratio”, meaning the ratio between the treatment effects in two subgroups. The 95% confidence interval of the indicator can also be calculated. As a numerical example, the hazard ratio between the survival times of postoperative gastric cancer patients treated by adjuvant immunochemotherapy and patients without adjuvant immunochemotherapy in a subgroup with high serum glycosidically bound sialic acid (SA) level was lower than that in a low-SA subgroup using an estimate for hazard ratio ratio of less than 0.5 with statistical significance. We propose this indicator be used as a “responder/non-responder ratio” of therapy effect. Received: 11 April 1995 / Accepted: 5 September 1995     

Growth and reproduction in higher plants I. Theoretical analysis by mathematical models

To analyse the whole life of higher plants, an attempt was made to describe their growth and reproduction by mathematical models based on the elements determining matter production and economy of the matter. A plant body was regarded as a compound system of two parts; “productive part” and “reproductive part”. A parameter (reproductive index) was introduced to connect these two parts, and a set of the mathematical models describing the quantitative growth of these two parts were established. Two basic patterns of reproduction in higher plants were distinguished into “D-reproduction” and “I-reproduction”. The state of matter production of the mother plant determined an initial size of the daughter plant in theD-reproduction, while, in theI-reproduction, it did not determine the initial size of the daughter, but determined the number of propagules. The model of each reproduction pattern was also constructed. A formula determining the initial size of a plant in a given generation was constructed as the model of theD-reproduction. The model for theI-reproduction described the number of propagules produced in a given generation. Some aspects of the plant life, e.g. the optimum reproductive index, the switch-over time from the vegetative to the reproductive growth phase, the seed number, types of expansive reproduction, were theoretically analysed and discussed under these mathematical models.     

Word organization in coding DNA: A mathematical model

This article deals with the relationship between vocabulary (total number of distinct oligomers or “words”) and text-length (total number of oligomers or “words”) for a coding DNA sequence (CDS). For natural human languages, Heaps established a mathematical formula known as Heaps' law, which relates vocabulary to text-length. Our analysis shows that Heaps' law fails to model this relationship for CDSs. Here we develop a mathematical model to establish the relationship between the number of type of words (vocabulary) and the number of words sampled (text-length) for CDSs, when non-overlapping nucleotide strings with the same length are treated as words. We use tangent-hyperbolic function, which captures the saturation property of vocabulary. Based on the parameters of the model, we formulate a mathematical equation, known as “equation of word organization”, whose parameters essentially indicate that nucleotide organization of coding sequences are different from one another. We also compare the word organization of CDSs with the random word distribution and conclude that a CDS is neither similar to a natural human language nor to a random one. Moreover, these sequences have their unique nucleotide organization and it is completely structured for specific biological functioning. IM and AS contributed equally to this work.     

A Contact-Network-Based Formulation of a Preferential Mixing Model

Heterogeneity in the number of potentially infectious contacts and connectivity correlations (“like attaches to like” i.e., assortatively mixed or “opposites attract” i.e., disassortatively mixed) have important implications for the value of the basic reproduction ratio R ₀ and final epidemic size. In this paper, we present a contact-network-based derivation of a simple differential equation model that accounts for preferential mixing based on the number of contacts. We show that results based on this model are in good qualitative agreement with results obtained from preferential mixing models used in the context of sexually transmitted diseases (STDs). This simple model can accommodate any mixing pattern ranging from completely disassortative to completely assortative and allows the derivation of a series of analytical results.     

Using sensory weighting to model the influence of canal,otolith and visual cues on spatial orientation and eye movements

 The sensory weighting model is a general model of sensory integration that consists of three processing layers. First, each sensor provides the central nervous system (CNS) with information regarding a specific physical variable. Due to sensor dynamics, this measure is only reliable for the frequency range over which the sensor is accurate. Therefore, we hypothesize that the CNS improves on the reliability of the individual sensor outside this frequency range by using information from other sensors, a process referred to as “frequency completion.” Frequency completion uses internal models of sensory dynamics. This “improved” sensory signal is designated as the “sensory estimate” of the physical variable. Second, before being combined, information with different physical meanings is first transformed into a common representation; sensory estimates are converted to intermediate estimates. This conversion uses internal models of body dynamics and physical relationships. Third, several sensory systems may provide information about the same physical variable (e.g., semicircular canals and vision both measure self-rotation). Therefore, we hypothesize that the “central estimate” of a physical variable is computed as a weighted sum of all available intermediate estimates of this physical variable, a process referred to as “multicue weighted averaging.” The resulting central estimate is fed back to the first two layers. The sensory weighting model is applied to three-dimensional (3D) visual–vestibular interactions and their associated eye movements and perceptual responses. The model inputs are 3D angular and translational stimuli. The sensory inputs are the 3D sensory signals coming from the semicircular canals, otolith organs, and the visual system. The angular and translational components of visual movement are assumed to be available as separate stimuli measured by the visual system using retinal slip and image deformation. In addition, both tonic (“regular”) and phasic (“irregular”) otolithic afferents are implemented. Whereas neither tonic nor phasic otolithic afferents distinguish gravity from linear acceleration, the model uses tonic afferents to estimate gravity and phasic afferents to estimate linear acceleration. The model outputs are the internal estimates of physical motion variables and 3D slow-phase eye movements. The model also includes a smooth pursuit module. The model matches eye responses and perceptual effects measured during various motion paradigms in darkness (e.g., centered and eccentric yaw rotation about an earth-vertical axis, yaw rotation about an earth-horizontal axis) and with visual cues (e.g., stabilized visual stimulation or optokinetic stimulation). Received: 20 September 2000 / Accepted in revised form: 28 September 2001     

Effects of genomic imprinting on quantitative traits

Standard Mendelian genetic processes incorporate several symmetries, one of which is that the level of expression of a gene inherited from an organism’s mother is identical to the level should that gene have been inherited paternally. For a small number of loci in a variety of taxa, this symmetry does not hold; such genes are said to be “genomically imprinted” (or simply “imprinted”). The best known examples of imprinted loci come from mammals and angiosperms, although there are also cases from several insects and some data suggesting that imprinting exists in zebra fish. Imprinting means that reciprocal heterozygotes need not be, on average, phenotypically identical. When this difference is incorporated into the standard quantitative-genetic model for two alleles at a single locus, a number of standard expressions are altered in fundamental ways. Most importantly, in contrast to the case with euMendelian expression, the additive and dominance deviations are correlated. It would clearly be of interest to be able to separate imprinting effects from maternal genetic effects, but when the latter are added to the model, the well-known generalized least-squares approach to deriving breeding values cannot be applied. Distinguishing these two types of parent-of-origin effects is not a simple problem and requires further research.     

Unreasonable implications of reasonable idiotypic network assumptions

We first analyse a simple symmetric model of the idiotypic network. In the model idiotypic interactions regulate B cell proliferation. Three non-idiotypic processes are incorporated: (1) influx of newborn cells; (2) turnover of cells: (3) antigen. Antigen also regulates proliferation. A model of 2 B cell populations has 3 stable equilibria: one virgin, two immune. The twodimensional system thus remembers antigens, i.e. accounts for immunity. By contrast, if an idiotypic clone proliferates (in response to antigen), its anti-idiotypic partner is unable to control this. Symmetric idiotypic networks thus fail to account for proliferation regulation. In high-D networks we run into two problems. Firstly, if the network accounts for memory, idiotypic activation always propagates very deeply into the network. This is very unrealistic, but is an implication of the “realistic” assumption that it should be easier to activate all cells of a small virgin clone than to maintain the activation of all cells of a large (immune) clone. Secondly, graph theory teaches us that if the (random) network connectance exceeds a threshold level of one interaction per clone, most clones are interconnected. We show that this theory is also applicable to immune networks based on complementary matching idiotypes. The combination of the first “percolation” result with the “interconnectancr” result means that the first stimulation of the network with antigen should eventually affect most of the clones. We think this is unreasonable. Another threshold property of the network connectivity is the existence of a virgin state. A gradual increase in network connectance eliminates the virgin state and thus causes an abrupt change in network behaviour. In contrast to weakly connected systems, highly connected networks display autonomous activity and are unresponsive to external antigens. Similar differences between neonatal and adult networks have been described by experimentalists. The robustness of these results is tested with a network in which idiotypic inactivation of a clone occurs more generally than activation. Such “long-range inhibition” is known to promote pattern formation. However, in our model it fails to reduce the percolation, and additionally, generates semi-chaotic behaviour. In our network, the inhibition of a clone that is inhibiting can alter this clone into a clone that is activating. Hence “long-range inhibition” implies “long-range activation”, and idiotypic activation fails to remain localized. We next complicate this model by incorporating antibody production. Although this “antibody” model statically accounts for the same set of equilibrium points, it dynamically fails to account for state switching (i.e. memory). The switching behaviour is disturbed by the autonomous slow decay of the (long-lived) antibodies. After antigenic triggering the system now performs complex cyclic behaviour. Finally, it is suggested that (idiotypic) formation of antibody complexes can play only a secondary role in the network. In conclusion, our results cast doubt on the functional role of a profound idiotypic network. The network fails to account for proliferation regulation, and if it accounts for memory phenomena, it “explodes” upon the first encounter with antigen due to extensive percolation.