Cross-Correlation Functions for a Neuronal Model

Cross-correlation functions, R_XY(t,τ), are obtained for a neuron model which is characterized by constant threshold θ, by resetting to resting level after an output, and by membrane potential U(t) which results from linear summation of excitatory postsynaptic potentials h(t). The results show that: (1) Near time lag τ = 0, R_XY(t,τ) = f_U [θ-h(τ), t + τ] {h′(τ) + E_U [u′(t + τ)]} for positive values of this quantity, where f_U(u,t) is the probability density function of U(t) and E_U [u′(t + τ)] is the mean value function of U′(t + τ). (2) Minima may appear in R_XY(t,τ) for a neuron subjected only to excitation. (3) For large τ, R_XY(t,τ) is given approximately by the convolution of the input autocorrelation function with the functional of point (1). (4) R_XY(t,τ) is a biased estimator of the shape of h(t), generally over-estimating both its time to peak and its rise time.     

Fractional exponential decay in the capture of ligands by randomly distributed traps in one dimension

In many biophysical and biochemical experiments one observes the decay of some ligand population by an appropriate system of traps. We analyse this decay for a one-dimensional system of randomly distributed traps, and show that one can distinguish three different regimes. The decay starts with a fractional exponential of the form exp[−(t/t ₀)^1/2], which changes into a fractional exponential of the form exp[−(t/t ₁)^1/3] for long times, which in its turn changes into a pure exponential time dependence, i.e. exp[−t/t ₂] for very long times. With these three regimes, we associate three time scales, related to the average trap density and the diffusion constant characterizing the motion of the ligands.     

Bounds on the total population for species governed by reaction-diffusion equations in arbitrary two-dimensional regions

Analysis based on the integration of differential inequalities is employed to derive upper and lower bounds on the total populationN(t) = ∫ _R θ(x ₁,x ₂,t) dx ₁ dx ₂ of a biological species with an area-density distribution function θ=θ(x ₁,x ₂,t) (≥0) governed by a reaction-diffusion equation of the form ∂θ/∂t =D∇²θ +fθ −gθ ⁿ⁺¹ whereD (>0),n (>0),f andg are constant parameters, θ=0 at all points on the boundary ∂R of an (arbitrary) two-dimensional regionR, and the initial distribution (θ(_{x 1},x ₂, 0) is such thatN(0) is finite. Forg≥0 withR the entire two-dimensional Euclidean space, a lower bound onN(t) is obtained, showing in particular thatN(∞) is bounded below by a finite positive quantity forf≥0 andn>1. An upper bound onN(t) is obtained for arbitrary bounded or unbounded)R withn=1,f andg negative, and ∫ _R θ(x ₁,x ₂, 0)² dx ₁ dx ₂ sufficiently small in magnitude, implying that the population goes to extinction with increasing values of the time,N(∞)=0. Forg≥0 andR of finite area, the analysis yields upper bounds onN(t), predicting eventual extinction of the population if eitherf≤0 or if the area ofR is less than a certain grouping of the parameters in cases for whichf is positive. These results are directly applicable to biological species with distributions satisfying the Fisher equation in two spatial dimensions and to species governed by certain specialized population models.     

Analysis of an SEIRS epidemic model with two delays

 A disease transmission model of SEIRS type with exponential demographic structure is formulated. All newborns are assumed susceptible, there is a natural death rate constant, and an excess death rate constant for infective individuals. Latent and immune periods are assumed to be constants, and the force of infection is assumed to be of the standard form, namely proportional to I(t)/N(t) where N(t) is the total (variable) population size and I(t) is the size of the infective population. The model consists of a set of integro-differential equations. Stability of the disease free proportion equilibrium, and existence, uniqueness, and stability of an endemic proportion equilibrium, are investigated. The stability results are stated in terms of a key threshold parameter. More detailed analyses are given for two cases, the SEIS model (with no immune period), and the SIRS model (with no latent period). Several threshold parameters quantify the two ways that the disease can be controlled, by forcing the number or the proportion of infectives to zero. Received 8 May 1995; received in revised form 7 November 1995     

AREA AND VISUAL THRESHOLD

1. The variation of threshold with field area was measured in fields homogeneous in rod-cone composition. At 15° above the fovea, an increase in field diameter from 1° to 5° reduces the threshold sevenfold, at 25° above the fovea tenfold. 2. These changes are shown to follow qualitatively from simple statistical properties of the retinal mosaic. Analytic treatment leads to the expression, (A – n_t)^k I = C, in which A = area, n_t = constant threshold number of elements, I = threshold intensity, and k and C are constants. This equation describes the available data accurately, and is the general form of previous empirical area-threshold formulae.     

Consistency of the Asymptotic Quasi-Likelihood Estimate on Linear Models

Consider a model y_t = f_t(θ) + M_t, 0 ⩽ t ⩽ T where θ∈ Θ in an unknown parameter, f_t(θ) is a linear predictable process, M_t is a martingale difference, and the nature of E(M²_t/ℱ_t—1) is unknown. This paper presents an estimating procedure for θ based on the asymptotic quasi-likelihood methodology. Conditions under which the asymptotic quasi-likelihood estimate converges to the true parameter θ₀ are discussed. This method is applied to several simulated examples, and estimates of the unknown parameter are obtained by means of a two-stage technique. Comparison is made between the estimates obtained via this method and those obtained via the ordinary least squares method. Discussion is provided on the application of the model.     

On enzymic clotting processes V: Rate equations for the case of arbitrary rate of production of the clotting species

The rate theory for enzyme-triggered coagulation reactions, such as the clotting of fibrin or casein, is extended to the case of an arbitrary rate of production of the clotting species. It is shown that the general expression for the growth of the weight-average molecular weight of the clotting product, -M_w, is given by -M_w = M₁{1 + k_s {∫₀^tP(t)² dt}/P(t)}, where M₁ is the “monomer” molecular weight, k_s the smoluchowskian flocculation rate constant and P(t) the total number of monomers produced by the enzyme in t. In the purely smoluchowskian case P(t) stands for the total number of monomers at the beginning of the clotting process. Numerical examples in which the rate of enzymic production is governed by complete Michaelis-Menten kinetics, are compared to cases in which this rate equals V_max- It is shown that after exhaustion of the substrate the system continues to coagulate in a purely smoluchowskian way. Turbidimetric experiments on the clotting of micelles of whole and κ-casein are presented which suggest inactivation of the enzyme by non-productive binding in the flocs formed.     

Generalized stable population theory

In generalizing stable population theory we give sufficient, then necessary conditions under which a population subject to time dependent vital rates reaches an asymptotic stable exponential equilibrium (as if mortality and fertility were constant). If x ₀(t) is the positive solution of the characteristic equation associated with the linear birth process at time t, then rapid convergence of x ₀(t) to x ₀ and convergence of mortality rates produce a stable exponential equilibrium with asymptotic growth rate x ₀–1. Convergence of x ₀(t) to x ₀ and convergence of mortality rates are necessary. Therefore the two sets of conditions are very close. Various implications of these results are discussed and a conjecture is made in the continuous case.     

Computer-determined rate constants of hemolysis induced by Prymnesium parvum toxin

Rates of hemolysis of rabbit erythrocyte suspensions induced by P. parvum (prymnesin) have been measured colorimetrically at 25.5°C and pH 5.5. The data have been treated previously as consecutive first-order rate processes associated with the prolytic and lytic periods from which two specific rate constants have been obtained, k′ and kψ, respectively. These constants have been related to those obtained by a computer-generated fit of the rate data (absorbance A_t, as a function of time t) with the rate equation $\text{Y =}\text{D}\text{[1 +}\text{exp}\text{((X ? B)}\text{C)}\text{]}\text{+ E}$. Here Y equals A_t, X = time, t; D is equal to a spread factor, A_i ? A_∞; C is the slope of the curve at the inflection point; B is the midpoint time value, i.e., the time at which $\text{A}{}_{\mathrm{t}}\text{=}\text{D}\text{2}$; E is termed the off-set constant and is equal to A_∞. Of these constants, B is directly related to the length of the prolytic period, and C^?1 is directly related to the specific first-order rate constant for hemolysis, k_ψ.     

Theoretical aspects of specific and non-specific equilibrium binding of proteins to DNA as studied by the nitrocellulose filter binding assay. Co-operative and non-co-operative binding to a one-dimensional lattice

The analysis of equilibrium binding isotherms obtained by methods such as the nitrocellulose filter binding assay, which measure the fraction. θ, of DNA to which at least one protein molecule is bound, as a function of the free protein concentration (L_F) require a different type of theoretical framework from that required for analysis of conventional equilibrium binding data, in which the number of moles of protein bound per mole of DNA, θ_c is measured as a function of L_F. The theoretical framework required to analyse equilibrium binding data generated by measuring θ(L_F) is developed for co-operative and non-co-operative binding of a protein to a large number of non-specific sites and to a specific sites(s) in the presence of a large number of non-specific sites on a DNA molecule. The theory is simple to apply, equations for θ(L_F) being easy to derive and evaluate, and is suitable for least-squares analysis. Two examples of the application of the theory to the analysis of experimental data are provided for the specific and non-specific binding of the EcoRI restriction endonuclease to bacteriophage λ DNA, and for the specific and non-specific binding of the enzyme dihydrofolate reductase from Lactobacillus casei to pBR322 and pWDLcB1 DNA, the latter differing from the former only in a 2.9 × 10³ base-pair insert containing the L. casei dihydrofolate reductase structural gene. The theoretical and experimental advantages and disadvantages of measuring θ(L_F) rather than θ_c(L_F) are discussed.     

Embryonic development of the European smelt Osmerus eperlanus eperlanus (L.) (Neva population)

The embryonic development of the European smelt Osmerus eperlanus eperlanus has been described. The whole period of the development has been subdivided into 6 subperiods. More than 80 stages may be described for all of these periods. The development of the European smelt embryo was studied in the 5 different regimes of a constant temperature in the range 9.5–18.3°C. The duration of different stages under different temperature regimes was expressed in the relative units of τ₀ (the time of a single cleavage) and τ_s (the time of formation of a single pair of somites). It was established that the τ_s value had proportionally changed by embryogenesis. Due to this fact, the duration of the stages was expressed only as τ_s. The dependency of the speed of the European smelt embryogenesis on temperature has been shown. This dependence was expressed using the standard interval τ_s and described using the equation log τ_s(t) = 3.22665?0.13876t + 0.00297t ², where t is the temperature.     

On washout in nonlinear compartmental systems

Let x(t) be a solution of a compartmental system. If, for some compartment j, x_j(t)→0 as t→∞, then we say that the compartment j washes out. We show that a compartment washes out if it always reaches (along a fixed path) either the environment or another compartment for which there is no return path. Additional criteria, particularly regarding exponential convergence, are also presented. Examples are drawn from tracer kinetics, enzyme reactions, and epidemic models.     

Synthesis, characterization, spectroscopic and electrochemical properties of trans,trans,trans-bis(triphenyl phosphine) bis(aroyl hydrazonato)ruthenium(II) complexes

Four ruthenium (II) complexes of general formula Ru(PPh₃)₂(L)₂ have been synthesized and characterized. The spectroscopic and cyclic voltammetric studies of these complexes are also reported. X-ray crystal structure determination of two of the complexes reveal that Ru(II) occupies trans,trans,trans-(t,t,t) N₂O₂P₂ centrosymmetric octahedral environments, with the ligand pair occupying the equatorial plane. ³¹P NMR confirms the presence of two trans-PPh₃ groups in all the complexes. The transformation of the complexes in dichloromethane solution is studied by spectrophotometry and ³¹P NMR spectroscopy.     

Ordering Dynamics in Neuron Activity Pattern Model: An Insight to Brain Functionality

We study the domain ordering kinetics in d = 2 ferromagnets which corresponds to populated neuron activities with both long-ranged interactions, V(r) ∼ r ⁻ⁿ and short-ranged interactions. We present the results from comprehensive Monte Carlo (MC) simulations for the nonconserved Ising model with n ≥ 2, interaction range considering near and far neighbors. Our model results could represent the long-ranged neuron kinetics (n ≤ 4) in consistent with the same dynamical behaviour of short-ranged case (n ≥ 4) at far below and near criticality. We found that emergence of fast and slow kinetics of long and short ranged case could imitate the formation of connections among near and distant neurons. The calculated characteristic length scale in long-ranged interaction is found to be n independent (L(t) ∼ t ^1/(n−2)), whereas short-ranged interaction follows L(t) ∼ t ^1/2 law and approximately preserve universality in domain kinetics. Further, we did the comparative study of phase ordering near the critical temperature which follows different behaviours of domain ordering near and far critical temperature but follows universal scaling law.     

Optimal investment for enhancing social concern about biodiversity conservation: A dynamic approach

To maintain biodiversity conservation areas, we need to invest in activities, such as monitoring the condition of the ecosystem, preventing illegal exploitation, and removing harmful alien species. These require a constant supply of resources, the level of which is determined by the concern of the society about biodiversity conservation. In this paper, we study the optimal fraction of the resources to invest in activities for enhancing the social concern y(t) by environmental education, museum displays, publications, and media exposure. We search for the strategy that maximizes the time-integral of the quality of the conservation area x(t) with temporal discounting. Analyses based on dynamic programming and Pontryagin’s maximum principle show that the optimal control consists of two phases: (1) in the first phase, the social concern level approaches to the final optimal value y^∗, (2) in the second phase, resources are allocated to both activities, and the social concern level is kept constant y(t)=y^∗. If the social concern starts from a low initial level, the optimal path includes a period in which the quality of the conservation area declines temporarily, because all the resources are invested to enhance the social concern. When the support rate increases with the quality of the conservation area itself x(t) as well as with the level of social concern y(t), both variables may increase simultaneously in the second phase. We discuss the implication of the results to good management of biodiversity conservation areas.     

Chaotic Dynamics of Neuromuscular System Parameters and the Problems of the Evolution of Complexity

The evolution rate v(t) varies among diverse biosystems, but a general theory can be formulated when the dynamics of the biosystem stater x = x(t) = (x₁, x₂, x _m ) ^T is considered in the m-dimensional space of states. A mathematical approach is proposed for evaluating such processes and describes the processes in terms of particular chaos of the statistical distribution functions f(x). In the case of complex multicomponent systems with a high dimension number m (m ?1) of the phase space of states, we propose using pairwise comparison matrices of samples x(t) when homeostasis is constant and calculating the parameters of quasiattractors. The Glensdorff–Prigogine thermodynamic approach to estimating evolution is inefficient in assessing the third-type systems, while it is applicable and the Prigogine theorem works at the level of molecular systems. Alterations in the state of the human neuromuscular system were found to lead to chaotic changes in the statistical functions f(x) in tremor recording samples, while quasiattractor parameters demonstrate a certain regularity.     

Modeling population dynamics of yeast-like symbionts (Ascomycota: Pyrenomycetes: Clavicipitaceae) of the planthopper <Emphasis Type="Italic">Delphacodes kuscheli</Emphasis> (Hemiptera: Delphacidae)

Delphacodes kuscheli establish mutualistic relationship with yeast-like symbionts (YLS) that live in the fat body and are necessary for host survival and reproduction. We estimated for a host of age t, its body weight, W_(t), and the number of YLS per host, YLS_(t). The host body weight was calculated as: W_(t)?=?Lm/[1+ e ^(d–kt)], (Lm?=?the maximum observed weight, and d and k are constants), and the fat body was considered a fixed proportion of W_(t). We calculated the number of YLS per unit host body mass: α_(t)?=?YLS_(t)/W_(t). We also calculated the number of YLS per host, cYLS_(t), and analyzed the pattern of variation in both sexes adapting the expression of the logistic model: cYLS_(t)?=?KN_oe^rt/K+(e^rt -1)N_o, (N_o?=?initial number of YLS, r?=?intrinsic per capita rate of natural increase, and K?=?variable carrying capacity). In females the carrying capacity varied according to a constant proportion of the host’s weight: K_(t)?=?αW_(t). In males α_(t) was considered a decreasing function of the host age: K_(t)?=?α_(t)W_(t). The coefficients No, α, and r were subjected to parameterization. We found that the patterns of W_(t) and YLS_(t) of D. kuscheli were similar to other planthoppers. In females YLS increased up to the adult stage and then remained almost constant, varying similarly to individual weight. In males YLS increased up to the 5th instar nymph as the individual weight did, but the number of YLS decreased in the adult stage and the correlation was not so good. The calculated number of YLS per host matches reasonably well with the number estimated experimentally both in females and males. This is the first study that quantified and modeled the dynamics of YLS endosymbionts in a Neotropical planthopper pest. The models will be used in future studies for better understand the experimental reduction of YLS in young nymphal stages.     

Dosimetric parameters of the new design 103Pd brachytherapy source based on Monte Carlo study

In this study version 5 of the MCNP photon transport simulation was used to calculate the dosimetric parameters for new palladium brachytherapy source design following AAPM Task Group No. 43U1 report. The internal source components include four resin beads of 0.6 mm diameters with ¹⁰³Pd uniformly absorbed inside and one cylindrical copper marker with 1.5 mm length. The resin beads and marker are then encapsulated within 0.8 mm in diameter and 4.5 mm long cylindrical capsule of titanium. The dose rate constant, Λ, line and point-source radial dose function, g_L(r) and g_P(r), and the anisotropy function, F(r,θ) of the IR01-¹⁰³Pd seed have been calculated at distances from 0.25 to 5 cm. All the results are in good agreement with previously published thermoluminescence-dosimeter measured values [3] for the source. The dosimetric parameters calculated in this work showed that in dosimetry point of view, the IR01-¹⁰³Pd seed is suitable for use in brachytherapy of prostate cancer.     

A theoretical explanation for some effects of calcium on the facilitation of neurotransmitter release

A quantal model developed earlier by the authors is recast in terms of common macroscopic variables and applied to the well-documented T, P, N/L, AE neuron network of a leech ganglion. The electrical potential of a neuron (φ) and the ion potentials (φ_a) for Na⁺, K⁺, Cl^? and Ca²⁺ are featured, though it proves possible to reduce the resulting set of coupled non-linear diffusion equations to a single pair whose admissible solutions are defined by a simple algebraic dispersion relation. Less than 30 s is required to solve the system for a functional interval of 2·25 s on a CYBER 175 computer using a modified Runge-Kutta algorithm, the program for which is given. Irreversible effects are included but reversibility is stressed, since the neurons are seen to exchange energy with their environment only in the immediate neighborhood of firing peaks. Plasma oscillations, resulting from a disruption of the Debye layer, offer a sound physical mechanism whereby transient currents and ion exchanges of the observed magnitudes may be generated.The total energy H and information rate Γ_I transferred across the neural membrane are also calculated in terms of φ and φ_a. It is shown that while φ is determined primarily by the K⁺ potential, Γ_I depends mainly on the Ca²⁺ potential together with its time derivatives, and H depends on both the K⁺ and Ca²⁺ potentials. This also makes it possible, not only to compare φ (t) solutions for each of the neurons in the incrementally-loaded network to experimental measurements of φ(t) made for similar stimulus levels, but also to trace the correlated flows of energy and information through the system. Nearly all of the distinctive features of the experimental curves are reproduced, despite the presence of such complexities as wide variations in pulse frequencies and amplitudes, sudden suppression of firing in one neuron when another begins to fire, refractory phases of different durations, and facilitation building to plateau values only slightly less than peak amplitudes for the sensory neurons.While both the energy and information curves possess sharp maxima which coincide with the firing pulses of the potential curves, those for Γ_I (t) are bimodal with rounded maxima that must represent transmissions associated with ion motions instead of polarization effects. In the case of the L and AE neurons, these curves exhibit a series of discrete energy/information packets that could easily produce the proportional increases in muscle tension actually observed.