A modified cable model for neuron processes with non-constant diameters.

The neuron axon cable model is expanded and developed to describe the linear subthreshold transmembrane potential of any circular cross section, thin membrane neuron fiber whose radius can be expressed as an analytic function of position, r(x). The transmembrane time constant is shown under the condition of space clamp to be independent of changes in geometry. Three typical neuron geometries are modeled (dendrite-soma, soma-axon, and dendrite-soma-axon) and the solutions to the resulting differential equations are numerically evaluated. The geometry-induced effects ate attributed to changes in current density and physiological correlates of the effects are proposed.     

A generalized tapering equivalent cable model for dendritic neurons

A mathematical model has been developed which collapses a dendritic neuron of complex geometry into a single electrotonically tapering equivalent cable. The modified cable equation governing the transient distribution of subthreshold membrane potential in a branching tree is transformed, becoming amenable to analytic solution. This transformation results in a Riccati differential equation whose six solutions (expressed in terms of elementary functions) control the amount and degree of taper found in the equivalent cable model. To illustrate the theory, an analytic solution (in series form) of the modified cable equation is obtained for a voltage-clamp present at the soma of a quadratically tapering equivalent cable whose distal end is sealed.     

A non-uniform equivalent cable model of membrane voltage changes in a passive dendritic tree

A non-uniform equivalent cable model of membrane voltage changes in a passive dendritic tree extending Rall's equivalent cylinder model is presented. It is obtained from a combination of cable theory with the continuum approach. Replacing the fine structure of the branching dendrites by an equivalent, conductive medium characterized by averaged electrical parameters, the one-dimensional cable equations with spatially varying parameters are derived. While these equations can be solved in general only numerically, we were able to formulate a general branching condition (comprising Rall's 3/2 power relationship as a special case) under which analytical solutions can be deduced from those of the equivalent cylinder model. This model allows dendritic trees with a greater variety of branching patterns than before to be analytically treated.     

Short-range axonal/dendritic transport by myosin-V: A model for vesicle delivery to the synapse

Myosin-V is a versatile motor involved in short-range axonal/dendritic transport of vesicles in the actin-rich cortex and synaptic regions of nerve cells. It binds to several different kinds of neuronal vesicles by its globular tail domain but the mechanism by which it is recruited to these vesicles is not known. In this study, we used an in vitro motility assay derived from axoplasm of the squid giant axon to study the effects of the globular tail domain on the transport of neuronal vesicles. We found that the globular tail fragment of myosin-V inhibited actin-based vesicle transport by displacing native myosin-V and binding to vesicles. The globular tail domain pulled down kinesin, a known binding partner of myosin-V, in affinity isolation experiments. These data confirmed earlier evidence that kinesin and myosin-V interact to form a hetero-motor complex. The formation of a kinesin/myosin-V hetero-motor complex on vesicles is thought to facilitate the coordination of long-range movement on microtubules and short-range movement on actin filaments. The direct interaction of motors from both filament systems may represent the mechanism by which the transition of vesicles from microtubules to actin filaments is regulated. These results are the first demonstration that the recombinant tail of myosin-V inhibits vesicle transport in an in vitro motility assay. Future experiments are designed to determine the functional significance of the interaction between myosin-V and kinesin and to identify other proteins that bind to the globular tail domain of myosin-V.     

A stochastic dynamical model for the characterization of the geometrical structure of dendritic processes

Under certain basic assumptions the branching pattern of dendrites can be modeled as a Galton-Watson process in varying environment. Using results from graph theory we compute the probability distributions, expectations and variances for biologically significant variables such as the number of (intermediate and terminal) branches, the maximum number of orders, etc., together with the limit behavior of these quantities. Furthermore, the probability measure induced by the Galton-Watson process on the set of all trees is calculated. The measure assigns to any set of branching patterns the probability that it is realized by a certain process, which is completely described through the bifurcation probabilities.     

A dendritic cable model for the amplification of synaptic potentials by an ensemble average of persistent sodium channels

The persistent sodium current density (I(NaP)) at the soma measured with the 'whole-cell' patch-clamp recording method is linearized about the resting state and used as a current source along the dendritic cable (depicting the spatial distribution of voltage-dependent persistent sodium ionic channels). This procedure allows time-dependent analytical solutions to be obtained for the membrane depolarization. Computer simulated response to a dendritic current injection in the form of synaptically-induced voltage change located at a distance from the recording site in a cable with unequally distributed persistent sodium ion channel densities per unit length of cable (the so-called 'hot-spots') is used to obtain conclusions on the density and distribution of persistent sodium ion channels. It is shown that the excitatory postsynaptic potentials (EPSPs) are amplified if hot-spots of persistent sodium ion channels are spatially distributed along the dendritic cable, with the local density of I(NaP) with respect to the recording site shown to specifically increase the peak amplitude of the EPSP for a proximally placed synaptic input, while the spatial distribution of I(NaP) serves to broaden the time course of the amplified EPSP. However, in the case of a distally positioned synaptic input, both local and nonlocal densities yield an approximately identical enhancement of EPSPs in contradiction to the computer simulations performed by Lipowsky et al. [J. Neurophysiol. 76 (1996) 2181]. The results indicate that persistent sodium channels produce EPSP amplification even when their distribution is relatively sparse (i.e. , approximately 1-2% of the transient sodium channels are found in dendrites of CA1 hippocampal pyramidal neurons). This gives a strong impetus for the use of the theory as a novel approach in the investigation of synaptic integration of signals in active dendrites represented as ionic cables.     

Conductance transients onto dendritic spines in a segmental cable model of hippocampal neurons.

Dendritic shaft (Zd) and spine (Zsp) input impedances were computed numerically for sites on hippocampal neurons, using a segmental format of cable calculations. The Zsp values for a typical spine appended onto a dendritic shaft averaged less than 2% higher than the Zd values for the adjacent dendritic shaft. Spine synaptic inputs were simulated by a brief conductance transient, which possessed a time integral of 12 X 10(-10)S X ms. This input resulted in an average peak spine response of 20 mV for both dentate granule neurons and CA1 pyramidal cells. The average spine transient was attenuated less than 2% in conduction across the spine neck, considering peak voltage, waveform parameters, and charge transfer. The spine conductance transient resulted in an average somatic response of 100 microV in the dentate granule neurons, because of passive electrotonic propagation. The same input transient was also applied to proximal and distal sites on CA1 pyramidal cells. The predicted responses at the soma demonstrated a clear difference between the proximal and distal inputs, in terms of both peak voltage and waveform parameters. Thus, the main determinant of the passive propagation of transient electrical signals in these neurons appears to be dendritic branching rather than signal attenuation through the spine neck.     

A mathematical model with a modified logistic approach for singly peaked population processes

When a small number of individuals of a single species are confined in a closed space with a limited amount of indispensable resources, breeding may start initially under suitable conditions, and after peaking, the population should go extinct as the resources are exhausted. Starting with the logistic equation and assuming that the carrying capacity of the environment is a function of the amount of resources, a mathematical model describing such a pattern of population change is obtained. An application of this model to a typical set of population records, that of deer herds by V. B. Scheffer (1951, Sci. Monthly 73, 356-362) and E. C. O'Roke and F. N. Hamerstrom (1948, J. Wildlife Management 12, 78-86), yields estimates of the initial amount of indispensable food and its availability or nutritional efficiency which were previously unspecified.     

Colour model analysis for microscopic image processing

This article presents a comparative study between different colour models (RGB, HSI and CIEL*a*b*) applied to a very large microscopic image analysis. Such analysis of different colour models is needed in order to carry out a successful detection and therefore a classification of different regions of interest (ROIs) within the image. This, in turn, allows both distinguishing possible ROIs and retrieving their proper colour for further ROI analysis. This analysis is not commonly done in many biomedical applications that deal with colour images. Other important aspects is the computational cost of the different processing algorithms according to the colour model. This work takes these aspects into consideration to choose the best colour model tailored to the microscopic stain and tissue type under consideration and to obtain a successful processing of the histological image.     

A coupled model of fast axonal transport of organelles and slow axonal transport of tau protein

We have developed a model that accounts for the effect of a non-uniform distribution of tau protein along the axon length on fast axonal transport of intracellular organelles. The tau distribution is simulated by using a slow axonal transport model; the numerically predicted tau distributions along the axon length were validated by comparing them with experimentally measured tau distributions reported in the literature. We then developed a fast axonal transport model for organelles that accounts for the reduction of kinesin attachment rate to microtubules by tau. We investigated organelle transport for two situations: (1) a uniform tau distribution and (2) a non-uniform tau distribution predicted by the slow axonal transport model. We found that non-uniform tau distributions observed in healthy axons (an increase in tau concentration towards the axon tip) result in a significant enhancement of organelle transport towards the synapse compared with the uniform tau distribution with the same average amount of tau. This suggests that tau may play the role of being an enhancer of organelle transport.     

A simple vector implementation of the Laplace-transformed cable equations in passive dendritic trees

Transient potentials in dendritic trees can be calculated by approximating the dendrite by a set of connected cylinders. The profiles for the currents and potentials in the whole system can then be obtained by imposing the proper boundary conditions and calculating these profiles along each individual cylinder. An elegant implementation of this method has been described by Holmes (1986), and is based on the Laplace transform of the cable equation. By calculating the currents and potentials only at the ends of the cylinders, the whole system of connected cylinders can be described by a set of n equations, where n denotes the number of internal and external nodes (points of connection and endpoints of the cylinders). The present study shows that the set of equations can be formulated by a simple vector equation which is essentially a generalization of Ohm's law for the whole system. The current and potential n-vectors are coupled by a n × n conductance matrix whose structure immediately reflects the connectivity pattern of the connected cylinders. The vector equation accounts for conductances, associated with driving potentials, which may be local or distributed over the membrane. It is shown that the vector equation can easily be adapted for the calculation of transients over a period in which stepwise changes in system parameters have occurred. In this adaptation it is assumed that the initial conditions for the potential profiles at the start of a new period after a stepwise change can be approximated by steady-state solutions. The vector representation of the Laplace-transformed equations is attractive because of its simplicity and because the structure of the conductance matrix directly corresponds to the connectivity pattern of the dendritic tree. Therefore it will facilitate the automatic generation of the equations once the geometry of the branching structure is known.     

cAMP differentially regulates axonal and dendritic development of dentate granule cells

Neurite polarity is a morphological characteristic of dentate gyrus granule cells, which extend axons to the hilar region and dendrites in the opposite direction, i.e. to the molecular layer. This remarkable polarity must require a differential system for axon and dendrite guidance. Here, we report that the axon and dendrites of a granule cell are differentially responsive to cAMP. In developing cultures of dispersed granule cells, dendritic growth cones were increased in number after pharmacological activation of cAMP signaling and decreased after blockade of cAMP signaling. Activation of cAMP signaling antagonized dendritic collapse induced by the potent repellents Sema3F and glutamate. In contrast to dendrites, axons were protected from Sema3F-induced collapse when cAMP signaling was inhibited. Axonal and dendritic growth cones both expressed type 1 adenylyl cyclase, but only axons showed a cAMP increase in response to Sema3F, and the elevated cAMP was sufficient to collapse axonal growth cones. Thus, the axons and dendrites of dentate granule cells differ in the regulation of cAMP levels as well as responsiveness to cAMP. cAMP may be crucial for shaping the information flow polarity in the dentate gyrus circuit.     

A modified wire-loop method for quantitative electron microscopic radioautography

Summary The distribution of reaction for acid and alkaline phosphatases in the proximal cartilage of the os penis and the mandibular condylar cartilage has been compared. The distribution of acid phosphatase in the two structures seems to be identical, whereas the distribution of alkaline phosphatase in the os penis cartilage seems to differ from that in the mandibular condylar cartilage and, by this, from all other studied growth cartilages.     

A microscopic model of enzyme kinetics.

Many in vivo enzymatic processes, such as those of the tissue factor pathway of blood coagulation, occur in environments with facilitated substrate delivery or enzymes bound to cellular or lipid surfaces, which are quite different from the ideal fluid environment for which the Michaelis-Menten equation was derived. To describe the kinetics of such reactions, we propose a microscopic model that focuses on the kinetics of a single-enzyme molecule. This model provides the foundation for macroscopic models of the system kinetics of reactions occurring in both ideal and nonideal environments. For ideal reaction systems, the corresponding macroscopic models thus derived are consistent with the Michaelis-Menten equation. It is shown that the apparent Km is in fact a function of the mechanism of substrate delivery and should be interpreted as the substrate level at which the enzyme vacancy time equals the residence time of ES-complexes; it is suggested that our microscopic model parameters characterize more accurately an enzyme and its catalytic efficiency than does the classical Km. This model can also be incorporated into computer simulations of more complex reactions as an alternative to explicit analytical formulation of a macroscopic model.     

A modified plastic culture flask for microscopic observation of fungi

A plastic culture flask modified for the direct microscopic observation of fungi is described. The use of this flask allows for in situ observation of fungal morphology and monitoring of development of different stages of fungal growth. The flask provides a safe, rapid, and inexpensive culture technique for the clinical laboratory.     

A modified cable formalism for modeling neuronal membranes at high frequencies

Intracellular recordings of cortical neurons in vivo display intense subthreshold membrane potential (V_m) activity. The power spectral density of the V_m displays a power-law structure at high frequencies (>50 Hz) with a slope of ∼−2.5. This type of frequency scaling cannot be accounted for by traditional models, as either single-compartment models or models based on reconstructed cell morphologies display a frequency scaling with a slope close to −4. This slope is due to the fact that the membrane resistance is short-circuited by the capacitance for high frequencies, a situation which may not be realistic. Here, we integrate nonideal capacitors in cable equations to reflect the fact that the capacitance cannot be charged instantaneously. We show that the resulting nonideal cable model can be solved analytically using Fourier transforms. Numerical simulations using a ball-and-stick model yield membrane potential activity with similar frequency scaling as in the experiments. We also discuss the consequences of using nonideal capacitors on other cellular properties such as the transmission of high frequencies, which is boosted in nonideal cables, or voltage attenuation in dendrites. These results suggest that cable equations based on nonideal capacitors should be used to capture the behavior of neuronal membranes at high frequencies.     

A mechanism for differential control of axonal and dendritic spiking underlying learning in a cerebellum-like circuit

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Analytical solutions to the multicylinder somatic shunt cable model for passive neurones with differing dendritic electrical parameters

The multicylinder somatic shunt cable model for passive neurones with differing time constants in each cylinder is considered in this paper. The solution to the model with general inputs is developed, and the parametric dependence of the voltage response is investigated. The method of analysis is straightforward and follows that laid out in Evans et al. (1992, 1994): (i) The dimensional problem is stated with general boundary and initial conditions, (ii) The model is fully non-dimensionalised, and a dimensionless parameter family which uniquely governs the behaviour of the dimensionless voltage response is obtained, (iii) The fundamental unit impulse problem is solved, and the solutions to problems involving general inputs are written in terms of the unit impulse solution, (iv) The large and small time behaviour of the unit impulse solution is examined, (v) The parametric dependence of the unit impulse upon the dimensionless parameter family is explored for two limits of practical interest. A simple expression for the principle relationship between the dimensionless parameter family is derived and provides insight into the interaction between soma and cylinders. A well-posed method for the solution of the dimensional inverse problem is presented.