A continuous cable method for determining the transient potential in passive dendritic trees of known geometry

Models using cable equations are increasingly employed in neurophysiological analyses, but the amount of computer time and memory required for their implementation are prohibitively large for many purposes and many laboratories. A mathematical procedure for determining the transient voltage response to injected current or synaptic input in a passive dendritic tree of known geometry is presented that is simple to implement since it is based on one equation. It proved to be highly accurate when results were compared to those obtained analytically for dendritic trees satisfying equivalent cylinder constraints. In this method the passive cable equation is used to express the potential for each interbranch segment of the dendritic tree. After applying boundary conditions at branch points and terminations, a system of equations for the Laplace transform of the potential at the ends of the segments can be readily obtained by inspection of the dendritic tree. Except for the starting equation, all of the equations have a simple format that varies only with the number of branches meeting at a branch point. The system of equations is solved in the Laplace domain, and the result is numerically inverted back to the time domain for each specified time point (the method is independent of any time increment t). The potential at any selected location in the dendritic tree can be obtained using this method. Since only one equation is required for each interbranch segment, this procedure uses far fewer equations than comparable compartmental approaches. By using significantly less computer memory and time than other methods to attain similar accuracy, this method permits extensive analyses to be performed rapidly on small computers. One hopes that this will involve more investigators in modeling studies and will facilitate their motivation to undertake realistically complex dendritic models.     

A deconvolution scheme

A deconvolution scheme is presented for the calculation of an input function given a system and its response—by converting the integral equation into an initial value problem. This technique is particularly simple and advantageous when dealing with compartmental systems and solving them numerically. The method also permits the imbedding of constraints and the estimation of confidence limits for the deconvoluted function.     

Beauty and the yeast: compartmental organization of the secretory pathway

Our perception of intracellular organelles and cellular architecture was initially based on striking light and electron micrographs of animal and plant cells. The high degree of compartmental organization within specalized mammalian secretory cells aided early efforts to track the movement of proteins through the organelles of the secretory pathway. In contrast, the morphological detail of the yeast Saccharomyces cerevisiae appeared superficially simple, even primitive, by comparison with the higher eukaryotic cells. However, the combination of genetic tools and the development of assays reconstituting vesicular traffic in yeast have facilitated the identification and characterization of individual proteins that function in the secretory pathway. Analogies between the function of yeast and mammalian proteins in vesicular traffic are being drawn with increasing frequency. In this review, the combination of genetic, biochemical, molecular and cell biological approaches used to study compartmental organization in the yeast secretory pathway will be discussed. The rapid progress in our understanding of yeast membrane traffic has revealed the beauty of working with this organism.     

Role of an A-Type K+ Conductance in the Back-Propagation of Action Potentials in the Dendrites of Hippocampal Pyramidal Neurons

Action potentials elicited in the axon actively back-propagate into the dendritic tree. During this process their amplitudes can be modulated by internal and external factors. We used a compartmental model of a hippocampal CA1 pyramidal neuron to illustrate how this modulation could depend on (1) the properties of an A-type K⁺ conductance that is expressed at high density in hippocampal dendrites and (2) the relative timing of synaptic activation. The simulations suggest that the time relationship between pre- and postsynaptic activity could help regulate the amplitude of back-propagating action potentials, especially in the distal portion of the dendritic tree.     

Whole tree xylem sap flow responses to multiple environmental variables in a wet tropical forest

In order to quantify and characterize the variance in rainforest tree physiology, whole tree sap flow responses to local environmental conditions were investigated in 10 species of trees with diverse traits at La Selva Biological Station, Costa Rica. A simple model was developed to predict tree sap flow responses to a synthetic environmental variable generated by a principle components analysis. The best fit was obtained with a sigmoid function which explained between 74 and 93% of the variation in sap flux of individual trees. Sap flow reached an asymptote where higher light and evaporative demand did not cause sap flux to increase further. Soil moisture had little influence on sap flux. The morphological characteristics of the trees significantly affected sap flow; taller trees responded to changes in environmental variables sooner than shorter trees and high liana cover buffered tree sap flow responses to weather. The effect of species‐specific differences on the model was small; the mean effectiveness of the model was reduced by 6% when parameters were estimated from a single pool of measurements taken from all individuals. The results indicate that sap flow response could be effectively estimated using a simple general model and composite environmental index for these 10 diverse tree species.     

A simple tree swaying model for forest motion in windstorm conditions

A simple tree swaying model, valid for windstorm conditions, has been developed for the purpose of simulating the effect of strong wind on the vulnerability of heterogeneous forest canopies. In this model the tree is represented as a flexible cantilever beam whose motion, induced by turbulent winds, is solved through a modal analysis. The geometric nonlinearities related to the tree curvature are accounted for through the formulation of the wind drag force. Furthermore, a breakage condition is considered at very large deflections. A variety of case studies is used to evaluate the present model. As compared to field data collected on three different tree species, and to the outputs of mechanistic models of wind damage, it appears to be able to predict accurately large tree deflections as well as tree breakage, using wind velocity at tree top as a forcing function. The instantaneous response of the modelled tree to a turbulent wind load shows very good agreement with a more complex tree model. The simplicity of the present model and its low computational time make it well adapted to future use in large-eddy simulation airflow models, aimed at simulating the complete interaction between turbulent wind fields and tree motion in fragmented forests.     

A phase transition for a random cluster model on phylogenetic trees

We investigate a simple model that generates random partitions of the leaf set of a tree. Of particular interest is the reconstruction question: what number k of independent samples (partitions) are required to correctly reconstruct the underlying tree (with high probability)? We demonstrate a phase transition for k as a function of the mutation rate, from logarithmic to polynomial dependence on the size of the tree. We also describe a simple polynomial-time tree reconstruction algorithm that applies in the logarithmic region. This model and the associated reconstruction questions are motivated by a Markov model for genomic evolution in molecular biology.     

Syntheses of spinal cord field potentials in the terrapin

A compartmental model of a terrapin motoneuron has been set up to compute membrane potential variations associated with synaptic input at different locations or with antidromic invasion. Membrane potential distributions obtained in that way were used to compute field potentials by means of a volume conduction formalism. The model was used to simulated field potentials measured in the spinal cord in response to stimulation of a muscle nerve with the intention to discriminate between different activation hypothesis for the generation of the spinal cord potential. Extracellular potentials calculated with an excitatory input distributed over the whole dorsal dendritic tree were found to give better reconstruction when compared with excitation restricted to the distal part of the dorsal dendrites, or with somatic inhibition.     

Direct response of tree growth to soil water and its implications for terrestrial carbon cycle modelling

Wood growth constitutes the main process for long‐term atmospheric carbon sequestration in vegetation. However, our understanding of the process of wood growth and its response to environmental drivers is limited. Current dynamic global vegetation models (DGVMs) are mainly photosynthesis‐driven and thus do not explicitly include a direct environmental effect on tree growth. However, physiological evidence suggests that, to realistically model vegetation carbon allocation under increased climatic stressors, it is crucial to treat growth responses independently from photosynthesis. A plausible growth response function suitable for global simulations in DGVMs has been lacking. Here, we present the first soil water‐growth response function and parameter range for deciduous and evergreen conifers. The response curve was calibrated against European larch and Norway spruce in a dry temperate forest in the Swiss Alps. We present a new data‐driven approach based on a combination of tree ring width (TRW) records, growing season length and simulated subdaily soil hydrology to parameterize ring width increment simulations. We found that a simple linear response function, with an intercept at zero moisture stress, used in growth simulations reproduced 62.3% and 59.4% of observed TRW variability for larch and spruce respectively and, importantly, the response function slope was much steeper than literature values for soil moisture effects on photosynthesis and stomatal conductance. Specifically, we found stem growth stops at soil moisture potentials of ?0.47 MPa for larch and ?0.66 MPa for spruce, whereas photosynthesis in trees continues down to ?1.2 MPa or lower, depending on species and measurement method. These results are strong evidence that the response functions of source and sink processes are indeed very different in trees, and need to be considered separately to correctly assess vegetation responses to environmental change. The results provide a parameterization for the explicit representation of growth responses to soil water in vegetation models.     

A dynamic model of angiotensin II infusion experiments

Applications of control theory in studies of biological system dynamics have come to be called compartmental modelling. A second order, nonlinear, compartmental model is developed which describes the dynamics of the hormone angiotensin II (AII) and arterial blood pressure (BP) during AII infusion experiments. The model is partially identified using dose response data for constant infusion rates between 0.01 and 0.10 μg/kg/min over a period of several minutes. This study represents a first step in understanding the dynamics of regulation of arterial blood pressure by the renin-angiotensin system. All is a vasoconstrictor and is known to participate in the natural regulation of BP. AII is also believed to be an agent in the development of hypertension and atherosclerosis. The model is used to identify causal mechanisms which are consistent both with the established correlation between plasma AII concentration and arterial BP and with current physiological knowledge. The study demonstrates how a simple state variable model can be used to provide guidance concerning the design of future infusion experiments.     

Membrane voltage changes in a compartmental chain model of a neurone

A mathematical model of the neurone has been developed using the method of subdivision of the neurone into a number of equivalent circuit compartments. Compartmental characteristics have been investigated by calculating the shape indices of the output produced in response to a given somatic input conductance change. A generalised form of compartmental chain has been chosen to allow calculation of the shape indices produced by a variety of geometrical configurations including the straight and tapering chain forms. Equations have been deduced from the computations made on a CDC 6600 computer relating the peak amplitude of the output response to the compartmental diameter for both the straight and tapering chain forms. The effect of variation in the location of the input conductance injection site has also been related to the peak amplitude of the somatic response. The optimum characteristics of the input conductance pulse shape have been computed initially using a rectangular pulse and later the more physiologically relevant double exponential shape. The effect of alteration in the end compartmental terminal impedances over the range from open to short circuit conditions was also calculated. The establishment of optimum single compartmental chain criteria allows the future investigation of multiple chain and pyramidal cell configurations.     

Measuring tissue pressure to assess the severity of burn-induced ischemia.

Circumferential burn injuries can produce severe ischemia in the underlying muscles by producing compartmental compression. The latter requires prompt decompression, but it is often difficult to diagnose. A simple, rapid, and accurate technique for the direct measurement of compartmental pressure is described. It may be used to determine the need for decompressive escharotomy or fasciotomy, and to intraoperatively assess its adequacy.     

Properties of the transfer functions of compartmental models

This paper is concerned with the nonlinear system of algebraic equations relating the positive parameters of a linear time-invariant compartmental model to its transfer function coefficients. The general form that these equations must take is shown, and simple necessary conditions for the existence of positive solutions are given. An immediate use of these conditions is the development of necessary conditions for a polynomial with positive coefficients to have negative roots. A method is then outlined which triangularizes the system and reduces the complete solution problem to one of finding and counting roots of a polynomial. Sufficient conditions for the existence of real and positive solutions are demonstrated.     

On the identification by filtering techniques of a biologicaln-compartment model in which the transport rate parameters are assumed to be stochastic processes

In this paper biological compartmental models are considered which take into account the intrinsic randomness of the transport rate parameters and their possible variability in time. An identification procedure is presented for the estimation of the stochastic processes representing the transport rate parameters of a compartmental model from a noisy input-output experiment. The problem is formulated in terms of nonlinear filtering. A simple model is discussed for the case in which the transport rate parameters are independent of each other. The possibility of testing possible important features of the behavior of the transport rate parameters is also evidenced.     

Qualitative theory of compartmental systems with lags

Dynamic models of many processes in the biological and physical sciences give systems of ordinary differential equations called compartmental systems. Often, these systems include time lags; in this context, continuous probability density functions (pdfs) of lags are far more important than discrete lags. There is a relatively complete theory of compartmental systems without lags, both linear and non-linear [SIAM Rev. 35 (1993) 43]. The authors extend their previous work on compartmental systems without lags to show that, for discrete lags and for a very large class of pdfs of continuous lags, compartmental systems with lags are equivalent to larger compartmental systems without lags. Consequently, the properties of compartmental systems with lags are the same as those of compartmental systems without lags. For a very large class of compartmental systems with time lags, one can show that the time lags themselves can be generated by compartmental systems without lags. Thus, such systems can be partitioned into a main system, which is the original system without the lags, plus compartmental subsystems without lags that generate the lags. The latter may be linear or non-linear and may be inserted into main systems that are linear or non-linear. The state variables of the compartmental lag subsystems are hidden variables in the formulation with explicit lags.     

Pyramidal neuron as two-layer neural network

The pyramidal neuron is the principal cell type in the mammalian forebrain, but its function remains poorly understood. Using a detailed compartmental model of a hippocampal CA1 pyramidal cell, we recorded responses to complex stimuli consisting of dozens of high-frequency activated synapses distributed throughout the apical dendrites. We found the cell's firing rate could be predicted by a simple formula that maps the physical components of the cell onto those of an abstract two-layer "neural network." In the first layer, synaptic inputs drive independent sigmoidal subunits corresponding to the cell's several dozen long, thin terminal dendrites. The subunit outputs are then summed within the main trunk and cell body prior to final thresholding. We conclude that insofar as the neural code is mediated by average firing rate, a two-layer neural network may provide a useful abstraction for the computing function of the individual pyramidal neuron.     

A compartmental model of iron regulation in the mouse

A simple compartmental model is developed for investigating the mechanism of iron homeostasis. In contrast to previous mathematical models of iron metabolism, the liver is included as a key site of iron regulation. Compartments for free iron in blood, diferric transferrin (Tf) in blood, hepatocytes, red blood cells, and macrophages are included, and their roles in iron regulation are explored. The function of hepcidin in regulating iron absorption is modeled through an inverse relationship between hepatocyte transferrin receptor 2 (TfR2) levels and the rate of iron export processes mediated by ferroportin (Fpn). Simulations of anemia and erythropoiesis stimulation support the idea that the iron demands of the erythroid compartment can be communicated through diferric Tf. The iron-responsive element of Fpn is found to be important for stabilizing intracellular iron stores in response to changing iron demands and allowing proper iron regulation through diferric Tf. The contribution of iron dysregulation to the pathogenesis of iron overload disorders is also investigated. It is shown that the characteristics of HFE hemochromatosis can be reproduced by increasing the setpoint of iron absorption in the duodenum to a level where the system cannot downregulate iron absorption to meet the iron excretion rate.     

Modeling the attenuation and failure of action potentials in the dendrites of hippocampal neurons.

We modeled two different mechanisms, a shunting conductance and a slow sodium inactivation, to test whether they could modulate the active propagation of a train of action potentials in a dendritic tree. Computer simulations, using a compartmental model of a pyramidal neuron, suggest that each of these two mechanisms could account for the activity-dependent attenuation and failure of the action potentials in the dendrites during the train. Each mechanism is shown to be in good qualitative agreement with experimental findings on somatic or dendritic stimulation and on the effects of hyperpolarization. The conditions under which branch point failures can be observed, and a few experimentally testable predictions, are presented and discussed.     

Optimal pharmacokinetic delivery of infused drugs: application to the treatment of cardiac arrhythmias

A common clinical goal of infusing drugs is to attain therapeutic steady-state concentrations as rapidly as possible. The desire is to closely approximate a step function in plasma concentration of the therapeutic agent. We have developed a novel approach to achieve this goal by using the principles of systems and compartmental analyses. The approach is to build a pharmacokinetic model for the disposition of the drug and then calculate backwards from the desired output function to derive the optimal input infusion function. We applied this technique to the infusion of Lidocaine, an antiarrhythmic agent which is often difficult to control. An optimal infusion function, used to drive a servo-controlled infusion pump, was derived to closely approximate a step-function response of drug levels. The efficacy of this infusion function was verified experimentally in dogs.     

Localization and interaction of N-methyl-D-aspartate and non-N-methyl-D-aspartate receptors of lamprey spinal neurons.

Small volumes of N-Methyl-D-Aspartate (NMDA) and non-NMDA excitatory amino acid receptor agonists were applied to localized regions of the dendritic trees of lamprey spinal neurons along their medial-lateral axis to obtain a spatial map of glutamate receptor distribution. Voltage clamp and frequency domain methods were used to obtain quantitative kinetic data of the voltage dependent ionic channels located both on the soma and on highly branched dendritic membranes. Pressure pulses of NMDA applied to the most peripheral regions of the dendritic tree elicited large somatic impedance increases, indicating that the most peripheral dendrites are well supplied with NMDA receptors. Experiments done with kainate did not elicit somatic responses to agonist applications on peripheral dendrites. The data obtained are consistent with the hypothesis that the activation of NMDA receptors by exogenous glutamate is significantly modified by the simultaneous activation of non-NMDA receptors, which shunts the NMDA response. The non-NMDA shunting hypothesis was tested by a combined application of kainate and NMDA to mimic the action of glutamate showing that the shunting effect of non-NMDA receptor activation virtually abolished the marked voltage dependency typical of NMDA receptor activation. These data were interpreted with a compartmental neuronal model having both NMDA and non-NMDA receptors.