The Green’s function formalism as a bridge between single- and multi-compartmental modeling

Neurons are spatially extended structures that receive and process inputs on their dendrites. It is generally accepted that neuronal computations arise from the active integration of synaptic inputs along a dendrite between the input location and the location of spike generation in the axon initial segment. However, many application such as simulations of brain networks use point-neurons—neurons without a morphological component—as computational units to keep the conceptual complexity and computational costs low. Inevitably, these applications thus omit a fundamental property of neuronal computation. In this work, we present an approach to model an artificial synapse that mimics dendritic processing without the need to explicitly simulate dendritic dynamics. The model synapse employs an analytic solution for the cable equation to compute the neuron’s membrane potential following dendritic inputs. Green’s function formalism is used to derive the closed version of the cable equation. We show that by using this synapse model, point-neurons can achieve results that were previously limited to the realms of multi-compartmental models. Moreover, a computational advantage is achieved when only a small number of simulated synapses impinge on a morphologically elaborate neuron. Opportunities and limitations are discussed.     

真菌深层培养过程的房室结构神经网络模型

在对横纹黑蛋巢菌深层培养过程进行分析的基础上,提出一种房室结构的神经网络模型,利用ＲＢＦ网络这各房室的输入,输出关系,并进一步对整个生化过程作了建模型研究,计算结果表明,所建模型性能较佳,对真菌培养过程的观测数据拟合结果令人满意。     

Mechanisms of magnetic stimulation of central nervous system neurons

Transcranial magnetic stimulation (TMS) is a stimulation method in which a magnetic coil generates a magnetic field in an area of interest in the brain. This magnetic field induces an electric field that modulates neuronal activity. The spatial distribution of the induced electric field is determined by the geometry and location of the coil relative to the brain. Although TMS has been used for several decades, the biophysical basis underlying the stimulation of neurons in the central nervous system (CNS) is still unknown. To address this problem we developed a numerical scheme enabling us to combine realistic magnetic stimulation (MS) with compartmental modeling of neurons with arbitrary morphology. The induced electric field for each location in space was combined with standard compartmental modeling software to calculate the membrane current generated by the electromagnetic field for each segment of the neuron. In agreement with previous studies, the simulations suggested that peripheral axons were excited by the spatial gradients of the induced electric field. In both peripheral and central neurons, MS amplitude required for action potential generation was inversely proportional to the square of the diameter of the stimulated compartment. Due to the importance of the fiber's diameter, magnetic stimulation of CNS neurons depolarized the soma followed by initiation of an action potential in the initial segment of the axon. Passive dendrites affect this process primarily as current sinks, not sources. The simulations predict that neurons with low current threshold are more susceptible to magnetic stimulation. Moreover, they suggest that MS does not directly trigger dendritic regenerative mechanisms. These insights into the mechanism of MS may be relevant for the design of multi-intensity TMS protocols, may facilitate the construction of magnetic stimulators, and may aid the interpretation of results of TMS of the CNS.     

A Stochastic Compartmental Model with Long Lasting Infusion

Most of the compartmental models in current use to model pharmacokinetic systems are deterministic. Stochastic formulations of pharmacokinetic compartmental models introduce stochasticity through either a probabilistic transfer mechanism or the randomization of the transfer rate constants. In this paper we consider a linear stochastic differential equation (LSDE) which represents a stochastic version of a one‐compartment linear model when input function undergoes random fluctuations. The solution of the LSDE, its mean value and covariance structure are derived. An explicit likelihood function is obtained either when the process is observed continuously over a period of time or when sampled data are available, as it is generally feasible. We discuss some asymptotic properties of the maximum likelihood estimators for the model parameters. Furthermore we develop expressions for two random variables of interest in pharmacokinetics: the area under the time‐concentration curve, M₀(T), and the plateau concentration, x_ss. Finally the estimation procedure is illustrated by an application to real data.     

A class of general stochastic compartmental systems

In this paper a general class of semi-Markov compartmental systems is studied. Two models for different input processes are analysed. Attention has been paid to the recurrence times associated with each compartment and to the distribution of the number of particles in each compartment. As an example, a three-compartment system is discussed to study the movement between three health states of patients with chronic diseases.     

An investigation into the influence of boundary condition specification in finite difference methods on the behaviour of passive and active neuronal models

Neuronal models provide a major aid to understanding the behaviour of individual neurons and networks of neurons. The solution of the model equations by finite difference methods is widespread because of the inherent simplicity of the technique. Error in the finite difference approach due to spatial and temporal discretisation is shown to be equivalent to a mis-specification of membrane current density. The effect of this mis-specification on the accuracy of the solution to the model equations is shown to depend on the structure of the model and its input, as well as the size of the discretisation intervals themselves. Through a theoretical analysis, illustrated by a number of examples on passive and active dendrites, this article demonstrates that the accuracy with which core current is implemented numerically at segment end-points in elementary models influences the behaviour of the numerical solution of these models, and consequently any physiological conclusions drawn from them.     

Ornstein-Uhlenbeck model neuron revisited

 The temporal patterns of action potentials fired by a two-point stochastic neuron model were investigated. In this model the membrane potential of the dendritic compartment follows the Orstein-Uhlenbeck process and is not affected by the spiking activity. The axonal compartment, corresponding to the spike initiation site, is described by a simplified RC circuit. Estimators of the mean and variance of the input, based on a sampling of the axonal membrane potential, were derived and applied to simulated data. The dependencies of the mean firing frequency and of the coefficient of variation and serial correlation of interspike intervals on the mean and variance of the input were also studied by computer simulation in both 1- and 2-point models. The main property distinguishing the 2-point model from the classical 1-point model is its ability to produce clusters of short (or long) intervals between spikes under conditions of constant stimulation, as often observed in real neurons. It is shown that the nearly linear frequency response of the neuron, starting with subthreshold values of the input, is accounted for by the variability of the input (noise), which indicates that noise can play a positive role in nervous systems. The linear response frequency with respect to noise of the models suggests that the neuron can function as a noise encoder. Received: 2 April 1993/Accepted in revised form: 15 September 1994     

Capillaries within compartments: microvascular interpretation of dynamic positron emission tomography data

Measurement of exchange of substances between blood and tissue has been a long-lasting challenge to physiologists, and considerable theoretical and experimental accomplishments were achieved before the development of the positron emission tomography (PET). Today, when modeling data from modern PET scanners, little use is made of earlier microvascular research in the compartmental models, which have become the standard model by which the vast majority of dynamic PET data are analysed. However, modern PET scanners provide data with a sufficient temporal resolution and good counting statistics to allow estimation of parameters in models with more physiological realism. We explore the standard compartmental model and find that incorporation of blood flow leads to paradoxes, such as kinetic rate constants being time-dependent, and tracers being cleared from a capillary faster than they can be supplied by blood flow. The inability of the standard model to incorporate blood flow consequently raises a need for models that include more physiology, and we develop microvascular models which remove the inconsistencies. The microvascular models can be regarded as a revision of the input function. Whereas the standard model uses the organ inlet concentration as the concentration throughout the vascular compartment, we consider models that make use of spatial averaging of the concentrations in the capillary volume, which is what the PET scanner actually registers. The microvascular models are developed for both single- and multi-capillary systems and include effects of non-exchanging vessels. They are suitable for analysing dynamic PET data from any capillary bed using either intravascular or diffusible tracers, in terms of physiological parameters which include regional blood flow.     

Structural identifiability of compartmental systems with respect to data obtained from constant-infusion tracer experiments

The problem of structural identifiability of compartmental systems receiving constant input rates of tracer material is studied, and the relationship between this steady-state problem and that of identification using the impulse response is sought. Input connectability of the compartmental system allows exogenous inputs to produce arbitrary steady-state values anywhere in state space, resulting in sufficient conditions for the structural identifiability of the system when direct measurements can be made for every compartment. Because of the steady-state nature of the problem, the systems concept of output connectability is shown to play no role in this identification scheme. The importance of constant-infusion tracer experiments is demonstrated for a compartment model describing volatile fatty acid production and conversion in ruminants.     

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.     

An integrated model of epidermal growth factor receptor trafficking and signal transduction

Endocytic trafficking of many types of receptors can have profound effects on subsequent signaling events. Quantitative models of these processes, however, have usually considered trafficking and signaling independently. Here, we present an integrated model of both the trafficking and signaling pathway of the epidermal growth factor receptor (EGFR) using a probability weighted-dynamic Monte Carlo simulation. Our model consists of hundreds of distinct endocytic compartments and approximately 13,000 reactions/events that occur over a broad spatio-temporal range. By using a realistic multicompartment model, we can investigate the distribution of the receptors among cellular compartments as well as their potential signal transduction characteristics. Our new model also allows the incorporation of physiochemical aspects of ligand-receptor interactions, such as pH-dependent binding in different endosomal compartments. To determine the utility of this approach, we simulated the differential activation of the EGFR by two of its ligands, epidermal growth factor (EGF) and transforming growth factor-alpha (TGF-alpha). Our simulations predict that when EGFR is activated with TGF-alpha, receptor activation is biased toward the cell surface whereas EGF produces a signaling bias toward the endosomal compartment. Experiments confirm these predictions from our model and simulations. Our model accurately predicts the kinetics and extent of receptor downregulation induced by either EGF or TGF-alpha. Our results suggest that receptor trafficking controls the compartmental bias of signal transduction, rather than simply modulating signal magnitude. Our model provides a new approach to evaluating the complex effect of receptor trafficking on signal transduction. Importantly, the stochastic and compartmental nature of the simulation allows these models to be directly tested by high-throughput approaches, such as quantitative image analysis.     

A new geometric-based model to accurately estimate arm and leg inertial estimates

Segment estimates of mass, center of mass and moment of inertia are required input parameters to analyze the forces and moments acting across the joints. The objectives of this study were to propose a new geometric model for limb segments, to evaluate it against criterion values obtained from DXA, and to compare its performance to five other popular models. Twenty five female and 24 male college students participated in the study. For the criterion measures, the participants underwent a whole body DXA scan, and estimates for segment mass, center of mass location, and moment of inertia (frontal plane) were directly computed from the DXA mass units. For the new model, the volume was determined from two standing frontal and sagittal photographs. Each segment was modeled as a stack of slices, the sections of which were ellipses if they are not adjoining another segment and sectioned ellipses if they were adjoining another segment (e.g. upper arm and trunk). Length of axes of the ellipses was obtained from the photographs. In addition, a sex-specific, non-uniform density function was developed for each segment. A series of anthropometric measurements were also taken by directly following the definitions provided of the different body segment models tested, and the same parameters determined for each model. Comparison of models showed that estimates from the new model were consistently closer to the DXA criterion than those from the other models, with an error of less than 5% for mass and moment of inertia and less than about 6% for center of mass location.     

Compartmental models with restricted movement

A restriction is imposed on the number of particles that can possibly move at any time from a compartment, so that any other particles present in the compartment must wait until such particles have moved out. The equations for such a system are formulated and the solution is given for a single compartment system; increased variability of the compartmental particle count is one effect of this restriction.     

A new class of biophysical models for predicting the probability of decompression sickness in scuba diving.

Interconnected compartmental models have been used for decades in physiology and medicine to account for the observed multi-exponential washout kinetics of a variety of solutes (including inert gases) both from single tissues and from the body as a whole. They are used here as the basis for a new class of biophysical probabilistic decompression models. These models are characterized by a relatively well-perfused, risk-bearing, central compartment and one or two non-risk-bearing, relatively poorly perfused, peripheral compartment(s). The peripheral compartments affect risk indirectly by diffusive exchange of dissolved inert gas with the central compartment. On the basis of the accuracy of their respective predictions beyond the calibration regime, the three-compartment interconnected models were found to be significantly better than the two-compartment interconnected models. The former, on the basis of a number of criteria, was also better than a two-compartment parallel model used for comparative purposes. In these latter comparisons, the models all had the same number of fitted parameters (four), were based on linear kinetics, had the same risk function, and were calibrated against the same dataset. The interconnected models predict that inert gas washout during decompression is relatively fast, initially, but slows rapidly with time compared with the more uniform washout rate predicted by an independent parallel compartment model. If empirically verified, this may have important implications for diving practice.     

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.     

On stochastic compartmental modeling

This communication contains a proof of the fact that the coefficient of variation of the contents of a compartment of a stochastic compartmental model with deterministic rate parameters is small for large populations. We can therefore conclude that the use of stochastic compartmental models is not of great consequence in the case of systems involving large populations when only the randomness of the transfer mechanism is considered.     

Modulatory effects of parallel fiber and molecular layer interneuron synaptic activity on purkinje cell responses to ascending segment input: a modeling study

Based on anatomical, physiological, and model-based studies, it has been proposed that synapses associated with the ascending segment of granule cell axons provide the principle excitatory drive on Purkinje cells which is then modulated by the more numerous parallel fiber synapses. In this study we have evaluated this idea using a detailed compartmental model of a cerebellar Purkinje cell by providing identical ascending segment synaptic inputs during different levels of random parallel fiber and molecular interneuron input. Results suggest that background inputs from parallel fibers and molecular layer interneurons can have a substantial effect on the response of Purkinje cells to ascending segment inputs. Interestingly, these effects are not reflected in the average firing rate of the Purkinje cell and are thus entirely dendritic in effect. These results are considered in the context of the known segregated spatial distribution of the parallel fibers and ascending segment synapses and a new hypothesis concerning the functional organization of cerebellar cortical circuitry.     

An environ analysis for nonlinear compartment models

Environ analysis, an input-output analysis for models of ecological systems, has been previously formulated for linear systems. This note has a twofold purpose: first, we indicate that a variation of parameters technique can be applied, at least in principle, to computeboth input and output environs; and second, we show that this technique may be used for computation of environs in nonautonomous, nonlinear compartment models. This nonlinear theory, obtained as a direct extension of dynamical system developments, allows the traditional environ partitioning of compartmental storages and flows. An example of a nonlinear nutrient-producer-consumer system whose output environs can be computed asymptotically is presented to illustrate these concepts. This research was supported by the U.S. Environmental Protection Agency under cooperative agreement R806727030.     

The relationship of intercompartmental excitation transfer rate constants to those of an underlying physical model

In studies on photosynthetic systems it is common practice to interpret the results of time-resolved fluorescence experiments on the basis of compartmental, or target, models. Each compartment represents a group of molecules with similar fluorescence characteristics. In cases of practical interest, the members of each compartment are spatially contiguous and make up part of an overall energy-transferring system. Since a rate constant describing the overall transfer between compartments is not that of any pair of molecules in the system, this question naturally rises: what do we learn about the microscopic structure from these data? In this note we introduce ‘compartment melting’, a smooth mathematical connection between the compartmental and microscopic levels. We then show, on the basis of model calculations on finite lattices in one, two, and three dimensions, that average microscopic rates at the interfaces between compartments may be estimated from observed intercompartmental rates. The estimate involves a modest number of structural assumptions about the system. As examples of the method, which is applicable mainly to systems containing homogeneous pigment pools, some recent chlorophyll-protein antenna studies are analyzed.     

Detecting autocatalytic dynamics in data modeled by a compartmental model

Modeling growth or reaction dynamics within a compartment in a compartmental model is often based on theoretical or first principle considerations. This approach is frequently applied due to the inability to observe or collect data directly from the compartment. When the internal dynamics are difficult to surmise, it is often the case that several competing models are constructed and compared in some way. In this paper, the dynamics which characterize the data of an autocatalytic process are used to describe a quantitative data analysis strategy to both recognize the presence of the autocatalytic process and to obtain some estimates of important parameters in the process. The compartmental model structure serves to communicate this dynamical information to the downstream compartments. This method has been applied to examine the dynamics of the engraftment of blood cells following hematopoietic stem cell transplantation in a clinical setting [Modeling the time to engraftment of white blood cells and platelets following autologous peripheral blood stem cell transplantation (2001)].