Exact epidemic models on graphs using graph-automorphism driven lumping

The dynamics of disease transmission strongly depends on the properties of the population contact network. Pair-approximation models and individual-based network simulation have been used extensively to model contact networks with non-trivial properties. In this paper, using a continuous time Markov chain, we start from the exact formulation of a simple epidemic model on an arbitrary contact network and rigorously derive and prove some known results that were previously mainly justified based on some biological hypotheses. The main result of the paper is the illustration of the link between graph automorphisms and the process of lumping whereby the number of equations in a system of linear differential equations can be significantly reduced. The main advantage of lumping is that the simplified lumped system is not an approximation of the original system but rather an exact version of this. For a special class of graphs, we show how the lumped system can be obtained by using graph automorphisms. Finally, we discuss the advantages and possible applications of exact epidemic models and lumping.     

The problem of time unit in Leslie’s population model

The conditions that will allow the lumping together of several age classes in the Leslie model are investigated. We show that if the lumping is to be valid for all population distributions, then the parameters of the model must be periodic. Lumping is valid when the population is in equilibrium, but equilibrium should be tested before the model is lumped.     

Neural activity states in different forms of physiological tremor. Facts and hypotheses

This paper attempts to combine (and thereby briefly review) various sets of physiological data in order to outline a qualitative model of the different states of stochastic neural activity underlying different forms of physiological tremor. Particular emphasis in put on spatial distributions of the properties of neural elements and their interconnexions, and on discharge characteristics of motor units and muscle spindle afferents including so-called early discharges and nonlinearities. It is argued that the wide variety of internal anatomical and functional structures of skeletal muscles and of their reflex organization must be considered when dealing with stability problems. Computer simulations of stochastic population models of the involved neuromuscular elements are advocated as means to investigate the relative importance of the many factors possibly contributing to stabilizing or de-stabilizing neuromuscular systems.     

A dynamical neural network model for motor cortical activity during movement: population coding of movement trajectories

As a dynamical model for motor cortical activity during hand movement we consider an artificial neural network that consists of extensively interconnected neuron-like units and performs the neuronal population vector operations. Local geometrical parameters of a desired curve are introduced into the network as an external input. The output of the model is a time-dependent direction and length of the neuronal population vector which is calculated as a sum of the activity of directionally tuned neurons in the ensemble. The main feature of the model is that dynamical behavior of the neuronal population vector is the result of connections between directionally tuned neurons rather than being imposed externally. The dynamics is governed by a system of coupled nonlinear differential equations. Connections between neurons are assigned in the simplest and most common way so as to fulfill basic requirements stemming from experimental findings concerning the directional tuning of individual neurons and the stabilization of the neuronal population vector, as well as from previous theoretical studies. The dynamical behavior of the model reveals a close similarity with the experimentally observed dynamics of the neuronal population vector. Specifically, in the framework of the model it is possible to describe a geometrical curve in terms of the time series of the population vector. A correlation between the dynamical behavior of the direction and the length of the population vector entails a dependence of the neural velocity on the curvature of the tracing trajectory that corresponds well to the experimentally measured covariation between tangential velocity and curvature in drawing tasks.On leave of absencefrom the Institute of Molecular Genetics, Russian Academy of Sciences, Moscow, Russia.     

Motor unit action potential field — Modelling results

The theoretical field of a motor unit (MU) action potential (MUP) was considered. It has been proved that in normal muscles the shape of a smooth threephasic MUP curve is determined mostly by the spatial distribution of MU muscle fibres. Phenomena called time dispersion are of prime importance in pathology, where they complicate normal threephasic MUP.Amplitudes and time parameters of model three-phasic MUP were analyzed as a function of the radial distance R from the geometrical centre of the motor unit territory (MUT) and approximated by mathematical expressions. It appeared that analysis of radial variability of MUP curve allows conclusions to be made about the MUT size and the spatial distribution of MU muscle fibres. These anatomical features of a MU are often changed in pathological muscles, thus the proposed methods of their evaluation could be helpful in diagnosis of neuromuscular diseases.     

Estimation of muscle spindle information rate by pattern matching and the effect of gamma system activity on parallel spindles

The information transmission properties of ensembles of MSs and the effect of the gamma system on these properties were studied. Three converging lines of research were taken: (1) the development of information theoretic estimation tools, and the formulation of an operational interpretation for the information rate; (2) animal experiments in which the mutual information rate was estimated and the effect of the gamma system was quantified; (3) simulation of a muscle spindle model with gamma activation in order to corroborate the results of the animal experiments. The main hypothesis was that the gamma system will enhance information theoretic measures that quantify the quality of the sensory neural channel comprised from an ensemble of primary muscle spindle afferents. A random stimulus was applied to a muscle in the hind limb of a cat, while spike trains from several primary MS afferents were recorded simultaneously. The stimulus was administered twice, with an operative and a disconnected gamma system. The mutual information rate between the stimulus and spike trains, as well as other information theoretic measures, was estimated. The information rate of ensembles of MSs increased with increasing ensemble size. However, with an operative gamma system the ensemble effect was much higher. In addition, the ensemble effect was influenced by the stimulus spectrum. A muscle spindle population model with gamma activation was simulated with stimuli that were identical to that of the animal experiments. The simulation results supported the experimental results and corroborated the main hypothesis. The results indicate that the gamma system has an important role in enhancing information transmission from ensembles of MSs to the spinal cord.     

The Influence of the Way the Muscle Force is Modeled on the Predicted Results Obtained by Solving Indeterminate Problems for a Fast Elbow Flexion

A critical point in models of the human limbs when the aim is to investigate the motor control is the muscle model. More often the mechanical output of a muscle is considered as one musculotendon force that is a design variable in optimization tasks solved predominantly by static optimization. For dynamic conditions, the relationship between the developed force, the length and the contraction velocity of a muscle becomes important and rheological muscle models can be incorporated in the optimization tasks. Here the muscle activation can be a design variable as well. Recently a new muscle model was proposed [22] Raikova R.T. Aladjov H.Ts. 2002 Hierarchical genetic algorithm versus static optimization–investigation of elbow flexion and extension movements Journal of Biomechanics 35 1123 1135  [Google Scholar]. A muscle is considered as a mixture of motor units (MUs) with different peculiarities and the muscle force is calculated as a sum of the MUs twitches. The aim of the paper is to compare these three ways for presenting the muscle force. Fast elbow flexion is investigated using a planar model with five muscles. It is concluded that the rheological models are suitable for calculation of the current maximal muscle forces that can be used as weight factors in the objective functions. The model based on MUs has many advantages for precise investigations of motor control. Such muscle presentation can explain the muscle co-contraction and the role of the fast and the slow MUs. The relationship between the MUs activation and the mechanical output is more clear and closer to the reality.     

Cybernetic models based on lumped elementary modes accurately predict strain‐specific metabolic function

In a recent article, Song and Ramkrishna (Song and Ramkrishna [2010]. Biotechnol Bioeng 106(2):271–284) proposed a lumped hybrid cybernetic model (L‐HCM) towards extracting maximum information about metabolic function from a minimum of data. This approach views the total uptake flux as distributed among lumped elementary modes (L‐EMs) so as to maximize a prescribed metabolic objective such as growth or uptake rate. L‐EM is computed as a weighted average of EMs where the weights are related to the yields of vital products (i.e., biomass and ATP). In this article, we further enhance the predictive power of L‐HCMs through modifications in lumping weights with additional parameters that can be tuned with data viewed to be critical. The resulting model is able to make predictions of diverse metabolic behaviors varying greatly with strain types as evidenced from case studies of anaerobic growth of various Escherichia coli strains. Incorporation of the new lumping formula into L‐HCM remarkably improves model predictions with a few critical data, thus presenting L‐HCM as a dynamic tool as being not only qualitatively correct but also quantitatively accurate. Biotechnol. Bioeng. 2011; 108:127–140. © 2010 Wiley Periodicals, Inc.     

Modeling motor cortical operations by an attractor network of stochastic neurons

Understanding the neural computations performed by the motor cortex requires biologically plausible models that account for cell discharge patterns revealed by neurophysiological recordings. In the present study the motor cortical activity underlying movement generation is modeled as the dynamic evolution of a large fully recurrent network of stochastic spiking neurons with noise superimposed on the synaptic transmission. We show that neural representations of the learned movement trajectories can be stored in the connectivity matrix in such a way that, when activated, a particular trajectory evolves in time as a dynamic attractor of the system while individual neurons fire irregularly with large variability in their interspike intervals. Moreover, the encoding of trajectories as attractors ensures high stability of the ensemble dynamics in the presence of synaptic noise. In agreement with neurophysiological findings, the suggested model can provide a wide repertoire of specific motor behaviors, whereas the number of specialized cells and specific connections may be negligibly small if compared with the whole population engaged in trajectory retrieving. To examine the applicability of the model we study quantitatively the relationship between local geometrical and kinematic characteristics of the trajectories generated by the network. The relationship obtained as a result of simulations is close to the 2/3 power law established by psychophysical and neurophysiological studies.     

Muscle Mitochondrial Uncoupling Dismantles Neuromuscular Junction and Triggers Distal Degeneration of Motor Neurons

Background

Amyotrophic lateral sclerosis (ALS), the most frequent adult onset motor neuron disease, is associated with hypermetabolism linked to defects in muscle mitochondrial energy metabolism such as ATP depletion and increased oxygen consumption. It remains unknown whether muscle abnormalities in energy metabolism are causally involved in the destruction of neuromuscular junction (NMJ) and subsequent motor neuron degeneration during ALS.

Methodology/Principal Findings

We studied transgenic mice with muscular overexpression of uncoupling protein 1 (UCP1), a potent mitochondrial uncoupler, as a model of muscle restricted hypermetabolism. These animals displayed age-dependent deterioration of the NMJ that correlated with progressive signs of denervation and a mild late-onset motor neuron pathology. NMJ regeneration and functional recovery were profoundly delayed following injury of the sciatic nerve and muscle mitochondrial uncoupling exacerbated the pathology of an ALS animal model.

Conclusions/Significance

These findings provide the proof of principle that a muscle restricted mitochondrial defect is sufficient to generate motor neuron degeneration and suggest that therapeutic strategies targeted at muscle metabolism might prove useful for motor neuron diseases.     

Motor Axon Pathfinding

Motor neurons are functionally related, but represent a diverse collection of cells that show strict preferences for specific axon pathways during embryonic development. In this article, we describe the ligands and receptors that guide motor axons as they extend toward their peripheral muscle targets. Motor neurons share similar guidance molecules with many other neuronal types, thus one challenge in the field of axon guidance has been to understand how the vast complexity of brain connections can be established with a relatively small number of factors. In the context of motor guidance, we highlight some of the temporal and spatial mechanisms used to optimize the fidelity of pathfinding and increase the functional diversity of the signaling proteins.Motor neurons residing in the brain stem and spinal cord extend axons into the periphery and are the final relay cells for locomotor commands. These cells are among the longest projection neurons in the body and their axons follow stereotypical pathways during embryogenesis to synapse with muscle and sympathetic/parasympathetic targets. Cellular studies of motor axon navigation in developing chick and zebrafish embryos have shown that motor neurons located at different rostrocaudal positions show specific preferences for axonal pathways (see Landmesser 2001; Lewis and Eisen 2003 for reviews). This early cellular research laid the foundation for molecular studies of motor axon guidance by establishing the concept that motor neurons are in fact a diverse cell population. The molecular studies covered in this article have sought to identify genetic differences between motor neurons and to characterize the signaling pathways that underlie the specificity of motor axon targeting.     

On the Evolution of Behavioral Heterogeneity in Individuals and Populations

A wide range of ecological and evolutionary models predict variety in phenotype or behavior when a population is at equilibrium. This heterogeneity can be realized in different ways. For example, it can be realized through a complex population of individuals exhibiting different simple behaviors, or through a simple population of individuals exhibiting complex, varying behaviors. In some theoretical frameworks these different realizations are treated as equivalent, but natural selection distinguishes between these two alternatives in subtle ways. By investigating an increasingly complex series of models, from a simple fluctuating selection model up to a finite population hawk/dove game, we explore the selective pressures which discriminate between pure strategists, mixed at the population level, and individual mixed strategists. Our analysis reveals some important limitations to the ESS framework often employed to investigate the evolution of complex behavior.     

The control of multi-muscle systems: human jaw and hyoid movements

A model is presented of sagittal plane jaw and hyoid motion based on the model of motor control. The model, which is implemented as a computer simulation, includes central neural control signals, position- and velocity-dependent reflexes, reflex delays, and muscle properties such as the dependence of force on muscle length and velocity. The model has seven muscles (or muscle groups) attached to the jaw and hyoid as well as separate jaw and hyoid bone dynamics. According to the model, movements result from changes in neurophysiological control variables which shift the equilibrium state of the motor system. One such control variable is an independent change in the membrane potential of -motoneurons (MNs); this variable establishes a threshold muscle length () at which MN recruitment begins. Motor functions may be specified by various combinations of s. One combination of s is associated with the level of coactivation of muscles. Others are associated with motions in specific degrees of freedom. Using the model, we study the mapping between control variables specified at the level of degrees of freedom and control variables corresponding to individual muscles. We demonstrate that commands can be defined involving linear combinations of change which produce essentially independent movements in each of the four kinematic degrees of freedom represented in the model (jaw orientation, jaw position, vertical and horizontal hyoid position). These linear combinations are represented by vectors in space which may be scaled in magnitude. The vector directions are constant over the jaw/hyoid workspace and result in essentially the same motion from any workspace position. The demonstration that it is not necessary to adjust control signals to produce the same movements in different parts of the workspace supports the idea that the nervous system need not take explicit account of musculo-skeletal geometry in planning movements.This article was processed by the author using the LAT_EX style file pljour2 from Springer-Verlag.     

Restoring motor function using optogenetics and neural engraftment

Schematic representation of a combinatorial closed-loop system to restore functionality to paralysed muscles, enabling control of specific motor functions. Briefly, intraneural grafts of optogenetically-engineered stem cell-derived motor neurons are placed closed to the motor-entry point of the target muscle, leading to its reinnervation (1) or, where intact, viral vectors are used to express opsins in endogenous motor axons. Next, a brain–machine interface embedded in the primary motor cortex (2), relays intended motor commands via an external neural decoding and processing unit (3), which then wirelessly transmits execution signals to an implanted optoelectronic stimulator (4) that activates the engrafted motor neurons (5) to induce muscle contraction (6). The implanted optoelectronic device can then relay feedback information from the muscle or nerve to the processing unit (not shown).

1. Download : Download high-res image (144KB)
2. Download : Download full-size image

     

A linear stochastic model of the single motor unit

The production of force and of the electrical signal by an active motor unit is theoretically described. Neural spikes are modelled using the Dirac delta function. Mechanisms for the generation of random impulse trains and the properties of the corresponding stochastic processes are discussed; the renewal model is proposed as the most appropriate. The possibility of using a linear model for the systems that produce force and electrical signal in the unit is examined. It is concluded that the linear assumption is justifiable during steady, constant-strength contractions of muscle. This linear stochastic model of the motor unit is used in two subsequent papers to study the muscle force and the electromyogram.     

An analog computer study of fast,isolated movements

The peripheral part of the motor control system is modelled on an analog computer. The model consists of an inertial load connected to an antagonistic muscle pair with their muscle spindles and neural connexions. It has inputs as well as inputs. The model is used for studying the hypothesis that simple on-off activities control fast isolated movements. It is found that the model responses on activation of the input alone are not realistic ones. A fair simulation may be obtained if the input and the input are activated simultaneously. Results suggest a diminished muscle spindle sensitivity during the movement.     

Evaluating model reduction under parameter uncertainty

Background

The dynamics of biochemical networks can be modelled by systems of ordinary differential equations. However, these networks are typically large and contain many parameters. Therefore model reduction procedures, such as lumping, sensitivity analysis and time-scale separation, are used to simplify models. Although there are many different model reduction procedures, the evaluation of reduced models is difficult and depends on the parameter values of the full model. There is a lack of a criteria for evaluating reduced models when the model parameters are uncertain.

Results

We developed a method to compare reduced models and select the model that results in similar dynamics and uncertainty as the original model. We simulated different parameter sets from the assumed parameter distributions. Then, we compared all reduced models for all parameter sets using cluster analysis. The clusters revealed which of the reduced models that were similar to the original model in dynamics and variability. This allowed us to select the smallest reduced model that best approximated the full model. Through examples we showed that when parameter uncertainty was large, the model should be reduced further and when parameter uncertainty was small, models should not be reduced much.

Conclusions

A method to compare different models under parameter uncertainty is developed. It can be applied to any model reduction method. We also showed that the amount of parameter uncertainty influences the choice of reduced models.

     

Kinetics of thermophilic,anaerobic oxidation of straight and branched chain butyrate and valerate

The degradation kinetics of normal and branched chain butyrate and valerate are important in protein-fed anaerobic systems, as a number of amino acids degrade to these organic acids. Including activated and primary wastewater sludge digesters, the majority of full-scale systems digest feeds with a significant or major fraction of COD as protein. This study assesses the validity of using a common kinetic parameter set and biological catalyst to represent butyrate, n-valerate, and i-valerate degradation in dynamic models. The i-valerate degradation stoichiometry in a continuous, mixed population system is also addressed, extending previous pure-culture and batch studies. A previously published mathematical model was modified to allow competitive uptake of i-valerate, and used to model a thermophilic manure digester operated over 180 days. The digester was periodically pulsed with straight and branched chain butyrate and valerate. Parameters were separately optimized to describe butyrate, i-valerate, and n-valerate degradation, as well as a lumped set optimized for all three substrates, and nonlinear, correlated parameter spaces estimated using an F distribution in the objective function (J). Each parameter set occupied mutually exclusive parameter spaces, indicating that all were statistically different from each other. However, qualitatively, the influence on model outputs was similar, and the lumped set would be reasonable for mixed acid digestion. The main characteristic not represented by Monod kinetics was a delay in i-valerate uptake, and was compensated for by a decreased maximum uptake rate (k(m)). Therefore, the kinetics need modification if fed predominantly i-valerate. Butyrate (i- and n-) and n-valerate could be modeled using stoichiometry consistent with beta-oxidation degradation pathways. However, i-valerate produced acetate only, supporting the stoichiometry of a reaction determined by other researchers in pure culture. Therefore, lumping i-valerate stoichiometry with that of n-valerate will not allow good system representation, especially when the feed consists of proteins high in leucine (which produces i-valerate), and the modified model structure and stoichiometry as proposed here should be used. This requires no additional kinetic parameters and one additional dynamic concentration state variable (i-valerate) in addition to the variables in the base model.     

The innervation pattern of crustacean skeletal muscle

Summary The innervation pattern of distal muscle fibers of the opener muscle of walking legs of crayfish (Astacus leptodactylus) was investigated using methylene-blue staining, cobalt infiltration, and electron microscopy. A quantitative analysis of the entire innervation of single muscle fibers was attempted.It was found that instead of the generally assumed parallel array of numerous excitatory and inhibitory terminals, innervation consists of only a few branched terminals. The branches of excitatory and inhibitory terminals lie side-by-side. Both types are characterized by numerous varicosities (see Fig. 9B). The aggregate length of excitatory as well as inhibitory terminals on one muscle fiber is, on the average, about 1,500 m with a total of 152 varicosities spaced about 10 m apart. The average diameter of the varicosities is 4.26 m, that of the connecting thin segments about 0.5 m. Total terminal surface of motor or inhibitory terminals amounts to about 10,000 m² per muscle fiber. There are approximately 2,000 motor synapses on each muscle fiber, but their average total area is only about 6% of the terminal membrane area, or 0.06% of the (idealized) muscle fiber surface.There are conspicuous differences in the postsynaptic specializations associated with excitatory and inhibitory terminals; these are described in detail.The results are discussed in a functional context and with regard to design and results of electrophysiological experiments.Supported by Sonderforschungsbereich 138 of the Deutsche Forschungsgemeinschaft