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1.  Dynamic Excitatory and Inhibitory Gain Modulation Can Produce Flexible, Robust and Optimal Decision-making 
PLoS Computational Biology  2013;9(6):e1003099.
Behavioural and neurophysiological studies in primates have increasingly shown the involvement of urgency signals during the temporal integration of sensory evidence in perceptual decision-making. Neuronal correlates of such signals have been found in the parietal cortex, and in separate studies, demonstrated attention-induced gain modulation of both excitatory and inhibitory neurons. Although previous computational models of decision-making have incorporated gain modulation, their abstract forms do not permit an understanding of the contribution of inhibitory gain modulation. Thus, the effects of co-modulating both excitatory and inhibitory neuronal gains on decision-making dynamics and behavioural performance remain unclear. In this work, we incorporate time-dependent co-modulation of the gains of both excitatory and inhibitory neurons into our previous biologically based decision circuit model. We base our computational study in the context of two classic motion-discrimination tasks performed in animals. Our model shows that by simultaneously increasing the gains of both excitatory and inhibitory neurons, a variety of the observed dynamic neuronal firing activities can be replicated. In particular, the model can exhibit winner-take-all decision-making behaviour with higher firing rates and within a significantly more robust model parameter range. It also exhibits short-tailed reaction time distributions even when operating near a dynamical bifurcation point. The model further shows that neuronal gain modulation can compensate for weaker recurrent excitation in a decision neural circuit, and support decision formation and storage. Higher neuronal gain is also suggested in the more cognitively demanding reaction time than in the fixed delay version of the task. Using the exact temporal delays from the animal experiments, fast recruitment of gain co-modulation is shown to maximize reward rate, with a timescale that is surprisingly near the experimentally fitted value. Our work provides insights into the simultaneous and rapid modulation of excitatory and inhibitory neuronal gains, which enables flexible, robust, and optimal decision-making.
Author Summary
Perceptual decision-making involves not only simple transformation of sensory information to a motor decision, but can also be modulated by high-level cognition. For example, the latter may include strategic allocation of limited attentional resources over time in a decision task to improve performance. At the neurophysiological level, there is evidence supporting attention-induced neuronal gain modulation of both excitatory and inhibitory cortical neurons. In the context of perceptual discrimination tasks performed by animals, we make use of a biologically inspired computational model of decision-making to understand the computational capabilities of such co-modulation of neuronal gains. We find that dynamic co-modulation of both excitatory and inhibitory neurons is important for flexible, and cognitively demanding decision-making while also enhancing robustness in the decision circuit's functions. Our model captures the neuronal activity and behavioural data in the animal experiments remarkably well. Decision performance in a reaction time task can be optimized, maximizing the rate of receiving reward by using fast gain recruitment. Our experimentally fitted timescale is near the optimal one, suggesting that the animals performed almost optimally. By providing both computational simulations and theoretical analyses, our computational model sheds light into the multiple functions of rapid co-modulation of neuronal gains during decision-making.
PMCID: PMC3694816  PMID: 23825935
2.  Modeling of brain metabolism and pyruvate compartmentation using 13C NMR in vivo: caution required 
Two variants of a widely used two-compartment model were prepared for fitting the time course of [1,6-13C2]glucose metabolism in rat brain. Features common to most models were included, but in one model the enrichment of the substrates entering the glia and neuronal citric acid cycles was allowed to differ. Furthermore, the models included the capacity to analyze multiplets arising from 13C spin-spin coupling, known to improve parameter estimates in heart. Data analyzed were from a literature report providing time courses of [1,6-13C2]glucose metabolism. Four analyses were used, two comparing the effect of different pyruvate enrichment in glia and neurons, and two for determining the effect of multiplets present in the data. When fit independently, the enrichment in glial pyruvate was less than in neurons. In the absence of multiplets, fit quality and parameter values were typical of those in the literature, whereas the multiplet curves were not modeled well. This prompted the use of robust statistical analysis (the Kolmogorov–Smirnov test of goodness of fit) to determine whether individual curves were modeled appropriately. At least 50% of the curves in each experiment were considered poorly fit. It was concluded that the model does not include all metabolic features required to analyze the data.
PMCID: PMC3734769  PMID: 23652627
cerebral energy metabolism; in vivo 13C NMR spectroscopy; mathematical modeling; rat; tricarboxylic acid cycle; [1,6-13C2]glucose
3.  A diffusion-based neurite length-sensing mechanism involved in neuronal symmetry breaking 
Shootin1, one of the earliest markers of neuronal symmetry breaking, accumulates in the neurite tips of polarizing neurons in a neurite length-dependent manner. Thus, neurons sense their neurites' length and translate this spatial information into a molecular signal, shootin1 concentration.Quantitative live cell imaging of shootin1 dynamics combined with mathematical modeling analyses reveals that its anterograde transport and retrograde diffusion in neurite shafts account for the neurite length-dependent accumulation of shootin1.The neurite length-dependent shootin1 accumulation and shootin1-induced neurite outgrowth constitute a positive feedback loop that amplifies stochastic shootin1 signals in neurite tips.Quantitative mathematical modeling shows that the above positive feedback loop, together with shootin1 upregulation, constitutes a core mechanism for neuronal symmetry breaking.
Cell morphology and size must be properly controlled to ensure cellular function. Although there has been significant progress in understanding the molecular signals that change cell morphology, the manner in which cells monitor their size and length to regulate their morphology is poorly understood. Cultured hippocampal neurons polarize by forming a single long axon and multiple short dendrites (Craig and Banker, 1994; Arimura and Kaibuchi, 2007), and symmetry breaking is the initial step of this process. This symmetry-breaking step reproduces even when the neuronal axon is transected; the longest neurite usually grows rapidly to become an axon after transection, regardless of whether it is the axonal stump or another neurite (Goslin and Banker, 1989). Elongation of an immature neurite by mechanical tension also leads to its axonal specification (Lamoureux et al, 2002). These results suggest that cultured hippocampal neurons can sense neurite length, identify the longest one, and induce its subsequent axonogenesis for symmetry breaking. However, little is known about the mechanism for this process.
Shootin1 is one of the earliest markers of neuronal symmetry breaking (Toriyama et al, 2006). During the symmetry-breaking step, it undergoes a stochastic accumulation in neurite tips, and eventually accumulates predominantly in a single neurite that subsequently grows to become an axon. In this study, we demonstrated that shootin1 accumulates in neurite tips in a neurite length-dependent manner, regardless of whether it is the axonal stump or another neurite (Figure 3A, C–F). Thus, morphological information (neurite length) is translated into a molecular signal (shootin1 concentration in neurite tips).
We previously reported that shootin1 is transported from the cell body to neurite tips as discrete boluses and diffuses back to the cell body (Toriyama et al, 2006). The boluses containing variable amounts of shootin1 traveled repeatedly but irregularly along neurites, and their arrival caused large stochastic fluctuations in shootin1 concentration in the neurite tips. To understand the mechanism of length-dependent shootin1 accumulation, we performed quantitative live cell imaging of the anterograde transport and retrograde diffusion of shootin1 and fitted the obtained data into mathematical models of the anterograde transport and retrograde diffusion. The parameters of these two models were derived entirely from quantitative experimental data, without any adjustment. Shootin1 concentration at neurite tips, calculated by integrating the two models, was neurite length dependent (Figure 3B) and showed good agreement with the experimental data (Figure 3A). These results suggest that the neurite length-dependent accumulation of shootin1 is quantitatively explained by its anterograde transport and retrograde diffusion.
This length-dependent shootin1 accumulation constitutes a positive feedback interaction with the previously reported shootin1-induced neurite outgrowth (Shimada et al, 2008). To analyze the functional role of this feedback loop, we quantified shootin1 upregulation (Toriyama et al, 2006) and shootin1-induced neurite outgrowth, and integrated them, together with the above model of length-dependent shootin1 accumulation, into a model neuron (Figure 7A). Furthermore, the parameters of the model components were chosen to give the best fit to the quantitative experimental data without any adjustment. Integrating the three components into a model neuron resulted in spontaneous symmetry breaking (Figure 7B and C). Furthermore, there are a total of 15 agreements between the model predictions and the experimental data, including the neurite length-dependent axon specification and regeneration (Goslin and Banker, 1989; Lamoureux et al, 2002). These data suggest that the three components in our model—namely, diffusion-based neurite length sensing system, shootin1-induced neurite outgrowth and shootin1 upregulation—are sufficient to induce neuronal symmetry breaking.
Bolus-like transport of shootin1 caused large stochastic fluctuations in shootin1 concentration in neurite tips. Interestingly, the generation of continuous shootin1 transport in our model neuron impaired the symmetry-breaking process (Figure 7D). This is consistent with theoretical models in which feedback amplification of fluctuations in signaling can give rise to robust patterns (Turing, 1952; Meinhardt and Gierer, 2000; Kondo, 2002), and underscores the importance of the stochastic fluctuating signals in spontaneous neuronal symmetry breaking.
The combination of quantitative experimentation and mathematical modeling is regarded as a powerful strategy for attaining a profound understanding of biological systems (Hodgkin and Huxley, 1952b; Lewis, 2008; Ferrell, 2009). By focusing on a simple system involving one of the earliest markers of neuronal symmetry breaking, shootin1, we were able to evaluate here the core components of neuronal symmetry breaking on the basis of quantitative experimental data. The present model may thus provide a core mechanism of neuronal symmetry breaking, to which other possible mechanisms can be added to increase the model's complexity in future studies.
Although there has been significant progress in understanding the molecular signals that change cell morphology, mechanisms that cells use to monitor their size and length to regulate their morphology remain elusive. Previous studies suggest that polarizing cultured hippocampal neurons can sense neurite length, identify the longest neurite, and induce its subsequent outgrowth for axonogenesis. We observed that shootin1, a key regulator of axon outgrowth and neuronal polarization, accumulates in neurite tips in a neurite length-dependent manner; here, the property of cell length is translated into shootin1 signals. Quantitative live cell imaging combined with modeling analyses revealed that intraneuritic anterograde transport and retrograde diffusion of shootin1 account for its neurite length-dependent accumulation. Our quantitative model further explains that the length-dependent shootin1 accumulation, together with shootin1-dependent neurite outgrowth, constitutes a positive feedback loop that amplifies stochastic fluctuations of shootin1 signals, thereby generating an asymmetric signal for axon specification and neuronal symmetry breaking.
PMCID: PMC2925530  PMID: 20664640
feedback loop; neuronal polarity; quantitative modeling; shootin1; stochasticity
4.  Inferring Nonlinear Neuronal Computation Based on Physiologically Plausible Inputs 
PLoS Computational Biology  2013;9(7):e1003143.
The computation represented by a sensory neuron's response to stimuli is constructed from an array of physiological processes both belonging to that neuron and inherited from its inputs. Although many of these physiological processes are known to be nonlinear, linear approximations are commonly used to describe the stimulus selectivity of sensory neurons (i.e., linear receptive fields). Here we present an approach for modeling sensory processing, termed the Nonlinear Input Model (NIM), which is based on the hypothesis that the dominant nonlinearities imposed by physiological mechanisms arise from rectification of a neuron's inputs. Incorporating such ‘upstream nonlinearities’ within the standard linear-nonlinear (LN) cascade modeling structure implicitly allows for the identification of multiple stimulus features driving a neuron's response, which become directly interpretable as either excitatory or inhibitory. Because its form is analogous to an integrate-and-fire neuron receiving excitatory and inhibitory inputs, model fitting can be guided by prior knowledge about the inputs to a given neuron, and elements of the resulting model can often result in specific physiological predictions. Furthermore, by providing an explicit probabilistic model with a relatively simple nonlinear structure, its parameters can be efficiently optimized and appropriately regularized. Parameter estimation is robust and efficient even with large numbers of model components and in the context of high-dimensional stimuli with complex statistical structure (e.g. natural stimuli). We describe detailed methods for estimating the model parameters, and illustrate the advantages of the NIM using a range of example sensory neurons in the visual and auditory systems. We thus present a modeling framework that can capture a broad range of nonlinear response functions while providing physiologically interpretable descriptions of neural computation.
Author Summary
Sensory neurons are capable of representing a wide array of computations on sensory stimuli. Such complex computations are thought to arise in large part from the accumulation of relatively simple nonlinear operations across the sensory processing hierarchies. However, models of sensory processing typically rely on mathematical approximations of the overall relationship between stimulus and response, such as linear or quadratic expansions, which can overlook critical elements of sensory computation and miss opportunities to reveal how the underlying inputs contribute to a neuron's response. Here we present a physiologically inspired nonlinear modeling framework, the ‘Nonlinear Input Model’ (NIM), which instead assumes that neuronal computation can be approximated as a sum of excitatory and suppressive ‘neuronal inputs’. We show that this structure is successful at explaining neuronal responses in a variety of sensory areas. Furthermore, model fitting can be guided by prior knowledge about the inputs to a given neuron, and its results can often suggest specific physiological predictions. We illustrate the advantages of the proposed model and demonstrate specific parameter estimation procedures using a range of example sensory neurons in both the visual and auditory systems.
PMCID: PMC3715434  PMID: 23874185
5.  Modeling Higher-Order Correlations within Cortical Microcolumns 
PLoS Computational Biology  2014;10(7):e1003684.
We statistically characterize the population spiking activity obtained from simultaneous recordings of neurons across all layers of a cortical microcolumn. Three types of models are compared: an Ising model which captures pairwise correlations between units, a Restricted Boltzmann Machine (RBM) which allows for modeling of higher-order correlations, and a semi-Restricted Boltzmann Machine which is a combination of Ising and RBM models. Model parameters were estimated in a fast and efficient manner using minimum probability flow, and log likelihoods were compared using annealed importance sampling. The higher-order models reveal localized activity patterns which reflect the laminar organization of neurons within a cortical column. The higher-order models also outperformed the Ising model in log-likelihood: On populations of 20 cells, the RBM had 10% higher log-likelihood (relative to an independent model) than a pairwise model, increasing to 45% gain in a larger network with 100 spatiotemporal elements, consisting of 10 neurons over 10 time steps. We further removed the need to model stimulus-induced correlations by incorporating a peri-stimulus time histogram term, in which case the higher order models continued to perform best. These results demonstrate the importance of higher-order interactions to describe the structure of correlated activity in cortical networks. Boltzmann Machines with hidden units provide a succinct and effective way to capture these dependencies without increasing the difficulty of model estimation and evaluation.
Author Summary
Communication between neurons underlies all perception and cognition. Hence, to understand how the brain's sensory systems such as the visual cortex work, we need to model how neurons encode and communicate information about the world. To this end, we simultaneously recorded the activity of many neurons in a cortical column, a fundamental building block of information processing in the brain. This allows us to discover statistical structure in their activity, a first step to uncovering communication pathways and coding principles. To capture the statistical structure of firing patterns, we fit models that assign a probability to each observed pattern. Fitting probability distributions is generally difficult because the model probabilities of all possible states have to sum to one, and enumerating all possible states in a large system is not possible. Making use of recent advances in parameter estimation, we are able to fit models and test the quality of the fit to the data. The resulting model parameters can be interpreted as the effective connectivity between groups of cells, thus revealing patterns of interaction between neurons in a cortical circuit.
PMCID: PMC4081002  PMID: 24991969
6.  Association of physical fitness with health-related quality of life in Finnish young men 
Currently, there is insufficient evidence available regarding the relationship between level of physical fitness and health-related quality of life (HRQoL) in younger adults. Therefore, the aim of the present study was to investigate the impact of measured cardiovascular and musculoskeletal physical fitness level on HRQoL in Finnish young men.
In a cross-sectional study, we collected data regarding the physical fitness index, including aerobic endurance and muscle fitness, leisure-time physical activity (LTPA), body composition, health, and HRQoL (RAND 36) for 727 men [mean (SD) age 25 (5) years]. Associations between HRQoL and the explanatory parameters were analyzed using the logistic regression analysis model.
Of the 727 participants who took part in the study, 45% were in the poor category of the physical fitness, while 37% and 18% were in the satisfactory and good fitness categories, respectively. A higher frequency of LTPA was associated with higher fitness (p < 0.001). Better HRQoL in terms of general health, physical functioning, mental health, and vitality were associated with better physical fitness. When the HRQoL of the study participants were compared with that of the age- and gender-weighted Finnish general population, both the good and satisfactory fitness groups had higher HRQoL in all areas other than bodily pain. In a regression analysis, higher LTPA was associated with three dimensions of HRQoL, higher physical fitness with two, and lower number of morbidities with all dimensions, while the effect of age was contradictory.
Our study of Finnish young men indicates that higher physical fitness and leisure-time physical activity level promotes certain dimensions of HRQoL, while morbidities impair them all. The results highlight the importance of health related physical fitness while promoting HRQoL.
PMCID: PMC2835678  PMID: 20109241
7.  Efficient context-dependent model building based on clustering posterior distributions for non-coding sequences 
Many recent studies that relax the assumption of independent evolution of sites have done so at the expense of a drastic increase in the number of substitution parameters. While additional parameters cannot be avoided to model context-dependent evolution, a large increase in model dimensionality is only justified when accompanied with careful model-building strategies that guard against overfitting. An increased dimensionality leads to increases in numerical computations of the models, increased convergence times in Bayesian Markov chain Monte Carlo algorithms and even more tedious Bayes Factor calculations.
We have developed two model-search algorithms which reduce the number of Bayes Factor calculations by clustering posterior densities to decide on the equality of substitution behavior in different contexts. The selected model's fit is evaluated using a Bayes Factor, which we calculate via model-switch thermodynamic integration. To reduce computation time and to increase the precision of this integration, we propose to split the calculations over different computers and to appropriately calibrate the individual runs. Using the proposed strategies, we find, in a dataset of primate Ancestral Repeats, that careful modeling of context-dependent evolution may increase model fit considerably and that the combination of a context-dependent model with the assumption of varying rates across sites offers even larger improvements in terms of model fit. Using a smaller nuclear SSU rRNA dataset, we show that context-dependence may only become detectable upon applying model-building strategies.
While context-dependent evolutionary models can increase the model fit over traditional independent evolutionary models, such complex models will often contain too many parameters. Justification for the added parameters is thus required so that only those parameters that model evolutionary processes previously unaccounted for are added to the evolutionary model. To obtain an optimal balance between the number of parameters in a context-dependent model and the performance in terms of model fit, we have designed two parameter-reduction strategies and we have shown that model fit can be greatly improved by reducing the number of parameters in a context-dependent evolutionary model.
PMCID: PMC2695821  PMID: 19405957
8.  Fitting Membrane Resistance along with Action Potential Shape in Cardiac Myocytes Improves Convergence: Application of a Multi-Objective Parallel Genetic Algorithm 
PLoS ONE  2014;9(9):e107984.
Fitting parameter sets of non-linear equations in cardiac single cell ionic models to reproduce experimental behavior is a time consuming process. The standard procedure is to adjust maximum channel conductances in ionic models to reproduce action potentials (APs) recorded in isolated cells. However, vastly different sets of parameters can produce similar APs. Furthermore, even with an excellent AP match in case of single cell, tissue behaviour may be very different. We hypothesize that this uncertainty can be reduced by additionally fitting membrane resistance (Rm). To investigate the importance of Rm, we developed a genetic algorithm approach which incorporated Rm data calculated at a few points in the cycle, in addition to AP morphology. Performance was compared to a genetic algorithm using only AP morphology data. The optimal parameter sets and goodness of fit as computed by the different methods were compared. First, we fit an ionic model to itself, starting from a random parameter set. Next, we fit the AP of one ionic model to that of another. Finally, we fit an ionic model to experimentally recorded rabbit action potentials. Adding the extra objective (Rm, at a few voltages) to the AP fit, lead to much better convergence. Typically, a smaller MSE (mean square error, defined as the average of the squared error between the target AP and AP that is to be fitted) was achieved in one fifth of the number of generations compared to using only AP data. Importantly, the variability in fit parameters was also greatly reduced, with many parameters showing an order of magnitude decrease in variability. Adding Rm to the objective function improves the robustness of fitting, better preserving tissue level behavior, and should be incorporated.
PMCID: PMC4176019  PMID: 25250956
9.  An Approximation to the Adaptive Exponential Integrate-and-Fire Neuron Model Allows Fast and Predictive Fitting to Physiological Data 
For large-scale network simulations, it is often desirable to have computationally tractable, yet in a defined sense still physiologically valid neuron models. In particular, these models should be able to reproduce physiological measurements, ideally in a predictive sense, and under different input regimes in which neurons may operate in vivo. Here we present an approach to parameter estimation for a simple spiking neuron model mainly based on standard f–I curves obtained from in vitro recordings. Such recordings are routinely obtained in standard protocols and assess a neuron’s response under a wide range of mean-input currents. Our fitting procedure makes use of closed-form expressions for the firing rate derived from an approximation to the adaptive exponential integrate-and-fire (AdEx) model. The resulting fitting process is simple and about two orders of magnitude faster compared to methods based on numerical integration of the differential equations. We probe this method on different cell types recorded from rodent prefrontal cortex. After fitting to the f–I current-clamp data, the model cells are tested on completely different sets of recordings obtained by fluctuating (“in vivo-like”) input currents. For a wide range of different input regimes, cell types, and cortical layers, the model could predict spike times on these test traces quite accurately within the bounds of physiological reliability, although no information from these distinct test sets was used for model fitting. Further analyses delineated some of the empirical factors constraining model fitting and the model’s generalization performance. An even simpler adaptive LIF neuron was also examined in this context. Hence, we have developed a “high-throughput” model fitting procedure which is simple and fast, with good prediction performance, and which relies only on firing rate information and standard physiological data widely and easily available.
PMCID: PMC3434419  PMID: 22973220
pyramidal cells; interneurons; f–I curve; adaptation; spike timing; temporal coding; prefrontal cortex
10.  Relationships among Attention Function, Exercise, and Body Mass Index: A Comparison between Young Breast Cancer Survivors and Acquaintance Controls 
Psycho-oncology  2014;24(3):325-332.
Although regular physical activity is associated with lower all-cause and disease-specific mortality among breast cancer survivors (BCS), most BCS do not meet its recommended guidelines. Attention function, a domain of cognition, is essential for daily tasks such as exercising, a form of planned physical activity. We tested the hypotheses that lower self-reported attention function in BCS would be associated with less exercise and higher body-mass index (BMI) by comparing a group of 505 young BCS (45 years or younger at diagnosis and 3–8 years post-treatment) to 466 acquaintance controls (AC).
The groups were compared on self-reported physical and psychological outcomes. Mplus software was used to perform confirmatory structural equation modeling (SEM) with a robust maximum likelihood estimator to evaluate hypothesized relationships among variables. The criteria for good model fit were: having RMSEA<.06, CFI>.95, and SRMR<.08. Modification indices were used to better fit the model.
The final model demonstrated good fit, with RMSEA=.05, CFI=.98, and SRMR=.03. After controlling for demographics, parameter estimates revealed that, compared to AC, young BCS reported worse attention function (p<.001), more depressive symptoms (p<.001), and more fatigue (p<.001). Controlling for fatigue, depression, and anxiety, better attention function was associated with a greater likelihood of exercise in the past 3 months (p=.039), which in turn was associated with a lower BMI (p<.001).
The significant association between attention function and physical activity, if confirmed in a longitudinal study, will provide new targets for interventions aimed at improving physical activity and decreasing BMI among BCS.
PMCID: PMC4269568  PMID: 24934396
11.  Adaptation to elastic loads and BMI robot controls during rat locomotion examined with point-process GLMs 
Currently little is known about how a mechanically coupled BMI system's actions are integrated into ongoing body dynamics. We tested a locomotor task augmented with a BMI system driving a robot mechanically interacting with a rat under three conditions: control locomotion (BL), “simple elastic load” (E) and “BMI with elastic load” (BMI/E). The effect of the BMI was to allow compensation of the elastic load as a function of the neural drive. Neurons recorded here were close to one another in cortex, all within a 200 micron diameter horizontal distance of one another. The interactions of these close assemblies of neurons may differ from those among neurons at longer distances in BMI tasks and thus are important to explore. A point process generalized linear model (GLM), was used to examine connectivity at two different binning timescales (1 ms vs. 10 ms). We used GLM models to fit non-Poisson neural dynamics solely using other neurons' prior neural activity as covariates. Models at different timescales were compared based on Kolmogorov-Smirnov (KS) goodness-of-fit and parsimony. About 15% of cells with non-Poisson firing were well fitted with the neuron-to-neuron models alone. More such cells were fitted at the 1 ms binning than 10 ms. Positive connection parameters (“excitation” ~70%) exceeded negative parameters (“inhibition” ~30%). Significant connectivity changes in the GLM determined networks of well-fitted neurons occurred between the conditions. However, a common core of connections comprising at least ~15% of connections persisted between any two of the three conditions. Significantly almost twice as many connections were in common between the two load conditions (~27%), compared to between either load condition and the baseline. This local point process GLM identified neural correlation structure and the changes seen across task conditions in the rats in this neural subset may be intrinsic to cortex or due to feedback and input reorganization in adaptation.
PMCID: PMC4411868  PMID: 25972789
augmenting BMI; elastic field; point-process general linear model; exoskeleton robot model system; motor adaptation
12.  Using Evolutionary Algorithms for Fitting High-Dimensional Models to Neuronal Data 
Neuroinformatics  2012;10(2):199-218.
In the study of neurosciences, and of complex biological systems in general, there is frequently a need to fit mathematical models with large numbers of parameters to highly complex datasets. Here we consider algorithms of two different classes, gradient following (GF) methods and evolutionary algorithms (EA) and examine their performance in fitting a 9-parameter model of a filter-based visual neuron to real data recorded from a sample of 107 neurons in macaque primary visual cortex (V1). Although the GF method converged very rapidly on a solution, it was highly susceptible to the effects of local minima in the error surface and produced relatively poor fits unless the initial estimates of the parameters were already very good. Conversely, although the EA required many more iterations of evaluating the model neuron’s response to a series of stimuli, it ultimately found better solutions in nearly all cases and its performance was independent of the starting parameters of the model. Thus, although the fitting process was lengthy in terms of processing time, the relative lack of human intervention in the evolutionary algorithm, and its ability ultimately to generate model fits that could be trusted as being close to optimal, made it far superior in this particular application than the gradient following methods. This is likely to be the case in many further complex systems, as are often found in neuroscience.
PMCID: PMC3272374  PMID: 22258828
Computational modelling; V1; Striate cortex; Filter-based; Model fitting; Optimisation methods; Evolutionary algorithms; Pareto optimality
13.  Using evolutionary algorithms for fitting high-dimensional models to neuronal data 
Neuroinformatics  2012;10(2):199-218.
In the study of neurosciences, and of complex biological systems in general, there is frequently a need to fit mathematical models with large numbers of parameters to highly complex datasets. Here we consider algorithms of two different classes, gradient following (GF) methods and evolutionary algorithms (EA) and examine their performance in fitting a 9-parameter model of a filter-based visual neuron to real data recorded from a sample of 107 neurons in macaque primary visual cortex (V1). Although the GF method converged very rapidly on a solution, it was highly susceptible to the effects of local minima in the error surface and produced relatively poor fits unless the initial estimates of the parameters were already very good. Conversely, although the EA required many more iterations of evaluating the model neuron’s response to a series of stimuli, it ultimately found better solutions in nearly all cases and its performance was independent of the starting parameters of the model. Thus, although the fitting process was lengthy in terms of processing time, the relative lack of human intervention in the evolutionary algorithm, and its ability ultimately to generate model fits that could be trusted as being close to optimal, made it far superior in this particular application than the gradient following methods. This is likely to be the case in many further complex systems, as are often found in neuroscience.
PMCID: PMC3272374  PMID: 22258828
14.  Modelling the Mechanical Properties of Single Suspension‐Cultured Tomato Cells 
Annals of Botany  2004;93(4):443-453.
• Background and Aims The relationship between composition and structure of plant primary cell walls, and cell mechanical properties is not fully understood, partly because intrinsic properties of walls such as Young’s modulus cannot be obtained readily. The aim of this work is to show that Young’s modulus of walls of single suspension‐cultured tomato cells can be determined by modelling force‐deformation data.
• Methods The model simulates the compression of a cell between two flat surfaces, with the cell treated as a liquid‐filled sphere with thin compressible walls. The cell wall and membrane were taken to be permeable, but the compression was so fast that water loss could be neglected in the simulations. Force‐deformation data were obtained by compressing the cells in micromanipulation experiments.
• Key Results Good fits were obtained between the model and low‐strain experimental data, using the modulus and initial inflation of the cell as adjustable parameters. The mean Young’s modulus for 2‐week‐old cells was found to be 2·3 ± 0·2 GPa at pH 5. This corresponds to an instantaneous bulk modulus of elasticity of approx. 7 MPa, similar to a value found by the pressure probe method. However, Young’s modulus is a better parameter, as it should depend only on the composition and structure of the cell wall, not on bulk cell behaviour. This new method has been used to show that Young’s modulus of cultured tomato cell walls is at its lowest at pH 4·5, the pH optimum for expansin activity.
• Conclusions The linear elastic model is very suitable for estimating wall Young’s modulus from micromanipulation experiments on single tomato cells. This is a powerful method for determining cell wall material properties.
PMCID: PMC4242341  PMID: 15023704
Tomato; cell wall; micromanipulation; Young’s modulus
15.  Parameter Estimation of a Spiking Silicon Neuron 
Spiking neuron models are used in a multitude of tasks ranging from understanding neural behavior at its most basic level to neuroprosthetics. Parameter estimation of a single neuron model, such that the model’s output matches that of a biological neuron is an extremely important task. Hand tuning of parameters to obtain such behaviors is a difficult and time consuming process. This is further complicated when the neuron is instantiated in silicon (an attractive medium in which to implement these models) as fabrication imperfections make the task of parameter configuration more complex. In this paper we show two methods to automate the configuration of a silicon (hardware) neuron’s parameters. First, we show how a Maximum Likelihood method can be applied to a leaky integrate and fire silicon neuron with spike induced currents to fit the neuron’s output to desired spike times. We then show how a distance based method which approximates the negative log likelihood of the lognormal distribution can also be used to tune the neuron’s parameters. We conclude that the distance based method is better suited for parameter configuration of silicon neurons due to its superior optimization speed.
PMCID: PMC3712290  PMID: 23852978
Neuromorphic; parameter estimation; silicon neuron
16.  Apps for IMproving FITness and Increasing Physical Activity Among Young People: The AIMFIT Pragmatic Randomized Controlled Trial 
Given the global prevalence of insufficient physical activity (PA), effective interventions that attenuate age-related decline in PA levels are needed. Mobile phone interventions that positively affect health (mHealth) show promise; however, their impact on PA levels and fitness in young people is unclear and little is known about what makes a good mHealth app.
The aim was to determine the effects of two commercially available smartphone apps (Zombies, Run and Get Running) on cardiorespiratory fitness and PA levels in insufficiently active healthy young people. A second aim was to identify the features of the app design that may contribute to improved fitness and PA levels.
Apps for IMproving FITness (AIMFIT) was a 3-arm, parallel, randomized controlled trial conducted in Auckland, New Zealand. Participants were recruited through advertisements in electronic mailing lists, local newspapers, flyers posted in community locations, and presentations at schools. Eligible young people aged 14-17 years were allocated at random to 1 of 3 conditions: (1) use of an immersive app (Zombies, Run), (2) use of a nonimmersive app (Get Running), or (3) usual behavior (control). Both smartphone apps consisted of a fully automated 8-week training program designed to improve fitness and ability to run 5 km; however, the immersive app featured a game-themed design and narrative. Intention-to-treat analysis was performed using data collected face-to-face at baseline and 8 weeks, and all regression models were adjusted for baseline outcome value and gender. The primary outcome was cardiorespiratory fitness, objectively assessed as time to complete the 1-mile run/walk test at 8 weeks. Secondary outcomes were PA levels (accelerometry and self-reported), enjoyment, psychological need satisfaction, self-efficacy, and acceptability and usability of the apps.
A total of 51 participants were randomized to the immersive app intervention (n=17), nonimmersive app intervention (n=16), or the control group (n=18). The mean age of participants was 15.7 (SD 1.2) years; participants were mostly NZ Europeans (61%, 31/51) and 57% (29/51) were female. Overall retention rate was 96% (49/51). There was no significant intervention effect on the primary outcome using either of the apps. Compared to the control, time to complete the fitness test was –28.4 seconds shorter (95% CI –66.5 to 9.82, P=.20) for the immersive app group and –24.7 seconds (95% CI –63.5 to 14.2, P=.32) for the nonimmersive app group. No significant intervention effects were found for secondary outcomes.
Although apps have the ability to increase reach at a low cost, our pragmatic approach using readily available commercial apps as a stand-alone instrument did not have a significant effect on fitness. However, interest in future use of PA apps is promising and highlights a potentially important role of these tools in a multifaceted approach to increase fitness, promote PA, and consequently reduce the adverse health outcomes associated with insufficient activity.
Trial Registration
Australian New Zealand Clinical Trials Registry: ACTRN12613001030763; (Archived by WebCite at
PMCID: PMC4642788  PMID: 26316499
physical fitness; motor activity; exercise; physical activity; adolescent; health promotion; telemedicine; mHealth; mobile applications; smartphone
17.  Human Annulus Fibrosus Material Properties from Biaxial Testing and Constitutive Modeling are Altered with Degeneration 
The annulus fibrosus (AF) of the intervertebral disc undergoes large and multidirectional stresses and strains. Uniaxial tensile tests are limited for measuring AF material properties, because freely contracting edges can prevent fiber stretch and are not representative of in situ boundary conditions. The objectives of this study were to measure human AF biaxial tensile mechanics and to apply and validate a constitutive model to determine material properties. Biaxial tensile tests were performed on samples oriented along the circumferential-axial and the radial-axial directions. Data were fit to a structurally-motivated anisotropic hyperelastic model composed of isotropic extrafibrillar matrix, nonlinear fibers, and fiber-matrix interactions (FMI) normal to the fibers. The validated model was used to simulate shear and uniaxial tensile behavior, to investigate AF structure-function, and to quantify the effect of degeneration. The biaxial stress-strain response was described well by the model (R2>0.9). The model showed that the parameters for fiber nonlinearity and the normal FMI correlated with degeneration, resulting in an elongated toe region and lower stiffness with degeneration. The model simulations in shear and uniaxial tension successfully matched previously published circumferential direction Young’s modulus, provided an explanation for the low values in previously published axial direction Young’s modulus, and was able to simulate shear mechanics. The normal FMI were important contributors to stress and changed with degeneration, therefore, their microstructural and compositional source should be investigated. Finally, the biaxial mechanical data and constitutive model can be incorporated into a disc finite element model to provide improved quantification of disc mechanics.
PMCID: PMC3500512  PMID: 21748426
biaxial tension; annulus fibrosus; continuum modeling; intervertebral disc; degeneration
18.  A model for spatial variations in life expectancy; mortality in Chinese regions in 2000 
Life expectancy in China has been improving markedly but health gains have been uneven and there is inequality in survival chances between regions and in rural as against urban areas. This paper applies a statistical modelling approach to mortality data collected in conjunction with the 2000 Census to formally assess spatial mortality contrasts in China. The modelling approach provides interpretable summary parameters (e.g. the relative mortality risk in rural as against urban areas) and is more parsimonious in terms of parameters than the conventional life table model.
Predictive fit is assessed both globally and at the level of individual five year age groups. A proportional model (age and area effects independent) has a worse fit than one allowing age-area interactions following a bilinear form. The best fit is obtained by allowing for child and oldest age mortality rates to vary spatially.
There is evidence that age (21 age groups) and area (31 Chinese administrative divisions) are not proportional (i.e. independent) mortality risk factors. In fact, spatial contrasts are greatest at young ages. There is a pronounced rural survival disadvantage, and large differences in life expectancy between provinces.
PMCID: PMC1876206  PMID: 17475003
19.  Smartphone apps to improve fitness and increase physical activity among young people: protocol of the Apps for IMproving FITness (AIMFIT) randomized controlled trial 
BMC Public Health  2015;15:635.
Physical activity is a modifiable behavior related to many preventable non-communicable diseases. There is an age-related decline in physical activity levels in young people, which tracks into adulthood. Common interactive technologies such as smartphones, particularly employing immersive features, may enhance the appeal and delivery of interventions to increase levels of physical activity in young people. The primary aim of the Apps for IMproving FITness (AIMFIT) trial is to evaluate the effectiveness of two popular “off-the-shelf” smartphone apps for improving cardiorespiratory fitness in young people.
A three-arm, parallel, randomized controlled trial will be conducted in Auckland, New Zealand. Fifty-one eligible young people aged 14–17 years will be randomized to one of three conditions: 1) use of an immersive smartphone app, 2) use of a non-immersive app, or 3) usual behavior (control). Both smartphone apps consist of an eight-week training program designed to improve fitness and ability to run 5 km, however, the immersive app features a game-themed design and adds a narrative. Data are collected at baseline and 8 weeks. The primary outcome is cardiorespiratory fitness, assessed as time to complete the one mile run/walk test at 8 weeks. Secondary outcomes are physical activity levels, self-efficacy, enjoyment, psychological need satisfaction, and acceptability and usability of the apps. Analysis using intention to treat principles will be performed using regression models.
Despite the proliferation of commercially available smartphone applications, there is a dearth of empirical evidence to support their effectiveness on the targeted health behavior. This pragmatic study will determine the effectiveness of two popular “off-the-shelf” apps as a stand-alone instrument for improving fitness and physical activity among young people. Adherence to app use will not be closely controlled; however, random allocation of participants, a heterogeneous group, and data analysis using intention to treat principles provide internal and external validity to the study. The primary outcome will be objectively assessed with a valid and reliable field-based test, as well as the secondary outcome of physical activity, via accelerometry. If effective, such applications could be used alongside existing interventions to promote fitness and physical activity in this population.
Trial Registration
Australian New Zealand Clinical Trials Registry: ACTRN12613001030763. Registered 16 September 2013.
Electronic supplementary material
The online version of this article (doi:10.1186/s12889-015-1968-y) contains supplementary material, which is available to authorized users.
PMCID: PMC4702326  PMID: 26159834
Physical activity; Fitness; Exercise adolescent; Health promotion; mHealth; Smartphone
20.  Functional consequences of age-related morphologic changes to pyramidal neurons of the rhesus monkey prefrontal cortex 
Layer 3 (L3) pyramidal neurons in the lateral prefrontal cortex (LPFC) of rhesus monkeys exhibit dendritic regression, spine loss and increased action potential (AP) firing rates during normal aging. The relationship between these structural and functional alterations, if any, is unknown. To address this issue, morphological and electrophysiological properties of L3 LPFC pyramidal neurons from young and aged rhesus monkeys were characterized using in vitro whole-cell patch-clamp recordings and high-resolution digital reconstruction of neurons. Consistent with our previous studies, aged neurons exhibited significantly reduced dendritic arbor length and spine density, as well as increased input resistance and firing rates. Computational models using the digital reconstructions with Hodgkin-Huxley and AMPA channels allowed us to assess relationships between demonstrated age-related changes and to predict physiological changes that have not yet been tested empirically. For example, the models predict that in both backpropagating APs and excitatory postsynaptic currents (EPSCs), attenuation is lower in aged versus young neurons. Importantly, when identical densities of passive parameters and voltage- and calcium-gated conductances were used in young and aged model neurons, neither input resistance nor firing rates differed between the two age groups. Tuning passive parameters for each model predicted significantly higher membrane resistance (Rm) in aged versus young neurons. This Rm increase alone did not account for increased firing rates in aged models, but coupling these Rm values with subtle differences in morphology and membrane capacitance did. The predicted differences in passive parameters (or parameters with similar effects) are mathematically plausible, but must be tested empirically.
PMCID: PMC4352129  PMID: 25527184
neuronal excitability; dendrites; spines; morphology; compartment model; aging; rhesus monkey; passive parameters
21.  Complex Parameter Landscape for a Complex Neuron Model 
PLoS Computational Biology  2006;2(7):e94.
The electrical activity of a neuron is strongly dependent on the ionic channels present in its membrane. Modifying the maximal conductances from these channels can have a dramatic impact on neuron behavior. But the effect of such modifications can also be cancelled out by compensatory mechanisms among different channels. We used an evolution strategy with a fitness function based on phase-plane analysis to obtain 20 very different computational models of the cerebellar Purkinje cell. All these models produced very similar outputs to current injections, including tiny details of the complex firing pattern. These models were not completely isolated in the parameter space, but neither did they belong to a large continuum of good models that would exist if weak compensations between channels were sufficient. The parameter landscape of good models can best be described as a set of loosely connected hyperplanes. Our method is efficient in finding good models in this complex landscape. Unraveling the landscape is an important step towards the understanding of functional homeostasis of neurons.
Neurons are believed to be electrical information processors. But how many models of a neuron can have similar input/output behavior? How precisely must the model parameters be tuned? These questions are crucial for models of the cerebellar Purkinje cell, a neuron with a huge dendritic arborization and a complex range of electrical outputs, for which recent experiments have demonstrated that dissimilar sets of ionic channel densities can produce similar activities. The authors have therefore used a detailed model of a Purkinje cell, released its 24 channel density parameters, and let them be optimized through an evolution strategy algorithm. They obtained 20 sets of parameters (20 models) that reproduce very precisely the original electrical waveforms. Therefore, model parameters are not uniquely identifiable. The parameters obtained vary several fold whereas small variations of these can also lead to drastically different results. Therefore, the authors have examined in more details the parameter space to gain better understanding of compensatory mechanisms in such complex models. They demonstrate that the 20 models are neither completely isolated nor fully connected, but rather, they belong to thin hyperplanes of good solutions that grid searches or random searches are likely to miss.
PMCID: PMC1513272  PMID: 16848639
22.  BAIAP2 Is Related to Emotional Modulation of Human Memory Strength 
PLoS ONE  2014;9(1):e83707.
Memory performance is the result of many distinct mental processes, such as memory encoding, forgetting, and modulation of memory strength by emotional arousal. These processes, which are subserved by partly distinct molecular profiles, are not always amenable to direct observation. Therefore, computational models can be used to make inferences about specific mental processes and to study their genetic underpinnings. Here we combined a computational model-based analysis of memory-related processes with high density genetic information derived from a genome-wide study in healthy young adults. After identifying the best-fitting model for a verbal memory task and estimating the best-fitting individual cognitive parameters, we found a common variant in the gene encoding the brain-specific angiogenesis inhibitor 1-associated protein 2 (BAIAP2) that was related to the model parameter reflecting modulation of verbal memory strength by negative valence. We also observed an association between the same genetic variant and a similar emotional modulation phenotype in a different population performing a picture memory task. Furthermore, using functional neuroimaging we found robust genotype-dependent differences in activity of the parahippocampal cortex that were specifically related to successful memory encoding of negative versus neutral information. Finally, we analyzed cortical gene expression data of 193 deceased subjects and detected significant BAIAP2 genotype-dependent differences in BAIAP2 mRNA levels. Our findings suggest that model-based dissociation of specific cognitive parameters can improve the understanding of genetic underpinnings of human learning and memory.
PMCID: PMC3879265  PMID: 24392092
23.  Potential application of item-response theory to interpretation of medical codes in electronic patient records 
Electronic patient records are generally coded using extensive sets of codes but the significance of the utilisation of individual codes may be unclear. Item response theory (IRT) models are used to characterise the psychometric properties of items included in tests and questionnaires. This study asked whether the properties of medical codes in electronic patient records may be characterised through the application of item response theory models.
Data were provided by a cohort of 47,845 participants from 414 family practices in the UK General Practice Research Database (GPRD) with a first stroke between 1997 and 2006. Each eligible stroke code, out of a set of 202 OXMIS and Read codes, was coded as either recorded or not recorded for each participant. A two parameter IRT model was fitted using marginal maximum likelihood estimation. Estimated parameters from the model were considered to characterise each code with respect to the latent trait of stroke diagnosis. The location parameter is referred to as a calibration parameter, while the slope parameter is referred to as a discrimination parameter.
There were 79,874 stroke code occurrences available for analysis. Utilisation of codes varied between family practices with intraclass correlation coefficients of up to 0.25 for the most frequently used codes. IRT analyses were restricted to 110 Read codes. Calibration and discrimination parameters were estimated for 77 (70%) codes that were endorsed for 1,942 stroke patients. Parameters were not estimated for the remaining more frequently used codes. Discrimination parameter values ranged from 0.67 to 2.78, while calibration parameters values ranged from 4.47 to 11.58. The two parameter model gave a better fit to the data than either the one- or three-parameter models. However, high chi-square values for about a fifth of the stroke codes were suggestive of poor item fit.
The application of item response theory models to coded electronic patient records might potentially contribute to identifying medical codes that offer poor discrimination or low calibration. This might indicate the need for improved coding sets or a requirement for improved clinical coding practice. However, in this study estimates were only obtained for a small proportion of participants and there was some evidence of poor model fit. There was also evidence of variation in the utilisation of codes between family practices raising the possibility that, in practice, properties of codes may vary for different coders.
PMCID: PMC3261214  PMID: 22176509
24.  Improved Quantitative Myocardial T2 Mapping 
To develop an improved T2 prepared (T2prep) balanced steady-state free-precession (bSSFP) sequence and signal relaxation curve fitting method for myocardial T2 mapping.
Myocardial T2 mapping is commonly performed by acquisition of multiple T2prep bSSFP images and estimating the voxel-wise T2 values using a 2-parameter fit for relaxation. However, a 2-parameter fit model does not take into account the effect of imaging pulses in a bSSFP sequence or other imperfections in T2prep RF pulses, which may decrease the robustness of T2 mapping. Therefore, we propose a novel T2 mapping sequence that incorporates an additional image acquired with saturation preparation, simulating a very long T2prep echo time. This enables the robust estimation of T2 maps using a 3-parameter fit model, which captures the effect of imaging pulses and other imperfections. Phantom imaging is performed to compare the T2 maps generated using the proposed 3-parameter model to the conventional 2-parameter model, as well as a spin echo reference. In-vivo imaging is performed on eight healthy subjects to compare the different fitting models.
Phantom and in-vivo data show that the T2 values generated by the proposed 3-parameter model fitting do not change with different choices of the T2prep echo times, and are not statistically different than the reference values for the phantom (P = 0.10 with three T2prep echoes). The 2-parameter model exhibits dependence on the choice of T2prep echo times and are significantly different than the reference values (P = 0.01 with three T2prep echoes).
The proposed imaging sequence in combination with a 3-parameter model allows accurate measurement of myocardial T2 values, which is independent of number and duration of T2prep echo times.
PMCID: PMC4320682  PMID: 25103908
Quantitative myocardial tissue characterization; myocardial T2 mapping; 3-parameter fit; myocardial inflammation
25.  Age- and sex-related regional compressive strength characteristics of human lumbar vertebrae in osteoporosis 
To obtain the compressive load bearing and energy absorption capacity of lumbar vertebrae of osteoporotic elderly for the everyday medical praxis in terms of the simple diagnostic data, like computed tomography (CT), densitometry, age, and sex.
Compressive test of 54 osteoporotic cadaver vertebrae L1 and L2, 16 males and 38 females (age range 43–93, mean age 71.6 ± 13.3 years, mean bone mineral density (BMD) 0.377 ± 0.089 g/cm2, mean T-score −5.57 ± 0.79, Z-score −4.05 ± 0.77) was investigated. Based on the load-displacement diagrams and the measured geometrical parameters of vertebral bodies, proportional, ultimate and yield stresses and strains, Young’s modulus, ductility and energy absorption capacity were determined. Three vertebral regions were distinguished: superior, central and inferior regions, but certain parameters were calculated for the upper/ lower intermediate layers, as well. Cross-sectional areas, and certain bone tissue parameters were determined by image analysis of CT pictures of vertebrae. Sex- and age-related decline functions and trends of strength characteristics were determined.
Size-corrected failure load was 15%–25% smaller in women, proportional and ultimate stresses were about 30%–35% smaller for women in any region, and 20%–25% higher in central regions for both sexes. Young’s moduli were about 30% smaller in women in any region, and 20%–25% smaller in the central region for both sexes. Small strains were higher in males, large strains were higher in females, namely, proportional strains were about 25% larger in men, yield and ultimate strains were quasi equal for sexes, break strains were 10% higher in women. Ultimate energy absorption capacity was 10%–20% higher in men; the final ductile energy absorption capacity was quasi equal for sexes in all levels. Age-dependence was stronger for men, mainly in central regions (ultimate load, male: r = −0.66, p < 0.01, female: r = −0.52, p < 0.005; ultimate stress, male: r = −0.69, p < 0.01, female: r = −0.50, p < 0.005; Young’s modulus, male: r = −0.55, p < 0.05, female: r = −0.52, p < 0.005, ultimate stiffness, male: r = −0.58, p < 0.05, female: r = −0.35, p < 0.03, central ultimate absorbed energy density, male: r = −0.59, p < 0.015, female: r = −0.29, p < 0.08).
For the strongly osteoporotic population (BMD < 0.4 g/cm2, T-score < −4) the statical variables (loads, stresses) showed significant correlation; mixed variables (stiffness, Young’s modulus, energy) showed moderate correlation; kinematical variables (displacements, strains) showed no correlation with age. The strong correlation of men between BMD and aging (r = −0.82, p < 0.001) and betwen BMD and strength parameters (r = 0.8–0.9, p < 0.001) indicated linear trends in age-related strength loss for men; however, the moderate correlation of women between BMD and aging (r = −0.47, p < 0.005) and between BMD and strength parameters (r = 0.4–0.5, p < 0.005) suggested the need of nonlinear (quadratic) approximation that provided the better fit in age-related strength functions of females modelling postmenopausal disproportionalities.
PMCID: PMC3004543  PMID: 21197342
osteoporosis; human lumbar vertebral body; regional compressive strength; load; stress; strain; young’s modulus; energy absorption capacity; age- and sex-dependence

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