Evaluation of Sampling-Based Methods for Sensitivity Analysis: Case Study for the E. coli Food Safety Process Risk Model

This article evaluates selected sensitivity analysis methods applicable to risk assessment models with two-dimensional probabilistic frameworks, using a microbial food safety process risk model as a test-bed. Six sampling-based sensitivity analysis methods were evaluated including Pearson and Spearman correlation, sample and rank linear regression, and sample and rank stepwise regression. In a two-dimensional risk model, the identification of key controllable inputs that can be priorities for risk management can be confounded by uncertainty. However, despite uncertainty, results show that key inputs can be distinguished from those that are unimportant, and inputs can be grouped into categories of similar levels of importance. All selected methods are capable of identifying unimportant inputs, which is helpful in that efforts to collect data to improve the assessment or to focus risk management strategies can be prioritized elsewhere. Rank-based methods provided more robust insights with respect to the key sources of variability in that they produced narrower ranges of uncertainty for sensitivity results and more clear distinctions when comparing the importance of inputs or groups of inputs. Regression-based methods have advantages over correlation approaches because they can be configured to provide insight regarding interactions and nonlinearities in the model.     

Neurobiological models of two-choice decision making can be reduced to a one-dimensional nonlinear diffusion equation

The response behaviors in many two-alternative choice tasks are well described by so-called sequential sampling models. In these models, the evidence for each one of the two alternatives accumulates over time until it reaches a threshold, at which point a response is made. At the neurophysiological level, single neuron data recorded while monkeys are engaged in two-alternative choice tasks are well described by winner-take-all network models in which the two choices are represented in the firing rates of separate populations of neurons. Here, we show that such nonlinear network models can generally be reduced to a one-dimensional nonlinear diffusion equation, which bears functional resemblance to standard sequential sampling models of behavior. This reduction gives the functional dependence of performance and reaction-times on external inputs in the original system, irrespective of the system details. What is more, the nonlinear diffusion equation can provide excellent fits to behavioral data from two-choice decision making tasks by varying these external inputs. This suggests that changes in behavior under various experimental conditions, e.g. changes in stimulus coherence or response deadline, are driven by internal modulation of afferent inputs to putative decision making circuits in the brain. For certain model systems one can analytically derive the nonlinear diffusion equation, thereby mapping the original system parameters onto the diffusion equation coefficients. Here, we illustrate this with three model systems including coupled rate equations and a network of spiking neurons.     

Slope-Based Stochastic Resonance: How Noise Enables Phasic Neurons to Encode Slow Signals

Fundamental properties of phasic firing neurons are usually characterized in a noise-free condition. In the absence of noise, phasic neurons exhibit Class 3 excitability, which is a lack of repetitive firing to steady current injections. For time-varying inputs, phasic neurons are band-pass filters or slope detectors, because they do not respond to inputs containing exclusively low frequencies or shallow slopes. However, we show that in noisy conditions, response properties of phasic neuron models are distinctly altered. Noise enables a phasic model to encode low-frequency inputs that are outside of the response range of the associated deterministic model. Interestingly, this seemingly stochastic-resonance (SR) like effect differs significantly from the classical SR behavior of spiking systems in both the signal-to-noise ratio and the temporal response pattern. Instead of being most sensitive to the peak of a subthreshold signal, as is typical in a classical SR system, phasic models are most sensitive to the signal''s rising and falling phases where the slopes are steep. This finding is consistent with the fact that there is not an absolute input threshold in terms of amplitude; rather, a response threshold is more properly defined as a stimulus slope/frequency. We call the encoding of low-frequency signals with noise by phasic models a slope-based SR, because noise can lower or diminish the slope threshold for ramp stimuli. We demonstrate here similar behaviors in three mechanistic models with Class 3 excitability in the presence of slow-varying noise and we suggest that the slope-based SR is a fundamental behavior associated with general phasic properties rather than with a particular biological mechanism.     

Improved neuronal models for studying neural networks

Previous neuronal models used for the study of neural networks are considered. Equations are developed for a model which includes: 1) a normalized range of firing rates with decreased sensitivity at large excitatory or large inhibitory input levels, 2) a single rate constant for the increase in firing rate following step changes in the input, 3) one or more rate constants, as required to fit experimental data for the adaptation of firing rates to maintained inputs. Computed responses compare well with the types of neuronal responses observed experimentally. Depending on the parameters, overdamped increases and decreases, damped oscillatory or maintained oscillatory changes in firing rate are observed to step changes in the input. The integrodifferential equations describing the neuronal models can be represented by a set of first-order differential equations. Steady-state solutions for these equations can be obtained for constant inputs, as well as the stability of the solutions to small perturbations. The linear frequency response function is derived for sufficiently small time-varying inputs. The linear responses are also compared with the computed solutions for larger non-linear responses.     

The Frequency Response, Coherence, and Information Capacity of Two Neuronal Models

Two neuronal models are analyzed in which subthreshold inputs are integrated either without loss (perfect integrator) or with a decay which follows an exponential time course (leaky integrator). Linear frequency response functions for these models are compared using sinusoids, Poisson-distributed impulses, or gaussian white noise as inputs. The responses of both models show the nonlinear behavior characteristic of a rectifier for sinusoidal inputs of sufficient amplitude. The leaky integrator shows another nonlinearity in which responses become phase locked to cyclic stimuli. Addition of white noise reduces the distortions due to phase locking. Both models also show selective attenuation of high-frequency components with white noise inputs. Input, output, and cross-spectra are computed using inputs having a broad frequency spectrum. Measures of the coherence and information transmission between the input and output of the models are also derived. Steady inputs, which produce a constant “carrier” rate, and intrinsic sources, which produce variability in the discharge of neurons, may either increase or decrease coherence; however, information transmission using inputs with a broad spectrum is generally increased by steady inputs and reduced by intrinsic variability.     

The relationship between neuronal calcium concentration and firing rate during stochastic synaptic inputs

We propose a novel, nonlinear theory about reading neuronal information using intracellular calcium concentrations, which includes the linear theory already developed in the literature as a special case. The theory is numerically confirmed using the Pinsky-Rinzel and integrate-and-fire models with constant rate Poisson inputs. Applying the theory to models with non-constant inputs, we find that there is a time lag equal to the calcium buffering time constant between the instantaneous firing rate and the firing rate estimated using calcium concentrations.     

Structure-preserving model reduction of passive and quasi-active neurons

The spatial component of input signals often carries information crucial to a neuron’s function, but models mapping synaptic inputs to the transmembrane potential can be computationally expensive. Existing reduced models of the neuron either merge compartments, thereby sacrificing the spatial specificity of inputs, or apply model reduction techniques that sacrifice the underlying electrophysiology of the model. We use Krylov subspace projection methods to construct reduced models of passive and quasi-active neurons that preserve both the spatial specificity of inputs and the electrophysiological interpretation as an RC and RLC circuit, respectively. Each reduced model accurately computes the potential at the spike initiation zone (SIZ) given a much smaller dimension and simulation time, as we show numerically and theoretically. The structure is preserved through the similarity in the circuit representations, for which we provide circuit diagrams and mathematical expressions for the circuit elements. Furthermore, the transformation from the full to the reduced system is straightforward and depends on intrinsic properties of the dendrite. As each reduced model is accurate and has a clear electrophysiological interpretation, the reduced models can be used not only to simulate morphologically accurate neurons but also to examine computations performed in dendrites.     

The response of cortical neurons to in vivo-like input current: theory and experiment

The study of several aspects of the collective dynamics of interacting neurons can be highly simplified if one assumes that the statistics of the synaptic input is the same for a large population of similarly behaving neurons (mean field approach). In particular, under such an assumption, it is possible to determine and study all the equilibrium points of the network dynamics when the neuronal response to noisy, in vivo-like, synaptic currents is known. The response function can be computed analytically for simple integrate-and-fire neuron models and it can be measured directly in experiments in vitro. Here we review theoretical and experimental results about the neural response to noisy inputs with stationary statistics. These response functions are important to characterize the collective neural dynamics that are proposed to be the neural substrate of working memory, decision making and other cognitive functions. Applications to the case of time-varying inputs are reviewed in a companion paper (Giugliano et al. in Biol Cybern, 2008). We conclude that modified integrate-and-fire neuron models are good enough to reproduce faithfully many of the relevant dynamical aspects of the neuronal response measured in experiments on real neurons in vitro.     

Estimating vital statistics and age distributions of measurable soil organic carbon fractions based on their pathway of formation and radiocarbon content

Radiocarbon measurements have been used in combination with "bomb 14C" models to estimate turnover of soil organic carbon fractions. However, the bomb 14C models assume that all SOC fractions are formed directly from external inputs of carbon, which is not always valid because some SOC fractions may receive carbon from other SOC fractions. Due to the continuous inputs of organic carbon, we argue that the most appropriate way to describe the age of SOC is by an age distribution. We developed age distributed models of SOC fractions and derived analytical solutions to them. The models all assume that SOC fraction decay can be described by first-order kinetics, but differ in their assumptions about the pathway of SOC fraction formation. The solutions can be used to estimate age distributions at steady state of different SOC fractions based on their radiocarbon content. These age distributions can be used to calculate the mean age, mean residence time, and other vital statistics of each measurable SOC fraction. Furthermore, if a sequential scheme is used to isolate the SOC fractions, an estimated age distribution of the total SOC can be obtained by adding the contributions of each soil fraction. The age distributions can be very helpful in interpretations of soil organic carbon dynamics in different soils.     

Kinetic models of spike-timing dependent plasticity and their functional consequences in detecting correlations

Spike-timing dependent plasticity (STDP) is a type of synaptic modification found relatively recently, but the underlying biophysical mechanisms are still unclear. Several models of STDP have been proposed, and differ by their implementation, and in particular how synaptic weights saturate to their minimal and maximal values. We analyze here kinetic models of transmitter-receptor interaction and derive a series of STDP models. In general, such kinetic models predict progressive saturation of the weights. Various forms can be obtained depending on the hypotheses made in the kinetic model, and these include a simple linear dependence on the value of the weight (“soft bounds”), mixed soft and abrupt saturation (“hard bound”), or more complex forms. We analyze in more detail simple soft-bound models of Hebbian and anti-Hebbian STDPs, in which nonlinear spike interactions (triplets) are taken into account. We show that Hebbian STDPs can be used to selectively potentiate synapses that are correlated in time, while anti-Hebbian STDPs depress correlated synapses, despite the presence of nonlinear spike interactions. This correlation detection enables neurons to develop a selectivity to correlated inputs. We also examine different versions of kinetics-based STDP models and compare their sensitivity to correlations. We conclude that kinetic models generally predict soft-bound dynamics, and that such models seem ideal for detecting correlations among large numbers of inputs.     

Does inhibition balance excitation in neocortex?

The distribution of inhibitory and excitatory synapses on neocortical neurons is at odds with a simple view that cortical functioning can persist by maintaining a balance between inhibitory and excitatory drives. Pyramidal cells can potentially be shut down by very powerful proximal inhibitory synapses, despite these accounting for perhaps less than 1% of their total number of synaptic inputs. Interneurons in contrast are dominated by excitatory inputs. These may be powerful enough to effect an apparent depolarizing block at the soma. In this extreme case though, models suggest that action potentials are generated down the axon, and the cells behave like integrate-and-fire neurons. We discuss possible network implications of these modelling studies.     

Modeling DNA sequence-based cis-regulatory gene networks

Gene network analysis requires computationally based models which represent the functional architecture of regulatory interactions, and which provide directly testable predictions. The type of model that is useful is constrained by the particular features of developmentally active cis-regulatory systems. These systems function by processing diverse regulatory inputs, generating novel regulatory outputs. A computational model which explicitly accommodates this basic concept was developed earlier for the cis-regulatory system of the endo16 gene of the sea urchin. This model represents the genetically mandated logic functions that the system executes, but also shows how time-varying kinetic inputs are processed in different circumstances into particular kinetic outputs. The same basic design features can be utilized to construct models that connect the large number of cis-regulatory elements constituting developmental gene networks. The ultimate aim of the network models discussed here is to represent the regulatory relationships among the genomic control systems of the genes in the network, and to state their functional meaning. The target site sequences of the cis-regulatory elements of these genes constitute the physical basis of the network architecture. Useful models for developmental regulatory networks must represent the genetic logic by which the system operates, but must also be capable of explaining the real time dynamics of cis-regulatory response as kinetic input and output data become available. Most importantly, however, such models must display in a direct and transparent manner fundamental network design features such as intra- and intercellular feedback circuitry; the sources of parallel inputs into each cis-regulatory element; gene battery organization; and use of repressive spatial inputs in specification and boundary formation. Successful network models lead to direct tests of key architectural features by targeted cis-regulatory analysis.     

The capabilities and limitations of conductance-based compartmental neuron models with reduced branched or unbranched morphologies and active dendrites

Conductance-based neuron models are frequently employed to study the dynamics of biological neural networks. For speed and ease of use, these models are often reduced in morphological complexity. Simplified dendritic branching structures may process inputs differently than full branching structures, however, and could thereby fail to reproduce important aspects of biological neural processing. It is not yet well understood which processing capabilities require detailed branching structures. Therefore, we analyzed the processing capabilities of full or partially branched reduced models. These models were created by collapsing the dendritic tree of a full morphological model of a globus pallidus (GP) neuron while preserving its total surface area and electrotonic length, as well as its passive and active parameters. Dendritic trees were either collapsed into single cables (unbranched models) or the full complement of branch points was preserved (branched models). Both reduction strategies allowed us to compare dynamics between all models using the same channel density settings. Full model responses to somatic inputs were generally preserved by both types of reduced model while dendritic input responses could be more closely preserved by branched than unbranched reduced models. However, features strongly influenced by local dendritic input resistance, such as active dendritic sodium spike generation and propagation, could not be accurately reproduced by any reduced model. Based on our analyses, we suggest that there are intrinsic differences in processing capabilities between unbranched and branched models. We also indicate suitable applications for different levels of reduction, including fast searches of full model parameter space.     

Comparing Families of Dynamic Causal Models

Mathematical models of scientific data can be formally compared using Bayesian model evidence. Previous applications in the biological sciences have mainly focussed on model selection in which one first selects the model with the highest evidence and then makes inferences based on the parameters of that model. This “best model” approach is very useful but can become brittle if there are a large number of models to compare, and if different subjects use different models. To overcome this shortcoming we propose the combination of two further approaches: (i) family level inference and (ii) Bayesian model averaging within families. Family level inference removes uncertainty about aspects of model structure other than the characteristic of interest. For example: What are the inputs to the system? Is processing serial or parallel? Is it linear or nonlinear? Is it mediated by a single, crucial connection? We apply Bayesian model averaging within families to provide inferences about parameters that are independent of further assumptions about model structure. We illustrate the methods using Dynamic Causal Models of brain imaging data.     

Extracting Information from the Power Spectrum of Synaptic Noise

In cortical neurons, synaptic "noise" is caused by the nearly random release of thousands of synapses. Few methods are presently available to analyze synaptic noise and deduce properties of the underlying synaptic inputs. We focus here on the power spectral density (PSD) of several models of synaptic noise. We examine different classes of analytically solvable kinetic models for synaptic currents, such as the "delta kinetic models," which use Dirac delta functions to represent the activation of the ion channel. We first show that, for this class of kinetic models, one can obtain an analytic expression for the PSD of the total synaptic conductance and derive equivalent stochastic models with only a few variables. This yields a method for constraining models of synaptic currents by analyzing voltage-clamp recordings of synaptic noise. Second, we show that a similar approach can be followed for the PSD of the the membrane potential (Vm) through an effective-leak approximation. Third, we show that this approach is also valid for inputs distributed in dendrites. In this case, the frequency scaling of the Vm PSD is preserved, suggesting that this approach may be applied to intracellular recordings of real neurons. In conclusion, using simple mathematical tools, we show that Vm recordings can be used to constrain kinetic models of synaptic currents, as well as to estimate equivalent stochastic models. This approach, therefore, provides a direct link between intracellular recordings in vivo and the design of models consistent with the dynamics and spectral structure of synaptic noise.     

The response of a nerve cylinder to spatially distributed white noise inputs

The response of a passive nerve cylinder (or dendritic tree in the equivalent cylinder representation) to random white noise input currents is determined. Results for the mean, variance and covariance of the depolarization are obtained for an arbitrary number of independent spatially distributed inputs. The case of a cylinder with sealed ends is considered in detail. The differences that arise when the input currents are distributed over a small but finite region of space instead of concentrated at a point are investigated. In the case of distributed inputs, the expectation is smoother near the stimulus and the variance becomes finite over the entire cable length including the region of the applied stimulus. Away from the stimulus, there are no appreciable differences between the responses for the two cases. The interaction between an excitatory input and an inhibitory input at various locations is examined and one case of more than two inputs is also analysed to study effects which could not have been discerned from point models for a neuron with random inputs.     

Low abundance of NPY in the hypothalamus can produce hyperphagia and obesity

States of increased metabolic demand are associated with up-regulation of NPY and hyperphagia. However, we present some instances of hyperphagia in which NPY is not up-regulated. Ablation or functional disruption of specific sites in the hypothalamus, such as the ventromedial or paraventricular nuclei, or transection of inputs to the hypothalamus from the hindbrain results in hyperphagia and excess body weight gain. However, NPY expression and concentration in these experimental models is either decreased or unchanged. While there is no up-regulation of NPY in these models, there is increased sensitivity to the orexigenic effects of NPY. This enhanced responsiveness to NPY may more than compensate for the reduced levels of NPY and result in hyperphagia and excess body weight gain. The hyper-responsiveness may be due either to an increase in NPY receptors or to other changes in target cells and response pathways that may result from the treatments used in these models.     

Integration of kinetic information on yeast sphingolipid metabolism in dynamical pathway models

For the first time, kinetic information from the literature was collected and used to construct integrative dynamical mathematical models of sphingolipid metabolism. One model was designed primarily with kinetic equations in the tradition of Michaelis and Menten whereas the other two models were designed as alternative power-law models within the framework of Biochemical Systems Theory. Each model contains about 50 variables, about a quarter of which are dependent (state) variables, while the others are independent inputs and enzyme activities that are considered constant. The models account for known regulatory signals that exert control over the pathway. Standard mathematical testing, repeated revisiting of the literature, and numerous rounds of amendments and refinements resulted in models that are stable and rather insensitive to perturbations in inputs or parameter values. The models also appear to be compatible with the modest amount of experimental experience that lends itself to direct comparisons. Even though the three models are based on different mathematical representations, they show dynamic responses to a variety of perturbations and changes in conditions that are essentially equivalent for small perturbations and similar for large perturbations. The kinetic information used for model construction and the models themselves can serve as a starting point for future analyses and refinements.     

Thermal sensation prediction by soft computing methodology

Thermal comfort in open urban areas is very factor based on environmental point of view. Therefore it is need to fulfill demands for suitable thermal comfort during urban planning and design. Thermal comfort can be modeled based on climatic parameters and other factors. The factors are variables and they are changed throughout the year and days. Therefore there is need to establish an algorithm for thermal comfort prediction according to the input variables. The prediction results could be used for planning of time of usage of urban areas. Since it is very nonlinear task, in this investigation was applied soft computing methodology in order to predict the thermal comfort. The main goal was to apply extreme leaning machine (ELM) for forecasting of physiological equivalent temperature (PET) values. Temperature, pressure, wind speed and irradiance were used as inputs. The prediction results are compared with some benchmark models. Based on the results ELM can be used effectively in forecasting of PET.     

Cellular Sensory Mechanisms for Detecting Specific Fold-Changes in Extracellular Cues

Cellular sensory systems often respond not to the absolute levels of inputs but to the fold-changes in inputs. Such a property is called fold-change detection (FCD) and is important for accurately sensing dynamic changes in environmental signals in the presence of fluctuations in their absolute levels. Previous studies defined FCD as input-scale invariance and proposed several biochemical models that achieve such a condition. Here, we prove that the previous FCD models can be approximated by a log-differentiator. Although the log-differentiator satisfies the input-scale invariance requirement, its response amplitude and response duration strongly depend on the input timescale. This creates limitations in the specificity and repeatability of detecting fold-changes in inputs. Nevertheless, FCD with specificity and repeatability by cells has been reported in the context of Drosophila wing development. Motivated by this fact and by extending previous FCD models, we here propose two possible mechanisms to achieve FCD with specificity and repeatability. One is the integrate-and-fire type: a system integrates the rate of temporal change in input and makes a response when the integrated value reaches a constant threshold, and this is followed by the reset of the integrated value. The other is the dynamic threshold type: a system response occurs when the input level reaches a threshold, whose value is multiplied by a certain constant after each response. These two mechanisms can be implemented biochemically by appropriately combining feed-forward and feedback loops. The main difference between the two models is their memory of input history; we discuss possible ways to distinguish between the two models experimentally.