Wiener analysis of functionals of a Markov chain: Application to neural transformations of random signals

The Wiener method of nonlinear system identification is extended to systems with a Markov chain input. Multivariate functionals are constructed that are orthonormal with respect to the probability measure of the Markov input. Any system operating on a Markov chain may be represented by an orthogonal expansion in these functionals. The coefficients of the orthogonal expansion may be evaluated by crosscorrelation. Application of this technique to nonlinear neural systems with a Markov actionpotential input are discussed.     

Bilinear state space systems for nonlinear dynamical modelling

Summary We discuss the identification of multiple input, multiple output, discrete-time bilinear state space systems. We consider two identification problems. In the first case, the input to the system is a measurable white noise sequence. We show that it is possible to identify the system by solving a nonlinear optimization problem. The number of parameters in this optimization problem can be reduced by exploiting the principle of separable least squares. A subspace-based algorithm can be used to generate initial estimates for this nonlinear identification procedure. In the second case, the input to the system is not measurable. This makes it a much more difficult identification problem than the case with known inputs. At present, we can only solve this problem for a certain class of single input, single output bilinear state space systems, namely bilinear systems in phase variable form.     

Finding complex oscillatory phenomena in biochemical systems. An empirical approach

Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmicity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the cAMP signalling system of Dictyostelium discoideum amoebae.     

Identification of nonlinear systems using random impulse train inputs

Nonlinear systems that require discrete inputs can be characterized by using random impulse train (Poisson process) inputs. The method is analagous to the Wiener method for continuous input systems, where Gaussian white-noise is the input. In place of the Wiener functional expansion for the output of a continuous input system, a new series for discrete input systems is created by making certain restrictions on the integrals in a Volterra series. The kernels in the new series differ from the Wiener kernels, but also serve to identify a system and are simpler to compute. For systems whose impulse responses vary in amplitude but maintain a similar shape, one argument may be held fixed in each kernel. This simplifies the identification problem. As a test of the theory presented, the output of a hypothetical second order nonlinear system in response to a random impulse train stimulus was computer simulated. Kernels calculated from the simulated data agreed with theoretical predictions. The Poisson impulse train method is applicable to any system whose input can be delivered in discrete pulses. It is particularly suited to neuronal synaptic systems when the pattern of input nerve impulses can be made random.     

Redundancy reducing processes in single neurons

Interaction mechanisms between excitatory and inhibitory impulse sequences operating on neurons play an important role for the processing of information by the nervous system. For instance, the convergence of excitatory and inhibitory influences on retinal ganglion cells to form their receptive fields has been taken as an example for the process of neuronal sharpening by lateral inhibition. In order to analyze quantitatively the functional behavior of such a system, Shannon's entropy method for multiple access channels has been applied to biological two-inputs-one-output systems using the theoretical model developed by Tsukada et al. (1979). Here we give an extension of this procedure from the point of view to reduce redundancy of information in the input signal space of single neurons and attempt to obtain a new interpretation for the information processing of the system. The concept for the redundancy reducing mechanism in single neurons is examined and discussed for the following two processes. The first process is concerned with a signal space formed by superposing two random sequences on the input of a neuron. In this process, we introduce a coding technique to encode the inhibitory sequence by using the timing of the excitatory sequence, which is closely related to an encoding technique of multiple access channels with a correlated source (Marko, 1966, 1970, 1973; Slepian and Wolf, 1973) and which is an invariant transformation in the input signal space without changing the information contents of the input. The second process is concerned with a procedure of reducing redundant signals in the signal space mentioned before. In this connection, it is an important point to see how single neurons reduce the dimensionality of the signal space via transformation with a minimum loss of effective information. For this purpose we introduce the criterion that average transmission of information from signal space to the output does not change when redundant signals are added. This assumption is based on the fact that two signals are equivalent if and only if they have identical input-output behavior. The mechanism is examined and estimated by using a computer-simulated model. As the result of such a simulation we can estimate the minimal segmentation in the signal space which is necessary and sufficient for temporal pattern sensitivity in neurons.     

The numerical solution of stiff differential equations

This paper first discusses the conditions in which a set of differential equations should give stable solutions, starting with linear systems assuming that these do not differ greatly in this respect from non-linear systems. Methods of investigating the stability of particular systems are briefly discussed. Most real biochemical systems are known from observation to be stable, but little is known of the regions over which stability persists; moreover, models of biochemical systems may not be stable, because of inaccurate choice of parameter values.The separate problem of stability and accuracy in numerical methods of approximating the solution of systems of non-linear equations is then treated. Stress is laid on the consistently unsatisfactory results given by explicit methods for systems containing "stiff" equations, and implicit multistep methods are particularly recommended for this class of problem, which is likely to include many biochemical model systems. Finally, an iteration procedure likely to give convergence both in multistep methods and in the steady-state approach is recommended, and areas in which improvement in methods is likely to occur are outlined.     

Optimal input design: case study of the metabolic process

An optimal input of nutrients into the metabolic process of the individual subject so as to effect desired therapeutic results is particularly important for the critically ill. A patient-related, individualized nutrients optimization procedure is proposed here. The procedure is applicable to any time-invariant and deterministic metabolic model that is bound by one or more quantitative limiting criteria. As a case in hand, the procedure is used to optimize the individual metabolic needs of critically ill patients. The results indicate that, given a proper metabolic model, the patient may be treated on an individual appropriateness basis rather than on the traditional statistical intuitive approach.     

人工神经网络模型在水稻群体分蘖动态模拟中外推能力的测试

研究利用人工神经网络模型 ,以水稻群体分蘖动态为例 ,采用交互验证和独立验证的方式 ,对水稻生长 BP网络模型进行了训练与模拟 ,其结果与水稻群体分蘖的积温统计模型、基本动力学模型和复合分蘖模型进行了比较。研究结果表明 ,神经网络模型具有一定的外推能力 ,但其外推能力依赖于大量的训练样本。神经网络模型具有较好的拟合能力 ,是因为有较多的模型参数 ,因此对神经网络模型的训练需要大量的参数来保证其参数不致过度吻合。具有外推能力神经网络模型的最少训练样本数应大于 6 .75倍于神经网络参数数目 ,小于 13.5倍于神经网络参数数目。因此在应用神经网络模型时 ,如果神经网络模型包括较多的输入变量时 ,可考虑采用主成分分析、对应分析等技术对输入变量进行信息综合 ,相应地减少网络模型的参数。另一方面 ,当训练样本不足时 ,最好只用神经网络模型对同一系统的情况进行模拟 ,应谨慎使用神经网络模型进行外推。神经网络模型给作物模拟研究的科学工作者提供了一个“傻瓜”式工具 ,对数学建模不熟悉的农业研究人员 ,人工神经网络可以替代数学建模进行仿真实验 ;对于精通数学建模的研究人员来说 ,它至少是一种补充和可作为比较的非线性数据处理方法     

Identification of nonlinear models of biological membranes using the voltage-clamp method

The method for identification of nonlinear systems proposed in 1952 by Hodgkin and Huxley is mathematically justified. A procedure for the application of this method is developed, including the development of the structure of a mathematical model, carrying out a series of tests with special chosen signals, and determination of unknown parameters. Basic requirements for the admissible sets of input and output signals and to the system operator have been determined. It is shown that this operator should be totally continuous and that the minimum number of unknown parameters and the minimum complexity of the operator structure should give an approximation of the necessary quality. The pros and cons of the Hodgkin-Huxley and Noble mathematical models and the methods used for their development are discussed. A structure for the operator for the identification of mathematical models of excitable membranes with a large number of membrane currents is proposed. It is found that the nonlinear electrical properties of biological membranes can be identified using tests with other types of “clamped” parameters, such as the current, ramp voltage, etc.     

Phenomenological definition of response times with application to metabolic reactions

The metabolic response time, i.e. the delay the system introduces in the response to an input flux, is considered. A novel phenomenological definition is presented, which is valid for any kind of behavior, including transitory or permanent oscillatory responses. In order to calculate the response time of single-input systems, output fluxes have to be deconvoluted with the input flux. The bases for this are established. The resulting function (unit impulse response in time-invariant linear systems) is transformed by subtracting its final state, taking the absolute value and normalizing by the resulting area, so that a norm can be applied that weights the response at every time. This response time can also be interpreted as an average. It coincides with the transition (characteristic) time of an output flux, provided that the input is performed instantaneously (step function). A strictly non-negative response function is needed for the response time to be interpreted as a mass balance. A simple example is used to study the deviation otherwise. The method is advantageous in that it provides clues on the phenomenological behavior of biochemical systems. For example, deconvolution reveals the intrinsic oscillation-generating mechanism of an allosteric enzyme, which becomes hidden when the input flux increases in a slow way. This is illustrated by means of a model.     

Sensitivity and Robustness in Covalent Modification Cycles with a Bifunctional Converter Enzyme

Regulation by covalent modification is a common mechanism to transmit signals in biological systems. The modifying reactions are catalyzed either by two distinct converter enzymes or by a single bifunctional enzyme (which may employ either one or two catalytic sites for its opposing activities). The reason for this diversification is unclear, but contemporary theoretical models predict that systems with distinct converter enzymes can exhibit enhanced sensitivity to input signals whereas bifunctional enzymes with two catalytic sites are believed to generate robustness against variations in system’s components. However, experiments indicate that bifunctional enzymes can also exhibit enhanced sensitivity due to the zero-order effect, raising the question whether both phenomena could be understood within a common mechanistic model. Here, I argue that this is, indeed, the case. Specifically, I show that bifunctional enzymes with two catalytic sites can exhibit both ultrasensitivity and concentration robustness, depending on the kinetic operating regime of the enzyme’s opposing activities. The model predictions are discussed in the context of experimental observations of ultrasensitivity and concentration robustness in the uridylylation cycle of the PII protein, and in the phosphorylation cycle of the isocitrate dehydrogenase, respectively.     

Sensitivity and Robustness in Covalent Modification Cycles with a Bifunctional Converter Enzyme

Regulation by covalent modification is a common mechanism to transmit signals in biological systems. The modifying reactions are catalyzed either by two distinct converter enzymes or by a single bifunctional enzyme (which may employ either one or two catalytic sites for its opposing activities). The reason for this diversification is unclear, but contemporary theoretical models predict that systems with distinct converter enzymes can exhibit enhanced sensitivity to input signals whereas bifunctional enzymes with two catalytic sites are believed to generate robustness against variations in system’s components. However, experiments indicate that bifunctional enzymes can also exhibit enhanced sensitivity due to the zero-order effect, raising the question whether both phenomena could be understood within a common mechanistic model. Here, I argue that this is, indeed, the case. Specifically, I show that bifunctional enzymes with two catalytic sites can exhibit both ultrasensitivity and concentration robustness, depending on the kinetic operating regime of the enzyme’s opposing activities. The model predictions are discussed in the context of experimental observations of ultrasensitivity and concentration robustness in the uridylylation cycle of the PII protein, and in the phosphorylation cycle of the isocitrate dehydrogenase, respectively.     

Using Uncertainty and Sensitivity Analyses in Socioecological Agent-Based Models to Improve Their Analytical Performance and Policy Relevance

Agent-based models (ABMs) have been widely used to study socioecological systems. They are useful for studying such systems because of their ability to incorporate micro-level behaviors among interacting agents, and to understand emergent phenomena due to these interactions. However, ABMs are inherently stochastic and require proper handling of uncertainty. We propose a simulation framework based on quantitative uncertainty and sensitivity analyses to build parsimonious ABMs that serve two purposes: exploration of the outcome space to simulate low-probability but high-consequence events that may have significant policy implications, and explanation of model behavior to describe the system with higher accuracy. The proposed framework is applied to the problem of modeling farmland conservation resulting in land use change. We employ output variance decomposition based on quasi-random sampling of the input space and perform three computational experiments. First, we perform uncertainty analysis to improve model legitimacy, where the distribution of results informs us about the expected value that can be validated against independent data, and provides information on the variance around this mean as well as the extreme results. In our last two computational experiments, we employ sensitivity analysis to produce two simpler versions of the ABM. First, input space is reduced only to inputs that produced the variance of the initial ABM, resulting in a model with output distribution similar to the initial model. Second, we refine the value of the most influential input, producing a model that maintains the mean of the output of initial ABM but with less spread. These simplifications can be used to 1) efficiently explore model outcomes, including outliers that may be important considerations in the design of robust policies, and 2) conduct explanatory analysis that exposes the smallest number of inputs influencing the steady state of the modeled system.     

Reciprocal Regulation as a Source of Ultrasensitivity in Two-Component Systems with a Bifunctional Sensor Kinase

Two-component signal transduction systems, where the phosphorylation state of a regulator protein is modulated by a sensor kinase, are common in bacteria and other microbes. In many of these systems, the sensor kinase is bifunctional catalyzing both, the phosphorylation and the dephosphorylation of the regulator protein in response to input signals. Previous studies have shown that systems with a bifunctional enzyme can adjust the phosphorylation level of the regulator protein independently of the total protein concentrations – a property known as concentration robustness. Here, I argue that two-component systems with a bifunctional enzyme may also exhibit ultrasensitivity if the input signal reciprocally affects multiple activities of the sensor kinase. To this end, I consider the case where an allosteric effector inhibits autophosphorylation and, concomitantly, activates the enzyme''s phosphatase activity, as observed experimentally in the PhoQ/PhoP and NRII/NRI systems. A theoretical analysis reveals two operating regimes under steady state conditions depending on the effector affinity: If the affinity is low the system produces a graded response with respect to input signals and exhibits stimulus-dependent concentration robustness – consistent with previous experiments. In contrast, a high-affinity effector may generate ultrasensitivity by a similar mechanism as phosphorylation-dephosphorylation cycles with distinct converter enzymes. The occurrence of ultrasensitivity requires saturation of the sensor kinase''s phosphatase activity, but is restricted to low effector concentrations, which suggests that this mode of operation might be employed for the detection and amplification of low abundant input signals. Interestingly, the same mechanism also applies to covalent modification cycles with a bifunctional converter enzyme, which suggests that reciprocal regulation, as a mechanism to generate ultrasensitivity, is not restricted to two-component systems, but may apply more generally to bifunctional enzyme systems.     

Minimal models of multidimensional computations

The multidimensional computations performed by many biological systems are often characterized with limited information about the correlations between inputs and outputs. Given this limitation, our approach is to construct the maximum noise entropy response function of the system, leading to a closed-form and minimally biased model consistent with a given set of constraints on the input/output moments; the result is equivalent to conditional random field models from machine learning. For systems with binary outputs, such as neurons encoding sensory stimuli, the maximum noise entropy models are logistic functions whose arguments depend on the constraints. A constraint on the average output turns the binary maximum noise entropy models into minimum mutual information models, allowing for the calculation of the information content of the constraints and an information theoretic characterization of the system's computations. We use this approach to analyze the nonlinear input/output functions in macaque retina and thalamus; although these systems have been previously shown to be responsive to two input dimensions, the functional form of the response function in this reduced space had not been unambiguously identified. A second order model based on the logistic function is found to be both necessary and sufficient to accurately describe the neural responses to naturalistic stimuli, accounting for an average of 93% of the mutual information with a small number of parameters. Thus, despite the fact that the stimulus is highly non-Gaussian, the vast majority of the information in the neural responses is related to first and second order correlations. Our results suggest a principled and unbiased way to model multidimensional computations and determine the statistics of the inputs that are being encoded in the outputs.     

Learning shapes spontaneous activity itinerating over memorized states

Learning is a process that helps create neural dynamical systems so that an appropriate output pattern is generated for a given input. Often, such a memory is considered to be included in one of the attractors in neural dynamical systems, depending on the initial neural state specified by an input. Neither neural activities observed in the absence of inputs nor changes caused in the neural activity when an input is provided were studied extensively in the past. However, recent experimental studies have reported existence of structured spontaneous neural activity and its changes when an input is provided. With this background, we propose that memory recall occurs when the spontaneous neural activity changes to an appropriate output activity upon the application of an input, and this phenomenon is known as bifurcation in the dynamical systems theory. We introduce a reinforcement-learning-based layered neural network model with two synaptic time scales; in this network, I/O relations are successively memorized when the difference between the time scales is appropriate. After the learning process is complete, the neural dynamics are shaped so that it changes appropriately with each input. As the number of memorized patterns is increased, the generated spontaneous neural activity after learning shows itineration over the previously learned output patterns. This theoretical finding also shows remarkable agreement with recent experimental reports, where spontaneous neural activity in the visual cortex without stimuli itinerate over evoked patterns by previously applied signals. Our results suggest that itinerant spontaneous activity can be a natural outcome of successive learning of several patterns, and it facilitates bifurcation of the network when an input is provided.     

Recognizing recurrent neural networks (rRNN): Bayesian inference for recurrent neural networks

Recurrent neural networks (RNNs) are widely used in computational neuroscience and machine learning applications. In an RNN, each neuron computes its output as a nonlinear function of its integrated input. While the importance of RNNs, especially as models of brain processing, is undisputed, it is also widely acknowledged that the computations in standard RNN models may be an over-simplification of what real neuronal networks compute. Here, we suggest that the RNN approach may be made computationally more powerful by its fusion with Bayesian inference techniques for nonlinear dynamical systems. In this scheme, we use an RNN as a generative model of dynamic input caused by the environment, e.g. of speech or kinematics. Given this generative RNN model, we derive Bayesian update equations that can decode its output. Critically, these updates define a 'recognizing RNN' (rRNN), in which neurons compute and exchange prediction and prediction error messages. The rRNN has several desirable features that a conventional RNN does not have, e.g. fast decoding of dynamic stimuli and robustness to initial conditions and noise. Furthermore, it implements a predictive coding scheme for dynamic inputs. We suggest that the Bayesian inversion of RNNs may be useful both as a model of brain function and as a machine learning tool. We illustrate the use of the rRNN by an application to the online decoding (i.e. recognition) of human kinematics.     

Some experiences in the estimation of parameters in non-linear differential equations

The author describes a procedure developed by himself and his colleagues for obtaining estimates of the parameters of rate equations, together with information about confidence regions for the estimates. The program has been used successfully for processing results from the chemical engineering industry, with highly non-linear model systems, particularly since temperature was a variable, and the "rate constants" were non-linear combinations of other constants. In biochemical situations, in which investigations are almost always at constant temperature, the non-linearity should not be so extreme, and the procedure may well be capable of dealing with more than 5 to 7 parameters for which it is recommended.     

Simultaneous determination of lot size and production rate at capacity-constrained multiple-product systems

In the capacity constrained manufacturing systems where multiple product types are manufactured, the products are often produced in lots. Although the lot production may increase the system throughput by reducing changeover times, it may also increase production lead time because each item in a large lot has a long waiting time. Hence, a production manager should consider both throughput and lead time at the same time when deciding production lot sizes. This paper, which is an extension to the previous work done in Koo et al. (2007) that assumes homogeneous setup times, addresses a lot sizing problem in the system with multiple product types and unequal setup times. We develop a non-linear optimization model for simultaneous determination of throughput rate and lot size for each product. Since this optimization model cannot be solved analytically, we propose a heuristic solution procedure by analyzing the characteristics of the problem. Some numerical examples are presented to validate the proposed model, and finally the performance of the heuristic procedure is evaluated by comparison with the results of simulation experiments.