Estimation of Instantaneous Gas Exchange in Flow-Through Respirometry Systems: A Modern Revision of Bartholomew's Z-Transform Method

Flow-through respirometry systems provide accurate measurement of gas exchange over long periods of time. However, these systems have limitations in tracking rapid changes. When an animal infuses a metabolic gas into the respirometry chamber in a short burst, diffusion and airflow in the chamber gradually alter the original signal before it arrives at the gas analyzer. For single or multiple bursts, the recorded signal is smeared or mixed, which may result in dramatically altered recordings compared to the emitted signal. Recovering the original metabolic signal is a difficult task because of the inherent ill conditioning problem. Here, we present two new methods to recover the fast dynamics of metabolic patterns from recorded data. We first re-derive the equations of the well-known Z-transform method (ZT method) to show the source of imprecision in this method. Then, we develop a new model of analysis for respirometry systems based on the experimentally determined impulse response, which is the response of the system to a very short unit input. As a result, we present a major modification of the ZT method (dubbed the ‘EZT method’) by using a new model for the impulse response, enhancing its precision to recover the true metabolic signals. The second method, the generalized Z-transform (GZT) method, was then developed by generalizing the EZT method; it can be applied to any flow-through respirometry system with any arbitrary impulse response. Experiments verified that the accuracy of recovering the true metabolic signals is significantly improved by the new methods. These new methods can be used more broadly for input estimation in variety of physiological systems.     

Probabilistic Firing of Neurons Considered as a First Passage Problem

A formalized neuron receiving unitary excitatory impulses at random is considered. Each impulse provokes an effect of equal magnitude and of a duration not constant for each impulse, but which varies according to an exponential distribution. The effects sum until a threshold is reached when a response occurs. The distribution of intervals between successive responses is computed and compared with those obtained from a model in which the effects decay exponentially with time. Upon introducing inhibitory impulses also, the theory is applied to data on discharge characteristics of driven and spontaneously active thalamic neurons reported in the literature.     

The response of a model neurone to a white noise input

We consider a simple model of a neurone in which the input voltage is integrated to form the somatic potential, and a pulse is emitted when this reaches a threshold; the somatic potential is then reset to its resting value. We subject this model to a white-noise input, and evaluate the cross correlation between input white noise and the output pulse train; this is proportional to the small-signal impulse response of the model. Some numerical estimations are presented.     

Effect of doublet impulse sequences in the crayfish claw opener muscles and the computer-simulated neuromuscular synapse

The effects of doublet impulse sequences of the excitatory motor axon on the movement of the claw opener muscles in the crayfish were examined. The excitatory motor axon was stimulated electrically with various patterns of doublet impulse sequences generated by a digital computer. Doublet impulse sequences of stimulation produced a larger sustained movement than an uniform impulse sequences at the same mean rate of stimulation. The movement was largest when the interval between the impulses of a doublet was about 5 ms. This interval generated a movement amplitude 25% greater than that for the uniform impulse sequence. A simple model was formulated to stimulate the neuromuscular synapse of the claw opener muscle. The relationship between stimulation sequences with alternating long and short intervals and responses (firing probabilities) of the neuromuscular synapse at the same mean rate was investigated. The responses was classified into two typical types which are noneffective Type I and effective Type II to the absolute refractory period (ARP). The characteristics which are larger responses with short intervals in Type I and reduction of responses in the ARP region of Type II formed a plateau peak of the experimental results. By incorporating the reduction of end-plate potential (EPP) as a property of nonlinear rule for temporal summation into the model, it was shown that Type I response is maximal with a plateau peak at short interval, agreeing well with the experimental results from the claw opener muscles.     

Pooling of impulse sequences,with emphasis on applications to neuronal spike data

Summary A mathematical model is presented that is supposed to describe those types of neuronal discharges which show a preponderance of short intervals, as well as one or more preferred intervals of a longer duration. It is assumed that via two channels impulses impinge upon a nerve cell and that each impulse gives rise to a response. The intervals between impulses in one channel are distributed according to an exponential, or an exponential-like, function; those in the other channel are distributed according to a monomodal, or a multimodal, function.The interval distributions and the expectation density (auto-correlation) functions of the model are in particular compared with data on thalamic neuron discharge patterns reported in the literature.The properties of superimposed time series of events would seem to be of a wider interest, stretching beyond the field of theoretical neurophysiology. It is indicated how the theory is of use in the detection of hidden rhythms in records which are composed of a mixture of different signals.     

Effect doublet impulse sequences on the transient and steady responses in the computer-simulated nerve cell

The effects of doublet impulse sequences of the excitatory axon on the output response as firing probability (pr.) in the computer-simulated nerve cell were examined. A simple model was formulated to simulate the nerve cell, including the property that the resetting potential is influenced by the final membrane potential in the previous stage before firing. The relationship between input sequences with alternating long and short interval at the same mean rate and the transient and steady responses of the nerve cell was investigated. In this simulation, three summarized results were obtained: i) The responses were very sensitive to changing small size of excitatory post-synaptic potential (EPSP), especially in the firing stage of the transient state. ii) In the transient state, the size of characteristic area of responses was depending upon the size of absolute refractory period (ARP). The rise for shorter intervals was faster than that for longer intervals, agreeing well with part of the experimental results from the crayfish claw opener muscles. The transient responses were almost finished before the fifth firing. iii) In the steady state, the doublet impulse sequences usually produced the minimum response or valley-like response at which the doublet interval T _dwas 20 and/or 25 ms. These effects related to the characteristic areas in the transient responses.     

Repetitive firing: quantitative analysis of encoder behavior of slowly adapting stretch receptor of crayfish and eccentric cell of Limulus

Techniques developed for determining summed encoder feedback in conjunction with the leaky integrator and variable-gamma models for repetitive firing are applied to spike train data obtained from the slowly adapting crustacean stretch receptor and the eccentric cell of Limulus. Input stimuli were intracellularly applied currents. Analysis of data from cells stringently selected by reproducibility criteria gave a consistent picture for the dynamics of repetitive firing. The variable-gamma model with appropriate summed feedback was most accurate for describing encoding behavior of both cell types. The leaky integrator model, while useful for determining summed feedback parameters, was inadequate to account for underlying mechanisms of encoder activity. For the stretch receptor, two summed feedback processes were detected: one had a short time constant; the other, a long one. Appropriate tests indicated that the short time constant effect was from an electrogenic sodium pump, and the same is presumed for the long time constant summed feedback. Both feedbacks show seasonal and/or species variations. Short hyperpolarizing pulses inhibited the feedback from the long time constant process. The eccentric cell also showed two summed feedback processes: one is due to self inhibition, the other is postulated to be a short time constant electrogenic sodium pump similar to that described in the stretch receptor.     

Neuron's firing time

The interspike interval distribution of neuronal firing is analyzed by a model that assumes unit effect EPSP's lasting an exponential length of time. The model allows a general interarrival distribution; this contrasts with the numerous models requiring Poisson arrivals. The Laplace transform of the time to firing, modelled as the first passage time to a fixed arbitrary threshold level, is found. Comparisons are made for exponential and regular interarrivals using the first two moments of the time to firing. Surprisingly, the mean and variance of the time to reach any threshold level greater than one is greater for regular arrivals for any ratio of mean interarrival intervals to mean EPSP duration greater than 0.6.     

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.     

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.     

基于物质流分析方法的唐山市经济与环境关系的协整检验和分解

利用物质流分析方法(MFA)建立物质流账户,分析唐山市在经济 环境系统运行中物质投入量与产出量的阶段特征及物质投入和产出强度对经济发展的影响程度;使用计量经济学模型,分别对国内生产总值(GDP)、直接物质投入(DMI)和直接废弃物排放(DPO)进行单位根检验、Johansen协整检验、向量误差修正分析、脉冲响应和方差分解分析,探索了各指标之间的双向作用机制和长期关系.结果表明: 1992—2011年,唐山市DMI和DPO 均呈增长趋势,DMI的增幅高于DPO.DMI投入强度呈增长趋势,DPO产出强度呈波动下降趋势.GDP与DMI、DPO之间存在着长期稳定协整关系,指标间的作用关系经历了由波动到逐步平稳的过程.DMI与DPO在短期内会对经济发展起到较强的正向冲击作用,但是经济-环境系统会逐步消化这些影响,并对系统内外指标进行短期的动态调整,最终使系统表现出一种长期的均衡关系;经济发展受到经济规模效应的影响逐步增加.将各指标对GDP的贡献度予以分解,其中,DMI的贡献度上升,GDP的贡献度下降,DPO的贡献度变化不大.总体上,唐山市的经济发展遵循了资源型城市的传统生产轨迹,较大程度上依赖于物质投入,高能源消耗又加剧了环境污染.     

Mathematical models of cochlear nucleus onset neurons: I. Point neuron with many weak synaptic inputs

The cochlear nucleus (CN) presents a unique opportunity for quantitatively studying input-output transformations by neurons because it gives rise to a variety of different response types from a relatively homogeneous input source, the auditory nerve (AN). Particularly interesting among CN neurons are Onset (On) neurons, which have a prominent response to the onset of sustained sounds followed by little or no response in the steady-state. On neurons contrast sharply with their AN inputs, which respond vigorously throughout stimuli. On neurons can entrain to stimuli (firing once per cycle of a periodic stimulus) at up to 1000 Hz, unlike their AN inputs. To understand the mechanisms underlying these response patterns, we tested whether an integrate-to-threshold point-neuron model with a fixed refractory period can account for On discharge patterns for tones, systematically examining the effect of membrane time constant and the number and strength of the exclusively excitatory AN synaptic inputs. To produce both onset responses to high-frequency tone bursts and entrainment to a broad range of low-frequency tones, the model must have a short time constant (0.125 ms) and a large number (>100) of weak synaptic inputs, properties that are consistent with the electrical properties and anatomy of On-responding cells. With these parameters, the model acts like a coincidence detector with a threshold-like relationship between the instantaneous discharge rates of the output and the inputs. Onset responses to high-frequency tone bursts result because the threshold effect enhances the initial response of the AN inputs and suppresses their relatively lower sustained response. However, when the model entrains across a broad range of frequencies, it also produces short interspike intervals at the onset of high-frequency tone bursts, a response pattern not found in all types of On neurons. These results show a tradeoff, that may be a general property of many neurons, between following rapid stimulus fluctuations and responding without short interspike intervals at the onset of sustained stimuli.     

Koinzidenzfilter mit kurzen Impulsen

Summary Coincidence filters consist of one or more threshold elements (e.g. neurons or monostable multivibrators, extended by multiple input gates). They permit the propagation of an impulse train applied to the input only if its repetition rate does not exceed an upper and a lower boundary value. The difference between the upper and the lower boundary value may be defined as the functional bandwidth of the coincidence filter. The functional bandwidth is one of the most interesting characteristic figures of a coincidence filter. By means of this definition, the coincidence filter may be described as a device selecting quickly those impulse trains the repetition rates of which lie within the functional bandwidthThe functional bandwidth depends on the parameters of the impulse trains and of the coincidence filter. This gives rise to the question, which minimal bandwidth could be realized by coincidence filters.On the initiation by Tischner the properties of coincidence filters operated by rectangular impulses have been investigated by Schie f and by Kosel. Rectangular impulses have the advantage, that moderate variations of the amplitudes do not disturb the coincidence. In this case however very small impulse durations are required for the realization of small bandwidth.In the present paper the operation of coincidence filters by non rectangular impulses has been considered. Having the shape of an excitatory post-synaptic potential of motoneurons, the impulses are completely determined by the rising phase and the falling phase. These impulses have been termed short impulses in contrast to the rectangular impulses, which are long, compared to the duration of their rising and falling phases. The width of the short impulses decreases with increasing measuring level. Close to the amplitude the width becomes very small, which theoretically provides a very small functional bandwidth. The practical realization of very small functional bandwidth is heavily limited by the big variations which will be caused by minute alterations of the amplitudes as introduced by noise. The variation of the functional bandwidth caused by 1% alteration of the amplitude has been termed the error factor. In the present paper some relationships between the following four quantities have been worked out: realizable functional bandwidth, tolerable variation of functional bandwidth, error factor, and given variation of the amplitudes and thresholds (noise).The short impulses have been piecewise approximated by analytical functions (parabolic and hyperbolic) which in general permits an analytical treatment of the problems.     

A model of the human visual system in its response to certain classes of moving stimuli

The purpose of this study is to construct a functional model of the human visual system in its response to certain classes of moving stimuli.Experimental data are presented describing the interdependence of the input variables, temporal frequency, spatial period, etc., for two constant response states, viz. threshold motion response and threshold flicker response. On the basis of these data, two basic units are isolated, a vertical (V) unit and a horizontal (H) unit. The H-unit is identified with the Reichardt multiplier (Reichardt and Varju, 1959), and the V-unit with the de Lange filter (de Lange, 1954).A definition of the general motion response of the H-units is obtained, and this is then reduced to an expression which may be applied directly to the observed motion response data. By this method, Thorson's simplification of the Reichardt scheme (Thorson, 1966) is adopted for the H-unit and total and relative (population) weighting factors, associated with the H-unit output, are defined.In order to reconcile the theoretical square-wave threshold motion response with the experimental data, Thorson's simplification is modified with the introduction of a low-pass filter on the output. The amended scheme is shown to predict a (temporal) frequency-dependent phase-sensitivity. This prediction is tested experimentally, and its validity indicated.     

Excitation properties of the squid axon membrane and model systems with current stimulation. Statistical evaluation and comparison.

The space-clamped squid axon membrane and two versions of the Hodgkin-Huxley model (the original, and a strongly adapting version) are subjected to a first order dynamic analysis. Stable, repetitive firing is induced by phase-locking nerve impulses to sinusoidal currents. The entrained impulses are then pulse position modulated by additional, small amplitude perturbation sinusoidal currents with respect to which the frequencies response of impulse density functions are measured. (Impulse density is defined as the number of impulses per unit time of an ensemble of membranes with each membrane subject to the same stimulus). Two categories of dynamic response are observed: one shows clear indications of a corner frequency, the other has the corner frequency obscured by dynamics associated with first order conductance perturbations in the interspike interval. The axon membrane responds with first order perturbations whereas the unmodified Hodgkin-Huxley model does not. Quantitative dynamic signatures suggest that the relaxation times of axonal recovery excitation variables are twice as long as those of the corresponding model variables. A number of other quantitative differences between axon and models, including the values of threshold stimuli are also observed.     

The spatial properties of a model neuron increase its coding range

The coding properties of one-compartment and two-compartment model neurons are compared. The membrane depolarization in both models is described as a deterministic leaky integrator. Interspike intervals are identified with the periods between reset of the depolarization after firing and consecutive crossing of a fixed firing threshold. The two-point model has an input in the dendritic compartment and an output in the trigger-zone compartment. It is shown that the sensitivity threshold for the two-point model is shifted to the larger values of the input intensity with respect to the sensitivity threshold of its single-point counterpart. Further, its coding range is substantially larger than the coding range of the single-point model.     

Phase-locked responses in the Limulus lateral eye. Theoretical and experimental investigation.

The 1:1 phase locking of the neural discharge to sinusoidally modulated stimuli was investigated both theoretically and experimentally. On the theoretical side, a neural encoder model, the self-inhibited leaky integrator, was considered, and the phase of the locked impulse was computed for each frequency in the locking range by imposing the condition that the "leaky integral" u(t) of the driving signal should reach the threshold for the first time one stimulus period after the preceding impulse. As u(t) can be a nonmonotonic function, this approach leads to results that sometimes differ from those reported in the literature. It turns out that the phase excursion is often much smaller than the values of about 180 degrees predicted from previous analysis. Moreover, our analysis shows a peculiar effect; the phase locking frequency range narrows when the input modulation depth increases. The theoretical predictions are then compared with phase-locked discharge patterns recorded from visual cells of the Limulus lateral eye, stimulated by sinusoidally modulated light or depolarizing current. The phases of the locked spikes at each of a number of modulation frequencies have been measured. The predictions offered by the model fit the experimental data, although there are some difficulties in determining the effective driving signal.     

The parameters of the stochastic leaky integrate-and-fire neuronal model

Five parameters of one of the most common neuronal models, the diffusion leaky integrate-and-fire model, also known as the Ornstein-Uhlenbeck neuronal model, were estimated on the basis of intracellular recording. These parameters can be classified into two categories. Three of them (the membrane time constant, the resting potential and the firing threshold) characterize the neuron itself. The remaining two characterize the neuronal input. The intracellular data were collected during spontaneous firing, which in this case is characterized by a Poisson process of interspike intervals. Two methods for the estimation were applied, the regression method and the maximum-likelihood method. Both methods permit to estimate the input parameters and the membrane time constant in a short time window (a single interspike interval). We found that, at least in our example, the regression method gave more consistent results than the maximum-likelihood method. The estimates of the input parameters show the asymptotical normality, which can be further used for statistical testing, under the condition that the data are collected in different experimental situations. The model neuron, as deduced from the determined parameters, works in a subthreshold regimen. This result was confirmed by both applied methods. The subthreshold regimen for this model is characterized by the Poissonian firing. This is in a complete agreement with the observed interspike interval data. Action Editor: Nicolas Brunel     

Growing of a Fuzzy Recurrent Artificial Neural Network (FRANN) for pattern classification

This paper describes a method for growing a recurrent neural network of fuzzy threshold units for the classification of feature vectors. Fuzzy networks seem natural for performing classification, since classification is concerned with set membership and objects generally belonging to sets of various degrees. A fuzzy unit in the architecture proposed here determines the degree to which the input vector lies in the fuzzy set associated with the fuzzy unit. This is in contrast to perceptrons that determine the correlation between input vector and a weighting vector. The resulting membership value, in the case of the fuzzy unit, is compared with a threshold, which is interpreted as a membership value. Training of a fuzzy unit is based on an algorithm for linear inequalities similar to Ho-Kashyap recording. These fuzzy threshold units are fully connected in a recurrent network. The network grows as it is trained. The advantages of the network and its training method are: (1) Allowing the network to grow to the required size which is generally much smaller than the size of the network which would be obtained otherwise, implying better generalization, smaller storage requirements and fewer calculations during classification; (2) The training time is extremely short; (3) Recurrent networks such as this one are generally readily implemented in hardware; (4) Classification accuracy obtained on several standard data sets is better than that obtained by the majority of other standard methods; and (5) The use of fuzzy logic is very intuitive since class membership is generally fuzzy.