Characterizing activity-dependent processes with a piecewise exponential model

The response of some biological processes is dependent on the frequency of stimulation. With first-order processes, the response is driven exponentially to an equilibrium determined by the value of the driving function. When the stimulus or driving function is viewed as switching between constant values the resulting response is piecewise exponential. With periodic excitation, the time course of a point fixed in time relative to the initiation time of each stimulus is shown to be exponential with a rate and steady state that are linearly dependent on the rates and equilibria associated with each component exponential. This linearity can be exploited and leads to a simple estimation procedure for the apparent state-dependent rates.     

A hierarchical neural network model for associative memory

A hierarchical neural network model with feedback interconnections, which has the function of associative memory and the ability to recognize patterns, is proposed. The model consists of a hierarchical multi-layered network to which efferent connections are added, so as to make positive feedback loops in pairs with afferent connections. The cell-layer at the initial stage of the network is the input layer which receives the stimulus input and at the same time works as an output layer for associative recall. The deepest layer is the output layer for pattern-recognition. Pattern-recognition is performed hierarchically by integrating information by converging afferent paths in the network. For the purpose of associative recall, the integrated information is again distributed to lower-order cells by diverging efferent paths. These two operations progress simultaneously in the network. If a fragment of a training pattern is presented to the network which has completed its self-organization, the entire pattern will gradually be recalled in the initial layer. If a stimulus consisting of a number of training patterns superposed is presented, one pattern gradually becomes predominant in the recalled output after competition between the patterns, and the others disappear. At about the same time when the recalled pattern reaches a steady state in he initial layer, in the deepest layer of the network, a response is elicited from the cell corresponding to the category of the finally-recalled pattern. Once a steady state has been reached, the response of the network is automatically extinguished by inhibitory signals from a steadiness-detecting cell. If the same stimulus is still presented after inhibition, a response for another pattern, formerly suppressed, will now appear, because the cells of the network have adaptation characteristics which makes the same response unlikely to recur. Since inhibition occurs repeatedly, the superposed input patterns are recalled one by one in turn.     

Properties of solutions for a chemotaxis system

 We investigate mathematically the system of equations proposed by Chaplain and Stuart [2], to describe the chemotactic response of endothelial cells under the angiogenesis stimulus. In particular, we characterize the steady state endothelial cell density function, and give conditions on the chemotactic parameter k and cell proliferation parameter b that ensure that migration/ proliferation either does or does not occur in steady state. The time dependent problem is also treated. Received 12 September 1995; received in revised form 6 August 1996     

Theoretical characterization of ion channel blockade: ligand binding to periodically accessible receptors

With repetitive stimulation, the time course of use-dependent blockade as assessed by peak membrane ion currents can be described by a sequence of blocking relationships that have the form of recurrence equations. The equations of the sequence describe blockade acquired during each interval of a stimulus where the possibly different binding and unbinding rates are assumed constant during each interval. The solution predicts that use-dependent uptake follows an exponential time course. Furthermore, the exponential uptake rate is a linear function of uptake rates associated with the stimulus time intervals. Similarly, the fraction of blocked channels at steady state is a linear function of the interval dependent blockade equilibria. Several novel tests of consistency between the model and observations are derived from these theoretical results. It is also shown that as the stimulus interval increases to infinity, steady state dissociation constants measured by peak membrane currents are theoretically equivalent to those measured with true equilibrium methods such as radioligand binding studies.     

A xylazine infusion regimen to provide analgesia in sheep.

The efficacy of continuous low-dose xylazine infusion following an initial loading dose in providing analgesia in sheep was examined using an algesimetry method based on a leg lifting response to an electrical stimulus. Sheep received a 5 mg intramuscular injection of xylazine followed by continuous infusion of intravenous xylazine (2mg/h) for 90 min. This treatment resulted in significant increases in the level of current required to elicit a leg lifting response (287% of baseline) and steady state analgesia was maintained from 10 min after the start of the infusion until the end of the experimental period. This protocol appears to be a simple and effective regimen for providing steady state analgesia in sheep.     

Frequency-dependent excitability of "membrane" slow responses of Rabbit left atrial trabeculae in the presence of Ba2+ and high K+

Small trabeculae of rabbit left atrium immersed in TKBa solution (Tyrode with 10 mM K+ and 1 mM Ba2+) were used to study frequency dependence of "membrane" slow response excitability at long cycle lengths (greater than 1 s). In TKBa, stimuli generate graded, low- amplitude (2-15 mV) subliminal responses of variable long duration (up to 450 ms). A full all-or-none slow response is generated when a subliminal response depolarizes the membrane to about--35 mV. Subliminal response amplitude and rate of rise augment with stimulus intensity-duration product. For a fixed stimulus, the subliminal response is larger and faster at higher frequencies. Sudden changes in stimulus frequency or time course induce changes in subliminal response tha take four to eight cycles to attain steady state. For a fixed stimulus, slow response latency shortens progressively during the first few cycles after a sudden increase in frequency or when a rested preparation is excited (latency adaptation phenomenon, LAP). Slow response threshold stimulus requirements decrease during LAP (excitability hysteresis). The degree of excitability hysteresis is dependent on stimulation frequency and is more pronounced at higher frequencies. Frequency sensitivity of subliminal response (which causes frequency sensitivity of slow response excitability) is explained in terms of a transient state of enhancement set up by each stimulus. The enhanced state decays between stimuli with a half-time of approximately 4 s, thus allowing cumulative effects to become evident at rates above 0.1 Hz.     

Modeling isotopomer distributions in biochemical networks using isotopomer mapping matrices

Within the last decades NMR spectroscopy has undergone tremendous development and has become a powerful analytical tool for the investigation of intracellular flux distributions in biochemical networks using (13)C-labeled substrates. Not only are the experiments much easier to conduct than experiments employing radioactive tracer elements, but NMR spectroscopy also provides additional information on the labeling pattern of the metabolites. Whereas the maximum amount of information obtainable with (14)C-labeled substrates is the fractional enrichment in the individual carbon atom positions, NMR spectroscopy can also provide information on the degree of labeling at neighboring carbon atom positions by analyzing multiplet patterns in NMR spectra or using 2-dimensional NMR spectra. It is possible to quantify the mole fractions of molecules that show a specific labeling pattern, i.e., information of the isotopomer distribution in metabolite pools can be obtained. The isotopomer distribution is the maximum amount of information that in theory can be obtained from (13)C-tracer studies. The wealth of information contained in NMR spectra frequently leads to overdetermined algebraic systems. Consequently, fluxes must be estimated by nonlinear least squares analysis, in which experimental labeling data is compared with simulated steady state isotopomer distributions. Hence, mathematical models are required to compute the steady state isotopomer distribution as a function of a given set of steady state fluxes. Because 2(n) possible labeling patterns exist in a molecule of n carbon atoms, and each pattern corresponds to a separate state in the isotopomer model, these models are inherently complex. Model complexity, so far, has restricted usage of isotopomer information to relatively small metabolic networks. A general methodology for the formulation of isotopomer models is described. The model complexity of isotopomer models is reduced to that of classical metabolic models by expressing the 2(n) isotopomer mass balances of a metabolite pool in a single matrix equation. Using this approach an isotopomer model has been implemented that describes label distribution in primary carbon metabolism, i.e., in a metabolic network including the Embden-Meyerhof-Parnas and pentose phosphate pathway, the tricarboxylic acid cycle, and selected anaplerotic reaction sequences. The model calculates the steady state label distribution in all metabolite pools as a function of the steady state fluxes and is applied to demonstrate the effect of selected anaplerotic fluxes on the labeling pattern of the pathway intermediates. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55:831-840, 1997.     

Intensity Characteristics of the Noctuid Acoustic Receptor

Spiking activity of the more sensitive acoustic receptor is described as a function of stimulus intensity. The form of the intensity characteristic depends strongly on stimulus duration. For very brief stimuli, the integral of stimulus power over stimulus duration determines the effectiveness. No response saturation is observed. With longer stimuli (50 msec), a steady firing rate is elicited. The response extends from the spontaneous rate of 20–40 spikes/sec to a saturated firing rate of nearly 700 spikes/sec. The characteristic is monotonic over more than 50 db in stimulus intensity. With very long stimuli (10 sec), the characteristics are nonmonotonic. Firing rates late in the stimulus decrease in response to an increase in stimulus intensity. The non-monotonic characteristics are attributed to intensity-related changes in response adaptation.     

Determination of the optimal priming dose for achieving an isotopic steady state in a two-pool system: application to the study of cholesterol metabolism

The daily administration of labeled cholesterol to humans or animals leads to an isotopic steady state. The specific activity of plasma cholesterol in the isotopic steady state gives information about the fraction of plasma cholesterol derived from endogenous and exogenous sources. A method, based on a two-pool model, is presented which allows the estimation of an optimal priming dose of labeled cholesterol whereby the time to reach the isotopic steady state is reduced to a minimum. A graphic procedure is presented which allows the estimation of an optimal priming dose for two-compartment systems with widely differing characteristics.     

Metabolomics: quantification of intracellular metabolite dynamics

The rational improvement of microbial strains for the production of primary and secondary metabolites ('metabolic engineering') requires a quantitative understanding of microbial metabolism. A process by which this information can be derived from dynamic fermentation experiments is presented. By applying a substrate pulse to a substrate-limited, steady state culture, cellular metabolism is shifted away from its metabolic steady state. With the aid of a rapid sampling and quenching routine it is possible to take 4-5 samples per second during this process, thus capturing the metabolic response to this stimulus. Over 30 metabolites, nucleotides and cofactors from Escherichia coli metabolism can be extracted and analysed using a range of different techniques, for example enzymatic assays, HPLC and LC-MS methods. Using different substrates as limiting and pulse-substrates (glucose, glycerol), different metabolic pathways and substrate uptake systems are investigated. The resulting plots of intracellular metabolite concentrations against time serve as a data basis for modelling microbial metabolic networks.     

Cycloheximide inhibits root water flow and stomatal conductance in aspen (Populus tremuloides) seedlings

Aspen (Populus tremuloides Michx.) roots were treated with cycloheximide, a protein synthesis inhibitor, to examine the role of protein synthesis in root water transport and plant water relations. Within less than 30 min following root application, cycloheximide inhibited steady‐state root water flow rates and 1 h after the application of 1 mm cycloheximide, root hydraulic conductivity had decreased by 85% compared with control roots. However, stomatal conductance showed a significant inhibition only after 2 h following cycloheximide treatment. The reduction in root hydraulic conductivity was accompanied by an almost three‐fold increase in the apoplastic water flow ratio as determined by the trisodium 3‐hydroxy‐5,8,10‐pyrenesulphonate tracer dye. Cycloheximide‐treated roots showed a decrease in the immunostaining intensity of a 32 kDa microsomal protein band that immunoreacted with the AnthPIP1; 1 antibody suggesting a decrease in the membrane aquaporin expression. These changes occurred without severe metabolic disruptions as measured by root respiration. The results point to the importance of protein‐mediated transport in roots and the rapidity of response suggests that protein synthesis may be used as a principal regulatory mechanism in root water transport in aspen.     

Isotopic composition of transpiration and rates of change in leaf water isotopologue storage in response to environmental variables

During daylight hours, the isotope composition of leaf water generally approximates steady‐state leaf water isotope enrichment model predictions. However, until very recently there was little direct confirmation that isotopic steady‐state (ISS) transpiration in fact exists. Using isotope ratio infrared spectroscopy (IRIS) and leaf gas exchange systems we evaluated the isotope composition of transpiration and the rate of change in leaf water isotopologue storage (isostorage) when leaves were exposed to variable environments. In doing so, we developed a method for controlling the absolute humidity entering the gas exchange cuvette for a wide range of concentrations without changing the isotope composition of water vapour. The measurement system allowed estimation of ¹⁸O enrichment both at the evaporation site and for bulk leaf water, in the steady state and the non‐steady state. We show that non–steady‐state effects dominate the transpiration isoflux even when leaves are at physiological steady state. Our results suggest that a variable environment likely prevents ISS transpiration from being achieved and that this effect may be exacerbated by lengthy leaf water turnover times due to high leaf water contents.     

An informational approach to reaction times

Simple reaction time is the minimum time required to respond to a signal such as a steady light or tone. Such a reaction time is taken to be the time required for transmission of a fixed quantity of information, ΔH, from stimulus to subject. That is, information summation replaces energy summation. This information is calculated from consideration of the quantum nature of the stimulus. The theoretically derived equation for reaction time is fitted to experimental data. Piéron's empirical law for reaction time is obtained as an approximation from a proposed informational equation. The exponent in Piéron's law is found to be the same as the exponent in the power law of sensation. Threshold appears to be the smallest stimulus capable of transmitting the quantity of information ΔH.     

Calculation of the Albumin Catabolic Rate in the Non-Steady State

Methods which are in current use for the calculation of the albumin breakdown rate apply only to the steady state animal. In this paper a simple but more general method based on analyses of I¹³¹-albumin tracer data is presented. It utilizes easily measured plasma specific activity and excretory data and is equally applicable to the steady and non-steady states.     

The refractory state of the generator and propagated potentials in a pacinian corpuscle

A propagated potential produced in the Pacinian corpuscle in response to mechanical stimuli leaves a refractory state of 7 to 10 msec. duration. The refractory state is presumably produced at the first intracorpuscular node of Ranvier. The recovery of receptor excitability for producing an all-or-none response to mechanical stimulation follows the same time course as that of the electrically excited axon. Upon progressive reduction of stimulus interval (mechanical), the propagated potential falls progressively to 75 per cent of its resting magnitude and becomes finally blocked within the corpuscle. A non-propagated all-or-none potential, presumably corresponding to activity of the first node, is then detected. The critical firing level for all-or-none potentials increases progressively during the relative refractory period of the all-or-none potential, as the stimulus interval is shortened. Thus generator potentials up to 85 per cent of a propagated potential can be produced in absence of all-or-none activity. Generator potentials show: gradual over-all increase in amplitude and rate of rise as a function of stimulus strength; constant latency; and spontaneous fluctuations in amplitude. A generator potential leaves a refractory state (presumably at the non-myelinated ending) so that the amplitude of a second generator response which falls on its refractory trail is directly related to the time elapsed after the first generator response and inversely to its amplitude. The generator potential develops independently of any refractory state left by a preceding all-or-none potential.     

Effects of pH probe lag on bioreactor mixing time estimation

Mixing time is one of the most important characteristics of bioreactors used to validate computational fluid dynamics (CFD) models in bioprocess. The pH tracer based method is commonly used to estimate mixing time by monitoring the pH change after perturbation to the system. However, pH probe lag is known to introduce errors in the test results. To determine the impact of probe lag on the measurement, pH probe response times were investigated. A second order mathematical model was established to describe the probe response profile and calculate the probe transfer function which was used to correct the empirical results. The corrected mixing time was used to validate the CFD models and also showed that the impact of pH probe response time on mixing time measurement is significant even in larger (pilot and manufacturing) scales. In addition, comparison with a conductance based tracer method is also discussed in this paper.     

Vestibular adaptation to long-term stimuli

Experimental procedures are described which respectively enlarge stimulus duration in vestibular peracceleratory tests, and allow to rule out direct thermal effects on the vestibular nerve during long term calorisations. First experimental results indicate that time course of nystagmus during prolonged stimulations differs markedly in rotational and caloric tests. Whereas there is a distinct decline of response during rotation (in accordance to the predictions of current mathematical models), in caloric tests nystagmus reaches a steady state level, maintained for at least 15 min.     

An integral equation approach to measuring turnover in nonsteady compartmental and distributed systems

Physiological systems are often modelled by a set of compartments. Alternatively they can be described by the diffusion-convection-reaction equations governing distributed systems. The problem considered here is that of identifying a continuously changing input of some metabolite )tracee), endogenous to the system and hence inaccessible, when a nonlinear or time-varying component is also introduced into the loss parameter, as for example through feedback mechanisms. A tracer is used to determine the steady-state impulse response under time-invariant, linear conditions. A known input of tracer is also administered when the system is driven out of steady state. The integral equations developed utilize the predetermined impulse response, the measured concentrations of both tracer and tracee (output) in some region of the system to estimate the changing loss parameter and the unknown input in a continuous fashion.     

Dynamics of turtle horizontal cell response

The small- and large-field (cone) horizontal cells produce similar dynamic responses to a stimulus whose mean luminance is modulated by a white-noise signal. Nonlinear components increase with an increase in the mean luminance and may produce a mean square error (MSE) of up to 15%. Increases in the mean luminance of the field stimulus bring about three major changes: the incremental sensitivity defined by the amplitude of the kernels decreases in a Weber-Fechner fashion; the waveforms of the kernels are transformed from monophasic (integrating) to biphasic (differentiating); the peak response time of the kernels becomes shorter and the cells respond to much higher-frequency inputs. The dynamics of the horizontal cell response also depend on the area of the retina stimulated. Smaller spots of light produce monophasic kernels of a longer peak response time. The presence of a steady background produces three major changes in the spot kernels: the kernel's amplitude becomes larger (incremental sensitivity increases); the peak response times become shorter; the waveform of the kernels changes in a fashion similar to that observed with an increase in the mean luminance of the field stimulus. A similar enhancement in the incremental sensitivity by a steady background has also been observed in catfish, which shows that this phenomenon is a common feature of the horizontal cells in the lower vertebrate retina.