Adiabatic solution for the Ohmic cable equation. An homogeneous cell, prospects for electronic measurements]

Exact and adiabatic electrotonic solutions [1] were calculated for reconstructed motoneurone and hippocampal interneurone in case of linear and exponential ramp stimulation by the fixed current, potential or homogenous electric field. For the rising exponential ramp the solutions are identical. In case of the decaying exponent the adiabatic solution becomes an asymptote for the exact one if the stimulus decays slower than relaxation of the initial conditions in the cell. If the stimulus decays faster, the asymptote is the current or potential axis, depending on the stimulation mode. For electrotonically short cell, the exact solution approaches the asymptote faster. The solution for the exponentially rising field does not depend on the dendritic tree configuration and depends only on the effective electrotonic length of the neurone. It could be useful to apply ramp stimulation, especially exponential ramp of the electric field, to estimate electrotonic parameters of cells.     

Determination of time-constants in cables of finite length

Current flow in cylindrical nerve and muscle fibre has been analysed in terms of a mathematical model leading to a linear partial differential equation for the voltage as a function of both position and time. In the case of a one-dimensional cable subject to a step input of current, the solution will consist of a steady-state behaviour preceded by an initial transient. The electrical properties of the fibre or cable itself determine a length-constant, λ, which can be determined experimentally from the steady-state response, and a time-constant, τ, which must be found from the initial transient. When the cable is infinite and when there is a single input electrode, an exact solution can be produced which enables ready determination of the time-constant τ. Two complications arise in experimental practice, however. In the first place, the fibre has finite length, and in the second, two spatially separated stimulation electrodes are often required. We thus analyse a more complicated and more general situation. The linearity of the membrane properties, however, allows the solution to the more general case to be built up by superposition of solutions from the simpler case (equivalent to the classical method of images). We also approximate the Hodgkin and Rushton solution by asymptotic formulae in order to allow more tractable expressions for the exact solution. We are thus able to give a method for the ready evaluation of the time constant τ under more general conditions.     

Cable Theory for Finite Length Dendritic Cylinders with Initial and Boundary Conditions

The cable equation is solved in the Laplace transform domain for arbitrary initial and boundary conditions. The cable potential is expressed directly in terms of the impedance of the terminations and the cable electrotonic length. A computer program is given to invert the transform. Numerical solutions may be obtained for any particular model by inserting expressions describing the terminations and parameter values into the program, without further computation by the modeler. For a finite length cable, sealed at one end, the solution is expressed in terms of the ratio of the termination impedance to the impedance of the finite length cable, a generalization of the steady-state conductance ratio. Analysis of a model of a soma with several primary dendrites shows that the dendrites may be lumped into one equivalent cylinder if they have the same electrotonic length, even though they may vary in diameter. Responses obtained under voltage clamp are conceptually predictable from measurements made under current clamp, and vice versa. The equalizing time constants of an infinite series expression of the solution are the negative reciprocals of the roots of the characteristic equation. Examination of computed solutions shows that solutions which differ theoretically may be indistinguishable experimentally.     

Random currents through nerve membranes

The linear cable equation with uniform Poisson or white noise input current is employed as a model for the voltage across the membrane of a onedimensional nerve cylinder, which may sometimes represent the dendritic tree of a nerve cell. From the Green's function representation of the solutions, the mean, variance and covariance of the voltage are found. At large times, the voltage becomes asymptotically wide-sense stationary and we find the spectral density functions for various cable lengths and boundary conditions. For large frequencies the voltage exhibits “1/f ^3/2 noise”. Using the Fourier series representation of the voltage we study the moments of the firing times for the diffusion model with numerical techniques, employing a simplified threshold criterion. We also simulate the solution of the stochastic cable equation by two different methods in order to estimate the moments and density of the firing time.     

Some exact solutions of a generalized Fisher equation related to the problem of biological invasion

The problem of biological invasion in a model single-species community is considered, the spatiotemporal dynamics of the system being described by a modified Fisher equation. For a special case, we obtain an exact solution describing self-similar growth of the initially inhabited domain. By comparison with numerical solutions, we show that this exact solution may be applicable to describe an early stage of a biological invasion preceding the propagation of the stationary travelling wave. Also, the exact solution is applied to the problem of critical aggregation to derive sufficient conditions of population extinction. Finally, we show that the solution we obtain is in agreement with some data from field observations.     

Note on the validity of constant field assumption in relation to the exact solution of Nernst--Planck equations

By means of the exact solution of the Nernst-Planck equations, it can be demonstrated that the constant electric field assumption is incompatible with the stationary flows of ions in the homogeneously charged membrane. The exact solution of the Nernst-Planck equation system is readily given by the power series expansion method. The constant field assumption is valid for membrane in which the fixed charge can move freely and neutralize the charge of permeating ions.The parabolical electric potential is shown to be incompatible with the homogeneously charged membranes including the neutral membranes.     

General solution of cable theory with both ends sealed

General solution of the cable theory with both ends sealed when injecting an arbitrary current at an arbitrary point of the cable is presented, which is a time-dependent transient solution. The solution is an infinite series, each term of which is the product of a cosine term including a position variable only and an exponential term including a time variable only. The general solution contains almost all solutions reported hitherto as particular cases and the mutual relations among the various solutions of quite different forms are clarified by this general solution. Moreover the shorter the cable becomes, the more rapidly this solution converges, therefore it is useful for an analysis of the short cable in the case where the relative deviation error may grow large. The truncation error can also be estimated as the solution is an infinite series of simple functions.     

A synthetic strand of cardiac muscle: its passive electrical properties

The passive electrical properties of synthetic strands of cardiac muscle, grown in tissue culture, were studied using two intracellular microelectrodes: one to inject a rectangular pulse of current and the other to record the resultant displacement of membrane potential at various distances from the current source. In all preparations, the potential displacement, instead of approaching a steady value as would be expected for a cell with constant electrical properties, increased slowly with time throughout the current step. In such circumstances, the specific electrical constants for the membrane and cytoplasm must not be obtained by applying the usual methods, which are based on the analytical solution of the partial differential equation describing a one-dimensional cell with constant electrical properties. A satisfactory fit of the potential waveforms was, however, obtained with numerical solutions of a modified form of this equation in which the membrane resistance increased linearly with time. Best fits of the waveforms from 12 preparations gave the following values for the membrane resistance times unit length, membrane capacitance per unit length, and for the myoplasmic resistance: 1.22 plus or minus 0.13 x 10-5 omegacm, 0.224 plus or minus 0.023 uF with cm-minus 1, and 1.37 plus or minus 0.13 x 10-7 omegacm-minus 1, respectively. The value of membrane capacitance per unit length was close to that obtained from the time constant of the foot of the action potential and was in keeping with the generally satisfactory fit of the recorded waveforms with solutions of the cable equation in which the membrane impedance is that of a single capacitor and resistor in parallel. The area of membrane per unit length and the cross-sectional area of myoplasm at any given length of the preparation were determined from light and composite electron micrographs, and these were used to calculate the following values for the specific electrical membrane resistance, membrane capacitance, and the resistivity of the cytoplasm: 20.5 plus or minus 3.0 x 10-3 omegacm-2, l.54 plus or minus 0.24 uFWITHcm-minus 2, and 180 plus or minus 34 omegacm, respectively.     

Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis

The existence of spatially localized solutions in neural networks is an important topic in neuroscience as these solutions are considered to characterize working (short-term) memory. We work with an unbounded neural network represented by the neural field equation with smooth firing rate function and a wizard hat spatial connectivity. Noting that stationary solutions of our neural field equation are equivalent to homoclinic orbits in a related fourth order ordinary differential equation, we apply normal form theory for a reversible Hopf bifurcation to prove the existence of localized solutions; further, we present results concerning their stability. Numerical continuation is used to compute branches of localized solution that exhibit snaking-type behaviour. We describe in terms of three parameters the exact regions for which localized solutions persist.     

An artificial membrane-cable: decay and delay of electrical potentials along a lipid bilayer with ion channels

A model membrane is described which exhibits the properties of a neurite with respect to passive propagation of electrical potentials. A groove in a glass plate is covered by a black lipid membrane of glycerol monooleate. Gramicidin is incorporated. The stationary and transient response of the assembly is tested by two experiments: (i) One end of the groove is clamped at a constant voltage. The voltage at the other end and the total electrical current are measured. (ii) A charge pulse is injected at one end of the groove. The time-dependent voltage at the other end is measured. The results with respect to the lateral decay and delay of voltage are in quantitative agreement with the stationary and transient solutions of Kelvin's equation for a homogeneous cable. If gramicidin is incorporated unevenly along the membrane, the lateral decay of voltage is found to be asymmetric with respect to both directions. The cable is a partial one-way transmission line.     

The cylindrical cell with a time-variant membrane resistance. Measuring passive parameters.

The passive electrical properties of a cable can be measured by injecting a step of current at a point and fitting the resulting potentials at several positions along the cable with analytic solutions of the cable equation. An error analysis is presented for this method (which is based on constant membrane resistance) when the membrane resistance is not constant, but increases linearly with time. The increase of rm produces a "creep" in the membrane potential at long times, as observed in cardiac, skeletal, and smooth muscle. The partial differential equation describing the time-varying cable was solved numberically for a step of current and these "data" were fit by standard constant-resistance methods. Comparing the resulting parameter values with the known true values, we suggest that a correction of the standard methods is not satisfactory for resistance changes of the kind observed; instead, the cable equation must be solved again for the particular form of rm(t). The practical implementation of a method by Adrian and Peachey for measuring the membrane capacitance and an approximate method for estimating the rate-of-change of membrane resistance are discussed in appendices.     

Temporal modulation of luminance adapts time constant of fly movement detectors

The time constant of movement detectors in the fly visual system has been proposed to adapt in response to moving stimuli (de Ruyter van Steveninck et al. 1986). The objective of the present study is to analyse, whether this adaptation can be induced as well, if the luminance of a stationary uniform field is modulated in time. The experiments were done on motion-sensitive wide-field neurones of the lobula plate, the posterior part of the third visual ganglion of the blowfly, calliphora erythrocephala. These cells are assumed to receive input from large retinotopic arrays of movement detectors. In order to demonstrate that our results concern the properties of the movement detectors rather than those of a particular wide-field cell we recorded from two different types of them, the H1- and the HSE-cell. Both cell types respond to a brief movement stimulus in their preferred direction with a transient excitation. This response decays about exponentially. The time constant of this decay reflects, in a first approximation, the time constant of the presynaptic movement detectors. It was determined after prestimulation of the cell by the following stimuli: (a) periodic stationary grating; (b) uniform field, the intensity of which was modulated sinusoidally in time (flicker stimulation); (c) periodic grating moving front-to-back; (d) periodic grating moving back-to-front. The decay of the response is significantly faster not only after movement but also after flicker stimulation as compared with pre-stimulation with a stationary stimulus. This is interpreted as an adaptation of the movement detector's time constant. The finding that flicker stimulation also leads to an adaptation shows that movement is not necessary for this process. Instead the adaptation of the time constant appears to be governed mainly by the temporal modulation (i.e., contrast frequency) of the signal in each visual channel.     

Cell-mass maximization in fed-batch cultures

Complete solutions are provided for cell-mass maximization for free and fixed final times and constant and variable yields. The optimal feed rate profile is a concatenation of maximum, minimum and singular feed rates. The exact sequence and duration of each feed rate depends primarily on the initial substrate concentration, and degenerate cases arise due to the magnitude constraint on the feed rate and the length of final time t _f. When the final time is free and not in the performance index, it is infinite for constant yield so that any form of feed rate leads to the same amount of cells, while for variable yield the singular feed rate is exponential and maximizes the yield. For fixed final time the singular feed rate for constant yield is exponential and maximizes the specific growth rate by maintaining the substrate concentration constant, while for variable yield, it is semi-exponential and the substrate concentration starts near the maximum specific growth rate and moves toward the maximum yield. A simple sufficient condition for existence of singular feed rate requires an existence of a region bounded by the maxima of specific growth and cellular yield. Otherwise, the optimal feed rate profile is a bang-bang type and the bioreactor operates in batch mode.     

A generalized tapering equivalent cable model for dendritic neurons

A mathematical model has been developed which collapses a dendritic neuron of complex geometry into a single electrotonically tapering equivalent cable. The modified cable equation governing the transient distribution of subthreshold membrane potential in a branching tree is transformed, becoming amenable to analytic solution. This transformation results in a Riccati differential equation whose six solutions (expressed in terms of elementary functions) control the amount and degree of taper found in the equivalent cable model. To illustrate the theory, an analytic solution (in series form) of the modified cable equation is obtained for a voltage-clamp present at the soma of a quadratically tapering equivalent cable whose distal end is sealed.     

Probabilistic approach to lysozyme crystal nucleation kinetics

Nucleation of lysozyme crystals in quiescent solutions at a regime of progressive nucleation is investigated under an optical microscope at conditions of constant supersaturation. A method based on the stochastic nature of crystal nucleation and using discrete time sampling of small solution volumes for the presence or absence of detectable crystals is developed. It allows probabilities for crystal detection to be experimentally estimated. One hundred single samplings were used for each probability determination for 18 time intervals and six lysozyme concentrations. Fitting of a particular probability function to experimentally obtained data made possible the direct evaluation of stationary rates for lysozyme crystal nucleation, the time for growth of supernuclei to a detectable size and probability distribution of nucleation times. Obtained stationary nucleation rates were then used for the calculation of other nucleation parameters, such as the kinetic nucleation factor, nucleus size, work for nucleus formation and effective specific surface energy of the nucleus. The experimental method itself is simple and adaptable and can be used for crystal nucleation studies of arbitrary soluble substances with known solubility at particular solution conditions.     

The bactericidal activity of a methyl and propyl parabens combination: isothermal and non-isothermal studies.

The effect of temperature on the kill rate of Escherichia coli by methyl and propyl parabens was studied. The kill kinetics was first order. It was shown that the Arrhenius equation provided a good model for describing the relationship between the first order rate constant and the temperature. The activation energy was found to be 274 kJ/mol for exponential phase cells and 168 kJ/mol for stationary phase cells. Exponential phase cells were much more susceptible to the lethal effects of the parabens than were the stationary phase cells. For example, at 34 degrees C stationary phase cells, in chemically defined media, had a kill rate constant of 0.072/h while the corresponding value for exponential phase cells was 0.238/h. In water the rate of kill for exponential phase cells was even faster giving a rate constant of 5.25/h at 34 degrees C. Non-isothermal kinetic testing was not found to be useful for modelling bacterial kill kinetics because we could not achieve the precision required in bacterial enumeration.     

Model transformations to evaluate transient thermal responses at a tissue surface

For a spatially distributed model describing the transient temperature response of a thermistor-tissue system, Wei et al. [J. Biomech. Eng., 117:74-85, 1995] obtained an approximate transformation for fast analysis of the temperature response at the tissue surface. This approximate transformation reduces the model to a single ordinary differential equation. Here, we present an exact transformation that yields a single differential-integral equation. Numerical solutions from the approximate and exact transformations were compared to evaluate the differences with several sets of parameter values. The maximum difference between the exact and approximate solutions did not exceed 15 percent and occurred for only a short time interval. The root-mean-square error of the approximate solution was no more than 5 percent and within the level of experimental noise. Under the experimental conditions used by Wei et al., the approximate transformation is justified for estimating model parameters from transient thermal responses.     

The bactericidal activity of a methyl and propyl parabens combination: isothermal and non-isothermal studies

D. GILLILAND, A. LI WAN PO AND E. SCOTT. 1992. The effect of temperature on the kill rate of Escherichia coli by methyl and propyl parabens was studied. The kill kinetics was first order. It was shown that the Arrhenius equation provided a good model for describing the relationship between the first order rate constant and the temperature. The activation energy was found to be 274 kJ/mol for exponential phase cells and 168 kJ/mol for stationary phase cells. Exponential phase cells were much more susceptible to the lethal effects of the parabens than were the stationary phase cells. For example, at 34°C stationary phase cells, in chemically defined media, had a kill rate constant of 0.072/h while the corresponding value for exponential phase cells was 0.238/h. In water the rate of kill for exponential phase cells was even faster giving a rate constant of 5.25/h at 34°C. Non-isothermal kinetic testing was not found to be useful for modelling bacterial kill kinetics because we could not achieve the precision required in bacterial enumeration.     

Probe diffusion in aqueous poly(vinyl alcohol) solutions studied by fluorescence correlation spectroscopy

We report fluorescence correlation spectroscopy measurements of the translational diffusion coefficient of various probe particles in dilute and semidilute aqueous poly(vinyl alcohol) solutions. The range of sizes of the particles (fluorescent molecules, proteins, and polymers) was chosen to explore various length scales of the polymer solutions as defined by the polymer-polymer correlation length. For particles larger than the correlation length, we find that the diffusion coefficient, D, decreases exponentially with the polymer concentration. This can be explained by an exponential increase in the solution viscosity, consistent with the Stokes-Einstein equation. For probes on the order of the correlation length, the decrease of the diffusion coefficient cannot be accounted for by the Stokes-Einstein equation, but can be fit by a stretched exponential, D approximately exp(-alphacn), where we find n = 0.73-0.84 and alpha is related to the probe size. These results are in accord with a diffusion model of Langevin and Rondelez (Polymer 1978, 19, 1875), where these values of n indicate a good solvent quality.     

Slow changes and wiener analysis of nonlinear summation in contractions of cat muscles

With regular trains of stimuli at a high frequency, the contribution of each stimulus to the force generated over time declines from the second to about the tenth stimulus, but then begins to increase again. This late increase is referred to as tetanic potentiation in analogy with the post-tetanic potentiation of the twitch after such a period of stimulation. With regular trains of stimuli at a low frequency, a progressive decrease in the essentially unfused twitches (negative staircase) is observed in the slow soleus muscle of the cat, while a progressive increase (positive staircase) is observed for the fast plantaris muscle. The time constant for the approximately exponential changes observed is on the order of 10 s. Random trains of stimuli were applied at intermediate frequencies and analyzed in terms of general methods of analysis for nonlinear systems. Systematic decreases in the magnitude and increases in the time course of the average tension per stimulus were observed with increasing mean rates of stimulation. Similar changes were observed for short intervals between stimuli within a given random train at a constant mean rate. These changes can be described in terms of an early depression and a later facilitation described in the previous papers in this series.