A model for evoked activity of hippocampal neuronal population

A system of equations governing the activity of hippocampal neuron populations is proposed. This continual firing-rate model is aimed to simulate evoked potentials and synchronous wave activity of the neural tissue. The populations of excitatory and inhibitory neurons and the types of synaptic receptors are distinguished. The model is based on the idea of control and averaging of Hodgkin-Huxley equations, a simple model of a threshold elicitation of population action potential bursts, approximations of synaptic currents by the second-order differential equations, and hyperbolic partial derivative equation of axonal excitation propagation. The model was fitted to intracellular cordings of postsynaptic potentials and postsynaptic currents in CA1 of rat hippocampal slices.     

Mathematical analysis of multienzyme systems. I. Modelling of the glycolysis of human erythrocytes.

A mathematical model for the glycolysis of human erythrocytes is presented which takes into account ATP-synthesis and -consumption. A set of three differential equations describes the steady states and the time-dependent changes of the metabolite concentrations under blood storage conditions. For a given parameter combination there are in general three stationary points of the system, one of which is unstable. At a low ATP-need the ATP-level is relatively constant for variations in the rate constant of the ATP-consuming processes. Above a critical level of the energy consumption the system breaks down. An important role of the 2.3P2G-bypass of the erythrocytes is its action as an "energy buffer", wasting ATP in case of ATP-overproduction and producing ATP in case of underproduction. A parameter combination consistent with the data on the isolated enzymes was found which gives a good agreement of theoretical predictions with the measured metabolite concentrations. Under blood preservation conditions the difference of the rates of ATP-production and -consumption is the most important factor for a high ATP-level over long periods.     

The singularly perturbed Hodgkin-Huxley equations as a tool for the analysis of repetitive nerve activity

A qualitative analysis of the Hodgkin-Huxley model (Hodgkin and Huxley 1952), which closely mimics the ionic processes at a real nerve membrane, is performed by means of a singular perturbation theory. This was achieved by introducing a perturbation parameter that, if decreased, speeds up the fast variables of the Hodgkin-Huxley equations (membrane potential and sodium activation), whereas it does not affect the slow variables (sodium inactivation and potassium activation). In the most extreme case, if the perturbation parameter is set to zero, the original four-dimensional system degenerates to a system with only two differential equations. This degenerate system is easier to analyze and much more intuitive than the original Hodgkin-Huxley equations. It shows, like the original model, an infinite train of action potentials if stimulated by an input current in a suitable range. Additionally, explanations for the increased sensitivity to depolarizing current steps that precedes an action potential can be found by analysis of the degenerate system. Using the theory of Mishchenko and Rozov (1980) it is shown that the degenerate system does not only represent a simplification of the original Hodgkin-Huxley equations but also gives a valid approximation of the original model at least for stimulating currents that are constant within a suitable range.     

Subunits in quantal transmission at the mouse neuromuscular junction: tests of peak intervals in amplitude distributions

The regular spacing of peaks throughout the amplitude distribution of miniature end-plate potentials, quantal evoked end-plate potentials and quantal currents was demonstrated using autocorrelations and power density spectra calculated from the number of events in the successive bins of the histograms built by Matteson et al. (1979), Kriebel & Florey (1983) and Erxleben & Kriebel (1984). At the same mouse neuromuscular junction, the calculated interpeak was constant for evoked and spontaneous quantal releases, throughout sequential sampling and after change of bin size. The presence of regular peak intervals supports the hypothesis that quantal potentials are composed of potential subunits the size of the smallest subminiature potential. Challenging the hypothesis of an acetylcholine quantum composed of acetylcholine subunits, a postsynaptic origin of the subunit is proposed on the basis of the spatial arrangement in rows of the ACh receptors. The ACh-saturating patch evoked by a quantum release (Land et al., 1980, 1981) activates 10-20 rows of receptors, which is roughly the number of subunits composing a quantal event. Therefore the position of the ACh patch or the continuous variations in its size might cause stepwise variations in the total number of ACh receptors activated by an ACh quantum.     

On the influence of rapid periodic parameter oscillations on the long-term behaviour of cell metabolism

A skeleton model of energy metabolism of the living cell is considered in the form of kinetic equations. It is assumed that one intermediate metabolite can be rapidly shifted into a buffer reservoir and that the respective affinity constant changes periodically so that the metabolite is periodically withdrawn from and injected into the system. Investigation of the nonlinear system of differential equations with the periodic parameter by way of the Bogolyubov averaging method shows that the long-time energy state of the system may be changed drastically when the period is sufficiently short as compared with the characteristic metabolic events. In particular, the stabilization of a high-energy state by appropriate buffer shuttling is possible which cannot be maintained when the oscillations are stopped at any fixed parameter value. The result bears on the understanding of the influence of short-period rhythms on long-time behaviour of biosystems.     

Dynamic model of neuronal response in the rat hippocampus

A linear lumped model was proposed for the hippocampal CA 1 region of anesthetized rats using differential equations of time-independent coefficients, the afferent and efferent fibers of the alveus as inputs and the averaged evoked potentials (AEPs) and poststimulus time histograms as outputs. The alvear tract, a major efferent path, was proposed to activate interneurons monosynaptically while the anterior alveus activated orthodromically pyramidal cells which then excited the interneurons. The interneurons then inhibited pyramidal cells. The observable field outputs were the excitatory postsynaptic potentials (EPSPs) of interneurons and the inhibitory postsynaptic potentials (IPSPs) of pyramidal cells. Positive neurophysiological feedbacks were proposed among interneurons and among pyramidal cells in order to account for the prolonged time courses of the interneuronal EPSPs and the pyramidal cell IPSPs. The parameters of the model were optimized by a nonlinear regression program which minimized the sum of squared deviations between the model-generated and actual AEPs. The parameters included the temporal dispersion of the input tract (about 3 ms) and the membrane time constant of interneuronal and pyramidal cell populations (4.8 ms). In anesthetized rats, positive feedback gain coefficients were 0.07 among interneurons and 0.85 among pyramidal cells. After a compound spike (I), two postsynaptic AEP components (II and III) of different time courses were detectable at all depths within CA 1 except at the turnover for each component. The hypothesis that the AEP component II was generated by interneurons was tested and confirmed. The quantitative model constitutes a concise construct of the functional organization of the hippocampal CA 1 region, which suggests further theoretical extensions and experimentation.     

Electrophysiology of identified neurosecretory and non-neurosecretory cells in the cockroach pars intercerebralis

Two cell types can be distinguished with intracellular recording from the pars intercerebralis of the American cockroach (Periplaneta americana). The first type, which corresponds morphologically to the medial neurosecretory cell, always had spontaneously occurring, overshooting action potentials. These action potentials are probably endogenously produced. Tetrodotoxin experiments revealed that sodium is the dominant ion of the action potential. The action potentials are followed by a relatively long after-hyperpolarization. The input resistance of these cells ranged from 120 to 390 M omega. A mathematical model, based on cellular morphology and response to current pulses, revealed a membrane time constant of about 100 msec and an axonal:somatic conductance ratio of approximately 13. Area-specific membrane resistance was estimated at 33 k omega cm2. These cells also often had reversible and spontaneous inhibitory postsynaptic potentials. The second cell type, which is non-neurosecretory, never produced spontaneous action potentials and rarely had synaptic potentials. Action potentials could be evoked by current injection into the cell body or by extracellular stimulation of their axons in the posteroventral portion of the the protocerebrum. These action potentials also depend on sodium ions. Their input resistance ranged from 16 to 35 M omega. They had a membrane time constant of approximately 15 msec and an axonal:somatic conductance ratio of about 9. Their area specific membrane resistance was estimated at 14 k omega cm2.     

Bioelectrical activity and neuromediator metabolism in different brain structures during chronic haloperidol administration]

First haloperidol administration is followed by the reorganization of evoked potentials in visual cortex. During haloperidol administration (10-12 days after the beginning) variations of evoked potentials is visual cortex and in subcortical structures uniform evoked potentials took place.     

Polymorphism maintenance in populations with mixed random mating and apomixis subjected to stabilizing and cyclical selection

We analysed a diploid population model with a mixed breeding system that includes panmixia and apomixis. Each individual produces a part (ss) of its progeny by random mating, the remainder (1-ss) being a result of precise copying (vegetative reproduction or apomixis) of the parental genotype. Both constant and periodically varying selection regimes were considered. In the main model, the selected trait was controlled by two diallelic additive or semidominant loci, A/a and B/b, whereas the parameter of breeding system (ss) was genotype-independent. A numerical iteration of the evolutionary equations were used to evaluate the proportion (V) of population trajectories converging to internal (polymorphic) fixed points. The results were the following. (a) A complex pattern of dependence of polymorphism stability on interaction among the breeding system, recombination rate, and the genetic architecture of the selected trait emerged. (b) The recombination provided some advantage to sex at intermediate period lengths and strong-to-moderate selection intensities. (c) The complex limiting behavior (CLB) was quite compatible with sexual reproduction, at least within the framework of pure genetic (not including variations in population density) models of multilocus varying selection.     

Reticular activation and the dynamics of neuronal networks

It is postulated that during arousal the cortical system is driven by a spatially and temporally noisy signal arising from non-specific reticulo-cortical pathways. An elementary unit of cortical neuroanatomy is assumed, which permits non-linear dynamics to be represented by stochastic linear equations. Under these assumptions the resonant modes of the system of cortical dendrites approach thermodynamic equilibrium. Specific sensory signals perturb the dendritic system about equilibrium, generate low frequency, linear, non-dispersive waves corresponding to the EEG, which in turn regulate action potential sequences, and instantiate internal inputs to the dendritic field. A large and distributed memory capacity in axo-synaptic couplings, resistance to interference between functionally separate logical operations, and a very large next-state function set emerge as properties of the network. The model is able to explain the close association of the EEG with cognition, the channel of low capacity corresponding to the field of immediate attention, the low overall correlation of action potentials with EEG, and specificity of action potentials in some neurons during particular cognitive activity. Predictions made from hypothesis include features of thermal equilibrium in EEG (determinable by autoregression) and expectation that the cortical evoked response can be accounted for as the response to a sensory impulse of specific time characteristics.     

Electrogenic absorption of sugars and amino acids in the small intestine of the human fetus

Some characteristics and the degree of intestinal absorption in the developing human fetus were examined by measuring solute evoked potentials and 14C-D-glucose uptake into the everted jejunal segments. In all segments, the Michaelis-Menten relationship was observed between D-glucose concentrations and the evoked potentials or D-glucose uptake. Increase of Na-ion concentrations enhanced both D-glucose evoked potentials and uptake. Both D-glucose and L-alpha-alanine evoked potentials increased in a significant correlation to the fetal age; however, the apparent Michaelis constants did not show any signficant change. The structural specificity of sugar for generating evoked potentials was the same as that reported for adult mammals. Among amino acids, only the L-form of neutral and acidic amino acids generated markedly high evoked potentials, but the basic ones hardly at all. Oligopeptides such as glycyl-glycine and glycyl-glycyl-glycine also generated high evoked potentials. Our results have indicated that the active transport system of sugars and amino acids in the human fetus have already developed by as early as the sixth month of gestation.     

Parameter estimation in Hodgkin-Huxley-Type equations for membrane action potentials in nerve and heart muscle

A method is presented which renders parameter estimation possible in systems of non-linear differential equations where normally no solution exists in terms of analytic functions and which have to be solved numerically. The method uses the concept of sensitivity equations. Two examples are given, taking mathematical models for membrane action potentials in nerve and heart muscle by Hodgkin and Huxley and by Beeler and Reuter. The model equations together with the corresponding system of sensitivity equations are given, which are necessary to estimate maximum conductivity coefficients defining the interactions of different ionic current components. A computer program is described and results of action potential numerical analysis are presented using simulated data. It can be seen, that even with superimposed simulated noise the real parameter values are estimated in an excellent manner. The method can be used to interpret observed changes in action potential time courses under physiological and pharmacological conditions.     

A numerical study of the Belousov-Zhabotinskii reaction using Galerkin finite element methods

The Belousov-Zhabotinskii reaction has been modelled by Field and Noyes [5] as a pair of nonlinear parabolic equations. Previous studies of these, both theoretical and numerical, have assumed wave solutions travelling with constant velocity leading to a simplification of the mathematical model in the form of a system of ordinary differential equations. In the present study a finite element Galerkin method is used directly on the original parabolic system for a range of parameter values.     

Transmission of electric fields in cortical tissue: A model for the origin of the alpha rhythm

A mathematical model of the cortex is developed, in which a great many independently-firing sources of voltage pulses are assumed to be imbedded in a diffusive medium of indefinite extent. In such a medium, characterized by a conductivity and a capacitivity (capacitance per unit volume), electrical disturbances diffuse in a manner analogous to the conduction of heat in solids. Assuming random firing of the individual pulse sources (which simulate neurons) and invoking Campbell's Theorem, the power spectrum of the voltage appearing at any point within the medium can be calculated. As a consequence of the positive after-potential in the neural discharge (which is shown to be a physical necessity), the power spectrum is found to contain a single low-frequency peak. For reasonable parameter values, this peak appears at very near the classical alpha frequency of ten cycles per second. The related phenomena of evoked potentials, alpha blocking, and photic driving are also discussed in terms of the model.     

Model for Transition from Waves to Synchrony in the Olfactory Lobe of Limax

A biophysical model for the interactions between bursting (B) cells and nonbursting (NB) cells in the procerebral lobe of Limax is developed and tested. Phase-sensitivity of the NB cells is exhibited due to the strong inhibition from the rhythmically bursting B cells. Electrical and chemical junctions coupled with a parameter gradient lead to sustained periodic waves in the lobe. Excitatory interactions between the NB cells, which rarely fire, lead to stimulus evoked synchrony in the lobe oscillations. A novel calcium current is suggested to explain the effects of nitric oxide (NO) on the lobe. Gap junctions are shown both experimentally and through simulations to be required for the oscillating field potentials.     

Cerebral chemosensory evoked potentials elicited by chemical stimulation of the human olfactory and respiratory nasal mucosa

A stimulation method was employed by which chemosensory evoked potentials were recorded without tactile somatosensory contamination. The purpose of the study was to determine whether potential components evoked by stimulation of the chemoreceptors of the trigeminal nerve can be distinguished from those of the olfactory nerve. The stimulants (vanillin, phenylethyl alcohol, limonene, menthol, anethol, benzaldehyde, carbon dioxide and a mixture of vanilin and carbon dioxide) were presented in a randomized order to 13 volunteers. Chemosensory evoked potentials to substances which anosmics are unable to perceive (vanillin, phenylethyl alcohol) were termed olfactory evoked potentials; potentials to CO₂, which effected no olfactory sensations were termed chemo-somatosensory potentials. Analysis of variance revealed that the different substances resulted in statistically significant changes in the amplitudes and latencies of the evoked potentials, and also in the subjective estimates of intensity. An increased excitation of the somatosensory system resulted in reduced latencies and enhanced amplitudes of the evoked potentials. Responses to the mixture of carbon dioxide and vanillin appeared significantly earlier (50–150 msec) than responses to either substance alone.     

Analytical solution of the cable equation with synaptic reversal potentials for passive neurons with tip-to-tip dendrodendritic coupling

A passive cable model is presented for a pair of electrotonically coupled neurons in order to investigate the effects of tip-to-tip dendrodendritic gap junctions on the interaction between excitation and either pre or postsynaptic inhibition. The model represents each dendritic tree by a tapered equivalent cylinder attached to an isopotential soma. Analytical solution of the cable equation with synaptic reversal potentials is considered for each neuron to yield a system of Volterra integral equations for the voltage. The solution to the system of linear integral equations (expressed as a Neumann series) is used to determine the current spread within the two coupled neurons, and to re-examine the sensitivity of the soma potentials (in particular) to the coupling resistance for various loci of synaptic inputs. The model is actually posed generally, so that active as well as passive properties could be considered. In the active case, a system of non-linear integral equations is derived for the voltage.     

The estimation of time varying dipoles on the basis of evoked potentials

When a stimulus is presented to a subject, different cortical areas are activated simultaneously. These cortical sources are the generators of evoked potentials. The geometry parameters and the amplitude time course of each source can be estimated on the basis of EP data and the appropriate mathematical model. In the present paper, a model is proposed which describes EPs as the result of a set of stagnant dipoles, with time varying amplitude. It is shown how the parameter estimation problem can be solved efficiently. The method automatically solves the problem of separating overlapping components. A lower limit for the residual is derived, which can be used to determine the minimum number of sources required to describe observed responses adequately.     

Change in the evoked potentials of the brain exposed to a constant magnetic field]

The influence of the constant magnetic field (CMF) upon the cerebral and cerebellar cortex potentials evoked by the sciatic stimulation was studied in rats. During the exposure to the CMF there occurred an increase of the evoked potentials in amplitude and the appearance of additional waves in their structure. The effect was enhanced with increase in the intensity of magnetic field within the range of 500-4000 oe.     

The Influence of Wavelets on Multiscale Analysis and Parametrization of Midlatency Auditory Evoked Potentials

This work shows methodological aspects of heuristic pattern recognition in auditory evoked potentials. A linear and a nonlinear transformation based on wavelet transform are presented. They result in a statistical error model and an entropy function related to the Gibbs function and describe changes in midlatency auditory evoked potentials induced by general anaesthesia. The same transformations were calculated using 12 common wavelets. We present a method to compare the two defined parametrizations with respect to their ability to discriminate two defined states which is responsive and unresponsive depending on the wavelet used for the analysis. Auditory evoked potentials of 60 patients undergoing general anaesthesia were analysed. We propose the defined statistical error model and the entropy function as a very robust measure of changes in auditory evoked potentials. The influence of the wavelets suggest that for each parametrization the goodness of the wavelet should be validated.