Theory of EEG potentials in a model of thin brain integuments. II. Model of a multilayer flat cable]

A model of a polylayer spheric cable for estimating EEG potentials was discussed earlier [1]. As far as the holes in the natural scull were ignored in that model, the contrary assumption is useful: the scull is regarded as an infinite plain. Among the solution of the equations of both models there exist uncomplicated relations.     

Further study of the two-dimensional cable theory: An electric model for a flat thin association of cells with a directional intercellular communication

Summary The two-dimensional cable theory originally presented in relation to the electrotonus along a flat tissue of the rat atrial appendage is improved by taking account of double space constants instead of a single one and an explicit boundary condition at the tip of the current injecting microelectrode. A differential equation is formulated for the membrane potential change which is produced along the tissue by the intracellular injection of a current. The solution is formally expressed in terms of the Green's function. Specific solutions corresponding to the injection of a unit current step or a linearly rising current are discussed in detail.     

The linear cable theory as a model of gill blood flow

The vascular organization of the teleost gill suggests that blood flow distribution from the filamental artery to the respiratory lamellae is governed by relationships analogous to the cable conduction properties of a nerve axon. The space constant (λ) by definition is the distance along the gill filament at which the in-series resistance of the afferent filament artery equals the in-parallel resistance of the afferent lamellar arteriolar, lamellar, efferent lamellar arteriolar (ALA-L-ELA) segments. Constriction of the afferent filamental artery or uniform dilation of the ALA-L-ELA will decrease λ. As λ decreases, flow through the proximal (basal) lamellae greatly increases at the expense of distal lamellar perfusion. When λ increases in a system of finite length the flow profile must account for reflected pressures within the main vessel. The λ calculated from corrosion casts of gill vasculature is $\text{1}\text{4}$ to $\text{1}\text{2}$ the filament length. This favors blood flow through the proximal lamellae and when cardiac output increases, the proportion of cardiac output perfusing the proximal areas increases at the expense of distal lamellar blood flow. To offset these changes it is proposed that increased distal lamellar perfusion is achieved by simultaneous vasodilatation of distal and constriction of proximal ALA-L-ELA segments and dilation of the afferent filamental artery.     

Off-center spherical model for dosimetry calculations in chick brain tissue

This paper presents calculations for the electric field and absorbed power density distribution in chick brain tissue inside a test tube, using an off-center spherical model. It is shown that the off-center spherical model overcomes many of the limitations of the concentric spherical model, and permits a more realistic modeling of the brain tissue as it sits in the bottom of the test tube surrounded by buffer solution. The effect of the unequal amount of buffer solution above the upper and below the lower surfaces of the brain is analyzed. The field distribution is obtained in terms of a rapidly converging series of zonal harmonics. A method that permits the expansion of spherical harmonics about an off-center origin in terms of spherical harmonics at the origin is developed to calculate in closed form the electric field distribution. Numerical results are presented for the absorbed power density distribution at a carrier frequency of 147 MHz. It is shown that the absorbed power density increases toward the bottom of the brain surface. Scaling relations are developed by keeping the electric field intensity in the brain tissue the same at two different frequencies. Scaling relations inside, as well as outside, the brain surface are given. The scaling relation distribution is calculated as a function of position, and compared to the scaling relations obtained in the concentric spherical model. It is shown that the off-center spherical model yields scaling ratios in the brain tissue that lie between the extreme values predicted by the concentric and isolated spherical models.     

Instability and excitation propagation in a catalytic reaction model. I. Model of a system with concentrated parameters]

Using peculiarities of oxidative-rastorative catalytic reactions the system of nonlinear differential equations has been obtained. Pecularities of the behaviour of the system on phase plane of variables at different parameters are considered. The system may be autooscillative, trigger, waiting. Problems of model application in membrane, quantum, enzymatic systems are discussed.     

Adiabatic solution of the Ohmic cable equation. General theory, homogeneous cylindrical fiber]

An adiabatic solution of the Ohmic cable equation is suggested, which reduces the non-stationary equation to a stationary form. The adiabatic length constant of the stationary equation is time-dependent. The adiabatic solutions for the boundary conditions that change in time linearly and exponentially were studied. In the latter case, the adiabatic length constant does not depend on time though it differs from the usual length constant. The cable input characteristics of exact and adiabatic solutions were compared in the cases of the voltage- and current-clamp, and electric field stimulation. The adiabatic and exact solutions are identical for the rising exponential stimuli. For the falling exponential stimuli, the adiabatic solution determines the exact asymptotic solution if the stimulus decays slower than the relaxation of initial conditions. It is propose to use linear and exponential ramp stimulation in electrotonic measurements.     

Theory of the EEG potential in models of the leptomeninges of the brain. IV. Radial dipoles and their double layers in the depths and on the surface of the brain]

The estimaiton of ECoG-potentials is fulfilled by means of a simple model of the isolated sphere, which is exact enough, as it was proved earlier [1]. A simple limit formula for the qualitative estimation of ECoG-potentials of a source situated in the cerebral cortex is obtained. The EEG as in [2] is obtained using the transformation of the first 20 spheric harmonics. The experimental facts of registration of the evoked potentials of comparatively little subcortical sources were explained theoretically. Numerical results of the model are used to estimate the intensity of the field induced in the brain by two electrodes placed on the scalp.     

Control of reduction potential by protein matrix: lesson from a spherical protein model

 Factors that contribute to the control of reduction potential by protein matrix are examined within a spherical protein model. These include the nonpolar nature of protein matrices, solvent accessibility of the redox center, and net charges and dipoles of surrounding amino acids. Simple rules on their effects are established. In particular, surface charges have little effect on the reduction potential, and polar groups may either increase or decrease the reduction potential, depending on their orientations relative to the redox center. The effects of complex formation, proton titration, and ionic strength are also discussed. Received, accepted: 26 November 1996     

Comparison of discrete models of charge transfer in thin membranes. I. Stationary regime]

The model of the ion transport through thin membranes is discussed. It is assumed that there exist some steps of the ion transfer in the membrane volume. Theory of absolute rate of processes is used to derive an equation for stationary transmembrane flow. As a result the thory parameters are connected with the form of potential energy curve of permeable ions into the membrane phase.     

Semiconductor theory of ion transport in thin lipid membranes: I. Potential and field distributions

In the theory as presented in this paper and the following one, we shall attempt to apply the semiconductor principles and methods to the study of ion transport in thin lipid membranes. Detailed formulations are given on the potential energy barriers at the interfaces, voltage drops in the polar and non-polar regions, and potential and field distributions in the diffuse double layer and within a charged membrane. These results will be used mainly as the boundary conditions for the solution of ion flow as to be given in the following paper. The analysis clearly indicates that the ion transport is interface-limited and is profoundly influenced by the presence of surface charges. An explanation of Na⁺ extrusion in nerve membrane is given based on the field distribution analysis. The theory also suggests that the “membrane potential” depends mainly on surface charges but not necessarily on ion permeation through the membrane.     

Unsteady beam-plasma discharge: I. Mathematical model and linear theory

A mathematical model is constructed that describes the development of the beam-plasma instability in a traveling-wave tube amplifier in the presence of a neutral gas. Steady solutions are derived for conditions of microwave discharges in a magnetized plasma-filled traveling-wave tube amplifier, and their stability is investigated. It is shown that the steady-state amplification regime may become unstable and change to the self-modulation regime. The relationships between the amplifier parameters at the instability threshold are obtained, and the frequencies of the excited ion acoustic waves are determined. The results of numerical modeling are found to agree well with the analytical results.     

Technical simulation of intensity-duration curves with a model based on the membrane theory]

In earlier studies using a simplified black-box model, new concepts about the shape of I-T curves following electrical stimulation were developed. This model, however, was incompatible with the modern membrane theory of Hodkin and Huxley. The present paper describes a computer-aided simulation of the excitation process, based exclusively on the experimental results of Hdokin and Huxley. The simulated I-T curves do confirm the conclusions of the black-box model.     

A modified cable model for neuron processes with non-constant diameters.

The neuron axon cable model is expanded and developed to describe the linear subthreshold transmembrane potential of any circular cross section, thin membrane neuron fiber whose radius can be expressed as an analytic function of position, r(x). The transmembrane time constant is shown under the condition of space clamp to be independent of changes in geometry. Three typical neuron geometries are modeled (dendrite-soma, soma-axon, and dendrite-soma-axon) and the solutions to the resulting differential equations are numerically evaluated. The geometry-induced effects ate attributed to changes in current density and physiological correlates of the effects are proposed.     

The propagation potential. An axonal response with implications for scalp-recorded EEG

An electrophysiological response of axons, referred to as the "propagation potential," was investigated. The propagation potential is a sustained voltage that lasts as long as an action potential propagates between two widely spaced electrodes. The sign of the potential depends on the direction of action potential propagation. The electrode towards which the action potential is propagating is positive with respect to the electrode from which it is receding. For normal frog sciatic nerves the magnitude of the propagation potential was 17% of the peak of the extracellular action potential; TEA increased it to 32%. For normal earthworm median or lateral giant fibers it was 30%. A ripple pattern on the propagation potential was attributed to variation in resistance along the length of the worm. Cooling increased the duration of the propagation potential and attenuated the higher frequency components of the ripple pattern. Differential records from two widely spaced intracellular microelectrodes in the same axon differed from the propagation potential. The amplitude of the plateau relative to the peak was smaller, it decreased as the action potential propagated from one electrode site to the other, and the potential did not return to zero as rapidly as for extracellular records. When propagation was blocked by heat, the propagation potential slowly decayed. There was no ripple pattern during the decay. In a volume conductor, electrodes contacting the worm did not show the typical propagation potential, but electrodes located a few centimeters away from the worm did. Simple core-conductor models based on classical action potential theory did not reproduce the propagation potential. More complex, modified core-conductor models were needed to accurately simulate it. The results suggest that long, slowly conducting fibers can contribute to the scalp-recorded EEG.     

Estimation of neurophysiological parameters from the waking EEG using a biophysical model of brain dynamics

This paper presents the results from using electroencephalographic (EEG) data to estimate the values of key neurophysiological parameters using a detailed biophysical model of brain activity. The model incorporates spatial and temporal aspects of cortical function including axonal transmission delays, synapto-dendritic rates, range-dependent connectivities, excitatory and inhibitory neural populations, and intrathalamic, intracortical, corticocortical and corticothalamic pathways. Parameter estimates were obtained by fitting the model's theoretical spectrum to EEG spectra from each of 100 healthy human subjects. Statistical analysis was used to infer significant parameter variations occurring between eyes-closed and eyes-open states, and a correlation matrix was used to investigate links between the parameter variations and traditional measures of quantitative EEG (qEEG). Accurate fits to all experimental spectra were observed, and both inter-subject and between-state variability were accounted for by the variance in the fitted biophysical parameters, which were in turn consistent with known independent experimental and theoretical estimates. These values thus provide physiological information regarding the state. transitions (eyes-closed vs. eyes-open) and phenomena including cortical idling and alpha desynchronization. The parameters are also consistent with traditional qEEG, but are more informative, since they provide links to underlying physiological processes. To our knowledge, this is the first study where a detailed biophysical model of the brain is used to estimate neurophysiological parameters underlying the transitions in a broad range (0.25-50 Hz) of EEG spectra obtained from a large set of human data.     

Linear model of brain electrical activity—EEG as a superposition of damped oscillatory modes

EEG time series were modeled as an output of the linear filter driven by white noise. Parameters describing the signal were determined in a way fulfilling the maximum entropy principle. Transfer function and the impulse response function were found. The solutions of the differential equations describing the system have the form of the damped oscillatory modes. The representation of the EEG time series as a superposition of the resonant modes with characteristic decay factors seems a valuable method of the analysis of the signal, since it offers high reduction of the data to the few parameters of a clear physiological meaning.