Electrodiffusion Models of Neurons and Extracellular Space Using the Poisson-Nernst-Planck Equations—Numerical Simulation of the Intra- and?Extracellular Potential for an Axon Model

In neurophysiology, extracellular signals—as measured by local field potentials (LFP) or electroencephalography—are of great significance. Their exact biophysical basis is, however, still not fully understood. We present a three-dimensional model exploiting the cylinder symmetry of a single axon in extracellular fluid based on the Poisson-Nernst-Planck equations of electrodiffusion. The propagation of an action potential along the axonal membrane is investigated by means of numerical simulations. Special attention is paid to the Debye layer, the region with strong concentration gradients close to the membrane, which is explicitly resolved by the computational mesh. We focus on the evolution of the extracellular electric potential. A characteristic up-down-up LFP waveform in the far-field is found. Close to the membrane, the potential shows a more intricate shape. A comparison with the widely used line source approximation reveals similarities and demonstrates the strong influence of membrane currents. However, the electrodiffusion model shows another signal component stemming directly from the intracellular electric field, called the action potential echo. Depending on the neuronal configuration, this might have a significant effect on the LFP. In these situations, electrodiffusion models should be used for quantitative comparisons with experimental data.     

Effect of microwave electromagnetic field on skeletal muscle fibre activity

The aim of the present study was to investigate the influence of microwave irradiation on fatiguing activity of isolated frog skeletal muscle fibres. The changes in the electrical and mechanical activity were used as criteria for the exposure effects. Repetitive suprathreshold stimulation with interstimulus interval of 200 ms for 3 min was applied. Intracellular (ICAP) and extracellular (ECAP) action potentials and twitch contractions (Tw) of muscle fibres after 1 hour microwave exposure (2.45 GHz, 20 mW/cm( 2) power density) were compared with those recorded after one hour sham exposure (control). The duration of uninterrupted activity in the trial (endurance time; ET) was not significantly affected by microwave field exposure. After microwave irradiation, the ICAP amplitude was higher, the rising time was shorter, and the resting membrane potential was more negative compared to controls. There was a slower rate of parameters changes during ET in potentials obtained from irradiated fibres. Microwave exposure increased the propagation velocity of excitation, the ECAP and Tw amplitudes, as well as shortened their time parameters. We concluded that a 2.45 GHz microwave field possesses a stimulating effect on muscle fibre activity, which is in part due to its specific, non-thermal properties. The microwave induced-changes in muscle fibre activity may reduce development of skeletal muscle fatigue.     

Modeling extracellular field potentials and the frequency-filtering properties of extracellular space

Extracellular local field potentials are usually modeled as arising from a set of current sources embedded in a homogeneous extracellular medium. Although this formalism can successfully model several properties of extracellular local field potentials, it does not account for their frequency-dependent attenuation with distance, a property essential to correctly model extracellular spikes. Here we derive expressions for the extracellular potential that include this frequency-dependent attenuation. We first show that, if the extracellular conductivity is nonhomogeneous, there is induction of nonhomogeneous charge densities that may result in a low-pass filter. We next derive a simplified model consisting of a punctual (or spherical) current source with spherically symmetric conductivity/permittivity gradients around the source. We analyze the effect of different radial profiles of conductivity and permittivity on the frequency-filtering behavior of this model. We show that this simple model generally displays low-pass filtering behavior, in which fast electrical events (such as Na(+)-mediated action potentials) attenuate very steeply with distance, whereas slower (K(+)-mediated) events propagate over larger distances in extracellular space, in qualitative agreement with experimental observations. This simple model can be used to obtain frequency-dependent extracellular field potentials without taking into account explicitly the complex folding of extracellular space.     

Extracellular potentials of a single myelinated nerve fiber in an unbounded volume conductor

The extracellular potentials of a single myelinated nerve fiber in an unbounded volume conductor were studied. The spatial distribution of the transmembrane potential was obtained by integrating the system of partial differential equations characterizing the electric processes in the active myelinated nerve fiber. The spatial distribution of the extracellular potentials at various radial distances in the volume conductor were calculated using the line source model. Up to a certain radial distance (500 m) the discontinuity of the action potential propagation is reflected in the extracellular potentials, while further in the volume conductor the potentials are smooth. The effect of the fiber diameter and the internodal distance on the volume conductor potentials as well as the changes in the magnitude of the extracellular potential (in the time domain) between two adjacent nodes at various radial distances were studied. The radial decline of the peak-to-peak amplitude of the extracellular potential depends on the radial coordinater of the field point and increases with the increase ofr.     

Active electric properties of cardiac muscle

A model of electrical activity in the heart has been developed that treats the intracellular domain and the extracellular domain as electrical syncytia with anisotropic resistivities (bi-syncytial model). At the microscopic level, propagation is assumed to proceed primarily along the axes of individual cells. Considerations at the macroscopic level relate the transmembrane current to the intracellular and extracellular resistivity and the transmembrane potential. The result is a relationship between instantaneous extracellular potentials and cardiac action potentials.     

The periodic axon membrane skeleton leads to Na nanodomains but does not impact action potentials

Recent work has established that axons have a periodic skeleton structure comprising of azimuthal actin rings connected via longitudinal spectrin tetramer filaments. This structure endows the axon with structural integrity and mechanical stability. Additionally, voltage-gated sodium channels follow the periodicity of the active-spectrin arrangement, spaced ～190 nm segments apart. The impact of this periodic arrangement of sodium channels on the generation and propagation of action potentials is unknown. To address this question, we simulated an action potential using the Hodgkin-Huxley formalism in a cylindrical compartment, but instead of using a homogeneous distribution of voltage-gated sodium channels in the membrane, we applied the experimentally determined periodic arrangement. We found that the periodic distribution of voltage-gated sodium channels does not significantly affect the generation or propagation of action potentials but instead leads to large, localized sodium action currents caused by high-density sodium nanodomains. Additionally, our simulations show that the distance between periodic sodium channel strips could control axonal excitability, suggesting a previously underappreciated mechanism to regulate neuronal firing properties. Together, this work provides a critical new insight into the role of the periodic arrangement of sodium channels in axons, providing a foundation for future experimental studies.     

Electrical Inexcitability of the Frog Neuromuscular Synapse

Frog muscle endplates were explored with an extracellular microelectrode. An intracellular microelectrode nearby simultaneously monitored invasion of the endplate by a spike directly evoked by a third microelectrode placed away from the endplate in the same fiber. External positivities were seen only at sites generating miniature endplate potentials. The external positivity reached a maximum prior to the internally recorded potential and was followed by a small late negativity. Small movements away from active synaptic sites resulted in positive-negative-positive potential sequences characteristic of activity and propagation. Since the external potential is a function of membrane current, the absence of negativity associated with the rising phase of the spike indicates the absence of inward current at synaptic sites. Thus, the synaptic membrane appears not to be excited by a depolarization of the magnitude of an action potential. In an Appendix it is shown that the late negativity and earlier maximum of the external potential can be accounted for by capacitative current through passive membrane.     

Effect of Bay K 8644 on tetraethylammonium-induced excitability of the rabbit pulmonary artery

The effect of Bay K 8644 on the electrical activity of the smooth muscle cells in the main pulmonary artery of the rabbit was examined. In normal physiological solution, the resting membrane potential was -56 +/- 0.6 mV, and the cells were electrically quiescent. Tetraethylammonium (5 mM) depolarized the membrane to about -45 mV, and electrical stimulation elicited action potentials. To suppress contractile responses and thereby facilitate sustained impalements, the muscle strips were bathed with a hypertonic solution containing sucrose. The mean amplitude of the tetraethylammonium-induced action potentials in the hypertonic solution was 35 +/- 0.9 mV. The action potentials were dependent upon the extracellular Ca2+ concentration and were abolished by diltiazem (10(-6) M). Spontaneous action potentials were occasionally generated in the presence of tetraethylammonium alone and could be induced by the further addition of Ba2+ (0.5 mM). The Ca2+ agonist Bay K 8644 (10(-8) to 10(-6) M) had no effect on the resting membrane potential or excitability in normal solution. However, in the hypertonic solution containing tetraethylammonium, Bay K 8644 caused a further depolarization and oscillatory potential changes, which were not prevented by tetrodotoxin. The oscillations were suppressed or abolished by diltiazem or nilvadipine. Thus, active responses can occur in the normally quiescent smooth muscle cells of the rabbit pulmonary artery when the outward K+ current(s) are suppressed.     

Dynamics of action potential initiation in the GABAergic thalamic reticular nucleus in vivo

Understanding the neural mechanisms of action potential generation is critical to establish the way neural circuits generate and coordinate activity. Accordingly, we investigated the dynamics of action potential initiation in the GABAergic thalamic reticular nucleus (TRN) using in vivo intracellular recordings in cats in order to preserve anatomically-intact axo-dendritic distributions and naturally-occurring spatiotemporal patterns of synaptic activity in this structure that regulates the thalamic relay to neocortex. We found a wide operational range of voltage thresholds for action potentials, mostly due to intrinsic voltage-gated conductances and not synaptic activity driven by network oscillations. Varying levels of synchronous synaptic inputs produced fast rates of membrane potential depolarization preceding the action potential onset that were associated with lower thresholds and increased excitability, consistent with TRN neurons performing as coincidence detectors. On the other hand the presence of action potentials preceding any given spike was associated with more depolarized thresholds. The phase-plane trajectory of the action potential showed somato-dendritic propagation, but no obvious axon initial segment component, prominent in other neuronal classes and allegedly responsible for the high onset speed. Overall, our results suggest that TRN neurons could flexibly integrate synaptic inputs to discharge action potentials over wide voltage ranges, and perform as coincidence detectors and temporal integrators, supported by a dynamic action potential threshold.     

Disopyramide phosphate effects on slow and depressed fast responses

We studied the effects of disopyramide phosphate on explanted neonatal rat ventricle cells exhibiting depressed fast responses or naturally occurring slow response action potentials together with automatic activity. Disopyramide suppressed the spontaneous activity at a concentration of 2.5 micrograms/mL with a half-maximal value of 10 micrograms/mL. Before spontaneous activity was lost, there was an increase in beating rate possibly related to membrane depolarization. In depressed fast and slow response action potentials there was an increase in action potential duration (APD) which was consistently found both at the level of the plateau and at 90% repolarization. Comparison of the APD increase observed after disopyramide treatment and that after exposure to 20 mM tetraethylammonium suggested a block of a potassium conductance as a possible cause underlying the change in APD. The Vmax values of the depressed fast response decreased at constant membrane potential and this was attributed to the local anesthetic effect of the drug. In addition, we report two novel findings: (i) a decrease of Vmax of the slow response action potentials which may be secondary to membrane depolarization, and (ii) an increase in the duration of slow action potentials, possibly caused by inhibition of a potassium conductance.     

High resolution magnetic images of planar wave fronts reveal bidomain properties of cardiac tissue

We magnetically imaged the magnetic action field and optically imaged the transmembrane potentials generated by planar wavefronts on the surface of the left ventricular wall of Langendorff-perfused isolated rabbit hearts. The magnetic action field images were used to produce a time series of two-dimensional action current maps. Overlaying epifluorescent images allowed us to identify a net current along the wavefront and perpendicular to gradients in the transmembrane potential. This is in contrast to a traditional uniform double-layer model where the net current flows along the gradient in the transmembrane potential. Our findings are supported by numerical simulations that treat cardiac tissue as a bidomain with unequal anisotropies in the intra- and extracellular spaces. Our measurements reveal the anisotropic bidomain nature of cardiac tissue during plane wave propagation. These bidomain effects play an important role in the generation of the whole-heart magnetocardiogram and cannot be ignored.     

Local development of action potentials in slow muscle fibres after complete or partial denervation.

Pyriformis muscles of Rana temporaria were completely or partially denervated by cutting the sciatic nerve or some of the small nerve branches entering the muscle. One stimulating and one to three recording microelectrodes were inserted along the fibres in order to compare the electrical activity at these points. In an early period following denervation action potentials of variable size and shape could be observed; these action potentials were often composed of two, sometimes of three or four, components. The size of individual components depended on the position of the recording microelectrode. Individual components could occasionally be triggered separately by adjusting the strength of the stimulating current pulse; propagation of these "all or none" responses was absent. In other fibres one component of the action potential could trigger another one several millimetres apart, thus indicating propagation. Conduction velocities were approximately 0.4 m/s. In partially denervated slow fibres, endplate potentials were confined to one lateral segment of the fibres, while the action potential occupied the denervated part of the membrane. The amplitudes of endplate and action potentials varied inversely with distance. Rough estimates of the length constant of the slow fibre membrane were calculated from the spatial decay of action potentials, endplate potentials and hyperpolarizing electrotonic potentials; mean values obtained were 2.5, 4.8 and 7.7 mm respectively. The results suggest that following denervation Na channels are built into discrete areas of the slow fibre membrane and that this process depends on the amount of denervation in individual fibres.     

Seasonal variation in hydrochloric acid, malic acid, and calciumions secreted by the trichomes of chickpea (Cicer arietinum)

A novel experimental set-up and method of recording of electrical potential differences in plants have been developed which enable continuous, 8-channel monitoring of electrical activity over extended periods of time using inserted, extracellular electrodes. The investigations were carried out on 21- to 23-day-old Helianthus annuus plants, and spontaneously-generated action potentials were recorded during monitoring sessions lasting for 3 days and nights. Characteristics of these spontaneous action potentials were elaborated, adopting as parameters their typical form, amplitude, duration, velocity, direction, and distance of propagation and frequency of occurrence in morphologically different parts of the plant, Variability, similarities, and interdependence of the above parameters in individual plants and in a group of 15 plants were determined. A hypothesis concerning propagation of action potentials in plants along specific impulse-propagating 'columns' is discussed. The frequency of generated impulses is highest at night and lowest in the day and also displays an apparent 24-h rhythm. Presumably this mechanism is under both endogenous and exogenous control and may be partly dependent on a biological clock.     

A study of the effects of externally applied sodium-ions and detection of spatial non-uniformity of the squid axon membrane under internal perfusion

Electrical properties of the axon membrane were examined under internal perfusion of squid giant axons with a dilute solution of NaF or CsF. The rate of propagation of the action potential was markedly enhanced when NaCl was added to the external CaCl₂ solution. The membrane conductance both at rest and during the action potential was increased with increasing Na-concentration in the external medium. In the perfusion zone of these axons, the action potentials in different parts of the membrane were found to terminate in a more-or-less spatially random and temporally irregular fashion. When the electric field outside the axon membrane was examined with hyperfine glass-pipette electrodes, small rectangular potential changes of uniform amplitude were observed. The small potential changes, which resemble those obtained by Bean in EIM-treated lipid bilayer, were interpreted as indicating spatial non-uniformity of the axon membrane during excitation. The importance of long-range electric interaction between different parts of the axon membrane is emphasized.     

Effects of bath resistance on action potentials in the squid giant axon: myocardial implications.

This study presents a simplified version of the quasi-one-dimensional theory (Wu, J., E. A. Johnson, and J. M. Kootsey. 1996. A quasi-one-dimensional theory for anisotropic propagation of excitation in cardiac muscle. Biophys. J. 71:2427-2439) with two components of the extracellular current, along and perpendicular to the axis, and a simulation and its experimental confirmation for the giant axon of the squid. By extending the one-dimensional core conductor cable equations, this theory predicts, as confirmed by the experiment, that the shapes of the intracellular and the extracellular action potentials are related to the resistance of the bath. Such a result was previously only expected by the field theories. The correlation between the shapes of the intracellular and the extracellular potentials of the giant axon of the squid resembles that observed during the anisotropic propagation of excitation in cardiac muscle. Therefore, this study not only develops a quasi-one-dimensional theory for a squid axon, but also provides one possible factor contributing to the anisotropic propagation of action potentials in cardiac muscle.     

Effects of external ions on membrane potentials of a lobster giant axon

The effects of varying external concentrations of normally occurring cations on membrane potentials in the lobster giant axon have been studied and compared with data presently available from the squid giant axon. A decrease in the external concentration of sodium ions causes a reversible reduction in the amplitude of the action potential and its rate of rise. No effect on the resting potential was detected. The changes are of the same order of magnitude, but greater than would be predicted for an ideal sodium electrode. Increase in external potassium causes a decrease in resting potential, and a decrease in potassium causes an increase in potential. The data so obtained are similar to those which have been reported for the squid giant axon, and cannot be exactly fitted to the Goldman constant field equation. Lowering external calcium below 25 mM causes a reduction in resting and action potentials, and the occasional occurrence of repetitive activity. The decrease in action potential is not solely attributable to a decrease in resting potential. Increase of external calcium from 25 to 50 mM causes no change in transmembrane potentials. Variations of external magnesium concentration between zero and 50 mM had no measurable effect on membrane potentials. These studies on membrane potentials do not indicate a clear choice between the use of sea water and Cole's perfusion solution as the better external medium for studies on lobster nerve.     

Roles of subcellular Na+ channel distributions in the mechanism of cardiac conduction

The gap junction and voltage-gated Na⁺ channel play an important role in the action potential propagation. The purpose of this study was to elucidate the roles of subcellular Na⁺ channel distribution in action potential propagation. To achieve this, we constructed the myocardial strand model, which can calculate the current via intercellular cleft (electric-field mechanism) together with gap-junctional current (gap-junctional mechanism). We conducted simulations of action potential propagation in a myofiber model where cardiomyocytes were electrically coupled with gap junctions alone or with both the gap junctions and the electric field mechanism. Then we found that the action potential propagation was greatly affected by the subcellular distribution of Na⁺ channels in the presence of the electric field mechanism. The presence of Na⁺ channels in the lateral membrane was important to ensure the stability of propagation under conditions of reduced gap-junctional coupling. In the poorly coupled tissue with sufficient Na⁺ channels in the lateral membrane, the slowing of action potential propagation resulted from the periodic and intermittent dysfunction of the electric field mechanism. The changes in the subcellular Na⁺ channel distribution might be in part responsible for the homeostatic excitation propagation in the diseased heart.     

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.     

A model for propagation of action potentials in smooth muscle

A modified Hodgkin & Huxley (1952) model for axons was used to simulate smooth muscle action potentials. The modifications were such as to match our own experimental results and published data on the passive and active behavior of smooth muscle.A brief account of the modifications introduced to the HH model is as follows. The resting ionic conductances were obtained from the data of Casteels (1969). Chloride conductance was replaced by an ad hoc leakage conductance ($\text{g}\text{?}{}_{\mathrm{L}}$) in order to obtain a resting membrane resistance of about 11 kΩcm². The ionic equilibrium potentials were according to Kao & Nishiyama (1969). The rate constants m, n and h have similar form to those in axons, but their different numerical values produce action potentials that match the duration of the smooth muscle action potential (about 16 ms) at half its maximum amplitude. The effective membrane capacitance was taken as 2.5 μF/cm².The results obtained by implementing those smooth muscle parameters in the HH formulation include: (a) a membrane potential that matches the main characteristics of experimentally recorded action potentials in uterine smooth muscle and guinea-pig taenia-coli, and (b) a propagated action potential which, on a cable diameter of 5 μm (similar to the diameter of a single smooth muscle cell), has a speed of propagation within the range of the values experimentally recorded in smooth muscle. This observed velocity of propagation is not compatible with a large cable and it is concluded that “functional units” are not required to sustain propagation of action potentials in smooth muscle.     

Effect of conditioning polarization of soma of giant neurons of mollusks on the mechanism of action-potential generation

To ascertain the properties of an excitable membrane of the soma of giant neurons of mollusks, experiments were carried out to study the effect of conditioning shift of the membrane potential on the mechanism of action-potential generation. The effect of conditioning was assessed from changes in the action-potential curve and its first derivative, as well as from the curve of transmembrane currents under voltage clamp conditions. It was found that a change in membrane potential evokes at least two reactions which have opposite effects on the mechanism of generation of action potentials. These reactions evidently have different time characteristics. One of these does not differ notably from the reaction recorded for other excitable structures, and is manifested in the activation (with hyperpolarization) or inactivation (with depolarization) of the mechanism generating action potentials. The other reaction contributes either to an increase (with depolarization) or a decrease (with hyperpolarization) in the efficiency of this mechanism. Conditioning polarization also has a marked effect on the system responsible for repolarization of the membrane during generation of action potentials. This effect is manifested in a change in the reaction of this system to tetraethylammonium ions. The specific membrane systems sustaining excitability and reacting to changes in the strength of the membrane's electrical field were found to be very inert. After a shift in the potential to a given stable level a rearrangement, lasting sometimes tens of seconds, takes place in the membrane.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 2, No. 1, pp. 91–99, January–February, 1970.