An electro-diffusion model for computing membrane potentials and ionic concentrations in branching dendrites,spines and axons

The Nernst-Planck equation for electrodiffusion was applied to axons, dendrites and spines. For thick processes (1 m) the results of computer simulation agreed accurately with the cable model for passive conduction and for propagating action potentials. For thin processes (0.1 m) and spines, however, the cable model may fail during transient events such as synaptic potentials. First, ionic concentrations can rapidly change in small compartments, altering ionic equilibrium potentials and the driving forces for movement of ions across the membrane. Second, longitudinal diffusion may dominate over electrical forces when ionic concentration gradients become large. We compare predictions of the cable model and the electro-diffusion model for excitatory postsynaptic potentials on spines and show that there are significant discrepancies for large conductance changes. The electro-diffusion model also predicts that inhibition on small structures such as spines and thin processes is ineffective. We suggest a modified cable model that gives better agreement with the electro-diffusion model.     

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.     

Sensitivity analysis of the Poisson Nernst–Planck equations: a finite element approximation for the sensitive analysis of an electrodiffusion model

Biological structures exhibiting electric potential fluctuations such as neuron and neural structures with complex geometries are modelled using an electrodiffusion or Poisson Nernst–Planck system of equations. These structures typically depend upon several parameters displaying a large degree of variation or that cannot be precisely inferred experimentally. It is crucial to understand how the mathematical model (and resulting simulations) depend on specific values of these parameters. Here we develop a rigorous approach based on the sensitivity equation for the electrodiffusion model. To illustrate the proposed methodology, we investigate the sensitivity of the electrical response of a node of Ranvier with respect to ionic diffusion coefficients and the membrane dielectric permittivity.

     

Prolonged action potentials from single nodes of Ranvier

The duration of action potentials from single nodes of Ranvier can be increased by several methods. Extraction of water from the node (e.g. by 2 to 3 M glycerin) causes increased durations up to 1000 msec. 1 to 5 min. after application of the glycerin the duration of the action potential again decreases to the normal value. Another type of prolonged action potential can be observed in solutions which contain K or Rb ions at concentrations between 50 mM and 2 M. The nodes respond only if the resting potential is restored by anodal current. The kinetics of these action potentials is slightly different. Their maximal durations are longer (up to 10 sec.). Like the normal action potential, they are initiated by cathodal make or anodal break. They also occur in external solutions which contain no sodium. The same type of action potentials as in KCl is found when the node is depolarized for some time (15 to 90 sec., 100 to 200 mv.) and is then stimulated by cathodal current. These action potentials require no K or Na ions in the external medium. Their maximal duration increases with the strength and duration of the preceding depolarization. The possible origin of the action potentials in KCl and after depolarization, and their relation to the normal action potentials and the negative after-potential are discussed.     

A distributed-parameter model of the myelinated nerve fiber

This paper presents a new model for the characterization of electrical activity in the nodal, paranodal and internodal regions of isolated amphibian and mammalian myelinated nerve fibers. It differs from previous models in the following ways: (1) in its ability to incorporate detailed anatomical and electrophysiological data; (2) in its approach to the myelinated nerve fiber as a multi-axial cable; and (3) in the numerical algorithm used to obtain distributed model equation solutions for potential and current. The morphometric properties are taken from detailed electron microscopic anatomical studies (Berthold & Rydmark, 1983a, Experientia 39, 964-976). The internodal axolemma is characterized as an excitable membrane and model-generated nodal and internodal membrane action potentials are presented. A system of describing equations for the equivalent network model is derived, based on the application of Kirchoff's Current Law, which take the form of multiple cross-coupled parabolic partial differential equations. An implicit numerical integration method is developed and the numerical solution implemented on a parallel processor. Non-uniform spatial step sizes are used, enabling detailed representation of the nodal region while minimizing the number of total segments necessary to represent the overall fiber. Conduction velocities of 20.2 m sec-1 at 20 degrees C for a 15 microns diameter amphibian fiber and 57.6 m sec-1 at 37 degrees C for a 17.5 microns diameter mammalian fiber are achieved, which agrees qualitatively with published experimental data at similar temperatures (Huxley & St?mpfli, 1949, J. Physiol., Lond. 108, 315-339; Rasminsky, 1973, Arch, Neurol. 28, 287-292). The simulation results demonstrate the ability of this model to produce detailed representations of the transaxonal, transmyelin and transfiber potentials and currents, as well as the longitudinal extra-axonal, periaxonal and intra-axonal currents. Also indicated is the potential contribution of the paranodal axolemma to nodal activity as well as the presence of significant longitudinal currents in the periaxonal space adjacent to the node of Ranvier.     

Effect of chromium ions on the function of neuromuscular junctions

It has been shown on neuro-muscular preparations of frog sartorius muscle that chromium ions in the concentrations 1-4 x 10(-6) g/ml strengthen spontaneous and evoked transmitter release. Cr3+ ions in the concentrations above 4 x 10(-6) g/ml decrease the membrane potential of muscle fibres, decrease the quantum content of the end plate potentials. Experiments on a single Ranvier node have shown that Cr3+ ions decrease the amplitude, increase the rate and duration of the action potential of a nerve fibre. It is concluded that chromium ions produce a pronounced effect on synaptic transmission, which differs significantly from the action of manganese, cobalt and nickel ions.     

Improved 3D continuum calculations of ion flux through membrane channels

A continuum model, based on the Poisson–Nernst–Planck (PNP) theory, is applied to simulate steady-state ion flux through protein channels. The PNP equations are modified to explicitly account (1) for the desolvation of mobile ions in the membrane pore and (2) for effects related to ion sizes. The proposed algorithm for a three-dimensional self-consistent solution of PNP equations, in which final results are refined by a focusing technique, is shown to be suitable for arbitrary channel geometry and arbitrary protein charge distribution. The role of the pore shape and protein charge distribution in formation of basic electrodiffusion properties, such as channel conductivity and selectivity, as well as concentration distributions of mobile ions in the pore region, are illustrated by simulations on model channels. The influence of the ionic strength in the bulk solution and of the externally applied electric field on channel properties are also discussed.     

First node of Ranvier facilitates high-frequency burst encoding

In central neurons the first node of Ranvier is located at the first axonal branchpoint, ～ 100 μm from the axon initial segment where synaptic inputs are integrated and converted into action potentials (APs). Whether the first node contributes to this signal transformation is not well understood. Here it was found that in neocortical layer 5 axons, the first branchpoint is required for intrinsic high-frequency (≥ 100 Hz) AP bursts. Furthermore, block of nodal Na(+) channels or axotomy of the first node in intrinsically bursting neurons depolarized the somatic AP voltage threshold (～ 5 mV) and eliminated APs selectively within a high-frequency cluster in response to steady currents or simulated synaptic inputs. These results indicate that nodal persistent Na(+) current exerts an anterograde influence on AP initiation in the axon initial segment, revealing a computational role of the first node of Ranvier beyond conduction of the propagating AP.     

Ion Channel Clustering at the Axon Initial Segment and Node of Ranvier Evolved Sequentially in Early Chordates

In many mammalian neurons, dense clusters of ion channels at the axonal initial segment and nodes of Ranvier underlie action potential generation and rapid conduction. Axonal clustering of mammalian voltage-gated sodium and KCNQ (Kv7) potassium channels is based on linkage to the actin–spectrin cytoskeleton, which is mediated by the adaptor protein ankyrin-G. We identified key steps in the evolution of this axonal channel clustering. The anchor motif for sodium channel clustering evolved early in the chordate lineage before the divergence of the wormlike cephalochordate, amphioxus. Axons of the lamprey, a very primitive vertebrate, exhibited some invertebrate features (lack of myelin, use of giant diameter to hasten conduction), but possessed narrow initial segments bearing sodium channel clusters like in more recently evolved vertebrates. The KCNQ potassium channel anchor motif evolved after the divergence of lampreys from other vertebrates, in a common ancestor of shark and humans. Thus, clustering of voltage-gated sodium channels was a pivotal early innovation of the chordates. Sodium channel clusters at the axon initial segment serving the generation of action potentials evolved long before the node of Ranvier. KCNQ channels acquired anchors allowing their integration into pre-existing sodium channel complexes at about the same time that ancient vertebrates acquired myelin, saltatory conduction, and hinged jaws. The early chordate refinements in action potential mechanisms we have elucidated appear essential to the complex neural signaling, active behavior, and evolutionary success of vertebrates.     

Diffusion of Ions in Myelinated Nerve Fibers

The diffusion of ions towards or away from the inner side of the nodal membrane in preparations, the cut ends of which are placed in various media, was investigated. The ion concentration changes were calculated by numerical solution of the unidimensional electrodiffusion equation under a variety of media compositions, axoplasmic diffusion coefficients, and internal anionic compositions. The potassium and cesium ion diffusion along the axon towards the node was determined experimentally by two different electrophysiological methods. On the basis of comparison between the experimental data and the computational predictions the axoplasmic potassium ion diffusion coefficient was determined to be almost equal to that in free aqueous solution, while that of cesium ion was close to one half of that in aqueous solution. Utilizing the values of diffusion parameters thus determined, we solved the electrodiffusion equation for a number of common experimental procedures. We found that in short fibers, cut 0.1-0.2 cm at each side of the node, the concentration approached values close to the new steady-state values within 5-30 min. In long fibers (over 1 cm long) steady-state concentrations were obtained only after a few hours. Under some conditions the internal concentrations transiently overshot the steady-state values. The diffusion potentials generated in the system were also evaluated. The ion concentration changes and generation of diffusion potential cannot be prevented by using side pools with cation content identical to that of the axoplasm.     

The extracellular potential of a myelinated nerve fiber in an unbounded medium and in nerve cuff models

A model is presented for the calculation of single myelinated fiber action potentials in an unbounded homogeneous medium and in nerve cuff electrodes. The model consists of a fiber model, used to calculate the action currents at the nodes of Ranvier, and a cylindrically symmetrical volume conductor model in which the fiber's nodes are represented as point current sources. The extracellular action potentials were shown to remain unchanged if the fiber diameter and the volume conductor geometry are scaled by the same factor (principle of corresponding states), both in an unbounded homogeneous medium and in an inhomogeneous volume conductor. The influence of several cuff electrode parameters, among others, cuff length and cuff diameter, were studied, and the results were compared, where possible, with theoretical and experimental results as reported in the literature.     

The nodal origin of intrinsic bursting

The output of cortical neurons in the form of bursts of action potentials was thought to be controlled solely by the dendrites. In this issue of Neuron, Maarten Kole reveals that axonal sodium channels at the first node of Ranvier are essential for neuronal burst firing.     

Intrinsic and extrinsic determinants of ion channel localization in neurons

Neurons are an extremely diverse group of excitable cells with a wide variety of morphologies including complex dendritic trees and very long axons. The electrical properties of neurons depend not only on the types of ion channels and receptors expressed, but also on where these channels are located in the cell. Two extreme examples that illustrate the subcellular polarized nature of neurons and the tight regulation of ion channel localization can be seen at the axon initial segment and the node of Ranvier. The axon initial segment is important for initiation of action potentials in the axon, whereas the node of Ranvier is required for the rapid, faithful and efficient propagation of action potentials along the axon. Given the similarity of their functions it is not surprising that nearly every protein component of the axon initial segment is also found at the node. However, there is one very important difference between these two sites: nodes require extrinsic, glial-derived factors in order to form, whereas the axon initial segment is intrinsically determined by the neuron. This mini-review discusses recent results that have begun to clarify the intrinsic and extrinsic mechanisms underlying formation of nodes and axon initial segments, and poses several important unanswered questions regarding their unique mechanisms of formation.     

Antidromic propagation of action potentials in branched axons: implications for the mechanisms of action of deep brain stimulation

Electrical stimulation of the central nervous system creates both orthodromically propagating action potentials, by stimulation of local cells and passing axons, and antidromically propagating action potentials, by stimulation of presynaptic axons and terminals. Our aim was to understand how antidromic action potentials navigate through complex arborizations, such as those of thalamic and basal ganglia afferents-sites of electrical activation during deep brain stimulation. We developed computational models to study the propagation of antidromic action potentials past the bifurcation in branched axons. In both unmyelinated and myelinated branched axons, when the diameters of each axon branch remained under a specific threshold (set by the antidromic geometric ratio), antidromic propagation occurred robustly; action potentials traveled both antidromically into the primary segment as well as "re-orthodromically" into the terminal secondary segment. Propagation occurred across a broad range of stimulation frequencies, axon segment geometries, and concentrations of extracellular potassium, but was strongly dependent on the geometry of the node of Ranvier at the axonal bifurcation. Thus, antidromic activation of axon terminals can, through axon collaterals, lead to widespread activation or inhibition of targets remote from the site of stimulation. These effects should be included when interpreting the results of functional imaging or evoked potential studies on the mechanisms of action of DBS.     

alphaII-spectrin is essential for assembly of the nodes of Ranvier in myelinated axons

Saltatory conduction in myelinated axons requires organization of the nodes of Ranvier, where voltage-gated sodium channels are prominently localized [1]. Previous results indicate that alphaII-spectrin, a component of the cortical cytoskeleton [2], is enriched at the paranodes [3, 4], which flank the node of Ranvier, but alphaII-spectrin's function has not been investigated. Starting with a genetic screen in zebrafish, we discovered in alphaII-spectrin (alphaII-spn) a mutation that disrupts nodal sodium-channel clusters in myelinated axons of the PNS and CNS. In alphaII-spn mutants, the nodal sodium-channel clusters are reduced in number and disrupted at early stages. Analysis of chimeric animals indicated that alphaII-spn functions autonomously in neurons. Ultrastructural studies show that myelin forms in the posterior lateral line nerve and in the ventral spinal cord in alphaII-spn mutants and that the node is abnormally long; these findings indicate that alphaII-spn is required for the assembly of a mature node of the correct length. We find that alphaII-spectrin is enriched in nodes and paranodes at early stages and that the nodal expression diminishes as nodes mature. Our results provide functional evidence that alphaII-spectrin in the axonal cytoskeleton is essential for stabilizing nascent sodium-channel clusters and assembling the mature node of Ranvier.     

Subtle Paranodal Injury Slows Impulse Conduction in a Mathematical Model of Myelinated Axons

This study explores in detail the functional consequences of subtle retraction and detachment of myelin around the nodes of Ranvier following mild-to-moderate crush or stretch mediated injury. An equivalent electrical circuit model for a series of equally spaced nodes of Ranvier was created incorporating extracellular and axonal resistances, paranodal resistances, nodal capacitances, time varying sodium and potassium currents, and realistic resting and threshold membrane potentials in a myelinated axon segment of 21 successive nodes. Differential equations describing membrane potentials at each nodal region were solved numerically. Subtle injury was simulated by increasing the width of exposed nodal membrane in nodes 8 through 20 of the model. Such injury diminishes action potential amplitude and slows conduction velocity from 19.1 m/sec in the normal region to 7.8 m/sec in the crushed region. Detachment of paranodal myelin, exposing juxtaparanodal potassium channels, decreases conduction velocity further to 6.6 m/sec, an effect that is partially reversible with potassium ion channel blockade. Conduction velocity decreases as node width increases or as paranodal resistance falls. The calculated changes in conduction velocity with subtle paranodal injury agree with experimental observations. Nodes of Ranvier are highly effective but somewhat fragile devices for increasing nerve conduction velocity and decreasing reaction time in vertebrate animals. Their fundamental design limitation is that even small mechanical retractions of myelin from very narrow nodes or slight loosening of paranodal myelin, which are difficult to notice at the light microscopic level of observation, can cause large changes in myelinated nerve conduction velocity.     

A critical role for Neurofascin in regulating action potential initiation through maintenance of the axon initial segment

The axon initial segment (AIS) is critical for the initiation and propagation of action potentials. Assembly of the AIS requires interactions between scaffolding molecules and voltage-gated sodium channels, but the molecular mechanisms that stabilize the AIS are poorly understood. The neuronal isoform of Neurofascin, Nfasc186, clusters voltage-gated sodium channels at nodes of Ranvier in myelinated nerves: here, we investigate its role in AIS assembly and stabilization. Inactivation of the Nfasc gene in cerebellar Purkinje cells of adult mice causes rapid loss of Nfasc186 from the AIS but not from nodes of Ranvier. This causes AIS disintegration, impairment of motor learning and the abolition of the spontaneous tonic discharge typical of Purkinje cells. Nevertheless, action potentials with a modified waveform can still be evoked and basic motor abilities remain intact. We propose that Nfasc186 optimizes communication between mature neurons by anchoring the key elements of the adult AIS complex.     

nsf is essential for organization of myelinated axons in zebrafish

BACKGROUND: Myelinated axons are essential for rapid conduction of action potentials in the vertebrate nervous system. Of particular importance are the nodes of Ranvier, sites of voltage-gated sodium channel clustering that allow action potentials to be propagated along myelinated axons by saltatory conduction. Despite their critical role in the function of myelinated axons, little is known about the mechanisms that organize the nodes of Ranvier. RESULTS: Starting with a forward genetic screen in zebrafish, we have identified an essential requirement for nsf (N-ethylmaleimide sensitive factor) in the organization of myelinated axons. Previous work has shown that NSF is essential for membrane fusion in eukaryotes and has a critical role in vesicle fusion at chemical synapses. Zebrafish nsf mutants are paralyzed and have impaired response to light, reflecting disrupted nsf function in synaptic transmission and neural activity. In addition, nsf mutants exhibit defects in Myelin basic protein expression and in localization of sodium channel proteins at nodes of Ranvier. Analysis of chimeric larvae indicates that nsf functions autonomously in neurons, such that sodium channel clusters are evident in wild-type neurons transplanted into the nsf mutant hosts. Through pharmacological analyses, we show that neural activity and function of chemical synapses are not required for sodium channel clustering and myelination in the larval nervous system. CONCLUSIONS: Zebrafish nsf mutants provide a novel vertebrate system to investigate Nsf function in vivo. Our results reveal a previously unknown role for nsf, independent of its function in synaptic vesicle fusion, in the formation of the nodes of Ranvier in the vertebrate nervous system.     

Ionic processes in excitable membranes

The dynamical theory of ionized media is applied to the semi-electrolyte component of an excitable cell membrane, and the adjacent electrolytes. The equations of conservation of charge and momentum for the ions, and Poisson's equation for the electrostatic potential, are applied first to investigate the steady states of the membrane, and then transient effects in the membrane. A dispersion equation is derived, and the characteristic modes of relaxation within the membrane are determined. Among these are oscillating modes whose frequencies and amplitudes are of the correct order of magnitude to explain the observed excitation phenomena.A pair of coupled non-linear equations in the ionic potentials, with action potential solutions, is derived from the time-dependent electrodiffusion equations, and calculations are presented which model the behaviour of the excitable membrane during the voltage clamp. It is not necessary to postulate large changes in the ionic permeabilities in the course of the action potential and the voltage clamp to account for the large transient membrane conductances. It is suggested that the sodium hypothesis be replaced by one which attributes the action potential to non-linear plasma oscillations.     

Identification of rat sinus node pacemaker cells by intracellular injection of lanthanum ions

Action potentials of pacemaker cells were recorded on an isolated preparation of rat right atrium. Lanthanum ions were injected via a recording microelectrode. The sinus node area was treated for electron microscopy. The electron-dense clusters of lanthanum ions were identified in pacemaker cell organelles, such as cytoplasmic and mitochondrial matrix, external membrane of mitochondria, and inner surface of cytolemma. Treatment with La3+ produced no significant alterations in cellular morphology.