D. I. Stephanova

A distributed-parameter model of the myelinated human motor nerve fibre: temporal and spatial distributions of electrotonic potentials and ionic currents

A double cable model of the myelinated human motor nerve fibre is presented. The model is based on the nodal and internodal channels in a previous, two-component model of human motor axons (Bostock et al. 1991), added to a complex extended cable structure of nodal, paranodal and internodal segments. The model assumes a high-resistance myelin sheath and a leakage pathway to the internodal axolemma via the paranodal seal resistance and periaxonal space. The parameter values of the model were adjusted to match the recordings of threshold electrotonus in human motor fibres from Bostock et al. (1991). Kirchoff s current law was used to derive a system of partial differential equations for the electrical equivalent circuit, and numerical integration was performed with a fixed time increment and non-uniform spatial step sizes, in accordance with the complex structure of the fibre. The model calculations provide estimates of the spatial and temporal distributions of action potentials and their transaxonal and transmyelin components, both in different segments of the fibre and at different moments during action potential propagation. The distribution of transaxonal and transmyelin currents along the fibre and their contributions from different ionic channels are also explored.

Myelin as longitudinal conductor: a multi-layered model of the myelinated human motor nerve fibre.

Abstract. The myelin sheath is normally regarded as an electrical insulator. Low values of radial conductance and capacitance have been measured, and in electrical models of myelinated axons the contribution of longitudinal conduction within the sheath has been ignored. According to X-ray diffraction studies, however, myelin sheaths comprise alternate lipid and aqueous layers, and the latter may be expected to have a low resistivity. We propose a new model of myelinated axons in which the aqueous layers within the myelin provide appreciable longitudinal and radial conductance, the latter via a spiral pathway. We have investigated the likely contribution of these conductive paths within the myelin to the electrical properties of a human motor nerve fibre by computer simulation, representing the myelin sheath as a series of interconnecting parallel lamellae. With this new model, action potential conduction has been simulated along a 20-node cable, and the electrotonic responses to 100-ms depolarizing and hyperpolarizing current pulses have been simulated for a uniformly polarized fibre. We have found that the hypothesis of a longitudinally conducting myelin sheath improves our previous model in two ways: it is no longer necessary to make implausible assumptions about the resistivity or width of the periaxonal space to simulate realistic electrotonus, and the conduction velocity is appreciably faster (by 8.6%).

Differences in potentials and excitability properties in simulated cases of demyelinating neuropathies. Part II. Paranodal demyelination.

OBJECTIVE The aim of this study is to investigate the potentials (intracellular, extracellular, electrotonic) and excitability properties (strength-duration and charge-duration curves, strength-duration time constants, rheobasic currents, recovery cycles) in progressively greater degrees of uniform reduction (20, 50 and 70%) of the paranodal seal resistance and myelin lamellae along the fibre length. METHODS Three paranodally internodally systematically demyelinated cases (termed as PISD1, PISD2 and PISD3, respectively) are simulated using our previous double cable model of human motor nerve fibres. RESULTS The results conform that in the more severely demyelinated cases, the intracellular potentials are with significantly reduced amplitude, prolonged duration and slowed conduction velocity, whereas the electrotonic potentials show abnormally greater increase in the early part of the hyperpolarizing responses. The extracellular potentials indicate increased polyphasia in the PISD3 case. The strength-duration time constants are shorter and the rheobasic currents higher in the demyelinated cases. In the recovery cycles, the demyelinated cases have less refractoriness, greater supernormality and less late subnormality than the normal case. CONCLUSIONS The uniform reduction of the paranodal seal resistance and myelin thickness along the fibre length has significant effects on the potentials and excitability properties of the simulated demyelinated human motor fibres. Unexpectedly, the PISD fibres behave like paranodally demyelinated ones, since the myelin reduction increases slightly the effect of the paranodal demyelination on the nerve membrane properties. The study shows that the excitability properties in demyelinating neuropathies are much more largely determined by the paranodal changes than by the internodal changes. SIGNIFICANCE The study provides new and important information about the pathophysiology of human demyelinating neuropathies.

Different effects of blocked potassium channels on action potentials, accommodation, adaptation and anode break excitation in human motor and sensory myelinated nerve fibres: computer simulations

Abstract. Action potentials and electrotonic responses to 300-ms depolarizing and hyperpolarizing currents for human motor and sensory myelinated nerve fibres have been simulated on the basis of double cable models. The effects of blocked nodal or internodal potassium (fast or slow) channels on the fibre action potentials, early and late adaptations to 30-ms suprathreshold slowly increasing depolarizing stimuli have been examined. The effects of the same channels on accommodation after the termination of a prolonged (100 ms) hyperpolarizing current pulse have also been investigated. By removing the nodal fast potassium conductance the action potentials of the sensory fibres are considerably broader than those of the motor neurons. For both types of fibres, the blocked nodal slow potassium channels have a substantially smaller effect on the action potential repolarization. When the suprathreshold depolarizing current intensity is increased, the onset of the spike burst occurs sooner, which is common in the behaviour of the fibres. The most striking differences in the burst activity during early adaptation have been found between the fibres when the nodal fast potassium channels are blocked. The results obtained confirm the fact that the motor fibres adapt more quickly to sustained depolarizing current pulses than the sensory ones. The results also show that normal human motor and sensory fibres cannot be excited by a 100-ms hyperpolarizing current pulse, even at the threshold level. When removing the potassium channels in the nodal or internodal axolemma, the posthyperpolarization increase in excitability is small, which is common in the behaviour of the fibres. However, anode break excitation can be simulated in the fibres with simultaneous removal of the potassium channels under the myelin sheath, and this is more pronounced in the human sensory fibres than in motor fibres. This phenomenon can also be found when the internodal and some of the nodal (fast or slow) potassium channels are simultaneously blocked.

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 μm) 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.

Differences in potentials and excitability properties in simulated cases of demyelinating neuropathies. Part I

OBJECTIVE The aim of this study is to investigate the potentials (intracellular, extracellular, electrotonic) and excitability properties (strength-duration and charge-duration curves, strength-duration time constants, rheobases, recovery cycles) in three cases of uniform myelin wrap reduction (20, 50 and 70%) along the fibre length. METHODS The internodally systematically demyelinated cases (termed as ISD1, ISD2 and ISD3) are simulated using our previous double cable model of human motor fibres. RESULTS In the more severely demyelinated cases, the intracellular potentials are with significantly reduced amplitude, prolonged duration and slowed conduction velocity, whereas the electrotonic potentials show greater increase in the early part of the hyperpolarizing responses. The radial decline of the extracellular potential amplitudes depends on the radial distance of the field point and increases with the increase of the distance and demyelination. The time constants and rheobasic currents increase with the increase of the degree of demyelination. In the recovery cycles, the more severely demyelinated cases have greater refractoriness (the increase in threshold current during the relative refractory period), supernormality and less late subnormality than the normal case. CONCLUSIONS The myelin thickness has significant effects on the potentials and axonal excitability properties of the simulated demyelinated human motor fibres. The obtained abnormalities in the potentials and excitability properties can be observed in Charcot-Marie-Tooth disease type 1A (CMT1A). SIGNIFICANCE The study provides new information about the pathophysiology of human demyelinating neuropathies.

Membrane property abnormalities in simulated cases of mild systematic and severe focal demyelinating neuropathies.

The investigation of multiple nerve membrane properties by mathematical models has become a new tool to study peripheral neuropathies. In demyelinating neuropathies, the membrane properties such as potentials (intracellular, extracellular, electrotonic) and indices of axonal excitability (strength-duration time constants, rheobases and recovery cycles) can now be measured at the peripheral nerves. This study provides numerical simulations of the membrane properties of human motor nerve fibre in cases of internodal, paranodal and simultaneously of paranodal internodal demyelinations, each of them mild systematic or severe focal. The computations use our previous multi-layered model of the fibre. The results show that the abnormally greater increase of the hyperpolarizing electrotonus, shorter strength-duration time constants and greater axonal superexcitability in the recovery cycles are the characteristic features of the mildly systematically demyelinated cases. The small decrease of the polarizing electrotonic responses in the demyelinated zone in turn leads to a compensatory small increase of these responses outside the demyelinated zone of all severely focally demyelinated cases. The paper summarizes the insights gained from these modeling studies on the membrane property abnormalities underlying the variation in clinical symptoms of demyelination in Charcot-Marie-Tooth disease type 1A, chronic inflammatory demyelinating polyneuropathy, Guillain-Barré syndrome and multifocal motor neuropathy. The model used provides an objective study of the mechanisms of these diseases which up till now have not been sufficiently well understood, because quite different assumptions have been given in the literature for the interpretation of the membrane property abnormalities obtained in hereditary, chronic and acquired demyelinating neuropathies.

Simulating mild systematic and focal demyelinating neuropathies: membrane property abnormalities.

This study provides numerical simulations of some of the abnormalities in the potentials and axonal excitability indices of human motor nerve fibers in simulated cases of internodal, paranodal and simultaneously of paranodal internodal demyelinations, each of them systematic or focal. A 70% reduction of the myelin lamellae (defining internodal demyelination), or of the paranodal seal resistance (defining paranodal demyelination), or simultaneously both of them (defining paranodal internodal demyelination) was uniform along the fiber length for the systematically demyelinated subtypes. These permutations were termed internodal systematic demyelination (ISD), paranodal systematic demyelination (PSD) and paranodal internodal systematic demyelination (PISD). In other tests, the same reductions of the myelin sheath parameters were used but restricted to only three (8th, 9th and 10th) consecutive internodes. Such fiber demyelinations were termed internodal focal demyelination (IFD), paranodal focal demyelination (PFD) and paranodal internodal focal demyelination (PIFD). The computations used our previous double cable model of the fibers. The axon model was comprised of 30 nodes and 29 internodes. The 70% reduction value was not sufficient to develop conduction block in all investigated demyelinations, which were regarded as mild. The membrane property abnormalities obtained in the ISD, PSD and PISD cases were quite different and abnormally greater than those in the IFD, PFD and PIFD cases. The changes in the excitability indices such as strength-duration time constants, rheobasic currents and recovery cycles in the focally demyelinated subtypes were so slight as to be essentially indistinguishable from normal values. Consequently, the excitability based approaches that have shown strong potential as diagnostic tools in systematically demyelinated conditions may not be useful in detecting mild focal demyelinations. The membrane property changes simulated in the systematically demyelinated subtypes are in good accordance with the data from patients with Charcot-Marie-Tooth disease type 1A (CMT1A) and chronic inflammatory demyelinating polyneuropathy (CIDP). The excitability abnormalities obtained in each focally demyelinated subtype match those observed in vivo in patients with demyelinating forms of Guillain-Barré syndrome (GBS). The results indicate that the model that was used is a rather promising tool in studying the membrane property abnormalities of hereditary, chronic and acquired demyelinating neuropathies, which up till now, have not been sufficiently well understood.

Conduction along myelinated and demyelinated nerve fibres during the recovery cycle: Model investigations

The membrane excitability changes as well as the underlying mechanisms of these changes in a normal and in a systematically paranodally demyelinated nerve fibre have been investigated by paired stimulation during the first 30 ms of the recovery cycle. The ionic current kinetics determining the observed changes in the action potential parameters are presented also. The simulation of the conduction in the normal fibre is based on the Frankenhaeuser and Huxley (1964) and Goldman and Albus (1968) equations, while in the case of a demyelinated fibre according to the same equations modified by Stephanova (1988a). It has been shown for the demyelinated membrane that increased demyelination increases both the threshold current for the second potential as well as the absolute refractory period. With increasing interpulse interval, the subnormality of the membrane excitability is followed by supernormality in the case of the demyelinated membrane. For the recovery cycle of 30 ms under consideration no supernormality of the normal membrane excitability is obtained. With interpulse interval from 8.8 to 10.9 ms, the highest degree of demyelination (l=30 μm) is accompanied by a refractory period of transmission. The membrane properties of the normal and demyelinated fibres recover 20 ms after the first pulse. For short interpulse intervals, the amplitude of the second action potential is decreased, and a slower propagation velocity is obtained. The most sensitive phenomenon is the excitability of the demyelinated membrane, which remains unrecovered 30 ms after the first pulses has been applied.

Differences in Membrane Properties in Simulated Cases of Demyelinating Neuropathies: Internodal Focal Demyelinations without Conduction Block

The aim of this study is to investigate the membrane properties (potentials and axonal excitability indices) in the case of myelin wrap reduction (96%) in one, two and three consecutive internodes along the length of human motor nerve fibre. The internodally focally demyelinated cases (termed as IFD1, IFD2 and IFD3, respectively, with one, two and three demyelinated internodes are simulated using our previous double cable model of the fibre. The progressively greater increase of focal loss of myelin lamellae blocks the invasion of the intracellular potentials into the demyelinated zones. For all investigated cases, the radial decline of the extracellular potential amplitudes increases with the increase of the radial distance and demyelination, whereas the electrotonic potentials show a decrease in the slow part of the depolarizing and hyperpolarizing responses. The time constants are shorter and the rheobases higher for the IFD2 and IFD3 cases than for the normal case. In the recovery cycles, the same cases have less refractoriness, greater supernormality and less late subnormality than the normal case. The simulated membrane abnormalities can be observed in vivo in patients with demyelinating forms of Guillain-Barré syndrome. The study provides new information about the pathophysiology of acquired demyelinating neuropathies.