Analytical theory for extracellular electrical stimulation of nerve with focal electrodes. II. Passive myelinated axon.

The cable model of a passive, myelinated fiber is derived using the theory of electromagnetic propagation in periodic structures. The cable may be excited by an intracellular source or by an arbitrary, time-varying, applied extracellular field. When the cable is stimulated by a distant source, its properties are qualitatively similar to an unmyelinated fiber. Under these conditions relative threshold is proportional to the cube of the source distance and inversely proportional to the square of the fiber diameter. Electrical parameters of the model are chosen where possible, from mammalian peripheral nerve and anatomic parameters from cat auditory nerve. Several anatomic representations of the paranodal region are analyzed for their effects on the length and time constants of the fibers. Sensitivity of the model to parameter changes is studied. The linear model reliably predicts the effects of fiber size and electrode-fiber separation on threshold of cat dorsal column fibers to extracellular electrical stimulation.     

A generalized activating function for predicting virtual electrodes in cardiac tissue.

To fully understand the mechanisms of defibrillation, it is critical to know how a given electrical stimulus causes membrane polarizations in cardiac tissue. We have extended the concept of the activating function, originally used to describe neuronal stimulation, to derive a new expression that identifies the sources that drive changes in transmembrane potential. Source terms, or virtual electrodes, consist of either second derivatives of extracellular potential weighted by intracellular conductivity or extracellular potential gradients weighted by derivatives of intracellular conductivity. The full response of passive tissue can be considered, in simple cases, to be a convolution of this "generalized activating function" with the impulse response of the tissue. Computer simulations of a two-dimensional sheet of passive myocardium under steady-state conditions demonstrate that this source term is useful for estimating the effects of applied electrical stimuli. The generalized activating function predicts oppositely polarized regions of tissue when unequally anisotropic tissue is point stimulated and a monopolar response when a point stimulus is applied to isotropic tissue. In the bulk of the myocardium, this new expression is helpful for understanding mechanisms by which virtual electrodes can be produced, such as the hypothetical "sawtooth" pattern of polarization, as well as polarization owing to regions of depressed conductivity, missing cells or clefts, changes in fiber diameter, or fiber curvature. In comparing solutions obtained with an assumed extracellular potential distribution to those with fully coupled intra- and extracellular domains, we find that the former provides a reliable estimate of the total solution. Thus the generalized activating function that we have derived provides a useful way of understanding virtual electrode effects in cardiac tissue.     

On the electrotonic spread in cardiac muscle of the mouse

As an appropriate model which can simulate the cardiac working muscle with respect to the passive electrical spread, a lattice whose sides have linear cable properties is presented, and the passive potential spread on the model is mathematically analyzed in the fiber direction. Distribution of electrotonic potential in the fiber direction was measured with a pair of intracellular microelectrodes in the cardiac muscle fiber of mouse. By employing “pencil type” microelectrodes potential distribution in the transverse direction within a fiber was also measured. This transverse effect was differentiated from the longitudinal potential distribution. A tonically applied potential at any point of a cell interior spreads continuously in a manner described by a Bessel function. Using appropriate electrical and morphological parameters the experimental results proved to fit the curve obtained from numerical calculation on the model. The apparent length constant obtained for smaller distances (less than 100 μ) from the current source was 70 μ, and it increases as the distance becomes larger. At a point inside the fiber the resistance to the extracellular fluid ranged from 200 to 600 KΩ. The influence of coupling resistance between current and recording electrodes on the measurement of electrotonic potential was examined for small interelectrode distance.     

Sensory stimulation-dependent plasticity in the cerebellar cortex of alert mice

In vitro studies have supported the occurrence of cerebellar long-term depression (LTD), an interaction between the parallel fibers and Purkinje cells (PCs) that requires the combined activation of the parallel and climbing fibers. To demonstrate the existence of LTD in alert animals, we investigated the plasticity of local field potentials (LFPs) evoked by electrical stimulation of the whisker pad. The recorded LFP showed two major negative waves corresponding to trigeminal (broken into the N2 and N3 components) and cortical responses. PC unitary extracellular recording showed that N2 and N3 occurred concurrently with PC evoked simple spikes, followed by an evoked complex spike. Polarity inversion of the N3 component at the PC level and N3 amplitude reduction after electrical stimulation of the parallel fiber volley applied on the surface of the cerebellum 2 ms earlier strongly suggest that N3 was related to the parallel fiber-PC synapse activity. LFP measurements elicited by single whisker pad stimulus were performed before and after trains of electrical stimuli given at a frequency of 8 Hz for 10 min. We demonstrated that during this later situation, the stimulation of the PC by parallel and climbing fibers was reinforced. After 8-Hz stimulation, we observed long-term modifications (lasting at least 30 min) characterized by a specific decrease of the N3 amplitude accompanied by an increase of the N2 and N3 latency peaks. These plastic modifications indicated the existence of cerebellar LTD in alert animals involving both timing and synaptic modulations. These results corroborate the idea that LTD may underlie basic physiological functions related to calcium-dependent synaptic plasticity in the cerebellum.     

Dielectric behavior of the frog lens in the 100 Hz to 500 MHz range. Simulation with an allocated ellipsoidal-shells model.

In an attempt to correlate the passive electrical properties of the lens tissue with its structure, we measured ac admittances for isolated frog lenses, lens nuclei, and homogenate of cortical fiber cells, over the frequency range 10(2)-5.10(8) Hz. The whole lenses molded into discoid shape show a characteristic "two-step" dielectric dispersion with a huge permittivity increment of the order of 10(5) at 1 kHz. Of the two subdispersions disclosed, dispersion 1 has a permittivity increment (delta epsilon) of 2.10(5) with a characteristic frequency (fc) of 2 kHz, and dispersion 2 has a delta epsilon of 400 with an fc of 2 MHz. In terms of loss tangent, these dispersions are more clearly located as two separate peaks. Data are analyzed using an allocated ellipsoidal-shells model which has been developed by taking into account fiber orientation inside the lens tissue. Dispersion 1 is assigned to the equatorial cortex, where fiber cells run parallel to the applied electric field, and dispersion 2 to the nucleus with a complex fiber arrangement and also to the polar cortex, in which the fiber alignment is predominantly perpendicular. In addition, the model analysis reveals that, in the frog lens, the nucleus occupies approximately 30% in volume and that relative permittivity and conductivity for the cell interior are, respectively, 45 and 3 mS/cm for the cortical cells, and 28 and 0.3 mS/cm for the nuclear cells.     

Tension in frog single muscle fibers while shortening actively and passively at velocities near Vu.

Experiments were undertaken to determine the contribution of passive tension to total tension during rapid shortening in a stimulated muscle fiber. Results were obtained by applying shortening movements at constant velocities slightly less than Vu (the velocity of unloaded shortening) to intact twitch fibers isolated from the frog (Rana temporaria). The tension maintained by unstimulated fibers during such shortening movements ("dynamic passive tension") from moderately long lengths was greater than zero but much less than the passive tension measured under static conditions ("static passive tension") at the same lengths. Fibers maximally activated by electrical stimulation and then shortened at the same velocity over the same range of average sarcomere lengths maintained tension that was greater than zero but less than the dynamic passive tension. For average sarcomere lengths up to approximately 3.1 microns, the dynamic passive tension appeared to be substantially abolished by activation. The onset of the apparent disappearance of dynamic passive tension was studied by initiating the stimulation and the shortening movement simultaneously. The resulting tension response exhibited a latency relaxation that was increased in amplitude compared with the isometric case, followed by a brief tension rise, giving way to a steady tension level equal to that expected if stimulation had been initiated well before the release. These changes are qualitatively explained in terms of the establishment of a steady state distribution of deformations of attached cross-bridges.     

Modeling the interactions between stimulation and physiologically induced APs in a mammalian nerve fiber: dependence on frequency and fiber diameter

Electrical stimulation of nerve fibers is used as a therapeutic tool to treat neurophysiological disorders. Despite efforts to model the effects of stimulation, its underlying mechanisms remain unclear. Current mechanistic models quantify the effects that the electrical field produces near the fiber but do not capture interactions between action potentials (APs) initiated by stimulus and APs initiated by underlying physiological activity. In this study, we aim to quantify the effects of stimulation frequency and fiber diameter on AP interactions involving collisions and loss of excitability. We constructed a mechanistic model of a myelinated nerve fiber receiving two inputs: the underlying physiological activity at the terminal end of the fiber, and an external stimulus applied to the middle of the fiber. We define conduction reliability as the percentage of physiological APs that make it to the somatic end of the nerve fiber. At low input frequencies, conduction reliability is greater than 95% and decreases with increasing frequency due to an increase in AP interactions. Conduction reliability is less sensitive to fiber diameter and only decreases slightly with increasing fiber diameter. Finally, both the number and type of AP interactions significantly vary with both input frequencies and fiber diameter. Modeling the interactions between APs initiated by stimulus and APs initiated by underlying physiological activity in a nerve fiber opens opportunities towards understanding mechanisms of electrical stimulation therapies.     

Relation of unit spike discharges and evoked potentials in the auditory cortex of cats.

Extracellular microelectrode recordings were made from the auditory cortex of anaesthetized cats during acoustic click stimulation. The microelectrode of low resistance allowed to record evoked field potentials and unit discharges simultaneously. In distant extracellular leads the relation of unit discharges and field potentials was equivocal. Near extracellular leads revealed that the antidromic invasion of the somadendritic membrane by excitation is a frequency dependent process (just as evoked field potentials) while spike potentials can reliably be elicited from the initial segment at high frequencies. It is assumed that the excitation spreading from the initial segment to the soma-dendritic membrane represents an important component of the evoked potentials, and their frequency dependence may be traced back to inhibitions activated by afferent impulses.     

Threshold fluctuations in an N sodium channel model of the node of Ranvier.

Computer simulations of stochastic single-channel open-close kinetics are applied to an N sodium channel model of a node of Ranvier. Up to 32,000 voltage-gated sodium channels have been simulated with modified amphibian sodium channel kinetics. Poststimulus time histograms are obtained with 1000 monophasic pulse stimuli, and measurements are made of changes in the relative spread of threshold (RS) with changes in the model parameters. RS is found to be invariant with pulse durations from 100 microseconds to 3 ms. RS is approximately of inverse proportion to square-root of N. It decreases with increasing temperature and is dependent on passive electrical properties of the membrane as well as the single-channel conductance. The simulated RS and its independence of pulse duration is consistent with experimental results from the literature. Thus, the microscopic fluctuations of single, voltage-sensitive sodium channels in the amphibian peripheral node of Ranvier are sufficient to account for the macroscopic fluctuation if threshold to electrical stimulation.     

Genetic Models in Applied Physiology. Merosin deficiency leads to alterations in passive and active skeletal muscle mechanics.

The role of extracellular elements on the mechanical properties of skeletal muscles is unknown. Merosin is an essential extracellular matrix protein that forms a mechanical junction between the sarcolemma and collagen. Therefore, it is possible that merosin plays a role in force transmission between muscle fibers and collagen. We hypothesized that deficiency in merosin may alter passive muscle stiffness, viscoelastic properties, and contractile muscle force in skeletal muscles. We used the dy/dy mouse, a merosin-deficient mouse model, to examine changes in passive and active muscle mechanics. After mice were anesthetized and the diaphragm or the biceps femoris hindlimb muscle was excised, passive length-tension relationships, stress-relaxation curves, or isometric contractile properties were determined with an in vitro biaxial mechanical testing apparatus. Compared with controls, extensibility was smaller in the muscle fiber direction and the transverse fiber direction of the mutant mice. The relaxed elastic modulus was smaller in merosin-deficient diaphragms compared with controls. Interestingly, maximal muscle tetanic stress was depressed in muscles from the mutant mice during uniaxial loading but not during biaxial loading. However, presence of transverse passive stretch increases maximal contractile stress in both the mutant and normal mice. Our data suggest that merosin contributes to muscle passive stiffness, viscoelasticity, and contractility and that the mechanism by which force is transmitted between adjacent myofibers via merosin possibly in shear.     

Effects of pH on the length-dependent twitch potentiation in skeletal muscle.

When muscle is elongated, there is a length dependence of twitch potentiation and an increased Ca(2+) sensitivity of the myofilaments. Changes in the charge potential of myofilaments, induced by a decrease in pH, are known to abolish the length dependence of Ca(2+) sensitivity. This study was aimed at testing the hypothesis that a decrease in pH, and the concomitant loss of length dependence of Ca(2+) sensitivity, depresses the length dependence of staircase potentiation. In vitro, isometric twitch contractions of fiber bundles dissected from the mouse extensor digitorum longus, performed before and after 10 s of 10-Hz stimulation (i.e., the staircase potentiation protocol) were analyzed at five different lengths, ranging from optimal length for maximal force production (L(o); = 12 +/- 0.7 mm) to L(o) + 1.2 mm (L(o) + 10%). These measurements were made at an extracellular pH of 6.6, 7.4, and 7.8 (pH changes induced by altering the CO(2) concentration of the bath solution). At pH 7.4 and 7.8, the degree of potentiation after 10-Hz stimulation showed a linear decrease with increased fiber bundle length (r(2) = 0.95 and r(2) = 0.99, respectively). At pH 6.6, the length dependence of potentiation was abolished, and the slope of the length-potentiation relationship was not different from zero (r(2) = 0.05). The results of this study indicate that length dependence of potentiation in intact skeletal muscle is abolished by lowering the pH. Because decreasing the pH decreases Ca(2+) sensitivity and changes the charge potential of the filaments, the mechanism of length-dependent potentiation may be closely related to the length dependence of Ca(2+) sensitivity, and changes in the charge potential of the myofilaments may be important in regulating this relationship.     

The “Mirror” Estimate: An Intuitive Predictor of Membrane Polarization during Extracellular Stimulation

Achieving controlled extracellular microstimulation of the central nervous system requires understanding the membrane response of a neuron to an applied electric field. The “activating function” has been proposed as an intuitive predictor of membrane polarization during stimulation, but subsequent literature raised several limitations of this estimate. In this study, we show that, depending on the space constant λ, the steady-state solution to the passive cable equation is theoretically well approximated by either the activating function when λ is small, or the “mirror” image of the extracellular potential when λ is large. Using simulations, we then explore the respective domain of both estimates as a function of λ, stimulus duration, fiber length, and electrode-fiber distance. For realistic λ (>50-100 μm), the mirror estimate is the best predictor for either long electrode-fiber distances or short distances (<20-30 μm) when stimulus durations exceed a few tens of microseconds. For intermediate distances, the mirror estimate is all the more valid that the stimulus duration is long and the fiber is short. We also illustrate that this estimate correctly predicts the steady-state membrane polarization of complex central nervous system arborizations. In conclusion, the mirror estimate can often be preferred to the activating function to intuitively predict membrane polarization during extracellular stimulation.     

Voltage dependence of proton pumping by bacteriorhodopsin is regulated by the voltage-sensitive ratio of M1 to M2.

The voltage dependence of light-induced proton pumping was studied with bacteriorhodopsin (bR) from Halobacterium salinarum, expressed in the plasma membrane of oocytes from Xenopus laevis in the range -160 mV to +60 mV at different light intensities. Depending on the applied field, the quenching effect by blue light, which bypasses the normal photo and transport cycle, is drastically increased at inhibiting (negative) potentials, and is diminished at pump current increasing (positive) potentials. At any potential, two processes with different time constants for the M --> bR decay of approximately 5 ms (tau1) and approximately 20 ms (tau2) are obtained. At pump-inhibiting potentials, a third, long-lasting process with tau3 approximately 300 ms at neutral pH is observed. The fast processes (tau1, tau2) can be assigned to the decay of M2 in the normal pump cycle, i.e., to the reprotonation of the Schiff base via the cytoplasmic side, whereas tau3 is due to the decay of M1 without net pumping, i.e., the reprotonation of the Schiff base via the extracellular side. The results are supported by determination of photocurrents induced by bR on planar lipid films. The pH dependence of the slow decay of M1 is fully in agreement with the interpretation that the reprotonation of the Schiff base occurs from the extracellular side. The results give strong evidence that an externally applied electrical field changes the ratio of the M1 and the M2 intermediate. As a consequence, the transport cycle branches into a nontransporting cycle at negative potentials. This interpretation explains the current-voltage behavior of bR on a new basis, but agrees with the isomerisation, switch, transfer model for vectorial transport.     

Reports of the length dependence of fatigue are greatly exaggerated.

Relative force depression associated with muscle fatigue is reported to be greater when assessed at short vs. long muscle lengths. This appears to be due to a rightward shift in the force-length relationship. This rightward shift may be caused by stretch of in-series structures, making sarcomere lengths shorter at any given muscle length. Submaximal force-length relationships (twitch, double pulse, 50 Hz) were evaluated before and after repetitive contractions (50 Hz, 300 ms, 1/s) in an in situ preparation of the rat medial gastrocnemius muscle. In some experiments, fascicle lengths were measured with sonomicrometry. Before repetitive stimulation, fascicle lengths were 11.3 +/- 0.8, 12.8 +/- 0.9, and 14.4 +/- 1.2 mm at lengths corresponding to -3.6, 0, and 3.6 mm where 0 is a reference length that corresponds with maximal active force for double-pulse stimulation. After repetitive stimulation, there was no change in fascicle lengths; these lengths were 11.4 +/- 0.8, 12.6 +/- 0.9, and 14.2 +/- 1.2 mm. The length dependence of fatigue was, therefore, not due to a stretch of in-series structures. Interestingly, the rightward shift that was evident when active force was calculated in the traditional way (subtraction of the passive force measured before contraction) was not seen when active force was calculated by subtracting the passive force that was associated with the fascicle length reached at the peak of the contraction. This calculation is based on the assumption that passive force decreases as the fascicles shorten during a fixed-end contraction. This alternative calculation revealed similar postfatigue absolute active force depression at all lengths. In relative terms, a length dependence of fatigue was still evident, but this was greatly diminished compared with that observed when active force was calculated with the traditional method.     

Segmental release of amino acid neurotransmitters from transcranial stimulation

The present study used microdialysis techniques in an intact rabbit model to measure the release of amino acids within the lumbar spinal cord in response to transcranial electrical stimulation. Dialysis samples from the extracellular space were obtained over a stimulation period of 90 minutes and were examined using high pressure liquid chromatography. Neuronal excitation was verified by recerding corticomotor evoked potentials (CMEPs) from the spinal cord. A significant increase in the release of glycine and taurine compared to sham animals was measured after 90 minutes of transcranial stimulation. Glutamate and aspartate release was not significantly elevated. GABA concentrations were consistently low. CMEP components repeatedly showed adequate activation of descending fiber pathways and segmental interneuron pools during dialysis sampling. Since glycine, and to a lesser extent taurine, have been shown to inhibit motor neuron activity and are closely associated with segmental interneuron pools, suprasegmental modulation of motor activity may be, in part, through these inhibitory amino acid neurotransmitters in the rabbit lumbar spinal cord.     

In vivo tracking of the human patella.

The purpose of this study was to describe the dynamic, in vivo, three-dimensional tracking pattern of the patella for one normal male subject. Intracortical pins were inserted into the patella, tibia, and femur. The subject performed seated and squatting knee flexion/extension, and maximum voluntary quadriceps contractions. In addition, the vastus medialis oblique was subjected to maximal electrical stimulation. Motions of the markers attached to the intracortical pins were analyzed using an automated video system. Patellar and tibial motions were determined relative to a femoral reference system. While the tibia flexed 50 degrees from full extension (seated condition), the patella flexed 30.3 degrees, tilted laterally 10.3 degrees, and shifted laterally 8.6 mm. In general, these results show qualitative agreement with the data collected from cadaveric specimens [van Kampen and Huiskes, J. orthop. Res. 8, 372-382 (1990)]. The differences present may reflect different passive constraints to patellar motions, and different relative loading of the individual quadriceps components, in our study compared to the cadaveric study. Only small differences were found between patellar motions in the seated and squatting conditions. Differences in patellar displacements produced by (1) maximal electrical stimulation of the vastus medialis oblique, and (2) maximum voluntary quadriceps contraction, at 30 degrees knee flexion and full extension, may reflect the dominant influence of passive constraints, and the vastus lateralis, on normal patellar motions. Further in vivo study of patellar tracking seems warranted to evaluate surgical and conservative interventions for patellofemoral disorders.     

Verapamil and zero Ca2+ alter responses of cat muscle to halothane and caffeine

Strips of soleus (slow twitch, oxidative) and gracilis (fast-twitch, glycolytic) muscle were obtained from 27 anesthetized cats and mounted in organ baths filled with oxygenated Krebs-Ringer solution (37 degrees C). The responses to caffeine, halothane (1%), caffeine in the presence of halothane, and electrical stimulation in the presence of halothane were examined in the two fiber types. These responses were compared with those observed in paired strips of muscle that had been treated with verapamil (10 or 28 microM), a slow calcium (Ca2+) channel blocker, with zero Ca2+, or with zero Ca2+ where magnesium (3.7 mM Ca2+) was added to replace the Ca2+. Halothane-induced contractures in the soleus were blocked by verapamil and zero Ca2+. Caffeine-induced contractures and tetanic contractions were attenuated in zero Ca2+ and by verapamil in both fiber types. Halothane overcame verapamil-induced reductions of caffeine contractures and tetanic contractions in both fiber types. In contrast, halothane did not overcome zero Ca2+-induced reductions in caffeine contractures or tetanic contractions in either fiber type. Furthermore, the addition of Mg2+ to the zero Ca2+ did not restore the responses. The findings with verapamil indicate that in cat muscle, both halothane- and caffeine-induced contractures and tetanic contractions are dependent on the influx of extracellular Ca2+. This extracellular Ca2+ may enter through the slow Ca2+ channels. However, because halothane in combination with caffeine or electrical stimulation overcame the effects of verapamil, there may be other sites involved.     

Selected contribution: effect of chronic passive length change on airway smooth muscle length-tension relationship.

The ability of rabbit trachealis to undergo plastic adaptation to chronic shortening or lengthening was assessed by setting the muscle preparations at three lengths for 24 h in relaxed state: a reference length in which applied force was approximately 1-2% of maximal active force (P(o)) and lengths considerably shorter and longer than the reference. Passive and active length-tension (L-T) curves for the preparations were then obtained by electrical field stimulation at progressively increasing muscle length. Classically shaped L-T curves were obtained with a distinct optimal length (L(o)) at which P(o) developed; however, both the active and passive L-T curves were shifted, whereas P(o) remained unchanged. L(o) was 72% and 148% that of the reference preparations for the passively shortened and lengthened muscles, respectively. The results suggest that chronic narrowing of the airways could induce a shift in the L-T relationship of smooth muscle, resulting in a maintained potential for maximal force production.