Frank Rattay

Analysis of Models for External Stimulation of Axons

Extracellular electrodes produce electrical fields at the outside of nerve fibers. Discretization of the axons length coordinate allows simulation of the excitation by a system of differential equations in time, and difference equations in space. For myelinated fibers this segmentation is naturally given by the nodes of Ranvier, whereas unmyelinated axons can be segmented arbitrarily. In both cases the equations are similar and can be treated in parallel. The activity of the axon depends on the second space derivative of the extracellular medium. The activating function is discussed for monopolar electrodes but the principle can be extended to arbitrary configurations of electrodes.

Analysis of models for extracellular fiber stimulation

The mathematical basis for analysis as well as for the computer simulation of the stimulus-response characteristics of nerve or muscle fibers is presented. The results follow from the extracellular potential along the fiber as a function of electrode geometry. The theory is of a general nature, but special investigations are made on monopolar, bipolar, and ring electrodes. Stimulation with monopolar electrodes shows better recruitment characteristics than with ring electrodes.<<ETX>>

The basic mechanism for the electrical stimulation of the nervous system

Neural signals can be generated or blocked by extracellular electrodes or magnetic coils. New results about artificial excitation are based on a compartmental model of a target neuron and its equivalent electrical network, as well as on the theory of the generalized activating function. The analysis shows that: (i) in most cases, the origin of artificial excitation is within the axon and the soma is much more difficult to excite; (ii) within the central nervous system, positive and negative threshold currents essentially depend on the position and orientation of the neurons relative to the applied electric field; (iii) in several cases, stimulation with positive currents is easier; and (iv) it should be possible to excite synaptic activity without the generation of propagating action potentials. Furthermore, the theory of the generalized activating function gives hints to understanding the blockage of neural activity.

A model of the electrically excited human cochlear neuron I. Contribution of neural substructures to the generation and propagation of spikes

Differences in neural geometry and the fact that the soma of the human cochlear neuron typically is not myelinated are reasons for disagreements between single fiber recordings in animals and the neural code evoked in cochlear implant patients. We introduce a compartment model of the human cochlear neuron to study the excitation and propagation process of action potentials. The model can be used to predict (i) the points of spike generation, (ii) the time difference between stimulation and the arrival of a spike at the proximal end of the central axon, (iii) the vanishing of peripherally evoked spikes at the soma region under specific conditions, (iv) the influence of electrode positions on spiking behavior, and (v) consequences of the loss of the peripheral axon. Every subunit of the cochlear neuron is separately modeled. Ion channel dynamics are described by a modified Hodgkin--Huxley model. Influence of membrane noise is taken into account. Additionally, the generalized activating function is introduced as a tool to give an envision of the origin of spikes in the peripheral and in the central axon without any knowledge of the gating processes in the active membranes. Comparing the reactions of a human and cat cochlear neuron, we find differences in spiking behavior, e.g. peripherally and centrally evoked spikes arrive with a time difference of about 400 mus in man and 200 mus in cat.

Ways to approximate current-distance relations for electrically stimulated fibers

The behaviour of myelinated and unmyelinated fibers is simulated by a model for extracellular stimulation in order to confirm current-distance phenomena known from experiments. Lower and upper limits for cells stimulated with cathode current are calculated for unmyelinated fibers. In the case of myelinated fibers the threshold depends on the distance from the axon and from the nodes.

A model of the electrically excited human cochlear neuron. II. Influence of the three-dimensional cochlear structure on neural excitability

A simplified spiraled model of the human cochlea is developed from a cross sectional photograph. The potential distribution within this model cochlea is calculated with the finite element technique for an active scala tympani implant. The method in the companion article [Rattay et al., 2001] allows for simulation of the excitation process of selected elements of the cochlear nerve. The bony boundary has an insulating influence along every nerve fiber which shifts the stimulation condition from that of a homogeneous extracellular medium towards constant field stimulation: for a target neuron which is stimulated by a ring electrode positioned just below the peripheral end of the fiber the extracellular voltage profile is rather linear. About half of the cochlear neurons of a completely innervated cochlea are excited with monopolar stimulation at three-fold threshold intensity, whereas bipolar and especially quadrupolar stimulation focuses the excited region even for stronger stimuli. In contrast to single fiber experiments with cats, the long peripheral processes in human cochlear neurons cause first excitation in the periphery and, consequently, neurons with lost dendrite need higher stimuli.

Posterior root-muscle reflexes elicited by transcutaneous stimulation of the human lumbosacral cord

Continuous epidural stimulation of lumbar posterior root afferents can modify the activity of lumbar cord networks and motoneurons, resulting in suppression of spasticity or elicitation of locomotor‐like movements in spinal cord–injured people. The aim of the present study was to demonstrate that posterior root afferents can also be depolarized by transcutaneous stimulation with moderate stimulus intensities. In healthy subjects, single stimuli applied through surface electrodes placed over the T11–T12 vertebrae with a mean intensity of 28.6 V elicited simultaneous, bilateral monosynaptic reflexes in quadriceps, hamstrings, tibialis anterior, and triceps surae by depolarization of lumbosacral posterior root fibers. The nature of these posterior root–muscle reflexes was demonstrated by the duration of the refractory period, and by modifying the responses with vibration and active and passive movements. Stimulation over the L4–L5 vertebrae selectively depolarized posterior root fibers or additionally activated anterior root fibers within the cauda equina depending on stimulus intensity. Transcutaneous posterior root stimulation with single pulses allows neurophysiological studies of state‐ and task‐dependent modulations of monosynaptic reflexes at multiple segmental levels. Continuous transcutaneous posterior root stimulation represents a novel, non‐invasive, neuromodulative approach for individuals with different neurological disorders. Muscle Nerve, 2006

Analysis of the electrical excitation of CNS neurons

The artificial excitation process of neurons of the central nervous system depends on the applied extracellular field, on the geometry of the neuron and on the electrical properties of the neural subunits. Results of computer simulations are based on a compartment model of the neuron and its equivalent electrical network. Furthermore, a theory is presented which generalizes the activating function concept known from peripheral nerve stimulation. The theory predicts the influence of electrical and geometrical parameters on the excitation threshold. Generally, the myelinated axon is the part of a neuron which is most excitable to a given applied field. An example demonstrates that for a target neuron the quotient (anodic threshold current)/(cathodic threshold current) essentially depends on the position and orientation of the neuron relative to the electrode.

Modeling the excitation of fibers under surface electrodes

It was demonstrated previously by the author (see J. Theor. Biol., vol.125, p.339-49, 1987) that the electrostimulation of nerve or muscle fibers is caused by the activating function, which is the second differential quotient of the extracellular potential along with fiber. To find the activating function, a simple model is used to approximate the potential distribution produced by a surface electrode in a homogeneous medium. Data from classical space-clamped experiments allow calculation of fiber response as a function of geometry and stimulating signal shape.<<ETX>>

Stimulation of the Human Lumbar Spinal Cord With Implanted and Surface Electrodes: A Computer Simulation Study

Human lumbar spinal cord networks controlling stepping and standing can be activated through posterior root stimulation using implanted electrodes. A new stimulation method utilizing surface electrodes has been shown to excite lumbar posterior root fibers similarly as with implants, an unexpected finding considering the distance to these target neurons. In the present study we apply computer modeling to compare the depolarization of posterior root fibers by both stimulation techniques. We further examine the potential for additional direct activation of motoneurons within the anterior roots. Using an implant, action potentials are initiated in the posterior root fibers at their entry into the spinal cord or along the longitudinal portions of the fiber trajectories, depending on the cathode position. For transcutaneous stimulation low threshold sites of the same fibers are identified at their exits from the spinal canal in addition to their spinal cord entries. In these exit regions anterior root fibers can also be activated. The simulation results provide a biophysical explanation for the electrophysiological findings of lower limb muscle responses induced by posterior root stimulation. Efficient excitation of afferent spinal cord structures with a simple noninvasive method can become a promising modality in the rehabilitation of people with motor disorders.