Pre And Postsomatic Segment
A breakthrough in our understanding of the physics of neural signals, which propagate as membrane voltage along a nerve fiber (axon), was achieved by the ingenious work of Hodgkin and Huxley1 on the nonmyelinated squid axon. This work helped explain the action potential or "spike" that conveys the all-or-none response of neurons. To explore the complicated gating mechanism of the ion channels, the stimulating electrode was a long uninsulated wire, thus every part of the neural membrane had to react in the same way; that is, propagation of signals was prevented (Figure 3.1A). Refinements of their method as well as the application of patch clamp techniques have supplied us with models for different neural cell membranes. Reliable prediction of membrane voltage V as a function of time is possible for arbitrary stimulating currents IsHmulus with proper membrane models.
Figure 3.1 Stimulation without signal propagation (space clamp). A stimulating electrode in the form of an uninsulated wire inserted along an unmyelinated nerve fiber (A) or current injection in a spherical cell (B) causes the same inside voltage, V, for every part of the cell membrane. (C) In the first subthreshold phase, the voltage across the cell membrane follows the simple constant membrane conductance model (dashed line). Figure shows simulation of a 30-|im spherical cell sheltered by a membrane with squid axon ion channel distribution (i.e., by solving the Hodgkin-Huxley model with original data). Membrane voltage V is the difference between internal and external voltage: V = V - Ve; Ve = 0.
Figure 3.1 Stimulation without signal propagation (space clamp). A stimulating electrode in the form of an uninsulated wire inserted along an unmyelinated nerve fiber (A) or current injection in a spherical cell (B) causes the same inside voltage, V, for every part of the cell membrane. (C) In the first subthreshold phase, the voltage across the cell membrane follows the simple constant membrane conductance model (dashed line). Figure shows simulation of a 30-|im spherical cell sheltered by a membrane with squid axon ion channel distribution (i.e., by solving the Hodgkin-Huxley model with original data). Membrane voltage V is the difference between internal and external voltage: V = V - Ve; Ve = 0.
The main equation for internal stimulation of the soma, or any other compartment where current flow to other processes or neighbored compartments is prevented, always has the same form: One part of the stimulating current is used to load the capacity Cm (in Farads) of the cell membrane and the other part passes through the ion channels; that is, dV
stimulus m d, wn
The rate of membrane voltage change, dV/dt, follows as:
where the ion currents I°m are calculated from appropriate membrane models. Usually, the membrane models are formulated for 1 cm2 of cell membrane and the currents in Eq. (3.1) become current densities.
A positive stimulating current applied at the inside of an axon or at any other part of a neuron will cause an increase of V according to Eq. (3.1), if the membrane has been in the resting state (Istimuius = 0 and dV/ dt = 0) before. In order to generate a spike, this positive stimulus current has to be strong enough for the membrane voltage to reach a threshold voltage, which causes many of the voltage-sensitive sodium channels to open. By sodium current influx, the transmembrane potential increases to an action potential without the need of further stimulating support. This means that as soon as the solid line in Figure 3.1C is some few millivolts above the dashed line we can switch off the stimulus without seeing any remarkable change in the shape of the action potential.
In general the excitation process along neural structures is more complicated than under space clamp conditions as shown in Figure 3.1. Current influx across the cell membrane in one region influences the neighboring sections and causes effects such as spike propagation (Figure 3.2). Besides modeling the natural signaling, the analysis of compartment models helps to explain the influences of applied electric or magnetic fields on representative target neurons. Typically, such a model neuron consists of functional subunits with different electrical membrane characteristics: dendrite, cell body (soma), initial segment, myelinated nerve fiber (axon), and nonmyelinated terminal. Plenty of literature exists on stimulated fibers, but little has been written about external stimulation of complete neurons.
In 1976, McNeal2 presented a compartment model for a myelinated nerve fiber and its response to external point-source stimulation. He inspired many authors to expand his model for functional electrical stimulation of the peripheral nerve system, such as analysis of external fiber stimulation by the activating function,3,4 unidirectional propagation of action potentials,5 stimulation of a nerve fiber within a6,7 selective axon stimulation,8,9,10 and influence of fiber ending.11 Of specific interest is simulation of the threshold and place of spike initiation generated with stimulating electrodes (e.g., by the near field of a point source or dipole) by finite element calculation for a specific implanted device or when the farfield influence from surface electrodes is approximated by a constant field.12 Stronger electric stimuli or application of alternating currents generate new effects in neural tissue. All effects depend essentially on the electric properties of the neural cell membrane and can be studied with compartment modeling. The ion channel dynamics can be neglected during the first response of the resting cell, but the complicated nonlinear membrane conductance becomes dominant in the supra-threshold phase (Figure 3.1C). Consequently, the behavior can be analyzed with simple linear models or with more computational effort by systems of differential equations that describe the ion channel mechanisms in every compartment individually.
Modeling the sub-threshold neural membrane with constant conductances allows the analysis of the first phase of the excitation process by the activating function as a rough approach. Without inclusion of the complicated ion channel dynamics, the activating function concept explains the
Figure 3.2 Simulation of the natural excitation of a human cochlear neuron by stimulation with a 50-pA, 250-^sec current pulse injected at the peripheral end. This bipolar cell is a non-typical neuron: (1) the dendrite is myelinated and often called peripheral axon; (2) in contrast to animal cochlear neurons, the soma and the pre-and post-somatic regions are unmyelinated; (3) a single synaptic input from the auditory receptor cell (inner hair cell) generates a spike that propagates with a remarkable delay across the current consuming somatic region. Note the decay of the action potential in every internode that again is amplified in the next node of Ranvier. Simulation uses internode with constant membrane conductance, a "warm" Hodgkin-Huxley model (k = 12) in the active membranes of the soma, and a 10-fold ion channel density in the peripheral terminal (node 0), all nodes, and pre- and post-somatic compartments. The lines are vertically shifted according to their real location of the rectified neuron. For details, see Rattay et al.35
Figure 3.2 Simulation of the natural excitation of a human cochlear neuron by stimulation with a 50-pA, 250-^sec current pulse injected at the peripheral end. This bipolar cell is a non-typical neuron: (1) the dendrite is myelinated and often called peripheral axon; (2) in contrast to animal cochlear neurons, the soma and the pre-and post-somatic regions are unmyelinated; (3) a single synaptic input from the auditory receptor cell (inner hair cell) generates a spike that propagates with a remarkable delay across the current consuming somatic region. Note the decay of the action potential in every internode that again is amplified in the next node of Ranvier. Simulation uses internode with constant membrane conductance, a "warm" Hodgkin-Huxley model (k = 12) in the active membranes of the soma, and a 10-fold ion channel density in the peripheral terminal (node 0), all nodes, and pre- and post-somatic compartments. The lines are vertically shifted according to their real location of the rectified neuron. For details, see Rattay et al.35
basic mechanism of external stimulation, the essential differences between anodic and cathodic threshold values, its dependence on the geometric situation, the mechanism of one side firing, and the blocking of spike propagation by hyperpolarized regions, as well as several other phenomena.13,14 Beside nerve fiber analysis, the activating function, which represents the direct influence of the electric field in every compartment, is useful for magnetic stimulation15,16 and for direct stimulation of denervated muscle fibers17 or, in generalized form, cardiac tissue18,19 and arbitrary neurons.20,12 In previous work we have shown how the shape of a neuron affects the excitation characteristics and that several surprising phenomena may occur.
This is demonstrated, for example, by comparing the threshold current for neurons with small and large soma. Whereas thick axons are easier to stimulate than thin ones (known as the inverse recruitment order21), this relation does not hold for the size of the soma: a 100-|s pulse from a point electrode 430 ||m above the soma requires -2.3 mA to stimulate a 30-|m-diameter soma neuron with a 2-|im axon but only -1.1 mA for a 10-|im soma. Increasing the electrode distance to 1000 |m results in the same -2.2-mA threshold for both cases; that is, the second surprise is that for the large soma the neuron excitation threshold increases slightly when the electrode is moved within a specific range toward the soma.12 The explanation is that the axon, which is generally more excitable than the soma, loses more current to load the larger capacity of the large soma; this effect is more pronounced when the place of spike initiation within the axon is close to the soma.
Comparing a pyramidal cell with a cochlear neuron demonstrates the variety in architecture and signal processing principles in the dendrite and soma region. Some neuron types, such as the afferent bipolar cochlear neurons, are effective transmission lines, where nearly all of the spikes initiated at the synaptic contact with an inner hair cell arrive with some delay and small temporal variation (jitter) in the terminal region (Figure 3.2). In contrast to the cochlear neuron, a single synaptic input at the dendritic tree of a pyramidal cell produces only minimal change in membrane voltage at the soma and at the initial segment, and usually the collective effect of many synaptic activities is necessary to influence spike initiation significantly. Typically, the pyramidal cell response is dominated by the internal calcium concentration, which depends on two types of voltage-dependent calcium channels: (1) low-voltage-activated channels respond in the subthreshold range and may include generation of low-threshold spikes, and (2) high-voltage-activated channels in the dendrites respond, for example, to backpropagating sodium spikes.22-24
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