As shown in Section 1.1, there are many different morphologies for neurons, depending on their location and function. However, neurosecretory cells, spiking interneu-rons with chemical synapses, and motor neurons generally follow a common plan:
1. They have a cell body or soma that contains the nucleus.
2. There is an axon that propagates nerve spikes regeneratively to the next site in the communications chain.
4. Also attached to the soma are branching, treelike structures called den-drites. The tubular cross section of dendrites becomes smaller as the number of branchings increases and the farther the branch is from the soma. Dendrites can provide an extensive, diverse contact surface for the synapses of other neurons. Some neurons have small "dendritic fields," while others, especially in the CNS, have huge dendritic trees, implying that there are many input synapses from a number of neurons. Some CNS interneurons have dendrites and axon terminal branches that are symmetrical with respect to the soma, and it is easy under the light microscope to confuse dendrites with terminal branches (see Kandel et al., 1991, Figure 50-6].
5. At the end of the axons of interneurons are terminal branches ending in synaptic boutons that make contact with the cell membrane of the soma of the next neuron, its axon hillock, or its dendrites. Or, if the neuron is a motoneuron, at the end if its axon motor end plates make intimate contact with muscle membrane.
The properties of the neuron membrane vary from passive (ionic conductances in the membrane are not voltage sensitive until an unrealistic 30 mV of depolarization is reached) to active, where sufficient depolarization (e.g., 10 mV) initiates a nerve action potential that can propagate over the active membrane. Dendrite membrane is generally thought to be passive. Axon hillock and axon membrane is active, and can propagate spikes. The cell body and terminal branches of the axon may be partially active; that is, there is a local response region where there is partial regenerative depolarization due to a low density of voltage-sensitive Na+ channels that reach a threshold depolarization voltage caused by summed epsps. This partial regeneration causes the transient membrane depolarization voltage to spread rapidly over the local response membrane and not be attenuated as the electrical activity spreads; a spike is not generated, however. In the purely passive membrane case, a local psp generated under a synapse is attenuated and delayed as it propagates down a dendritic branch, and over the soma.
The sections below consider some mathematical models that can be used to describe psp propagation in a dendritic tree toward the SGL. First considered are the electrical properties of a uniform tube of passive membrane. This tube is in effect, an electrical transmission line without inductance. It is, however, a linear circuit, so superposition applies, and Fourier and Laplace transforms can be used to describe its behavior. In general, the mathematical models for transmission lines are very complex, and detailed analysis that includes their taper and branching properties is left for computer simulation. The basics of dendrite behavior are examined below.
If one assumes that a tube of constant-diameter passive membrane is immersed in a conductive saline solution, the bulk properties of the tube compose what is called a core-conductor, illustrated in Figure 4.2-1. The specific parallel transmembrane ionic conductances (for Na+, K+, Cl-, Ca++, Mg++, etc.) are assumed to be constant and are summed to form a net gm, in S/cm tube length. Similarly, the lipid bilayers in the membrane form a transmembrane capacitance, cm, in F/cm tube length. (Passive nerve membrane has a capacitance of about 1 |F/ cm2, which must be changed to cm in F/cm by using the tube diameter. See Section 1.2.1) There is an external spreading resistance, ro ohm/cm. ro is determined by the resistivity of the external medium and the tube diameter. The homogeneous gel that fills the dendrite tube (the axoplasm) also has a conductivity pi ohm cm that determines an internal resistance, ri ohm/cm. The entire dendrite can be modeled by linking a large number of incremental RCR sections in series, each of length Ax, and writing a set of partial differential equations that describes the electrical behavior of the core-conductor.
The next step is to write voltage and current relations for the model based on Kirchoff current and voltage laws. Then, let Ax ^ dx, and differential equations are obtained that permit general solutions. Note that each section of the discrete model has a transmembrane depolarization voltage, vm, a transmembrane current, im (x, t), through cm in parallel with gm, and internal and external axial (longitudinal) currents, iL(x, t). (Although these derivations were introduced in Section 1.2.2, they are repeated here for clarity.)
The first relation formed on Kirchoff voltage law (KVL):
vm (x + Ax) = vm (x) - i(x)[ro + ri ]Ax 4.2-1A i v (x + Ax) - v (x) , . r T
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