Introduction

Why is it important to simulate the electrical and molecular behavior of individual neurons and small neural networks? There are several answers to this question: (1) To prove that how they work is truly understood. (2) To predict neural behavior not yet seen in nature by altering ionic conductances in membranes with models of drug action. (3) To verify connectivity structures in small biological neural networks that exhibit unique firing behavior (e.g., two-phase bursting). (4) To model CNS functions (on a greatly reduced scale) such as the motor control of eye movements, the detection of objects by electric fish, or a basic learning behavior. Biological neural networks (BNNs), such as in the retina or cochlear nucleus, are too complex and nonlinear to permit their neurophysiological behavior to be predicted from anatomy alone. Neuroanatomists have the tools to identify the neurotransmitters in synapses, and enough neuropharmacology is known to identify whether a given synapse is excitatory or inhibitory, fast or slow, and what ions are gated, etc. Such details can be inserted in the detailed, conductance-based dynamic models to approach verisimilitude.

In the past 10 years, as personal computers and desktop workstations have become ever more powerful, there has been a proliferation of specialized software applications designed to simulate the electrical behavior of individual neurons and the information-processing properties of assemblies of biological neurons. As will be seen, some of these programs deal best with the molecular and ionic events in and around the cell membranes of individual neurons and their dendrites, while other programs appear to be better used to investigate the properties of BNNs. There is a trend to sacrifice molecular and ionic details of neuron function as the number of neurons in the BNN increases. For example, instead of relying on the Hodgin-Huxley (HH) model or one of its variations for spike generation, the spike generator locus (SGL) can be modeled by the RPFM (leaky integrator) spike generator (see Section 4.3.2). A more primitive SGL can use the IPFM (integrate-fire-reset) voltage-to-frequency converter (see Section 4.3.1).

Most of the neural modeling (NM) applications described below are freeware, downloadable from Web sites. Also, most of them have been designed to run on workstations having UNIX operating systems (DECOSF, Ultrix, AIX, SunOS, HPux, etc.) using Xwindows, or on systems using LINUX. Some NM applications have been modified to run on personal computers using Microsoft Windows™ or Windows NT™.

There are many Web sites from which the interested reader can download descriptions of NM software and, in most cases, the software itself. The reader should be warned that information on the World-Wide Web is ephemeral; Web sites can move and also can close down. Information on the Web does not have the same permanence as books and journals on library shelves, and CDs.

A summary listing of Computational Neuroscience Software can be found at wysiwyg://2/http://home.earthlink.net/~perlewitz/sftwr.html. Under the heading, Compartmental Modeling, this site lists and has hot links to the following NM programs: CONICAL, EONS, GENESIS, NEURON, NeuronC, NODUS, NSL, SNNAP, the Surf-Hippo Neuron Simulation System, and XPP. (Note that "compart-mental," used in an NM context means a closed membrane volume over which the same electrical potential exists, which is quite different from "compartment" as used in pharmacokinetics.) Realistic Network Modeling includes BIOSIM, SONN, and XNBC. The GENESIS and NSL programs are also supported by detailed textbooks: Bower and Beeman (1998) wrote The Book of GENESIS, and Weitzenfeld et al. (1999) wrote The Neural Simulation Language (NSL). As will be seen below, some programs such as NSL and NEURON can easily model networks, and probably should have been listed under both categories above.

Figure 9.1 illustrates the evolution of the compartmental model of a typical neuron having dendrites, a soma, and axon. One chemical synapse is shown at the tip of the top dendrite. The first step in the compartmental modeling approach is to subdivide the features of the neuron into cylindrical "compartments," each of which has an area of membrane, Ak = dknLk cm2; dk is the diameter of the kth cylinder, and Lk is its length. The modeler must choose Lk small enough to give an accurate lumped-parameter model, and large enough to keep a reasonable number of ODEs and auxiliary equations.

Each compartment is characterized by a total transmembrane capacitance, cmk, in farads, and one or more specific ionic conductance parameters, gjk; some may be fixed, or functions of the membrane voltage of that compartment, Vmk. Other conductances can depend on calcium ion concentration, or the local concentration of neurotransmitter. Dendrite tips are modeled by cones of area Adk = dknLk/2. Most simulation programs using the compartmental modeling architecture allow the user to specify input currents to individual compartments, and also to place certain compartments under voltage clamp conditions. Note that every compartment is joined to its neighbors by longitudinal spreading resistances based on local axoplasm resistivity and the length of the compartment(s) involved.

One method to simulate the action of chemical synapses is to treat the arrival of the presynaptic spike as a delta function, U_1(t), which acts as the input to a pair of concatenated, first-order ODEs, the output of which is the postsynaptic ion conductivity function in time that generates the epsp or ipsp at the site of the synapse. For the so-called alpha function governing Na+ conductance, the defining ODEs are x j = -ax1 + U_j(t) a = 1/Tj 9.1

Neuron

Neuron

FIGURE 9.0-1 (A) A bipolar neuron with axon, soma, and dendrites. Thin lines denote compartment boundaries. One synapse is shown. (B) The neuron of A is modeled with linked compartments made up from cylinders of nerve membrane. The area of the kth cylinder is Ak = n DkLk cm2. (D, diameter; L, length.) (C) Each compartment is characterized by (1) a transmembrane capacitance, (2) a transmembrane conductance that can be fixed (in the case of passive membrane on dendrites and soma), or voltage dependent (gK and gNa in the axon), or chemically dependent, as in the compartments receiving synapses. Connecting adjoining compartments are the two ri/2 of the adjoining compartments (ri is the axonal longitudinal resistance.)

FIGURE 9.0-1 (A) A bipolar neuron with axon, soma, and dendrites. Thin lines denote compartment boundaries. One synapse is shown. (B) The neuron of A is modeled with linked compartments made up from cylinders of nerve membrane. The area of the kth cylinder is Ak = n DkLk cm2. (D, diameter; L, length.) (C) Each compartment is characterized by (1) a transmembrane capacitance, (2) a transmembrane conductance that can be fixed (in the case of passive membrane on dendrites and soma), or voltage dependent (gK and gNa in the axon), or chemically dependent, as in the compartments receiving synapses. Connecting adjoining compartments are the two ri/2 of the adjoining compartments (ri is the axonal longitudinal resistance.)

For added flexibility, one sometimes uses a two-time-constant model for postsynaptic conductivity increase.

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