There are two main ways to approach the dynamic modeling of small biological neural networks (BNNs). The first is at the so-called compartmental level; the second, the locus method, uses a more heuristic, phenomenological approach. Both approaches have advantages and disadvantages.
In the compartmental method, the membrane of each of the neurons to be modeled is subdivided into passive, local response, and active areas. The passive areas generally cover most dendrites and parts of the soma. Passive membrane is described by constant-parameter, linear RCG modules that resemble the elements approximating an RC transmission line when connected together. Each dendrite is subdivided into short cylinders of membrane of known length and diameter. Each cylinder is a "compartment" with a total shunting conductance in siemens and a capacitance and farads, derived from the per-unit area Cm and Gm. Each compartment is connected to its nearest neighbors by resistances representing the internal and external axial (longitudinal) resistance based on the compartment diameter and the resistivities of the axoplasm and the extracellular fluid. Refer back to Figure 1.2-1 for the compartmental circuit model of three dendrite sections. Dendrite taper is handled by changing the section diameters, hence the numerical values of cm, gm, ro, and ri. Branches in dendrites are handled by joining three (or more) ris and ros at common junction nodes (see Figure 9.0-1).
Modeling local response membrane and active (spike-generating) membrane in the compartmental context allows the modeler to add specific, voltage-dependent conductances (i.e., for Na+, K+, Ca++) in parallel with the constant general leakage conductance and cm for each compartment. This approach allows the Hodgkin-Hux-ley (HH) (1952) model for spike generation to be put in the model, and also local response transients and spike propagation on an axon to be emulated.
The action of chemical synapses in a compartmental model is generally done by using a comparator operator to detect the arrival of the propagated spike at the boutons or motor end plate at the end of the axon. This event is used to trigger a transient sodium conductance increase in the subsynaptic membrane (SSM, on an otherwise passive dendrite compartment). This conductance increase may be of the form: gNa(t) = GNamax a2 t exp(-at). This sodium conductance transient causes an epsp to be generated and propagate in the dendritic tree model.
The compartmental approach allows the insertion of a great deal of detail about voltage- and chemical-dependent conductances to be included in the model. Such detail may be useful if long-term, plastic changes in neuron behavior are of interest. However, in many cases this amount of detail is unwarranted when one is only interested in the input/output behavior of a BNN.
In the locus approach, a neuron is again subdivided into regions: synapses, passive dendrites and soma, spike generation, and axon. Instead of directly modeling neurotransmitter-induced conductance changes in the SSM, the locus approach directly models the epsp and ipsp at the SSM. Nerve axons are not modeled, but the spike propagation delay from the spike generator locus (SGL) to the synapse is simulated. The psp voltage transients at synapses are passed through high-order, low-pass filters to model their propagation from SSM to SGL on dendrites. All propagated psps are summed at the SGL.
In the locus approach, spike generation is modeled phenomenologically. No consideration is given to the HH formalism or voltage-gated conductances. Instead, the relaxation pulse frequency modulation (RPFM) model (also known by some as the leaky integrator spike generator) is generally used. In RPFM, the voltage at the SGL node is acted on by a simple RC low-pass filter (LPF). When the LPF output reaches a preset firing threshold voltage, V^, the RPFM spike generator puts out a unit impulse, and simultaneously resets its output voltage to zero. The locus approach is computationally simpler than the compartmental method. It is not necessary to simulate the dendritic tree, only its net effect in conditioning psps as they impinge on the SGL node. No conductances are required in a locus model; however, the HH spike generation process can be substituted for the RPFM SGL, if so desired.
The following sections examine the details of synaptic behavior, dendrite behavior, and RPFM spike generation in the locus context, and examine the behavior of simple BNNs simulated with locus model neurons.
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