Necessary Attributes Of Small And Mediumscale Neural Models

An important question to ask when one is contemplating modeling the behavior of a small assembly of neurons is what level of detail to pursue. Common sense tells one to use as little detail as possible to still obtain an accurate emulation of known biological behavior. Because an action potential travels at constant velocity and has a constant shape [Vm(x, t)] as it propagates along an axon, it can be described in summary by a unit impulse delayed by a simple transport lag between the spike initiation event and the MEPP, epsp, or ipsp. One does not need to model the axon by a linked series of HH modules, or be concerned with sodium, potassium, and calcium ion currents.

Detail is needed, however, to accurately model the spike generation process and absolute and relative refractory periods. The origin of a spike generator potential

(SGP) should reflect the low-pass filtering propagation delays (if relevant) and summation inherent in the electrotonic conduction of epsps and ipsps over various dendritic branches to the soma and the SGL. It is also necessary to model the effect of axosomatic inhibitory chloride synapses if they exist. Their effect is to clamp the SGP to the chloride Nernst potential through a high conductance that effectively attenuates the pooled epsp and ipsp inputs from the dendrites. Thus, the axosomatic inhibitory synapse has a powerful action in preventing the SGP from reaching the firing threshold.

In an analog neuromime simulation of dendritic inputs, it is easy to emulate the epsp ballistic potentials, attenuate and low-pass filter them, then delay them (if appropriate) before summing to form the SGP. The generation of the epsp or ipsp, its attenuation, and low-pass filtering can all be combined and modeled with two or three nonlinear low-pass filters (see Figure 3.2-2).

There are several ways to model the spike generation process electronically:

1. The simplest model for spike generation is integral pulse frequency modulation (IPFM) (see Section 4.3.1 for details). In IPFM, the positive SGP is integrated. When the integrator output voltage reaches a threshold voltage with positive slope, an output impulse is produced. This output impulse causes the integrator to be reset to zero. The process repeats, and it is easy to show that the IPFM process is an ideal, linear voltage-to-frequency converter for constant positive inputs.

2. The relaxation pulse frequency modulator (RPFM) is very similar to the IPFM system except that, instead of an integrator with infinite "memory," the RPFM system inputs the SGP into a simple low-pass filter (LPF) with the transfer function, H(s) = a/(s + a). The output of the LPF then must exceed a firing threshold voltage, Vr with positive slope to initiate an output spike. As in the case of IPFM, the output spike resets the LPF output to zero. The RPFM system is not a linear voltage-to-frequency converter, and it has finite memory to transient SGP inputs.

The RPFM system is generally more realistic neurobiologically than is the IPFM spike generator (see Section 4.3.2 for details). Early neuromime circuits generally used IPFM or RPFM spike generation methods. IPFM was used in the neuromime of Figure 3.2-1.

Transport lags are a feature useful for realism in modeling any neural system in which the conduction time (8 = axon length /conduction velocity) is an appreciable fraction of the epsp or ipsp time constants. Thus, if a peripheral sensory neuron in the foot has a 1.5 m axon and a conduction velocity of 20 m/s, it will take 75 ms for information to reach the spinal cord. Thus, a transport lag of 75 ms should be put in the path between the sensory transduction process (including spike generation) model and a spinal reflex model. Many interneuron axons in the CNS are sufficiently short that their axonal delays are negligible compared with their dendritic integration times and psp time constants. Thus, in dense, compact neuropile such as the retina or olfactory system, axonal transport lags probably do not need to be included in a modeling scenario.

One way to realize transport lags with analog neuromimes is to use an acoustic delay line, or a spinning magnetic drum with write, read, and erase heads. (For example, if the drum is spinning at 300 rpm, this is equivalent to 1800°/s. If the read head is spaced 36° from the write head, then the output pulse will occur 20 ms after the input pulse.) The delay drum was used by the author with his early neuromime simulations.

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