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FIGURE 2.1-3 Plot of a typical cost function, as given by Equations 2.1-10 and 2.1-11, vs. the firing threshold. Note that there is a threshold value, yo, where C has a true minimum.

FIGURE 2.1-3 Plot of a typical cost function, as given by Equations 2.1-10 and 2.1-11, vs. the firing threshold. Note that there is a threshold value, yo, where C has a true minimum.

2.1.5 Simulation of a Model Receptor with a Continuously Variable Firing Threshold

Imagine a sensory receptor that must sense infrequent, random inputs of the quantity under measurement. The QUM events are considered to be a random, point process. Their times of arrival can be described by a Poisson random process. All the QUM events have the same amplitude. (For example, the events could be photons absorbed by a photoreceptor.) Ideally, the model receptor should fire 1:1 with the input events. Such output pulses are called true positives = TPs. However, because of random noise associated with the transmembrane (generator) potential, the receptor SGL

can produce output spikes in the absence of input (false positives = FPs) and may not fire when a true input is present (false negatives = FNs). The biological cost to the animal of FPs and FNs can be varied. If a sensory neuron output occurs, it may trigger an energy-intensive behavior, such as escape swimming. If no predator was actually present (an FP event), the animal is not eaten, but expends energy it must eventually replace. When an FN event occurs, the animal does not respond, and a predator may eat it, an extreme cost.

In another scenario, the receptor may detect single pheromone molecules in the air. Now FN events can lead to the animal sitting still and not finding a mate, hence the failure to reproduce. FP events can send the animal on a high-energy-cost, random search for the apparent pheromone source that may lead it to a mate by chance, or no mate. Thus, the operating cost, C, of the receptor depends on the sensory modality to be detected and its importance in the animal species' survival. Thus C = RFP + RFN - RTP may be a good starting point in evaluating neural receptor operation, but the cost function eventually used should reflect the animal's breeding success as governed by the receptor.

Figure 2.1-4 illustrates a model for spike generation in which the output spikes act to raise the firing threshold, <2, causing a reduction in the RFP and the rate of true positives (RTP). Raising <2 raises the RFN. To examine the behavior of this model, which involves nonlinear dynamics and nonstationary statistical behavior, the author has written a Simnon program, ADTHRESH.t. Two independent Gaussian noise sources are used. One drives an IPFM, voltage-to-frequency converter, the output of which is random impulses representing the events the sensory neuron is to detect. The impulses are passed through a single-time constant low-pass filter to generate exponential pulses, ein, representing generator potential transients. Also added to ein is bandwidth-limited Gaussian noise, Vmn. (Vmn + ein) are acted on by a second low-pass filter representing the membrane low-pass characteristic on the neuron cell body (soma). The output of this filter, v2, is the input to a simple threshold pulse generator, which generates a sensory neuron output pulse when v2 > <2, and dv2/dt > 0. This model for neural spike generation is called an RPFM (leaky integrator) neuron; it is described in detail in Section 4.3.2. The firing threshold for the RPFM neuron, <2, is given by

q2 is the output of a two-pole low-pass ("ballistic") filter whose input is the sensory neuron output spikes, y2. Thus, every time the RPFM neuron fires, either from sensory input or Vmn, the threshold is raised, making it less sensitive to noise and sensory input. What will be shown is that this manipulation of <2 reduces RFP at the expense of raising RFN. The Simnon program follows:

continuous system ADTHRESH " 7/02/99. System uses NFB to raise phi2 of

" RPFM sensory neuron to reduce false positives.

STATE v1 v2 r1 r2 p2 q2 Vmn in ein

DER dv1 dv2 dr1 dr2 dp2 dq2 dVmn din dein

TIME t

" Use Euler Integration w/ dt = tau.

RPFM Sensory Receptor

Generator potential yi

Random QLM events

Membrane noise

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