2. Neurons

We have seen that it is not necessary to have an explicit plan or representation of the situation in order to accomplish a task. This greatly reduces the complexity of the nervous system required. As the snail example showed, a set of ordered reflexes is adequate for simple navigation. Still, there is the question, "What are these reflexes made of?" In animals the answer is, of course, neurons. So let us take a look at how real neurons work.

Over long distances neurons communicate via electrical impulses. However, when these signals reach the inputs (called "dendrites") of another neuron, every input pulse causes a small puff of a specific chemical to be released into the gap (called a "synapse") between neurons. This substance briefly opens a number of ion channels in a nearby patch of the receiving neuron's cell membrane, and the resulting flow of charge carriers causes this area to act like a miniature battery. The same thing happens throughout the neuron's tree-like collection of inputs and the combined charges are funnelled back to the body of the cell (known as the "soma"). Thus, the neuron gradually becomes more and more electrically charged. When there is enough accumulated potential, the neuron spontaneously generates its own series of impulses which travel down the output fiber (the "axon"). Eventually this signal impinges on the inputs of succeeding neurons where a similar series of events then takes place.

Our model of a neuron takes into account all of these phenomena. As shown in this figure, there are a number of inputs which converge on a single summation node. The combined value is then slowly integrated over time and passed to a threshold unit which determines whether any output will be produced. Our model also incorporates some other salient features of real neurons not yet mentioned. For instance, all inputs are not treated the same. Each connection (the triangles on the left) has a particular "weight" associated with it. These weights multiply the value of the input signal, which is normally either one or zero, thereby allowing the input to have a stronger influence on the neuron. This reflects the anatomical fact that certain synapses in animals are physically larger or more transmissive than others. In addition, we also allow inputs with negative weights (bottom of the diagram). These correspond to the inhibitory connections often observed between actual neurons. You can think of a neuron as a tub being filled with water. There are a number of hoses of different sizes feeding into the tub as well as a number of drain spouts. If more water comes in that goes out, the tub will eventually fill up and slop water over its edge.

In both the real and modelled cases, combining the effects of all the inputs is a little more complicated than previously described. While we have neglected several interesting effects based on the actual geometry of the input structure, we do not overlook the fact that real neurons are "leaky". In a primitive model, the presence of a "one" at some excitatory input would cause the neuron to charge up to the threshold and then set its own output to "one". Now, suppose that this input goes to "zero". Since the integral of zero is zero, the potential of the modelled neuron would not change and the output would remain active. This is not the case for a real neuron; its charge eventually decays over time. In terms of the previous analogy, this corresponds to a hole in the bottom of the tub. To capture this effect, we have improved our model by adding an extra "-1" term to the summation. This is essentially an inhibitory input that is always on. When none of the other inputs are active, the sum is now negative so the integrated excitation always decreases toward zero.

However, we never let the value of the integral go negative. Although not strictly correct biologically, we restrict the integrated excitation to be between 0 and 1 at all times. This reflects the fact that we can not fill a tub above its rim or drain it below its bottom. While real neurons also have limits to the voltages that can be achieved, the resting state does not necessarily correspond to the maximum depolarization. In animals, it is possible, and sometimes computationally useful, to discharge a cell below this neutral level and thereby make it harder for later inputs to trigger the neuron. The final twist on our neural model is the nature of the threshold. We use a device known as a "Schmitt trigger" instead of a simple comparator. This device has two thresholds, a high one for rising signals and a lower one for falling signals. In our modelled neuron, the output switches on when the value of the integrated excitation reaches 1. However, the neuron will then continue to generate a signal until the integral descends to 0 again. This is much like the thermostat in a typical house. If you set the temperature to 70 degrees, the furnace will not turn on until it gets as cold as 68 degrees or so. Then, it will proceed to warm the house until the temperature passes the setpoint, often reaching 72 degrees before shutting off. Including this feature in our model is convenient, but, again, not totally accurate from a neurological point of view.

To see how such simulated neurons are used, consider the example shown. Here, we depict the body of the neuron as a circle with a line coming out of it to represent the output. Input terminals are shaped like the bell of a trumpet and have their associated weight written next to them. White terminals are excitatory whereas black ones are inhibitory. Since the structural details do not matter, we show all the inputs impinging directly on the cell body.

This neuron will only generate an output if input A is active and B is not. First, suppose neither A nor B is active. The input sum is 0*2 - 0*3 - 1 = -1 (the last term comes from the leaking property of the model). This negative result causes the accumulated value inside the neuron to decay until the output switches to zero. Now suppose A comes on. The new sum is 1*2 - 0*2 - 1 = +1. If we set the time constant of the neuron to be very short, the output will almost immediately switch to one. Finally, imagine that B comes on as well. The computed sum for this case is 1*2 - 1*2 - 1 = -1, which forces the neuron to turn off instead.

A more complicated example is the oscillator shown above. The central part of this configuration is similar to the AND-NOT gate. Here, the neuron itself supplies the inhibitory input while another neuron provides excitation. The asterisk inside this auxiliary neuron indicates that it is of a special type which is always on. Assume that the central neuron's internal potential starts off at zero and that its output is off. The input sum is 1*2 - 0*4 - 1 = +1 in this case. Thus, the integrated input value starts climbing slowly as shown in the plot at the top right. The value written inside the neuron tells how long it takes for a sum of zero to fully charge the neuron. As can be seen, it takes 3 seconds for the integrated value to reach the value one. For a higher sum, less time would have been required.

Once the integral has reached the prescribed threshold level, the output of the neuron comes on. This is shown in the lower right hand plot. However, this changes the overall sum sent to the integrator. It is now 1*2 - 1*4 - 1 = -3, so the neuron's internal value starts to decay. Yet, because of our special threshold stage, the neuron's output value remains at one until the integral again reaches 0. This happens one second after the output comes on: a sum of 3 means it takes only 1/3 of the specified time for the internal value to swing across its full range. At this point, the output turns off and the whole cycle repeats. As can be seen, the result is a pulse train with a period of 4 seconds. The frequency, as well as the relative width of the on and off portions, can be changed by adjusting the input weights.

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