[From Bill Powers (2004.09.16.1037 MDT)]

SteveO 09/16/2004 8:13 CST –

Hello, Steve. Always glad to get basic questions, especially about those

chapters that nobody reads!

I’m confused about the difference

between the adder (Figure 3.5, pg 30) and

the multiplier (Figure 3.6, pg 30). I believe the discussion is on

page

- Are we saying the receiving cells are different between the

two

functions?

Yes, I think the difference would have to be there. The critical

properties would be the cell-wall capacitance and the time constant of

the post-synaptic potential that builds up inside the cell as each jolt

of neurotransmitter arrives. I’m a bit hazy about the details here; it’s

been some years since I reviewed the facts. At any rate, it takes a

certain amount of charge to change the potential enough to cause an

output spike to be generated, and generating that spike reduces the

potential so it has to build up for a while to generate the next output

spike. The calcium and potassium pumps in the cell wall reset the

threshold potential after each discharge.

If the capacitance of the cell walls is very large, the potential doesn’t

change much for a single synaptic event. If the input is blip and the

output of the cell is BANG, the process goes

blip-blip-blip-blip-blip-blip-blip-blip-BANG-blip-blip-blip-blip-blip-blip-blip-blip-BANG-blip-blip-blip-blip-blip-blip-blip-blip-BANG.

The faster the input blips arrive, the faster the output bangs occur. In

this case I’ve said it takes 8 blips to produce one output pulse, so the

output frequency is one eighth of the input frequency.

Now suppose we have two synapses on the same high-capacitance cell, both

providing blips of neurotransmitter at the same rate. Obviously it will

now take only half as long for the postsynaptic potential to change

enough to create an output spike. Half the interval means twice the

frequency. With three inputs it would take 1/3 as long and the output

frequency would be three times as high as with only one input. So the

frequency of the output spikes is the input frequency times the number of

inputs receiving impulses at the same rate.

Now, what if the frequency of a single input varies? If twice as many

impulses arrive per second, clearly it will take only half as long

to reach the firing threshold, and the output frequency will double. So

the output frequency is proportional to the input frequency. That gives

us all we need to say that this sort of high-capacitance cell will be an

adder: output frequency is the sum of all input frequencies.

Now what about the multiplier or logic gate? This, of course, was the

initial model inspired by the digital computer, which treated nerve cells

as logical elements and assumed that a cell fired only when two input

spikes occurred at the same time. To implement this sort of cell, the

capacitance would have to be made very low, so that a single input event

could bring the postsynaptic potential a substantial fraction of the way

to the firing threshold, but not all the way. Also, to make a

time-coincidence of inputs necessary, it would be necessary that the

capacitor be rather leaky, so the potential from a single input spike

would die out rapidly. You can see that with a non-leaky capacitor, an

input impulse would change the potential and then the potential would

just remain, so that a second input event occurring at *any* later

time could cause enough additional change in potential to produce an

output spike. This would be a different kind of semi-logical element in

which coincidences are not necessary – it would be interesting to see

what could be built out of such elements, which are half digital and half

analog.

At any rate, if the capacitance is small and the leakage is significant,

a second input event would have to occur before the change in potential

produced by the first input had died away too much. That would be the

classical “coincidence detector” form of AND gate. Input A AND

input B would have to occur within some small time slice in order to

generate an output spike.

Finally we get to the multiplier, which grows out of the

coincidence-detector kind of gate. Suppose that the input events occur at

random with some uniform distribution. The probability that two events

will occur within a given time is the product of the probabilities that

either event will occur. Suppose that one of the events occurs in

10 percent of the time intervals, at random. Let the second event occur

at a higher average frequency, say in 50% of the samples. If both events

must occur for an output spike to be generated, then clearly the

probability of a coincidence per sample time will be (0.1)*(0.5) or 0.05.

The average output frequency will be the product of the two average input

frequencies.

In neural processes, even when we consider logic, more than just

occurrance and non-occurrance of input impulses matters. If the threshold

is set high, then it could take three simultaneous inputs to create an

output event, which is like a three-input AND gate. But if another

high-frequency input arrived in a way that contributed only a very small

change in potential per impulse, the frequency of this input could

effectively change the threshold, turning the gate into a four-way

AND gate if the new input were inhibitory, or a two-way

AND gate or even into a simple OR gate if excitatory. If the threshold

were made low enough, the AND gate would become an OR gate since any of

the multiple inputs would be sufficient to produce an output. Actually,

we’re talking about an “any N out of M” gate, where N depends

on the firing threshold and M is the number of independent inputs. We

don’t see this sort of variable-logic element in any of our artificial

devices. Such devices might prove to do interesting things once we worked

out the methods of analysis.

We’ve come a long way from the old idea that synapses were simply places

where neural impuses were “relayed” to the next neuron in a

chain.

Best,

Bill P.