Free C++ source code for the firing rate neuron model is available.
So far there has been a good deal of discussion about the properties and behaviors
of the different neuron types. However, unless there is a way to connect
these different neurons together into a network, then they are useless.
It is only through the complex, dynamic interactions of the entire
neural network that the biomechanical organism can generate survival behaviors. This section
attempts to explain the types of synapses that are used to connect the
model neurons into networks. There are three basic types of synapses,
but the first one will be looked at in two different ways. The first is
the regular synapse as it is used to excite other neurons. The other
aspect of the regular synapse is when it is used to inhibit other
neurons. The second major type of synapse is the gated synapse. And
finally there is the modulatory synapse.
The excitatory synapse is a regular synapse. It causes the post-synaptic neuron to be depolarized by injecting a positive current into
it. The strength of the synapse is given below the synapse line as
5 na. This means that if neuron 1 is firing at its peak value of 1 Hz then it will
be injecting 5 na of current into neuron 2. Lower values for the
firing frequency will inject proportionally less current. So if the
firing rate is 0.5 Hz then it will be injecting 2.5 na.
In figure 2 the first two axis of this graph show the currents for neuron 1 and then
the currents for neuron 2. On the voltage and firing frequency axis,
neuron 1's output is in red, and neuron 2's output is in green.
The graph shows that current is injected into neuron 1 causing it to fire.
This is transmitted through the axon and injects current into neuron B.
Neuron 2's firing frequency lags slightly behind the
firing frequency for 1. This is because it takes time to for current injected into
neuron 1 to begin having an effect. Once it does this is propogated to neuron 2. This is not
due to any synaptic transmission delay. Synaptic delays are not modeled in this system.
The inhibitory synapse is also a regular synapse. However, it causes other neurons
to be less likely to fire. It uses a filled circle on the target neuron to tell which direction the
current will flow. For the example above the negative current will go from neuron 3 to neuron 4.
This synapse works exactly like the excitatory synapse except it
injects negative currents instead of positive currents.
In figure 3 the first two axis of this graph show the currents for neuron 3 and then
the currents for neuron 4. On the voltage and firing frequency axis,
neuron 3's output is in red, and neuron 4's output is in green.
Neuron 4 is a tonic neuron that maintians a constant firing frequency.
Neuron 3 is then injected with a series of different currents that cause
it to fire. This is transmitted through the axon and negative current is injected into neuron 4.
The negative current pulls the membrane potential down and reduces the
firing frequency of neuron 4. The thing to notice here is that as the
firing frequency of 3 increases the firing rate of 4 decreases.
All the synapses in this module are derived from the Link object and share the properties of that item.
For a description of these properties please see the text that
discusses the Link properties.
The gated synapse works like a basic switch or transistor. A gated synapse is always associated with one or more other
synapses that are entering the target neuron. In the example above, the gated synapses are between neuron 6 and 7 and neuron 8 and 9.
They are symbolized using a triangle with an attached line.
The synapse that is actually gated is associated with the other synapse that goes from
neuron 5 to neuron 6 and 8. The gate can either be on by default or off by default. If it is on by default
then when neuron 9 fires it will block the current of neuron 5 from entering neuron 8. But if neuron 9 is
silent then the current from neuron 5 is free to enter neuron 8. If the gated synapse is off by default then
the above works in just the opposite manner. The first example uses a gated synapse that is off by default and the second example uses a gated synapses that is on by default.
In figure 4 The first two axis are for neurons 5,6, and 7. The second two axis are for neuron 5, 8, and 9.
The first axis in each pair shows relevant currents and the second axis shows firing frequencies.
A series of currents are applied to neuron 5 to stimulate it. The important thing to see in this graph is
what happens when neurons 7 and 9 are on. In axis 2 you should notice that neuron 6 does not fire until neuron
7 is firing strongly. When 7 is firing the gate between neurons 5 and 6 is open allowing synaptic current to flow.
When neuron 7 is not firing this gate is closed and synaptic current is blocked even though neuron 5 is firing.
The important thing to notice for the second portion of the graph is that when neuron 9 fires the synaptic current
from 5 to 8 dries up and neuron 8 quits firing. When 9 begins firing it closes its gate and stops the current from flowing.
Once it quits firing the gate is reopened and current can flow between 5 and 8 again.
If there were another neuron 20 that was firing and injecting
current into 6 it would cause 6 to fire regardless of whether neuron 7 was firing or not. The gated synapse
simply works like a water faucet to turn on and off some of the other synapses. The equation that
describes the behavior for the gated synapses is I = (U + Sign(IG)) * IS . Where
U is the un-gated state of the synapse, IG is the synaptic current of
the gating synapse, and IS is the current of the gated synapse. So in the synapse above
U is one. When neuron A is not firing then I = (1 + 0) * IC = IC, and the
current from neuron C is passed through to B. But since the weight of the gated synapse is
negative one, when neuron A is firing then I = (1 - 1) * IC = 0, and the
current from neuron C is blocked.
The modulatory synapse works more like a dimmer switch or an amplifier.
Like the gated synapse, it is always associated with one or more other
synapses that are entering the target neuron. In the example above, the
modulatory synapse is symbolized using a diamond. The equation for the modulated synapse is:
(1 + IM) * IS if IM > 0
IS / (1+ |IM|) if IM < 0
So if the modulating current is greater than zero then it amplifies
the synaptic current. And if the modulating current is less than zero then it
reduces the synaptic current. In the example network from above the
weight of the modulating synapse is -2 between neuron 13 and 14. If neuron 10 is firing at a rate of 1
Hz then IM = -2 for neuron 14. Since IM < 0, I = IC / (1
+ 2) = IC / 3. So the current going into neuron 13 from 10 is reduced by a third.
It is important to understand that the modulatory synapse is not
inhibiting neuron 13 and causing it not to fire. It is simply amplifying
or reducing the amount of current that is being injected from other neurons.
Real neurons in living organisms are capable of
a wide diversity of actions on other neurons. At their simplest, they
just depolarize or hyperpolarize other neurons. These types of interactions are
modeled nicely with a regular excitatory or inhibitory synapse connection.
However, real neurons are also capable of modulating other connections to the
neuron they are synapsed with. This is where the gated and modulatory synapse
types come into play. Their main purpose is to allow one neuron to manipulate
the connection between two other neurons. This is very important later on
because it allows the higher decision functions to shut off or modulate other
This project was supported by: