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I have now finished work on a much more advanced version of the insect simulator named AnimatLab. AnimatLab is a software tool that combines biomechanical simulation and biologically realistic neural networks. You can build the body of an animal, robot, or other machine and place it in a virtual world where the physics of its interaction with the environment are accurate and realistic. You can then design a nervous system that controls the behavior of the body in the environment. The software currently has support for simple firing rate neuron models and leaky integrate and fire spiking neural models. In addition, there a number of different synapse model types that can be used to connect the various neural models to produce your nervous system. On the biomechanics side there is support for a variety of different rigid body types, including custom meshes that can be made to match skeletal structures exactly. The biomechanics system also has hill-based muscle and muscle spindle models. These muscle models allow the nervous system to produce movements around joints. In addition, there are also motorized joints for those interested in controlling robots or other biomimetic machines. This allows the user to generate incredibly complicated artificial lifeforms that are based on real biological systems. Best of all AnimatLab is completely free and it includes free C++ source code!

The page that corresponds to this one on the AnimatLab site is " Synapse models for the firing rate neurons in AnimatLab"

4.1.9 Synapses

1. Introduction

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 network that the 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.


2. Excitatory Synapse

Excitatory Synapse Analysis
Figure 1. This is a graph of the output from two neurons with externally applied input currents of -4 na, 2 na, 4 na, 6 na, and 8 na to neuron A in 100 ms intervals with the following neuron properties:
Neuron A
Cn = 10 nf Gn = 0.5 uS
Vth = 0 mv Fmin = 0 Hz
Gain = 50  
Neuron B
Cn = 10 nf Gn = 0.5 uS
Vth = 0 mv Fmin = 0 Hz
Gain = 50  

The excitatory synapse is a regular synapse. It causes other neurons to be more likely to fire and it uses an arrow on the end to show the direction that the action potential travels. So for the example above, the action potential would travel from neuron A to neuron B. The strength of the synapse is given below the synapse line as 10 na. This means that if neuron A is firing at its peak value of one then it will be injecting 10 na of current into neuron B. 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 5 na. In graph 1 the first two axis of this graph show the currents for neuron A and then the currents for neuron B. On the voltage and firing frequency axis, neuron A's output is in red, and neuron B's output is in blue. The graph shows that current is injected into neuron A causing it to fire. This is transmitted through the axon and injects current into neuron B. It can be seen that neuron B's firing frequency lags slightly behind the firing frequency for A. This is because of the time it takes for what is happening in neuron A to be transmitted to neuron B and begin affecting it. Also, in this graph both neurons reach the same steady state values. This only happens because both neurons have the same property settings and because the strength of the axon is 10 na. So when neuron A is firing at 0.80 Hz it is injecting 8 na into neuron B, which is exactly the same amount of currently being externally injected into neuron A.


3. Inhibitory Synapse

Inhibitory Synapse Analysis
Figure 2. This is a graph of the output from two neurons with externally applied input currents of -4 na, 2 na, 4 na, 6 na, and 8 na to neuron A and 10 na into neuron B in 100 ms intervals with the following neuron properties:
Neuron A
Cn = 10 nf Gn = 0.5 uS
Vth = 0 mv Fmin = 0 Hz
Gain = 50  
Neuron B
Cn = 10 nf Gn = 0.5 uS
Vth = 0 mv Fmin = 0 Hz
Gain = 50  

The inhibitory synapse is also a regular synapse. However, it causes other neurons to be less likely to fire. It uses a circle on the target neuron to tell which direction the action potential will flow. So for the example above the negative current will go from neuron A to neuron B. This synapse works exactly like the excitatory synapse except it injects negative currents instead of positive currents. In graph 2 the first two axis of this graph show the currents for neuron A and then the currents for neuron B. On the voltage and firing frequency axis, neuron A's output is in red, and neuron B's output is in blue. The graph shows that current is injected into both neurons A and B. Neuron B is injected with a constant 10 na current to make it fire continuously. Neuron A is then injected with a series of different currents that cause it to fire. This is transmitted through the axon and injects negative current into neuron B. The negative current pulls down the membrane potential and reduces the firing frequency of neuron B. The thing to notice here is that as the firing frequency of A increases the firing rate of B decreases.


4. Gated Synapse

Gated Synapse Analysis
Figure 3. This is a graph of the output from three neurons with externally applied input currents of -4 na, 2 na, 4 na, 6 na, and 8 na to neuron C in 100 ms intervals and 10 na into neuron A with the following neuron properties:
Neuron A
Cn = 10 nf Gn = 0.5 uS
Vth = 0 mv Fmin = 0 Hz
Gain = 50  
Neuron B
Cn = 10 nf Gn = 0.5 uS
Vth = 0 mv Fmin = 0 Hz
Gain = 50  
Neuron C
Cn = 10 nf Gn = 0.5 uS
Vth = 0 mv Fmin = 0 Hz
Gain = 50  

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 synapse uses a diamond on the target neuron. The gated synapse is associated with the other synapse that goes from neuron C to neuron B. It can either be on by default or off by default. If it is on by default then when neuron A fires it will block the current of neuron C from entering neuron B. But if neuron A is silent then the current from neuron C is free to enter neuron B. If the gated synapse is off by default then the above works in just the opposite manner. The example above uses a gated synapse that is on by default. In graph 3 The first three axis of the graph are the currents for neurons A, B, and C. On the voltage and firing frequency axis, neuron A's output is in red, neuron B's output is in green and neuron C's output is in blue. It shows that a series of currents are injected into neuron C causing it to fire. At around 550 ms a 10 na current in injected into neuron A that causes it to fire. This causes the gated synapse to block the current of neuron C from entering neuron B. Neuron C is still firing just as before, and any synapses to other neurons will work just fine. Also, it is important to understand that the gated synapse is not inhibiting neuron B and causing it not to fire. If there were another neuron D that was firing and injecting current into B it would cause B to fire regardless of whether neuron A 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.


5. Modulatory Synapse

Modulatory Synapse Analysis
Figure 4. This is a graph of the output from three neurons with externally applied input currents of -4 na, 2 na, 4 na, 6 na, and 8 na to neuron C in 100 ms intervals and 10 na into neuron A with the following neuron properties:
Neuron A
Cn = 10 nf Gn = 0.5 uS
Vth = 0 mv Fmin = 0 Hz
Gain = 50  
Neuron B
Cn = 10 nf Gn = 0.5 uS
Vth = 0 mv Fmin = 0 Hz
Gain = 50  
Neuron C
Cn = 10 nf Gn = 0.5 uS
Vth = 0 mv Fmin = 0 Hz
Gain = 50  

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 uses a filled fork symbol on the target neuron. Its synapse is associated with the other synapse that goes from neuron C to neuron B. 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 -1.5. If neuron A is firing at a rate of 1 Hz then IM = -1.5. Since IM < 0, I = IC / (1 + 1.5) = IC / 2.5. When neuron C is being injected with 8 na it is firing at a rate of 0.797 Hz. So neuron C attempts to inject IC = (10 na * 0.797) = 7.97 na. This then gives IC = 7.97 / 2.5 = 3.188 na. Again, it is important to understand that the modulatory synapse is not inhibiting neuron B and causing it not to fire. It is simply amplifying or reducing the amount of current that is being injected from other neurons.

6. Synapse Overview

Real neurons in living organisms are capable of taking 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 subsystems.


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