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4.1.5 Pacemaker Neuron

Update Alert!

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 " Pacemaker firing rate neuron model"


1. Pacemaker Properties

Research into the details of locomotion and the rhythmic properties of some neural systems has shown that there is usually a subsystem that is able to produce rhythmic bursting to drive those behaviors. For example, the motion to move a leg would be controlled by one of these pacemaker units. Another good example would be in the heart, where the pacemaker would control the contraction of the heart muscle to pump blood. The pacemaker is usually a group of neurons that interact in a really complex way to produce this bursting property. However, this behavior can be modeled by a single neuron. Kandel (4.1.5.1) described five properties for pacemakers that have been discovered through research. These properties are:

  • When the pacemaker is sufficiently hyperpolarized, it is silent.
  • When the pacemaker is sufficiently depolarized, it fires continuously.
  • Between these two extremes, it rhythmically produces a series of relatively fixed-duration bursts, and the length of the interval between bursts is a continuous function of the injected current.
  • A transient depolarization which causes the cell to fire between bursts can reset the bursting rhythm.
  • A transient hyperpolarization which prematurely terminates a burst can also reset the bursting rhythm.
  • Dr. Randall Beer(4.1.5.2) then used these findings to come up with a set of rules that could produce that behavior in a neuron using two intrinsic currents. Ih is the high, depolarizing current and would tend to pull the membrane potential above threshold. Il is the low, hyperpolarizing current and would pull the membrane potential below threshold. The rules that use these two currents to produce the pacemaker behavior are:

  • Ih is triggered whenever the cell goes above threshold or Il terminates, and it then remains active for a fixed length of time.
  • Il is triggered whenever Ih terminates, and it then remains active for a variable amount of time whose duration is a function of the steady state membrane potential.
  • Il is triggered whenever the membrane potential goes below a given lower threshold, and stays on until the threshold is exceeded.
  • Only one of the intrinsic currents is active at any given time.
  • The pacemaker neuron has all of the properties associated with a regular neuron: Cn, Gn, Vth, Fmin, and Gain. For a description of them please see the text that discusses the regular neuron. The properties that are unique for the pacemaker are listed below with a description of each.

  • Il: This is the hyperpolarizing current that brings the membrane potential back down after it has been firing.
  • Ih: This is the depolarizing current that raises the membrane potential and causes the neuron to fire.
  • Vssm: This is a lower steady state threshold. If the steady state voltage of the neuron goes below this value then the Il current is locked on until that voltage rises above this threshold.
  • Mtl: This is the slope of the linear function that is used to calculate the length of time that Il current remains active.
  • Btl: This is the intercept of the linear function that is used to calculate the length of time that Il current remains active.
  • Th: This is the length of time that the Ih current remains active.
  • The steady state voltages mentioned above are calculated using Vss = (Iinput / Gn(1). Iinput is the input and external currents only. It does not include the intrinsic currents themselves. Also, The length of time that the Il current remains active is determined by this equation. Tl = (Mtl * Vss) + Btl  (2). So increasing Btl will increase the amount of time between pulses irregardless of the steady state voltage. Mtl determines in what way, and how much, the steady state voltage affects the length of time between pulses. What is desired is that by increasing the input current it will increase the pulse frequency. So if Vss is positive then Mtl should be negative in order to decrease the time that Il is active, and thus the length between pulses. Also, if Vss is negative then it means that the pacemaker neuron is being inhibited. If it is being inhibited strongly enough that Vss < Vssm then the pacemaker should be kept from firing at all. However, if enough depolarizing current is input into the pacemaker then the Il intrinsic current will no longer be strong enough to be able to pull the membrane voltage down enough to disable firing. This will mean that the neuron will fire continuously at that point until the input current is lessened.

    Pacemaker neuron Dialog
    Figure 1. This is a dialog for setting the properties for a pacemaker neuron.

    2. Neuron Output

    Pacemaker neuron Output Sample
    Figure 2. This is a graph of the output from a pacemaker neuron with the following neuron properties:
    Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz
    Gain = 100 Il =-15 na Ih = 15 na Vssm = -50 mV
    Mtl = -5 Btl = 200 ms Th = 100 ms  

    Figure 2 shows the output of a simple pacemaker cell that has no input current. It can be seen that the time between bursts is 200 ms and that the time of the burst is 100 ms. This matches with the values for Btl and Th that were used for this example. Also notice the way the intrinsic current oscillates between 10 na and -10 na. This is what is causing the rhythmic bursting that is characteristic of the pacemaker neuron.

    3. Increasing The Bursting Frequency

    Pacemaker neuron Output Sample. Faster Bursting
    Figure 3. This is a graph of the output from a pacemaker neuron that is being injected with 10 na of current. It has the following neuron properties:
    Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz
    Gain = 100 Il =-15 na Ih = 15 na Vssm = -50 mV
    Mtl = -5 Btl = 200 ms Th = 100 ms  

    Figure 3 shows the output for the same neuron as in figure 2, except it is being injected with 10 na of current. This causes the steady state voltage to be Vss = (10 na / 0.5 uS) = 20 mV. Plugging this into equation 2 gives Tl = (-5 * 20 mV) + 200 ms = 100 ms. And by looking at the oscillation of the intrinsic current, it can be seen that this is roughly what the output is doing. So by injecting 10 na of current into the cell it has increased the bursting frequency from 3 bursts / 1.1 seconds to 5 bursts / 1.1 seconds.

    4. Decreasing The Bursting Frequency

    Pacemaker neuron Output Sample. Slower Bursting
    Figure 4. This is a graph of the output from a pacemaker neuron that is being injected with -10 na of current. It has the following neuron properties:
    Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz
    Gain = 100 Il =-15 na Ih = 15 na Vssm = -50 mV
    Mtl = -5 Btl = 200 ms Th = 100 ms  

    Figure 4 shows the output for the same neuron as in figure 2, except it is being injected with -10 na of current. This causes the steady state voltage to be Vss = (-10 na / 0.5 uS) = -20 mV. Plugging this into equation 2 gives Tl = (-5 * -20 mV) + 200 ms = 300 ms. And by looking at the oscillation of the intrinsic current, it can be seen that this is roughly what the output is doing. So by injecting -10 na of current into the cell it has decreased the bursting frequency from 3 bursts / 1.1 seconds to 2 bursts / 1.1 seconds.

    5. Resetting The Bursting Frequency

    Pacemaker neuron Output Sample. Resetting Bursting High
    Figure 5. This is a graph of the output from a pacemaker neuron that is hit with a brief injection of 40 na of current for 20 ms. It has the following neuron properties:
    Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz
    Gain = 100 Il =-15 na Ih = 15 na Vssm = -50 mV
    Mtl = -5 Btl = 200 ms Th = 100 ms  

    Pacemaker neuron Output Sample. Resetting Bursting Low
    Figure 6. This is a graph of the output from a pacemaker neuron that is hit with a brief injection of -40 na of current for 20 ms. It has the following neuron properties:
    Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz
    Gain = 100 Il =-15 na Ih = 15 na Vssm = -50 mV
    Mtl = -5 Btl = 200 ms Th = 100 ms  

    Figure 5 shows what happens when a depolarizing current enters the neuron in between bursts. As the membrane voltage goes above the Vth threshold it causes the neuron to turn on the Ih intrinsic current and reset itself. Figure 6, on the other hand, shows what happens if a hyperpolarizing current is injected during a burst. It causes the membrane voltage to drop and turns on Il. This resets the pacemaker neuron. So when a pacemaker neuron is injected with a depolarizing current between bursts, it causes a new burst and resets the timing. But when a pacemaker is injected with a hyperpolarizing current during a pulse it terminates the pulse and resets the timing.

    6. Inhibiting Bursting

    Pacemaker neuron Output Sample. Inhibiting Bursting
    Figure 7. This is a graph of the output from a pacemaker neuron that is being injected with -20 na of current. It has the following neuron properties:
    Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz
    Gain = 100 Il =-15 na Ih = 15 na Vssm = -50 mV
    Mtl = -5 Btl = 200 ms Th = 100 ms  

    Figure 7 shows the output for the same neuron as in figure 2, except it is being injected with -20 na of current after the 400 ms mark. This pulls the membrane voltage down low enough so that the Ih current is not strong enough to bring the potential back above the Vth threshold value and cause the neuron to fire. So the bursting property of the pacemaker is shutdown and it does not fire when sufficiently hyperpolarized.

    7. Continuous Bursting

    Pacemaker neuron Output Sample. Continuous Bursting
    Figure 8. This is a graph of the output from a pacemaker neuron that is being injected with 20 na of current. It has the following neuron properties:
    Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz
    Gain = 100 Il =-15 na Ih = 15 na Vssm = -50 mV
    Mtl = -5 Btl = 200 ms Th = 100 ms  

    Figure 8 shows the output for the same neuron as in figure 2, except it is being injected with 20 na of current after the 400 ms mark. This causes the neuron to fire, which triggers the Ih intrinsic current to activate. That current stays on for Th seconds and then the intrinsic current is reset back to the Il value. Normally, an injected current sufficiently large to insure that it would continue firing even with Il active would be required. However, in this instance something else happened so that this was not necessary. The input current was large enough that the length of time that Il is active was zero. That is why the intrinsic current drops in spikes in figure 8. It is being reset and then immediately being set back to Ih in the next iteration. This was a fluke, but it was in here to demonstrate that these systems are pretty complex and sometimes even the people who build them are surprised by what they find it doing. Also, it would have been a real pain in the butt to go back and redo the parameters for all of the pacemaker graphs just because the numbers ended up odd in this instance.

    8. Pacemaker Overview

    The pacemaker neuron is the most complex neuron that is contained within the library of model neurons. It has a variety of complex behaviors that were quite difficult to try and model even after reading Dr. Beer's descriptions on how he was able to do so. However, this neuron, and all of it's main features play a key role in allowing the simulated insect be able to move its legs in a synchronized manner that allows it to walk, instead of simply flailing its legs about in opposition to one another.


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