Free C++ source code for the firing rate neuron model is available.
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 described five properties for
pacemakers that have
been discovered through research. These properties are:
These findings were used to come
up with a set of rules that could produce that
behavior in a neuron using two intrinsic currents (Beer, 1990). 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
The pacemaker neuron has all of the properties associated with a regular neuron:
Cm, Gm, Vth, Fmin, and Gain. For a description of them please see the text that
discusses the normal neuron. The properties
that are unique for the pacemaker are listed below with a description of each.
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.
Figure 2 demonstrates all the properties of a pacemaker neuron that were listed above. We will go through each one.
First notice that the initial 0.5 na stimulation causes the pacemaker to fire at a slow freqency. When the
current is increased at 10 seconds the burst frequency of the neuron increases visibly. This demonstrates point 3 above.
The firing frequency is a function of the current entering the cell.
Lets look at the equations to try and understand how the interburst interval is calculated. We have
Gm = 100 nS and an input current of 0.5 na. This gives us a steady state voltage
of Vss = (0.5 na / 100 nS) = 5 mV. Plugging this
into equation 2 gives Tl = (-100 * 5 mV) + 2 s = 1.5 s.
Looking at the graph, two bursts occur within the first five seconds. Each burst lasts one second for a total of
two seconds in burst mode. This leaves a total of three seconds between bursts. Divide 3 in half and you end up with
1.5 seconds between bursts. Also, The calculated interburst interval is displayed on the graph so you can see how
it changes for different input current levels.
At 10 seconds we increase the input current to 1.5 na. This increases the steady state voltage to
Vss = (1.5 na / 100 nS) = 15 mV. This gives us an interburst interval of
Tl = (-100 * 15 mV) + 2 s = 0.5 s. You can see that the calculated interval drops to
0.5 at 10 seconds.
Bursting is reset in three places in figure 2. The first occurs when a hyper-polarizing
pulse is injected at 15 s. This pulls the membrane voltage below Vssm.
We still get an intrinsic current oscillation, but it is no longer large enough to
move the membrane voltage above its threshold and we do not get any firing while this
hyper-polarizing current is being injected. Once this current is removed a new firing pattern
with the same frequency emerges. The second place where the bursting is reset is when a
depolarizing current is injected at 28.5 ms. A burst just finished when this new
current was injected and this caused another burst to be fired immediately and reset the
pattern. The third time was after the bout of continuous firing. You can also see that
the interburst interval drops to zero during this period. 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.
The pacemaker neuron is the most complex neuron that is contained within this neural plug-in module.
It has a variety of complex behaviors and this model only represents these properties in a very abstract manner.
But this neuron, and all of it's features, play a key role in allowing
simulated organisms be able to move their limbs in a synchronized manner.
This project was supported by: