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Relaxation Oscillators:
 
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The Fitzhugh-Nagumo equations were derived simultaneoulsly by Fitzhugh
and Nagumo et.al.
-
R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membranes,
1182-Biophys. J., 1 (1961), pp. 445--466. and
- J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating
1214-nerve axons, Proc. IRL, 50 (1960), pp. 2061--2070.
These equations provide a fairly detailed picture of the action potential
of a neuron.
Here is the system of first-order differential equations which describes
the Fitzhugh-Nagumo oscillator:
Like all relaxation oscillators, this oscillator has a slow
accrual phase and a fast release phase. Here, v is voltage
and w is recovery of voltage. Note that gamma is a shunting
variable, and that theta is a thresholding variable. An explanation
of shunting and thresholding is outside of the scope of Vibe.
A very important variable epsilon represents
the coupling between the slow and fast phases. As epsilon increases,
so does frequency. The variable omega
is constant voltage and is held constant in Vibe at .112. Some voltage is required
at all times to keep this oscillator going else the system becomes a point-attractor.
What is a neuron?
A neuron is a nerve cell, and is the primary building block of
our nervous systems.
Neurons have branches called axons which communicate with other neurons by
sending them pulse-like electrical signals. Thes signals are picked up by the
receiving neuron on small nodules called dendrites. When a neuron receives
enough electrical input from its neigbors it will fire, sending off
its own electrial pulse. Once fired, the neuron must recover before firing
again. The slow collection and quick release of voltage is called
integrate and fire behavior. The inability for the neuron to
collect voltage immediatly after firing is called depolarization.
This entire sequence (integrate, fire, rest) is called a neuron's action potential.
With enough voltage and the right settings, the Fitzhugh-Nagumo model creates
nice action potentials.
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