More About the Math:
Geometry of the Spike
The mechanics of neurons is certainly a complicated subject. The success
of the Hodgkin-Huxley (HH) model of an action potential is that it captures
the essence of generating a neural spike in a simple model of only four
nonlinear differential equations. But to be fair, the combined dynamics
of four variables is still a tough thing to imagine. Is there a simpler
model, with fewer differential equations, that can describe an action
potential? Can such a model be principally derived from the original
HH model? This was the question that mathematicians Richard Fitzhugh
and J. Nagumo/Arimoto/Yoshizawa independently asked in the early 1960s.
Their basic strategy in deriving reductions of the HH model starts with
making three observations about the solutions of the HH model.
1. There is a basic processes that makes the action potential rise,
or depolarize, from negative to positive potentials. This gives the upstroke
of the action potential. Let's call this process activation.
2. There is a basic process that makes the action potential fall, or
hyperpolarize, from positive to negative potentials. This gives the downstroke
of the action potential. Let's call this process recovery.
3. Activation happens much faster than recovery does. To be technical
there is a time- scale separation between activation and recovery.
Let's look at each separately.
Activation
Biophysically, the activation phase of an action potential is a positive
feedback interaction between the Na+ gate and the membrane potential V.
The more Na+ there is inside the cell, the larger V is. The
larger V is, the more open Na+ gates there are thus letting
more Na+ inside the cell. This interaction can explode V to
large values. The HH model has the two variables V and m to
create this dynamic; however, the simplest differential equation for
explosion is
\begin{eqnarray} \frac{dV}{dt}=V. \end{eqnarray}
Na+ channels will reverse their flow if the potential goes to high
(the cell can only hold so much Na+). This means V has to be
constrained to be between reasonable values. An easy way to do this is
to add to our explosion term:
\begin{eqnarray} \frac{dV}{dt}=V-\frac{V^3}{3}. \end{eqnarray}
This last equation models the explosion for 0 < V < \sqrt{3}
(\frac{dV}{dt} > 0 ), and outside of this region keeps V from
becoming to positive or negative, and does it all with only one differential
equation!
Recovery
In real neurons when V is large, like at the top of an action
potential, two things happen. First, the K+ gates start to open and let
K+ flow from the inside to the outside of the cell. Second, the Na+ gates
start to close, or inactivate, reducing the flow of Na+ within the cell.
These two process conspire to reduce the number of positive ions inside
the cell, making V recover, back down to hyperpolarized values.
The HH model has two variables, n and h, to account
for recovery. But, if the two variables are doing the same job, then
it is reasonable to try to use only one variable for the job. This is
exactly what the FN model does, modeling recovery with the single variable W:
\begin{eqnarray} \frac{dW}{dt}=V+0.7-0.8W. \end{eqnarray}
Here when V gets large, then W grows. But if W gets
to big then it turns itself off —- just like the HH recovery variables n and h.
Time-scale separation
Now it is time to merge the equations for activation and recovery and
make a spike. The equation for dW / dt already depends on V .
The recovery variable should make V recover back to low values,
to have this let us just subtract it from the right hand side of the dV
/ dt equation. This will give use the coupled system of differential
equations.
\begin{eqnarray} \epsilon\frac{dV}{dt}&=&V-\frac{V^3}{3}-W,
\end{eqnarray}
\begin{eqnarray} \frac{dW}{dt}&=&0.8(V+0.7-0.8W). \end{eqnarray}
You will notice the right hand side of the equation has the derivative
term multiplied by 0.08. This means that overall the W dynamics
are much slower than the V dynamics, modeling that activation
happens on a faster time scale than recovery.
Phase Space of the FN Model
Now we are ready to model a spike with two variables, V and W.
What is the advantage of having only two variables? Two variables are
not that much less than the HH model's four variables. The advantage
is that with two variables we can plot the whole system in the (V,W)
plane - called the phase space of the system. The HH model has a (V,m,h,n)
phase space, and visualizing in four dimensions is pretty rough. In short,
when working in phase space the drop from four to two variables makes
all the difference.
A solution to FN model, V (t) and W(t), evolves as t grows.
As we had hoped, the FN model gives robust spiking in V(t),
as shown in the right hand movies above. Let's study spiking in the (V,W)
phase space. To help organize the plots we also plot the (V,W)
points where either dV/dt = 0 or dW/dt = 0,
called nullclines. The V nullcline is simply the cubic
equation V - V 3=3 - W = 0, while
the W nullcline is the linear equation V + 0.7 - 0.8W
= 0; these are marked in blue in the movie on the left. We start the
evolution of (V,W) at a low point on the W nullcline,
in the movie this is where the green curve starts from. Right away we
see the green curve zips off to the right and then `hugs' the V nullcline.
The curve moves fast in the horizontal direction because V is
much faster then W (recall ε
<< 1), and it moves to the right since dV/dt > 0 below
the cubic nullcline. However, once V gets near the V nullcline
then dV/dt \approx 0 and V must
slow down. Along the V nullcline the motion is primarily upward,
since to the right of the linear nullcline dW/dt > 0.
These parts of the green trajectory are the activation of the spike.
Once the the green curve reaches the top of the right `knee' in the cubic
nullcline, it must zip off to the left since \frac{dV}{dt} < 0 above
the cubic nullcline. After the green trajectory reaches the left part
of the cubic nullcline it moves downward since \frac{dV}{dt} < 0.
This is the recovery part of the spike. Finally, the green curve hits
the bottom knee of the cubic nullcline and the trajectory again zips
off to the right to start a new spike. Putting this all together, a
spike produces a closed curve in (V,W) phase space. Thinking
of a spike as a geometric curve helps understand how different inputs
can cause spikes. Also, it links spiking in the
brain to other kinds of oscillations which also produce curves in their
respective phase space descriptions.
