Understanding bifurcations: Saddle Node on a Invarient Circle (SNIC bifurcation)

 
Today I am going to describe a type of bifurcation called a saddle-node on a invariant circle, or SNIC bifurcation. This is very similar to the saddle node bifurcation that we have discussed previously in previous posts, here, here and here. However, this particular bifurcation happens on a periodic orbit called a limit cycle.

Figure 1a: a stable limit cycle.

Figure 1b: a unstable limit cycle.

Simply put, a limit cycle is any periodic attractor in a dynamical system. Just
like equilibrium points limit cycles can be stable, or unstable. Figure 1 shows an example of a stable limit cycle (solid line) and unstable limit cycle (dashed line). Any initial condition that starts off the limit cycle, will either exponentially approach or be repelled from the limit cycles depending on the stability of the limit cycle. For some more discussion of limit cycle stability in various contexts here, here, and here. I am not going to discuss to much about how to generate a limit-cycle mathematically, they can be done either using a Hopf-bifurcation, or relaxation oscillator. Regardless of the mechanism of generation, a SNIC occurs on the limit cycle.

So, a natural question arises, what happens if we put a equilibrium point on a limit cycle. Interestingly, it turns out that in order to do this, you need to have two equilibrium points, one stable and one unstable, and this is what is meant by a saddle-node bifurcation. Just like in regular saddle node bifurcation, equilibrium points are always generated (or annihilated) in pairs, where on stable equilibrium point and equilibrium point appear (or disappear) suddenly at the bifurcation point. Figure 2a shows the standard saddle-node bifurcation, and Figure 2b shows the SNIC bifurcation. Notice that, the SNIC bifurcation is literally, just the Saddle-Node, but constrained to lie on the limit cycle. 

 

Figure 2a: A saddle node bifurcation. The solid circle is the stable equilibrium, and the open circle is the unstable equilibrium.

Figure 2b: A saddle node on an invariant circle bifurcation. The solid circle is the stable equilibrium, and the open circle is the unstable equilibrium.

 

There is a nice way to combine the two bifurcations into a common framework by considering the equivalence between two mathematical models of neurons, namely the quadratic integrate and fire neuron (QIF) and the theta neuron. It turns out that the QIF has the same equation as the normal form of a saddle node bifurcation, plus a reset, and that the theta neuron undergoes a SNIC bifurcation.

Recall that the normal form of a saddle node bifurcation is
$$
\frac{dV}{dt} = V^2 + I
$$
where $V$ is our membrane potential and $I$ is some constant injected current. It is not hard to see that the system has two equilibrium points when $I<0$ Corresponding to $V = \pm \sqrt{I}$, where $V = - \sqrt{I}$ is the stable equilibrium, and $V = +\sqrt{I} $ is the unstable equilibrium. Interestingly any solution with initial condition $V_0> +\sqrt{I} $ will explode to infinity. Likewise, there is a saddle node bifurcation whenever $I = 0$ and whenever $I >0$ there are no equilibrium points

The QIF considers this a spike and adds a reset condition
$$
\frac{dV}{dt} = V^2 + I
$$
$$
\text{ if } V = +\infty \text{ then } V \rightarrow V_{\text{reset}}
$$
Thus our system can fire a continuous spike train if $I>0$, and can be bistable if $V_{reset} > +\sqrt{I}$. Interestingly, if $V_{reset} < 0$ we have a monostable system that can be converted into the theta neuron. Notice that in this case, the system appears to somehow wrap around infinity, and is periodic (see figure 3). This is somewhat akin to a limit cycle if we can somehow project this onto a circle via a coordinate transform. 

Figure 3: A QIF neuron with a reset. The Red line will be the invariant circle after the coordinate transform.

 

Lets motivate this a bit, notice that if we solve $\frac{dV}{dt} = V^2 + I $ explicitly, we can show that
$$
V = \sqrt{I} \tan{( C +t \sqrt{I })}
$$
 
Also recall that if $I<0$  we have complex values that gives us
$$\sqrt{-I} \tanh{ (C +t \sqrt{-I })}$$
or
$$\sqrt{-I} \coth{( C +t \sqrt{-I })}$$
depending on the initial condition.
Given these solutions are related to $\tan$ lets guess the coordinate transform
$$V = \tan{\frac{\theta} {2}} $$.
Here $\theta$ will be the phase coordinate of our limit cylce. The reason for the $\frac{\theta}{2}$ will become apparent later on. Doing a bit of calculus and rearranging we can get
$$
\frac{d\tan{\frac{\theta} {2}}}{dt} = \tan^2{\frac{\theta} {2}}+ I
$$
$$
 \frac{1}{2} \times \frac{d\theta}{dt}\times\sec^2{\frac{\theta }{2}} = \tan^2{\frac{\theta} {2}}+ I
$$
$$
 \frac{1}{2}\frac{d\theta}{dt} =\frac{\tan^2{\frac{\theta }{2}}}{ \sec^2{\frac{\theta} {2}}}+ \frac{I}{\sec^2{\frac{\theta }{2}}}
$$
$$
 \frac{1}{2}\frac{d\theta}{dt} =\sin^2{\frac{\theta }{2}} + I \cos^2{\frac{\theta}{2}}
$$

Then we can take advatage of the $\frac{\theta}{2}$ and use the half-angle formulas to get  
$$
 \frac{1}{2}\frac{d\theta}{dt} =\frac{1-cos\theta}{2} + I\frac{1+cos\theta}{2}
$$

$$
\frac{d\theta}{dt} =1-cos\theta + I(1+cos\theta)
$$

The above equation is known as the theta neuron (or Ermentrout–Kopell canonical model). What is great about this neuron model is that it gives a nice easy mathematical equation that undergoes the SNIC bifurcation at $I = 0$ (figure 2b).

Interestingly one can transform the quadratic integrate and fire model into the theta neuron if $V_{reset} < +\sqrt{I}$, if $V_{reset} < - \sqrt{I}$ then the system is actually equivalent to a homoclinic bifurcation. Perhaps I will discuss this bifurcation at a later date.

Author: Alexander White