Synchronize or Antisynchronize, a Hidden Decision in 4-Dimensions

Today want to talk about a very interesting paper called "The uncoupling limit of identical Hopf bifurcations with an application to perceptual bistability" by  Alberto Pérez-Cervera, Peter Ashwin, Gemma Huguet, Tere M. Seara, and James Rankin. In this paper they study double Hopf bifurcations, and the synchronization properties of two oscillators connected together. What they find is that depending on the parameters of the system, there can exist two stable synchronization states. The two oscillators can be in phase and fully synchronized, or they can be exactly anti-synchronized, where the two oscillators are perfectly offset from one another. Because, both states exist this means that the input into the system can push the the system to be either synchronized or anti-synchronized.

An example of synchronized (top) and anti-synchronized (bottom) neurons. Generated using the Izhikevich model.

So lets quickly describe what all that highly technical vocabulary means. A Hopf bifurcation is one of the bifurcation structures that describes a limit cycle. Recall, that a limit cycle is a periodic solution to an ordinary differential equation, and as such can be represented by polar coordinates, or complex coordinates. It can be described by the equations
$$\frac{dz}{dt} = (\lambda + i \omega)z - z|z|^2 $$
in complex coordinates,
$$\frac{dr}{dt}= \lambda r - r^3$$
$$\frac{d\phi}{dt}=\omega$$
in polar coordinates. Here the system has a limit cycle for $\lambda > 0$, and when $\lambda = 0$ the system undergoes a Hopf bifurcation. Then the equilibrium  point $r=0$ loses stability, and a limit cycle with $r=\sqrt{\lambda}$ becomes stable.  Now the interesting thing here is that the variable $\phi$ represents the phase of the oscillating limit cycle. Next they couple these two of these Hopf bifurcations together, but rather than using a simple linear coupling they use the highly nonlinear double Hopf bifurcation including nonlinear terms up to order $O(z^4)$.
$$\frac{dz_1}{dt} = (\lambda + i \omega)z_1 - a_{01} z_1|z_1|^2 + \epsilon[z_1(a_0+a_1|z_1|^2+a_2|z_2|^2+a_3\bar{z_2}z_1)+z_2(b_0+b_1||z_2|^2+b_2|z_1|^2+b_3\bar{z_1}z_1^2)] $$
$$\frac{dz_2}{dt} = (\lambda + i \omega)z_2 - a_{01} z_2|z_2|^2 + \epsilon[z_2(a_0+a_1|z_2|^2+a_2|z_1|^2+a_3\bar{z_1}z_2)+z_1(b_0+b_1||z_1|^2+b_2|z_2|^2+b_3\bar{z_2}z_1^2)] $$

Now that may be a lot of terms, but we can clean up the notation a bit, convert to polar coordinates and take all of those complicated nonlinear terms and represent them by the functions $f_r$ and $f_\phi$.

$$\frac{dr_1}{dt}= \lambda r_1 - a_{01R} r_1^3+ \epsilon f_r(r_1,r_2,\Delta\phi)$$
$$\frac{dr_2}{dt}= \lambda r_2 - a_{01R} r_2^3+ \epsilon f_r(r_1,r_2,\Delta\phi)$$
$$\frac{d\Delta\phi_1}{dt}=a_{01I}(r_2^2-r_1^2)+\epsilon f_{\Delta \phi}(r_1,r_2,\Delta\phi)$$
$$\frac{d\phi_1}{dt} = \omega + a_{01I} r_1^2 +\frac{\epsilon}{r_1} f_{ \phi}(r_1,r_2,\Delta\phi) $$

Now that still looks like a intimidating wall of math. However if you break it down term by term its not scary. There are two oscillators, with radii, $r_1$ and $r_2$. These two oscillators are coupled together in a highly nonlinear way, which is represented here as the functions $f_r$, and $f_\phi$. However, because the two oscillators, have the same frequency (as they do in this paper), then you will have two phases $\phi_1$ and $\phi_2$, and we can speak of the difference in phase $\Delta \phi= \phi_1 -\phi_2$. We can then describe how this phase difference evolves in time with the term $\frac{d \Delta \phi}{d t}$.

Example of stable and unstable radii during the double Hopf bifurcation. The limit cycles can either be unstable or stable. This depends on the parameters $\lambda$ and $\epsilon$. From figure 2 of the paper.

Now the question is how do we find these two synchronized and anti-synchronized states. We know at least have a way to tell if the systems are phase-locked. If $\frac{d \Delta \phi}{dt} = 0$ then we are at a equilibrium for the phase difference. As we will see there are two potential equilibria states $\Delta \phi = 0$ (synchronized) and $\Delta \phi = \pi$ (anti-synchronized). 

So how do we know there are only two potential equilibria? Well, the authors have a ingenious coordinate transformation be defining $s = r_1 + r_2$ and $d = r_1 -r_2$. Applying this transform to the above equations, and it gives us

$$\frac{ds}{dt}  s(\lambda + \frac{a_{01R}}{4}(s^2+3 d^2)) +\epsilon g_s (s,d,\Delta \phi)$$
$$\frac{d d}{dt} = d(\lambda + \frac{a_{01R}}{4}(3s^2+ d^2)) +\epsilon g_d (s,d,\Delta \phi)$$
$$\frac{d \Delta \phi}{dt} = -a_{01I} s d $$

here the $g$ functions just track the complicated nonlinear terms that we don't actually care about. The great thing about this coordinate transform is that $\Delta \phi$ no longer depends on its self and the system is now 2-dimensional. What We will notice here, is that if we assume $\epsilon = 0$ that the system undergoes a pitchfork bifurcation! This pitchfork continues to exist for $\epsilon>0$. As we can see in figure 5 from the paper, we see that in the $s$-direction that these equilibria are always attractive. Whats more, when the value these two equilibrium $s_{osc}^+$ and $s_{osc}^-$ are plugged into the expression of $\Delta \phi$ we get
$$s_{osc}^+ \rightarrow \Delta \phi = 0$$
$$s_{osc}^- \rightarrow \Delta \phi = \pi$$

Here is the hidden pitchfork bifurcation in the variable $s$. The left equilibrium corresponds to $s_{osc}^-$ limit cycle while the right equilibrium corresponds to $s_{osc}^+$ cycle. This figure is from figure 5 in the paper.

However the system is actually 3-dimensional reduction of a 4 dimensional equation. This  means that while in the s-direction they are stable, they are not required to be stable. In fact for $\epsilon =0$ They are saddle nodes and are unstable in both the $d$-direction and the $\Delta \phi$-direction. Thus the authors consider all of the various possible combinations of parameters, and analysis how the stability of the $d$-direction and the $\Delta \phi$-direction change. For a majority of the parameters, only the $s_{osc}^+$ equilibrium is stable. However, for some values of parameters $s_{osc}^-$  is also stable at the same time. For this special region of parameter space that means that the system is bistable, and both synchronized and anti-synchronized states exist! Thus the authors proved what they set out to prove. A double Hopf bifurcation can support both synchronized and anti synchronized states at the same time!

 

Here is an example of some of the different parameter sets from the paper. Top is $s_{osc}^+$, and it shows that it is mostly stable for a majority of parameters except for region A. Bottom is $s_{osc}^-$.  It can be stable or unstable. If both $s_{osc}^+$ and $s_{osc}^+$ are stable then the system is bistable, and the input determines whether it is synchronized or anti-synchronized. If both are in region $A$ then there is no synchronization. Furthermore in the region above $A$ (above line $C^+_{HB}$) there are not two limit cycles, so synchronization is impossible.  These are figures 9 and 10 in the paper. They are for the parameter set when $b_{\epsilon 0 R}>0$ and $C_{det}+b_{\epsilon 0 R}>0$.

That may seem like a lot of work for such a simple conclusion. Why is this important? Well they go on to show how this bifurcation can exist in 4 neuron networks. In fact a large number of these networks exist on a microcircuit level in many brains. This means that many neural microcircuits could be processing different stimuli based on we their or not the microcircuit is synchronized or anti-synchronized. More and more studies are realizing that computation in the brain is working through phase computation, and this study puts it on firmer ground.  


Author: Alexander White


Source: Alberto Pérez-Cervera, Peter Ashwin, Gemma Huguet, Tere M. Seara, and James Rankin. The uncoupling limit of identical Hopf bifurcations with an application to perceptual bistability. The Journal of Mathematical Neuroscience, 9(7), (2019)