A model close to my Heart
今天我們要討論一個在數學神經科學領域很著名的模型，菲茨休-
$$\epsilon \frac{dV}{dt}=V-\frac{V^3}{3}-W+I$$
$$\frac{dW}{dt} = a + V - b W$$
$$\frac{dW}{dt} = a + V - b W$$
一年之後南雲博士等人發表了一個與此數學模型相同微分方程的電路
然而簡化的模型仍可能非常複雜，
首先我們來定義他們的模型。
$$\epsilon\frac{dV}{dt}=\
$$\frac{dW}{dt} = a + V - b W$$
在我重製的GIF圖中我使用傳統的立方零斜率線$F[V]=V-
下一步是將FHN心臟投射到一個奇異擾動的FHN（
卷繞數定義為初始條件在$t=0$時圍繞黑點（或$t=\
當$\epsilon\neq0$時意味著什麼？他們猜測（
撰稿：Alex White
翻譯：高暐哲
Source:
source : Eric N. Cytrynbaum and Kelly M. Paton (2017). An Invariant Winding Number for the FitzHugh–Nagumo System with Applications to Cardiac Dynamics. SIAM J. APPLIED DYNAMICAL SYSTEMS 2017 Vol. 16, No. 4, pp. 1893–1922
$$\epsilon \frac{dV}{dt}=V-\frac{V^3}{3}-W+I$$
$$\frac{dW}{dt} = a + V - b W $$
A year later Dr. Nagumo et al. published a electrical circuit that had the same differential equation as the mathematical model. This model and the corresponding electrical circuit quickly became extremely popular in the neuroscience community. Many computational neuroscience students, including myself, cut their mathematical teeth on this model. However, the cardiac electrophysiologists were paying attention as well, and today the FHN model doubles as a simplified model of a heart cell!
However, simple models can be extremely complicated, so much so that in 2017 we are still asking questions about the FHN model. A paper by Dr. Eric Cytrynbaum and his Master's student Kelly Paton have an article in SIAM about a simplified “heart” composed of FHN cells. They ask the seemingly easy question, which initial conditions gives rise to a beating heart (stable wave train), and which initial conditions flatline (go to rest). They’re central proof is that the winding number is in a singular FHN, and they conjecture that is that if the winding number is ever $0$, then the heart will flatline. However, the converse is not necessarily true. It is always astounding proving something so “easy” tends to be quite difficult.
First Let’s define their model. Their model heart is a 1-Dimensional ring of FHN cells. In other words, the heart is a one-dimensional partial differential equation with periodic boundary conditions. They prove their results for a more general FHN model, one where the voltage nullcline $F[V]$ has the stereotypical N-shape, however they walk through the gory details with the linear piecewise version. Here is the general model for their FHN heart.
$$\epsilon \frac{dV}{dt}=\epsilon^2 \frac{d^2V}{dx^2}+F[V]-W+I$$
$$\frac{dW}{dt} = a + V - b W $$
In my reproduced GIFs I use the traditional cubic nullcline $F[V] = V- \frac{V^3}{3}$. The proofs hold for any N shaped nullcline.
The next step is to project the FHN heart into a singularly perturbed FHN (or SFHN) heart. Recall the $\epsilon$ in front of the derivatives? As $\epsilon\rightarrow 0$ the original FHN equation reduces to an singular FHN (SFHN) equation where red dots jump instantaneously between locations on the blue N-shaped nullcline. Each red dot represents a individual Heart cell. See the frozen images below (a reproduction of figure 8 with easier initial conditions). (Note, that for simulation purposes I cheat and set $\epsilon = .005$ which is small enough for our purposes). What does it mean for the red dot to jump to a location on the blue nullcline? In the SFHN model ($\epsilon = 0$), the blue nullcline is a slow manifold, meaning that all solutions are attracted infinitely quickly to a location on the N-shaped nullcline. Specifically, the outer part of the N (parts with negative slope) are the attractive manifolds and the inner part (positive slope) is the repelling, meaning all initial conditions must find a location on the blue nullcline that has negative slope. Once on the attracting slow manifold the red dots will slowly move towards the the local extrema. Once they hit the local extrema they are now "on" the repelling slow manifold, and thus "jump away" to the other attracting slow manifold. This process repeats over and over again, thus creating a stable wave train.
Now we can define winding number. Notice that after jumping to a location on the N-shaped nullcline there is a resulting closed curve. The winding number is defined as the number of times that the red curve wraps around the green line after 1 time step. An equivalent statement is whether the initial conditions wrap around the black dot. In fact one can think if a initial condition started at the black dot, it would get pulled along the green line in both directions. It wants to be on both attracting slow manifolds. Thus the green line "represents" the black dot after 1 time step.
The winding number is defined by how many times the inital conditions wrap around the black dot at $t=0$, or the green line at $t = \Delta t$. If it doesn't wrap around it, then the winding number is $0$. If it wraps once in the counter-clockwise direction it is $+1$, if it wraps in the clockwise direction it is $-1$. The paper proves that in the SFHN the winding number is conserved, meaning it never changes and only initial conditions with winding number $0$ can go to rest.
撰稿：Alex White
翻譯：高暐哲
Source:
source : Eric N. Cytrynbaum and Kelly M. Paton (2017). An Invariant Winding Number for the FitzHugh–Nagumo System with Applications to Cardiac Dynamics. SIAM J. APPLIED DYNAMICAL SYSTEMS 2017 Vol. 16, No. 4, pp. 1893–1922
Simple Stable Wave Train Example:
Simple Flatline Example:
FHN Flatline but SFHN Stable Wave Train Example:
Complicated Stable Wave Train:
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