Characterizing mixed-mode oscillations shaped by canard and bifurcation structure in a three-dimensional cardiac cell model

Abstract

This paper investigates mixed-mode oscillations (MMOs) with a three-dimensional conductance-based cardiac action potential model, which makes the heart beat in a nonrenewable way. The 3D model was entailed by utilizing voltage-dependent timescales to describe the mechanism in which MMOs are generated. As expected, motivated by geometric singular perturbation theory, our analysis explains in detail the geometric mechanisms that there is a range of parameters under which the cardiac model highlights that the presence of MMOs is induced by the intrinsical canard phenomenon. Much is currently known about the geometric mechanisms, for a folded saddle, the two singular canards perturb to maximal canards. Characteristics of the stimulus current such as frequency and duration determine which early afterdepolarizations (EADs), as a special case of MMOs bears, as well as the article compares the detailed manifold structures of original and dimensionless systems with square wave pulses by setting the pacing cycle length. An exceedingly vital technique of the analysis is the slow–fast dynamics analysis by which the system governs multiple timescale structures analytically. A more novel and successful multiple-timescale approach divides the system so that there is only one fast variable and demonstrates that the MMOs arise from canard dynamics, such as using a three-variable model in which two variables are treated as “slow” and one treated as “fast”, which the layer problem and the reduced problem are considered to explain the trajectory on the critical manifold. Meanwhile, if one variable was regarded as the single slow variable, substantial bifurcation properties are discovered for slow–fast system, as well as general one-parameter bifurcation type is discussed for the whole system similarly. By focusing on the first Lyapunov coefficient of the Hopf bifurcation, which decides whether the bifurcation is supercritical or subcritical, it was shown that an unstable limit cycle can arise via a delayed subcritical Hopf bifurcation for the original system. Meanwhile, the dynamical studies of cardiac model have major implications for further elaborating the complex dynamic behaviors, such as EADs, which can lead to tissue-level arrhythmias. Ultimately, it has turned these researches into a considerable player in the signal and information transmission for underlying nervous systems.

Appendix

The first step, the paper rewrites the system for Eq. (1) as

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {\dot{V}}=F_1(V,f,x),\\ {\dot{f}}=F_2(V,f),\\ {\dot{x}}=F_3(V,x) \end{array}\right. \end{aligned}$$

(26)

where

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle F_1=\frac{1}{C}[-g_{Ca}d_{\infty }(V)f(V-E_{Ca})\\ \quad -g_Kx(V-E_K)+I_{\mathrm{sti}}],\\ F_2=\frac{f_{\infty }(V)-f}{\tau _{f}},\\ F_3=\frac{x_{\infty }(V)-x}{\tau _{x}} \end{array}\right. \end{aligned}$$

(27)

where $d_\infty (V)$, $f_\infty (V)$, $x_\infty (V)$, $\tau _{f}$ and $\tau _{x}$ are detailed in Table 1.

Additionally, the Jacobian matrix can be denoted as

$$\begin{aligned} A = \left( \begin{array}{ccc} \displaystyle \frac{\partial F_1}{\partial V} &{} \displaystyle \frac{\partial F_1}{\partial f} &{} \displaystyle \frac{\partial F_1}{\partial x} \\ \displaystyle \frac{\partial F_2}{\partial V} &{} \displaystyle \frac{\partial F_2}{\partial f} &{}\displaystyle \frac{\partial F_2}{\partial x} \\ \displaystyle \frac{\partial F_3}{\partial V} &{} \displaystyle \frac{\partial F_3}{\partial f} &{}\displaystyle \frac{\partial F_3}{\partial x} \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned} \displaystyle \frac{\partial F_1}{\partial V}= & {} \displaystyle -0.4006410258e-2f(V-100)\\&e^{(-.1602564103V-5.608974360)}/(1+e^{(-.1602564103V-5.608974360)})^2\\&-0.025f/(1+e^{(-0.1602564103V-5.608974360)})-0.046x ,\\ \frac{\partial F_1}{\partial f}= & {} \displaystyle -(0.025(V-100))/(1+e^{(-0.1602564103V-5.608974360)}),\\ \frac{\partial F_1}{\partial x}= & {} -0.046V-3.680,\\ \frac{\partial F_2}{\partial V}= & {} -0.001453488372\\&\times e^{(0.1162790698V+2.325581396)}/(1+e^{(0.1162790698V+2.325581396)})^2 ,\\ \frac{\partial F_2}{\partial f}= & {} -1/80,\\ \frac{\partial F_2}{\partial x}= & {} 0,\\ \frac{\partial F_3}{\partial V}= & {} (1/1500)\\&e^{(-(1/5)V-8)}/(1+e^{(-(1/5)V-8)})^2,\\ \frac{\partial F_3}{\partial f}= & {} 0,\\ \frac{\partial F_3}{\partial x}= & {} -1/300,\\ \end{aligned}$$

Ulteriorly, the paper calculates the equilibrium of the system for Eq. (1) at point $H_1$ (when $g_{K} =0.034428$) is ($V_0, {f_0},x_0) = (-28.896476 0.737782 0.902093)$. Meanwhile, the corresponding matrix via taking into the specific values at that point $H_1$ is

$$\begin{aligned} A|_{H_1} = \left( \begin{array}{ccc} 0.02076226414 &{} 2.341845250 &{} -2.350762104 \\ -0.2811912152\cdot 10^{-3} &{} -1/80 &{}0 \\ 0.00005888056529&{} 0&{} -1/300 \end{array} \right) , \end{aligned}$$

which has one pair of conjugate eigenvalues $\lambda$ and  ${\bar{\lambda }}$, where $\lambda =iw$, $w= 0.02321742$.

Typically, the Hopf bifurcation occurred at $H_1$ point as shown in Fig. 9. Mathematically, set

$$\begin{aligned} q= & {} \left( \begin{array}{c} 0.99995072 \\ -0.00583631 + 0.00766909i \\ 0.00081884 - 0.00223607i \end{array} \right) , \\ p_0= & {} \left( \begin{array}{c} 0.99995072\\ -0.00583631 - 0.00766909i \\ 0.00081884 + 0.00223607i \end{array} \right) , \end{aligned}$$

which satisfy that $Aq=iwq$, $Ap_0=-iwp_0$, $A^Tp'=-iwp'$,

After calculating, we obtain the following

$$\begin{aligned} \begin{array}{lll} p'=\left( \begin{array}{c} 0.00276352- 0.00754651i\\ 0.61635485 - 0.19031157i \\ -0.76408352 \end{array} \right) , \end{array} \end{aligned}$$

In order to render $\langle p,q \rangle =1$, there is

$$\begin{aligned} \begin{array}{lll} p=\left( \begin{array}{c} -0.603956560697810 - 0.077737986299549i\\ -24.39227406 -42.35565830i \\ 12.80510500 +56.46133930i \end{array} \right) , \end{array} \end{aligned}$$

Specially note that,  $\langle p, q \rangle ={\bar{p}}_1q_1+{\bar{p}}_2q_2+{\bar{p}}_3q_3$ is the standard scalar product in $\mathbf{C }^3$.

To begin with the computation of first Lyapunov coefficient, the equilibrium of the original system is firstly moved to the origin of coordinate by taking the following transformation

$$\begin{aligned} \left\{ \begin{array}{l} V=\xi _1+V_0,\\ f=\xi _2+{f_{0}},\\ x=\xi _3+x_0. \end{array}\right. \end{aligned}$$

(28)

where $(V_0, f_0,x_0) = (-28.896476, 0.737782, 0.902093)$.

Through this transformation, system for Eq. (19) changes into

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \frac{\mathrm{d}\xi _1}{\mathrm{d}t}=\frac{1}{C}[-g_{Ca}d_{\infty }(\xi _1+V_0)(\xi _2+{f_{0}})(\xi _1+V_0-E_{Ca})\\ \quad -g_K(\xi _3+x_0)(\xi _1+V_0-E_K)+I_{\mathrm{sti}}]\\ \displaystyle \frac{\mathrm{d}\xi _2}{\mathrm{d}t}=\frac{f_{\infty }(\xi _1+V_0)-(\xi _2+{f_{0}})}{\tau _{f}}\\ \displaystyle \frac{\mathrm{d}\xi _3}{\mathrm{d}t}=\frac{x_{\infty }(\xi _1+V_0)-(\xi _3+x_0)}{\tau _{x}} \end{array}\right. \end{aligned}$$

(29)

This system (19) can also be expressed as

$$\begin{aligned} {\dot{x}}=Ax+F(x),~x\in \mathbf{R }^3, \end{aligned}$$

(30)

where $A=A|_{H_1}$, $F(x)=\displaystyle \frac{1}{2}B(x,x)+\displaystyle \frac{1}{6}C(x,x,x)+O(\Vert x\Vert ^4)$, B(x, y) and C(x, y, z) are both symmetric and multilinear vector functions by which we pick up the planar vectors $x=(x_1,x_2,x_3)^T$, $y=(y_1,y_2,y_3)^T$, $z=(z_1,z_2,z_3)^T$. Mathematically, there are

$$\begin{aligned}&B_i(x,y)=\left. \sum \limits _{j,k=1}^3\frac{\partial ^2F_i(\xi )}{\partial \xi _j \partial \xi _k}\right| _{\xi =0}x_j y_k,i=1,2,3 \end{aligned}$$

(31)

$$\begin{aligned}&C_i(x,y,z)=\left. \sum \limits _{j,k,l=1}^3\frac{\partial ^3F_i(\xi )}{\partial \xi _j \partial \xi _k \partial \xi _l}\right| _{\xi =0}x_j y_k z_l,i=1,2,3\nonumber \\ \end{aligned}$$

(32)

where $\xi =(\xi _1,\xi _2,\xi _3)^T$.

It is not hard to calculate

$$\begin{aligned} B_1(x,y)= & {} -0.00667259x_1y_1+0.08438610x_1y_2\\&-0.046x_1y_3+0.08438610x_2y_1-0.046x_3y_1 \\ B_2(x,y)= & {} -0.1554937366\cdot 10^{-4}x_1y_1,\\ B_3(x,y)= & {} -0.9470196136\cdot 10^{-5}x_1y_1 \\ C_1(x,y,z)= & {} -0.244225714\cdot 10^{-3}x_1y_1z_1\\&-0.9044115820\cdot 10^{-2}x_1y_1z_2 \\&-0.9044115820\cdot 10^{-2}x_1y_2z_1\\&-0.9044115820\cdot 10^{-2}x_2y_1z_1, \\ C_2(x,y,z)= & {} 6.11189169\cdot 10^{-7}x_1y_1z_1, \\ C_3(x,y,z)= & {} 0.1107131062\cdot 10^{-5}x_1y_1z_1 \end{aligned}$$

Moreover, if we take $\xi =(\xi _1,\xi _2,\xi _3)^T=\mathbf{0 }$, there are

$$\begin{aligned} B(x,y)= & {} \left( \begin{array}{c} -0.00667259x_1y_1+0.08438610(x_1y_2+x_2y_1)-0.046(x_1y_3+x_3y_1) \\ -0.1554937366\cdot 10^{-4}x_1y_1 \\ -0.9470196136\cdot 10^{-5}x_1y_1 \end{array} \right) ,\\ C(x,y,z)= & {} \left( \begin{array}{c} -0.244225714\cdot 10^{-3}x_1y_1z_1-0.9044115820\cdot 10^{-2}(x_1y_1z_2+x_1y_2z_1+x_2y_1z_1) \\ 6.11189169\cdot 10^{-7}x_1y_1z_1 \\ 0.1107131062\cdot 10^{-5}x_1y_1z_1 \end{array} \right) . \end{aligned}$$

Accordingly, we calculate

$$\begin{aligned} C(q,q,{\bar{q}})= & {} \left( \begin{array}{c} -8.585248088515901 \cdot 10^{-5} - 6.935329019316943 \cdot 10^{-5}i \\ 6.110988241\cdot 10^{-7} \\ 1.106967408 \cdot 10^{-6} \end{array} \right) . \\ \langle p,C(q,q,{\bar{q}})\rangle= & {} 5.651129809645031\cdot 10^{-5} - 1.404993804116788\cdot 10^{-6}i \\ B(q,{\bar{q}})= & {} \left( \begin{array}{c} -0.007732216224 \\ -0.1554784130\cdot 10^{-4} \\ -0.9469262867\cdot 10^{-5} \end{array} \right) . \\ A^{-1}B(q,{\bar{q}})= & {} \left( \begin{array}{c} 0.054016025185156 \\ 0.000028720763033 \\ 0.003794927089386 \end{array} \right) . \end{aligned}$$

$$\begin{aligned} \begin{array}{l} B(q,A^{-1}B(q,{\bar{q}}))\\ \quad =\left( \begin{array}{c} -0.5611810173 \cdot 10^{-3}+0.4051328738 \cdot 10^{-4}i \\ -8.398739722\cdot 10^{-7} \\ -5.118340071\cdot 10^{-7} \end{array} \right) . \end{array} \end{aligned}$$

$\langle p,B(q,A^{-1}B(q,{\bar{q}}))\rangle = 3.497118837 \cdot 10^{-4}-7.476792938\cdot 10^{-5}i$

$$\begin{aligned}&B(q,q)\\&\quad =\left( \begin{array}{c} -0.7732216224 \cdot 10^{-2}+0.1499973052 \cdot 10^{-2}i \\ -0.1554784130 \cdot 10^{-4} \\ -0.9469262867 \cdot 10^{-5} \end{array} \right) . \\&(2iwE-A)^{-1}B(q,q)\\&\quad =\left( \begin{array}{c} 0.155538578616271 + 0.173649101485302i \\ -0.001300959741788 + 0.000926500924901i \\ 0.000218584210219 + 0.000022390027940i \end{array} \right) . \end{aligned}$$

$$\begin{aligned}&B({\bar{q}},(2iwE-A)^{-1}B(q,q)) \\&\quad =\left( \begin{array}{c} -0.001109846259466 - 0.001290202538074i \\ -0.2418408304 \cdot 10^{-5} -0.2700001715 \cdot 10^{-5}i \\ -0.1472908265 \cdot 10^{-5} -0.1644410017 \cdot 10^{-5}i \end{array} \right) . \end{aligned}$$

$\langle p,B({\bar{q}},(2iwE-A)^{-1}B(q,q))\rangle = 8.3224116828 \cdot 10^{-4}+ 7.1848051062 \cdot 10^{-4}i$.

The first Lyapunov coefficient is a classical index to distinguish the stability of the Hopf equilibrium, which is first applicable to the low dimensional system such as two-dimensional system. Nevertheless, for high-dimensional systems, the paper gives another expression as follows [58]:

$$\begin{aligned}&l_1(0)=\displaystyle \frac{1}{2w}\mathrm{Re}\{\langle p,C(q,q,{\bar{q}})\rangle -2\langle p,B(q,A^{-1}B(q,{\bar{q}}))\rangle \nonumber \\&\qquad +\,\langle p,B({\bar{q}},(2iwE-A)^{-1}B(q,q))\rangle \}\\&\quad =0.004077298001>0. \end{aligned}$$