Multirhythmicity for a Time-Delayed FitzHugh-Nagumo System with Threshold Nonlinearity

Abstract

A time-delayed FitzHugh-Nagumo (FHN) system exhibiting a threshold nonlinearity is studied both experimentally and theoretically. The basic steady state is stable but distinct stable oscillatory regimes may coexist for the same values of parameters (multirhythmicity). They are characterized by periods close to an integer fraction of the delay. From an asymptotic analysis of the FHN equations, we show that the mechanism leading to those oscillations corresponds to a limit-point of limit-cycles. In order to investigate their robustness with respect to noise, we study experimentally an electrical circuit that is modeled mathematically by the same delay differential equations. We obtain quantitative agreements between numerical and experimental bifurcation diagrams for the different coexisting time-periodic regimes.

Appendices

Appendix

A Connection at $t=t_{1}$ and $t=t_{2}$

We first require that the solutions (17.9)–(17.10) and (17.13)–(17.14) are equal at the critical times $t=t_{1}$ and $t=t_{2}$. This leads to the following four equations

$$\begin{aligned} C+D&=Ae^{\lambda _{+}t_{1}}+Be^{\lambda _{-}t_{1}}, \end{aligned}$$

(17.31)

$$\begin{aligned} 1-(1+\varepsilon \lambda _{+})Ae^{\lambda _{+}t_{1}}-(1+\varepsilon \lambda _{-})Be^{\lambda _{-}t_{1}}&=-(1+\varepsilon \lambda _{+})C-(1+\varepsilon \lambda _{-})D, \end{aligned}$$

(17.32)

$$\begin{aligned} Ce^{\lambda _{+}t_{21}}+De^{\lambda _{-}t_{21}}&=A+B, \end{aligned}$$

(17.33)

$$\begin{aligned} -(1+\varepsilon \lambda _{+})Ce^{\lambda _{+}t_{21}}-(1+\varepsilon \lambda _{-})De^{\lambda _{-}t_{21}}&=1-(1+\varepsilon \lambda _{+})A-(1+\varepsilon \lambda _{-})B, \end{aligned}$$

(17.34)

where $t_{21}\equiv t_{2}-t_{1}$. We now determine the constants A, B, C, and D as functions of $t_{1}$ and $t_{2}$.

From Eqs. (17.31) and (17.33), we determine

$$\begin{aligned} Ae^{\lambda _{+}t_{1}}&=C+D-Be^{\lambda _{-}t_{1}}, \end{aligned}$$

(17.35)

$$\begin{aligned} Ce^{\lambda _{+}t_{21}}&=A+B-De^{\lambda _{-}t_{21}}. \end{aligned}$$

(17.36)

Inserting these expressions of $A\exp (\lambda _{+}t_{1})$ and $C\exp (\lambda _{+}t_{21})$ into Eqs. (17.32) and (17.34), respectively, leads to two coupled equations for B and D

$$\begin{aligned} Be^{\lambda _{-}t_{1}}&=D-\frac{1}{\varepsilon \left( \lambda _{+}-\lambda _{-}\right) }, \end{aligned}$$

(17.37)

$$\begin{aligned} De^{\lambda _{-}t_{21}}&=\frac{1}{\varepsilon \left( \lambda _{+}-\lambda _{-}\right) }+B. \end{aligned}$$

(17.38)

By using (17.37), we eliminate D into Eq. (17.38) and find

$$\begin{aligned} B=\frac{1-e^{\lambda _{-}t_{21}}}{\varepsilon \left( \lambda _{+}-\lambda _{-}\right) \left[ e^{\lambda _{-}t_{2}}-1\right] }. \end{aligned}$$

(17.39)

Introducing then B given by (17.39) into Eq. (17.37), we obtain D as

$$\begin{aligned} D=\frac{e^{\lambda _{-}t_{1}}-1}{\varepsilon \left( \lambda _{+}-\lambda _{-}\right) \left[ e^{\lambda _{-}t_{2}}-1\right] }. \end{aligned}$$

(17.40)

Inserting (17.37) into (17.35) and (17.38) into (17.36) provides two coupled equations for A and C given by

$$\begin{aligned} Ae^{\lambda _{+}t_{1}}&=C+\frac{1}{\varepsilon \left( \lambda _{+}-\lambda _{-}\right) }, \end{aligned}$$

(17.41)

$$\begin{aligned} Ce^{\lambda _{+}t_{21}}&=A-\frac{1}{\varepsilon \left( \lambda _{+}-\lambda _{-}\right) }. \end{aligned}$$

(17.42)

Using (17.41), we eliminate C in Eq. (17.42) and find

$$\begin{aligned} A=\frac{e^{\lambda _{+}t_{21}}-1}{\varepsilon \left( \lambda _{+}-\lambda _{-}\right) \left[ e^{\lambda _{+}t_{2}}-1\right] }. \end{aligned}$$

(17.43)

Finally, introducing A given by (17.43) into Eq. (17.41) provides C as

$$\begin{aligned} C=\frac{1-e^{\lambda _{+}t_{1}}}{\varepsilon \left( \lambda _{+}-\lambda _{-}\right) \left[ e^{\lambda _{+}t_{2}}-1\right] }. \end{aligned}$$

(17.44)

From Fig. 17.4b, we note that x increases at time $t=0$ when $x(t-\tau )=a$. At time $t=\delta$, it is the turn of x to equal a. From Fig. 17.4c, we note that $t=t_{1}$ and $t=t_{1}+\delta$ mark the times where $x(t-\tau )$ and then x are equal to a. Using (17.9) with $x(\delta )=a$ and (17.13) with $x(t_{1}+\delta )=a$, we obtain

$$\begin{aligned} Ae^{\lambda _{+}\delta }+Be^{\lambda _{-}\delta }&=a, \end{aligned}$$

(17.45)

$$\begin{aligned} Ce^{\lambda _{+}\delta }+De^{\lambda _{-}\delta }&=a. \end{aligned}$$

(17.46)

Equations (17.31)–(17.46) with

$$\begin{aligned} t_{2}=T_{n}, \end{aligned}$$

(17.47)

defined by (17.6), are six equations for seven unknowns, namely A, B, C, D, $t_{1}$, and $\delta$. We introduce the expressions of A, B, C, and D into Eqs. (17.45) and (17.46), and find

$$\begin{aligned} a\varepsilon \left( \lambda _{+}-\lambda _{-}\right)&=\frac{1-e^{\lambda _{+}t_{21}}}{1-e^{\lambda _{+}t_{2}}}e^{\lambda _{+}\delta }+\frac{ e^{\lambda _{-}t_{21}}-1}{1-e^{\lambda _{-}t_{2}}}e^{\lambda _{-}\delta }, \end{aligned}$$

(17.48)

$$\begin{aligned} a\varepsilon \left( \lambda _{+}-\lambda _{-}\right)&=\frac{e^{\lambda _{+}t_{1}}-1}{1-e^{\lambda _{+}t_{2}}}e^{\lambda _{+}\delta }+\frac{ 1-e^{\lambda _{-}t_{1}}}{1-e^{\lambda _{-}t_{2}}}e^{\lambda _{-}\delta }. \end{aligned}$$

(17.49)

Substracting side by side, we eliminate $a\varepsilon \left( \lambda _{+}-\lambda _{-}\right)$. Multiplying then by $e^{-\lambda _{+}\delta }$, we have

$$\begin{aligned} 0=\frac{2-e^{\lambda _{+}t_{21}}-e^{\lambda _{+}t_{1}}}{1-e^{\lambda _{+}t_{2}}}+\left( \frac{e^{\lambda _{-}t_{21}}-2+e^{\lambda _{-}t_{1}}}{ 1-e^{\lambda _{-}t_{2}}}\right) e^{\left( \lambda _{-}-\lambda _{+}\right) \delta }. \end{aligned}$$

(17.50)