# Entrainment of Weakly Coupled Canonical Oscillators with Applications in Gradient Frequency Neural Networks Using Approximating Analytical Methods

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. The Canonical Model of Single Oscillators

#### 2.1.1. Wilson–Cowan Model

#### 2.1.2. The Normal Form

#### 2.1.3. The Normal Form about a Hopf Bifurcation Point: The Canonical Model

**Theorem**

**1.**

**Theorem**

**2.**

#### 2.2. The Canonical Model of Two Coupled Oscillators

#### 2.2.1. The General Normal Form

#### 2.2.2. Hopf Normal Form: The Canonical Model

#### 2.2.3. Radial and Angular Equations

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

GFNNs | Gradient Frequency Neural Networks |

CNF | Conventional Normal Form |

## Appendix A

- $case\left(I\right):(p,q)=(1,2)$

- $case\left(II\right):(p,q)=(2,1)$

- $case\left(I\right):(p,q)=(1,2)\Rightarrow {\dot{z}}_{s}={A}_{1}z+{A}_{2}\overline{z}+{A}_{21}{\left|z\right|}^{2}z$.
- $case\left(II\right):(p,q)=(2,1)\Rightarrow {\dot{z}}_{s}={A}_{1}z+{A}_{2}\overline{z}+{A}_{12}{\left|z\right|}^{2}\overline{z}$.

## Appendix B

- $case\left(I\right):(p,q)=(1,2),(m,n)=(1,2)$${\dot{{z}_{1}}}_{c}={B}_{1}{z}_{2}+{B}_{2}\overline{{z}_{2}}+{B}_{5}{z}_{1}^{2}{z}_{2}+{B}_{6}{z}_{1}^{2}\overline{{z}_{2}}+({B}_{7}-2{\kappa}_{1}\overline{{B}_{2}}){z}_{1}{z}_{2}\overline{{z}_{1}}+({B}_{3}-{B}_{1}{\kappa}_{1}){z}_{2}{\overline{{z}_{1}}}^{2}\phantom{\rule{0ex}{0ex}}+({B}_{8}-2{\kappa}_{1}\overline{{B}_{1}}){z}_{1}\overline{{z}_{1}}\overline{{z}_{2}}+({B}_{4}-{B}_{2}{\kappa}_{1}){\overline{{z}_{1}}}^{2}\overline{{z}_{2}}+{B}_{1}{z}_{2}{\kappa}_{2}{\overline{{z}_{2}}}^{2}+{B}_{2}\overline{{\kappa}_{2}}{z}_{2}^{2}\overline{{z}_{2}}$
- $case\left(II\right):(p,q)=(2,1),(m,n)=(1,2)$${\dot{{z}_{1}}}_{c}={B}_{1}{z}_{2}+{B}_{2}\overline{{z}_{2}}+{B}_{3}{z}_{2}{\overline{{z}_{1}}}^{2}+{B}_{4}{\overline{{z}_{1}}}^{2}\overline{{z}_{2}}+({B}_{5}-{\kappa}_{1}\overline{{B}_{2}}){z}_{1}^{2}{z}_{2}+({B}_{6}-{\kappa}_{1}\overline{{B}_{1}}){z}_{1}^{2}\overline{{z}_{2}}+({B}_{7}-2{\kappa}_{1}\overline{{B}_{1}}){z}_{1}{z}_{2}\overline{{z}_{1}}\phantom{\rule{0ex}{0ex}}+({B}_{8}-2{\kappa}_{1}{B}_{2}){z}_{1}\overline{{z}_{1}}\overline{{z}_{2}}+{B}_{1}{z}_{2}{\kappa}_{2}{\overline{{z}_{2}}}^{2}+{B}_{2}\overline{{\kappa}_{2}}{z}_{2}^{2}\overline{{z}_{2}}$
- $case\left(III\right):(p,q)=(1,2),(m,n)=(2,1)$${\dot{{z}_{1}}}_{c}={B}_{1}{z}_{2}+{B}_{5}{z}_{1}^{2}{z}_{2}+{B}_{2}\overline{{z}_{1}}+{B}_{6}{z}_{1}^{2}\overline{{z}_{2}}+({B}_{3}-{B}_{1}{\kappa}_{1}){z}_{2}{\overline{{z}_{1}}}^{2}+({B}_{4}-{B}_{2}{\kappa}_{1})\overline{{z}_{2}}{\overline{{z}_{1}}}^{2}+({B}_{7}-2\overline{{B}_{2}}{\kappa}_{1}){z}_{1}{z}_{2}\overline{{z}_{1}}\phantom{\rule{0ex}{0ex}}+({B}_{8}-2\overline{{B}_{1}}{\kappa}_{1}){z}_{1}\overline{{z}_{1}}\overline{{z}_{2}}+{B}_{1}{\kappa}_{2}{z}_{2}^{2}\overline{{z}_{2}}+{B}_{2}\overline{{\kappa}_{2}}{z}_{2}{\overline{{z}_{2}}}^{2}$
- $case\left(IV\right):(p,q)=(2,1),(m,n)=(2,1)$${\dot{{z}_{1}}}_{c}={B}_{1}{z}_{2}+{B}_{3}{\overline{{z}_{1}}}^{2}{z}_{2}+{B}_{2}\overline{{z}_{2}}+{B}_{4}{\overline{{z}_{1}}}^{2}\overline{{z}_{2}}+({B}_{5}-\overline{{B}_{2}}{\kappa}_{1}){z}_{2}{z}_{1}^{2}+({B}_{6}-\overline{{B}_{1}}{\kappa}_{1})\overline{{z}_{2}}{z}_{1}^{2}+({B}_{7}-2{B}_{1}{\kappa}_{1}){z}_{1}{z}_{2}\overline{{z}_{1}}\phantom{\rule{0ex}{0ex}}+({B}_{8}-2{B}_{2}{\kappa}_{1}){z}_{1}\overline{{z}_{1}}\overline{{z}_{2}}+{B}_{1}{\kappa}_{2}{z}_{2}^{2}\overline{{z}_{2}}+{B}_{2}\overline{{\kappa}_{2}}{z}_{2}{\overline{{z}_{2}}}^{2}$

## Appendix C

- ${n}_{1}=Re\left({B}_{1}\right),{n}_{2}=Re\left({B}_{2}\right),{n}_{3}=Re({B}_{8}+{B}_{6}-2{\kappa}_{1}\overline{{B}_{1}}),\phantom{\rule{0ex}{0ex}}{n}_{4}=Re({B}_{5}+{B}_{7}-2{\kappa}_{1}\overline{{B}_{2}}),{n}_{5}=Re({B}_{3}-{\kappa}_{1}{B}_{1}),\phantom{\rule{0ex}{0ex}}{n}_{6}=Re({B}_{4}-{\kappa}_{1}{B}_{2}),{n}_{7}=Re\left(\overline{{\kappa}_{2}}{B}_{2}\right),{n}_{8}=Re\left({\kappa}_{2}{B}_{1}\right),$
- ${l}_{1}=Im\left({B}_{1}\right),{l}_{2}=Im\left({B}_{2}\right),\phantom{\rule{0ex}{0ex}}{l}_{3}=Im({B}_{8}+{B}_{6}-2{\kappa}_{1}\overline{{B}_{1}}),{l}_{4}=Im({B}_{5}+{B}_{7}-2{\kappa}_{1}\overline{{B}_{2}}),\phantom{\rule{0ex}{0ex}}{l}_{5}=Im({B}_{3}-{\kappa}_{1}{B}_{1}),{l}_{6}=Im({B}_{4}-{\kappa}_{1}{B}_{2}),\phantom{\rule{0ex}{0ex}}{l}_{7}=Im\left(\overline{{\kappa}_{2}}{B}_{2}\right),{l}_{8}=Im\left({\kappa}_{2}{B}_{1}\right).$

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**Figure 1.**The nullclines. Nullclines of both Wilson–Cowan (black) and canonical (blue) models. Here $a=1$, $b=c=\omega =1.7$, $d=-1$, $\alpha =0$, $\beta =-0.75$.

**Figure 2.**The solutions of r for different values of the bifurcation parameter $\alpha $. These solutions exhibit different behaviors of the coupled oscillators, as we changed the bifurcation parameter $\alpha $. For $\alpha =-1<0$ we have damped oscillations, and for $\alpha =0$ we faced critical ones. For $\alpha =1$ spontaneous oscillation was the result. In this figure $a=\alpha +1$, $b=c=\omega =5$, $d=a-2$, $\u03f5=0.1$, ${c}_{1}={c}_{2}={c}_{3}={c}_{4}=\u03f5$, $\beta ={A}_{21}$ and $\delta =0.1$. The initial conditions were ${r}_{1}={r}_{2}=1$, ${\varphi}_{1}=\pi $ and ${\varphi}_{2}=\pi /6$.

**Figure 3.**The solutions of $\varphi $ for different values of the bifurcation parameter $\alpha $. For the two cases of $\alpha <0$ (damped) and $\alpha =0$ (critical) the two oscillators become in-phase with the relative phase of $\varphi =0$. Obviously, the rate of change of $\varphi $ for the damped oscillation is greater than the critical one. When $\alpha >0$ (spontaneous), the two oscillators become anti-phase. In this figure $a=\alpha +1$, $b=c=\omega =10$, $d=a-2$, $\u03f5=0.1$, ${c}_{1}={c}_{2}={c}_{3}={c}_{4}=\u03f5$, $\beta ={A}_{21}$ and $\delta =0.1$. The initial conditions were ${r}_{1}={r}_{2}=1$, ${\varphi}_{1}=\pi $ and ${\varphi}_{2}=\pi /6$.

**Figure 4.**The evolution of relative phase when ${\omega}_{1}={\omega}_{2}$. The relative phase $\left(\varphi \right)$ as a function of time for different initial conditions when $C=0.02$. This is a special case when the relative frequency of the oscillators $\omega $ is zero, which means they oscillate with the same frequency. One can see that they become phase-locked as time increases. In this special case, since $\omega =0$, the fixed relative phase will be $\pi $.

**Figure 5.**The evolution of relative phase when ${\omega}_{1}\ne {\omega}_{2}$. The relative phase $\left(\varphi \right)$ is a function of time for different initial conditions while $C=2$ and $\omega =1$, so we can see the convergence of the solutions and phase-locking phenomenon. In this special case, since $\omega =1$ and $C=2$, the fixed relative phase will be $\frac{\pi}{6}$.

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**MDPI and ACS Style**

Farokhniaee, A.; Almonte, F.V.; Yelin, S.; Large, E.W.
Entrainment of Weakly Coupled Canonical Oscillators with Applications in Gradient Frequency Neural Networks Using Approximating Analytical Methods. *Mathematics* **2020**, *8*, 1312.
https://doi.org/10.3390/math8081312

**AMA Style**

Farokhniaee A, Almonte FV, Yelin S, Large EW.
Entrainment of Weakly Coupled Canonical Oscillators with Applications in Gradient Frequency Neural Networks Using Approximating Analytical Methods. *Mathematics*. 2020; 8(8):1312.
https://doi.org/10.3390/math8081312

**Chicago/Turabian Style**

Farokhniaee, AmirAli, Felix V. Almonte, Susanne Yelin, and Edward W. Large.
2020. "Entrainment of Weakly Coupled Canonical Oscillators with Applications in Gradient Frequency Neural Networks Using Approximating Analytical Methods" *Mathematics* 8, no. 8: 1312.
https://doi.org/10.3390/math8081312