# Recurrence-Based Synchronization Analysis of Weakly Coupled Bursting Neurons under External ELF Fields

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## Abstract

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## 1. Introduction

## 2. Model

#### 2.1. Morris–Lecar Neuron Model under an Extremely Low Frequency Electric Field

#### 2.2. Bursting Patterns of a Neuron

## 3. Recurrence Quantification Analysis (RQA)

## 4. Coupling of Two Bursting Neurons

#### 4.1. Model and Numerical Simulation

#### 4.2. Phase Synchronization Analysis Using Recurrence Features

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CPR | correlation of probability of recurrence |

EF | electric field |

ELF | extremely low frequency |

HH | Hodgkin–Huxley model |

ML | Morris–Lecar model |

RP | recurrence plot |

RQA | recurrence quantification analysis |

RR | recurrence rate |

## References

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**Figure 1.**Spiking patterns of ML neuron membrane voltage under an external EF for different frequencies: (

**A**) $\omega =0.02$ rad/ms, (

**B**) $\omega =0.05$ rad/ms, (

**C**) $\omega =0.06$ rad/ms, (

**D**) $\omega =0.10$ rad/ms, (

**E**) $\omega =0.286$ rad/ms, (

**F**) $\omega =0.32$ rad/ms, and (

**G**) $\omega =0.5$ rad/ms.

**Figure 2.**Bifurcation diagram of ML neuron dynamics under an external EF for varying frequencies $\omega $ (based on interspike intervals of the membrane voltage). For better visibility of the dynamics for larger $\omega $, the y-axis was bounded to 250 ms.

**Figure 3.**RPs of the membrane voltage v of selected bursting neurons: (

**A**) 4 spike burst ($\omega =0.05$ rad/ms), (

**B**) 2 spike burst ($\omega =0.10$ rad/ms), (

**C**) chaos ($\omega =0.286$ rad/ms), and (

**D**) 1 spike ($\omega =0.5$ rad/ms). Diagonal lines represent the spikes, the larger extended structures represent the “silent” epochs, and the structures perpendicular to the diagonal lines and small thickenings represent the slow–fast dynamics (blue circle in (

**D**)). The orange boxes in (

**A**,

**B**) mark a sequence of diagonal lines. The number of diagonal lines counted from the main diagonal of such a box towards the corner of this box represents the number of spikes within this period. Recurrence threshold $\epsilon $ is selected to ensure a recurrence point density of 0.15.

**Figure 4.**Probability of recurrence after time $\tau $ ($\tau $-recurrence rate) for the bursting neurons as shown in Figure 3: (

**A**) 4 spike burst, (

**B**) 2 spike burst, (

**C**) chaos, and (

**D**) 1 spike. The n bursts are visible as the rather thin side peaks of the main peaks (in addition to the main peak).

**Figure 5.**Spiking pattern in the membrane voltage of two weakly coupled neurons in an ELF EF in chaotic regime ($\omega =0.286$) with (

**A**) no synchronization with $g=0.01$ and (

**B**) phase synchronization with $g=0.04$.

**Figure 6.**Frequency mismatch between chaotic coupled neurons for increasing coupling g without external EF (red) and with external EF where $A=0.1$ and $\omega =0.286$ (blue).

**Figure 7.**Frequency mismatch between chaotic coupled neurons for increasing amplitude A and for different coupling strengths g.

**Figure 8.**Recurrence plots of the membrane voltage for weakly coupled neurons as shown in Figure 5 for (

**A**,

**B**) no synchronization, $g=0.01$, and (

**C**,

**D**) phase synchronization, $g=0.04$. Embedding parameters were estimated using the PECUZAL method [70]; the recurrence threshold is selected to ensure a recurrence rate of $RR=0.1$.

**Figure 10.**Correlation of probability of recurrence CPR based on Pearson (dotted) and Spearman (line) correlations indicating the onset of phase synchronization between chaotic coupled neurons without external EF (red) and with external EF where $A=0.1$ and $\omega =0.286$ (blue).

**Figure 11.**Hellinger distance of the $\tau $-recurrence rate indicating the onset of phase synchronization between chaotic coupled neurons without external EF (red) and with external EF where $A=0.1$ and $\omega =0.286$ (blue). A drop of H below the confidence limit of 95% (dotted line) represents the significance of this finding.

${\mathit{u}}_{1}$ | $-1.2$ mV | ${\mathit{g}}_{\mathbf{fast}}$ | 20 mS/cm${}^{2}$ | ${\mathit{e}}_{\mathbf{Na}}$ | 50 mV | $\mathit{\phi}$ | 0.15 |
---|---|---|---|---|---|---|---|

${u}_{2}$ | 18 mV | ${g}_{\mathrm{slow}}$ | 20 mS/cm${}^{2}$ | ${e}_{\mathrm{K}}$ | $-100$ mV | c | 2 $\mu $ |

${u}_{3}$ | $-13$ mV | ${g}_{\mathrm{leak}}$ | 2 mS/cm${}^{2}$ | ${e}_{\mathrm{leak}}$ | $-70$ mV | ${V}_{\mathrm{E}}$ | $-17.63$ mV |

${u}_{4}$ | 10 mV |

Time Series | Dimension | Delay |
---|---|---|

4 spike burst ($\omega =0.05$ rad/ms) | 3 | 17, 22 |

2 spike burst ($\omega =0.10$ rad/ms) | 2 | 16, 20 |

chaos ($\omega =0.286$ rad/ms) | 2 | 20 |

1 spike ($\omega =0.5$ rad/ms | 2 | 19 |

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**MDPI and ACS Style**

Nkomidio, A.M.; Ngamga, E.K.; Nbendjo, B.R.N.; Kurths, J.; Marwan, N.
Recurrence-Based Synchronization Analysis of Weakly Coupled Bursting Neurons under External ELF Fields. *Entropy* **2022**, *24*, 235.
https://doi.org/10.3390/e24020235

**AMA Style**

Nkomidio AM, Ngamga EK, Nbendjo BRN, Kurths J, Marwan N.
Recurrence-Based Synchronization Analysis of Weakly Coupled Bursting Neurons under External ELF Fields. *Entropy*. 2022; 24(2):235.
https://doi.org/10.3390/e24020235

**Chicago/Turabian Style**

Nkomidio, Aissatou Mboussi, Eulalie Ketchamen Ngamga, Blaise Romeo Nana Nbendjo, Jürgen Kurths, and Norbert Marwan.
2022. "Recurrence-Based Synchronization Analysis of Weakly Coupled Bursting Neurons under External ELF Fields" *Entropy* 24, no. 2: 235.
https://doi.org/10.3390/e24020235