# Synchronization of a Network Composed of Stochastic Hindmarsh–Rose Neurons

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Synchronization of Complex Networks and Multi-Agent Systems

#### 1.2. Synchronization of Neurons

#### 1.3. Purpose and Outline of This Paper

#### 1.4. Notation

- The Kronecker product is denoted by the symbol ⊗.
- The expected value of a random variable $\phi $ is denoted by $\mathbb{E}\left(\phi \right)$.
- If $A,B$ are matrices, then $\mathrm{diag}(A,B)$ is a block-diagonal matrix with blocks $A,B$ on the diagonal.
- The symbol ${}^{T}$ denotes the transposed matrix.
- The time argument t is often omitted: $f\left(t\right)=f$. However, if dependence on this time argument needs to be emphasized or the time argument is different from t, it is written in full.
- The time delay is written in the subscript: $f(t-\tau \left(t\right))=f(t-\tau )={f}_{\tau}\left(t\right)={f}_{\tau}$.
- The N-dimensional identity matrix is denoted by I.

## 2. Graph Theory

## 3. Synchronization of Stochastic Multi-Agent Systems

## 4. The Hindmarsh–Rose Neuronal Model

**Remark**

**1.**

**Remark**

**2.**

## 5. Synchronization of the Membrane Potential in the Stochastic Neuronal Networks

**Assumption**

**A1.**

**Theorem**

**1.**

**Remark**

**3.**

**Theorem**

**2.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proof of Theorem 2.**

**Remark**

**4.**

## 6. Synchronization of the Adaptation and Recovery Variables

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Remark**

**5.**

## 7. Example

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**State of the leader neuron. Blue line: ${x}_{0,1}$, green line: ${x}_{0,2}$, red line: ${x}_{0,3}$.

**Figure 3.**State of neuron 1. Blue line: ${x}_{1,1}$, green line: ${x}_{1,2}$, red line: ${x}_{1,3}$.

**Figure 4.**State of neuron 3. Blue line: ${x}_{3,1}$, green line: ${x}_{3,2}$, red line: ${x}_{3,3}$.

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**MDPI and ACS Style**

Rehák, B.; Lynnyk, V.
Synchronization of a Network Composed of Stochastic Hindmarsh–Rose Neurons. *Mathematics* **2021**, *9*, 2625.
https://doi.org/10.3390/math9202625

**AMA Style**

Rehák B, Lynnyk V.
Synchronization of a Network Composed of Stochastic Hindmarsh–Rose Neurons. *Mathematics*. 2021; 9(20):2625.
https://doi.org/10.3390/math9202625

**Chicago/Turabian Style**

Rehák, Branislav, and Volodymyr Lynnyk.
2021. "Synchronization of a Network Composed of Stochastic Hindmarsh–Rose Neurons" *Mathematics* 9, no. 20: 2625.
https://doi.org/10.3390/math9202625