# A Rectified Linear Unit-Based Memristor-Enhanced Morris–Lecar Neuron Model

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## Abstract

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## 1. Introduction

## 2. Model

^{++}and K

^{+}channels at steady state, and ${\tau}_{W}$ denotes the opening rate constant of K

^{+}channels, which are defined as follows:

^{+}current. The maximal conductances associated with the three transmembranar currents are shown by ${g}_{Ca}$, ${g}_{K}$, and ${g}_{L}$, and the steady-state Nernst potentials of calcium ions, potassium ions, and leak channels are shown by ${V}_{Ca}$, ${V}_{K}$, and ${V}_{L}$. The input current is denoted by ${I}_{0}$, and $C$ is the capacitance of the membrane. The parameters of the model can be set to $C=1,{V}_{Ca}=1,{V}_{L}=-0.5,{g}_{Ca}=1.2,{g}_{K}=2,{g}_{L}=0.5,{V}_{1}=-0.01,{V}_{2}=0.15,{V}_{3}=0.1,\mathrm{a}\mathrm{n}\mathrm{d}{V}_{4}=0.05,\sigma =3.$ Note that ${V}_{K}$ and ${I}_{0}$ are chosen as the varying parameters for creating abundant firing patterns.

## 3. The Magnetic Induction Effect on the ML Model Dynamics

## 4. The Magnetic Induction Effect on the Synchronization

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The bifurcation diagram of the ReLU-based memristor ML model for ${I}_{0}=0$, according to ${V}_{K}$. The corresponding MLE diagram is shown below the bifurcation diagram. (

**a**) $k=0.5$. (

**b**) $k=3$. (

**c**) $k=5$. (

**d**) $k=8$.

**Figure 2.**The bifurcation diagram of the ReLU-based memristor ML model for ${V}_{k}=-220$, according to ${I}_{0}$. The corresponding MLE diagram is shown below the bifurcation diagram. (

**a**) $k=2$. (

**b**) $k=5$.

**Figure 3.**The bifurcation diagram of the ReLU-based memristor ML model for ${V}_{k}=-220$, according to $k$. The corresponding MLE diagram is shown below the bifurcation diagram. (

**a**) ${I}_{0}=0$. (

**b**) ${I}_{0}=0.01$.

**Figure 4.**The phase space of the ReLU-based memristor ML model in different parameters of $k$ and ${V}_{K}$ and ${I}_{0}=0$. The time series of each attractor is shown in its subset. (

**a**) $k=0.5,{V}_{K}=-400.$ (

**b**) $k=0.5,{V}_{K}=-300$. (

**c**) $k=0.5,{V}_{K}=-220$. (

**d**) $k=0.5,{V}_{K}=-200.$ (

**e**) $k=3,{V}_{K}=-200$. (

**f**) $k=3,{V}_{K}=-220$. (

**g**) $k=5,{V}_{K}=-300.$ (

**h**) $k=5,{V}_{K}=-220$.

**Figure 5.**Dynamical map of the ReLU-based memristor ML model in two-dimensional planes of $({V}_{K},k)$ in parts (

**a**,

**b**) and $({I}_{0},k)$ in parts (

**c**,

**d**). The period of oscillation is shown in the left column and the maximum Lyapunov exponent is shown in the right column, where the yellow color shows the chaotic region.

**Figure 6.**Multistability emerges in the ReLU-based memristor ML model. (

**a**) The parameters are $k=0.6$, ${V}_{K}=-220$, and ${I}_{0}=0$ and the initial conditions of the orange and red firings are $[-0.16,0,-187.11]$ and $\left[-0.19,0,2.97\right]$, respectively. (

**b**) The parameters are $k=6.5$, ${V}_{K}=-220$, and ${I}_{0}=0$ and the initial conditions of the orange and red firings are $[-0.22,0,4.8]$ and $[-0.24,0,4.78]$, respectively.

**Figure 7.**(

**a**) The synchronization error ($E$) of the network of memristive ML neurons (Equation (5)) according to the coupling strength ($\u03f5$) and the magnetic induction strength ($k$). (

**b**) One-dimensional synchronization error according to the coupling strength ($\u03f5$) for $k=3$, $5,$ and $8$.

**Figure 8.**Temporal evolution of the network of memristive ML neurons (Equation (5)). (

**a**) For $k=8$ and $\u03f5=1.4$, a traveling chimera is formed. (

**b**) For $k=4$ and $\u03f5=0.5$, a traveling chimera is formed. (

**c**) For $k=1$ and $\u03f5=0.5$, an imperfect traveling chimera is formed. (

**d**) For $k=1$ and $\u03f5=1$, a non-stationary chimera is formed.

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**MDPI and ACS Style**

Almatroud, O.A.; Pham, V.-T.; Rajagopal, K.
A Rectified Linear Unit-Based Memristor-Enhanced Morris–Lecar Neuron Model. *Mathematics* **2024**, *12*, 2970.
https://doi.org/10.3390/math12192970

**AMA Style**

Almatroud OA, Pham V-T, Rajagopal K.
A Rectified Linear Unit-Based Memristor-Enhanced Morris–Lecar Neuron Model. *Mathematics*. 2024; 12(19):2970.
https://doi.org/10.3390/math12192970

**Chicago/Turabian Style**

Almatroud, Othman Abdullah, Viet-Thanh Pham, and Karthikeyan Rajagopal.
2024. "A Rectified Linear Unit-Based Memristor-Enhanced Morris–Lecar Neuron Model" *Mathematics* 12, no. 19: 2970.
https://doi.org/10.3390/math12192970