# A New Memristive Neuron Map Model and Its Network’s Dynamics under Electrochemical Coupling

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## Abstract

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## 1. Introduction

## 2. Memristive Neuron Map Model

## 3. Network’s Dynamics

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The phase diagram in the x-$\phi $ plane (left panel) and the time series of $x$ (navy blue) and $\phi $ (orange) variables of the memristive map model (right panel) for different $\mathsf{\mu}$ values. (

**a**,

**b**) Spiking behavior for $\mu =0.1$. (

**c**,

**d**) Periodic bursting for $\mu =0.25$. (

**e**,

**f**) Chaotic bursting for $\mu =0.225$. Other parameters are ${k}_{1}=0.03$, ${k}_{2}=0.15$, ${k}_{3}={k}_{4}=0.00001$, $I=1$, $v{r}_{1}=-55$, $v{r}_{2}=-3$, $v{c}_{1}=-59$, $v{c}_{2}=-3$, ${v}_{t{h}_{1}}=-30$, ${v}_{t{h}_{2}}=-20$, ${v}_{rest}=-75$, ${v}_{s}=0$, $\theta =-40$, $r=0.95,s$ and $\epsilon =0.2$. The used initial condition is $\left(x\left(1\right),\phi \left(1\right)\right)=\left(0.1,-0.1\right)$.

**Figure 2.**(

**a**) The bifurcation diagram (navy blue and orange colors refer to the $x$ and $\phi $ variables) and (

**b**) the Lyapunov exponents’ diagram of the memristive map model by the variation of $\mu $ with considering ${k}_{1}=0.03$, ${k}_{2}=0.15$, ${k}_{3}={k}_{4}=0.00001$, $I=1$, $v{r}_{1}=-55$, $v{r}_{2}=-3$, $v{c}_{1}=-59$, $v{c}_{2}=-3$, ${v}_{t{h}_{1}}=-30$, ${v}_{t{h}_{2}}=-20$, ${v}_{rest}=-75$, ${v}_{s}=0$, $\theta =-40$, $r=0.95$, and $\epsilon =0.2$. The initial condition is $\left(x\left(1\right),\phi \left(1\right)\right)=\left(0,0\right)$. Periodic behavior is the most noticeable behavior of the neuron; however, for $\mu \in \left[0.1836,0.1862\right]{{\displaystyle \cup}}^{\text{}}\left[0.1884,0.1901\right]{{\displaystyle \cup}}^{\text{}}\left[0.208,0.2117\right]{{\displaystyle \cup}}^{\text{}}\left[0.2172,0.2339\right]{{\displaystyle \cup}}^{\text{}}\left[0.2393,2437\right]$, the periodic bursting can also be observed.

**Figure 3.**(

**a**) The bifurcation diagram (navy blue and orange colors refer to the $x$ and $\phi $ variables) and (

**b**) the Lyapunov exponents’ diagram of the memristive map model by the variation of $r$ with considering ${k}_{1}=0.03$, ${k}_{2}=0.15$, ${k}_{3}={k}_{4}=0.00001$, $I=1$, $v{r}_{1}=-55$, $v{r}_{2}=-3$, $v{c}_{1}=-59$, $v{c}_{2}=-3$, ${v}_{t{h}_{1}}=-30$, ${v}_{t{h}_{2}}=-20$, ${v}_{rest}=-75$, ${v}_{s}=0$, $\theta =-40$, $\mu =0.225$, and $\epsilon =0.2$. The initial condition is $\left(x\left(1\right),\phi \left(1\right)\right)=\left(0,0\right)$. The neuron has periodic behavior for $0\le r<0.3783$; however, by increasing the $r$ parameter, the most significant behavior of the neuron is chaotic bursting.

**Figure 4.**Synchronization error of the network of 100 memristive neuron maps connected through (

**a**) chemical and (

**b**) electrical synapses by the variation of coupling strengths. The network parameters are ${\Theta}_{s}=-40$, $\beta =50$, and ${v}_{s}^{*}=-40$. Complete synchronization can be observed in chemically coupled neurons for $0.0427\le {g}_{c}\le 0.0463$.

**Figure 5.**Synchronization error of the ring network of 100 memristive maps connected through both chemical and electrical synaptic couplings by the variation of coupling strengths. The network parameters are ${\Theta}_{s}=-40$, $\beta =50$, and ${v}_{s}^{*}=-40$. Complete synchronization can be observed for a small range of chemical coupling strength.

**Figure 6.**The spatiotemporal patterns (left panel) and the time series and snapshots (right panel) of the network of 100 memristive neuron maps coupled via chemical synapses. (

**a**,

**b**) For ${g}_{c}=0.04$, imperfect synchronization is observed. (

**c**,

**d**) For ${g}_{c}=0.044$, complete synchronization is achieved. (

**e**,

**f**) For ${g}_{c}=0.06$, the solitary state appears.

**Figure 7.**The spatiotemporal patterns (left panel) and the time series and snapshots (right panel) of the network of 100 memristive neuron maps coupled via electrochemical synapses. (

**a**,

**b**) For ${g}_{c}=0.07$ and $\u03f5=0.01$, imperfect synchronization is formed. (

**c**,

**d**) For ${g}_{c}=0.05$ and $\u03f5=0.05$, the chimera state is observed. (

**e**,

**f**) For ${g}_{c}=0.52$ and $\u03f5=0.11$, the nonstationary chimera appears.

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**MDPI and ACS Style**

Ramakrishnan, B.; Mehrabbeik, M.; Parastesh, F.; Rajagopal, K.; Jafari, S.
A New Memristive Neuron Map Model and Its Network’s Dynamics under Electrochemical Coupling. *Electronics* **2022**, *11*, 153.
https://doi.org/10.3390/electronics11010153

**AMA Style**

Ramakrishnan B, Mehrabbeik M, Parastesh F, Rajagopal K, Jafari S.
A New Memristive Neuron Map Model and Its Network’s Dynamics under Electrochemical Coupling. *Electronics*. 2022; 11(1):153.
https://doi.org/10.3390/electronics11010153

**Chicago/Turabian Style**

Ramakrishnan, Balamurali, Mahtab Mehrabbeik, Fatemeh Parastesh, Karthikeyan Rajagopal, and Sajad Jafari.
2022. "A New Memristive Neuron Map Model and Its Network’s Dynamics under Electrochemical Coupling" *Electronics* 11, no. 1: 153.
https://doi.org/10.3390/electronics11010153