## A mean-field model of memristive circuit interaction

## Abstract

We construct an exactly solvable circuit of interacting memristors and study its dynamics and fixed points. This simple circuit model interpolates between decoupled circuits of isolated memristors, and memristors in series, for which exact fixed points can be obtained. Here, we introduce a Lyapunov functional that is found to be minimized along the non-equilibrium dynamics and which resembles a long-range Ising Hamiltonian with non-linear self-interactions. We use the Lyapunov functional as a Hamiltonian to calculate, in the mean-field theory approximation, the average asymptotic behavior of the circuit given a random initialization, yielding exact predictions for the case of decay to the lower resistance state, and reasonable predictions for the case of a decay to the higher resistance state.

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Univ. College London, London (United Kingdom). Dept. of Computer Science; London Inst. for Mathematical Sciences, London (United Kingdom)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1481128

- Report Number(s):
- LA-UR-17-23729

Journal ID: ISSN 1286-4854

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Europhysics Letters (Online)

- Additional Journal Information:
- Journal Name: Europhysics Letters (Online); Journal Volume: 122; Journal Issue: 4; Journal ID: ISSN 1286-4854

- Publisher:
- IOP Publishing

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; memristors

### Citation Formats

```
Caravelli, Francesco, and Barucca, Paolo. A mean-field model of memristive circuit interaction. United States: N. p., 2018.
Web. doi:10.1209/0295-5075/122/40008.
```

```
Caravelli, Francesco, & Barucca, Paolo. A mean-field model of memristive circuit interaction. United States. doi:10.1209/0295-5075/122/40008.
```

```
Caravelli, Francesco, and Barucca, Paolo. Wed .
"A mean-field model of memristive circuit interaction". United States. doi:10.1209/0295-5075/122/40008. https://www.osti.gov/servlets/purl/1481128.
```

```
@article{osti_1481128,
```

title = {A mean-field model of memristive circuit interaction},

author = {Caravelli, Francesco and Barucca, Paolo},

abstractNote = {We construct an exactly solvable circuit of interacting memristors and study its dynamics and fixed points. This simple circuit model interpolates between decoupled circuits of isolated memristors, and memristors in series, for which exact fixed points can be obtained. Here, we introduce a Lyapunov functional that is found to be minimized along the non-equilibrium dynamics and which resembles a long-range Ising Hamiltonian with non-linear self-interactions. We use the Lyapunov functional as a Hamiltonian to calculate, in the mean-field theory approximation, the average asymptotic behavior of the circuit given a random initialization, yielding exact predictions for the case of decay to the lower resistance state, and reasonable predictions for the case of a decay to the higher resistance state.},

doi = {10.1209/0295-5075/122/40008},

journal = {Europhysics Letters (Online)},

number = 4,

volume = 122,

place = {United States},

year = {2018},

month = {7}

}

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