# Asymptotic Behavior of Memristive Circuits

## Abstract

The interest in memristors has risen due to their possible application both as memory units and as computational devices in combination with CMOS. This is in part due to their nonlinear dynamics, and a strong dependence on the circuit topology. We provide evidence that also purely memristive circuits can be employed for computational purposes. In the present paper we show that a polynomial Lyapunov function in the memory parameters exists for the case of DC controlled memristors. Such a Lyapunov function can be asymptotically approximated with binary variables, and mapped to quadratic combinatorial optimization problems. This also shows a direct parallel between memristive circuits and the Hopfield-Little model. In the case of Erdos-Renyi random circuits, we show numerically that the distribution of the matrix elements of the projectors can be roughly approximated with a Gaussian distribution, and that it scales with the inverse square root of the number of elements. This provides an approximated but direct connection with the physics of disordered system and, in particular, of mean field spin glasses. Using this and the fact that the interaction is controlled by a projector operator on the loop space of the circuit. We estimate the number of stationary points ofmore »

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1557331

- Alternate Identifier(s):
- OSTI ID: 1558046

- Report Number(s):
- LA-UR-18-24748

Journal ID: ISSN 1099-4300; ENTRFG; PII: e21080789

- Grant/Contract Number:
- AC52-06NA25396; 89233218CNA000001

- Resource Type:
- Published Article

- Journal Name:
- Entropy

- Additional Journal Information:
- Journal Name: Entropy Journal Volume: 21 Journal Issue: 8; Journal ID: ISSN 1099-4300

- Publisher:
- MDPI

- Country of Publication:
- Switzerland

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; memristive circuits; spin models; disordered systems

### Citation Formats

```
Caravelli, Francesco. Asymptotic Behavior of Memristive Circuits. Switzerland: N. p., 2019.
Web. doi:10.3390/e21080789.
```

```
Caravelli, Francesco. Asymptotic Behavior of Memristive Circuits. Switzerland. doi:10.3390/e21080789.
```

```
Caravelli, Francesco. Thu .
"Asymptotic Behavior of Memristive Circuits". Switzerland. doi:10.3390/e21080789.
```

```
@article{osti_1557331,
```

title = {Asymptotic Behavior of Memristive Circuits},

author = {Caravelli, Francesco},

abstractNote = {The interest in memristors has risen due to their possible application both as memory units and as computational devices in combination with CMOS. This is in part due to their nonlinear dynamics, and a strong dependence on the circuit topology. We provide evidence that also purely memristive circuits can be employed for computational purposes. In the present paper we show that a polynomial Lyapunov function in the memory parameters exists for the case of DC controlled memristors. Such a Lyapunov function can be asymptotically approximated with binary variables, and mapped to quadratic combinatorial optimization problems. This also shows a direct parallel between memristive circuits and the Hopfield-Little model. In the case of Erdos-Renyi random circuits, we show numerically that the distribution of the matrix elements of the projectors can be roughly approximated with a Gaussian distribution, and that it scales with the inverse square root of the number of elements. This provides an approximated but direct connection with the physics of disordered system and, in particular, of mean field spin glasses. Using this and the fact that the interaction is controlled by a projector operator on the loop space of the circuit. We estimate the number of stationary points of the approximate Lyapunov function and provide a scaling formula as an upper bound in terms of the circuit topology only.},

doi = {10.3390/e21080789},

journal = {Entropy},

number = 8,

volume = 21,

place = {Switzerland},

year = {2019},

month = {8}

}

DOI: 10.3390/e21080789

#### Figures / Tables:

**a**) and the corresponding evolution of the Lyapunov function of Equation (A1) ((

**b**), continuous line) for 8750 memristors and for the asymptotic function of Equation (6) Figure ((

**b**)-dashed line). We considered α = 0.1, β = 1 and ξ =more »

Works referenced in this record:

##
The missing memristor found

journal, May 2008

- Strukov, Dmitri B.; Snider, Gregory S.; Stewart, Duncan R.
- Nature, Vol. 453, Issue 7191

##
Neural networks and physical systems with emergent collective computational abilities.

journal, April 1982

- Hopfield, J. J.
- Proceedings of the National Academy of Sciences, Vol. 79, Issue 8

##
Universal Memcomputing Machines

journal, November 2015

- Traversa, Fabio Lorenzo; Di Ventra, Massimiliano
- IEEE Transactions on Neural Networks and Learning Systems, Vol. 26, Issue 11

##
Complex dynamics of memristive circuits: Analytical results and universal slow relaxation

journal, February 2017

- Caravelli, F.; Traversa, F. L.; Di Ventra, M.
- Physical Review E, Vol. 95, Issue 2

##
Sensory and short-term memory formations observed in a Ag _{2} S gap-type atomic switch

journal, November 2011

- Ohno, Takeo; Hasegawa, Tsuyoshi; Nayak, Alpana
- Applied Physics Letters, Vol. 99, Issue 20

##
Solving mazes with memristors: A massively parallel approach

journal, October 2011

- Pershin, Yuriy V.; Di Ventra, Massimiliano
- Physical Review E, Vol. 84, Issue 4

##
Memristive devices and systems

journal, January 1976

- Chua, L. O.
- Proceedings of the IEEE, Vol. 64, Issue 2

##
Memristive networks: From graph theory to statistical physics

journal, January 2019

- Zegarac, A.; Caravelli, F.
- EPL (Europhysics Letters), Vol. 125, Issue 1

##
Emergent Criticality in Complex Turing B-Type Atomic Switch Networks

journal, October 2011

- Stieg, Adam Z.; Avizienis, Audrius V.; Sillin, Henry O.
- Advanced Materials, Vol. 24, Issue 2

##
The *mise en scéne* of memristive networks: effective memory, dynamics and learning

journal, February 2017

- Caravelli, Francesco
- International Journal of Parallel, Emergent and Distributed Systems, Vol. 33, Issue 4

##
Chaotic dynamics in nanoscale NbO2 Mott memristors for analogue computing

journal, August 2017

- Kumar, Suhas; Strachan, John Paul; Williams, R. Stanley
- Nature, Vol. 548, Issue 7667

##
A sequence of approximated solutions to the S-K model for spin glasses

journal, April 1980

- Parisi, G.
- Journal of Physics A: Mathematical and General, Vol. 13, Issue 4

##
Cycle bases in graphs characterization, algorithms, complexity, and applications

journal, November 2009

- Kavitha, Telikepalli; Liebchen, Christian; Mehlhorn, Kurt
- Computer Science Review, Vol. 3, Issue 4

##
Self-organization and solution of shortest-path optimization problems with memristive networks

journal, July 2013

- Pershin, Yuriy V.; Di Ventra, Massimiliano
- Physical Review E, Vol. 88, Issue 1

##
Heuristics for cardinality constrained portfolio optimisation

journal, November 2000

- Chang, T. -J.; Meade, N.; Beasley, J. E.
- Computers & Operations Research, Vol. 27, Issue 13

##
Ant colonies for the travelling salesman problem

journal, July 1997

- Dorigo, Marco; Gambardella, Luca Maria
- Biosystems, Vol. 43, Issue 2

##
Spin-glass models of neural networks

journal, August 1985

- Amit, Daniel J.; Gutfreund, Hanoch; Sompolinsky, H.
- Physical Review A, Vol. 32, Issue 2

##
Memory and Information Processing in Neuromorphic Systems

journal, August 2015

- Indiveri, Giacomo; Liu, Shih-Chii
- Proceedings of the IEEE, Vol. 103, Issue 8

##
Memristors with diffusive dynamics as synaptic emulators for neuromorphic computing

journal, September 2016

- Wang, Zhongrui; Joshi, Saumil; Savel’ev, Sergey E.
- Nature Materials, Vol. 16, Issue 1

##
Trajectories Entropy in Dynamical Graphs with Memory

journal, April 2016

- Caravelli, Francesco
- Frontiers in Robotics and AI, Vol. 3

##
Associative memory realized by a reconfigurable memristive Hopfield neural network

journal, June 2015

- Hu, S. G.; Liu, Y.; Liu, Z.
- Nature Communications, Vol. 6, Issue 1

##
Memristors for the Curious Outsiders

journal, December 2018

- Caravelli, Francesco; Carbajal, Juan
- Technologies, Vol. 6, Issue 4

##
Optimization by Simulated Annealing

journal, May 1983

- Kirkpatrick, S.; Gelatt, C. D.; Vecchi, M. P.
- Science, Vol. 220, Issue 4598

##
Scale-free networks as an epiphenomenon of memory

journal, January 2015

- Caravelli, F.; Hamma, A.; Di Ventra, M.
- EPL (Europhysics Letters), Vol. 109, Issue 2

##
Neuromorphic Atomic Switch Networks

journal, August 2012

- Avizienis, Audrius V.; Sillin, Henry O.; Martin-Olmos, Cristina
- PLoS ONE, Vol. 7, Issue 8

##
Memristive devices for computing

journal, January 2013

- Yang, J. Joshua; Strukov, Dmitri B.; Stewart, Duncan R.
- Nature Nanotechnology, Vol. 8, Issue 1, p. 13-24

##
Memcomputing *NP* -complete problems in polynomial time using polynomial resources and collective states

journal, July 2015

- Traversa, Fabio Lorenzo; Ramella, Chiara; Bonani, Fabrizio
- Science Advances, Vol. 1, Issue 6

##
The existence of persistent states in the brain

journal, February 1974

- Little, W. A.
- Mathematical Biosciences, Vol. 19, Issue 1-2

##
Locality of interactions for planar memristive circuits

journal, November 2017

- Caravelli, F.
- Physical Review E, Vol. 96, Issue 5

##
Computing with neural circuits: a model

journal, August 1986

- Hopfield, J.; Tank, D.
- Science, Vol. 233, Issue 4764

Figures / Tables found in this record:

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*