Networks of memristors and the effective memristor
Anne-Men Huijzer1 , Arjan van der Schaft1 , Bart Besselink1
1. Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence and CogniGron - Groningen
Cognitive Systems and Materials Center, University of Groningen, Groningen, The Netherlands.

Summary. Memristors, originally introduced by L.O. Chua in 1971, are resistors with a memorystorage. They are characterized as a relation between electrical charge and magnetic flux. In thiswork, we show that a network of memristors attached to an external source can be described bya single memristor. Moreover, we will see that this memristor can be characterized based on thedynamics of the individual memristors in the network and their orientation. This also enables thedesign of networks with desired port behavior.

1Introduction

The emerging field of neuromorphic computing aims to reduce the energy requirements of computing
platforms. It draws inspiration from the functioning of the brain and wants to learn from its energy
efficiency [2]. For these new neuromorphic technologies dedicated hardware needs to be developed. It issuggested that memristors will play an important role within this new hardware. Memristors, originallyintroduced by Chua in [1], are resistors with a memory storage that can act as non-volatile memory. Inaddition, because of their dynamical nature, memristors have high potential of mimicking the behaviorof certain elements of the brain.
In this work, we consider networks of memristors attached to an external source, and we show thatsuch networks can be replaced by a so-called effective memristor. In addition, this effective memristor canbe characterized based on the dynamics and orientation of individual memristors in the network. This
result provides a stepping stone for the analysis of neuromorphic computing systems with memristors.

2Main results

We will consider memristor networks that are attached to an external source through a port, see Figure 1
for an example. We denote the current through and the voltage across the port by iP (t) and vP (t),respectively. In addition, we define qP (t) and φP (t) as the charge and flux at the port which satisfy
ddt qP (t) = iP (t),ddt φP (t) = vP (t).

In this work, we want to show that the relation between qP (t) and φP (t) can equivalently be describedby a single effective memristor.
iP (t)
pk (t)

im (t)pℓ (t)

Figure 1: Left, a depiction of a memristor between a node k an ℓ. The voltage through this device isgiven by vm (t) = pk (t) − pℓ (t). Here, pk (t) and pℓ (t) denote the voltage potentials at node k and ℓ. Right,
a network of memristors attached to a source applying an input iP (t) to the circuit.
Before we can do so, we provide the description of a single device. A memristor on a branch m in anelectrical circuit can be depicted as in Figure 1. Here, we let im (t) ∈ R and vm (t) ∈ R denote the current
through and voltage across the device, respectively. The device can be characterized as a relationshipbetween charge and flux, see [1]. Therefore, we define qm (t) ∈ R and φm (t) ∈ R as the charge and flux
at branch m, which satisfy
ddt qm (t) = im (t),ddt φm (t) = vm (t).

(1)

Then, a flux-controlled memristor is modeled as φm (t) = hm (qm (t)) for some (strongly) monotone function hm : R → R. We observe that, using (1), its behavior can be modeled as the dynamical systemvm (t) =dhm (qm (t))
im (t),dqmddt qm (t) = im (t).

(2)m (qm )represents the memristance function of the device. It has the unit Ohm and its value varies
Here, dhdqm
depending on the history of the current im applied to the device. A typical example of a memristancefunction is depicted in Figure 2.
Now, we are able to derive the following result.

Theorem 1 Consider a connected memristor network attached to an external source through a port. If
all memristors in the network are described by (2) for some (strongly) monotone function hm : R → R,then there exists a (strongly) monotone functions hM : R → R such thatdhM (qM (t))
diM (t),dt qM (t) = iM (t).
dqM
Theorem 1 shows that there exists a relation between the charge and flux at the port of a network ofmemristors. We interpreted this results as that a network of memristors can be replaced by a so-called
effective memristor. The behavior of this effective memristor is described by the function hM . The
observation that the port behavior of a network of memristors can be described as a single memristor was
already made in [1, Theorem 2]. However, Theorem 1 provides an explicit description of this relation.
This enables us to provide simulations of the behavior of the effective memristor of a network, for example
in Figure 2 we show the memristance curve of the effective memristor of the network depicted in Figure 1.
MemristancevM (t) =3Single memristor
Effective memristor

21−2−1012qm (qm )Figure 2: Depiction of a memristance function dhdq
for a single device, and that of the correspondingmeffective memristor of the network in Figure 1.

3Discussion

The above result shows that the memristive behavior of a network of memristors can be described by asingle effective memristor. This result is of interest when investigating materials that host memristors.
These materials often include large networks of memristors instead of single devices. The small scale ofthese materials makes it complicated to measure the behavior of a single memristor while it is possible
to measure the behavior of the network between certain points, e.g. by connecting these points to an
external source. In addition to the analyses of the port behavior of networks of memristors our result
can be used in the synthesis of networks of memristors in order to achieve desired port behavior.
These results can be a starting point for the analysis of network including both memristors andcapacitors, which are of interest since materials often not only include memristors but also other elementssuch as capacitors.

References[1] L. Chua. Memristor-the missing circuit element. IEEE Transactions on Circuit Theory, 18(5):507–519,1971.
[2] C. D. James, J. B. Aimone, N. E. Miner, C. M. Vineyard, F. H. Rothganger, K. D. Carlson, S. A.
Mulder, T. J. Draelos, A. Faust, and M. J. Marinella. A historical survey of algorithms and hardware architectures for neural-inspired and neuromorphic computing applications. Biologically Inspired
Cognitive Architectures, 19:49–64, 2017.