BioNanoScience

, Volume 5, Issue 1, pp 1–8 | Cite as

Slime Mould Memristors

  • Ella Gale
  • Andrew Adamatzky
  • Ben de Lacy Costello
Article

Abstract

In laboratory experiments, we demonstrate that protoplasmic tubes of the acellular slime mould Physarum polycephalum show current versus voltage profiles consistent with memristive systems, and that this effect is due to the living protoplasm of the mould. This complements previous findings on memristive properties of other living systems (human skin and blood) and contributes to the development of self-growing bio-electronic circuits. Distinctive asymmetric VI curves which were occasionally observed when the internal current is on the same order as the driven current, are well-modelled by the concept of active memristors.

Keywords

Memristor Slime mould Bioelectronics Active memristor Physarum 

1 Introduction

Memristors [8] have revolutionised the material basis of computation [29, 30] and neuromorphic architectures [17, 19, 24, 27]. Since the announcement of the first documented two-terminal memristor [29], researchers have been eager to experiment with memristors, but they are difficult to synthesize and are not yet commercially available. Few looked beyond standard electronic engineering approaches, but those who did uncovered a promising behaviour of living systems. Johnsen et al [20] found that conductance properties of sweat ducts in human skin are well approximated by a memristive model [20], and experimental evidence that flowing blood [21] and leaves [22] exhibits memristive properties was provided by Kosta et al. In 2008, Pershin et al. [25] described an adaptive ‘learning’ behaviour [31] of slime mould Physarum polycephalum in terms of a memristor model. Memristor theory has been applied to neurons [9, 10] and synapses [35], suggesting that memristance might be useful to explain the process of learning.

The plasmodium of P. polycephalum (order Physarales, class Myxomecetes, subclass Myxogastromycetidae) is a single cell, visible with an unaided eye. The plasmodium behaves and moves as a giant amoeba. It feeds on bacteria, other microbial creatures, spores and micro-particles [28]. Structurally, Physarum is composed of a semi-rigid gel put down by the living protoplasm, a type of cytosol, within it, and this gel is covered with a protective ‘slime’ which gives rise to the plasmodium’s colloquial name ‘slime mould’. This protoplasm contains many nuclei which can be described as interacting oscillators. Furthermore, the protoplasm undergoes shuttle transport, switching direction approximately every 50 s. Thus, when measuring electrical properties of Physarum, we must take care to separate the effects of the living protoplasm, and non-living exterior gel and slime layer, as well as being aware that the cytosol is a moving and living system which will change over time.

In an environment with distributed sources of nutrients, the plasmodium forms a network of protoplasmic tubes connecting food sources. The network of protoplasmic tubes developed by Physarum shows some signs of optimality in terms of shortest path [23] and proximity graphs [1]. In [2], we have used Physarum to make prototypes of massively parallel amorphous computers—Physarum machines—capable of solving problems of computational geometry, graph theory and logic. Physarum machines implement morphological computation: given a problem represented by a spatial configuration of attractants and repellents the Physarum gives a solution by patterns of its protoplasmic network. This limits the application domain of Physarum processors to computational tasks with natural parallelism. If we were able to make electronic devices from living Physarum, we would be able to construct a full spectrum of computing devices with a conventional architecture. Furthermore, to make a set of ‘all possible computing devices’, it is enough to make a material implication gate and memristors naturally implement material implication [7, 12]. Thus answering the question ‘Is Physarum a memristor?’ is a task of upmost priority.

Physarum’s growth can be directed with chemo-attractants and repellants [11] and it chooses efficient paths between food sources. Thus, Physarum could be used to ‘design’ efficient circuits. Previous preliminary work [13, 27] has shown that Physarum can take up iron-based magnetic particles, so it could be used to lay down efficient circuits and, if the magnetic effects were detrimental to the Physarum, we might expect it to lay down circuits with a good electromagnetic profile; thus, we also investigated the electrical properties of the Physarum tubes with and without these particles.

2 Methods

Plasmodium of P. polycephalum were cultivated on wet absorbent paper (to keep the humidity level high) in an aerated, dark environment and fed with oat flakes. The culture was periodically replanted to a fresh substrate.

The experimental set-up is shown in Fig. 1. Two electrodes (Fig. 1c) were stuck to a plastic Petri dish ≈ 10 mm apart and two islands of 2 ml agar (Fig. 1b) were placed on each electrode. To perform the experiments, a Physarum-colonised oat flake was inoculated on one island with a fresh oat flake on the other; Physarum would then colonise the other island (Fig. 1a), linking both electrodes with a single protoplasmic tube (Fig. 1d). This experimental set-up has proved to be efficient in uncovering patterns of electrical activity of Physarum [6] and Physarum’s response to chemical, optical and tactile stimulation [3, 4, 5, 34].
Fig. 1

A scheme of experimental setup: aPhysarum. b Agar islands. c Electrodes. d Protoplasmic tube. All parts of Physarum shown in dark grey form a single cell

Electrical measurements were performed with a Keithley 617 programmable electrometer which allows the measurement of currents from pA–3.5 mA. Measurements were performed with a voltage range of ±50 mV (samples 1–13) or ±100 mV (samples 15–22), a triangular voltage waveform and a measurement rate of 0.5, 1 or 2 s; this is the D.C. equivalent to an A.C. voltage frequency of 2, 1 or 0.5 mHz. As Physarum is a living system and can respond, we compared the first and second runs across different samples. The tests were divided into two batches: batch 1 was measured with timesteps of 0.5, 1 or 2 s and a voltage range of ±50 and ±100 mV; batch 2 was measured only with the 2-s time step and a voltage range of ±100 to ±250 mV.

Three different electrode set-ups were tested: thick (2 mm) aluminium wire, thin (0.5 mm) silver wire and thin aluminium mesh. Physarum was also tested for the electrical effect of the uptake of magnetic particles (fluidMAG-D,100 nm, 25m g/ml, Chemicell) by inoculating the source or target oat flakes with particles. As Physarum is sensitive to light, all tests were run in the dark.

The starting resistance, R0, the hysteresis H (calculated as in [16]), and scaled hysteresis, \(\bar {H}\), (calculated as a ratio of R0 as in [16]) were calculated for the Physarum that exhibited hysteresis. To control for experimental set-up variation, R0, H and \(\bar {H}\) values were compared to the length of the tube L, and the electrode separation. The tube lengths were measured from photographs of the first batch of measurements. For completeness, the calculation of hysteresis was done in the following manner. The memristor plot is split into four branches: (1) 0 < V < +Vmax, (2) +Vmax < V < 0, (3) 0 < V < −Vmax and (4) −Vmax < V < 0. The hysteresis is calculated from the work done by the device as the difference between the upper (2 and 4) and lower (1 and 3) branches in the integration of the instantaneous power consumed by the device over the course of an input cycle. We use a discretised form: A the work is
$$ W_{A} = \sum\limits_{n=1}^{n=1+N/4} I(n) V(n) {\Delta} t, $$
(1)
where I(n) and V(n) are the current and voltage values at that data point, and Δt is the time between measurements and N is the total number of measured points. The measured hysteresis is the difference between the work for the upper and lower branches of the I–V curve, H = (W2 + W3) − (W1 + W4). The scaled hysteresis is \(\bar {H} = \frac {H}{R_{0}}\) .

3 Results

Twenty-three samples were tested; in one sample, Physarum did not form a tube across the electrodes and in another, the Physarum grew, formed and abandoned the tube before it could be measured; this sample was used as a control for the gel part of the tube. The remaining 21 samples were measured and these comprised of two samples with magnetic nanoparticles on the inoculation electrode, two samples with magnetic nanoparticles on the target electrode and the rest were normal Physarum. Three samples were electrically unconnected.

3.1 Memristive Effects

Of the 11 samples in batch 1, 2 exhibited good memristance curves (as in they had pinched hysteresis), the other 9 exhibited open curves as shown in Figs. 2 and 3 (these open curves are very similar to those seen in low-voltage TiO2 sol-gel memristor measurements which are indicative of memristive results at high voltage measurements [15]). For batch 2, which were measured at a larger voltage range (between ±100 and ±250 mV), eight out of eight samples showed good memristive curves as shown in Fig. 2 (note that these four results from two different samples: a and b are the different runs on sample 18 and c and d are different runs on sample 19; the variance in response is due to the fact that the Physarum is alive and continuously changing).
Fig. 2

Typical current versus voltage profiles recorded in laboratory experiments with slime mould P. polycephalum

Fig. 3

Open-curves measured under different frequencies: Δt = 1 s corresponds to a frequency of 1 mHz; Δt = 2 s corresponds to a frequency of 0.5 mHz

Comparison between the memristance curves shown in Fig. 2 and the results for an abandoned tube shows that the memristance is due to the living Physarum protoplasm. Similar to inorganic memristors, this could be due to voltage-driven charge transport. Shuttle transport reverses direction around every 50 s and this could give rise to a measurable hysteresis; however, because these data were measured at Δt ≈ 2 s for 160 steps, the period is 320 s which is over three times the period of the shuttle transport and, as such, is not the cause of the measured effect. Longer time-scale current responses have been observed in D.C. experiments [14] which could be related to the memristive effect. Finally, it could be due to a voltage-mediated change in material properties of the protoplasm which has a relaxation time, leading to a memristive hysteresis—this seems to be the mostly likely explanation because repeated applications of voltage increased the resistance (see Fig. 3).

No electrical effect of the different electrode types was seen beyond the mechanical difficulties: the thicker aluminium electrodes put strain on the tube when measured leading to breakages or short circuits. No discernible electrical effect was seen from the presence of magnetic nanoparticles. The Physarum picked them up and internalised them, but the measured resistance curves were qualitatively the same as those without the particles and within the same range. Our other work [26] shows that the nanoparticles are localised within the gel part of the Physarum so this lack of effect is not because the nanoparticles are not present. One possibility could be that the Physarum might internalise or biofoul the nanoparticles (biofouling is when a biological entity grows on or over a non-biological component inserted into it).

This effect whereby open loops at low voltages show memristance at high voltages suggests that, like TiO2 sol-gel memristors, the low-voltage open-loop behaviour is related to memristance. These results show that relatively large voltages are needed for the measurement of memristance in Physarum.

3.2 Repeatability and the Effect of Frequency

The testing process may have resulted in behavioural modification of the Physarum; successive applications of voltage caused the Physarum to abandon the tube as observed by a thinning and lightening of the tube. However, Physarum was still alive and active after testing, and the tube abandonment could be due to it exploring the environment for other food sources (the time spent connecting the two oat flakes was commensurate with that observed without electrical testing, but due to the high variance in behaviour, we cannot say if the applied voltage harmed the Physarum).

As frequency can effect the size of memristor hysteresis (due to natural response speed of the system), the voltage waveform frequency was altered. As increasing the voltage range can turn the ‘open loop’ type of memristor into a pinched hysteresis loop, the lack of pinched hysteresis on the open-loop memristor responses could be due to the chosen voltage range. The effect of frequency is tested in Fig. 3, where three repeats of the same ±V range is tried at the standard and half the frequency and one larger ±V range is tried at the standard frequency. The shape is qualitatively similar over this voltage range and unaffected by frequency over this voltage range. Figure 3 shows that repeated applications of voltage causes the resistance of the tube to increase (similar results were observed on the two repeats with other samples).

As Fig. 4 shows no repeated resistance change for an empty tube, this suggests that the protoplasm part of Physarum is the material responsible for the observed memristance rather than a chemical or physical change in the structure of the outer parts of the tube.
Fig. 4

Repeated I–V runs performed on an abandoned tube. No memristance or increasing resistance effect is seen

3.3 Study of Memristor Properties

3.3.1 The Effect of Electrode Separation and Tube Length

The range of tube lengths found was 6.25 to 43 mm, with a mean of 19.71 mm and standard deviation of 10.64 mm. This high deviation is because the Physarum initially explores the space following a chemical gradient to connect the oat flakes. The tube length shortens over time as the Physarum increases the efficiency of connection between food sources; for example, the length of one protoplasmic tube went from 5 to 4.26 mm over a day. No correlation was found between R0 and L, \(\bar {H}\) and L, H and L, H and R0 and R0 and electrode separation (graphs not shown), this demonstrates that the tube length (L) or electrode separation cannot be used to control the electrical properties and that the starting resistance (R0) is not a predictor for the hysteresis (H and \(\bar {H}\)): as a comparison, results for sol-gel memristors are given in [13]. The total power used over an IV loop and the average power used were also calculated for 11 samples; no correlation was found between the power and R0, or the power and electrode separation (graphs not shown). Thus, we can conclude that the variation in electrical response is due to the variation between individual samples of Physarum and not the variation in set up or tube length.
Fig. 5

Instantaneous resistance-time plots. a With tangents, b As a closed loop in resistance-time space

4 Theoretical Analysis of the Slime Mould as an Active Memristor

The shape of the curve in Fig. 2d is interesting, as it shows asymmetry between in the resistance change rate as is shown in the resistance-time curves plotted in Fig. 5, and has not been observed in our inorganicmemristors. Thememory conservation theory of memristance [13] explains memristive effects in terms of an interaction between state-carrying ions, q, and conduction (state-sampling) electrons e. To undergo locomotion, Physarum exhibits shuttle transport where the protoplasm is moved backwards and forwards; ions in the protoplasm would be moved around by this motion and this could give rise to a background current. Thus, this background current should be included and we investigated this in order to try and understand the shape of the distinctive Physarum memristor (see Fig. 2d). A similar approach has been used to model ReRAM, where the electromotive force associated with a ‘nanobattery’ is added to a memristor circuit [33].

Figure 6 shows a circuit which could be used to model the situation: this circuit contains a 2-port ‘black box’ which we measure. We assume that this 2-port contains a current source, which we are symbolising as a battery because Physarum is alive and uses chemical energy to produce reactions and the motion of the membrane, and a memristor (or a memristor-resistor in series). In fact, from long-term experiments, we have seen a slow oscillation with a half-period of around 700 s [14] that could fit the description of such a current source, especially as it was observed at ≈ 2 × 10−9 A, so it is on the same order as our lower current VI curve measurements. With the addition of an internal current source, we are now modelling the Physarum as an active memristor [18] (standard memristors are passive components).
Fig. 6

The equivalent circuit for the Physarum memristor, the shaded area represents the middle active memristor 2-port

The current, iq is the background current from the living Physarum. From the circuit in Fig. 6, we can write the following expression for the measured current, Itot:
$$ I_{\text{tot}} = i_{d} \pm i_{q} \, $$
(2)
where id is the current that is driven by the external voltage, V. The background battery can either add to or oppose the external power source, and thus, the background current is either in the same or opposite direction to the driven current, as we use + and − to represent this, where it is understood that the internal ions may not have the same charge as the electrons and we take +iq to be the direction of increase in total current. Vq is the voltage associated with the internal ‘battery’.

If we have a current that adds to our driven current at one point in time and subtracts at another, we would expect to see a non-rotationally symmetric memristor curve.

From Eq. 2 , we can write the memristive response, R, as:
$$ R = M \pm \frac{V_{q}}{i_{q}} \;, $$
(3)
where M is the memristance due to the motion of ions under the applied voltage—that expected from the memory-conservation theory of memristance—and the second term is the internal resistance response due to the background current, which we label as Mint.

We don’t know what the form of iq is, but there are two options: we can model it as a sine wave with a period of roughly 700 s or we can model it as a bipolar piece-wise linear (BPWL) waveform, which corresponds more to what is observed down a microscope when watching Physarum shuttle transport. As the IV curves took a total of ≈ 350 s to run, we can model the current as being constant over this period, especially if it is the long-time oscillations observed in [14] and thus additive to the memristance current in one direction and subtractive in the other.

We can discretise Eq. 3, to get an expression for the discretised rate of change of memristance, \(\frac {\Delta M}{\Delta t}\) as:
$$ \frac{\Delta R}{\Delta t} (+) = \frac{\Delta M}{\Delta t} + \frac{\Delta M_{\text{int}}}{\Delta t} \; , $$
(4)
for the positive lobe of the plot and
$$ \frac{\Delta R}{\Delta t} (-) = \frac{\Delta M}{\Delta t} - \frac{\Delta M_{\text{int}}}{\Delta t} \; , $$
(5)
for the negative lobe of the plot, see Fig. 5.
Assuming that the rate of change of the M(t) does not change over the memristors range, although it does change direction, (which is an approximation), we can substitute for \(\frac {\Delta M}{\Delta t}\) and write
$$ g \frac{\Delta R}{\Delta t}(+) + \frac{\Delta M_{\text{int}}}{\Delta t} = \frac{\Delta R}{\Delta T}(+) - \frac{\Delta M_{\text{int}}}{\Delta t} \:, $$
(6)
where g is the factor that \(\frac {\Delta R}{\Delta t} (-)\) is bigger than \(\frac {\Delta R}{\Delta T} (+)\) by and it is equal to 2.88, i.e. the rate of change of resistance is around three times faster on the negative lobe compared to the positive, leading to a non-rotationally symmetric pinched hysteresis loop.
We can calculate the actual rates \(\frac {\Delta R}{\Delta T} (+)\) and \(\frac {\Delta R}{\Delta t} (-)\) from the measured current I, which we do by calculating the ‘instantaneous’ memristance Mt) at each measurement point:
$$ M({\Delta} t) = \frac{\Delta V_{s}}{\Delta I} \;, $$
(7)
and this is shown in Fig. 5. Around zero and small values of V, we get large discrepancies, due to the method we’re using to calculate the memristance, but over most of the curve, we can see that the straight-line approximation of the change in memristance holds pretty well. Figure 5b shows that there are two gradients: a shallower one for the positive loop and a steeper one for the negative loop. If these gradients were equal, we would have a standard (ideal) memristor curve.

The memristor curve is commonly broken up into four segments: (1) 0V → + max(Vs), (2) + max(Vs) → 0V, (3) 0V → −max(Vs) and (4) −max(Vs) → 0V. We chose to fit a straight line to the first and third segments as they start from the same place (0V), these lines are shown on the curve and their equations are gradients of 3.1009 and 8.9348 × 105 Ω s−1, y-intercepts of 4.9696 × 105 Ω and −9.9034 × 107 Ω (the negative intercept is obviously unphysical and is a result of approximating and changing \(\frac {\Delta R}{\Delta t}\) by a tangent) with a norm of residuals of 5.7685 and 5.8028 × 106 Ω segments 1 and 3, respectively.

We can get a measure of the memristance of the cell’s ‘internal battery’ from rearranging Eq. 6:
$$ \frac{\Delta M_{\text{int}}}{\Delta t} = \frac{\left( g-1 \right) \frac{\Delta R}{\Delta t}(+)}{-2} \; , $$
(8)
this gives us a negative slope of −2.91485 × 105 Ω s−1 with a negative resistance intercept whose modulus is 94 % R0 (where we are taking R0 as the y-intercept from the fitted tangent for the first segment). Negative resistance implies the presence of active components in our test circuit, verifying our approach of treating the cells as possessing an internal battery. This shows that at these voltages (which are close to physiological voltages), the cell’s internal battery gives physiological currents close to our driven current. Thus, to model living cells over physiological ranges, it seems that active memristors are a better approximation than passive memristors.

5 Discussion

The results clearly show hysteresis and memristive effects in P. polycephalum. The frequency and voltage range choice affected the results; we found that a time step of δt = 2s and a ± of over 200 mV gave the best results. At low voltages, an open-curve shape was measured instead, which we suspect is the memristance effect when measured at below a threshold voltage. As the memristive effect disappeared when the Physarum moved elsewhere, and an abandoned tube showed a high linear resistance, we conclude that the memristive response is due to the living protoplasm. Active memristor models show a promising explanation for the asymmetric shape seen when the memristor current response is below 10−8 A; for higher current responses, the internal current is small enough that ignoring it and modelling the Physarum as an ideal memristor is a valid approximation.

It is intriguing that Physarum exhibits memristive ability, given that it is a simple biological system that is nonetheless capable of habituation and learning and that neurological components (synapses, ion pumps) also exhibit memristance and learning abilities. This could suggest that evolution may have made use of memristance in learning systems. The presence of biological active memristors suggests that biological chaotic circuits could be possible (active memristors are a common component of chaotic circuits [18]), and even that they may have been utilised by evolution.

Current versus voltage profiles measured for protoplasmic tubes of P. polycephalum exhibit great variability in the magnitude of hysteresis and the location of pinch points. This is to be expected given the fact that slime mould is an ever-changing living entity and although attempts were made to standardise the experimental set-up such as the measurement of single protoplasmic tubes across a known electrode gap, it proved difficult to precisely control the morphology of the tubes. For example, even though the electrode distance can be controlled, this does not ensure standardisation of the protoplasmic tube length or the width. It also proves difficult to control the position and numbers of small sub-branches which may arise during experimental measurements. Although these do not usually contact the electrode except for sub-branching at the terminal ends, this alteration in morphology is likely to affect the conductivity. Thus, future research would focus on the stabilisation of protoplasmic tubes.

Stabilisation could be achieved by employing the Physarum’s potential for internalisation and re-distribution of conductive and magnetic particles [26] or at least constraining the growing with some kind of scaffolding [32]. There is also the potential that the slime mould could construct an internal scaffold or that this could be induced by appropriately applied external fields; in fact, morphological control could be accomplished by application of appropriate fields per se [11]. Despite the complications of living electronics, we have demonstrated that it is feasible to implement living memristive devices from slime moulds of P. polycephalum.

We believe that future electronic designs will incorporate growing slime mould networks capable of forming a skeleton of conductive information processing elements as part of integrated computing circuits. The slime mould circuits will allow for a high density of computing elements and very low power consumption. To date, the useful lifetime of a slime mould memristor is 3–5 days. However, future studies on loading and coating of the tubes with functional materials with a dual role of structural re-enforcement such as nano-metallic, nano-magnetic or nano-structured semiconducting particles, conducting polymers, etc. should enable us to increase the lifespan substantially whilst also imparting a diverse range of tunable electronic characteristics. If electronic circuits can be ‘grown’ or laid down from Physarum, it would be very useful for reducing the water requirements and poisonous waste of the electronics industry.

Notes

Acknowledgments

This work was supported by the European Research Council on grant FP7-ICT-2011-8, Physarum Chip project

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Ella Gale
    • 1
  • Andrew Adamatzky
    • 1
  • Ben de Lacy Costello
    • 1
  1. 1.University of the West of EnglandBristolUK
  2. 2.Applied Maths and ScienceKhalifa University of Science, Technology and ResearchAbu DhabiUnited Arab Emirates

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