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# On Boolean gates in fungal colony

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A fungal colony maintains its integrity via flow of cytoplasm along mycelium network. This flow, together with possible coordination of mycelium tips propagation, is controlled by calcium waves and associated waves of electrical potential changes. We propose that these excitation waves can be employed to implement a computation in the mycelium networks. We use FitzHugh-Nagumo model to imitate propagation of excitation in a single colony of Aspergillus niger. Boolean values are encoded by spikes of extracellular potential. We represent binary inputs by electrical impulses on a pair of selected electrodes and we record responses of the colony from sixteen electrodes. We derive sets of two-inputs-on-output logical gates implementable the fungal colony and analyse distributions of the gates.
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On Boolean gates in fungal colony
Andrew Adamatzkya,b, Martin Tegelaarc, Han A. B. Wostenc,
Anna L. Powella, Alexander E. Beasleya, Richard Maynea,d
aUnconventional Computing Laboratory, UWE, Bristol, UK
bCorresponding author: andrew.adamatzky@uwe.ac.uk
cMicrobiology Department, University of Utrecht, Utrecht, The Netherlands
dDepartment of Applied Sciences, University of the West of England, UWE, Bristol, UK
Abstract
A fungal colony maintains its integrity via ﬂow of cytoplasm along mycelium
network. This ﬂow, together with possible coordination of mycelium tips
propagation, is controlled by calcium waves and associated waves of elec-
trical potential changes. We propose that these excitation waves can be
employed to implement a computation in the mycelium networks. We use
FitzHugh-Nagumo model to imitate propagation of excitation in a single
colony of Aspergillus niger. Boolean values are encoded by spikes of extra-
cellular potential. We represent binary inputs by electrical impulses on a pair
of selected electrodes and we record responses of the colony from sixteen elec-
trodes. We derive sets of two-inputs-on-output logical gates implementable
the fungal colony and analyse distributions of the gates.
Keywords: mycelium network, Boolean gates, unconventional computing
1. Introduction
A vibrant ﬁeld of unconventional computing aims to employ space-time
dynamics of physical, chemical and biological media to design novel com-
putational techniques, architectures and working prototypes of embedded
computing substrates and devices. Interaction-based computing devices, is
one of the most diverse and promising families of the unconventional com-
puting structures. They are based on interactions of ﬂuid streams, sig-
nals propagating along conductors or excitation wave-fronts, see e.g. [67,
64, 10, 10, 81, 76, 37, 29, 30, 70, 27, 73, 70, 31]. Typically, logical gates
and their cascade implemented in an excitable medium are ‘handcrafted’
Preprint submitted to Elsevier February 25, 2020
arXiv:2002.09680v1 [cs.ET] 22 Feb 2020
Potential, V
0.001
0
0.001
0.002
0.003
0.004
0.005
0.006
Time, sec
9×10510×105
Figure 1: Exemplar spikes of extracellular electrical potential propagating in fungal
mycelium.
to address exact timing and type of interactions between colliding wave-
fronts [67, 64, 1, 9, 74, 8, 20, 71, 82, 72, 32, 7, 69]. The artiﬁcial design of
logical circuits might be suitable when chemical media or functional materi-
als are used. However, the approach might be not feasible when embedding
computation in living systems, where the architecture of conductive path-
ways may be diﬃcult to alter or control. In such situations an opportunistic
approach to outsourcing computation can be adopted. The system is per-
turbed via two or more input loci and its dynamics if recorded at one or
more output loci. A wave-front appearing at one of the output loci is in-
terpreted as logical truth or ‘1’. Thus the system with relatively unknown
structure implements a mapping {0,1}n→ {0,1}m, where nis a number of
input loci and mis a number of output loci, n, m > 0 [12, 6]. The approach
belong to same family of computation outsourcing techniques as in materio
computing [47, 48, 68, 49, 50] and reservoir computing [77, 43, 18, 42, 19].
Fungal colonies are characterised by rich typology of mycelium networks [36,
28, 23, 24, 38] in some cases aﬃne to fractal structures [53, 56, 16, 46, 15, 55].
Rich morphological features might imply rich computational abilities and
thus worse to analyse from realising Boolean functions point of view. In
numerical experiments we study implementation of logical gates via inter-
action of numerous travelling excitation waves, seen as as action potentials,
on an image of a real fungal colony. Action potential-like spikes of electrical
potential have been discovered using intra-cellular recording of mycelium of
Neurospora crassa [66] and further conﬁrmed in intra-cellular recordings of
action potential in hypha of Pleurotus ostreatus and Armillaria bulbosa [54]
2
and in extra-cellular recordings of fruit bodies of and substrates colonized
by mycelium of Pleurotus ostreatus [4] (Fig. 1). While the exact nature of
the travelling spikes remains uncertain we can speculate, by drawing analo-
gies with oscillations of electrical potential of slime mould Physarum poly-
cephalum [39, 40, 41, 45], that the spikes in fungi are triggered by calcium
waves, reversing of cytoplasmic ﬂow, translocation of nutrients and metabo-
lites. Studies of electrical activity of higher plants can brings us even more
clues. Thus, the plants use the electrical spikes for a long-distance communi-
cation aimed to coordinate an activity of their bodies [75, 26, 83]. The spikes
of electrical potential in plants relate to a motor activity [65, 25, 63, 80],
responses to changes in temperature [51], osmotic environment [79] and me-
chanical stimulation [59, 58]. The paper is structured as follows. Colony
imaging and numerical solutions of FitzHugh-Nagumo equations are intro-
duced in Sect. 2. Section 3 studies a role of excitability on the coverage of
the network by travelling waves of excitation and exempliﬁes distributions of
Boolean computable on the given mycelium network. We discuss operation
characteristics of the mycelium computer in Sect. 4.
2. Methods
2.1. Colony imaging
Aspergillus niger strain AR9#2 [78], expressing Green Fluorescent Pro-
tein (GFP) from the glucoamylase (glaA) promoter, was grown at 30oC on
minimal medium (MM) [21] with 25 mM xylose and 1.5% agarose (MMXA).
MMXA cultures were grown for three days, after which conidia were har-
vested using saline-Tween (0.8% NaCl and 0.005% Tween-80). 250 ml liquid
cultures were inoculated with 1.25·109freshly harvested conidia and grown at
200 rpm and 30oC in 1 L Erlenmeyer ﬂasks in complete medium (CM) (MM
containing 0.5% yeast extract and 0.2% enzymatically hydrolyzed casein)
supplemented with 25 mM xylose (repressing glaA expression). Mycelium
was harvested after 16 h and washed twice with PBS. Ten g of biomass (wet
weight) was transferred to MM supplemented with 25 mM maltose (inducing
glaA expression).
Fluorescence of GFP was localised in micro-colonies using a DMI 6000
CS AFC confocal microscope (Leica, Mannheim, Germany). Micro-colonies
were ﬁxed overnight at 4oC in 4% paraformaldehyde in PBS, washed twice
with PBS and taken up in 50 ml PBS supplemented with 150 mM glycine to
3
quench autoﬂuorescence. Micro-colonies were then transferred to a glass bot-
tom dish (Cellview, Greiner Bio-One, Frickenhausen, Germany, PS, 35/10
MM) and embedded in 1% low melting point agarose at 45oC. Micro-colonies
were imaged at 20×magniﬁcation (HC PL FLUOTAR L 20 ×0.40 DRY).
GFP was excited by white light laser at 472 nm using 50% laser intensity
(0.1 kW/cm2) and a pixel dwell time of 72 ns. Fluorescent light emission was
detected with hybrid detectors in the range of 490–525 nm. Pinhole size was
1 Airy unit. Z-stacks of imaged micro-colonies were made using 100 slices
with a slice thickness of 8.35 µm. 3D projections were made with Fiji [62].
2.2. Numerical modelling
We used still image of the colony as a conductive template. The image of
the fungal colony (Fig. 2a) was projected onto a 1000 ×960 nodes grid. The
original image M= (mij )1jni,1jnj,mij ∈ {rij , gij , bij }, where ni= 1000
and nj= 960, and 1 r, g, b 255 (Fig. 2a), was converted to a conductive
matrix C= (mij )1i,jn(Fig. 2b) derived from the image as follows: mij = 1
if rij <20, (gij >40) and bij <20; a dilution operation was applied to C.
FitzHugh-Nagumo (FHN) equations [22, 52, 57] is a qualitative approxi-
mation of the Hodgkin-Huxley model [14] of electrical activity of living cells:
∂v
∂t =c1u(ua)(1 u)c2uv +I+Du2(1)
∂v
∂t =b(uv),(2)
where uis a value of a trans-membrane potential, va variable accountable
for a total slow ionic current, or a recovery variable responsible for a slow
negative feedback, Iis a value of an external stimulation current. The cur-
rent through intra-cellular spaces is approximated by Du2, where Duis a
conductance. Detailed explanations of the ‘mechanics’ of the model are pro-
vided in [60], here we shortly repeat some insights. The term Du2ugoverns
a passive spread of the current. The terms c2u(ua)(1 u) and b(uv)
describe the ionic currents. The term u(ua)(1 u) has two stable ﬁxed
points u= 0 and u= 1 and one unstable point u=a, where ais a threshold
of an excitation.
We integrated the system using the Euler method with the ﬁve-node
Laplace operator, a time step ∆t= 0.015 and a grid point spacing ∆x= 2,
while other parameters were Du= 1, a= 0.13, b= 0.013, c1= 0.26. We
controlled excitability of the medium by varying c2from 0.05 (fully excitable)
4
(a) (b)
Ratio ν
0
0.05
0.10
0.15
0.20
Neighbourhood size k
12345678
(c) (d)
Figure 2: Image of the fungal colony, 1000 ×960 pixels used as a template conductive
for FHN. (a) Original image, mycelium is seen as green pixels. (b) Conductive matrix C,
conductive pixels are black. (c) Distribution of neigbourhood sizes. (d) Conﬁguration of
electrodes.
5
to 0.015 (non excitable). Boundaries are considered to be impermeable:
∂u/∂n= 0, where nis a vector normal to the boundary.
The waves of excitation propagated on conductive nodes of the grid of C,
in addition to the parameter c2, excitability of each conductive node was de-
pendent on a number kof its immediate conductive neighbours. Distribution
of neighbourhood sizes are shown in Fig. 2c.
To show dynamics of excitation in the network we simulated electrodes by
calculating a potential pt
xat an electrode location xas px=Py:|xy|<2(ux
vx). Conﬁguration of electrodes 1,· · · ,16 is shown in Fig. 2d. The numerical
integration code written in Processing was inspired by previous methods of
numerical integration of FHN and our own computational studies of the
impulse propagation in biological networks [33, 57, 60, 5, 6]. Time-lapse
snapshots provided in the paper were recorded at every 100th time step, and
we display sites with u > 0.04; videos and ﬁgures were produced by saving
a frame of the simulation every 100th step of the numerical integration and
assembling the saved frames into the video with a play rate of 30 fps. Videos
are available at [13].
3. Results
While implementing numerical experiments were selected a range the net-
work excitability (Subsect. 3.1) and then realised sets of logical gates for
excitability values selected (Subsect. 3.2).
3.1. Eﬀect of excitability on overall activity
For c2<0.0945 any source of excitation triggers excitation dynamics
which occupies all parts of the network accessible, via mycelial strands, from
the source. Due to the high level of excitability the network remains in the
excitable state (Fig. 3a). For values c2from 0.094 to 0.00965 we observe
propagation of ‘classical’ excitation wave-fronts resembling circular, target
and spiral waves in a continuous medium. Examples of wave-fronts propa-
gating in networks with excitability levels c2= 0.095 and c2= 0.096, excited
at the same loci shown in Fig. 4a, are shown in Figs.4 and 5. In the network
with c2there are many pathways for propagation of the excitation wave-
fronts, therefore, despite being fully deterministic, the network exhibit disor-
dered oscillations of its activity (Fig. 3c). A number of conductive pathways
decreases when c2increases from 0.095 to 0.096. Thus many propagating
wave-fronts become, relatively, quickly conﬁned to a limited domains of the
6
Activity α, ratio
0
0.01
0.02
0.03
0.04
Time, iterations
0 2×105
(a) c2= 0.094
Activity α, ratio
0
0.0005
0.0010
0.0015
0.0020
Time, iterations
0 20,000
(b) c2= 0.097
(c) c2= 0.095
Activity α, ratio
0
0.001
0.002
0.003
0.004
Time, iterations
0 1×1052×1053×1054×105
0.0035
0.0040
0.0045
2.0×1052.2×105
(d) c2= 0.096
Figure 3: Dynamics of the activity αfor various values of excitability c2, the values are
shown in sub-captions. For every iteration twe measured the activity of the network as a
number of conductive nodes xwith ut
x>0.1.
7
(a) t= 200 (b) t= 1900 (c) t= 40500
(d) t= 70400 (e) t= 104800 (f) t= 115850
Figure 4: Snapshots of excitation dynamics for c2= 0.095.
8
(a) t= 1900 (b) t= 40500 (c) t= 70400
(d) t= 104800 (e) t= 115850 (f) t= 155000
Figure 5: Snapshots of excitation dynamics for c2= 0.096. Compare (c) and (e): the
pattern of excitation returns to the exact point of the cycle.
(a) c2= 0.095 (b) c2= 0.096 (c) c2= 0.097
Figure 6: Coverage of the network for excitability c2(a) 0.095, (b) 0.096, (c) 0.097. If the
a pixel pof the image was excited, ut
p>0.1, it is assumed to be covered and coloured red
in the pictures (abc); the pixels which never were excited are coloured gray.
9
Sy Sy Sy Sx x+y
xy xy x y
Potential, units
50
0
50
Time, iterations
30,000 40,000 50,000 60,000
Figure 7: Fragment of electrical potential record on electrode 7 in response to inputs (01),
black dashed line, (10), red dotted line, (11), solid green line, entered as impulses via
electrodes Ex= 5 and Ey= 15. See locations of electrodes in Fig. 2d. To make the
individual plots visible in places of exact overlapping, we added potential 5 to recording
in response to input (01) and and potential 5 to recording in response to input (11).
network, where they continue ‘circling’ indeﬁnitely. A set of regular oscil-
lations of activity becomes evidence after a number of iterations (Fig. 3d).
The coverage of the network by excitation wave-fronts reduced with increase
of c2from 0.095 to 0.096 (Fig. 6ab) and becomes localised when c2reaches
0.097 (Figs. 6c and 3b). Thus we used networks with c2= 0.095 or 0.096 for
implementation of Boolean functions.
3.2. Distribution of Boolean gates
Input Boolean values are encoded as follows. We earmark two sites of the
network as dedicated inputs, xand y, and represent logical True, or ‘1’, as
an excitation, or an impulse injected injected in the network via electrodes.
If x= 1 then the site corresponding to xis excited, if x= 0 the site is not
excited.
We here assume that each spike represents logical True and that spikes
occurring within less than 2·102iterations happen simultaneously. By select-
ing speciﬁc intervals of recordings we can realise several gates in a single site
of recording. In this particular case we assumed that spikes are separated if
their occurrences lie more than 103iterations apart. An example is shown in
Fig. 7.
10
Figure 8: Recording of electrical potential from all electrodes in responses to inputs in
response to inputs (01), black line, (10), red line, (11), green line, injected as spikes via
electrodes Ex= 5 and Ey= 15.
11
(a) Ex= 3, Ey= 13, c2= 0.095
E x +y Sy x y Sx xy xy xy Total
0 0 0 0 0 0 0 0 0
1 0 0 0 2 0 0 0 2
2 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0
4 1 0 0 7 1 0 0 9
5 0 0 0 2 2 0 0 4
6 0 0 0 2 0 0 0 2
7 1 0 0 8 2 0 0 11
8 1 0 0 6 1 0 0 8
9 0 0 0 0 1 0 0 1
10 0 1 1 2 0 1 2 7
11 0 0 0 4 2 0 0 6
12 0 0 0 3 2 0 0 5
13 1 5 0 0 0 1 0 7
14 2 5 0 1 0 1 0 9
15 0 1 0 5 2 0 0 8
Total 6 12 1 42 13 3 2 79
(b) Ex= 7, Ey= 14, c2= 0.095
E x +y Sy x y Sx xy xy xy Total
0 0 0 0 0 0 0 0 0
1 0 0 0 5 0 0 0 5
2 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0
4 0 0 0 6 1 0 0 7
5 0 0 0 6 0 0 0 6
6 0 0 0 2 0 0 0 2
7 1 0 0 4 2 0 0 7
8 0 0 0 7 0 0 0 7
9 0 0 0 0 0 0 0 0
10 1 0 1 5 0 0 0 7
11 1 0 0 4 3 0 0 8
12 2 0 0 3 0 0 0 5
13 1 5 0 1 0 2 0 9
14 0 4 0 2 1 2 1 10
15 0 0 0 8 2 0 0 10
Total 6 9 1 53 9 4 1 83
(c) Ex= 7, Ey= 14, c2= 0.094
E x +y Sy x y Sx xy xy xy Total
0 0 0 0 0 0 0 0 0
1 0 0 0 2 1 0 0 3
2 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0
4 0 0 0 1 3 0 0 4
5 0 0 0 1 1 0 0 2
6 0 0 0 1 1 0 0 2
7 0 0 0 0 2 0 0 2
8 0 0 0 3 4 0 0 7
9 0 0 0 0 1 2 0 3
10 0 0 0 1 1 0 0 2
11 0 0 0 0 6 0 0 6
12 1 0 1 2 3 1 0 8
13 0 2 0 2 0 1 0 5
14 1 3 0 0 3 0 2 9
15 0 2 0 0 0 1 0 3
Total 2 7 1 13 26 5 2 56
(d) Ex= 5, Ey= 15, c2= 0.094
E x +y Sy x y Sx xy xy xy Total
0 0 0 0 0 0 0 0 0
1 0 0 0 1 1 0 0 2
2 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0
4 0 1 0 0 2 1 1 5
5 0 0 0 0 1 0 0 1
6 0 0 0 0 1 0 0 1
7 1 2 0 0 0 4 0 7
8 0 1 0 1 2 1 0 5
9 0 0 0 1 2 0 0 3
10 0 0 0 2 0 0 1 3
11 1 5 0 0 1 1 1 9
12 0 6 0 0 1 2 0 9
13 2 0 2 1 1 1 1 8
14 0 1 0 1 0 5 0 7
15 0 0 0 0 0 1 0 1
Total 4 16 2 7 12 16 4 61
(e) Ex= 5, Ey= 15, c2= 0.095
E x +y Sy x y Sx xy xy xy Total
0 0 0 0 0 0 0 0 0
1 0 0 0 8 0 0 0 8
2 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0
4 1 4 0 0 0 2 0 7
5 3 0 0 4 0 0 0 7
6 0 0 0 0 1 0 0 1
7 1 3 1 1 0 1 1 8
8 0 5 0 1 0 2 0 8
9 0 0 0 3 0 0 0 3
10 1 0 2 4 2 0 2 11
11 2 4 0 2 2 0 1 11
12 1 7 0 0 0 3 0 11
13 3 1 0 2 0 1 0 7
14 1 5 0 0 0 6 0 12
15 1 3 1 2 2 1 1 11
Total 14 32 4 27 7 16 5 105
(f) Ex= 5, Ey= 15, c2= 0.096
E x +y Sy x y Sx xy xy xy Total
0 0 0 0 0 0 0 0 0
1 0 0 0 2 1 0 0 3
2 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0
4 1 3 0 0 0 1 0 5
5 0 0 0 4 2 0 0 6
6 0 0 0 2 1 0 0 3
7 1 4 0 1 0 1 0 7
8 1 5 0 0 0 1 0 7
9 0 0 0 7 0 0 0 7
10 0 1 0 5 1 1 0 8
11 2 3 0 1 1 1 1 9
12 0 4 0 0 0 0 0 4
13 0 2 0 2 0 2 0 6
14 1 3 0 2 0 3 0 9
15 0 3 0 2 0 2 0 7
Total 6 28 0 28 6 12 1 81
Table 1: Numbers of Boolean gates detected for selected pairs of input electrodes Exand
Ey.
12
x+y Sx Syxy xy xy xy
Figure 9: Comparative ratios of Boolean gates discovered in mycelium network in
present analysed in present paper, black disc and solid line; slime mould Physarum poly-
cephalum [34], black circle and dotted line; succulent plant [11], red snowﬂake and dashed
line; single molecule of protein verotoxin [2], light blue ‘+’ and dash-dot line; actin bundles
network [12], green triangle pointing right and dash-dot-dot line; actin monomer [3], ma-
genta triangle pointing left and dashed line. Area of xor gate is magniﬁed in the insert.
Lines are to guide eye only.
13
Numbers of Boolean gates detected on the electrodes for selected pairs
of input electrodes are shown in Tab. 1. We see that select xand select y
gates, Sx and Sy are most frequent. They usually are detected with the same
frequency (Tab. 1d–f), however there are examples of input electrode pairs,
where one of the select gates is found much more often than another. This
is most visible for the pair (Ex, Ey) = (3,13) where Sx dominates (Tab. 1a),
and the pair (7,14) and (Tab. 1bc) where Sy dominates. Next common gates
in the hierarchy are xy and xy. The gates xy and x+yare detected with
nearly the same frequency with gate x+ybeing slightly more common. The
xyis the most rare gate.
The sub-tables Tab. 1d, Tab. 1e and Tab. 1f show how excitability of the
network aﬀects numbers of gates detected. The networks with high excitabil-
ity, c2= 0.094, and low excitability, c2= 0.0096 realise smaller number of
gates then that realised by sub-excitable network, c2= 0.095.
Overall distribution (average of outputs of input electrode pairs (3,13),
(5,15), (7,14), (4,13), (13,7)) of a ratio of gates discovered is shown in Fig. 9.
This is accompanied by distributions of gates discovered in experimental lab-
oratory reservoir computing with slime mould Physarum polycephalum [34],
succulent plant [11] and numerical modelling experiments on computing with
protein verotoxin [2], actin bundles network [12], and actin monomer [3]. All
the listed distributions have very similar structure with gates selecting one
of the inputs in majority, followed by or gate, not-and an and-not gates.
The gate and is usually underrepresented in experimental and modelling
experiments. The gate xor is a rare ﬁnd.
4. Discussion
We have demonstrated how sets of logical gates can be implemented in
single colony mycelium networks via initiation of electrical impulses. The
impulses travel in the network, interact with each other (annihilate, reﬂect,
change their phase). Thus for diﬀerent combinations of input impulses and
record diﬀerent combinations of output impulses, which in some cases can be
interpreted as representing two-inputs-one-output functions.
To estimate a speed of computation we refer to Olsson and Hansson’s [54]
original study, in which they proposed that electrical activity in fungi could
be used for communication with message propagation speed 0.5 mm/sec.
Diameter of the colony (Fig. 2a), which experimental laboratory images has
been used to run FHN model, is c. 1.7 mm. Thus, it takes the excitation
14
waves initiated at a boundary of the colony up to 3-4 sec to span the whole
mycelium network (this time is equivalent to c. 70K iterations of the numer-
ical integration model). In 3-4 sec the mycelium network can compute up to
a hundred logical gates. This gives us the rate of a gate per 0.03 sec, or, in
terms of frequency this will be c. 30 Hz. The mycelium network computing
can not compete with existing silicon architecture however its application
domain can be a unique of living biosensors (a distribution of gates realised
might be aﬀected by environmental conditions) [44] and computation em-
bedded into structural elements where fungal materials are used [61, 35, 17].
5. Acknowledgement
This project has received funding from the European Union’s Horizon
2020 research and innovation programme FET OPEN “Challenging current
thinking” under grant agreement No 858132.
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