# Memory capacity of adaptive flow networks

Komal Bhattacharyya  Max Planck Institute for Dynamics and Self-Organisation,
37077 Göttingen, Germany  David Zwicker  Max Planck Institute for Dynamics and
Self-Organisation, 37077 Göttingen, Germany  Karen Alim<EMAIL_ADDRESS>Max
Planck Institute for Dynamics and Self-Organisation, 37077 Göttingen, Germany
Physik-Department and Center for Protein Assemblies, Technische Universität
München, 85748 Garching, Germany

###### Abstract

Biological flow networks adapt their network morphology to optimise flow while
being exposed to external stimuli from different spatial locations in their
environment. These adaptive flow networks retain a memory of the stimulus
location in the network morphology. Yet, what limits this memory and how many
stimuli can be stored is unknown. Here, we study a numerical model of adaptive
flow networks by applying multiple stimuli subsequently. We find strong memory
signals for stimuli imprinted for a long time into young networks.
Consequently, networks can store many stimuli for intermediate stimulus
duration, which balance imprinting and ageing.

##  I Introduction

Biological flow networks, like vasculature [1], fungal mycelium [2], or slime
mold [3, 4], optimise their function by remodelling the network morphology in
response to internal and external stimuli [5, 6, 7, 8, 9, 3, 10]. In
particular, the slime mold Physarum polycephalum reorganises its network
morphology during foraging and migration [11, 12], or as responses to
environmental influences [13], by adapting tubes to flow [8, 9, 14]. Although
the organism only consist of a single cell, it processes information [15, 16,
9, 17, 8, 18] and stores memory of external stimuli in the network morphology
[13, 19]. Yet, the information processing capabilities of Physarum in
particular, and adaptive flow networks more generally, are so far unclear.

The self-organised information processing of Physarum is reminiscent of other
physical learning systems [20]: Physical networks can be trained to have
unusual mechanical properties [21, 22] and functionalities [23, 24] either by
modifying microscopic properties by global optimisation [25] or as local
responses [26]. Such networks can also learn multiple states [27], which is
key for obtaining multi-functionality [27] and multi-stability [28, 29, 30,
31, 32, 33, 34, 35, 36, 37] in physical systems, and for performing complex
tasks like image classification [38, 39, 40] using such physical networks. The
multiple states can be either imprinted simultaneously [27] or learned
subsequently [38]. In both cases, there is a maximal number of states that can
be learned, which is the learning capacity of the system [27, 38].

Although memory is essential for learning [20], the memory capacity of flow
networks remains unexplored. We, here, investigate this question
theoretically, by analysing memory in a model of adaptive flow network, which
is subjected to various external stimuli, similar to natural flow networks
[41, 42]. We identify that a stimulus is stored more robustly, and can, thus,
be retrieved more easily, when networks are young and are exposed to a
stimulus for a long time. Since these two criteria are incommensurable for
multiple stimuli, a trade-off determines the memory capacity of these adaptive
flow networks.

##  II Model

Figure 1: Stimulus locations are retained in previously stimulated networks.
(a) Schematic of adaptive flow networks with a central outlet (red dot),
fluctuating inflows at all other nodes, and additional inflow for stimuli
(blue dots). The temporal sequence shows snapshots of the training protocol,
where $N$ stimuli (purple boxes) are applied sequentially by stimulating for a
period $t^{\mathrm{train}}_{n}$ followed by relaxation period
$t^{\mathrm{wait}}_{n}$ for each stimulus. Finally, the last stimulus is
probed (pink dots) to determine the signal $S_{N}$ according to Eq. (4). (b)
$S_{N}$ as a function of the training time
$t^{\mathrm{train}}_{n}=t^{\mathrm{train}}_{\mathrm{pre}}$ and waiting time
$t^{\mathrm{wait}}_{n}=t^{\mathrm{wait}}_{\mathrm{pre}}$ for most stimuli
($n<N$) with $N=5$. (c) $S_{N}$ as a function of
$t^{\mathrm{train}}_{\mathrm{pre}}$ and $N$ for
$t^{\mathrm{wait}}_{\mathrm{pre}}=5\delta t$. (d) $S_{N}$ as a function of
$t^{\mathrm{wait}}_{\mathrm{pre}}$ and $N$ for
$t^{\mathrm{train}}_{\mathrm{pre}}=5\delta t$. (a–d) Model parameters are
$t^{\mathrm{train}}_{N}=10\delta t$, $t^{\mathrm{wait}}_{N}=5\delta t$,
$N_{\mathrm{nodes}}=1100$, $q^{\mathrm{add}}=2000\,q^{(0)}$, $q^{(0)}=1$,
$\mathcal{K}=1600$, and $T=30\delta t$. Data shows mean from $1500$
independent simulations.

We use the standard model for adaptive flow networks that minimises energy
dissipation in order to maximise transport through the network, for fixed
network building material [1, 43, 44, 45, 46, 47]. These flow networks are
modelled as a graph of $N_{\mathrm{nodes}}$ nodes connected by links ${ij}$,
where $i,j\in\\{1,\ldots,N_{\mathrm{nodes}}\\}$. The links have length
$l_{ij}$ and time-dependent conductances $C_{ij}(t)$. We consider a network of
cylindrical hollow tubes with conductances $C_{ij}=\pi r_{ij}(t)^{4}/8\mu
l_{ij}$ according to Hagen-Poiseuille’s law, where $\mu$ is the viscosity of
the enclosed fluid. In our case, node $i=1$ serves as the sole outlet, while
all other nodes are inlets with fluctuating inflows $q_{i}(t)$, which are
either $q_{i}=0$ or $q_{i}=2q^{(0)}$ with equal probability. Conservation of
total flow implies $q_{1}(t)=-\sum_{i>1}q_{i}(t)$. We chose a disk-shaped
network geometry with the outlet in the centre; see Fig. 1(a). Conservation of
flow at every node, as described by Kirchhoff’s law, then uniquely determines
the flow $Q_{ij}(t)$ in all links, given the entire networks conductances
$C_{ij}(t)$ and the inflows $q_{i}(t)$; see Appendix A. The adaptive dynamics
follow from the assumption that networks minimise dissipation [1]

$E(t)=\sum_{<ij>}\frac{Q_{ij}(t)^{2}}{C_{ij}(t)}\;,$ (1)

while obeying the constraint

$\mathcal{K}^{\frac{1}{2}}=\sum_{<ij>}C_{ij}(t)^{\frac{1}{2}}l_{ij}^{\frac{3}{2}}\;,$
(2)

where $\mathcal{K}^{\frac{1}{2}}$ is proportional to the fixed overall volume
of all links. We follow an iterative relaxation algorithm [44], where the
conductances at the next time step, $C_{ij}(t+\delta t)$, adapt to minimise
$E(t)$ while obeying Eq. (2), implying

$C_{ij}(t+\delta t)=\frac{\mathcal{K}\langle
Q_{ij}(t)^{2}\rangle_{T}^{\frac{2}{3}}}{\left(\sum_{<ij>}\langle
Q_{ij}(t)^{2}\rangle_{T}^{\frac{1}{3}}l_{ij}\right)^{2}l_{ij}}\;,$ (3)

where we average the flow over a duration $T$, $\langle
Q_{ij}^{2}\rangle_{T}$, since the inflows at every node fluctuate over time.

To probe for memory, we initiate networks with conductances $C_{ij}$ chosen
uniformly from the interval $[0,1]$, which are then rescaled, so they obey the
constraint given by Eq. (2). We then stimulate the networks using an
additional inflow $q^{\mathrm{add}}$ at the outer rim at a specific angular
location; see Fig. 1(a). We distribute the additional inflow over a few nodes
to avoid artefacts from the symmetries of the underlying networks. The
adaptation dynamics then imprint the stimulus in a treelike structure from the
nodes of additional inflow to the centred outlet [19]; see Fig. 1(a). Once the
additional inflow is withdrawn, networks return to seemingly isotropic
morphologies. Yet, when probing networks by re-applying an additional load at
exactly the same location, the power loss of previously stimulated, and thus
trained, networks, $E_{\mathrm{trained}}$, is distinctively less than if
probed at any other location. In particular, $E_{\mathrm{trained}}$ is less
than the power loss $E_{\mathrm{untrained}}$ for probing untrained networks
that evolved for the same total time, but did not see the stimulus [19]. To
quantify this memory, we established the normalised difference in power loss
between trained and untrained networks as a measure of the memory readout
signal $S$, [19],

$S=1-\frac{\langle E_{\mathrm{trained}}\rangle}{\langle
E_{\mathrm{untrained}}\rangle}\;,$ (4)

where brackets indicate ensemble averages over initial configurations and
positions of the additional loads. We used this quantification to show that
freshly initiated networks memorise single stimuli in the spatial location and
orientation of the vanishing links [19]. The stimulus can be read out by
probing the network again after the stimulus is withdrawn. Yet, it is unclear
how well already stimulated networks can store stimuli and whether networks
can store multiple stimuli simulatenously.

##  III Results

###  III.1  Pre-stimulated networks can memorise stimuli

We start by asking whether previously evolved networks can store a stimulus
reliably. We evolve networks with a stimulation protocol by consecutively
applying $N$ stimuli, distinguished by the angle of the additional inflow. We
choose the angles randomly from $10$ possibilities,
$\\{0,\frac{\pi}{5},\ldots,\frac{9\pi}{5}\\}$, and we set the angular range of
each stimulus to $\frac{\pi}{6}$ to avoid stimuli overlap. Starting with a
randomly initialised network, we apply one stimulus after the other. The
$n$-th stimulus is imprinted on the network by iterating Eq. (3) with the
additional load corresponding to the stimulus for a duration
$t^{\mathrm{train}}_{n}$ and then without load for a duration
$t^{\mathrm{wait}}_{n}$. Taken together, the network evolved to time

$t^{\mathrm{age}}_{n}=\sum_{m=1}^{n}\left(t^{\mathrm{train}}_{m}+t^{\mathrm{wait}}_{m}\right)$
(5)

after the $n$-th stimulus has been applied.

To test whether a stimulated network can memorise an additional stimulus, we
apply $N-1$ stimuli with identical properties and then probe the signal of a
final stimulus; see Fig. 1(a). We thus have
$t^{\mathrm{train}}_{n}=t^{\mathrm{train}}_{\mathrm{pre}}$ and
$t^{\mathrm{wait}}_{n}=t^{\mathrm{wait}}_{\mathrm{pre}}$ for $n<N$, while the
final stimulus can have different parameters. The signal $S_{N}$ quantifies
the dissipation difference of applying the $N$-th stimulus, analogously to Eq.
(4). For constant parameters of the pre-stimulation protocol, we observe that
$S_{N}$ increases with $t^{\mathrm{train}}_{N}$ and decays with
$t^{\mathrm{wait}}_{N}$; see Appendix B. This behaviour closely resembles
memory formation in a freshly initiated network [19], even though we here use
pre-stimulated networks.

We next test the influence of the precise pre-stimulation protocol by varying
the number of applied stimuli, $N$, the training time,
$t^{\mathrm{train}}_{\mathrm{pre}}$, and the relaxation time,
$t^{\mathrm{wait}}_{\mathrm{pre}}$. Fig. 1(b–d) shows that the signal $S_{N}$
of the final stimulus decreases when increasing any of these parameters, so
the pre-stimulation protocol affects how well additional memories can be
stored. However, our simulations demonstrated that pre-stimulated adaptive
networks can store information about additional stimuli.

###  III.2 Memory capacity reduces with age

Figure 2:  Memory signal of final stimulus reduces with network age. (a)
Fraction of vanishing links (blue symbols) as a function of network iterations
averaged over $80$ independent runs. Model parameters $N_{\mathrm{nodes}}$,
$q^{\mathrm{add}}$, $q^{(0)}$, and $T$ are given in Fig. 1. Red line indicates
an exponential fit. (b, c) Signal $S_{N}$ of final stimulus as a function of
age $t^{\mathrm{age}}_{N-1}$ before stimulus was applied. Panels b and c show
data of Fig. 1(b) and Fig. 1(c), respectively. Blue lines indicate exponential
fits.  Figure 3:  Signal $S_{1}$ of first stimulus increases with training
time and decreases with network age. (a) Snapshots of network, which is
subjected to the first stimulus for $t^{\mathrm{train}}_{1}$, relaxed for
$t^{\mathrm{wait}}_{1}$, and then $N-1$ stimuli are applied with
$t^{\mathrm{train}}_{n}=t^{\mathrm{train}}_{\mathrm{post}}$ and
$t^{\mathrm{wait}}_{n}=t^{\mathrm{wait}}_{\mathrm{post}}$ ($n>1$, dotted box),
until the first stimulus is probed. (b) $S_{1}$ as a function of
$t^{\mathrm{train}}_{1}$ for $N=5$ and
$t^{\mathrm{train}}_{\mathrm{post}}=5\delta t$. (c) $S_{1}$ as a function of
$t^{\mathrm{age}}_{\mathrm{post}}=t^{\mathrm{age}}_{N}-t^{\mathrm{train}}_{1}$
for various $N$ at $t^{\mathrm{train}}_{1}=10\delta t$. (b–c) Blue lines
indicate exponential fits. Parameters are
$t^{\mathrm{wait}}_{1}=t^{\mathrm{wait}}_{\mathrm{post}}=5\delta t$ and given
in Fig. 1.

We next investigate how the pre-stimulation protocol affects the memory of the
final stimulus. Since information about stimuli locations are stored in the
orientation and location of irreversibly decaying links [19], we first
determine how the micro-structure of the network evolves with time. Fig. 2(a)
shows that the average fraction of vanishing links saturates exponentially
with time, which suggest that the memory capacity of adaptive flow networks
decreases with the time $t^{\mathrm{age}}_{N-1}$, given by Eq. (5), that the
network evolved for before the stimulus is applied. Re-plotting the memory
signal $S_{N}$ of the final stimulus as a function of $t^{\mathrm{age}}_{N-1}$
leads to a data collapse for various values of $N$,
$t^{\mathrm{train}}_{\mathrm{pre}}$, and $t^{\mathrm{wait}}_{\mathrm{pre}}$;
see Fig. 2(b, c). The two panels differ in whether $N$ (panel b) or
$t^{\mathrm{wait}}_{\mathrm{pre}}$ (panel c) are kept fixed while the other
parameters are varied. In both cases, the data collapse is well-described by
an exponential decay

$S_{N}(t^{\mathrm{age}}_{N-1})\approx
S^{\infty}_{N}+A_{N}\exp\left(-\frac{t^{\mathrm{age}}_{N-1}}{\tau_{\mathrm{pre}}}\right)\;,$
(6)

where $\tau_{\mathrm{pre}}\approx 52\,\delta t$ denotes the time scale, with
which pre-stimulation reduces the memory capacity of the final stimulus. The
maximal memory capacity, $S^{\infty}_{N}+A_{N}\approx 0.22$ for
$t^{\mathrm{age}}_{N-1}=0$, is significantly larger than the residual
capacity, $S^{\infty}_{N}\approx 0.03$, consistent with the fact that pre-
stimulation of the networks reduces the memory capacity. The fact that the
exponential decay adequately describes the decreasing capacity suggests that
only the total duration of pre-stimulation is important, while the details of
the protocol are irrelevant. Consequently, younger networks allow for a larger
memory signal of the final stimulus.

###  III.3 Training time dominates signal of first stimulus

Figure 4: Memory capacity depends on stimulation protocol parameters. (a)
Numerically obtained signals $S_{n}$ for all $N=5$ stimuli as functions of
training times $t^{\mathrm{train}}$ and waiting times $t^{\mathrm{wait}}$. (b)
Analytical prediction of $S_{n}$ given by Eq. (9) for all stimuli as functions
of $t^{\mathrm{train}}$ and $t^{\mathrm{wait}}$ for $N=5$. (c) Fraction of
stimuli with strong signal ($S_{n}>0.04$) as a function of
$t^{\mathrm{train}}$ and $N$ for $t^{\mathrm{wait}}=5\delta t$. (d) Number of
stimuli with $S_{n}>0.04$ as a function of $t^{\mathrm{train}}$ and
$t^{\mathrm{wait}}$ for $N=5$. (e) Minimal signal $\min_{n}S_{n}$ of $N=5$
stimuli as a function of $t^{\mathrm{train}}$ and $t^{\mathrm{wait}}$. (a–e)
Model parameters are given in Fig. 1.

To retain multiple memories, adaptive networks need to store information about
all stimuli. We, thus, next investigate how information about earlier stimuli
are retained and particularly focus on the first stimulus. To investigate the
first stimulus in detail, we change the protocol to control the training
parameters of the first stimulus separately from all the other stimuli; see
Fig. 3(a). For simplicity, we use identical parameters for the other stimuli,
$t^{\mathrm{train}}_{n}=t^{\mathrm{train}}_{\mathrm{post}}$ and
$t^{\mathrm{wait}}_{n}=t^{\mathrm{wait}}_{\mathrm{post}}$ for $n=2,..,N$. The
network is probed at the same location as the first stimulus to obtain the
memory signal $S_{1}$ of the first stimulus. Fig. 3(b) shows that $S_{1}$
increases with the training time of the first stimulus,
$t^{\mathrm{train}}_{1}$, and approaches zero for $t^{\mathrm{train}}_{1}=0$.
$S_{1}$ again shows an exponential saturation,

$S_{1}(t^{\mathrm{train}}_{1})\approx
B_{1}\left[1-\exp\left(-\frac{t^{\mathrm{train}}_{1}}{\tau_{\mathrm{train}}}\right)\right]\;,$
(7)

where $\tau_{\mathrm{train}}$ is the training time scale and $B_{1}$ denotes
the maximal signal for $t^{\mathrm{train}}_{1}\rightarrow\infty$. Similar to
our previous work [19], longer training leads to a stronger signal.

We next investigate how the signal of the first stimulus depends on
subsequently applied stimuli. Fig. 3(c) indicates that $S_{1}$ decays as the
networks evolves further, similar to our previous study [19]. We find that
$S_{1}$ only depends on the duration of evolution after the first training
period,
$t^{\mathrm{age}}_{\mathrm{post}}=t^{\mathrm{age}}_{N}-t^{\mathrm{train}}_{1}$,
and not the precise details of the protocol. Moreover, $S_{1}$ again decays
exponentially,

$S_{1}(t^{\mathrm{age}}_{\mathrm{post}})\approx
S^{\infty}_{1}+A_{1}\exp\left(-\frac{t^{\mathrm{age}}_{\mathrm{post}}}{\tau_{\mathrm{post}}}\right)\;,$
(8)

where the coefficients have the same interpretation as in Eq. (6). Our fits
indicate that $\tau_{\mathrm{post}}\approx\tau_{\mathrm{pre}}$, consistent
with an intrinsic time scale of memory formation. Note that the residual
memory capacity $S^{\infty}_{1}\approx 0.1$ is large, implying that subsequent
training does not affect the signal very strongly. This is consistent with the
picture that memory is stored by vanishing links that cannot be revived; see
analytical and numerical observations of the transitions phase between stimuli
in Appendix C and D. Taken together, we find that adaptive flow networks can
store multiple memories.

###  III.4 Trade off between age and training time limits memory capacity

We found that stimuli are imprinted most strongly when they are trained for a
long time on a young network. These goals of long training times and young
networks are contradictory for late stimuli, suggesting there must be a trade-
off for best performance of imprinting multiple stimuli. To understand how
many stimuli can be imprinted in a network, we next consider $N$ non-
overlapping stimuli with identical stimulation parameters,
$t^{\mathrm{train}}_{n}=t^{\mathrm{train}}$ and
$t^{\mathrm{wait}}_{n}=t^{\mathrm{wait}}$ for $n=1,..,N$. We now also probe
all stimuli locations to obtain a signal $S_{n}$ for each stimulus. Fig. 4(a)
shows data for five stimuli as a function of $t^{\mathrm{train}}$ and
$t^{\mathrm{wait}}$. We recover that the signal $S_{1}$ of the first stimulus
mainly depends on the training time $t^{\mathrm{train}}$ and is barely
affected by the subsequent dynamics. Conversely, the signal of all other
stimuli decreases with network age, i.e., with increasing $t^{\mathrm{train}}$
and $t^{\mathrm{wait}}$. In particular, mid-timed stimuli have the weakest
signals, suggesting that they are affected by both pre-stimuli ageing as well
as subsequent degradation.

We next develop an analytical prediction of the signal of all stimuli,
motivated by the successful description of the signals of the first and last
stimulus demonstrated above. We hypothesise that the signal of the $n$-th
stimulus is a combination of the pre-stimulus ageing, described by Eq. (6),
and the actual training, described by Eq. (7), while we neglect the small
effect of the post-stimulus signal degradation. We show in Section V of the
Supplementary Information that a weighted sum of the two effects adequately
describes the data, which results in the prediction

$S_{n}\approx\frac{1}{2}\begin{cases}1-e^{-\frac{t^{\mathrm{train}}}{\tau_{\mathrm{train}}}}&n=1\\\
e^{-\frac{t^{\mathrm{age}}_{n-1}}{\tau_{\mathrm{pre}}}}+\left(\frac{n}{2}\right)^{1-n}\left(1-e^{-\frac{t_{\mathrm{train}}}{\tau_{\mathrm{train}}}}\right)&n>1\;,\end{cases}$
(9)

where $t^{\mathrm{age}}_{n}$ is given by Eq. (5). This equation correctly
captures that $S_{n}$ decreases with the $n-1$ previously applied stimuli.
Fig. 4(b) shows that Eq. (9) also captures the qualitative features of the
dependence on $t^{\mathrm{train}}$ and $t^{\mathrm{wait}}$. However, the
prediction overestimates the signal of mid-timed stimuli, likely because we
neglect the post-stimuli degradation. We also note that this analytical
description does not reproduce the quantitative features of the signal
observed numerically because the signal is not just a linear superposition of
the training and age impact; see Appendix E. Even though more research is
needed to obtain the exact dependency of the signal on training and age, we
choose to use the simple function to draw insights about the system. For
instance, we observe that the ratio of the coefficient of training and the
coefficient of age reduces with $n$, indicating that with every new stimulus
application the impact of age on memory formation becomes stronger. The
advantage of the prediction is its simplicity. Moreover, the prediction is an
ad-hoc description of the parameter dependence and other choices are possible;
see Appendix E. The stimulation protocol is characterised by the three
parameters $n$, $t^{\mathrm{train}}$, and $t^{\mathrm{wait}}$, while the
almost identical $\tau_{\mathrm{pre}}$ and $\tau_{\mathrm{train}}$ capture the
characteristic time scale of network adaptation.

Finally, we investigate how many stimuli an adaptive network can store. We
demand that a stored stimulus can be read out at a later time, implying that
its signal exceeds a given threshold $S_{\mathrm{thresh}}$, which captures
uncertainties in the read-out apparatus as well as intrinsic noise. Fig. 4(c)
shows the fraction of stimuli that can be retrieved (where
$S_{n}>S_{\mathrm{thresh}}$) as a function of the total number of stimuli,
$N$, and the training time $t^{\mathrm{train}}$. In this case, large training
times are detrimental since they age the network too much for later stimuli to
be retrieved. Fig. 4(d) shows the number of stimuli that can be retrieved as a
function of the training and waiting time. The largest number of stimuli is
stored for smaller waiting time, as this reduces the age of the network. To
find the optimal parameters for storing memory independent of a read-out
apparatus specific threshold $S_{\mathrm{thresh}}$, we quantify the minimum
signal out of the five stimuli’s signals for varying training and waiting
time, see Fig. 4(e). We observe that while the optimal waiting time is $0$, a
non-zero optimal training time exists for storing memory. Taken together, our
analysis reveals the strong trade-off between writing stimuli for a sufficient
duration and the resulting inevitable ageing of the network that suppresses
signals of subsequent stimuli.

##  IV Discussion

We showed that adaptive flow networks can store memory of multiple stimuli in
the morphology of weak links, which cannot be revived in our model.
Consequently, signatures of earlier stimuli are not destroyed by subsequent
evolution, in contrast to the behaviour of typical mechanical networks [48].
Since older networks contain fewer strong links, which could shrink to store
memory, the readout signal of each stimulus strongly decreases with the age of
the network before the stimulus was written, which is similar to the memory
plasticity observed in disordered system [49, 50]. Conversely, the signal
strength increases with its training time, i.e., the duration the stimulus is
presented, similar to memory formation by directed aging in mechanical
networks [24, 22]. Taken together, we showed that adaptive flow networks reach
maximal capacity at an intermediate training time, which compromises between
imprinting sufficiently and ageing minimally.

Our work focuses on the simple situation that non-overlapping stimuli are
subsequently applied at the edge of flow networks of similar morphology. To
describe, realistic living flow networks, like Physarum or our vasculature,
our work will need to be extended in multiple directions: First, the overall
network geometry will have an impact on how stimuli are stored. Work in
mechanical networks [27, 48] suggests that the internal timescales of flow
networks and their memory capacity will depend on network size. Second,
realistic systems deal with time-varying and potentially overlapping stimuli
of various strengths. Third, living systems can grow and expand [51], implying
that links can possibly regrow from their minimal size and new links can be
added to the network. Taken together, it is likely that realistic adaptive
flow networks show a dynamic behaviour, storing information about stimuli on
various time scales.

Taken together, our work quantifies the memory capacity of adaptive flow
networks for varying parameters, crucial for understanding the emergent
behaviour of non-neuronal organism P. polycephalum.

###### Acknowledgements.

This work was supported by the Max Planck Society. This project has received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation program (Grant Agreement No. 947630,
FlowMem).

##  Appendix A Model of adaptive flow network

In this section. we give a detailed derivation of the iteration rule for our
model of adaptive networks. A networks is modelled as a connected set of
straight tubes described by a graph with $N_{\mathrm{nodes}}$ nodes enumerated
by $i=1,2,\ldots,N_{\mathrm{nodes}}$ and tube $ij$ of length $l_{ij}$, which
connect node $i$ and $j$. Assuming Poiseuille flow in all tubes, conductances
$C_{ij}$ are assigned to all tubes. To study the flow in the network, we
designate the first node ($i=1$) as the outlet, while all other nodes ($i>1$)
are inlets with inflow $q_{i}(t)$. Kirchhoff’s current law then reads
$L(t)\vec{p}(t)=\vec{q}(t)$, where
$\vec{p}(t)=\\{p_{1}(t),p_{2}(t),\ldots,p_{N_{\mathrm{nodes}}}(t)\\}$
represents the pressures at all nodes,
$\vec{q}(t)=\\{q_{1}(t),q_{2}(t),\ldots,q_{N_{\mathrm{nodes}}}(t)\\}$
represents the inflows at all nodes, and $L$ is the graph Laplacian matrix
with elements $L_{ij}=\delta_{ij}\sum_{n}C_{in}-C_{ij}$, where $C_{ij}=0$ if
tube $ij$ is not present. We use Kirchoff’s current law to find the
instantaneous flow rate $Q_{ij}(t)=C_{ij}(t)(p_{i}(t)-p_{j}(t))$ through tube
$ij$ at every time step $t$. Following [1, 45, 46, 44], we evolve the network
by adapting its conductances to minimise the power loss of the network,

$E(t)=\sum_{<ij>}\frac{\langle Q_{ij}(t)^{2}\rangle_{T}}{C_{ij}(t)}\;,$ (10)

while obeying the constraint of the total network volume,

$\mathcal{K}^{1/2}=\sum_{<ij>}(C_{ij}(t)l_{ij})^{1/2}l_{ij}\;.$ (11)

The optimal conductance $C_{ij}^{\star}(t)$ that minimise Eq. (10) at $t$,
maintaining the constraint of available material given by Eq. (11),

$C_{ij}^{\star}(t)=\frac{\langle
Q_{ij}(t)^{2}\rangle_{T}^{2/3}\mathcal{K}}{\left(\sum_{<ij>}\langle
Q_{ij}(t)^{2}\rangle_{T}^{1/3}l_{ij}\right)^{2}l_{ij}}\;.$ (12)

The conductances $C_{ij}(t+\delta t)$ at time step $t+\delta t$ adapt to the
optimal conductance $C_{ij}^{\star}(t)$

$C_{ij}(t+\delta t)\leftarrow C_{ij}^{\star}(t)\;.$ (13)

We then repeat this rule iteratively over many time steps. Taken together,
this iterative rule for adapting conductances follows from independently
optimising conductances $C_{ij}$ to minimises the power loss of the network
while keeping the total network volume fixed.

In this work we model the network as an hexagonal network, with an addition of
noise to the node positions to reduce artifacts generated from the
regularities of the network. The node at the center of the network, $i=1$, is
the sole outlet, with all the nodes in the network are sources of inflows
which fluctuate by either turning on $q_{i>1}=2q^{(0)}$ or off $q_{i>1}=0$
with equal probability. Meaning the average base inflow through the source
nodes are $<q_{i>1}>=q^{(0)}$.

##  Appendix B Signal of last stimulus changes with its training and
relaxation time

Figure 5: Signal $S_{N}$ of final stimulus depend on its training time
$t^{\mathrm{train}}_{N}$ and waiting time $t^{\mathrm{wait}}_{N}$. (a–c)
$S_{N}$ for varying of $t^{\mathrm{train}}_{N}$ and $t^{\mathrm{wait}}_{N}$
for three different parameter sets. (d–f) The same data $S_{N}$ is plotted
against $t^{\mathrm{wait}}_{N}$ for different $t^{\mathrm{train}}_{N}$
(colors). Lines represent exponential fits. (a–f) Model parameters:
$t^{\mathrm{wait}}_{\mathrm{pre}}=5\delta
t$,$N_{\mathrm{node}}=760$,$q^{\mathrm{add}}=2000q^{(0)}$, $q^{(0)}=1$,
$T=30$, $\mathcal{K}=1600$.

We observe that the memory signal $S_{N}$ of the last stimulus increases with
its training time $t^{\mathrm{train}}_{N}$ and decreases with its waiting time
$t^{\mathrm{wait}}_{N}$ independent of the pre-stimulation protocol, see Fig
5. This behaviour is similar to the signal of a stimulus written in a freshly
initiated untrained network [19]. In particular,
$S_{N}(t^{\mathrm{wait}}_{N})$ can be fitted with exponential decay functions,
which is similar to our observation in the main manuscript of $S_{1}$ decaying
with the age after the stimulus application,
$t_{\mathrm{post}}^{\mathrm{age}}$, following an exponential decay, while
$S_{1}$ increases with the training time $t_{1}^{\mathrm{train}}$. Since, here
the age after stimulus application is
$t_{\mathrm{post}}^{\mathrm{age}}=t_{N}^{\mathrm{wait}}$. These observations
imply a general dependency of signal $S_{n}$ on age, post stimulus application
$t_{\mathrm{post}}^{\mathrm{age}}$, and training time of that particular
stimulus, $t_{n}^{\mathrm{train}}$, for all $n$.

##  Appendix C Stable thick tubes do not decay irreversibly between stimuli

Figure 6: Dynamics of networks when a stimulus is enabled or disabled. (a–c)
Network morphology at three consecutive time steps during which a stimulus is
switched off. Color code indicates relative change of conductances,
$C(t+\delta t)/C(t)$. (d, e) $C(t+\delta t)/C(t)$ as a function of $C(t)$ for
$6$ different time point around the transition time $t=21$ where the stimulus
is switch off (panel d) or switch on (panel e). Model parameters are
$N_{\mathrm{nodes}}=560$, $T=30\delta t$, and $q^{(0)}=1$.

To understand how adaptive networks evolve between stimuli, we here study the
dynamics of conductances by measuring $C(t+\delta t)/C(t)$ for all the tubes
of the network before and after the transition of switching the additional
load from $q^{\mathrm{add}}=1000q^{(0)}$ to $q^{\mathrm{add}}=0$. We map the
the dynamics of conductances on the network at $3$ different time step,
$t=t_{\mathrm{trans}}-\delta t,t_{\mathrm{trans}},t_{\mathrm{trans}}+\delta
t$, where $t_{\mathrm{trans}}$ is the transition time point when the stimulus
is switched off. Fig. 6(a) shows that before the transition, the adaptive
network shows a hierarchical network structure with thick tubes connecting the
nodes where the stimulus is applied (blue nodes) to the outlet of the network
(red node). In particular, $C(t+\delta t)/C(t)$ of the thick tubes are
approximately $1$, indicating that they do not evolve further. After switching
the stimulus off at the transition time at $t_{\mathrm{trans}}$, we notice
that the conductances of the thick tubes start decaying, while other tubes
grow; see Fig. 6(b). Already one step later, at $t_{\mathrm{trans}}+\delta t$,
the network assumes an isotropic, tree-like structure; see Fig. 6(c). The
quantification in Fig. 6(d, e) indicates that the conductances follow the
universal dynamics of adaptive networks [19], where conductances below a
threshold conductance $C_{\mathrm{th}}$ decay, while larger conductances
fluctuate over time. The threshold value reads [19]

$C_{\mathrm{th}}=\frac{\mathcal{K}}{\left(1+\frac{\delta
q}{q^{(0)}}\right)^{4}\left(N_{\mathrm{nodes}}+\frac{q^{\mathrm{add}}}{q^{(0)}}\right)^{4/3}}\;,$
(14)

where $N_{\mathrm{nodes}}$ denotes the number of nodes in a network, and
$\delta q$ quantifies load fluctuations.

The conductances show an unique dynamics at the exact time step when the
stimulus is switched, at $t=t_{\mathrm{trans}}=21$. As an example, we observe
a cluster of tubes with high conductances, either decaying see Fig. 6(d) or
growing see Fig. 6(e). As we know from [19] that all the conductance below the
threshold $C_{\mathrm{th}}$ given by Eq. (14) will decay irreversibly, we plot
the $C(t+\delta t)=C_{\mathrm{th}}$ line (the blue line in Fig. 6(d) and (e))
to track the tubes that will decay irreversibly. Here, $C_{th}$ is obtained
from fitting the dynamics of thin tubes to,

$\frac{C(t+\delta t)}{C(t)}=\Big{[}\frac{1}{C_{th}}\Big{]}^{1/3}C(t)^{1/3},$
(15)

for a constant $q^{add}$ following [19]. In Fig. 6(d), the line corresponds to
$C_{th}(q^{add}=0)/C(t)$ vs $C(t)$, and in Fig. 6(e) the line corresponds to
$C_{th}(q^{add}=1000)/C(t)$ vs $C(t)$. Indicating all the data points with
$C(t+\delta t)/C(t)$ higher than this line have $C(t+\delta t)>C_{th}$ after
the transition.

We observe that the data points belonging to the cluster of high conductance
tubes (at $t=21$) are above of the line, indicating although these tubes decay
during the transition of switching the stimulus to $0$, yet does not decay
irreversibly, see Fig. 6(d).

These observations imply that, the tubes with very large conductances do not
decay irreversibly due to the transitions of stimulus strength ($q^{add}$),
and are retained in the network.

##  Appendix D Thin tubes cannot be recovered between stimuli

We next attempt to understand the dynamics of smaller tubes during the
transition between stimuli.

For this, we calculate the tubes’ dynamics in the minimal triangular network
shown in Fig. 7(a), which follows the same adaptation rule as the simulated
adaptive networks,

$\displaystyle C_{ij}(t+\delta t)$
$\displaystyle=Q_{ij}(t)^{4/3}.\frac{\mathcal{K}}{A(t)^{2}}$ with
$\displaystyle A(t)$ $\displaystyle=\sum_{<i,j>}Q_{ij}(t)^{2/3}\;.$ (16)

We focus on the dynamics when $q^{\mathrm{add}}=1000\rightarrow 0$, but we
expect the opposite case of $q^{\mathrm{add}}=0\rightarrow 1000$ to exhibit
similar behaviour.

Figure 7: Minimal asymmetric network during the transition step, where the
stimulus of addition is inflow switched off. (a) Minimal network at $t-\delta
t$ where the inflow at node 2 is $q^{(0)}$ and at node 3 is
$N_{\mathrm{nodes}}q^{(0)}+q^{\mathrm{add}}$. (b) Minimal network at time $t$,
where the inflow at node 2 is $q^{(0)}+\delta q$ and node 3 is
$N_{\mathrm{nodes}}q^{(0)}$.

For the minimal network, we can express Kirchhoff’s current law,
$L\vec{p}=\vec{q}$, explicitely

$\left[{\begin{array}[]{c}q_{1}\\\ q_{2}\\\ q_{3}\\\
\end{array}}\right]=\left[{\begin{array}[]{ccc}C_{13}+C_{12}&-C_{12}&-C_{13}\\\
-C_{12}&C_{12}+C_{23}&-C_{23}\\\ -C_{13}&-C_{23}&C_{13}+C_{23}\\\
\end{array}}\right]\left[{\begin{array}[]{c}p_{1}\\\ p_{2}\\\ p_{3}\\\
\end{array}}\right]\;;$ (17)

see Section A. Setting $p_{1}=0$ because node 1 is a sink, we have

$\left[{\begin{array}[]{cc}C_{12}+C_{23}&-C_{23}\\\ -C_{23}&C_{13}+C_{23}\\\
\end{array}}\right]^{-1}\left[{\begin{array}[]{c}q_{2}\\\ q_{3}\\\
\end{array}}\right]=\left[{\begin{array}[]{c}p_{2}\\\ p_{3}\\\
\end{array}}\right]$ (18)

and thus

$\frac{1}{C_{13}(C_{12}+C_{23})+C_{12}C_{23}}\left[{\begin{array}[]{cc}C_{13}+C_{23}&C_{23}\\\
C_{23}&C_{12}+C_{23}\\\ \end{array}}\right]\\\
\left[{\begin{array}[]{c}q_{2}\\\ q_{3}\\\
\end{array}}\right]=\left[{\begin{array}[]{c}p_{2}\\\ p_{3}\\\
\end{array}}\right]\;.$ (19)

Solving for $p_{2}$ and $p_{3}$, we find

$\begin{split}p_{2}&=\frac{1}{C_{13}(C_{12}+C_{23})+C_{12}C_{23}}\Big{(}(C_{13}+C_{23})q_{2}+C_{23}q_{3}\Big{)}\\\
p_{3}&=\frac{1}{C_{13}(C_{12}+C_{23})+C_{12}C_{23}}\Big{(}C_{23}q_{2}+(C_{12}+C_{23})q_{3}\Big{)}\;.\end{split}$
(20)

We assume the network has been adapted with a stimulus of an additional inflow
and reached a local energy minimum at $t-\delta t$, implying a tree morphology
which connects all inlets directly to the outlet. We can then assume the
inflow on node $2$ and $3$ is distributed through tubes ${12}$ and ${13}$,
respectively. Without loss of generality, we assume the additional load has
been applied to node 3, implying $Q_{21}(t-\delta t)=q_{2}(t-\delta t)=q_{2}$
and $Q_{31}(t-\delta t)=q_{3}(t-\delta t)=q_{3}+q^{\mathrm{add}}$. Using (16),
we can then approximate the conductances of tube $12$ and $13$ as
$C_{12}(t)=q_{2}(t-\delta t)^{4/3}\mathcal{K}/A(t-\delta t)^{2}$ and
$C_{13}(t)=q_{3}(t-\delta t)^{4/3}\mathcal{K}/A(t-\delta t)^{2}$,
respectively, where we assumed $C_{23}(t)/C_{12}(t)\ll 1$ and
$C_{23}(t)/C_{13}(t)\ll 1$ since $Q_{23}(t-\delta t)\ll Q_{21}(t-\delta t)$
and $Q_{23}(t-\delta t)\ll Q_{31}(t-\delta t)$.

During the transition from $t-\delta t$ to $t$, the stimulus is switched from
a non-zero value to $0$, implying the load at node $3$ reduces to
$q_{3}(t)=q_{3}$. The load at node $2$ also changes to $q_{2}(t)=q_{2}+\delta
q$ due to the background fluctuation of magnitude $\delta q$, where we assume
this perturbation to be small compared to the load at node 2, $\delta
q/q_{2}\ll 1$. In general, the minimal network is connected with a bigger
network with $N_{\mathrm{nodes}}$ nodes, so the calculation we present in the
following subsections considers the simple case of an asymmetric connection,
with highest inflow difference, as shown in the Fig. 7(a),

$q_{2}=q^{(0)},\>q_{3}\approx N_{\mathrm{nodes}}q^{(0)}\;,$ (21)

which implies tube $12$ is thinner than $13$, $C_{12}(t)<C_{13}(t)$, in the
steady state network.

###  D.1 Dynamics of medium thick tubes

The flow rate $Q_{12}$ through tube ${12}$ reads

$Q_{12}(t)=C_{12}(p_{2}-p_{1})\\\
=C_{12}(t)\frac{[C_{13}(t)+C_{23}(t)]q_{2}(t)+C_{23}(t)q_{3}(t)}{C_{13}(t)(C_{12}(t)+C_{23}(t))+C_{12}(t)C_{23}(t)}$
(22)

where we used Eq. (20). Inserting the values of $C_{13}(t)$, $C_{12}(t)$,
$q_{2}(t)$, and $q_{3}(t)$ obtained above, we find

$Q_{21}(t)=C_{12}(t)\frac{(q_{3}+q^{\mathrm{add}})^{4/3}(q_{2}+\delta
q)}{\Big{(}q_{2}(q_{3}+q^{\mathrm{add}})\Big{)}^{4/3}}\cdot\frac{A(t-\delta
t)^{2}}{\mathcal{K}}\;.$ (23)

Using Eq. (23) in Eq. (16), we can write

$C_{12}(t+\delta
t)=C_{12}(t)^{4/3}\frac{(q_{3}+q^{\mathrm{add}})^{16/9}(q_{2}+\delta
q)^{4/3}}{\Big{(}q_{2}(q_{3}+q^{\mathrm{add}})\Big{)}^{16/9}}\\\
\cdot\frac{A(t-\delta
t)^{8/3}}{\mathcal{K}^{4/3}}\frac{\mathcal{K}}{A(t)^{2}}$ (24)

and

$\frac{C_{12}(t+\delta t)}{C_{12}(t)}=\left(1+\frac{\delta
q}{q_{2}}\right)^{4/3}\frac{\Big{(}q_{2}^{2/3}+(q_{3}+q^{\mathrm{add}})^{2/3}\Big{)}^{2}}{\Big{(}(q_{2}+\delta
q)^{2/3}+q_{3}^{2/3}\Big{)}^{2}}\;.$ (25)

Expanding in the small parameter $q^{(0)}/q^{\mathrm{add}}$ results in

$\frac{C_{12}(t+\delta
t)}{C_{12}(t)}=\frac{(N_{\mathrm{nodes}}+\frac{q^{\mathrm{add}}}{q^{(0)}})^{4/3}}{N_{\mathrm{nodes}}^{4/3}}=\left(1+\frac{q^{\mathrm{add}}}{N_{\mathrm{nodes}}q^{(0)}}\right)^{4/3}\;,$
(26)

which implies the medium thick tubes $12$ will grow when the stimulus is
switched off.

###  D.2 Dynamics of thin tubes with largest inflow difference

We next calculate the dynamics of the thin tube $23$ of the minimal network
shown in Fig. 7. The flow rate $Q_{23}$ through tube ${23}$ follows from Eq.
(20),

$Q_{23}(t)=C_{23}(p_{2}-p_{3})=C_{23}\frac{C_{13}q_{2}-C_{12}q_{3}}{C_{13}(C_{12}+C_{23})+C_{12}C_{23}}\;.$
(27)

Inserting the values of $C_{13}(t)$, $C_{12}(t)$, $q_{2}(t)$, $q_{3}(t)$, we
find

$Q_{23}(t)=C_{23}(t)\frac{(q_{3}+q^{\mathrm{add}})^{4/3}(q_{2}+\delta
q)-(q_{2})^{4/3}(q_{3})}{\Big{(}q_{2}(q_{3}+q^{\mathrm{add}})\Big{)}^{4/3}}\\\
\cdot\frac{A(t-\delta t)^{2}}{\mathcal{K}}\;.$ (28)

Using Eq. (16), we write the conductance $C_{23}(t+\delta t)$ at time
$t+\delta t$ as a function of conductance $C(t)$ at time $t$,

$\frac{C_{23}(t+\delta
t)}{C_{23}(t)}=\mathcal{S}_{\mathrm{disable}}C_{23}(t)^{1/3}\;,$ (29)

where the prefactor reads

$\mathcal{S}_{\mathrm{disable}}=\frac{\Big{(}(q_{3}+q^{\mathrm{add}})^{4/3}(q_{2}+\delta
q)-(q_{2})^{4/3}(q_{3})\Big{)}^{4/3}}{\Big{(}q_{2}(q_{3}+q^{\mathrm{add}})\Big{)}^{16/9}}\\\
\cdot\frac{A(t-\delta t)^{8/3}\mathcal{K}}{\mathcal{K}^{4/3}A(t)^{2}}\;.$ (30)

Since $A(t-\delta t)=\Big{(}q_{2}^{2/3}+(q_{3}+q^{\mathrm{add}})^{2/3}\Big{)}$
and $A(t)=\Big{(}(q_{2}+\delta q)^{2/3}+q_{3}^{2/3}\Big{)}$, the prefactor
becomes

$\mathcal{S}_{\mathrm{disable}}=\frac{\Big{(}(q_{3}+q^{\mathrm{add}})^{4/3}(q_{2}+\delta
q)-(q_{2})^{4/3}(q_{3})\Big{)}^{4/3}}{\Big{(}q_{2}(q_{3}+q^{\mathrm{add}})\Big{)}^{16/9}}\\\
\cdot\frac{\Big{(}q_{2}^{2/3}+(q_{3}+q^{\mathrm{add}})^{2/3}\Big{)}^{8/3}}{\mathcal{K}^{1/3}\Big{(}(q_{2}+\delta
q)^{2/3}+q_{3}^{2/3}\Big{)}^{2}}\;.$ (31)

Inserting the values of $q_{2}$ and $q_{3}$ fromEq. (21), we obtain

$\mathcal{S}_{\mathrm{disable}}=\frac{\Big{(}\Big{[}\frac{1}{N_{\mathrm{nodes}}+\frac{q^{\mathrm{add}}}{q^{(0)}}}\Bigr{]}^{2/3}+1\Big{)}^{8/3}}{\mathcal{K}^{1/3}}\\\
\cdot\frac{\Big{(}\frac{1}{N_{\mathrm{nodes}}}(N_{\mathrm{nodes}}+\frac{q^{\mathrm{add}}}{q^{(0)}})^{4/3}(1+\frac{\delta
q}{q^{(0)}})-1\Big{)}^{4/3}}{\Big{(}(\frac{1}{N_{\mathrm{nodes}}}(1+\frac{\delta
q}{q^{(0)}}))^{2/3}+1\Big{)}^{2}}\;.$ (32)

Assuming a large network, $N_{\mathrm{nodes}}q^{(0)}+q^{\mathrm{add}}\gg
q^{(0)}$, we find

$\mathcal{S}_{\mathrm{disable}}\approx\frac{(1+\frac{\delta
q}{q^{(0)}})^{4/3}\Big{(}N_{\mathrm{nodes}}+\frac{q^{\mathrm{add}}}{q^{(0)}}\Big{)}^{16/9}}{N_{\mathrm{nodes}}^{4/3}\mathcal{K}^{1/3}}\\\
=\frac{(1+\frac{\delta
q}{q^{(0)}})^{4/3}\Big{(}N_{\mathrm{nodes}}+\frac{q^{\mathrm{add}}}{q^{(0)}}\Big{)}^{4/9}\Big{(}1+\frac{q^{\mathrm{add}}}{N_{\mathrm{nodes}}q^{(0)}}\Big{)}^{4/3}}{\mathcal{K}^{1/3}}\;.$
(33)

Since tubes with $C(t+\delta t)<C_{\mathrm{th}}(q^{\mathrm{add}}=0)$ will
decay irreversibly after the stimulus is disabled, we find

$\mathcal{S}_{\mathrm{disable}}C(t)^{4/3}<C_{\mathrm{th}}(q^{\mathrm{add}}=0)$
(34)

and thus

$C(t)<\left(\frac{C_{\mathrm{th}}(q^{\mathrm{add}}=0)}{S_{\mathrm{disable}}}\right)^{3/4}\;.$
(35)

Using $C_{\mathrm{th}}$ from Eq. (14) and $\mathcal{S}_{\mathrm{disable}}$
from Eq. (33), we find

$\left(\frac{C_{\mathrm{th}}(q^{\mathrm{add}}=0)}{S_{\mathrm{disable}}}\right)^{3/4}=\Big{(}\frac{\mathcal{K}}{(1+\frac{\delta
q}{q^{(0)}})^{4}N_{\mathrm{nodes}}^{4/3}}\\\
\cdot\frac{\mathcal{K}^{1/3}}{(1+\frac{\delta
q}{q^{(0)}})^{4/3}\Big{(}N_{\mathrm{nodes}}+\frac{q^{\mathrm{add}}}{q^{(0)}}\Big{)}^{4/9}\Big{(}1+\frac{q^{\mathrm{add}}}{N_{\mathrm{nodes}}q^{(0)}}\Big{)}^{4/3}}\Big{)}^{3/4},\\\
=\frac{K}{(1+\frac{\delta
q}{q^{(0)}})^{4}(N_{\mathrm{nodes}}+\frac{q^{\mathrm{add}}}{q^{(0)}})^{4/3}}=C_{\mathrm{th}}(q^{\mathrm{add}})\;.$
(36)

This implies tubes with $C(t)>C_{\mathrm{th}}(q^{\mathrm{add}})$ will be
retained in the network even after disabling the stimulus. Moreover, tubes
that were decaying in the presence of the stimulus will continue decaying even
when the stimulus is disabled.

Figure 8: Dynamics of tube conductances during the transition of disabling a
stimulus matches analytical approximations. (a) Average conductance changes
$\langle C(t+\delta t)/C(t)\rangle$ of medium thick tubes in networks as a
function of the corresponding analytical prediction given by Eq. (26). The
solid line is a power-law fit. (b) The pre-factor
$\mathcal{S}_{\mathrm{disable}}$, obtained from fitting the dynamics of thin
tubes to $C(t+\delta t)/C(t)=\mathcal{S}C(t)^{1/3}$, as a function of the
prediction given by Eq. (33). , including the correction of the power obtained
from (a), shows the data follows a straight line with the corrected
prediction. (a–b) For varying $q^{(0)}$ and $q^{\mathrm{add}}$ and fixed model
parameters $N_{\mathrm{nodes}}=760$, $T=30\delta t$.

Fig. 8 shows that our analytical prediction is reasonably correct. However, we
observe that the measured value of the dynamics of the medium thick tubes are
slightly lower than that of the analytical prediction, probably because the
calculation only considers the most extreme case of highest inflow difference.
Taken together, we show the established hierarchy of tube conductances cannot
be erased by switching the stimulus from a non-zero value to $0$ or vice
versa.

##  Appendix E Memory signal from combination of age and training components

Figure 9: Memory signal $S_{n}$ depends on training time $t^{\mathrm{train}}$
and age of the network before training. (a) Analytical prediction assuming
$S_{n}$ is a product of the impact of age and training. (b) $S_{n}$ as a
function of $t^{\mathrm{train}}$ for various $n$. (c) Fit parameters
(coefficient of training and age component) obtained while fitting memory
readout signal as a sum of training and age component, plotted against the
probed stimulus. (d) $R^{2}$ of the fit obtained from fitting the signal as
various combination of training and age with respect to the probed stimulus.

In the main text, we observed the signal $S_{1}$ of the first stimulus
increases with the training time $t^{\mathrm{train}}$,

$S_{1}(t^{\mathrm{train}})\sim(1-e^{-t^{\mathrm{train}}/\tau_{\mathrm{train}}})\;.$
(37)

We also observed that the signal $S_{N}$ of the last stimulus decays with the
age $t^{\mathrm{age}}_{N-1}$ before the stimulus application,

$S_{N}(t^{\mathrm{age}}_{N-1})\sim
e^{-t^{\mathrm{age}}_{N-1}/\tau_{\mathrm{pre}}}\;.$ (38)

Combining these observations, we hypothesise that the signals $S_{n}$ of
general stimuli between the first and last stimulus are affected by both
effects. $S_{n}$ should thus be a function of
$S^{\mathrm{train}}_{n}=(1-e^{-t^{\mathrm{train}}_{n}/\tau_{\mathrm{train}}})$
and $S^{\mathrm{age}}_{n}=e^{-t^{\mathrm{age}}_{n-1}/\tau_{\mathrm{pre}}}$.

We start by asking whether a simple product is an adequate description,

$S_{n}=S^{\mathrm{age}}_{n}S^{\mathrm{train}}_{n}=e^{-t^{\mathrm{age}}_{n-1}/\tau_{\mathrm{pre}}}(1-e^{-t^{\mathrm{train}}_{n}/\tau_{\mathrm{train}}})\;.$
(39)

Fig. 9(a) and Fig. 9(b) shows that this prediction cannot reproduce the
qualitative dependency of the signal on $t^{\mathrm{train}}$ and
$t^{\mathrm{wait}}$. In particular, Eq. (39) suggests a maximal signal for a
non-zero training time for any $n$, while the numerical data shown in Fig. 4
of the main text shows that the signal is strongest for very small
$t^{\mathrm{train}}$ and $t^{\mathrm{wait}}$, specifically for $n>3$.

Figure 10: Analytical approximation of signal w.r.t training and waiting time
when nth stimulus is probed (a) Numerically obtained memory read out signal
with respect to training and waiting time as shown in Fig. 4(a) (b) Signal
$S_{n}$ approximated as a product of $S_{n}^{\mathrm{age}}$ and
$S_{n}^{\mathrm{train}}$, in Eq. (39), with parameters
$\tau_{\mathrm{pre}}=50\delta t$ and $\tau_{\mathrm{train}}=50\delta t$. (c)
Signal $S_{n}$ approximated as a linear sum of $S_{n}^{\mathrm{age}}$ and
$S_{n}^{\mathrm{train}}$ as shown in Eq. (40) using the functional form found
from fitting the data Eq. (41), with parameters $\tau_{\mathrm{pre}}=50\delta
t$ and $\tau_{\mathrm{train}}=50\delta t$. (d)
$S_{n}=\frac{1}{2}\big{(}e^{-t^{\mathrm{age}}_{n-1}/\tau_{\mathrm{pre}}}+\frac{n}{2}^{-n+1}(1-e^{-t^{\mathrm{train}}/\tau_{\mathrm{train}}})\big{)}$,
with parameter $\tau_{\mathrm{pre}}=50\delta t$ and
$\tau_{\mathrm{train}}=62\delta t$ (e)
$S_{n}=(1-e^{-(n+1)/1.5})e^{-t^{\mathrm{age}}_{n-1}/\tau_{\mathrm{pre}}}+e^{-(n-1)/1.5}(1-e^{-t^{\mathrm{train}}/\tau_{\mathrm{train}}})$
with parameters $\tau_{\mathrm{pre}}=50\delta t$ and
$\tau_{\mathrm{train}}=50\delta t$.

We next hypothesis that $S_{n}$ is a linear combination of
$S^{\mathrm{age}}_{n}$ and $S^{\mathrm{train}}_{n}$,

$S_{n}=f^{\mathrm{train}}_{n}S^{\mathrm{age}}_{n}+f^{\mathrm{age}}_{n}S^{\mathrm{train}}_{n}\;,$
(40)

where we allow different coefficients $f^{\mathrm{train}}_{n}$ and
$f^{\mathrm{age}}_{n}$ for each stimulus. We determine these by fitting
$S_{n}/S^{\mathrm{age}}_{n}(t^{\mathrm{train}},t^{\mathrm{wait}})$ to
$f^{\mathrm{age}}_{n}+f^{\mathrm{train}}_{n}S^{\mathrm{train}}_{n}/S^{\mathrm{age}}_{n}(t^{\mathrm{train}},t^{\mathrm{wait}})$
for different values of $n$ over a range of $t^{\mathrm{train}}/\delta
t\in[1,40]$ and $t^{\mathrm{wait}}/\delta t\in[1,40]$. Fig. 9(c) shows that a
suitable fit is given by

$\begin{split}f^{\mathrm{train}}_{n}&=0.14+0.386e^{-n/1.75}\\\
f^{\mathrm{age}}_{n}&=0.0475+0.0023e^{n/0.99}\end{split}$ (41)

We note in Fig. 9(d) that the $R^{2}$ error of the fit to compute the
coefficients of each component is $\approx 0.4$, indicating that the signal is
not just a linear superposition of the components. We use a few different
models to fit the data. As example, we fit the signal to
$S_{n}=f_{n}^{\mathrm{train}}S_{n}^{\mathrm{train}}+f_{n}^{\mathrm{age}}(S_{n}^{\mathrm{age}})^{3}$
as we do not expect the power of the age component to change with n.
Additionally we fit the signal to some other simple non-linear composition of
age and training component, as example
$S_{n}=f_{n}^{\mathrm{train}}(S_{n}^{\textrm{train}})^{n}+f_{n}^{\mathrm{age}}S_{n}^{\textrm{age}}$
and
$S_{n}=f_{n}^{\mathrm{train}}(S_{n}^{\textrm{train}})^{1/n}+f_{n}^{\mathrm{age}}S_{n}^{\textrm{age}}$.
However, none of these models increase the $R^{2}$ of the fit significantly.
So for simplicity, we choose to approximate the signal as a linear sum of
$S^{\mathrm{train}}_{n}$ and $S^{\mathrm{age}}_{n}$. We observe, if we choose
the coefficients of $S^{\mathrm{train}}_{n}$ and $S^{\mathrm{age}}_{n}$
following (41), the signal of a stimulus obtained using this approximation
qualitatively agrees with the numerical observation, see Fig. 10(c).

Now to obtain a generic analytical approximation of the memory read-out signal
without including many different parameters, that also describes the signal as
a linear superposition of the age and training impact, we assume that the
coefficient of $S^{\mathrm{age}}_{n}$ does not change and the coefficient of
$S^{\mathrm{train}}_{n}$ decreases of n. We assume these coefficients because,
Eq. (41) suggests that the coefficient of the training component reduces with
$n$ and the coefficient of age component increase with $n$. Moreover, the fit
parameters show that the change of the coefficient of age component over n
($\sim 0.0023$) is much smaller than the change of the coefficient of the
training component over n ($\sim 0.36$), which agrees with our observation in
Fig. 2(b).

Using these observations we now choose the following bounded function which is
a sum of $S_{n}^{\mathrm{age}}$ and $S_{n}^{\mathrm{train}}$ to approximate
the signal of $n$th stimulus when $n>1$.

$\begin{split}S_{n}&=e^{-t^{\mathrm{age}}_{n-1}/\tau_{\mathrm{pre}}}+f^{\mathrm{train}}_{n}(1-e^{-t^{\mathrm{train}}/\tau_{\mathrm{train}}}),\\\
&=\frac{1}{2}\big{(}e^{-t^{\mathrm{age}}_{n-1}/\tau_{\mathrm{pre}}}+(\frac{n}{2})^{-n+1}(1-e^{-t^{\mathrm{train}}/\tau_{\mathrm{train}}})\big{)}.\end{split}$
(42)

Additionally, from the numerical observation, we approximate for $n=1$,

$S_{1}=\frac{1}{2}(1-e^{-t^{\mathrm{train}}/\tau_{\mathrm{train}}}).$ (43)

Although Eq. (42) reproduces, the dependency of signal on training and waiting
time qualitatively (see, Fig. 4(b) and Fig. 10(d)), this does not reproduce
the dependency quantitatively. However, we show that the same qualitative
dependency on training and waiting time is robust to small changes of
parameters. As example, when Eq. (42) is plotted assuming
$\tau_{\mathrm{pre}}=50\delta t$ and $\tau_{\mathrm{train}}=62\delta t$, see
Fig. 10(d), the qualitative feature does not change from the observation in
Fig. 4(b). Similarly the qualitative dependency does not change, when
different functions $f^{\mathrm{train}}_{n}$ and $f^{\mathrm{age}}_{n}$ are
chosen as the coefficients of the training and age impact. As an example, if
the chosen functions are,

$\begin{split}f^{\mathrm{train}}_{n}=e^{-(n-1)/1.5},\\\
f^{\mathrm{age}}_{n}=(1-e^{-(n-1)/1.5}),\end{split}$ (44)

consequently, the signal of the nth stimulus is approximated as,

$S_{n}=(1-e^{-(n-1)/1.5})e^{-t^{\mathrm{age}}_{n-1}/\tau_{\mathrm{pre}}}\\\
+e^{-(n-1)/1.5}(1-e^{-t^{\mathrm{train}}/\tau_{\mathrm{train}}}),$ (45)

the qualitative dependency of approximated signal on $t^{\mathrm{train}}$ and
$t^{\mathrm{wait}}$ is similar to Fig. 4(b), see Fig. 10(e). Note that in both
approximations Eq. (42) and Eq. (45), the ratio of the coefficients of age and
training impact increases with n, meaning after a certain number of pre-
stimuli the impact of age dominates over the impact of training.

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Applying Lemma <ref>, we have $H^1(G_S,\hat{T}) \isom \ker \varphi$, where $\varphi$ is as defined in the exact sequence (<ref>). As discussed in the paragraphs preceding Lemma <ref>, $\phi$ sends cyclic subextensions $M/K$ of $K_S$ (with a given choice of generator for $\Gal(M/K)$) to their compositum $LM/L$, and so elements of $\ker \phi$ correspond to cyclic subextensions of $L/K$. However, since $L'/K$ is the maximal abelian subextension of $L/K$, it contains all cyclic subextensions of $L/K$, and so the map $H^1(G_S, \hat{T'}) \ra H^1(G_S, \hat{T})$ is surjective. Since $\Q/\Z$ is a divisible group, it is an injective object in the category of abelian groups, and so the contravariant functor $\Hom(-,\Q/\Z)$ is exact. We deduce that $\theta$ is an injection. Therefore, if $\xi'_S((b_v)_{v\in S})=0$, then $\xi_S((b_v)_{v\in S})=0$, so $(b_v)_{v\in S} \in \ker \xi'_S \implies (b_v)_{v \in S} \in \ker \xi_S$. Recalling (<ref>), this means that $(2) \implies (1)$.

§ THE BRAUER GROUP FOR THE EQUATION $F(T) = \BN(\BX) \NEQ 0$

This appendix will be concerned with the Brauer group of a smooth projective model $X$ of the equation $f(t) = \bN(\bx) \neq 0$. In particular, we prove that in the setting of Corollary <ref>, we have $\Br(X) = \Br(\Q)$ whenever $n\geq 3$. We are grateful to $\CT$ for providing the arguments presented in this appendix.

§.§ Main results

Let $k$ be a field of characteristic zero. Let $K/k$ be an extension of degree $n$, and let $L/k$ be the Galois closure. Suppose that $\Gal(L/k) = S_n$. Let $f(t) \in k[t]$ be a squarefree polynomial. Let $Y/k$ be the affine variety given by the equation $f(t)= \bN(x_1, \ldots, x_n) \neq 0$, and $Y \ra \A^1_k$ its projection onto $t$. Let $\pi:X\ra \PP_k^1$ be a smooth projective birational model of $Y \ra \A^1_k$. Suppose that $L$ and the number field generated by $f$ are linearly disjoint over $k$. Then $\Br(k) = \Br(X)$.

Let $k$ be a field of characteristic zero. Let $K/k$ be a finite extension of degree $n\geq 3$, such that the Galois closure $L/k$ satisfies $\Gal(L/k) = S_n$. Let $c\in k^{\times}$, and let $Z$ be a smooth projective model of $\bN(x_1, \ldots, x_n) = c$. Then $\Br(k) \ra \Br(Z)$ is surjective.

§.§ Proof of Theorem <ref>

The key ideas of the proof are discussed in detail by Bayer-Fluckiger and Parimala <cit.>, and so here we just give a sketch. We would like to show that $\Br(Z)/\op{Im}(\Br(k))=0$. We begin by reducing to the case $c=1$. Suppose that $T$ is the norm one torus given by $\bN(x_1, \ldots, x_n) = 1$, and let $T^c$ denote a smooth compactification of $T$. Let $k_s$ denote the separable closure of $k$, and let $\overline{Z} = Z\times_{k}k_s$, and $\overline{T} = T \times _k k_s$. By <cit.>, we have an isomorphism $H^1(k,\Pic \overline{Z}) \isom H^1(k, \Pic \overline{T}^c)$. Combining this with <cit.>, we have

\begin{equation}
\Br(Z)/\op{Im}(\Br(k)) \hookrightarrow H^1(k, \Pic \overline{Z}) \isom H^1(k, \Pic \overline{T^c}) \isom \Br(T^{c})/\Br(k),
\end{equation}

and hence it suffices to show that $\Br(T^c)/ \Br(k) = 0$.

Let $G = \Gal(L/k)$. The character group $\hat{T} = \Hom(T_{\overline{k}}, \G_{m, \overline{k}})$ can be viewed as a $G$-lattice. By <cit.>, we have an isomorphism

$$\Br(T^c)/\Br(k) \isom \Sh^2_{\op{cycl}}(G, \hat{T}),$$

$$\Sh^2_{\op{cycl}}(G, \hat{T}) = \ker\l[H^2(G,\hat{T}) \ra \prod_{g\in G} H^2(\langle g \rangle, \hat{T}) \r].$$

Let $M/k$ be a finite extension with $M$ linearly disjoint from $L$, and let $L'=LM, K'=KM, k'=kM$. Then the extension $K'/k'$ has degree $n$ and Galois closure $L'$, with $\Gal(L'/k') = G$. Moreover, by a construction of Frölich <cit.>, we may choose $M$ in such a way that $L'/k'$ is unramified.

Using <cit.>, we have $\Sh^2_{\op{cycl}}(G, \hat{T}) \isom \Sh^2(k', \hat{T})$, where $\hat{T}$ is regarded as a $\Gal(k'_s/k')$-module via the surjection $\Gal(k'_s/k') \ra G$. In turn, this is isomorphic to $\Sh^1(k', T)^{\vee}$ by Poitou–Tate duality <cit.>. To summarise, we have isomorphisms

\begin{equation}
\Br(T^c)/\Br(k) \isom \Sh^2_{\op{cycl}}(G, \hat{T}) \isom \Sh^2(k', \hat{T}) \isom \Sh^1(k', T)^{\vee},
\end{equation}

and so it suffices to show that $\Sh^1(k', T)=0$. However, $\Sh^1(k', T)$ is isomorhpic to the knot group $\kappa(K'/k') = \f{k'^{\times}\cap N_{K'/k'}(\A_{K'}^{\times})}{N_{K'/k'}(k'^{\times})}$. Due to the assumption $\Gal(L'/k') = G = S_n$, we may apply the result of Kunyavskiĭ and Voskresenskiĭ <cit.> to deduce that the Hasse norm principle holds for the extension $K'/k'$, and hence $\kappa(K'/k')=0$.

§.§ Proof of Theorem <ref>

Before commencing with the proof, we require one more fact about the smooth projective model $X$ from the statement of Theorem <ref>.

In the notation of Theorem <ref>, the base change $X_K = X \times_k K$ is a $K$-rational variety.

Since $X$ is a smooth projective model of $Y$, it suffices to show that $Y_K$ is $K$-rational. Let $K = k[x]/(p(x))$, where $p(x)$ is an irreducible polynomial over $k$. Let $a$ denote the class of $x$. Over $K$, the polynomial $p(x)$ factorises as $p(x) =\prod_{i=0}^r q_i(x)$, where $q_0(x), \ldots, q_r(x) \in K[x]$ are distinct and irreducible, and $q_0(x)  = x-a$. Let $K_i = K[x]/(q_i(x))$. We shall construct a birational map of the form

\begin{equation}\label{the birat map}
\begin{split}
\phi: Y_K &\ra \A^1_K \times \prod_{i=1}^r R_{K_i/K}\A^1\\
(t,x_0, \ldots, x_{n-1}) &\mapsto (t,z_1, \ldots, z_r),
\end{split}
\end{equation}

where $R_{K_i/K}$ denotes the Weil restriction. Since $R_{K_i/K}\A^1 \isom \A^{\deg q_i}_K$, the right hand side of (<ref>) is isomorphic to $\A^n_K$, which is a Zariski open subset of $\PP^n_K$. Therefore, $\phi$ induces a birational map $Y_K \birat \PP^n_K$, as desired.

Let $\overline{k}$ denote an algebraic closure of $k$. We denote by $\Emb_k(K, \overline{k})$ the embeddings $K\hookrightarrow \overline{k}$ fixing $k$, or in other words, the conjugates of $K/k$ in $\overline{k}$. Over $\overline{k}$, the polynomials $p(x), q_0(x), \ldots, q_r(x)$ split as

\begin{equation}\label{factorising the factors}
p(x) = \prod_{\sigma \in \Emb_k(K,\overline{k})}(x-\sigma(a)), \qquad q_i(x) = \prod_{\ss{\sigma \in \Emb_k(K,\overline{k})\\ q_i(\sigma(a))=0}}(x-\sigma(a)).
\end{equation}

For each $i$, we fix an isomorphism $K_i \isom K(\sigma_i(a))$ for some $\sigma_i \in \Emb_k(K,\overline{k})$ satisfying $q_i(\sigma_i(a))=0$, and view $\sigma_i(a)$ as the class of $x$ in $K_i/K$. (The particular choice of representative $\sigma_i$ does not matter.) Since $q_i(x)$ is the minimum polynomial of $\sigma_i(a)$ over $K$, it splits over $\overline{k}$ as the product of the conjugates of $\sigma_i(a)$, and so

$$ q_i(x) = \prod_{\sigma' \in \Emb_K(K_i, \overline{k})}(x - \sigma'\sigma_i(a)).$$

For $i\in \{0,\ldots, r\}$, we define $z_i \in R_{K_i/K}\A^1$ as

\begin{equation}\label{definition of zi}
z_i = x_0+\sigma_i(a)x_1 + \cdots + \sigma_i(a)^{n-1}x_{n-1}.
\end{equation}

the polynomial $\sum_{j=0}^{\deg q_i -1}z_i^{(j)}x^j$ representing $z_i$ is the reduction of $x_0 + x_1x + \cdots + x_{n-1}x^{n-1}$ modulo $q_i$. Consequently, by the Chinese remainder theorem, $z_0, \ldots, z_r \in \prod_{i=0}^r R_{K_i/K}\A^1$ uniquely determine $x_0, \ldots, x_{n-1} \in \A^1_K$.

For any number field extension $E/M$, and any $y \in E$, we have

$$N_{E/M}(y)  = \prod_{\sigma \in \Emb_M(E,\overline{M})} \sigma (y).$$

\begin{align*}
N_{K/k}(y) = \prod_{\sigma \in \Emb_k(K,\overline{k})}\sigma(y) = \prod_{i=0}^r \prod_{\sigma' \in \Emb_{K}(K_i, \overline{k})}\sigma'\sigma_i(y) = \prod_{i=0}^r N_{K_i/K}(\sigma_i(y)),
\end{align*}

and so

\begin{equation}\label{rewriting the norm form}
\bN(x_0, \ldots, x_{n-1}) = \prod_{i=0}^r N_{K_i/K}(z_i).
\end{equation}

We deduce that the equations (<ref>) define an isomorphism from $Y_K$ to the variety $V\subseteq \A^1_K \times \prod_{i=0}^rR_{K_i/K}\A^1$ given by $z_0\prod_{i=1}^r N_{K_i/K}(z_i) = f(t)\neq 0$.
For $t,z_1, \ldots, z_r$ satisfying the Zariski open condition $\prod_{i=1}^r N_{K_i/K}(z_i) \neq 0$, we have $z_0 = f(t)/\prod_{i=1}^n N_{K_i/K}(z_i)$. Therefore, the projection of $V$ onto $\A^1_K \times \prod_{i=1}^r R_{K_i/K}\A^1$ is birational. We conclude that the map $\phi$ from (<ref>) is birational.

We now commence with the proof of Theorem <ref>. Let $k$ be a field of characteristic zero. For a smooth irreducible variety $X/k$ with function field $\kappa(x)$, we recall that $\Br(X)$ consists of all elements of $\Br(\kappa(X))$ which are unramified everywhere on $X$. Constant classes are unramified, and so we have $\Br(k) \subseteq \Br(X) \subseteq \Br(\kappa(X))$. By the purity theorem <cit.>, the ramification locus of $\mc{A}\in \Br(\kappa(X))$ is pure of codimension one. Consequently, to check $\mc{A} \in \Br(X)$, it suffices to check it is  unramified at all codimension one points of $X$.

Let $C$ be a codimension one point. We recall from <cit.> the residue map

$$\del_C: \Br(\kappa (X)) \ra H^1(\kappa(C), \Q/\Z)$$

is such that $\mc{A}$ is unramified at $C$ if and only if $\del_C(\mc{A})$ is trivial.

In our setting, codimension one points of $X$ come in two types:

* Irreducible components of fibres $X_c = \pi^{-1}(c)$ above codimension one points $c$ of $\PP_k^1$,

* Codimension one points on the generic fibre $X_{\eta}$ of $\pi:X\ra \PP_k^1$.

We recall that $\kappa(X) = \kappa(X_\eta)$. The codimension one points of $X_{\eta}$ are a subset of the codimension one points of $X$, and so we have an inclusion $\Br(X) \inj \Br(X_{\eta})$. Since  $X_{\eta}$ is a smooth projective model of $\bN_{K(t)/k(t)}(x_1, \ldots, x_n) = f(t)$ over $k(t)$, it follows from Theorem <ref>, applied to the extension $K(t)/k(t)$ and with $Z=X_{\eta}$, that $\Br(k(t)) \ra \Br(X_{\eta})$ is surjective. Putting everything together, we obtain a commutative diagram

\[\begin{tikzcd}
{\Br(X)} && {\Br(X_{\eta})} & {\Br(\kappa(X_{\eta})) = \Br(\kappa(X))} \\
{\Br(k)} && {\Br(k(t))}
\arrow[hook, from=2-1, to=2-3]
\arrow[hook', from=2-1, to=1-1]
\arrow[hook, from=1-1, to=1-3]
\arrow[two heads, from=2-3, to=1-3]
\arrow[hook, from=1-3, to=1-4]
\end{tikzcd}\]

Let $\alpha \in \Br(X)$. By the above diagram, we can find $\beta \in \Br(k(t))$ whose image in $\Br(X_{\eta})$ is equal to the image of $\alpha$ in $\Br(X_{\eta})$. We want to show that $\beta$ is the image of an element of $\Br(k)$, because then it follows from commutativity of the diagram that $\alpha$ is the image of an element of $\Br(k)$.

For any $n\geq 1$ and any field $k$, we have $\Br(\PP_k^n) = \Br(k)$ <cit.>. In particular, we have $\Br(k) = \Br(\PP_k^1)$. Also, $k(t) = \kappa(\PP_k^1)$, so $\Br(k(t)) = \Br(\kappa(\PP_k^1))$. Therefore, as discussed above, to prove that $\beta$ is in the image of $\Br(k)$, it suffices to show $\beta$ is unramified at every codimension one point of $\PP^1_k$. This is formalised by the Faddeev exact sequence <cit.>, which is the exact sequence

\begin{equation}
0 \ra \Br(k) \inj \Br(k(t)) \ra \bigoplus_{Q\in (\PP^1_k)^{(1)}} H^1(k_Q , \Q/\Z) \surj H^1(k, \Q/\Z) \ra 0,
\end{equation}

where $(\PP^1_k)^{(1)}$ denotes the codimension one points of $\PP_k^1$ and the third map is the direct sum of the residue maps $\del_Q$. In other words, to show that $\beta \in \Br(k(t))$ is actually in $\Br(k)$, it suffices to show that $\del_Q(\beta) = 0$ for all $Q \in (\PP_k^1)^{(1)}$. We have $\del_Q(\beta) = 0$ unless $Q$ is an irreducible factor of $f(t)$ by <cit.>, so we suppose from now on that $Q$ is an irreducible factor of $f(t)$.

By Lemma <ref>, the base change $X_K = X \times_{k} K$ is birational to $\PP_K^n$. Since the Brauer group is a birational invariant on smooth projective varieties <cit.>, it follows that $\Br(X_K) = \Br(\PP^n_K) = \Br(K)$. Therefore, we obtain the following commutative diagram:

(X_K)        (X_K,η)

(X)        (X_η)

(K)        (K(t))     ⊕_Q ∈(^1_k)^(1)H^1(Kk_Q, /)

(k)        (k(t))     ⊕_Q ∈(^1_k)^(1)H^1(k_Q, /)
["ψ_K"'pos=0.3, two heads, from=3-4, to=1-4]
[two heads, from=3-4, to=3-5]
[""'pos=0.3, from=3-2, to=1-2]
[hook, from=3-2, to=3-4]
[from=4-3, to=3-2]
["ι", hook, from=2-3, to=2-5, crossing over]
[hook, from=4-3, to=2-3, crossing over]
[hook, from=4-3, to=4-5]
["ψ"'pos=0.3, two heads, from=4-5, to=2-5]
[two heads, from=4-5, to=4-6]
["ι_K", hook, from=1-2, to=1-4]
[from=4-5, to=3-4]
[from=2-3, to=1-2]
[from=2-5, to=1-4]
["φ"', from=4-6, to=3-5]

The map $\phi$ is a direct sum over the restriction maps $\phi_Q:H^1(k_Q, \Q/\Z)\ra H^1(Kk_Q, \Q/\Z)$ for each $Q \in (\PP^1_k)^{(1)}$.

We have $\del_Q(\beta) \in \ker \varphi_Q$, where $\varphi_Q$ is as defined above.

Let $\beta_K$ denote the image of $\beta$ in $\Br(K(t))$, and $\del_{K,Q}$ the residue map at $Q$ on $\Br(K(t))$. We want to show that $\del_{K,Q}(\beta_K) = 0$. By exactness of the Faddeev exact sequence over $K$, for this it suffices to show $\beta_K$ is in the image $\Br(K) \ra \Br(K(t))$. We know that $\psi(\beta) = \iota(\alpha)$. Applying a base change to $K$, we see that $\psi_K(\beta_K) = \iota_K(\alpha_K)$, where $\alpha_K, \beta_K$ are the images of $\alpha, \beta$ under base change. Since $\Br(K) \isom \Br(X_K)$, we have that $\alpha_K$ is in the image of $\Br(K) \ra \Br(X_K)$, and hence by commutativity of the diagram, $\beta_K$ is in the image of $\Br(K) \ra \Br(K(t))$, as required.

Let $M$ be a number field, and let $G_M = \Gal(\overline{M}/M)$. For a finite Galois extension $M'/M$, we consider $\Gal(M'/M)$ as a topological space with the discrete toplology. We can then put the profinite topology on $G_M$, which is defined as the inverse limit

$$ G_M = \varprojlim_{M'/M \textrm{ Galois}} \Gal(M'/M). $$

We recall that $H^1(M,\Q/\Z) = \Hom_{\textrm{cont}}(G_M, \Q/\Z)$, the continuous group homomorphisms $G_M \ra \Q/\Z$ <cit.>. Suppose that $\theta \in \Hom_{\textrm{cont}}(G_M, \Q/\Z)$. Then $\ker \theta$ is an open subgroup of $G_M$. Since $G_M$ is a profinite group, this implies that $\ker \theta$ has finite index in $G_M$. By the fundamental theorem of Galois theory, $\op{im}\theta \isom G_M/\ker \theta \isom \Gal(M'/M)$, for some finite Galois extension $M'/M$. Moreover, $\op{im}\theta$ is a finite subgroup of $\Q/\Z$. All finite subgroups of $\Q/\Z$ are cyclic groups of the form $\f{1}{n}\Z/\Z$ for some positive integer $n$. Consequently, $\ker \theta = \Gal(M'/M)$ for a cyclic extension $M'/M$. To summarise, we have the identification

$$ H^1(M, \Q/\Z) = \{M'/M \textrm{ cyclic, with a given map }\gamma:\Gal(M'/M) \inj \Q/\Z\}.$$

We now describe $\phi_Q: H^1(k_Q, \Q/\Z) \ra H^1(Kk_Q, \Q/\Z)$ explicitly. Using the above identification, we view an element $\theta \in H^1(k_Q, \Q/\Z)$ as a pair $(M'/k_Q, \gamma)$. The map $\phi_Q$ is given by taking the compositum with $K$. More precisely, it sends the above pair to $(KM'/Kk_Q, \gamma)$, where now $\gamma$ is viewed as a map $\Gal(KM'/KK_Q) \hookrightarrow \Q/\Z$ via the natural identification of $\Gal(KM'/Kk_Q)$ as a subgroup of $\Gal(M'/k_Q)$. Therefore, $(M'/k_Q, \gamma) \in \ker \phi_Q$ if and only if $M'/k_Q$ is a cyclic subextension of $Kk_Q/k_Q$.

Due to the assumption that $k_Q$ and $L$ are linearly disjoint over $k$, we have $\Gal(Kk_Q/k_Q) \isom \Gal(K/k) \isom S_n$. We now complete the proof of Theorem <ref> with the following elementary group theory fact.

Suppose that $K/k$ is a finite extension of degree $n\geq 3$, and the Galois group of the Galois closure $\Gal(L/k)$ is isomorphic to $S_n$. Then there are no nontrivial cyclic extensions $M/k$ with $M\subseteq K$.

By the fundamental theorem of Galois theory, if $M/k$ is a subextension of $K/k$, then $\Gal(L/K) \leq \Gal(L/M) \leq \Gal(L/k) = S_n$. However, $\Gal(L/K) \isom S_{n-1}$. (More explicitly, if $K=k(\alpha_1)$ and $\alpha_1, \ldots, \alpha_n$ are the roots of the minimum polynomial of $\alpha_1$ over $k$, then $\Gal(L/K)$ consists of all permutations of $\{\alpha_1, \ldots, \alpha_n\}$ which fix $\alpha_1$.) However, $S_{n-1}$ is a maximal subgroup of $S_n$, and so $M=k$ or $M=K$. Since $n\geq 3$, the extension $K/k$ is not cyclic. Therefore, $M=k$.

In conclusion, the map $\phi_Q$ is injective by Lemma <ref>, and $\del_Q(\beta) \in \ker \phi_Q$ by Lemma <ref>, and hence $\del_Q(\beta)=0$. This means that all the residue maps of $\beta$ are trivial, so $\beta$ is in the image of $\Br(k) \ra \Br(k(t))$. Hence $\alpha$ is in the image of $\Br(k) \ra \Br(X)$, and so $\Br(X) = \Br(k)$. This completes the proof of Theorem <ref>.

We return to Example <ref>. Choosing the height $H$ given in <ref>, we obtain the counting problem

\begin{equation}
N(B) = \f{1}{2}\#\{(x_0,x_1)\in \Z^2_{\prim}: |x_0|,|x_1|\leq B, \, x_0 \textrm{ m-full}\}.
\end{equation}

The factor $\f{1}{2}$ comes from the fact that there are two representations of each point $x \in \PP^n(\Q)$ by a vector $\bx \in (\Z_{\neq 0})^{n+1}_{\prim}$.

By Möbius inversion, we have

\begin{equation}\label{toy example mobius}
N(B) = 2\sum_{d=1}^{\infty}\mu(d)(N_d(B))\#\{1 \leq x_1 \leq B: d\mid x_1\},
\end{equation}

\begin{equation}\label{definition NdB}
N_d(B) = \#\{x \in \Z: 1\leq x \leq B, x \textrm{ squareful}, d
\mid x\}.
\end{equation}

The factor $2$ in (<ref>) combines the factor $1/2$ in (<ref>) with a factor $2^2$ arising from passing from $(\Z_{\neq 0})^{2}$ to $\N^{2}$. The quantity $N_d(B)$ is studied in <cit.>, where Pieropan and Schindler establish the following result.
Let $m\geq 2$ and let $d\geq 1$ be squarefree. Then

\begin{equation}
N_d(B) = C_{d,m}B^{\f{1}{m}} +O\l(B^{\f{1}{m+1}}d^{-\f{m}{m+1} + \eps}\r),
\end{equation}

where the implied constant depends only on $m$ and $\eps$,

$$C_{d,m} = C_m\prod_{p\mid d}\l(p-p^{\f{m-1}{m}} +1\r)^{-1},$$

and $C_m$ is given in (<ref>).

We apply Lemma <ref>, together with the estimate
$$\#\{1 \leq x_1 \leq B: d\mid x_1\} =
\f{B}{d} + O(1).$$

We also note that we may restrict the $d$-sum in (<ref>) to $d\leq B^{1/m}$. This is because if $d$ is squarefree and $x_1$ is enumerated by $N_d(B)$, then $d^m \mid x_1$, so $d^m \leq x_1 \leq B$. We obtain

\begin{align*}
N(B) &= 2\sum_{d\leq B^{1/m}}\l(\mu(d)C_{d,m}B^{\f{1}{m}} + O\l(B^{\f{1}{m+1}}d^{-\f{m}{m+1} + \eps}\r)\r)\l(\f{B}{d}+O(1)\r)\\
&=2B^{1+1/m}{\sum_{d=1}^{\infty}\f{\mu(d)C_{d,m}}{d}} + O(B^{1+1/(m+1)+\eps}).
\end{align*}

We now find an Euler product for the leading constant $c$ in the above asymptotic for $N(B)$. Using the fact that $C_{d,m}$ is multiplicative as a function of $d$, we have

\begin{align*}
c &=2\sum_{d=1}^{\infty}\f{C_{d,m}\mu(d)}{d}\\
\end{align*}

We compare our asymptotic formula for $N(B)$ with the prediction from the PSTV-A conjecture.

We recall we are working with $X=\PP^1$ and the divisor $(1-1/m)D$, where $D=[0:1]$, and the degree of this divisor is $1-1/m$. We have $\Pic(X) \isom \Z$, with the isomorphism given by the degree. Under this isomorphism, we identify the cone of effective divisors $\Lambda_{\op{eff}}$ with $\R_{\geq 0}$. The canonical line bundle is $\omega_X \isom \OO_{\PP^1}(2)$, and so $[K_X] = -2[D]$. The height $H$ we have used comes from the line bundle $\mc{L} = \OO_{\PP^1}(1)$, and so $[L] = [D]$. Therefore,

\begin{align}
a &= \inf\{t\in \R: tD -2D + (1-1/m)D \geq 0\}\\
&= 1+1/m.
\end{align}

The minimal supported face of $\Lambda_{\op{eff}}$ which contains $a[L] + [K_X] + (1-1/m)[D] = 0$ is $\{0\}$, which has codimension one in $\Lambda_{\op{eff}}$, and so $b = 1$. Therefore, the exponents of $B$ and $\log B$ predicted by the PSTV-A conjecture are consistent with our asymptotic formula for $N(B)$.

The leading constant $c$ is defined as

$$c = \frac{\alpha\beta\tau}{ab!}=\frac{\alpha\beta\tau}{1+1/m}.$$

We now compute the constants $\alpha, \beta, \tau$.

§.§ $\alpha$-Constant

In the notation from <cit.>, $E$ is a rigid effective divisor linearly equivalent to $aL+K_x + (1-1/m)D$ and so $E = 0$ and $U = X\bs \supp{E} = \Proj^1$. The set $\mathcal{A}(L)$ is the set $\{D\}$. The projection map $\rho: \Pic(X) \ra \Pic(U)$ is just the identity map, so $\alpha$ is the usual $\alpha$-constant of $\Proj^1$ from Manin's conjecture (as defined in <cit.>), times the factor $1-1/m$. The usual usual $\alpha$-constant of $\Proj^1$ is given by

$$\int_{\Lambda^{*}} e^{-\langle 1, x\rangle} \,\textrm{d}x = \int_{\R_{\geq 0}} e^{-x} \,\textrm{d}x = 1.$$

Here $\Lambda^{*}$ is the dual of $\Lambda_{\op{eff}} \tensor_{\Z}\R$, which is identified with $\R_{\geq 0}$. Therefore we have $\alpha = (1-1/m)$, and so $c = \beta\tau$.

§.§ $\beta$-Constant

We have $\beta=1$. Therefore $c=\tau$.

§.§ Tamagawa number