Pulsatile Driving Stabilizes Loops in Elastic Flow Networks

Abstract

Existing models of adaptation in biological flow networks consider their constituent vessels (e.g. veins and arteries) to be rigid, thus predicting a non physiological response when the drive (e.g. the heart) is dynamic. Here we show that incorporating pulsatile driving and properties such as fluid inertia and vessel compliance into a general adaptation framework fundamentally changes the expected structure at steady state of a minimal one-loop network. In particular, pulsatility is observed to give rise to resonances which can stabilize loops for a much broader class of metabolic cost functions than predicted by existing theories. Our work points to the need for a more realistic treatment of adaptation in biological flow networks, especially those driven by a pulsatile source, and provides insights into pathologies that emerge when such pulsatility is disrupted in human beings.

Full text

Pulsatile Driving Stabilizes Loops in Elastic Flow Networks
Purba Chatterjee, Sean Fancher, and Eleni Katifori
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
Existing models of adaptation in biological flow networks consider their constituent vessels (e.g.
veins and arteries) to be rigid, thus predicting a non physiological response when the drive (e.g
the heart) is dynamic. Here we show that incorporating pulsatile driving and properties such as
fluid inertia and vessel compliance into a general adaptation framework fundamentally changes
the expected structure at steady state of a minimal one-loop network. In particular, pulsatility is
observed to give rise to resonances which can stabilize loops for a much broader class of metabolic
cost functions than predicted by existing theories. Our work points to the need for a more realistic
treatment of adaptation in biological flow networks, especially those driven by a pulsatile source, and
provides insights into pathologies that emerge when such pulsatility is disrupted in human beings.
I. INTRODUCTION
The structure of physiological transport networks such
as animal vasculature and leaf venation, has important
consequences for biological functionality, and as such has
elicited considerable scientific interest over the years. In
particular, looped network architectures, ubiquitous in
biology, are beneficial for mitigating vessel damage and
optimizing responses to source fluctuations [1–3]. Net-
work remodeling [4, 5], also known as adaptation, is now
understood to proceed by optimizing the total energy dis-
sipation in the network, subject to some metabolic cost
[6–9]. Such metabolic costs can generally be described
by a power law (Kσ), where Kis vessel conductivity and
σis a system specific parameter. Existing theories of
adaptation in flow networks predict a critical transition
at σ= 1, with a structure with many loops for σ < 1 and
one which is a loop-less tree for σ > 1 [1, 2, 10–12]. How-
ever the scope and generality of this prediction remains
to be investigated in the light of biologically relevant dy-
namical considerations.
Previous studies on adaptation consider vessels to be
rigid, leading to the assumption that modulations in flow
boundary conditions are instantaneously propagated to
individual network elements at all times. However, vessel
compliance and fluid inertia have been shown to gener-
ate a finite timescale of information transfer from the
sources to the bulk [13, 14], leading to trade-offs between
energy efficiency on one hand and mechanical response
to sudden changes in the steady state dynamics on the
other [13]. Moreover, while fluctuating sources and sinks
have been implemented within the framework of network
adaptation [1, 6, 15, 16], the effect of deterministic pul-
satile driving at the source is largely unexplored . This is
an important consideration for biological transport net-
works, many of which rely on pulsatility to maintain fluid
pressure. The most prominent example of this is mam-
malian vasculature, with the periodic beating of the heart
muscle introducing pulsatile components into blood flow.
In this Letter, we investigate the effect of both pul-
satile driving and the internal spatio-temporal dynamics
of elastic vessels on the adaptation of a simple one-loop
flow network. Depending on the lengths of the vessels in
FIG. 1. (color online) (a) Minimal model with L1=L2=L,
and the vessel mean-squared current in the absence of fluid
inertia and vessel compliance. (b,c) Flow diagrams for AE
in R1−R2space, a=b=L= 1. Insets show radii at
steady state for the chosen initial condition (green solid circle)
depicted in (a). Sinks at steady state denoted by solid orange
circles, and saddle points by hollow orange circles.
relation to each other and to a characteristic length scale
over which pulsatility is damped, resonant frequencies
are shown to exist, which amplify energy dissipation and
stabilize loops for a much broader class of metabolic cost
functions than predicted by existing theories of adap-
tation. Our results emphasize the need for a more so-
phisticated treatment of adaptation in order to correctly
predict the steady state structure of more complicated
biological transport networks and might be key in ex-
plaining the development of vascular malformations in
patients with artificial hearts [17, 18].
arXiv:2210.06557v2 [nlin.AO] 19 Oct 2022

2
II. THE ADAPTATION EQUATION
Our minimal network consists of two vessels connecting
the nodes N1and N2(Fig. 1(a)), subject to pulsatile
driving at the frequency ω. The vessel radii Rµ,µ∈[1,2],
change in response to the local current Qµover time t0,
according to the Adaptation Equation (AE)
dRµ
dt0=ahQ2
µiγ
R3
µ
−bRµ,(1)
where aand bare constants, and γis a parameter as-
sociated with the metabolic cost. Assuming Poiseuille
flow (Kµ∝R4
µ), this AE is identical to a general adap-
tation rule that has been used to model the dynamics of
hydraulic vessel conductivities (Kµ) during animal vas-
cular development and the slime mold Physarum poly-
cephalum [6, 18–23]. Each vessel adapts through a local
positive feedback, expanding in radius when the current
through it is large, and shrinking at the characteristic
timescale b−1when it is small. The steady states of the
AE correspond to the critical points of the optimization
functional
E=X
µ
Lµ
Q2
µ
Kµ
+β X
µ
LµKσ
µ−C!,(2)
where, Lµis the vessel length, βis a Lagrange multiplier
and Cis a constant [6]. The first term corresponds to
the total power dissipated in the network, and the second
term imposes a metabolic or material cost characterized
by σ= 1/γ −1. For γ= 2/3, this material cost is equal
to the total volume of flow in the network, which is an
important constraint for animal vasculature.
We define the vessel mean-squared current at a given
time t0of adaptation as
hQ2
µi=1
TZT
0
dt h1
LµZLµ
0
dz Qµ(z, t)2i,(3)
where T= 2π/ω is the time-period of pulsatility. Note
that we distinguish the adaptation time t0in the AE
(Eq. 1) from the time used to calculate the vessel mean-
squared current t, because adaptation typically occurs on
much longer timescales than that of local modulations of
flow in individual vessels (i.e. t0t).
When fluid inertia and vessel compliance are neglected,
the current at node N1splits proportionally between the
two vessels depending on their conductance, and the ves-
sel mean-squared current has the form given in Fig. 1(a).
The critical transition of the AE for this two-vessel net-
work ∀ωcan be analytically shown to occur at γAE
c= 1/2
(see supplemental materials). This is illustrated by the
steady state flow-diagrams in R1−R2space (Fig. 1(b,c)).
For γ < γAE
c, the diagonal has a stable fixed point (sink),
which corresponds to a stabilized loop with vessels of
equal radius at steady steady, irrespective of their initial
sizes (Fig. 1(b)). For γ > γAE
c, there exist two stable
fixed points at the boundaries with large basins of at-
traction, indicating that for most initial conditions, one
or the other vessel is lost. The diagonal has an unsta-
ble fixed point (saddle), indicating that the loop is stable
only for a narrow range of initial conditions correspond-
ing to exactly equal starting radii (Fig. 1(b)).
III. COMPLIANT VESSELS
The treatment above does not consider the opposi-
tion to changes in flow pressure due to fluid mass and
the resulting fluid inertia. Moreover, biological networks
are composed of compliant vessels, which can change
in radius reversibly at short timescales to accommodate
changes in flow volume. As shown in [13, 24], inertia and
compliance generates a finite time lag in flow propagation
from the sources to the bulk of the transport network.
Thus, in addition to the flow resistance (or conductance),
the combined contribution of fluid inertia and compliance
can be expected to alter the vessel mean-squared current
on the timescale of adaptation. We follow the treatment
of compliant vessels in [13], and assuming an incompress-
ible, laminar flow with rotational symmetry, the axial
current Q(z, t) and pressure P(z, t) in each vessel satisfy
∂Q
∂z +c∂P
∂t = 0,(4)
∂P
∂z +l∂Q
∂t +rQ = 0.(5)
The cross-section of the vessel changes as A(z, t) =
A0+cP (z, t) in response to wall pressure, where cis
the vessel compliance. We assume such changes in cross-
section to be small in magnitude, i.e. A0cP (z, t).
These vessel parameters can be combined to construct
the characteristic length (λ), time (τ) and admittance (α)
scales, which all vary proportional to the area of cross-
section, as
λ=λ0(R/R0)2=2
rrl
c,
τ=τ0(R/R0)2=2l
r,
α=α0(R/R0)2=rc
l,(6)
where R0is a typical radius. In particular, increasing λ0
at constant radius, with αλ and τheld fixed, reflects a de-
crease in the compliance cof the vessel. Generalizing the
single compliant vessel to a network of compliant vessels
is straightforward, and following [13], the vessel mean-
squared current hQ2
µi(Eq. 3) can be calculated for each
vessel. For the two-vessel network, this mean-squared
current has a more complex dependence (see supplemen-
tal materials) on the vessel radius than the form given
in Fig. 1(a), and when used to drive the AE, results in a
more realistic description of the evolution of the network

3
structure, as we show below. For convenience, we will
refer to the new framework of adaptation with fluid iner-
tia and vessel compliance taken into consideration as the
Modified Adaptation Equation (MAE), to distinguish it
from the AE.
IV. RESULTS
The consequences of periodic driving of the MAE for
the steady state structure of the two-vessel network are
significant. As an example, Fig. 2 shows the steady state
flow diagrams in R1−R2space with vessels of equal
but small effective lengths (L/λ0<1), for which com-
pliance is relatively low and pulsatile components of the
flow are non-negligible. Like in the AE, the stable fixed
point for γ= 1/3 and all values of ω, lies on the di-
agonal (R1=R2), indicating a stable symmetric loop.
For γ= 2/3, the non-pulsatile (ω= 0) flow diagram
has sinks on the boundaries, meaning the loss of one or
more vessel, once again similar to the AE result shown in
Fig. 1(c).For ω= 1.8πhowever, while stable steady states
exist on both the diagonal as well as the boundaries, the
former has a much broader basin of attraction than the
latter. This implies that unlike the AE, for most initial
conditions of the MAE loops can be stabilized for pul-
satile driving at this frequency. Even more interestingly,
for ω= 2.6π, new steady states with vessels of finite but
unequal radii emerge and are stable, with basins of at-
traction larger than that of the boundary sinks. Clearly,
at this frequency loops exist at steady state for a broad
range of initial conditions, albeit with asymmetric flow
distribution between the two vessels.
In general with pulsatility, the energy dissipation
through vessel µis amplified at special resonant frequen-
cies depending on the value of Lµ/λ0, causing it to ex-
pand even for γ > γAE
c. For each initial condition, we can
calculate the quantity Z= min hR1/R2, R2/R1it→∞ , the
minimum of the ratio of the radii of the vessels at steady
state. For Z= 0, the steady state is loopless, and for
0< Z ≤1 the steady state is looped, with Z= 1 corre-
sponding to vessels of equal radius. The critical transi-
tion from a looped to a loop-less structure in the MAE
as a function of the driving frequency can then be under-
pinned by the order parameter hZi, averaged over many
initial conditions. Fig. 3(a) shows the phase-diagram of
hZiin the γ−ωphase space, for the low compliance case
depicted in Fig. 2. The AE in this case would yield loops
(hZi>0) below γAE
c= 1/2 and no loops (hZi= 0) above
it (dashed red line). In contrast, the phase boundary
between looped and loop-less steady states in the MAE
has periodic modulations with respect to ω, with loops
stabilized for all physiologically relevant values of γat
resonant frequencies (dashed blue lines). Moreover there
is a significant spread of frequencies around the resonant
values, for which loops are stable above γAE
c. Also, stable
loops tend to be increasingly more asymmetric in radius
for higher values of γ, even at resonant frequencies.
Increasing vessel lengths to be greater than λ0, but
still equal, decreases the looped hZi>0 phase in area,
as shown in Fig. 3(b). The periodic modulations with
frequency in the critical transition are however retained,
with shorter and more frequent peaks. Thus, even in
the case of high relative compliance (L/λ0>1), where
the effect of pulsatile driving is significantly damped, the
MAE is able to stabilize loops for a larger range of γ
values than the AE.
For L1=L2=L, an on-diagonal steady state always
exists (Fig. 2), but is stable for each ωvalue only be-
low a critical value γ=γM AE
c(ω). Above γMAE
c(ω) the
MAE generates either a loop-less network or an asym-
metric loop at steady state (0 <hZi<1). Fig 3(c)
shows the analytical phase diagram of γMAE
c(see supple-
mental materials for derivation) in the ωτ −L/λ space.
Here τand λare the full radius dependent time and
length scales (Eq. 6) corresponding to the on-diagonal
fixed point for the given ωand for γ=γMAE
c. Clearly,
the value of γMAE
coscillates with the frequency, with
γAE
c< γM AE
c≤1. The trajectories in black and red
in Fig 3(c) correspond to the trajectories of the critical
transition in Fig 3(a,b), showing excellent agreement be-
tween the predictions of the numerical phase-diagram of
hZi= 1 and the analytical phase-diagram of γMAE
c(ω).
The differences between the steady sate structures of the
MAE and the AE are most pronounced for shorter vessels
with L/λ0<1, and the upper bound of oscillations in
γMAE
c(ω) decreases monotonically with increasing L/λ0.
Lastly, Fig. 4(a,b) shows two cases with L16=L2, one
where both vessels are shorter, and another where only
one vessel is shorter than λ0. Here, unlike in the L1=L2
case, the on-diagonal steady state does not exist for all
values of γand ω. This explains the decrease in area of
the hZi= 1 phase from Fig. 3 to Fig. 4. However, while
the AE would predict hZi= 0 for all γ > γAE
c(dashed
green line), clearly the MAE supports resonance frequen-
cies (dashed blue lines) for which asymmetric loops are
stabilized for a broad range of γvalues (see supplemental
materials for a more detailed discussion).
V. DISCUSSION
In summary, our simple model confirms that through
the interplay between the time and length scales of pul-
satile driving and those generated by network properties
such as fluid inertia and vessel compliance, the modi-
fied adaptation framework (MAE) displays rich behavior
that would be missed if the bulk flow was considered to
instantaneously reflect modulations at the source, as in
the AE. Loops (symmetric or asymmetric) are stable in
the MAE for a much broader ranger of γvalues than
in the AE, more so for less compliant vessels, that are
shorter than the characteristic length scale λ0. This is
true even when the two vessels comprising the loop have
unequal lengths, and importantly also when only one is

4
FIG. 2. (color online) Flow Diagrams in R1−R2space with L1=L2=Land L/λ0<1. Sinks denoted by solid circles
and saddle points by hollow circles. For γ= 1/3, the steady state is a stable loop with R1=R2for all values of ω. For
(γ= 2/3, ω = 0) loops are unstable for all initial conditions not on the diagonal. For (γ= 2/3, ω = 1.8π) loops with vessels
of equal radius are stable for almost all initial conditions not on the boundaries. For (γ= 2/3, ω = 2.6π) loops with unequal
vessel radii are stable for a broad range of intermediate initial conditions. Here a=b=L= 1 and λ0= 2.
shorter than the damping length scale.
It is crucial to understand the role of pulsatility in
adaptation, because of its important consequences for the
proper functioning and maintenance of biological trans-
port networks. For instance, continuous-flow Left Ven-
tricular Assist Devices (LVADs), used as a Bridge-To-
Transplant therapy for advanced congestive heart failure,
has been associated with detrimental pathology such as
gastrointestinal bleeding, arterio-venous malformations
and other complications, thought to stem from decreased
arterial pulsatility [25, 26]. Our finding that pulsatile
driving can prevent the loss of vessels adaptation is at
the very least consistent with such claims, even though
our minimal model does not presume to capture the vast
complexity of the human circulatory system.
Lastly, our results should be generalizable to mechani-
cal networks [17, 18] or bigger networks with many inter-
nal loops, for which the average energy dissipation over
the network has been shown to be amplified at special
resonant frequencies of the pulsatile driving [13]. The
full analysis of the distribution of loops at steady state
for such large, hierarchical networks is the scope of future
work.
ACKNOWLEDGMENTS
The authors acknowledge support from the NSF
Award PHY-1554887, the Simons Foundation through
Award 568888 and the University of Pennsylvania Mate-
rials Research Science and Engineering Center (MRSEC)
through Award DMR-1720530.
[1] E. Katifori, G. J. Szollosi, and M. O. Magnasco, Phys.
Rev. Lett. 104 (4), 048704 (2010).
[2] F. Corson, Phys. Rev. Lett. 104 (4), 048703 (2010).
[3] F. Kaiser, H. Ronellenfitsch, and D. Witthaut, Nat.
Commun. 11 (1), 1-11 (2020).
[4] A. R. Pries, T. W. Secomb, and P. Gaehtgens, Am. J.
Physiol. 275 (2), H349-H360 (1998).
[5] A. R. Pries and T. W. Secomb, Microcirculation 15 (8),
753-764 (2008).
[6] D. Hu and D. Cai, Phys. Rev. Lett. 111 (13), 138701
(2013).
[7] S. S. Chang and M. Roper, J. Theor. Biol. 462, 48-64
(2019).
[8] G. Gounaris, M. R. Garcia, and E. Katifori,

5
FIG. 3. The critical transition for L1=L2. (a,b) Phase-
diagram of hZiin the γ−ωphase-space over the 144 initial
conditions in the range R1, R2∈(0.1,1.75) depicted in Fig. 2.
L1=L2= 1 in (a), L1=L2= 2√2 in (b), and λ0= 2 in
both cases. Dashed green lines mark the critical transition
in the AE, and dashed blue lines depict resonant frequencies.
The black trajectory in (a) and the red trajectory in (b) fol-
lows γMAE
cas a function of increasing ω. (c) Phase-diagram
of γMAE
cin the ωτ −L/λ phase-space. The black and red
trajectories in (c) correspond to those in (a) and (b) respec-
tively.
FIG. 4. Phase-diagram of hZifor L16=L2.L1= 1 and
L2=√2 in (a), L1 = 1 and L2= 2√2 in (b), and λ0= 2 in
both cases. Dashed red lines mark the critical transition in
the AE.
arXiv:2111.04657 (2021).
[9] F. Kramer and C. D. Modes, arXiv:2111.04326. (2021).
[10] J. R. Banavar, F. Colaiori, A. Flammini, A. Maritan,
and A. Rinaldo, Phys. Rev. Lett. 84 (20), 4745 (2000).
[11] M. Durand, Phys. Rev. Lett. 98 (8), 088701 (2007).
[12] S. Bohn and M. O. Magnasco, Phys. Rev. Lett. 98 (8),
088702 (2007).
[13] S. Fancher and E. Katifori, Phys. Rev. Fluids 7 (1),
013101 (2022).
[14] S. Fancher and E. Katifori, arXiv:2102.13197 (2021),
arXiv:2102.13197.
[15] D. Hu, D. Cai, and A. Rangan, PLoS One 7, e45444
(2012).
[16] J. Gr¨awer, C. D. Modes, M. O. Magnasco, and
E. Katifori, Phys. Rev. E 92 (1), 012801 (2015),
arXiv:1405.7870.
[17] D. Hexner, A. J. Liu, and S. R. Nagel, Proc. Natl. Acad.
Sci. U.S.A 117 (50), 31690-31695 (2020).
[18] F. K. B¨auerle, S. Karpitschka, and K. Alim, Phys. Rev.
Lett. 124 (9), 098102 (2020).
[19] W. J. Hacking, E. VanBavel, and J. A. E. Spaan, Am.
J. Physiol. 270 (1), H364 (1996).
[20] A. G. Rolland-Lagan and P. Prusinkiewicz, Plant J. 44
(5), 854 (2005).
[21] K. van Berkel, R. J. de Boer, B. Scheres, and K. ten
Tusscher, Development 140 (11), 2253 (2013).
[22] H. Ronellenfitsch and E. Katifori, Phys. Rev. Lett. 117
(13), 138301 (2016).
[23] H. Ronellenfitsch and E. Katifori, Phys. Rev. Lett., 123
(24), 248101 (2019).
[24] A. C. L. Barnard, W. A. Hunt, W. P. Timlake, and
E. Varley, Biophys. J. 6 (6), 717-724 (1966).
[25] A. Cheng, C. A. Williamitis, and M. A. Slaughter, Ann.
Cardiothorac. Surg. 3 (6), 573 (2014).
[26] C. R. Bartoli, G. A. Giridharan, K. N. Litwak, M. So-
bieski, S. D. Prabhu, M. S. Slaughter, and S. C. Koenig,
ASAIO J. 56 (5), 410 (2010).

6
APPENDIX
In this appendix, we will derive the expression for the vessel mean squared current hQ2
µifor the Modified Adaptation
Equation (MAE), analyze the stability of the diagonal steady state in the case of vessels with equal lengths, and expand
the discussion from the main text on the steady state structure for vessels with unequal lengths.
A1. Vessel mean squared current
Let Hin(t) and Hout (t) be the flow that goes into node N1and comes out from node N2respectively. Then
Hin =−Hout =Q0+Asin ωt, where ω= 2π/T is the frequency of the pulsatile driving. This allows for three Fourier
modes with ˜
H0=Q0,˜
H1=˜
H−1=A/2. For each vessel e, the solutions of Eq. 4 and Eq. 5 can thus be written as
the discrete sum
Qµ(z, t) =
1
X
n=−1
einω0t˜
Q(n)
µ(z),(A1)
Pµ(z, t) =
1
X
n=−1
einω0t˜
P(n)
µ(z),(A2)
where
˜
Q(n)
µ(z) = inωτµαµ
k(nωτµ)
˜
P(n)
in cosh Lµ−z
λµk(nωτµ)−˜
P(n)
out cosh z
λµk(nωτµ)
sinh Lµ
λµk(nωτµ),(A3)
˜
P(n)
µ(z) =
˜
P(n)
in sinh Lµ−z
λµk(nωτµ)+˜
P(n)
out sinh z
λµk(nωτµ)
sinh Lµ
λµk(nωτµ).(A4)
Here ˜
P(n)
in =˜
P(n)
µ(0) and ˜
P(n)
out =˜
P(n)
µ(Lµ) are the boundary pressures, identical for each of the two vessels
irrespective of their lengths. Eqs. 4 and 5 can be rearranged to take the form
∂
∂z (P Q) + 1
2
∂
∂t `Q2+cP 2+rQ2= 0.(A5)
The mean squared current in each vessel is given by
hQ2
µi=1
rLµTZT
0
dt ZLµ
0
dz Qµ(z, t)2
=1
rLµTZT
0
dt ZLµ
0
dz ∂
∂z Pµ(z , t)Qµ(z, t)+1
2
∂
∂t lQµ(z , t)2+cPµ(z, t)2,
=1
rLµTZT
0
dt Pµ(0, t)Qµ(0, t)−Pµ(Lµ, t)Qµ(Lµ, t)
−1
rLµTZT
0
dt ∂
∂t ZLµ
0
dz 1
2
∂
∂t lQµ(z , t)2+cPµ(z, t)2!.(A6)
Noting that all quantities are identical at the start and end of each period, the second integral over time in Eq. A6
completely vanishes. We can then write the vessel mean squared current as a sum of Fourier modes as

7
hQ2
µi=1
rLµTZT
0
dt 



1
X
n0=−1
ein0ωt ˜
P(n0)
µ(0)


1
X
n=−1
einωt ˜
Q(n)
µ(0)

−

1
X
n0=−1
ein0ωt ˜
P(n0)
µ(Lµ)


1
X
n=−1
einωt ˜
Q(n)
µ(Lµ)


,
=1
rLµ
1
X
n=−1˜
P(−n)
µ(0) ˜
Q(n)
µ(0) −˜
P(−n)
µ(Lµ)˜
Q(n)
µ(Lµ).(A7)
Eq. A7 can be expanded using Eq. A3 and noting that ˜
P(−n)(z)=(˜
P(n)(z))∗. With this, we get the expression
for the vessel mean squared current in terms of the boundary pressures as
hQ2
µi=1
rLµ
1
X
n=−1
inωτµαµ
k(nωτµ)
˜
P(n)
in 
2+
˜
P(n)
out 
2cosh Lµ
λµk(nωτµ)−˜
P(n)
in ˜
P(n)
out ∗−˜
P(n)
out ˜
P(n)
in ∗
sinh Lµ
λµk(nωτµ),
= αµλµ
2Lµ!2
˜
P(0)
in −˜
P(0)
out2+αµλµ
Lµ
˜
P(1)
in 
2+
˜
P(1)
out
2Re 

iωτµαµ
k(ωτµ)coth Lµ
λµ
k(ωτµ)!

−2αµλµ
Lµ
Re ˜
P(1)
in ˜
P(1)
out∗Re 

iωτµαµ
k(ωτµ)csch Lµ
λµ
k(ωτµ)!
.(A8)
Thus, if we can calculate the boundary pressures ˜
P(n)
in and ˜
P(n)
out for the modes n= 0,1, we can determine the mean
squared current in each vessel at every step on the adaptation timescale. Denoting by These boundary pressures can
be calculated by noting that
"˜
H(n)
in
˜
H(n)
out#="PµQ(n)
µ(0)
−PµQ(n)
µ(Lµ)#=Lµ(ω)"˜
P(n)
in (ω)
˜
P(n)
out (ω)#,(A9)
where Lµ(ω) is the network Laplacian, that takes the form
Lµ(ω) = 

Pµ
iωτµαµ
k(ωτµ)coth Lµ
λµk(ωτµ)−Pµ
iωτµαµ
k(ωτµ)csch Lµ
λµk(ωτµ)
−Pµ
iωτµαµ
k(ωτµ)csch Lµ
λµk(ωτµ)Pµ
iωτµαµ
k(ωτµ)coth Lµ
λµk(ωτµ)

.(A10)
In the ω= 0 case, the flow is steady and uniform, obeying
Q(0)
µ=αµλµ
2Lµ˜
P(0)
in −˜
P(0)
out=αµλµ
2Lµα1λ1
2L1
+α2λ2
2L2−1
˜
H(0) =R4
µ/Lµ
R4
1/L1+R4
2/L2
Q0.(A11)
In the pulsatile case, the imposed symmetry in the boundary currents along with the explicit form of the Laplacian
also forces the relation ˜
P(n)
in (ω) = −˜
P(n)
out (ω). This readily admits the solution
˜
P(n)
in (ω) = −˜
P(n)
out (ω) = 

X
µ
inωτµαµ
k(nωτµ)
coth Lµ
λµ
k(nωτµ)!+ csch Lµ
λµ
k(nωτµ)!



−1
˜
H(n),
=
X
µ
inωτµαµ
k(nωτµ)coth Lµ
2λµ
k(nωτµ)!

−1
˜
H(n),(A12)

8
where we have used the relation (coth(x) + csch(x)) = coth(x/2). Substituting Eq. A11 and Eq. A12 into Eq. A8, we
obtain the vessel mean squared current
hQ2
µi=Q2
0 αµλµ
2Lµ!2α1λ1
2L1
+α2λ2
2L2−2
+2αµλµ
Lµ
˜
P(1)
in (ω)
2Re 


iωτµαµ
k(ωτµ)
coth Lµ
λµ
k(ωτµ)!+ csch Lµ
λµ
k(ωτµ)!



=Q2
0 αµλµ
2Lµ!2α1λ1
2L1
+α2λ2
2L2−2
+A2
2
αµλµ
LµRe iωτµαµ
k(ωτµ)coth Lµ
2λµk(ωτµ)
Pµ
iωτµαµ
k(nωτµ)coth Lµ
2λµk(nωτµ)
2,(A13)
Eq. A13 can be simplified in notation by considering it as a function of R1and R2. We first note that when Eq. 6
of the main text is used to rewrite Eq. A13, the factors of α0cancel out. We can then define the unitless function,
g(ω, Rµ), and its zero frequency limit,
g(ω, Rµ) = iωτ0Lµ
k(ωτ0(Rµ/R0)2)λ0
coth LµR2
0
2λ0R2
e
k(ωτ0(Rµ/R0)2)!=⇒lim
ω→0g(ω, Rµ) = 1.(A14)
With this notation, Eq. A13 can be written as
hQ2
µi=Q2
0 R4
µ/Lµ
R4
1/L1+R4
2/L2!2
+A2
2
(R8
µ/L2
µ)Re g(ω, Rµ)
(R4
1/L1)g(ω, R1)+(R4
2/L2)g(ω, R2)
2.(A15)
The MAE then takes the form
dRµ
dt0=1
R3
µ
Q2
0 R4
µ/Lµ
R4
1/L1+R4
2/L2!2
+A2
2
(R8
µ/L2
µ)Re g(ω, Rµ)
(R4
1/L1)g(ω, R1)+(R4
2/L2)g(ω, R2)
2

γ
−bRµ.(A16)
A2. Stability of the Diagonal Fixed Point
In this section, we will derive the critical value γMAE
c(ω) above which the diagonal steady state in the case of
vessels with equal length (L1=L2=L) becomes unstable. To analyze the stability of the diagonal fixed point, i.e. to
determine whether it is a sink or a saddle, we need merely to look at the eigenvalues of the exterior derivative matrix
Mµν =∂(dRµ/dt)/∂Rν, where µ, ν = 1,2. To begin constructing such a matrix, we first note that the derivative of
k(y) = piy(2 + iy) with respect to its argument is
d
dy k(y)=i1 + iy
k(y)=⇒d
dy 1
k(y)=−i(1 + iy)
k(y)3=−1 + iy
y(2 + iy)k(y).(A17)
This allows for the derivative of g(ω, Rµ) with respect to each radius, which we denote as j(ω, Rµ), to be evaluated as

9
jω, Rµ=∂
∂Rµgω, Rµ
=−iωτ0Lµ
λ0
·
1 + iωτ0Rµ
R02
ωτ0Rµ
R022 + iωτ0Rµ
R02kωτ0Rµ
R02·2ωτ0Rµ
R2
0
·coth 

La2
0
2λ0R2
µ
k ωτ0Rµ
R02!

−iωτ0Lµ
kωτ0Rµ
R02λ0





La2
0
2λ0R2
µ
·i
1 + iωτ0Rµ
R02
kωτ0Rµ
R02·2ωτ0Rµ
R2
0
−La2
0
λ0R3
µ
k ωτ0Rµ
R02!




csch2

La2
0
2λ0R2
µ
k ωτ0Rµ
R02!

=−2
Rµ
·
1 + iωτ0Rµ
R02
2 + iωτ0Rµ
R02·iωτ0Lµ
kωτ0Rµ
R02λ0
coth 

La2
0
2λ0R2
µ
k ωτ0Rµ
R02!

+1
Rµ





iωτ0Lµ
kωτ0Rµ
R02λ0





2
csch2

La2
0
2λ0R2
µ
k ωτ0Rµ
R02!

=1
Rµ


gω, Rµ2−2gω, Rµ1 + iωτ0Rµ
R02
2 + iωτ0Rµ
R02− La2
0
λ0R2
µ!2iωτ0Rµ
R02
2 + iωτ0Rµ
R02



(A18)
In the ω→0 limit, j(ω, Rµ) vanishes, as can be readily determined by differentiating the ω→0 limit of g(ω, Rµ)
given in Eq. A14. Given this notation, we can readily differentiate hQ2
µiwith respect to Rµto produce
∂
∂RµDQ2
µE=Q2
0
R8
µ
R4
1+R4
22 8
Rµ
−8R3
µ
R4
1+R4
2!+A2
2


8R7
µRe gω, Rµ+R8
µRe jω, Rµ
R4
1g(ω, R1) + R4
2g(ω, R2)
2
−
2R8
µRe gω, RµRe 4R3
µgω, Rµ+R4
µjω, Rµ∗R4
1g(ω, R1) + R4
2g(ω, R2)
R4
1g(ω, R1) + R4
2g(ω, R2)
4




=Q2
0
R8
µ
R4
1+R4
22 8
Rµ
−8R3
µ
R4
1+R4
2!+A2
2
R8
µRe gω, Rµ
R4
1g(ω, R1) + R4
2g(ω, R2)
2


8
Rµ
+
Re jω, Rµ
Re gω, Rµ
−
2Re 4R3
µgω, Rµ+R4
µjω, Rµ∗R4
1g(ω, R1) + R4
2g(ω, R2)
R4
1g(ω, R1) + R4
2g(ω, R2)
2




.(A19)
Conversely, the derivative of hQ2
µiwith respect to Rν, where ν6=µ, takes the form

10
∂
∂RνDQ2
µE=−Q2
0
R8
µ
R4
1+R4
22
8R3
ν
R4
1+R4
2
−A2R8
µRe gω, RµRe 4R3
νg(ω, Rν) + R4
νj(ω, Rν)∗R4
1g(ω, R1) + R4
2g(ω, R2)
R4
1g(ω, R1) + R4
2g(ω, R2)
4
=−Q2
0
R8
µ
R4
1+R4
22
8R3
ν
R4
1+R4
2
−A2R8
µRe gω, Rµ
R4
1g(ω, R1) + R4
2g(ω, R2)
2
Re 4R3
νg(ω, Rν) + R4
νj(ω, Rν)∗R4
1g(ω, R1) + R4
2g(ω, R2)
R4
1g(ω, R1) + R4
2g(ω, R2)
2.
(A20)
Eqs. A19 and A20 then allow us to calculate the on diagonal and off diagonal matrix components Mµµ and Mµν
Mµµ =γDQ2
µEγ−1
R3
µ




Q2
0
R8
µ
R4
1+R4
22 8
Rµ
−8R3
µ
R4
1+R4
2!+A2
2
R8
µRe gω, Rµ
R4
1g(ω, R1) + R4
2g(ω, R2)
2


8
Rµ
+
Re jω, Rµ
Re gω, Rµ
−
2Re 4R3
µgω, Rµ+R4
µjω, Rµ∗R4
1g(ω, R1) + R4
2g(ω, R2)
R4
1g(ω, R1) + R4
2g(ω, R2)
2










−3DQ2
µEγ
R4
µ
−b, (A21a)
Mµν =γDQ2
µEγ−1
R3
µ
−Q2
0
R8
µ
R4
1+R4
22
8R3
ν
R4
1+R4
2
−A2R8
µRe gω, Rµ
R4
1g(ω, R1) + R4
2g(ω, R2)
2
Re 4R3
νg(ω, Rν) + R4
νj(ω, Rν)∗R4
1g(ω, R1) + R4
2g(ω, R2)
R4
1g(ω, R1) + R4
2g(ω, R2)
2

.
(A21b)
We are particularly interested in the case where Rµ=Rν=R. This causes each mean squared current to reduce
to
DQ2E=Q2
0
R8
(2R4)2+A2
2
R8Re g(ω, R)
2R4g(ω, R)
2=1
4Q2
0+1
2
A2
2Re 1
g(ω, R).(A22)
Additionally, this imposed symmetry forces M11 =M22 and M12 =M21 . This particular form of Malways has [1,1]
and [1,−1] as eigenvectors with eigenvalues we will denote as kand ⊥respectively. It is the sign of ⊥at the diagonal
fixed point that dictates whether such a point is a stable basin or saddle. To investigate this, let R∗be defined such
that Eq. A16 vanishes when Rµ=Rν=R∗, thus allowing for the relation hQ2iγ/R∗4=b. At this point, ⊥takes
the form

11
⊥=

γQ2γ−1
R3
Q2
0
R8
(2R4)2 8
R−8a3
2R4!+A2
2
R8Re g(ω, R)
2R4g(ω, R)
28
R
+Re j(ω, R)
Re g(ω, R)−
2Re 4R3g(ω, R) + R4j(ω, R)∗2R4g(ω, R)
2R4g(ω, R)
2





−3Q2γ
R4−b




R=R∗
−
γQ2γ−1
R3 −Q2
0
R8
(2R4)2
8R3
2R4
−A2R8Re g(ω, R)
2R4g(ω, R)
2
Re 4R3g(ω, R) + R4j(ω, R)∗2R4g(ω, R)
2R4g(ω, R)
2





R=R∗
=b


8γ
hQ2i

1
4Q2
0+A2
8Re 1
g(ω, R∗) 1 + Re R∗j(ω, R∗)
8Re g(ω, R∗)!
−4


= 4b






2γ
Q2
0+A2
2Re 1
g(ω,R∗) 1 + ReR∗j(ω,R∗)
8Re(g(ω,R∗))!
Q2
0+A2
2Re 1
g(ω,R∗)−1






.(A23)
From Eq. A23 it is clear that the value of γdirectly determines the sign of ⊥. The transition value of γat which
⊥= 0 takes the form
γMAE
c(ω) = 1
2
Q2
0+A2
2Re 1
g(ω,R∗)
Q2
0+A2
2Re 1
g(ω,R∗)1 + Re(R∗j(ω,R∗))
8Re(g(ω,R∗)).(A24)
The case of the traditional adaptation equation (AE) occurs when A= 0, which in turn causes Eq. A24 to simply
become γAE
c= 1/2. For the modified adaption equation considered in this work with nonzero ω, the steady state is
a symmetric loop with equal vessel radii for γ < γM AE
c(ω), and either loopless or an asymmetric loop with unequal
vessel radii for γ > γMAE
c(ω). Fig. 3(c) of the main text can be constructed by plotting the value of γMAE
cgiven in
Eq. A24 for each combination of ωτ =ωτ0(R∗/R0)2and L/λ =L/(λ0(R∗/R0)2), where R∗is the steady state radius
of each vessel at that specific value of ωand at γ=γM AE
c(ω).
A3. Vessels of unequal lengths
In this section we will discuss the steady state behavior obtained for cases where the two vessels have unequal
lengths. Fig. A1 shows the phase-diagram of the order parameter hZiobtained with the AE (top panels) and the
MAE (bottom panels) for the two cases discussed in Fig. 4 of the main text. While in both cases, the second vessel is
longer that the first vessel, in the first case both vessels are shorter than the characteristic length scale λ0(Fig. A1(a,c)),
whereas in the second case, the first vessel is shorter and the second vessel longer than λ0(Fig. A1(b,d)). It is clear
that in either scenario, the MAE stabilizes loops for a larger range of γvalues than the AE. We also observe that as
the length of the second vessel is increased in comparison to the first, resonances become weaker, and the peaks in
hZivalues become shorter in height.
In contrast to the situation where vessels have equal lengths the transition from a looped to a loopless steady state
is not sharp either the AE for unequal vessel lengths. Instead we observe a more gradual transition from hZi= 1 to
hZi= 0 in Fig. A1(a,b). This is because unlike the special symmetry furnished by the equal vessel case, vessels of

12
FIG. A1. (color online) Comparison of the steady state structures in the AE and MAE for vessels of unequal lengths. Top:
Phase-diagrams in the AE, with L1= 1 and L2=√2 in (a), and L1= 1 and L2= 2√2 in (b). Bottom: Phase-diagrams in
the MAE, with L1= 1 and L2=√2 in (c), and L1= 1 and L2= 2√2 in (d). λ0= 2 in all four cases, and dashed red lines
depict the critical transition in the AE for each case.
different lengths donot always support a steady state on the diagonal of the R1−R2space, for which hZiwould be
unity. Instead, asymmetric loops with 0 <hZi<1, i.e. with different but non-zero radii for each vessel, are stabilized
below γAE
cin the AE. This also explains why the MAE phase-diagram of hZi(Fig. A1(c,d)) has less sharp boundaries
between the yellow (hZi= 1) and dark blue regions (hZi= 0) than in the equal vessel cases shown in Fig. 3 of the
main text.
The qualitative features of the phase-diagram of hZifor unequal vessel lengths L16=L2can be understood by
looking at the superposition of the cases where both vessels have length L=L1or L=L2. Fig. A2 shows such a
comparison for vessels shorter than λ0. In the equal vessel length cases (Fig. A2(a,b)) we have marked with solid
lines the values of γ(ω) above which symmetric loops (hZi= 1) become unstable, and with dashed lines the values
of γ(ω) above which even asymmetric loops become unstable. Firstly, it is clear from Fig. A2(c), that the resonances
of the shorter vessel dominates the phase diagram in the unequal vessel lengths case, generating peaks of hZiat the
same frequencies as in Fig. A2(a). Secondly, the regions of the ω−γphase space which supports the highest values
of hZi ≈ 1, is the region for which the steady state is looped, i.e hZi>0 for each of the equal vessel length cases
(demarcated by dashed black lines). In regions where only one or the other equal vessel length cases is looped, a sort
of destructive interference leads to a reduction in the value of the order parameter hZifor the unequal vessel length
case. Thus, even without calculating the full phase-diagram for a network with a mixture of short and long vessels,
we should be able to qualitatively predict its steady state structure from the corresponding equal vessel length cases.

13
FIG. A2. (color online) Steady state structure for unequal vessel lengths compared to the corresponding equal vessel length
cases. (a,b) Phase-diagram for equal vessel lengths, with L= 1 in (a) and L=√2 in (b), and λ0= 2. Solid lines depict the
values of γ(ω) above which symmetric loops (hZi= 1) become unstable. Dashed lines depict the values of γ(ω) above which
asymmetric loops (hZi>0) become unstable. (c) Phase-diagram for the unequal vessel length case with L1= 1 and L2=√2.
The colored dashed lines correspond to those in (a,b). Vertical black dashed lines demarcate regions of the ω−γphase space
that have the maximum values of the order parameter hZi, that is the most symmetric loops at steady state.