By Miguel Ruiz-García and Eleni Katifori
Flow networks comprise a set of connections that carry and transport fluid. In linear networks, the current that circulates through each “duct” increases proportionally with the pressure difference between the inlet and outlet. But in animal and plant circulatory systems, each element’s resistance can be highly nonlinear. Due to the active nature of animal vessels [1, 6] or the presence of passive valves in plant vasculature [4], certain situations may manifest wherein the current falls as the pressure difference increases; such behavior is known as negative differential resistance.
Blood vessels are more like active organs than rigid ducts. Specifically, vascular musculature covers the arteries and allows them to contract or expand in response to different stimuli. For example, when a blood vessel that feeds an organ detects an increase in pressure at its inlet, it can respond by contracting (compressing its muscles) to reduce flow and protect the organ. This effect is called the myogenic mechanism [1], and similar circumstances can cause a negative differential resistance in the flow through a blood vessel [6]. In plant vasculature, the xylem transports water from the roots to the leaves, ultimately forming a network of channels and valves (pit pores) that control the flow. These valves also present a highly nonlinear conductance with negative differential resistance [4]. Figures 1a and 1b depict schematics of both systems.
Figure 1. Nonlinear flow in plant and animal vasculature. 1a. Zooming into the plant stem shows xylem tracheids and pit pores, which transport water from the roots to the leaves. 1b. Zooming into the mammalian brain reveals the arteries that feed it. 1c. When a pit pore [4] or cerebral artery [6] is isolated and connected to a pump, the flow \((I)\) that is measured as a function of pressure difference \((\Delta P)\) follows a nonlinear curve that reveals a region of negative slope (negative differential resistance). Figure courtesy of Miguel Ruiz-García.
Inspired by the behavior of animal and plant vasculature, we recently proposed a model that accounts for these nonlinearities as well as volume accumulation/depletion in flow networks of any topology [5]. A set of nodes that are interconnected by edges form the network. Pressure is defined at each node \((P_i)\), and the flow through every edge depends on the pressure difference between the nodes that it connects:
\[I_{ij} \propto \Gamma(P_i-P_j).\]
Here, \(\Gamma\) is a nonlinear function of \(\Delta P\) that may present a negative slope (see Figure 1c). In the biological systems that we model, the volume inside the network can change; for the sake of simplicity, we account for this volume capacity at the nodes \((V_i)\). If the volume increases in one part of the network within the biological vasculature, it deforms the surrounding tissue and affects the pressure in nearby regions. We model this effect by coupling volume accumulation and pressure:
\[V_i-1 \propto \sum_k L_{ik}P_k,\]
where \(L_{ik}\) is the network’s graph Laplacian. Finally, we must impose mass conservation at every node:
\[\frac{dV_i}{dt}=-\sum_k I_{ik}.\]
The previous equations form a system of nonlinear differential equations that we can apply to networks of any topology — i.e., each edge can connect any two nodes. Complex dynamical phenomena may thus emerge once we integrate the equations over time. When the network is large enough, it can behave like an excitable system. Animation 1 demonstrates an example of this behavior, wherein we apply a constant direct current (DC) pump to the two sides of a rectangular network and allow the system to relax to a stationary situation (where all of the flows are constant). We then introduce a perturbation to the pressure boundary conditions, prompting the appearance of a travelling wave that moves from one side of the network to the other. We are unable to excite another wave in the presence of a previous one, which leads to a refractory time.
Animation 1. Excitability in nonlinear flow networks. The network is a rectangular grid with five rows and 100 columns of nodes. Upper panels color every node’s surroundings according to its pressure (\(P_i\), upper left) or volume (\(V_i\), upper right); darker colors correspond to higher values of pressure or volume. We impose pressure boundary conditions on the left and right sides of the network: the pressure of the red nodes in the right column (the “out” nodes) remains at zero, whereas the pressure of the blue nodes in the left column (the “in” nodes) follows the protocol that is displayed in blue in the bottom panel. The middle panel illustrates the total current that travels from the pump into the blue nodes and the total current that travels from the red nodes out to the pump. These quantities do not have to be equal because the volume inside the system can change. Animation courtesy of Miguel Ruiz-García.
Yet contrary to other network models—in which the elements (edges or nodes) are intrinsically excitable—excitability emerges in our scenario as a collective phenomenon, given that our edges and nodes are simply passive elements that respectively conduct or store volume. Such behavior would be forbidden in a network of linear resistors because the flows within the network are uniquely determined by the boundary conditions; a perturbation at the boundary would instantaneously change the flows in the whole network, negating the possibility of observing travelling waves.
Another phenomenon that arises in these nonlinear flow networks is the appearance of self-sustained travelling waves (see Animation 2). Under the right circumstance (a certain range of model parameters) and with constant pressure boundary conditions (equivalent to a DC pump), the system does not find a stable stationary solution. Travelling waves nucleate at one contact and travel to the other, recycling every time; these actions induce periodic oscillations in the current that runs through the pump [5]. Animation 2 depicts this behavior in a planar network with random connections and periodic spatial boundary conditions. The pump is connected to the blue and red nodes and imposes a constant pressure difference between them. Travelling waves nucleate around the blue node and move towards the red one until they disappear and recycle, generating periodic oscillations in the current (see the bottom panel of Animation 2). This behavior has an analog—called the Gunn effect—in the semiconductor realm, wherein negative differential resistance also leads to self-sustained oscillations of the current; in that case, however, the system is continuous and effectively one dimensional [2, 3].
Animation 2. Travelling waves in a nonlinear flow network. The planar network has random connections, an average node degree of \(5.5\), and periodic spatial boundary conditions. Upper panels color every node’s surroundings according to its pressure (\(P_i\), upper left) or volume (\(V_i\), upper right); darker colors correspond to higher values of pressure or volume. The red and blue nodes maintain a constant pressure difference because they are connected to a direct current pump. Self-sustained waves emerge spontaneously, nucleating in the blue node and travelling to the red one to recycle again. The bottom panel depicts the current that travels from the pump into the blue node and the current that travels from the red node out to the pump. These quantities do not have to be equal because the volume inside the system can change. Animation courtesy of Miguel Ruiz-García.
To summarize, the nonlinear behavior of plant and animal vasculature inspired us to create a model for nonlinear flow networks of arbitrary topology. Our model displays a rich phenomenology that includes memory effects, excitability, and self-sustained oscillations, and we hope that it will enable further understanding of complex phenomena in the living realm. In the case of animal vasculature, the spontaneous emergence of hemodynamic fluctuations in the mammalian brain is particularly intriguing. When mice—or any mammal, for that matter—are resting or taking medication that suppresses brain activity, traveling waves of volume accumulation materialize in the arteries that surround the brain. Some researchers have suggested that a portion of these oscillations might have a non-neural origin [7]. Although this proposal is still under debate, it would be interesting to determine whether these waves are related to the negative differential resistance of the brain arteries, which would be reminiscent of the traveling waves that emerge in our theoretical model.
In the case of plant vasculature, it is especially striking that pit pores in the xylem transport network comprise a large region of negative differential resistance [4]. An exciting future research direction could investigate whether plants harness any of the collective phenomena that emerge when multiple nonlinear resistors are connected. For instance, we know that a pressure range wherein the flow is almost constant—independent of the amount of applied pressure—is present when many of these valves are connected in a series.
Ultimately, we hope that these results will motivate experimental researchers to build such systems in the laboratory. Harnessing the complex phenomenology of our model in practice could give rise to new and exciting technologies.
References
[1] Bayliss, W.M. (1902). On the local reactions of the arterial wall to changes of internal pressure. J. Physiol., 28(3), 220-231.
[2] Gunn, J.B. (1963). Microwave oscillations of current in III-V semiconductors. Solid State Commun., 1(4), 88-91.
[3] Kroemer, H. (1964). Theory of the Gunn effect. Proc. IEEE, 52(12), 1736.
[4] Park, K., Tixier, A., Paludan, M., Østergaard, E., Zwieniecki, M., & Jensen, K.H. (2021). Fluid-structure interactions enable passive flow control in real and biomimetic plants. Phys. Rev. Fluids, 6(12), 123102.
[5] Ruiz-García, M., & Katifori, E. (2021). Emergent dynamics in excitable flow systems. Phys. Rev. E, 103(6), 062301.
[6] Thorin-Trescases, N., & Bevan, J.A. (1998). High levels of myogenic tone antagonize the dilator response to flow of small rabbit cerebral arteries. Stroke, 29(6), 1194-1201.
[7] Winder, A.T., Echagarruga, C., Zhang, Q., & Drew, P.J. (2017). Weak correlations between hemodynamic signals and ongoing neural activity during the resting state. Nat. Neurosci., 20(12), 1761-1769.
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Miguel Ruiz-García is a CONEX-Plus Marie Curie independent researcher at Universidad Carlos III de Madrid with broad interests in soft matter physics and applied mathematics. He studies flow networks and theoretical machine learning, and uses neural networks to tackle systems like active matter and social relationships. |
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Eleni Katifori is an associate professor in the Department of Physics and Astronomy at the University of Pennsylvania. Her research falls at the interface of complex systems, soft matter, fluid dynamics, and biophysics. Katifori has worked extensively on problems that are inspired by (and related to) biological flow networks in animals and plants, as well as thin shell elasticity. |