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Self-Organized Stationary Patterns in Networks of Bistable Chemical Reactions

Experiments with networks of discrete reactive bistable electrochemical elements organized in regular and nonregular tree networks are presented to confirm an alternative to the Turing mechanism for the formation of self-organized stationary patterns. The results show that the pattern formation can be described by the identification of domains that can be activated individually or in combinations. The method also enabled the localization of chemical reactions to network substructures and the identification of critical sites whose activation results in complete activation of the system. Although the experiments were performed with a specific nickel electrodissolution system, they reproduced all the salient dynamic behavior of a general network model with a single nonlinearity parameter. Thus, the considered pattern-formation mechanism is very robust, and similar behavior can be expected in other natural or engineered networked systems that exhibit, at least locally, a treelike structure.


I. INTRODUCTION
Complex networks can be formed by chemical reactors [1][2][3], biological cells [4], or engineered units [5][6][7]. An interplay between local dynamics at the nodes and architecture of the links in such systems leads to a wealth of self-organization phenomena, including synchronization [3,8,9], stationary Turing and oscillatory patterns [10][11][12][13][14], or excitation waves [15][16][17]. The Turing mechanism [18] requires a difference in diffusion rates of inhibitor and activator species, which destabilizes a spatially homogeneous steady state and gives rise to the formation of new spatial patterns [19,20]. Here, an alternative mechanism for emergence of stationary patterns in networks of coupled bistable units is experimentally explored. Bistable units can be found in a broad class of chemical reactions, see e.g. [21], but also in cellular [22], social [23], and engineered systems [24]. Such units can have local or diffusive connections between them.
In continuous bistable media, traveling or retreating activation fronts are observed [25]. In chains and lattices of diffusively coupled one-component bistable elements, traveling fronts can become pinned if coupling is sufficiently weak [26][27][28][29] and thus stationary patterns are formed. Recently, the theory has been extended to networks representing trees and a rich variety of stationary patterns, sensitive to the topology of the networks, has been predicted [30,31]. However, experimental confirmation of the theory has been so far missing. Now, in the experiments with specially constructed tree networks of electrochemical units, we not only confirm the theoretical predictions, but also show that they hold well beyond the simplifying conditions under which the theory has been * Electronic address: nkouba@gmail.com developed.

II. THEORY OF BISTABLE TREE NETWORKS
Suppose that some substance can undergo chemical reactions in reactors occupying nodes i (i = 1, . . . , N ) of a network and this substance can spread diffusively from one node to another. Such a network-organized reactiondiffusion system is generally described by equationṡ where u i is the chemical concentration in the node i, the function f (u) specifies the local dynamics at the nodes, and the coefficient K characterizes the strength of diffusive coupling. The function f (u) can be chosen in such a way that individual elements are bistable. In our theoretical analysis, we employed a classical one-component bistable system [32] to describe the dynamics in the network nodes. Specifically, the dynamics was characterized by the cubic function where u represented the chemical concentration in a node. Function f (u) had three roots which corresponded to the fixed points of the local dynamics: two stable fixed points with real negative eigenvalues (i.e., stable nodes) at u = 0 and u = a, and one unstable fixed point with two real eigenvalues of opposite signs (i.e., a saddle point) at u = h. The network structure is determined by a symmetric adjacency matrix whose elements are A ij = 1, if there is a connection between nodes i and j (i = j), and A ij = 0 otherwise. For such models an approximate analytical theory is available [30] for special networks representing regular trees. In a regular tree with the branching ratio r, each node in the layer l is connected to r nodes in the next layer l + 1 and dynamics can be described by equationṡ where u l is the chemical concentration in the layer l. Previously, it was theoretically shown [30] that such equations support traveling and pinned (stationary) fronts, and the transition between them occurs via saddle-node bifurcations. Here we determine these bifurcations as a function of the parameters r and K. In the parameter plane r − K (see FIG. 1a), fronts which correspond to center activation get pinned via a saddle-node bifurcation at (red curves) and merge in the cusp point (black point). They separate regions II (pinned, stationary fronts) from region III (propagating fronts) as well as region II from region IV (retreating fronts). Fronts can also turn from propagating to retreating ones without getting pinned, when the curve (green curve that separates regions III and IV ) is traversed. Fronts that correspond to peripheral activation get pinned via a saddle-node bifurcation at (blue curve). While they are pinned in region I, the fronts begin to propagate towards the center of the tree when the bifurcation boundary is crossed. Peripheral activation cannot retreat. The bifurcation diagram in Under given nonlinearity factors (that determine the range of bistability and the transition boundary between the two stable states), the two major parameters are the coupling strength (K) and the branching ratio of the regular tree (r). For weak coupling, the activation is pinned (i.e., it does not propagate) in region I. At the intermediate coupling strength there is a range of branching ratios (region II), where the center activation is pinned while the periphery activation propagates towards the center. At sufficiently strong coupling and relatively low branching ratios (region III), the center activation spreads towards to periphery, whereas the periphery activation propagates towards the center. At strong coupling and large branching ratios (region IV ), the center activation retreats and the periphery activation propagates towards the center. The behavior of a regular tree with a fixed branching ratio can be fully interpreted in terms of these four regimes. Moreover, these results can also be applied to analyze the behavior in more complex, non-regular tree networks. Numerical simulations for this network were performed with the coupling constant K = 0.04, so that these domains corresponded to regions II, III or IV in the bifurcation diagram. Simulations were done by applying initial activation to one of the nodes and following the subsequent evolution until a stationary state was reached. Stationary patterns were then classified by identifying all the possible stationary states that could be obtained by such an initial activation.
The developed patterns can be interpreted by the formation of domains activated individually or in combinations. The domains consist of groups of nodes that correspond to the same dynamical region and thus behave similarly. For example, domains of elements in region III support spreading of activation towards both the center and the periphery. When such domains are adjacent to, or surrounded by nodes that are amenable to pinning or to retreating (e.g., in region I, II, or IV ), the propagating fronts get pinned and stationary structures develop. Therefore, the observed structures consist of uniform domains separated by nodes amenable to pinning or to retreating. The exact configuration of the patterns depends on the architecture of the network, the applied coupling strength, and the nonlinearity of the reaction that altogether determine the assignment of the nodes to the different regions and the configuration of the domains.  (1) connects the first to the second electrode and Rc (2) connects the second to the third electrode. Similarly, the connection may be non-local by connecting a coupling resistance between two non-local electrodes.

III. EXPERIMENTAL OBSERVATION OF STATIONARY DOMAINS IN ELECTROCHEMICAL BISTABLE NETWORKS
Experiments with networks of bistable electrochemical reactions were performed. Each element (see FIG. 2) represented a corroding metal (nickel) wire that accommodated a complex reaction system (including, e.g., formation of multiple forms of metal oxides, bisulfate adsorption, oxygen evolution and metal dissolution) that exhibited bistable behavior. Moreover, coupling was established in the form of the charge flow between the wires (due to difference in electrode potential) which affected the rate of metal dissolution of the coupled electrodes.

A. Experimental apparatus
We performed experiments using an electrochemical cell where a platinum coated titanium rod is the counter electrode, Hg/Hg 2 SO 4 /saturated K 2 SO 4 is the reference electrode, an electrode array of 72 nickel, 1.00 mm diameter, wires embedded in epoxy is the working electrode and the electrolyte is 3M H 2 SO 4 (Macron Chemicals, Lot# K39005) held at 10 o C. Each electrode in the array is connected to a potentiostat (ACM Instruments GillAC) through an individual resistance (R ind ) of 1 kΩ and an individual capacitance (C ind ) of 4.7 mF in parallel (see FIG. 2a). We apply a constant circuit potential (V ) and the current of each electrode in the array is measured at 50 Hz. All electrodes exhibit bistability for a range of circuit potential [V min , V max ]. The C ind serves to prevent the oscillations which naturally occur at the applied potential region of interest. For each experiment, we set the circuit potential V such that V = V min +(V max −V min )/4; this procedure ensures that the dynamics is similar in the repeated experiments. Desired active (high current) or passive state (low current) is achieved by superimposing a locally applied potential sweep on the circuit potential.
Interactions between electrodes occur through external resistance connections. The strength of the interactions are controlled by the coupling resistance (R c ); the coupling strength (K) is the inverse of R c . When the electrodes are connected, a network topology, determined by the external connections, is established. Electrodes are the nodes and the external connections are the links between nodes in all network diagrams. A simple three electrode schematics is shown in FIG. 2b where the electrodes are coupled linearly.
Bistable region is not exactly the same for every electrode, therefore the experimental results exhibit some minor deviation from numerical simulations and theoretical predictions.

B. Experimental results
It should be noted that, although our experimental systems indeed represented a network of coupled bistable elements, it was not accurately described by a simple one-component model used in the theoretical analysis. A realistic quantitative model for such electrochemical elements is not yet available, but it should definitely include many chemical variables. While coupling was pairwise, it could not be interpreted as corresponding to diffusion of a certain chemical species. Because of such differences, the validity of theoretical predictions was not obvious.
First, experiments with regular trees were undertaken. Figure 3 shows that center activation in a four-layer tree with the branching ratio 3 could result in spreading fronts (FIG. 3b, region III) for weak coupling, pinned fronts (FIG. 3c, region II) for moderate coupling, and retreating fronts (FIG. 3d, region IV ) for strong coupling. Similarly, periphery activation (FIG. 3e) yielded either pinned fronts (FIG. 3e, region I) at very weak coupling or spreading fronts (FIG. 3f) at stronger coupling; such behavior was found in all regions II, III, or IV. Figure 4 shows the evolution observed in the same network after activation of an intermediate node at different coupling strengths. In region I (FIG. 4a) the fronts are pinned on both sides. In region II (FIG. 4b) the center facing front spreads towards the center and the periphery facing front is pinned. In region III fronts spread in both directions and finally activate the entire network (FIG. 4c). In region IV the center facing front propagates towards the center but the periphery facing front is retreated from the periphery finally establishing the passive state in the entire network (FIG. 4d) Furthermore, we could also build the same non-regular tree as in the simulations in FIG. 1b. By performing experiments with regular trees under the same experimental setup, we could find (see Supplemental Figure  S1) that at K ≈ 1 mS regions II, III, and IV correspond to the branching ratios r = 3, 4, r = 1, 2, and r ≥ 5, respectively. Similar to theoretical predictions, we could see that activations with a single (FIG. 5a and Supplemental Video S1) or multiple nodes (FIG. 5b . 5c and Supplemental Video S3). To achieve complete activation of the network, peripheral nodes of the branches with the highest branching ratios r = 3, 4 and 5 had to be initially activated (see FIG. 5d and Supplemental Video S4). When a root node of a branch with high branching ratios r = 4 or r = 5 was initially activated, the activation could not spread and died out (FIG. 5e) of more central node in the branch with r = 3 resulted in a pinned front (FIG. 5f).

IV. DISCUSSION
We have experimentally demonstrated that bistable tree networks can support a rich variety of stationary domain patterns, determined both by the network architecture and initial activation conditions. A surprisingly good agreement with theoretical predictions has been found despite the fact that our experimental system did not perfectly meet the idealizations made in the theory [30], i.e. the state of a network element was described by more than a single variable and coupling between the elements was local, but not exactly of the diffusive form. This indicates that the considered behavior is generic and robust; therefore it can be expected in various natural and engineered bistable networks with tree-like structure, such as, e.g., nephrons of the kidney [33] or supercomputer networks [34].
While only (regular or non-regular) tree networks were considered, our results are also relevant for large random networks. Indeed, it is known [35] that locally such networks possess the tree structure. Hence, formation of stationary domains should also be characteristic for random networks provided that such domains remain sufficiently small. Finally, it should be noted that, as has been theoretically shown [31], self-organized stationary domains can be controlled by introducing global feedbacks and experimental realization of such control would further facilitate the design of complex stationary structures on networks.