Detecting coexisting oscillatory patterns in delay coupled Lur'e systems.

This work addresses the problem of pattern analysis in networks consisting of delay-coupled identical Lur'e systems. We study a class of nonlinear systems, which, being isolated, are globally asymptotically stable. Assembling such systems into a network via time-delayed coupling may result in the change of network equilibrium stability under parameter variation in the coupling. In this work, we focus on cases where a Hopf bifurcation causes the change of stability of the network equilibrium and leads to the occurrence of oscillatory modes (patterns). Moreover, some of these patterns can co-exist for the same set of coupling parameters, which makes the analysis by means of common methods, such as the Lyapunov-Krasovskii method or the analysis of Poincaré maps, cumbersome. A numerically efficient algorithm, aiming at the computation of the oscillatory patterns occurring in such networks, is presented. Moreover, we show that our approach is able to deal with co-existing patterns, and both stable and unstable regimes can be simultaneously computed, which gives deep insight into the network dynamics. In order to illustrate the efficiency of the method, we present two examples in which the instability of the network equilibria is caused by a subcritical and a supercritical Hopf bifurcation. In addition, a bifurcation analysis of the subcritical case is performed in order to further explain the occurrence of the detected coexisting modes.

Robust partial synchronization of delay-coupled networks.

Networks of coupled systems may exhibit a form of incomplete synchronization called partial synchronization or cluster synchronization, which refers to the situation where only some, but not all, systems exhibit synchronous behavior. Moreover, due to perturbations or uncertainties in the network, exact partial synchronization in the sense that the states of the systems within each cluster become identical, cannot be achieved. Instead, an approximate synchronization may be observed, where the states of the systems within each cluster converge up to some bound, and this bound tends to zero if (the size of) the perturbations tends to zero. In order to derive sufficient conditions for this robustified notion of synchronization, which we refer to as practical partial synchronization, first, we separate the synchronization error dynamics from the network dynamics and interpret them in terms of a nonautonomous system of delay differential equations with a bounded additive perturbation. Second, by assessing the practical stability of this error system, conditions for practical partial synchronization are derived and formulated in terms of linear matrix inequalities. In addition, an explicit relation between the size of perturbation and the bound of the synchronization error is provided.

The challenge of catalyst prediction.

New insights and successful use of computational catalysis are highlighted. This is within the context of remaining issues that prevent theoretical catalysis to be fully predictive of catalyst performance. A major challenge is to include in modelling studies the transient initiation as well as deactivation processes of the catalyst. We will illustrate this using as an example for solid acid catalysis, the alkylation process, and for transition metal catalysis, the Fischer-Tropsch reaction. For the alkylation reaction of isobutane and alkene, an important reaction for high octane gasoline, we will present a deactivation model. For the Fischer-Tropsch reaction, which converts synthesis gas into gasoline grade molecules, we discuss structural reorganization of the catalyst induced by reaction.

A neural mass model of phase-amplitude coupling.

Brain activity shows phase-amplitude coupling between its slow and fast oscillatory components. We study phase-amplitude coupling as recorded at individual sites, using a modified version of the well-known Wendling neural mass model. To the population of fast inhibitory interneurons of this model, we added external modulatory input and dynamic self-feedback. These two modifications together are sufficient to let the inhibitory population serve as a limit-cycle oscillator, with frequency characteristics comparable to the beta and gamma bands. The frequency and power of these oscillations can be tuned through the time constant of the dynamic and modulatory input. Alpha band activity is generated, as is usual in such models, as a result of interactions of pyramidal neurons and a population of slow inhibitory interneurons. The slow inhibitory population activity directly influences the fast oscillations via the synaptic gain between slow and fast inhibitory populations. As a result, the amplitude envelope of the fast oscillation is coupled to the phase of the slow activity; this result is consistent with the notion that phase-amplitude coupling is effectuated by interactions between inhibitory interneurons.

Programmable metachronal motion of closely packed magnetic artificial cilia.

Despite recent advances in artificial cilia technologies, the application of metachrony, which is the collective wavelike motion by cilia moving out-of-phase, has been severely hampered by difficulties in controlling closely packed artificial cilia at micrometer length scales. Moreover, there has been no direct experimental proof yet that a metachronal wave in combination with fully reciprocal ciliary motion can generate significant microfluidic flow on a micrometer scale as theoretically predicted. In this study, using an in-house developed precise micro-molding technique, we have fabricated closely packed magnetic artificial cilia that can generate well-controlled metachronal waves. We studied the effect of pure metachrony on fluid flow by excluding all symmetry-breaking ciliary features. Experimental and simulation results prove that net fluid transport can be generated by metachronal motion alone, and the effectiveness is strongly dependent on cilia spacing. This technique not only offers a biomimetic experimental platform to better understand the mechanisms underlying metachrony, it also opens new pathways towards advanced industrial applications.

Partial synchronization in diffusively time-delay coupled oscillator networks.

We study networks of diffusively time-delay coupled oscillatory units and we show that networks with certain symmetries can exhibit a form of incomplete synchronization called partial synchronization. We present conditions for the existence and stability of partial synchronization modes in networks of oscillatory units that satisfy a semipassivity property and have convergent internal dynamics.

Deactivation Kinetics of Solid Acid Catalyst with Laterally Interacting Protons.

Differences in catalyst deactivation kinetics in solid acid catalysis are studied with catalyst models that allow for lateral interaction between protons. Deactivation of a solid acid catalyst with laterally interacting protons induces inhomogeneity of proton reactivity that develops with time. As a consequence, product selectivity changes and deactivation will accelerate. This is demonstrated by simulations of the deactivation kinetics of the alkylation reaction of propylene with isobutane. The effect of lateral interactions between protons arises because initial catalyst deactivation is not caused by pore blocking or coke deposition but by a molecular mechanism where protons are consumed due to the formation of stable nonreactive carbenium ions. High selectivity to alkylate requires a catalyst with protons of high reactivity. When protons become consumed by formation of stable deactivating carbenium ions, initially reactive protons are converted into protons of lower reactivity. The latter only catalyze deactivating oligomerization reactions. Simulations that compare the deactivation kinetics of a catalyst model with laterally interacting protons and a catalyst model that contains protons of similar but different reactivity, but that do not laterally interact, are compared. These simulations demonstrate that the lateral interaction catalyst model is initially more selective but also has a lower stability. Catalyst deactivation of the alkylation reaction occurs through two reaction channels. One reaction channel is due to oligomerization of reactant propylene. The other deactivation reaction channel is initiated by deprotonation of intermediate carbenium ions that increase alkene concentration. By consecutive reactions, this also leads to deactivation. The hydride transfer reaction competes with oligomerization reactions. It is favored by strongly acid sites that also suppress the deprotonation reaction. Within the laterally interacting proton catalyst model, when reactive protons become deactivated, weakly reactive protons are generated that only catalyze the deactivating alkene oligomerization and consecutive reactions. This rapid formation of the weakly reactive protons is the cause of decreasing selectivity with reaction time and increased rate of deactivation. Solutions of the mean field kinetic equations as well as stochastic simulations are presented. Comparative simulations with a reduced number of neighbors of the protons illustrate decreased deactivation rates when the proton density decreases. Island formation of adsorbed reaction intermediates on the catalyst surface is observed in stochastic kinetics simulations. When alkylation selectivity is high, this island formation increases the rate of catalyst deactivation in comparison to the rate of deactivation according to the mean field studies. A nonlinear dynamics model of proton dynamics is provided, which shows that the differences between stochastic and mean field simulations are due to frustrated proton state percolation.

A neural mass model of cross frequency coupling.

Electrophysiological signals of cortical activity show a range of possible frequency and amplitude modulations, both within and across regions, collectively known as cross-frequency coupling. To investigate whether these modulations could be considered as manifestations of the same underlying mechanism, we developed a neural mass model. The model provides five out of the theoretically proposed six different coupling types. Within model components, slow and fast activity engage in phase-frequency coupling in conditions of low ambient noise level and with high noise level engage in phase-amplitude coupling. Between model components, these couplings can be coordinated via slow activity, giving rise to more complex modulations. The model, thus, provides a coherent account of cross-frequency coupling, both within and between components, with which regional and cross-regional frequency and amplitude modulations could be addressed.

Self-organisation of small-world networks by adaptive rewiring in response to graph diffusion.

Complex networks emerging in natural and human-made systems tend to assume small-world structure. Is there a common mechanism underlying their self-organisation? Our computational simulations show that network diffusion (traffic flow or information transfer) steers network evolution towards emergence of complex network structures. The emergence is effectuated through adaptive rewiring: progressive adaptation of structure to use, creating short-cuts where network diffusion is intensive while annihilating underused connections. With adaptive rewiring as the engine of universal small-worldness, overall diffusion rate tunes the systems' adaptation, biasing local or global connectivity patterns. Whereas the former leads to modularity, the latter provides a preferential attachment regime. As the latter sets in, the resulting small-world structures undergo a critical shift from modular (decentralised) to centralised ones. At the transition point, network structure is hierarchical, balancing modularity and centrality - a characteristic feature found in, for instance, the human brain.

Hierarchical control of enzymatic actuators using DNA-based switchable memories.

Inspired by signaling networks in living cells, DNA-based programming aims for the engineering of biochemical networks capable of advanced regulatory and computational functions under controlled cell-free conditions. While regulatory circuits in cells control downstream processes through hierarchical layers of signal processing, coupling of enzymatically driven DNA-based networks to downstream processes has rarely been reported. Here, we expand the scope of molecular programming by engineering hierarchical control of enzymatic actuators using feedback-controlled DNA-circuits capable of advanced regulatory dynamics. We developed a translator module that converts signaling molecules from the upstream network to unique DNA strands driving downstream actuators with minimal retroactivity and support these findings with a detailed computational analysis. We show our modular approach by coupling of a previously engineered switchable memories circuit to downstream actuators based on ß-lactamase and luciferase. To the best of our knowledge, our work demonstrates one of the most advanced DNA-based circuits regarding complexity and versatility.

Spatially constrained adaptive rewiring in cortical networks creates spatially modular small world architectures.

A modular small-world topology in functional and anatomical networks of the cortex is eminently suitable as an information processing architecture. This structure was shown in model studies to arise adaptively; it emerges through rewiring of network connections according to patterns of synchrony in ongoing oscillatory neural activity. However, in order to improve the applicability of such models to the cortex, spatial characteristics of cortical connectivity need to be respected, which were previously neglected. For this purpose we consider networks endowed with a metric by embedding them into a physical space. We provide an adaptive rewiring model with a spatial distance function and a corresponding spatially local rewiring bias. The spatially constrained adaptive rewiring principle is able to steer the evolving network topology to small world status, even more consistently so than without spatial constraints. Locally biased adaptive rewiring results in a spatial layout of the connectivity structure, in which topologically segregated modules correspond to spatially segregated regions, and these regions are linked by long-range connections. The principle of locally biased adaptive rewiring, thus, may explain both the topological connectivity structure and spatial distribution of connections between neuronal units in a large-scale cortical architecture.

State and parameter estimation for canonic models of neural oscillators.

We consider the problem of how to recover the state and parameter values of typical model neurons, such as Hindmarsh-Rose, FitzHugh-Nagumo, Morris-Lecar, from in-vitro measurements of membrane potentials. In control theory, in terms of observer design, model neurons qualify as locally observable. However, unlike most models traditionally addressed in control theory, no parameter-independent diffeomorphism exists, such that the original model equations can be transformed into adaptive canonic observer form. For a large class of model neurons, however, state and parameter reconstruction is possible nevertheless. We propose a method which, subject to mild conditions on the richness of the measured signal, allows model parameters and state variables to be reconstructed up to an equivalence class.