Malbor Asllani

Turing patterns in multiplex networks.

The theory of patterns formation for a reaction-diffusion system defined on a multiplex is developed by means of a perturbative approach. The interlayer diffusion constants act as a small parameter in the expansion and the unperturbed state coincides with the limiting setting where the multiplex layers are decoupled. The interaction between adjacent layers can seed the instability of a homogeneous fixed point, yielding self-organized patterns which are instead impeded in the limit of decoupled layers. Patterns on individual layers can also fade away due to cross-talking between layers. Analytical results are compared to direct simulations.

The theory of pattern formation on directed networks

Dynamical processes on networks have generated widespread interest in recent years. The theory of pattern formation in reaction-diffusion systems defined on symmetric networks has often been investigated, due to its applications in a wide range of disciplines. Here we extend the theory to the case of directed networks, which are found in a number of different fields, such as neuroscience, computer networks and traffic systems. Owing to the structure of the network Laplacian, the dispersion relation has both real and imaginary parts, at variance with the case for a symmetric, undirected network. The homogeneous fixed point can become unstable due to the topology of the network, resulting in a new class of instabilities, which cannot be induced on undirected graphs. Results from a linear stability analysis allow the instability region to be analytically traced. Numerical simulations show travelling waves, or quasi-stationary patterns, depending on the characteristics of the underlying graph.

The linear noise approximation for reaction-diffusion systems on networks

Stochastic reaction-diffusion models can be analytically studied on complex networks using the linear noise approximation. This is illustrated through the use of a specific stochastic model, which displays travelling waves in its deterministic limit. The role of stochastic fluctuations is investigated and shown to drive the emergence of stochastic waves beyond the region of the instability predicted from the deterministic theory. Simulations are performed to test the theoretical results and are analyzed via a generalized Fourier transform algorithm. This transform is defined using the eigenvectors of the discrete Laplacian defined on the network. A peak in the numerical power spectrum of the fluctuations is observed in quantitative agreement with the theoretical predictions.

Turing instabilities on Cartesian product networks.

The problem of Turing instabilities for a reaction-diffusion system defined on a complex Cartesian product network is considered. To this end we operate in the linear regime and expand the time dependent perturbation on a basis formed by the tensor product of the eigenvectors of the discrete Laplacian operators, associated to each of the individual networks that build the Cartesian product. The dispersion relation which controls the onset of the instability depends on a set of discrete wavelengths, the eigenvalues of the aforementioned Laplacians. Patterns can develop on the Cartesian network, if they are supported on at least one of its constitutive sub-graphs. Multiplex networks are also obtained under specific prescriptions. In this case, the criteria for the instability reduce to compact explicit formulae. Numerical simulations carried out for the Mimura-Murray reaction kinetics confirm the adequacy of the proposed theory.

Delay-induced Turing-like waves for one-species reaction-diffusion model on a network

A one-species time-delay reaction-diffusion system defined on a complex network is studied. Traveling waves are predicted to occur following a symmetry-breaking instability of a homogeneous stationary stable solution, subject to an external nonhomogeneous perturbation. These are generalized Turing-like waves that materialize in a single-species populations dynamics model, as the unexpected byproduct of the imposed delay in the diffusion part. Sufficient conditions for the onset of the instability are mathematically provided by performing a linear stability analysis adapted to time-delayed differential equations. The method here developed exploits the properties of the Lambert W-function. The prediction of the theory are confirmed by direct numerical simulation carried out for a modified version of the classical Fisher model, defined on a Watts-Strogatz network and with the inclusion of the delay.

Using Hamiltonian control to desynchronize Kuramoto oscillators

Many coordination phenomena are based on a synchronisation process, whose global behaviour emerges from the interactions among the individual parts. Often in Nature, such self-organising mechanism allows the system to behave as a whole and thus grounding its very first existence, or expected functioning, on such process. There are however cases where synchronisation acts against the stability of the system; for instance in the case of engineered structures, resonances among sub parts can destabilise the whole system. In this Letter we propose an innovative control method to tackle the synchronisation process based on the use of the Hamiltonian control theory, by adding a small control term to the system we are able to impede the onset of the synchronisation. We present our results on the paradigmatic Kuramoto model but the applicability domain is far more large.Many coordination phenomena are based on a synchronization process, whose global behavior emerges from the interactions among the individual parts. Often in nature, such self-organized mechanism allows the system to behave as a whole and thus grounding its very first existence, or expected functioning, on such process. There are, however, cases where synchronization acts against the stability of the system; for instance in some neurodegenerative diseases or epilepsy or the famous case of Millennium Bridge where the crowd synchronization of the pedestrians seriously endangered the stability of the structure. In this paper we propose an innovative control method to tackle the synchronization process based on the application of the Hamiltonian control theory, by adding a small control term to the system we are able to impede the onset of the synchronization. We present our results on a generalized class of the paradigmatic Kuramoto model.

Pattern formation in a two-component reaction–diffusion system with delayed processes on a network

Reaction–diffusion systems with time-delay defined on complex networks have been studied in the framework of the emergence of Turing instabilities. The use of the Lambert W-function allowed us to get explicit analytic conditions for the onset of patterns as a function of the main involved parameters, the time-delay, the network topology and the diffusion coefficients. Depending on these parameters, the analysis predicts whether the system will evolve towards a stationary Turing pattern or rather to a wave pattern associated to a Hopf bifurcation. The possible outcomes of the linear analysis overcome the respective limitations of the single-species case with delay, and that of the classical activator–inhibitor variant without delay. Numerical results gained from the Mimura–Murray model support the theoretical approach.

Pattern formation for reactive species undergoing anisotropic diffusion

AbstractTuring instabilities for a two species reaction-diffusion system is studied under anisotropic diffusion. More specifically, the diffusion constants which characterize the ability of the species to relocate in space are direction sensitive. Under this working hypothesis, the conditions for the onset of the instability are mathematically derived and numerically validated. Patterns which closely resemble those obtained in the classical context of isotropic diffusion, develop when the usual Turing condition is violated, along one of the two accessible directions of migration. Remarkably, the instability can also set in when the activator diffuses faster than the inhibitor, along the direction for which the usual Turing conditions are not matched.

A minimally invasive neurostimulation method for controlling abnormal synchronisation in the neuronal activity

Many collective phenomena in Nature emerge from the -partial- synchronisation of the units comprising a system. In the case of the brain, this self-organised process allows groups of neurons to fire in highly intricate partially synchronised patterns and eventually lead to high level cognitive outputs and control over the human body. However, when the synchronisation patterns are altered and hypersynchronisation occurs, undesirable effects can occur. This is particularly striking and well documented in the case of epileptic seizures and tremors in neurodegenerative diseases such as Parkinson’s disease. In this paper, we propose an innovative, minimally invasive, control method that can effectively desynchronise misfiring brain regions and thus mitigate and even eliminate the symptoms of the diseases. The control strategy, grounded in the Hamiltonian control theory, is applied to ensembles of neurons modelled via the Kuramoto or the Stuart-Landau models and allows for heterogeneous coupling among the interacting unities. The theory has been complemented with dedicated numerical simulations performed using the small-world Newman-Watts network and the random Erdős-Rényi network. Finally the method has been compared with the gold-standard Proportional-Differential Feedback control technique. Our method is shown to achieve equivalent levels of desynchronisation using lesser control strength and/or fewer controllers, being thus minimally invasive.Many coordination phenomena in Nature are grounded on a synchronisation regime. In the case of brain dynamics, such self-organised process allows the neurons of particular brain regions to behave as a whole and thus directly controlling the neural activity, the muscles and finally the whole human body. However, not always such synchronised collective behaviour is the desired one, this is the case of neurodegenerative diseases such as Parkinson’s or epilepsy where abnormal synchronisation induces undesired effects such as tremors and epileptic seizures. In this paper we propose an innovative, minimally invasive, control method able to effectively desynchronise the interested brain zones and thus to reduce the onset of undesired behaviour.

Tune the topology to create or destroy patterns

Abstract We consider the dynamics of a reaction-diffusion system on a multigraph. The species share the same set of nodes but can access different links to explore the embedding spatial support. By acting on the topology of the networks we can control the ability of the system to self-organise in macroscopic patterns, emerging as a symmetry breaking instability of an homogeneous fixed point. Two different cases study are considered: on the one side, we produce a global modification of the networks, starting from the limiting setting where species are hosted on the same graph. On the other, we consider the effect of inserting just one additional single link to differentiate the two graphs. In both cases, patterns can be generated or destroyed, as follows the imposed, small, topological perturbation. Approximate analytical formulae allow to grasp the essence of the phenomenon and can potentially inspire innovative control strategies to shape the macroscopic dynamics on multigraph networks.