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BIOE 402. Medical Technology Assessment. 2 or 3 hours. Bioentrepreneur course. Assessment of medical technology in the context of commercialization. Objectives, competition, market share, funding, pricing, manufacturing, growth, and intellectual property; many issues unique to biomedical products. Course Information: 2 undergraduate hours. 3 graduate hours. Prerequisite(s): Junior standing or above and consent of the instructor.

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Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

Evolutionary games on graphs

Complex systems are very often organized under the form of networks where nodes and edges are embedded in space Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks, neural networks, are all examples where space is relevant and where topology alone does not contain all the information Characterizing and understanding the structure and the evolution of spatial networks is thus crucial for many different fields ranging from urbanism to epidemiology An important consequence of space on networks is that there is a cost associated to the length of edges which in turn has dramatic effects on the topological structure of these networks We will expose thoroughly the current state of our understanding of how the spatial constraints affect the structure and properties of these networks We will review the most recent empirical observations and the most important models of spatial networks We will also discuss various processes which take place on these spatial networks, such as phase transitions, random walks, synchronization, navigation, resilience, and disease spread

Spatial Networks

Chimera states are complex spatio-temporal patterns that consist of coexisting domains of spatially coherent and incoherent dynamics. This counterintuitive phenomenon was first observed in systems of identical oscillators with symmetric coupling topology. Can one overcome these limitations? To address this question, we discuss the robustness of chimera states in networks of FitzHugh-Nagumo oscillators. Considering networks of inhomogeneous elements with regular coupling topology, and networks of identical elements with irregular coupling topologies, we demonstrate that chimera states are robust with respect to these perturbations and analyze their properties as the inhomogeneities increase. We find that modifications of coupling topologies cause qualitative changes of chimera states: additional random links induce a shift of the stability regions in the system parameter plane, gaps in the connectivity matrix result in a change of the multiplicity of incoherent regions of the chimera state, and hierarchical geometry in the connectivity matrix induces nested coherent and incoherent regions.

Robustness of chimera states for coupled FitzHugh-Nagumo oscillators.

We discuss the superdiffusive motion of a random walk in a medium containing random velocity fields. For a two-dimensional layered medium with y-dependent random velocities in the x direction u(y), 〈${\mathit{x}}^{2}$(t)〉\ensuremath{\sim}${\mathit{t}}^{2\ensuremath{\nu}}$, with 2\ensuremath{\nu}=3/2, and with strong sample-to-sample fluctuations. The probability distribution of displacements, averaged over environments, takes a non-Gaussian scaling form at large time, 〈P(x,t)〉\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}3/4}$f(x/${\mathit{t}}^{3/4}$), where v(u)\ensuremath{\sim}exp(-${\mathit{u}}^{\mathrm{\ensuremath{\delta}}}$) for u\ensuremath{\gg}1, with \ensuremath{\delta}=4/3. For an isotropic two-dimensional medium with ${\mathit{u}}_{\mathit{x}}$(y)=f(y) and ${\mathit{u}}_{\mathit{y}}$(x)=f(x), we find \ensuremath{\nu}=2/3 and \ensuremath{\delta}=1-\ensuremath{\nu}${)}^{\mathrm{\ensuremath{-}}1}$=3.

Superdiffusion in random velocity fields.

The effects of low-dimensional supports (one and two dimensions) on the steady state and the dynamics of open reactive systems capable of giving rise to oscillatory behavior are studied. A lattice Lotka–Volterra model involving reaction, adsorption, and desorption mechanisms is developed for which mean-field behavior predicts a continuum of closed trajectories around a center. It is shown that the spatial constraints of the support radically change this behavior. Specifically, while in one dimension, oscillations are suppressed, in two dimensions, the system selects a preferred oscillation frequency depending on the intrinsic parameters and the lattice geometry.

Oscillatory dynamics in low-dimensional supports: A lattice Lotka–Volterra model

We study numerically the development of chimera states in networks of nonlocally coupled oscillators whose limit cycles emerge from a Hopf bifurcation. This dynamical system is inspired from population dynamics and consists of three interacting species in cyclic reactions. The complexity of the dynamics arises from the presence of a limit cycle and four fixed points. When the bifurcation parameter increases away from the Hopf bifurcation the trajectory approaches the heteroclinic invariant manifolds of the fixed points producing spikes, followed by long resting periods. We observe chimera states in this spiking regime as a coexistence of coherence (synchronization) and incoherence (desynchronization) in a one-dimensional ring with nonlocal coupling and demonstrate that their multiplicity depends on both the system and the coupling parameters. We also show that hierarchical (fractal) coupling topologies induce traveling multichimera states. The speed of motion of the coherent and incoherent parts along the ring is computed through the Fourier spectra of the corresponding dynamics.

/pdf/chimera-states-in-population-dynamics-networks-with-4mvld4p4ui.pdf

Chimera states in population dynamics: networks with fragmented and hierarchical connectivities

We discuss synchronization patterns in networks of FitzHugh-Nagumo and leaky integrate-and-fire oscillators coupled in a two-dimensional toroidal geometry. A common feature between the two models is the presence of fast and slow dynamics, a typical characteristic of neurons. Earlier studies have demonstrated that both models when coupled nonlocally in one-dimensional ring networks produce chimera states for a large range of parameter values. In this study, we give evidence of a plethora of two-dimensional chimera patterns of various shapes, including spots, rings, stripes, and grids, observed in both models, as well as additional patterns found mainly in the FitzHugh-Nagumo system. Both systems exhibit multistability: For the same parameter values, different initial conditions give rise to different dynamical states. Transitions occur between various patterns when the parameters (coupling range, coupling strength, refractory period, and coupling phase) are varied. Many patterns observed in the two models follow similar rules. For example, the diameter of the rings grows linearly with the coupling radius.

Astero Provata

Papers

Robustness of chimera states for coupled FitzHugh-Nagumo oscillators.

Superdiffusion in random velocity fields.

Oscillatory dynamics in low-dimensional supports: A lattice Lotka–Volterra model

Chimera states in population dynamics: networks with fragmented and hierarchical connectivities

Chimera patterns in two-dimensional networks of coupled neurons