Over the past 20 years, neuroimaging has become a predominant technique in systems neuroscience. One might envisage that over the next 20 years the neuroimaging of distributed processing and connectivity will play a major role in disclosing the brain's functional architecture and operational principles. The inception of this journal has been foreshadowed by an ever-increasing number of publications on functional connectivity, causal modeling, connectomics, and multivariate analyses of distributed patterns of brain responses. I accepted the invitation to write this review with great pleasure and hope to celebrate and critique the achievements to date, while addressing the challenges ahead.

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Functional and effective connectivity: a review.

We give an overview of all codim 1 bifurcations in generic planar discontinuous piecewise smooth autonomous systems, here called Filippov systems. Bifurcations are defined using the classical approach of topological equivalence. This allows the development of a simple geometric criterion for classifying sliding bifurcations, i.e. bifurcations in which some sliding on the discontinuity boundary is critically involved. The full catalog of local and global bifurcations is given, together with explicit topological normal forms for the local ones. Moreover, for each bifurcation, a defining system is proposed that can be used to numerically compute the corresponding bifurcation curve with standard continuation techniques. A problem of exploitation of a predator–prey community is analyzed with the proposed methods.

https://www.staff.science.uu.nl/~kouzn101/MiniFilippov/Filippov2D.pdf

One-Parameter Bifurcations in Planar Filippov Systems

BSPlib is a small communications library for bulk synchronous parallel (BSP) programming which consists of only 20 basic operations. This paper presents the full definition of BSPlib in C, motivates the design of its basic operations, and gives examples of their use. The library enables programming in two distinct styles: direct remote memory access (DRMA) using put or get operations, and bulk synchronous message passing (BSMP). Currently, implementations of BSPlib exist for a variety of modern architectures, including massively parallel computers with distributed memory, shared memory multiprocessors, and networks of workstations. BSPlib has been used in several scientific and industrial applications; this paper briefly describes applications in benchmarking, Fast Fourier Transforms (FFTs), sorting, and molecular dynamics.

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BSPlib: The BSP programming library

We present a bibliography on moving and free boundary problems for the heatdiffusion equation, particularly regarding the Stefan and related problems. It contains 5869 titles referring to 588 scientific Journals, 122 books, 88 symposia (having at least 3 contributions on the subject), 30 collections, 59 thesis and 247 technical reports. It tries to give a comprehensive account of the western existing mathematicalphysical-engineering literature on this research field. RESUMEN Se presenta una bibliografía sobre problemas de frontera móvil y libre para la ecuación del calor-difusión, en particular sobre el problema de Stefan y problemas relacionados. Contiene 5869 títulos distribuidos en 588 revistas científicas, 122 libros, 88 simposios (teniendo al menos 3 contribuciones en el tema), 30 colecciones, 59 tesis y 247 informes técnicos o prepublicaciones. Se da un informe amplio de la bibliografía matemática, física y de las ingenierías existente en occidente sobre este tema de investigación. Primary Mathematics Subject Classification Number (*): 35R35, 80A22 Secondary Mathematics Subject Classification Number (*): 35B40, 35C05, 35C15, 35Kxx, 35R30, 46N20, 49J20, 65Mxx, 65Nxx, 76R50, 76S05, 76T05, 93C20. (*) Following the 1991 Mathematics Subject Classification compiled by Mathematical Reviews and Zentralblatt fur Mathematik. Primary key words: Enthalpy formulation or method, Filtration, Free boundary problems, Freezing, Melting, Moving boundary problems, Mushy region, Phase-change problem, Solidification, Stefan problem. Secondary key words: Continuous mechanics, Diffusion process, Functional analysis, Heat conduction, Mathematical methods, Numerical methods, Partial differential equations, Variational inequalities, Weak solutions. Palabras claves primarias: Método o formulación en entalpía, Filtración, Problemas de frontera libre, Congelación, Derretimiento, Problemas de frontera móvil, Región pastosa, Problema de cambio de fase, Solidificación, Problema de Stefan. Palabras claves secundarias: Mecánica del continuo, Procesos difusivos, Análisis funcional, Conducción del calor, Métodos matemáticos, Métodos numéricos, Ecuaciones diferenciales a derivadas parciales, Inecuaciones variacionales, Soluciones débiles. El manuscrito fue recibido y aceptado en octubre de 1999. D.A. Tarzia, A bibliography on FBP. The Stefan problem, MAT Serie A, # 2 (2000). 3

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A bibliography on moving-free boundary problems for the heat-diffusion equation. The stefan and related problems

/pdf/normal-form-maps-for-grazing-bifurcations-in-n-dimensional-4eh5zpqcjs.pdf

Normal form maps for grazing bifurcations in n -dimensional piecewise-smooth dynamical systems

The effect of harmonic excitation on suspension bridges is examined as a first step towards the understanding of the effect of wind, and possibly certain kinds of earthquake, excitation on such structures. The Lazer-McKenna suspension bridge model is studied completely for the first time by using a methodology that has been successfully applied to models of rocking blocks and other free-standing rigid structures. An unexpectedly rich dynamical structure is revealed in this way. Conditions for the existence of asymptotic periodic responses are established, via a complicated nonlinear transcen- dental equation. A two-part Poincare map is derived to study the orbital stability of such solutions. Numerical results are presented which illustrate the application of the analytical procedure to find and classify stable and unstable solutions, as well as determine bifurcation points accurately. The richness of the possible dynamics is then illustrated by a menagerie of solutions which exhibit fold and flip bifurca...

A piecewise linear suspension bridge model : nonlinear dynamics and orbital continuation

We examine the dynamics of two simple coupled non-linear ordinary differential equations (ODEs) first introduced by Lazer and McKenna (SIAM Review 32 (4), 537-578, 1990). Using the numerical continuation package AUTO, we obtain multiple coexistence of periodic motions, period-doubling sequences and the onset of 'beats' solutions via torus bifurcations. We discuss the implications of these results for the modelling of suspension bridge dynamics.

Non-linear dynamics of the extended Lazer-McKenna bridge oscillation model

The analysis of networks of time-summating binary neural networks is relevant to the study of coherent oscillatory behaviour in neuronal populations. A class of networks based on a discrete time version of leaky integrator networks has recently been extended to include the effects of hyperpolarisationactivated inward currents [S. Coombes and S. H. Doole, Dynamics and Stability of Systems 11, 36, (1996)]. Such rebound currents are important for central pattern generation in neuronal circuits with reciprocal inhibition. In this paper, we incorporate models of intrinsic synaptic and threshold noise into the above neural system. The macroscopic behaviour of time summating networks with rebound currents and random thresholds is analysed in the thermodynamic limit. Mean field equations are derived for the average network activity in a homogeneous network with inhibitory synaptic connections. Periodic and chaotic solutions are shown to exist, together with hysteretic transitions between periodic orbits. This hysteresis is observed between particular periodic orbit branches, as well as more globally with respect to variations in external input or threshold noise. Moreover, rebound currents are shown to suppress chaotic network response to external input, in favour of low order periodic responses, which in turn define well ordered coherent macroscopic oscillatory 1 states for the system. The response characteristic of a single neuron in the presence of synaptic multiplicative noise is also considered and compared to its zero noise limit. In this latter case, the dynamics is reduced to a piecewise linear discontinuous circle map, whilst the former is expressed in terms of a random iterated function system. 87.10+e, 02.50.Ey Typeset using REVTEX 2

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Neuronal populations with reciprocal inhibition and rebound currents: effects of synaptic and threshold noise

Post inhibitory rebound (PIR) is a nonlinear phenomenon present in a variety of nerve cells. It is an important mechanism underlying central pattern generation for heartbeat, swimming and other motor patterns in many neuronal systems. In this paper, we propose an extension of the binary threshold neuron model to incorporate the effects of PIR. For a single neuron, the dynamics can be described by a piecewise linear circle map with two discontinuities. Both frequency-locking and chaos can occur. The Liapunov exponent of the map is evaluated and used to define transitions between these two distinct types of asymptotic behaviour. Hysteresis between periodic orbits is also observed. A small network of these model neurons, with reciprocal inhibition, is shown to exhibit 'self-sustained' anti-phase oscillations, making PIR a plausible mechanism for central pattern generation in neuronal systems. Unlike coupled oscillator theories, network oscillations emerge naturally as a consequence of the biological descript...

/pdf/neuronal-population-dynamics-with-post-inhibitory-rebound-a-5bmxoi9eq5.pdf

SH Doole

Papers

A piecewise linear suspension bridge model : nonlinear dynamics and orbital continuation

Non-linear dynamics of the extended Lazer-McKenna bridge oscillation model

Neuronal populations with reciprocal inhibition and rebound currents: effects of synaptic and threshold noise

Neuronal population dynamics with post inhibitory rebound: A reduction to piecewise linear discontinuous circle maps

The nonlinear dynamics of suspension bridges under harmonic forcing