61 research outputs found
On the non-global linear stability and spurious fixed points of MPRK schemes with negative RK parameters
Recently, a stability theory has been developed to study the linear stability
of modified Patankar--Runge--Kutta (MPRK) schemes. This stability theory
provides sufficient conditions for a fixed point of an MPRK scheme to be stable
as well as for the convergence of an MPRK scheme towards the steady state of
the corresponding initial value problem, whereas the main assumption is that
the initial value is sufficiently close to the steady state. Initially,
numerical experiments in several publications indicated that these linear
stability properties are not only local, but even global, as is the case for
general linear methods. Recently, however, it was discovered that the linear
stability of the MPDeC(8) scheme is indeed only local in nature. Our conjecture
is that this is a result of negative Runge--Kutta (RK) parameters of MPDeC(8)
and that linear stability is indeed global, if the RK parameters are
nonnegative. To support this conjecture, we examine the family of
MPRK22() methods with negative RK parameters and show that even among
these methods there are methods for which the stability properties are only
local. However, this local linear stability is not observed for
MPRK22() schemes with nonnegative Runge-Kutta parameters.Comment: 19 pages, 3 figure
Lyapunov Stability of First and Second Order GeCo and gBBKS Schemes
In this paper we investigate the stability properties of fixed points of the
so-called gBBKS and GeCo methods, which belong to the class of non-standard
schemes and preserve the positivity as well as all linear invariants of the
underlying system of ordinary differential equations for any step size. The
schemes are applied to general linear test equations and proven to be generated
by -maps with locally Lipschitz continuous first derivatives. As
a result, a recently developed stability theorem can be applied to investigate
the Lyapunov stability of non-hyperbolic fixed points of the numerical method
by analyzing the spectrum of the corresponding Jacobian of the generating map.
In addition, if a fixed point is proven to be stable, the theorem guarantees
the local convergence of the iterates towards it.
In the case of first and second order gBBKS schemes the stability domain
coincides with that of the underlying Runge--Kutta method. Furthermore, while
the first order GeCo scheme converts steady states to stable fixed points for
all step sizes and all linear test problems of finite size, the second order
GeCo scheme has a bounded stability region for the considered test problems.
Finally, all theoretical predictions from the stability analysis are validated
numerically.Comment: 31 pages, 7 figure
Mechanical response of a thick poroelastic gel in contactless colloidal-probe rheology
When a rigid object approaches a soft material in a viscous fluid,
hydrodynamic stresses arise in the lubricated contact region and deform the
soft material. The elastic deformation modifies in turn the flow, hence
generating a soft-lubrication coupling. Moreover, soft elastomers and gels are
often porous. These materials may be filled with solvent or uncrosslinked
polymer chains, and might be permeable to the surrounding fluid, which
complexifies further the description. Here, we derive the point-force response
of a semi-infinite and permeable poroelastic substrate. Then, we use this
fundamental solution in order to address the specific poroelastic lubrication
coupling associated with contactless colloidal-probe methods. In particular, we
derive the conservative and dissipative components of the force associated with
the oscillating vertical motion of a sphere close to the poroelastic substrate.
Our results may be relevant for dynamic surface force apparatus and contactless
colloidal-probe atomic force microscopy experiments on soft, living and/or
fragile materials, such as swollen hydrogels and biological membranes
A dynamic model for action understanding and goal-directed imitation
The understanding of other individuals' actions is a fundamental cognitive skill for all
species living in social groups. Recent neurophysiological evidence suggests that an
observer may achieve the understanding by mapping visual information onto his own
motor repertoire to reproduce the action effect. However, due to differences in embodiment, environmental constraints or motor skills, this mapping very often cannot be direct. In this paper, we present a dynamic network model which represents in its layers the functionality of neurons in different interconnected brain areas known to be involved in action observation/execution tasks. The model aims at substantiating the idea that action understanding is a continuous process which combines sensory evidence, prior task knowledge and a goal-directed matching of action observation and action execution. The model is tested in variations of an imitation task in which an observer with dissimilar embodiment tries to reproduce the perceived or inferred end-state of a grasping-placing sequence. We also propose and test a biologically plausible learning scheme which allows establishing during practice a goal-directed organization of the distributed network. The modeling results are discussed with respect to recent experimental findings in action observation/execution studies.European Commission JAST project
IST-2-003747-I
Model-based functional neuroimaging using dynamic neural fields: An integrative cognitive neuroscience approach
A fundamental challenge in cognitive neuroscience is to develop theoretical frameworks that effectively span the gap between brain and behavior, between neuroscience and psychology. Here, we attempt to bridge this divide by formalizing an integrative cognitive neuroscience approach using dynamic field theory (DFT). We begin by providing an overview of how DFT seeks to understand the neural population dynamics that underlie cognitive processes through previous applications and comparisons to other modeling approaches. We then use previously published behavioral and neural data from a response selection Go/Nogo task as a case study for model simulations. Results from this study served as the âstandardâ for comparisons with a model-based fMRI approach using dynamic neural fields (DNF). The tutorial explains the rationale and hypotheses involved in the process of creating the DNF architecture and fitting model parameters. Two DNF models, with similar structure and parameter sets, are then compared. Both models effectively simulated reaction times from the task as we varied the number of stimulusâresponse mappings and the proportion of Go trials. Next, we directly simulated hemodynamic predictions from the neural activation patterns from each model. These predictions were tested using general linear models (GLMs). Results showed that the DNF model that was created by tuning parameters to capture simultaneously trends in neural activation and behavioral data quantitatively outperformed a Standard GLM analysis of the same dataset. Further, by using the GLM results to assign functional roles to particular clusters in the brain, we illustrate how DNF models shed new light on the neural populationsâ dynamics within particular brain regions. Thus, the present study illustrates how an interactive cognitive neuroscience model can be used in practice to bridge the gap between brain and behavior
Distractor effects on saccade trajectories: A comparison of prosaccades, antisaccades, and memory-guided saccades
A competitive integration model of exogenous and endogenous eye movements
We present a model of the eye movement system in which the programming of an eye movement is the result of the competitive integration of information in the superior colliculi (SC). This brain area receives input from occipital cortex, the frontal eye fields, and the dorsolateral prefrontal cortex, on the basis of which it computes the location of the next saccadic target. Two critical assumptions in the model are that cortical inputs are not only excitatory, but can also inhibit saccades to specific locations, and that the SC continue to influence the trajectory of a saccade while it is being executed. With these assumptions, we account for many neurophysiological and behavioral findings from eye movement research. Interactions within the saccade map are shown to account for effects of distractors on saccadic reaction time (SRT) and saccade trajectory, including the global effect and oculomotor capture. In addition, the model accounts for express saccades, the gap effect, saccadic reaction times for antisaccades, and recorded responses from neurons in the SC and frontal eye fields in these tasks. © The Author(s) 2010
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