8 research outputs found
Mass concentration in a nonlocal model of clonal selection
Self-renewal is a constitutive property of stem cells. Testing the cancer
stem cell hypothesis requires investigation of the impact of self-renewal on
cancer expansion. To understand better this impact, we propose a mathematical
model describing dynamics of a continuum of cell clones structured by the
self-renewal potential. The model is an extension of the finite
multi-compartment models of interactions between normal and cancer cells in
acute leukemias. It takes a form of a system of integro-differential equations
with a nonlinear and nonlocal coupling, which describes regulatory feedback
loops in cell proliferation and differentiation process. We show that such
coupling leads to mass concentration in points corresponding to maximum of the
self-renewal potential and the model solutions tend asymptotically to a linear
combination of Dirac measures. Furthermore, using a Lyapunov function
constructed for a finite dimensional counterpart of the model, we prove that
the total mass of the solution converges to a globally stable equilibrium.
Additionally, we show stability of the model in space of positive Radon
measures equipped with flat metric. The analytical results are illustrated by
numerical simulations
Fast-slow bursters in the unfolding of a high codimension singularity and the ultra-slow transitions of classes
Bursting is a phenomenon found in a variety of physical and biological
systems. For example, in neuroscience, bursting is believed to play a key role
in the way information is transferred in the nervous system. In this work, we
propose a model that, appropriately tuned, can display several types of
bursting behaviors. The model contains two subsystems acting at different
timescales. For the fast subsystem we use the planar unfolding of a high
codimension singularity. In its bifurcation diagram, we locate paths that
underly the right sequence of bifurcations necessary for bursting. The slow
subsystem steers the fast one back and forth along these paths leading to
bursting behavior. The model is able to produce almost all the classes of
bursting predicted for systems with a planar fast subsystems. Transitions
between classes can be obtained through an ultra-slow modulation of the model's
parameters. A detailed exploration of the parameter space allows predicting
possible transitions. This provides a single framework to understand the
coexistence of diverse bursting patterns in physical and biological systems or
in models.Comment: 22 pages, 15 figure
Normal form analysis of a mean-field inhibitory neuron model
In neuroscience one of the open problems is the creation of the alpha rhythm detected
by the electroencephalogram (EEG). One hypothesis is that the alpha rhythm
is created by the inhibitory neurons only. The mesoscopic approach to understand
the brain is the most appropriate to mathematically modelize the EEG records of the
human scalp. In this thesis we use a local, mean-field potential model restricted to
the inhibitory neuron population only to reproduce the alpha rhythm. We perform
extensive bifurcation analysis of the system using AUTO.We use Kuznetsov’s method
that combines the center manifold reduction and normal form theory to analytically
compute the normal form coefficients of the model. The bifurcation diagram is largely
organised around a codimension 3 degenerate Bogdanov-Takens point. Alpha rhythm
oscillations are detected as periodic solutions
Multiparametric bifurcation analysis of a basic two stage population model.
In this paper we investigate long-term dynamics of the most basic model for stage-structured populations, in which the per capita transition from the juvenile into the adult class is density dependent. The model is represented by an autonomous system of two nonlinear differential equations with four parameters for a single population. We find that the interaction of intra-adult competition and intra-juvenile competition gives rise to multiple attractors, one of which can be oscillatory. A detailed numerical study reveals a rich bifurcation structure for this two-dimensional system, originating from a degenerate Bogdanov-Takens (BT) bifurcation point when one parameter is kept constant. Depending on the value of this fixed parameter, the corresponding triple critical equilibrium has either an elliptic sector or it is a topological focus, which is demonstrated by the numerical normal form analysis. It is shown that the canonical unfolding of the codimension-three BT point reveals the underlying dynamics of the model. Certain new features of this unfolding in the elliptic case, which are important in applications but have been overlooked in available theoretical studies, are established. Various three-, two-, and one-parameter bifurcation diagrams of the model are presented and interpreted in biological terms. © 2006 Society for Industrial and Applied Mathematics
Analysis of toxic effects and nutrient stress in aquatic ecosystems
Kooijman, S.A.L.M. [Promotor]Kooi, B.W. [Copromotor