146 research outputs found
Dynamics of neural systems with discrete and distributed time delays
In real-world systems, interactions between elements do not happen instantaneously, due to the time
required for a signal to propagate, reaction times of individual elements, and so forth. Moreover,
time delays are normally nonconstant and may vary with time. This means that it is vital to introduce
time delays in any realistic model of neural networks. In order to analyze the fundamental
properties of neural networks with time-delayed connections, we consider a system of two coupled
two-dimensional nonlinear delay differential equations. This model represents a neural network,
where one subsystem receives a delayed input from another subsystem. An exciting feature of the
model under consideration is the combination of both discrete and distributed delays, where distributed
time delays represent the neural feedback between the two subsystems, and the discrete
delays describe the neural interaction within each of the two subsystems. Stability properties are
investigated for different commonly used distribution kernels, and the results are compared to the
corresponding results on stability for networks with no distributed delays. It is shown how approximations
of the boundary of the stability region of a trivial equilibrium can be obtained analytically
for the cases of delta, uniform, and weak gamma delay distributions. Numerical techniques are used
to investigate stability properties of the fully nonlinear system, and they fully confirm all analytical
findings
Stability and bifurcations in an epidemic model with varying immunity period
An epidemic model with distributed time delay is derived to describe the
dynamics of infectious diseases with varying immunity. It is shown that
solutions are always positive, and the model has at most two steady states:
disease-free and endemic. It is proved that the disease-free equilibrium is
locally and globally asymptotically stable. When an endemic equilibrium exists,
it is possible to analytically prove its local and global stability using
Lyapunov functionals. Bifurcation analysis is performed using DDE-BIFTOOL and
traceDDE to investigate different dynamical regimes in the model using
numerical continuation for different values of system parameters and different
integral kernels.Comment: 16 pages, 5 figure
Control of unstable steady states in neutral time-delayed systems
We present an analysis of time-delayed feedback control used to stabilize an
unstable steady state of a neutral delay differential equation. Stability of
the controlled system is addressed by studying the eigenvalue spectrum of a
corresponding characteristic equation with two time delays. An analytic
expression for the stabilizing control strength is derived in terms of original
system parameters and the time delay of the control. Theoretical and numerical
results show that the interplay between the control strength and two time
delays provides a number of regions in the parameter space where the
time-delayed feedback control can successfully stabilize an otherwise unstable
steady state.Comment: 11 pages, 8 figure
Asymptotic properties of the spectrum of neutral delay differential equations
Spectral properties and transition to instability in neutral delay
differential equations are investigated in the limit of large delay. An
approximation of the upper boundary of stability is found and compared to an
analytically derived exact stability boundary. The approximate and exact
stability borders agree quite well for the large time delay, and the inclusion
of a time-delayed velocity feedback improves this agreement for small delays.
Theoretical results are complemented by a numerically computed spectrum of the
corresponding characteristic equations.Comment: 14 pages, 6 figure
Time-delayed models of gene regulatory networks
We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternativemodelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems
Enhancing noise-induced switching times in systems with distributed delays
The paper addresses the problem of calculating the noise-induced switching rates in systems with
delay-distributed kernels and Gaussian noise. A general variational formulation for the switching
rate is derived for any distribution kernel, and the obtained equations of motion and boundary conditions
represent the most probable, or optimal, path, which maximizes the probability of escape.
Explicit analytical results for the switching rates for small mean time delays are obtained for the
uniform and bi-modal (or two-peak) distributions. They suggest that increasing the width of the distribution
leads to an increase in the switching times even for longer values of mean time delays for
both examples of the distribution kernel, and the increase is higher in the case of the two-peak distribution.
Analytical predictions are compared to the direct numerical simulations and show excellent
agreement between theory and numerical experiment
Effects of Viral and cytokine delays on dynamics of autoimmunity
A major contribution to the onset and development of autoimmune disease is known to come from infections. An important practical problem is identifying the precise mechanism by which the breakdown of immune tolerance as a result of immune response to infection leads to autoimmunity. In this paper, we develop a mathematical model of immune response to a viral infection, which includes T cells with different activation thresholds, regulatory T cells (Tregs), and~a cytokine mediating immune dynamics. Particular emphasis is made on the role of time delays associated with the processes of infection and mounting the immune response. Stability analysis of various steady states of the model allows us to identify parameter regions associated with different types of immune behaviour, such as, normal clearance of infection, chronic infection, and autoimmune dynamics. Numerical simulations are used to illustrate different dynamical regimes, and to identify basins of attraction of different dynamical states. An important result of the analysis is that not only the parameters of the system, but also the initial level of infection and the initial state of the immune system determine the progress and outcome of the dynamics
Dynamics of vaccination in a time-delayed epidemic model with awareness
This paper investigates the effects of vaccination on the dynamics of infectious disease, which is spreading in a population concurrently with awareness. The model considers contributions to the overall awareness from a global information campaign, direct contacts between unaware and aware individuals, and reported cases of infection. It is assumed that there is some time delay between individuals becoming aware and modifying their behaviour. Vaccination is administered to newborns, as well as to aware individuals, and it is further assumed that vaccine-induced immunity may wane with time. Feasibility and stability of the disease-free and endemic equilibria are studied analytically, and conditions for the Hopf bifurcation of the endemic steady state are found in terms of system parameters and the time delay. Analytical results are supported by numerical continuation of the Hopf bifurcation and numerical simulations of the model to illustrate different types of dynamical behaviour
Time-delayed model of autoimmune dynamics
Among various environmental factors associated with triggering or exacerbating autoimmune response, an important role is played by infections. A breakdown of immune tolerance as a byproduct of immune response against these infections is one of the major causes of autoimmune disease. In this paper we analyse the dynamics of immune response with particular emphasis on the role of time delays characterising the infection and the immune response, as well as on interactions between different types of T cells and cytokines that mediate their behaviour. Stability analysis of the model provides insights into how different model parameters affect the dynamics. Numerical stability analysis and simulations are performed to identify basins of attraction of different dynamical states, and to illustrate the behaviour of the model in different regime
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