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1.
We present necessary and sufficient conditions on the stability matrix of a general n(≥2)-dimensional reaction-diffusion system which guarantee that its uniform steady state can undergo a Turing bifurcation.
The necessary (kinetic) condition, requiring that the system be composed of an unstable (or activator) and a stable (or inhibitor)
subsystem, and the sufficient condition of sufficiently rapid inhibitor diffusion relative to the activator subsystem are
established in three theorems which form the core of our results. Given the possibility that the unstable (activator) subsystem
involves several species (dimensions), we present a classification of the analytically deduced Turing bifurcations into p (1 ≤p≤ (n− 1)) different classes. For n = 3 dimensions we illustrate numerically that two types of steady Turing pattern arise in one spatial dimension in a generic
reaction-diffusion system. The results confirm the validity of an earlier conjecture [12] and they also characterise the class
of so-called strongly stable matrices for which only necessary conditions have been known before [23, 24]. One of the main consequences of the present
work is that biological morphogens, which have so far been expected to be single chemical species [1–9], may instead be composed
of two or more interacting species forming an unstable subsystem.
Received: 21 September 1999 / Revised version: 21 June 2000 / Published online: 24 November 2000 相似文献
2.
Spatial pattern formation is one of the key issues in developmental biology. Some patterns arising in early development have
a very small spatial scale and a natural explanation is that they arise by direct cell—cell signalling in epithelia. This
necessitates the use of a spatially discrete model, in contrast to the continuum-based approach of the widely studied Turing
and mechanochemical models. In this work, we consider the pattern-forming potential of a model for juxtacrine communication,
in which signalling molecules anchored in the cell membrane bind to and activate receptors on the surface of immediately neighbouring
cells. The key assumption is that ligand and receptor production are both up-regulated by binding. By linear analysis, we
show that conditions for pattern formation are dependent on the feedback functions of the model. We investigate the form of
the pattern: specifically, we look at how the range of unstable wavenumbers varies with the parameter regime and find an estimate
for the wavenumber associated with the fastest growing mode. A previous juxtacrine model for Delta-Notch signalling studied
by Collier et al. (1996, J. Theor. Biol.
183, 429–446) only gives rise to patterning with a length scale of one or two cells, consistent with the fine-grained patterns
seen in a number of developmental processes. However, there is evidence of longer range patterns in early development of the
fruit fly Drosophila. The analysis we carry out predicts that patterns longer than one or two cell lengths are possible with our positive feedback
mechanism, and numerical simulations confirm this. Our work shows that juxtacrine signalling provides a novel and robust mechanism
for the generation of spatial patterns. 相似文献
3.
An efficient new method for the exact digital simulation of time-invariant linear systems is presented. Such systems are
frequently encountered as models for neuronal systems, or as submodules of such systems. The matrix exponential is used to
construct a matrix iteration, which propagates the dynamic state of the system step by step on a regular time grid. A large
and general class of dynamic inputs to the system, including trains of δ-pulses, can be incorporated into the exact simulation
scheme. An extension of the proposed scheme presents an attractive alternative for the approximate simulation of networks
of integrate-and-fire neurons with linear sub-threshold integration and non-linear spike generation. The performance of the
proposed method is analyzed in comparison with a number of multi-purpose solvers. In simulations of integrate-and-fire neurons,
Exact Integration systematically generates the smallest error with respect to both sub-threshold dynamics and spike timing.
For the simulation of systems where precise spike timing is important, this results in a practical advantage in particular
at moderate integration step sizes.
Received: 3 October 1998 / Accepted in revised form: 19 March 1999 相似文献
4.
5.
In this paper we examine spatio-temporal pattern formation in reaction-diffusion systems on the surface of the unit sphere
in 3D. We first generalise the usual linear stability analysis for a two-chemical system to this geometrical context. Noting
the limitations of this approach (in terms of rigorous prediction of spatially heterogeneous steady-states) leads us to develop,
as an alternative, a novel numerical method which can be applied to systems of any dimension with any reaction kinetics. This
numerical method is based on the method of lines with spherical harmonics and uses fast Fourier transforms to expedite the
computation of the reaction kinetics. Numerical experiments show that this method efficiently computes the evolution of spatial
patterns and yields numerical results which coincide with those predicted by linear stability analysis when the latter is
known. Using these tools, we then investigate the r?le that pre-pattern (Turing) theory may play in the growth and development
of solid tumours. The theoretical steady-state distributions of two chemicals (one a growth activating factor, the other a
growth inhibitory factor) are compared with the experimentally and clinically observed spatial heterogeneity of cancer cells
in small, solid spherical tumours such as multicell spheroids and carcinomas. Moreover, we suggest a number of chemicals which
are known to be produced by tumour cells (autocrine growth factors), and are also known to interact with one another, as possible
growth promoting and growth inhibiting factors respectively. In order to connect more concretely the numerical method to this
application, we compute spatially heterogeneous patterns on the surface of a growing spherical tumour, modelled as a moving-boundary
problem. The numerical results strongly support the theoretical expectations in this case. Finally in an appendix we give
a brief analysis of the numerical method.
Received: 27 July 2000 / Revised version: 15 August 2000 / Published online: 16 February 2001 相似文献
6.
Christoph Berding 《Bulletin of mathematical biology》1987,49(2):233-252
Several current reaction-diffusion mechanisms have been proposed as models for morphogenesis in the Turing (1952,Phil. Trans. R. Soc. Lond. B
237, 37–72) sense. We introduce and exploit a quantity, we have termed heterogeneity, which allows us to elaborate the differences
between the various models with regard to spatial pattern formation. It is shown that this quantity provides a concise view
for the comparison of theoretical models with experimental observations. Two model mechanisms are treated explicitly both
for linear and for biased diffusion. 相似文献
7.
Dynamic models of many processes in the biological and physical sciences give systems of ordinary differential equations called compartmental systems. Often, these systems include time lags; in this context, continuous probability density functions (pdfs) of lags are far more important than discrete lags. There is a relatively complete theory of compartmental systems without lags, both linear and non-linear [SIAM Rev. 35 (1993) 43]. The authors extend their previous work on compartmental systems without lags to show that, for discrete lags and for a very large class of pdfs of continuous lags, compartmental systems with lags are equivalent to larger compartmental systems without lags. Consequently, the properties of compartmental systems with lags are the same as those of compartmental systems without lags. For a very large class of compartmental systems with time lags, one can show that the time lags themselves can be generated by compartmental systems without lags. Thus, such systems can be partitioned into a main system, which is the original system without the lags, plus compartmental subsystems without lags that generate the lags. The latter may be linear or non-linear and may be inserted into main systems that are linear or non-linear. The state variables of the compartmental lag subsystems are hidden variables in the formulation with explicit lags. 相似文献
8.
The idea of evolutionary game theory is to relate the payoff of a game to reproductive success (= fitness). An underlying
assumption in most models is that fitness is a linear function of the payoff. For stochastic evolutionary dynamics in finite
populations, this leads to analytical results in the limit of weak selection, where the game has a small effect on overall
fitness. But this linear function makes the analysis of strong selection difficult. Here, we show that analytical results
can be obtained for any intensity of selection, if fitness is defined as an exponential function of payoff. This approach
also works for group selection (= multi-level selection). We discuss the difference between our approach and that of inclusive
fitness theory. 相似文献
9.
Debbie L. Benson Philip K. Maini Jonathan A. Sherratt 《Journal of mathematical biology》1998,37(5):381-417
. The Turing bifurcation is the basic bifurcation generating spatial pattern, and lies at the heart of almost all mathematical
models for patterning in biology and chemistry. In this paper the authors determine the structure of this bifurcation for
two coupled reaction diffusion equations on a two-dimensional square spatial domain when the diffusion coefficients have a
small explicit variation in space across the domain. In the case of homogeneous diffusivities, the Turing bifurcation is highly
degenerate. Using a two variable perturbation method, the authors show that the small explicit spatial inhomogeneity splits
the bifurcation into two separate primary and two separate secondary bifurcations, with all solution branches distinct. This
splitting of the bifurcation is more effective than that given by making the domain slightly rectangular, and shows clearly
the structure of the Turing bifurcation and the way in which the!
var
ious solution branches collapse together as the spatial variation is reduced. The authors determine the stability of the solution
branches, which indicates that several new phenomena are introduced by the spatial variation, including stable subcritical
striped patterns, and the possibility that stable stripes lose stability supercritically to give stable spotted patterns..
Received: 10 January 1996/Revised version: 3 July 1996 相似文献
10.
Under certain controllability and observability restrictions, two different parameterisations for a non-linear compartmental model can only have the same input-output behaviour if they differ by a locally diffeomorphic change of basis for the state space. With further restrictions, it is possible to gain valuable information with respect to identifiability via a linear analysis. Examples are presented where non-linear identifiability analyses are substantially simplified by means of an initial linear analysis. For complex models, with four or more compartments, this linear analysis can prove lengthy to perform by hand and so symbolic computation has been employed to aid this procedure. 相似文献
11.
12.
Voit EO 《Systems biology》2005,152(4):207-213
S-systems have been used as models of biochemical systems for over 30 years. One of their hallmarks is that, although they are highly non-linear, their steady states are characterised by linear equations. This allows streamlined analyses of stability, sensitivities and gains as well as objective, mathematically controlled comparisons of similar model designs. Regular S-systems have a unique steady state at which none of the system variables is zero. This makes it difficult to represent switching phenomena, as they occur, for instance, in the expression of genes, cell cycle phenomena and signal transduction. Previously, two strategies were proposed to account for switches. One was based on a technique called recasting, which permits the modelling of any differentiable non-linearities, including bistability, but typically does not allow steady-state analyses based on linear equations. The second strategy formulated the switching system in a piece-wise fashion, where each piece consisted of a regular S-system. A representation gleaned from a simplified form of recasting is proposed and it is possible to divide the characterisation of the steady states into two phases, the first of which is linear, whereas the other is non-linear, but easy to execute. The article discusses a representative pathway with two stable states and one unstable state. The pathway model exhibits strong separation between the stable states as well as hysteresis. 相似文献
13.
R. Candau A. Belli G. Y. Millet D. Georges B. Barbier J. D. Rouillon 《European journal of applied physiology and occupational physiology》1998,77(6):479-485
The aim of the present study was to examine the physiological and mechanical factors which may be concerned in the increase
in energy cost during running in a fatigued state. A group of 15 trained triathletes ran on a treadmill at velocities corresponding
to their personal records over 3000m [mean 4.53 (SD 0.28) m · s−1] until they felt exhausted. The energy cost of running (C
R) was quantified from the net O2 uptake and the elevation of blood lactate concentration. Gas exchange was measured over 1 min firstly during the 3rd–4th min
and secondly during the last minute of the run. Blood samples were collected before and after the completion of the run. Mechanical
changes of the centre of mass were quantified using a kinematic arm. A significant mean increase [6.9 (SD 3.5)%, P < 0.001] in C
R from a mean of 4.4 (SD 0.4) J · kg−1 · m−1 to a mean of 4.7 (SD 0.4) J · kg−1 · m−1 was observed. The increase in the O2 demand of the respiratory muscles estimated from the increase in ventilation accounted for a considerable proportion [mean
25.2 (SD 10.4)%] of the increase in CR. A mean increase [17.0 (SD 26.0)%, P < 0.05] in the mechanical cost (C
M) from a mean of 2.36 (SD 0.23) J · kg−1 · m−1 to a mean of 2.74 (SD 0.55) J · kg−1 · m−1 was also noted. A significant correlation was found between C
R and C
M in the non-fatigued state (r = 0.68, P < 0.01), but not in the fatigued state (r = 0.25, NS). Furthermore, no correlations were found between the changes (from non-fatigued to fatigued state) in C
R and the changes in C
M suggesting that the increase in C
R is not solely dependent on the external work done per unit of distance. Since step frequency decreased slightly in the fatigued
state, the internal work would have tended to decrease slightly which would not be compatible with an increase in C
R. A stepwise regressions showed that the changes in C
R were linked (r = 0.77, P < 0.01) to the changes in the variability of step frequency and in the variability of potential cost suggesting that a large
proportion of the increase in C
R was due to an increase in the step variability. The underlying mechanisms of the relationship between C
R and step variability remains unclear.
Accepted: 15 September 1997 相似文献
14.
Takashi Kitagawa Shingo Kimura Hideaki Nakata Harumi Yamada Akira Nitta Yoshikazu Sasai Hideharu Sasaki 《Environmental Biology of Fishes》2009,84(2):193-196
The habitat and movements of a Pacific bluefin tuna were investigated by reanalyzing archival tag data with sea surface temperature
data. During its trans-Pacific migration to the eastern Pacific, the fish took a direct path and primarily utilized waters,
in the Subarctic Frontal Zone (SFZ). Mean ambient temperature during the trans-Pacific migration was 14.5 ± 2.9 (°C ± SD),
which is significantly colder than the waters typically inhabited by bluefin tuna in their primary feeding grounds in the
western and eastern Pacific (17.6 ± 2.1). The fish moved rapidly through the colder water, and the heat produced during swimming
and the thermoconservation ability of bluefin tuna likely enabled it to migrate through the cold waters of the SFZ. 相似文献
15.
According to the kinetic theory for the build-up and elimination of haemoglobin (Hb) adducts, unstable Hb adducts are simultaneously eliminated by zero-order Hb turnover and first-order chemical instability. Thus, the elimination of unstable Hb adducts is non-linear with respect to time. Nonetheless, many studies of Hb adduct stability have characterized the elimination of Hb adducts using linear zero-order or linear first-order models. This paper demonstrates the use of non-linear regression to estimate the first-order rate constant of Hb adduct instability (k) using data on the elimination of Hb adducts in rats dosed with benzene or ortho -toluidine. Results obtained using non-linear regression models are compared with results from the more commonly employed zero- and first-order linear models. It is shown that exposure estimates based on measured levels of unstable Hb adducts can be severely biased if zero-order turnover is assumed. Furthermore, based on published data, estimates of k are subject to estimated relative biases in the range of -4% to 96% when first-order linear models are used to characterize Hb adduct instability. 相似文献
16.
A correlation-based learning (CBL) neural network model is proposed, which simulates the emergence of grating cells as well
as some of their response characteristics to periodic pattern stimuli. These cells, found in areas V1 and V2 of the visual
cortex of monkeys, respond vigorously and exclusively to bar gratings of a preferred orientation and periodicity. Their non-linear
behaviour differentiates grating cells from other orientation-selective cells, which show linear spatial frequency filtering.
Received: 9 June 1997 / Accepted in revised form: 9 February 1998 相似文献
17.
Intercellular signalling is key in determining cell fate. In closely packed tissues such as epithelia, juxtacrine signalling
is thought to be a mechanism for the generation of fine-grained spatial patterns in cell differentiation commonly observed
in early development. Theoretical studies of such signalling processes have shown that negative feedback between receptor
activation and ligand production is a robust mechanism for fine-grained pattern generation and that cell shape is an important
factor in the resulting pattern type. It has previously been assumed that such patterns can be analysed only with discrete
models since significant variation occurs over a lengthscale concomitant with an individual cell; however, considering a generic
juxtacrine signalling model in square cells, in O’Dea and King (Math Biosci 231(2):172–185 2011), a systematic method for the derivation of a continuum model capturing such phenomena due to variations in a model parameter
associated with signalling feedback strength was presented. Here, we extend this work to derive continuum models of the more
complex fine-grained patterning in hexagonal cells, constructing individual models for the generation of patterns from the
homogeneous state and for the transition between patterning modes. In addition, by considering patterning behaviour under
the influence of simultaneous variation of feedback parameters, we construct a more general continuum representation, capturing
the emergence of the patterning bifurcation structure. Comparison with the steady-state and dynamic behaviour of the underlying
discrete system is made; in particular, we consider pattern-generating travelling waves and the competition between various
stable patterning modes, through which we highlight an important deficiency in the ability of continuum representations to
accommodate certain dynamics associated with discrete systems. 相似文献
18.
Anotida Madzvamuse Eamonn A. Gaffney Philip K. Maini 《Journal of mathematical biology》2010,61(1):133-164
By using asymptotic theory, we generalise the Turing diffusively-driven instability conditions for reaction-diffusion systems
with slow, isotropic domain growth. There are two fundamental biological differences between the Turing conditions on fixed
and growing domains, namely: (i) we need not enforce cross nor pure kinetic conditions and (ii) the restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Our theoretical findings
are confirmed and reinforced by numerical simulations for the special cases of isotropic linear, exponential and logistic
growth profiles. In particular we illustrate an example of a reaction-diffusion system which cannot exhibit a diffusively-driven
instability on a fixed domain but is unstable in the presence of slow growth. 相似文献
19.
The well-known neural mass model described by Lopes da Silva et al. (1976) and Zetterberg et al. (1978) is fitted to actual
EEG data. This is achieved by reformulating the original set of integral equations as a continuous-discrete state space model.
The local linearization approach is then used to discretize the state equation and to construct a nonlinear Kalman filter.
On this basis, a maximum likelihood procedure is used for estimating the model parameters for several EEG recordings. The
analysis of the noise-free differential equations of the estimated models suggests that there are two different types of alpha
rhythms: those with a point attractor and others with a limit cycle attractor. These attractors are also found by means of
a nonlinear time series analysis of the EEG recordings. We conclude that the Hopf bifurcation described by Zetterberg et al.
(1978) is present in actual brain dynamics.
Received: 11 August 1997 / Accepted in revised form: 20 April 1999 相似文献
20.
Luiz Alberto Díaz Rodrigues Diomar Cristina Mistro Sergei Petrovskii 《Theoretical Ecology》2012,5(3):341-362
The spatiotemporal dynamics of a space- and time-discrete predator–prey system is considered theoretically using both analytical
methods and computer simulations. The prey is assumed to be affected by the strong Allee effect. We reveal a rich variety
of pattern formation scenarios. In particular, we show that, in a predator–prey system with the strong Allee effect for prey,
the role of space is crucial for species survival. Pattern formation is observed both inside and outside of the Turing domain.
For parameters when the local kinetics is oscillatory, the system typically evolves to spatiotemporal chaos. We also consider
the effect of different initial conditions and show that the system exhibits a spatiotemporal multistability. In a certain
parameter range, the system dynamics is not self-organized but remembers the details of the initial conditions, which evokes
the concept of long-living ecological transients. Finally, we show that our findings have important implications for the understanding
of population dynamics on a fragmented habitat. 相似文献