2,384 research outputs found
Waves and propagation failure in discrete space models with nonlinear coupling and feedback
Many developmental processes involve a wave of initiation of pattern formation,
behind which a uniform layer of discrete cells develops a regular pattern that determines
cell fates. This paper focuses on the initiation of such waves, and then on
the emergence of patterns behind the wavefront. I study waves in discrete space
differential equation models where the coupling between sites is nonlinear. Such
systems represent juxtacrine cell signalling, where cells communicate via membrane
bound molecules binding to their receptors. In this way, the signal at cell j is a
nonlinear function of the average signal on neighbouring cells. Whilst considerable
progress has been made in the analysis of discrete reaction-diffusion systems, this
paper presents a novel and detailed study of waves in juxtacrine systems.
I analyse travelling wave solutions in such systems with a single variable representing
activity in each cell. When there is a single stable homogeneous steady
state, the wave speed is governed by the linearisation ahead of the wave front. Wave
propagation (and failure) is studied when the homogeneous dynamics are bistable.
Simulations show that waves may propagate in either direction, or may be pinned.
A Lyapunov function is used to determine the direction of propagation of travelling
waves. Pinning is studied by calculating the boundaries for propagation failure for
sigmoidal and piecewise linear feedback functions, using analysis of 2 active sites and
exact stationary solutions respectively. I then explore the calculation of travelling
waves as the solution of an associated n-dimensional boundary value problem posed
on [0, 1], using continuation to determine the dependence of speed on model parameters.
This method is shown to be very accurate, by comparison with numerical
simulations. Furthermore, the method is also applicable to other discrete systems
on a regular lattice, such as the discrete bistable reaction-diffusion equation.
Finally, I extend the study to more detailed models including the reaction kinetics
of signalling, and demonstrate the same features of wave propagation. I discuss how
such waves may initiate pattern formation, and the role of such mechanisms in
morphogenesis
Bumps, breathers, and waves in a neural network with spike frequency adaptation
In this Letter we introduce a continuum model of neural tissue that include the effects of so-called spike frequency adaptation (SFA). The basic model is an integral equation for synaptic activity that depends upon the non-local network connectivity, synaptic response, and firing rate of a single neuron. A phenomenological model of SFA is examined whereby the firing rate is taken to be a simple state-dependent threshold function. As in the case without SFA classical Mexican-Hat connectivity is shown to allow for the existence of spatially localized states (bumps). Importantly an analysis of bump stability using recent Evans function techniques shows that bumps may undergo instabilities leading to the emergence of both breathers and traveling waves. Moreover, a similar analysis for traveling pulses leads to the conditions necessary to observe a stable traveling breather. Direct numerical simulations both confirm our theoretical predictions and illustrate the rich dynamic behavior of this model, including the appearance of self-replicating bumps
The spatiotemporal order of signaling events unveils the logic of development signaling
Motivation: Animals from worms and insects to birds and mammals show distinct body plans; however, the embryonic development of diverse body plans with tissues and organs within is controlled by a surprisingly few signaling pathways. It is well recognized that combinatorial use of and dynamic interactions among signaling pathways follow specific logic to control complex and accurate developmental signaling and patterning, but it remains elusive what such logic is, or even, what it looks like.
Results: We have developed a computational model for Drosophila eye development with innovated methods to reveal how interactions among multiple pathways control the dynamically generated hexagonal array of R8 cells. We obtained two novel findings. First, the coupling between the long-range inductive signals produced by the proneural Hh signaling and the short-range restrictive signals produced by the antineural Notch and EGFR signaling is essential for generating accurately spaced R8s. Second, the spatiotemporal orders of key signaling events reveal a robust pattern of lateral inhibition conducted by Ato-coordinated Notch and EGFR signaling to collectively determine R8 patterning. This pattern, stipulating the orders of signaling and comparable to the protocols of communication, may help decipher the well-appreciated but poorly-defined logic of developmental signaling
How predation can slow, stop or reverse a prey invasion
How predation can slow, stop or reverse a prey invasio
Oscillations and patterns in spatially discrete models for developmental intercellular signalling
We extend previous models for nearest neighbour ligand-receptor binding to include both lateral induction and inhibition of ligand and receptor production, and different geometries (strings of cells and hexagonal arrays, in addition to
square arrays). We demonstrate the possibility of lateral inhibition giving patterns
with a characteristic length scale of many cell diameters, when receptor production is included. In contrast, lateral induction combined with inhibition of receptor synthesis cannot give rise to a patterning instability under any circumstances.
Interesting new dynamics include the analytical prediction and consequent numerical observation of spatiotemporal oscillationsāthis depends crucially on the
production terms and on the relationship between the decay rates of ligand and
free receptor.
Our approach allows for a detailed comparison with the model for Delta-Notch
interactions of Collier et al. [4], and we find that a formal reduction may be made
only when the ligand receptor binding kinetics are very slow. Without such very
slow receptor kinetics, spatial pattern formation via lateral inhibition in hexagonal
cellular arrays requires significant activation of receptor production, a feature that
is not apparent from previous analyses
Influence of slow oscillation on hippocampal activity and ripples through cortico-hippocampal synaptic interactions, analyzed by a cortical-CA3-CA1 network model
Hippocampal sharp wave-ripple complexes (SWRs) involve the synchronous discharge of thousands of cells throughout the CA3-CA1-subiculum-entorhinal cortex axis. Their strong
transient output affects cortical targets, rendering SWRs a possible means for memory transfer from the hippocampus to the neocortex for long-term storage. Neurophysiological
observations of hippocampal activity modulation by the cortical slow oscillation (SO) during deep sleep and anesthesia, and correlations between ripples and UP states, support the role of SWRs in memory consolidation through a cortico-hippocampal feedback loop. We couple a cortical network exhibiting SO with a hippocampal CA3-CA1 computational network model exhibiting SWRs, in order to model such cortico-hippocampal correlations and uncover important parameters and coupling mechanisms controlling them. The cortical oscillatory output entrains the CA3 network via connections representing the mossy ļ¬ber input, and the CA1 network via the temporoammonic pathway (TA). The
spiking activity in CA3 and CA1 is shown to depend on the excitation-to-inhibition ratio, induced by combining the two hippocampal inputs, with mossy ļ¬ber input controlling the UP-state correlation of CA3 population bursts and corresponding SWRs, whereas the temporoammonic input affects the overall CA1 spiking activity. Ripple characteristics and pyramidal spiking participation to SWRs are shaped by the strength of the Schaffer collateral drive. A set of in vivo recordings from the rat hippocampus conļ¬rms a model-predicted segregation of pyramidal cells into subgroups according to the SO state where they preferentially ļ¬re and their response to SWRs. These groups can potentially play distinct functional roles in the replay of spike sequences
Modelling and analysis of planar cell polarity
Planar cell polarity (PCP) occurs in the epithelia of many animals and can lead to the alignment of hairs, bristles and feathers; physiologically, it can organise ciliary beating. Here we present two approaches to modelling this phenomenon. The aim is to discover the basic mechanisms that drive PCP, while keeping the models mathematically tractable. We present a feedback and diffusion model, in which adjacent cell sides of neighbouring cells are coupled by a negative feedback loop and diffusion acts within the cell. This approach can give rise to polarity, but also to period two patterns. Polarisation arises via an instability provided a sufficiently strong feedback and sufficiently weak diffusion. Moreover, we discuss a conservative model in which proteins within a cell are redistributed depending on the amount of proteins in the neighbouring cells, coupled with intracellular diffusion. In this case polarity can arise from weakly polarised initial conditions or via a wave provided the diffusion is weak enough. Both models can overcome small anomalies in the initial conditions. Furthermore, the range of the effects of groups of cells with different properties than the surrounding cells depends on the strength of the initial global cue and the intracellular diffusion
Is a persistent global bias necessary for the establishment of planar cell polarity?
Planar cell polarity (PCP) ā the coordinated polarisation of a whole field of cells within the plane of a tissue ā relies on the interaction of three modules: a global module that couples individual cellular polarity to the tissue axis, a local module that aligns the axis of polarisation of neighbouring cells, and a readout module that directs the correct outgrowth of PCP-regulated structures such as hairs and bristles. While much is known about the molecular components that are required for PCP, the functional details ofāand interactions betweenāthe modules remain unclear. In this work, we perform a mathematical analysis of two previously proposed computational models of the local module (Amonlirdviman et al., Science, 307, 2005; Le Garrec et al., Dev. Dyn., 235, 2006). Both models can reproduce wild-type and mutant phenotypes of PCP observed in the Drosophila wing under the assumption that a tissue-wide polarity cue from the global module persists throughout the development of PCP. We demonstrate that both models can also generate tissue-level PCP when provided with only a transient initial polarity cue. However, such transient cues are not sufficient to ensure robustness of the resulting cellular polarisation
Long-distance hormone transport via the phloem
Several key plant hormones are synthesised in the shoot and are advected within the phloem to the root tip. In the root tip, these hormones regulate growth and developmental processes, and responses to environmental cues. However, we lack understanding of how environmental factors and biological parameters affect the delivery of hormones to the root tip. In this study, we build on existing models of phloem flow to develop a mathematical model of sugar transport alongside the transport of a generic hormone. We derive the equations for osmotically driven flow in a long, thin pipe with spatially varying membrane properties to capture the phloem loading and unloading zones. Motivated by experimental findings, we formulate solute membrane transport in terms of passive and active components, and incorporate solute unloading via bulk flow (i.e. advection with the water efflux) by including the Staverman reflection coefficient. We use the model to investigate the coupling between the sugar and hormone dynamics. The model predicts that environmental cues that lead to an increase in active sugar loading, an increase in bulk flow sugar unloading or a decrease in the relative root sugar concentration result in an increase in phloem transport velocity. Furthermore, the model reveals that such increases in phloem transport velocity result in an increase in hormone delivery to the root tip for passively loaded hormones
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