23 research outputs found
Existence and stability analysis of spiky solutions for the Gierer-Meinhardt system with large reaction rates
No full textWe study the Gierer-Meinhardt system in one dimension in the limit of large reaction rates. First we construct three types of solutions: (i) an interior spike; (ii) a boundary spike and (iii) two boundary spikes. Second we prove results on their stability. It is found that an interior spike is always unstable; a boundary spike is always stable. The two boundary spike configuration can be either stable or unstable, depending on the parameters. We fully classify the stability in this case. We characterise the destabilizing eigenfunctions in all cases. Numerical simulations are shown which are in full agreement with the analytical results
Dynamics of pulse solutions in Gierer-Meinhardt model with time dependent diffusivity
No full textDispersive processes with a time dependent diffusivity appear in a plethora of physical systems. Most often a solution is attained for a predefined form of diffusion coefficient D(t). Here existence of pulse solutions with an arbitrary time dependence thereof is proved for the Gierer-Meinhardt model with three types of transport: regular diffusion, sub-diffusion and L´evy flights. Admission of a solution of the classical pulse shape, but for an unencumbered form of D(t) is a valuable property that allows to study phenomena of the ilk observed in various ostensibly unrelated applications. Closed form solutions are obtained for some pulse constellations. Transitions between periods of nearly constant diffusivities trigger respective cross-over between counterpart solutions known for a constant diffusivity, thereupon exhibiting otherwise unattainable behaviour, qualitatively reconstructing observable evolution peculiarities of tagged molecular structures, such as essential slowing down or speeding up during various stages of motion, inexplicable with a single constant diffusion coefficient.Peer reviewedanomalous diffusionnon-linear pattern formationreaction - diffusionGierer-Meinhardt modelpulse solutiontransient diffusion regime
On the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt system
No full textThis is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://iopscience.iop.org/article/10.1088/1361-6544/aa64b2/metaThe purpose of the current paper is to contribute to the comprehension of the dynamics of the shadow system of an activator-inhibitor system known as a Gierer-Meinhardt model. Shadow systems are intended to work as an intermediate step between single equations and reaction-diffusion systems. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the Gierer-Meinhardt model is reduced to a single though non-local equation whose dynamics will be investigated. We mainly focus on the derivation of blow-up results for this non-local equation which can be seen as instability patterns of the shadow system. In particular, a {\it diffusion driven instability (DDI)}, or {\it Turing instability}, in the neighbourhood of a constant stationary solution, which it is destabilised via diffusion-driven blow-up, is obtained. The latter actually indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns
Dynamics of pulse solutions in Gierer-Meinhardt model with time dependent diffusivity
No full textDispersive processes with a time dependent diffusivity appear in a plethora of physical systems. Most often a solution is attained for a predefined form of diffusion coefficient D(t). Here existence of pulse solutions with an arbitrary time dependence thereof is proved for the Gierer-Meinhardt model with three types of transport: regular diffusion, sub-diffusion and L´evy flights. Admission of a solution of the classical pulse shape, but for an unencumbered form of D(t) is a valuable property that allows to study phenomena of the ilk observed in various ostensibly unrelated applications. Closed form solutions are obtained for some pulse constellations. Transitions between periods of nearly constant diffusivities trigger respective cross-over between counterpart solutions known for a constant diffusivity, thereupon exhibiting otherwise unattainable behaviour, qualitatively reconstructing observable evolution peculiarities of tagged molecular structures, such as essential slowing down or speeding up during various stages of motion, inexplicable with a single constant diffusion coefficient.Peer reviewedanomalous diffusionnon-linear pattern formationreaction - diffusionGierer-Meinhardt modelpulse solutiontransient diffusion regime
Spike solutions in Gierer-Meinhardt model with a time dependent anomaly exponent
No full textExperimental evidence of complex dispersion regimes in natural systems, where the growth of the 7 mean square displacement in time cannot be characterised by a single power, has been accruing 8 for the past two decades. In such processes the exponent γ(t) in hr2i ∼ tγ(t) at times might be 9 approximated by a piecewise constant function, or it can be a continuous function. Variable order 10 differential equations are an emerging mathematical tool with a strong potential to model these 11 systems. However, variable order differential equations are not tractable by the classic differential 12 equations theory. This contribution illustrates how a classic method can be adapted to gain insight 13 into a system of this type. Herein a variable order Gierer-Meinhardt model is posed, a generic 14 reaction– diffusion system of a chemical origin. With a fixed order this system possesses a solution 15 in the form of a constellation of arbitrarily situated localised pulses, when the components’ diffu- 16 sivity ratio is asymptotically small. The pattern was shown to exist subject to multiple step-like 17 transitions between normal diffusion and sub-diffusion, as well as between distinct sub-diffusive 18 regimes. The analytical approximation obtained permits qualitative analysis of the impact thereof. 19 Numerical solution for typical cross-over scenarios revealed such features as earlier equilibration 20 and non-monotonic excursions before attainment of equilibrium. The method is general and allows 21 for an approximate numerical solution with any reasonably behaved γ(t).Peer reviewednumerical estimates of memory integralfractional differential equationsmatched asymptotic expansionsvariable order differential equation
Investigation occurrences of turing pattern in Schnakenberg and Gierer-Meinhardt equation
No full textAbnormal Paraplegin Expression in Swollen Neurites, tau- and alpha-Synuclein Pathology in a Case of Hereditary Spastic Paraplegia SPG7 with an Ala510Val Mutation
No full textMutations in the SPG7 gene are the most frequent cause of autosomal recessive hereditary spastic paraplegias and spastic ataxias. Ala510Val is the most common SPG7 mutation, with a frequency of up to 1% in the general population. Here we report the clinical, genetic, and neuropathological findings in a homozygous Ala510Val SPG7 case with spastic ataxia. Neuron loss with associated gliosis was found in the inferior olivary nucleus, the dentate nucleus of the cerebellum, the substantia nigra and the basal nucleus of Meynert. Neurofilament and/or paraplegin accumulation was observed in swollen neurites in the cerebellar and cerebral cortex. This case also showed subcortical τ-pathology in an unique distribution pattern largely restricted to the brainstem. α-synuclein containing Lewy bodies (LBs) were observed in the brainstem and the cortex, compatible with a limbic pattern of Braak LB-Disease stage 4. Taken together, this case shows that the spectrum of pathologies in SPG7 can include neuron loss of the dentate nucleus and the inferior olivary nucleus as well as neuritic pathology. The progressive supranuclear palsy-like brainstem predominant pattern of τ pathology and α-synuclein containing Lewy bodies in our SPG7 cases may be either coincidental or related to SPG7 in addition to neuron loss and neuritic pathology.sponsorship: Dietmar R. Thal received consultant honorary from GE-Healthcare, and Covance Laboratories, speaker honorary from GE-Healthcare and collaborated with Novartis Pharma AG. Matthis Synofzik received consulting fees from Actelion Pharmaceuticals Ltd. Dietmar R. Thal received research funds from Alzheimer Forschung Initiative (AFI) Grant No.: #13803. Stephan Zuchner was supported by NIH R01NS075764, 5R01NS072248, and U54NS065712. Rebecca Schule, Ludger Schols and Matthis Synofzik were funded by the Interdisciplinary Center for Clinical Research IZKF Tubingen (grant 2191-0-0 to Matthis Synofzik, grant 1970-0-0 to Rebecca Schule) and the European Union (grant F5-2012-305121 "NEUROMICS" to Ludger Schols and grant PIOF-GA-2012-326681 "HSP/CMT genetics" to Rebecca Schule). (Actelion Pharmaceuticals Ltd., Alzheimer Forschung Initiative (AFI) Grant|13803, NIH|R01NS075764, NIH|5R01NS072248, NIH|U54NS065712, Interdisciplinary Center for Clinical Research IZKF Tubingen|2191-0-0, Interdisciplinary Center for Clinical Research IZKF Tubingen|1970-0-0, European Union|F5-2012-305121, European Union|PIOF-GA-2012-326681)status: Publishe
