nikola popovic - Profile on Academia.edu (original) (raw)

Papers by nikola popovic

Physica D-nonlinear Phenomena, 2010

a b s t r a c t 'Cut-offs' were introduced to model front propagation in reaction-diffusion syste... more a b s t r a c t 'Cut-offs' were introduced to model front propagation in reaction-diffusion systems in which the reaction is effectively deactivated at points where the concentration lies below some threshold. In this article, we investigate the effects of a cut-off on fronts propagating into metastable states in a class of bistable scalar equations. We apply the method of geometric desingularization from dynamical systems theory to calculate explicitly the change in front propagation speed that is induced by the cut-off. We prove that the asymptotics of this correction scales with fractional powers of the cut-off parameter, and we identify the source of these exponents, thus explaining the structure of the resulting expansion. In particular, we show geometrically that the speed of bistable fronts increases in the presence of a cut-off, in agreement with results obtained previously via a variational principle. We first discuss the classical Nagumo equation as a prototypical example of bistable front propagation. Then, we present corresponding results for the (equivalent) cut-off Schlögl equation. Finally, we extend our analysis to a general family of reaction-diffusion equations that support bistable fronts, and we show that knowledge of an explicit front solution to the associated problem without cut-off is necessary for the correction induced by the cut-off to be computable in closed form.

Journal of Dynamics and Differential Equations, 2006

We investigate traveling wave solutions in a family of reaction-diffusion equations which include... more We investigate traveling wave solutions in a family of reaction-diffusion equations which includes the Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with quadratic nonlinearity and a bistable equation with degenerate cubic nonlinearity. It is known that, for each equation in this family, there is a critical wave speed which separates waves of exponential decay from those of algebraic decay at one of the end states. We derive rigorous asymptotic expansions for these critical speeds by perturbing off the classical FKPP and bistable cases. Our approach uses geometric singular perturbation theory and the blow-up technique, as well as a variant of the Melnikov method, and confirms the results previously obtained through asymptotic analysis in [J.H. Merkin and D.

Nonlinearity, 2007

The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with cut-off was introduced in [E. Bru... more The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with cut-off was introduced in [E. Brunet and B. Derrida, Shift in the velocity of a front due to a cutoff, Phys. Rev. E 56 , 2597-2604 (1997)] to model N -particle systems in which concentrations less than ε = 1 N are not attainable. It was conjectured that the cut-off function, which sets the reaction terms to zero if the concentration is below the small threshold ε, introduces a substantial shift in the propagation speed of the corresponding traveling waves. In this article, we prove the conjecture of Brunet and Derrida, showing that the speed of propagation is given by ccrit(ε) = 2 − π 2 (ln ε) 2 + O((ln ε) −3 ), as ε → 0, for a large class of cut-off functions. Moreover, we extend this result to a more general family of scalar reaction-diffusion equations with cut-off. The main mathematical techniques used in our proof are geometric singular perturbation theory and the blow-up method, which lead naturally to the identification of the reasons for the logarithmic dependence of ccrit on ε as well as for the universality of the corresponding leading-order coefficient (π 2 ).

Journal of Mathematical Analysis and Applications, 2007

We study traveling wave solutions for the class of scalar reaction-diffusion equations ∂u ∂t = ∂ ... more We study traveling wave solutions for the class of scalar reaction-diffusion equations ∂u ∂t = ∂ 2 u ∂x 2 + fm(u), where the family of potential functions {fm} is given by fm(u) = 2u m (1 − u). For each m ≥ 1 real, there is a critical wave speed ccrit(m) that separates waves of exponential structure from those which decay only algebraically. We derive a rigorous asymptotic expansion for ccrit(m) in the limit as m → ∞. This expansion also seems to provide a useful approximation to ccrit(m) over a wide range of m-values. Moreover, we prove that ccrit(m) is C ∞ -smooth as a function of m −1 . Our analysis relies on geometric singular perturbation theory, as well as on the blow-up technique, and confirms the results obtained by means of asymptotic methods in [D.

Siam Journal on Applied Dynamical Systems, 2008

We present a mathematical study of some aspects of mixed-mode oscillation (MMO) dynamics in a thr... more We present a mathematical study of some aspects of mixed-mode oscillation (MMO) dynamics in a three time scale system of ODEs as well as analyze related features of a biophysical model of a neuron from the entorhinal cortex that serves as a motivation for our study. The neuronal model includes standard spiking currents (sodium and potassium) that play a critical role in the analysis of the interspike interval as well as persistent sodium and slow potassium (M) currents. We reduce the dimensionality of the neuronal model from six to three dimensions in order to investigate a regime in which MMOs are generated and to motivate the three time scale model system upon which we focus our study. We further analyze in detail the mechanism of the transition from MMOs to spiking in our model system. In particular, we prove the existence of a special solution, a singular primary canard, that serves as a transition between mixed-mode oscillations and spiking in the singular limit by employing appropriate rescalings and center manifold reductions. Additionally, we conjecture that the singular canard solution is the limit of a family of canards and provide numerical evidence for the conjecture.

Chaos, 2008

Mixed-mode dynamics is a complex type of dynamical behavior that has been observed both numerical... more Mixed-mode dynamics is a complex type of dynamical behavior that has been observed both numerically and experimentally in numerous prototypical systems in the natural sciences. The compartmental Wilson-Callaway model for the dopaminergic neuron is an example of a system that exhibits a wide variety of mixed-mode patterns upon variation of a control parameter. One characteristic feature of this system is the presence of multiple time-scales. In this article, we study the Wilson-Callaway model from a geometric point of view. We show that the observed mixed-mode dynamics is caused by a slowly varying canard structure. By appropriately transforming the model equations, we reduce them to an underlying three-dimensional canonical form that can be analyzed via a slight adaptation of the approach developed in [1].

Homozygous mutations incaveolin-3 cause a severe form of rippling muscle disease

Annals of Neurology, 2003

Heterozygous missense mutations in the caveolin-3 gene (CAV3) cause different muscle disorders. M... more Heterozygous missense mutations in the caveolin-3 gene (CAV3) cause different muscle disorders. Most patients with CAV3 alterations present with rippling muscle disease (RMD) characterized by signs of increased muscle irritability without muscle weakness. In some patients, CAV3 mutations underlie the progressive limb-girdle muscular dystrophy type 1C (LGMD1C). Here, we report two unrelated patients with novel homozygous mutations (L86P and A92T) in CAV3. Both presented with a more severe clinical phenotype than usually seen in RMD. Immunohistochemical and immunoblot analyses of muscle biopsies showed a strong reduction of caveolin-3 in both homozygous RMD patients similar to the findings in heterozygous RMD. Electron microscopy studies showed a nearly complete absence of caveolae in the sarcolemma in all RMD patients analyzed. Additional plasma membrane irregularities (small plasmalemmal discontinuities, subsarcolemmal vacuoles, abnormal papillary projections) were more pronounced in homozygous than in heterozygous RMD patients. A stronger activation of nitric oxide synthase was observed in both homozygous patients compared with heterozygous RMD. Like in LGMD1C, dysferlin immunoreactivity is reduced in RMD but more pronounced in homozygous as compared with heterozygous RMD. Thus, we further extend the phenotypic variability of muscle caveolinopathies by identification of a severe form of RMD associated with homozygous CAV3 mutations. Ann Neurol 2003

Rippling Muscle Disease in Childhood

Journal of Child Neurology, 2002

Rippling muscle disease is a rare autosomal dominant disorder first described in 1975. Recently, ... more Rippling muscle disease is a rare autosomal dominant disorder first described in 1975. Recently, it could be classified as a caveolinopathy; in European families, mutations in the caveolin-3 gene were revealed as causing this disease. Although clinical symptoms were almost all described in adulthood, we are now reporting clinical data of seven children with rippling muscle disease owing to mutations in the caveolin-3 gene. Initial symptoms were frequent falls, inability to walk on heels, tiptoe walking with pain and a warm-up phenomenon, calf hypertrophy, and an elevated serum creatine kinase level. Percussion-/pressure-induced rapid contractions, painful muscle mounding, and rippling could be observed even in early childhood. The diagnosis can be confirmed by molecular genetic analysis. Muscle biopsy must be considered in patients without muscle weakness or mechanical hyperirritability to differentiate between rippling muscle disease and limb-girdle muscular dystrophy 1C.

Journal of Physics: Conference Series, 2005

One common characteristic of many classical singular perturbation problems is the occurrence of l... more One common characteristic of many classical singular perturbation problems is the occurrence of logarithmic (switchback) terms in the corresponding asymptotic expansions. We discuss two such problems well known to give rise to logarithmic switchback: first, Lagerstrom's equation, a model related to the asymptotic treatment of low Reynolds number flow from fluid mechanics, and second, the Evans function approach to the stability of degenerate shock waves in (scalar) reaction-diffusion equations. We show how asymptotic expansions for these two problems can be obtained by means of methods from dynamical systems theory as well as of the blow-up technique. We identify the structure of these expansions and demonstrate that the occurrence of the logarithmic switchback terms therein is in fact caused by a resonance phenomenon.