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Revisiting the Fisher-Kolmogorov-Petrovsky-Piskunov equation to interpret the spreading-extinction dichotomy.

ABSTRACT: The Fisher-Kolmogorov-Petrovsky-Piskunov model, also known as the Fisher-KPP model, supports travelling wave solutions that are successfully used to model numerous invasive phenomena with applications in biology, ecology and combustion theory. However, there are certain phenomena that the Fisher-KPP model cannot replicate, such as the extinction of invasive populations. The Fisher-Stefan model is an adaptation of the Fisher-KPP model to include a moving boundary whose evolution is governed by a Stefan condition. The Fisher-Stefan model also supports travelling wave solutions; however, a key additional feature of the Fisher-Stefan model is that it is able to simulate population extinction, giving rise to a spreading-extinction dichotomy. In this work, we revisit travelling wave solutions of the Fisher-KPP model and show that these results provide new insight into travelling wave solutions of the Fisher-Stefan model and the spreading-extinction dichotomy. Using a combination of phase plane analysis, perturbation analysis and linearization, we establish a concrete relationship between travelling wave solutions of the Fisher-Stefan model and often-neglected travelling wave solutions of the Fisher-KPP model. Furthermore, we give closed-form approximate expressions for the shape of the travelling wave solutions of the Fisher-Stefan model in the limit of slow travelling wave speeds, c?1.

SUBMITTER: El-Hachem M

PROVIDER: S-EPMC6784399 | biostudies-literature | 2019 Sep

REPOSITORIES: biostudies-literature

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Revisiting the Fisher-Kolmogorov-Petrovsky-Piskunov equation to interpret the spreading-extinction dichotomy.

El-Hachem Maud M McCue Scott W SW Jin Wang W Du Yihong Y Simpson Matthew J MJ

Proceedings. Mathematical, physical, and engineering sciences 20190904 2229

The Fisher-Kolmogorov-Petrovsky-Piskunov model, also known as the Fisher-KPP model, supports travelling wave solutions that are successfully used to model numerous invasive phenomena with applications in biology, ecology and combustion theory. However, there are certain phenomena that the Fisher-KPP model cannot replicate, such as the extinction of invasive populations. The Fisher-Stefan model is an adaptation of the Fisher-KPP model to include a moving boundary whose evolution is governed by a ...[more]

PMID: 31611732

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Project description:Extinction learning refers to the phenomenon that a previously learned response to an environmental stimulus, for example the expression of an aversive behavior upon exposure to a specific context, is reduced when the stimulus is repeatedly presented in the absence of a previously paired aversive event. Extinction of fear memories has been implicated with the treatment of anxiety disease but the molecular processes that underlie fear extinctionare only beginning to emerge. Here we show that fear extinction initiates up-regulation of hippocampal insulin-growth factor 2 (Igf2) and down-regulation of insulin-growth factor binding protein 7 (Igfbp7). In line with this observation we demonstrate that IGF2 facilitates fear extinction, while IGFBP7 impairs fear extinction in an IGF2-dependent manner. Furthermore, we identify one cellular substrate of altered IGF2-signaling during fear extinction. To this end we show that fear extinction-induced IGF2/IGFBP7-signaling promotes the survival of 17-19 day-old newborn hippocampal neurons. In conclusion, our data suggests that therapeutic strategies that enhance IGF2-signaling and adult neurogenesis might be suitable to treat disease linked to excessive fear memory. We employed mice to investigate fear extinction in the hippocampus-dependent contextual fear conditioning paradigm. To this end, male C57BL/6J mice were exposed to the fear conditioning box (context) followed by an electric foot-shock which elicits the acquisition of conditioned contextual fear. For extinction training animals were repeatedly reexposed to the conditioned context on consecutive days (24h interval) without receiving the footshockagain (extinction trial, E). This procedure eventually results in the decline of the aversive freezing behavior. Mice that were exposed to the conditioning context without receiving fear conditioning training served as control groups. To gain a better understanding of the molecular processes underlying fear extinction we performed a genome-wide analysis of the hippocampal transcriptome during fear extinction. In the employed paradigm fear extinction is a gradual process. To capture the longitudinal course of fear extinction we decided to perform hippocampal microarray analysis at two time points: (1) After the first extinction trial (E1) when animals display high levels of aversive freezing behavior and (2) at the extinction trial on which the freezing behavior was significantly reduced when compared to E1. This extinction trial, in the case of this experiment E5, we termed “extinction trial low freezing” (ELF). Mice that were exposed to the conditioning context without receiving fear conditioning training served as control groups (3). For all three groups we hybridized 5 samples (biological replicates).

Dataset Information

Revisiting the Fisher-Kolmogorov-Petrovsky-Piskunov equation to interpret the spreading-extinction dichotomy.

Publications

Revisiting the Fisher-Kolmogorov-Petrovsky-Piskunov equation to interpret the spreading-extinction dichotomy.

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