We consider the effect of a small cut-off ε on the velocity of a traveling wave in one dimension. Simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cut-off ε is to select a single velocity which converges when ε → 0 to the one predicted by the marginal stability argument. For small ε, the shift in velocity has the form K(log ε) −2 and our prediction for the constant K agrees very well with the results of our simulations. A very similar logarithmic shift appears in more complicated situations, in particular in finite size effects of some microscopic stochastic systems. Our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the Fisher-Kolmogorov or similar equations.
Perspective-taking is a stepping stone to human empathy. When empathizing with another individual, one can imagine how the other perceives the situation and feels as a result. To what extent does imagining the other differs from imagining oneself in similar painful situations? In this functional magnetic resonance imaging experiment, participants were shown pictures of people with their hands or feet in painful or non-painful situations and instructed to imagine and rate the level of pain perceived from different perspectives. Both the Self's and the Other's perspectives were associated with activation in the neural network involved in pain processing, including the parietal operculum, anterior cingulate cortex (ACC; BA32) and anterior insula. However, the Self-perspective yielded higher pain ratings and involved the pain matrix more extensively in the secondary somatosensory cortex, the ACC (BA 24a /24b ), and the insula proper. Adopting the perspective of the Other was associated with specific increase in the posterior cingulate/precuneus and the right temporo-parietal junction. These results show the similarities between Self-and Other-pain representation, but most interestingly they also highlight some distinctiveness between these two representations, which is a crucial aspect of human empathy. It may be what allows us to distinguish empathic responses to others versus our own personal distress. These findings are consistent with the view that empathy does not involve a complete Self-Other merging.
We propose a phenomenological description for the effect of a weak noise on the position of a front described by the Fisher-Kolmogorov-Petrovsky-Piscounov equation or any other travelling wave equation in the same class. Our scenario is based on four hypotheses on the relevant mechanism for the diffusion of the front. Our parameter-free analytical predictions for the velocity of the front, its diffusion constant and higher cumulants of its position agree with numerical simulations.
We use traveling-wave theory to derive expressions for the rate of accumulation of deleterious mutations under Muller's ratchet and the speed of adaptation under positive selection in asexual populations. Traveling-wave theory is a semi-deterministic description of an evolving population, where the bulk of the population is modeled using deterministic equations, but the class of the highestfitness genotypes, whose evolution over time determines loss or gain of fitness in the population, is given proper stochastic treatment. We derive improved methods to model the highest-fitness class (the stochastic edge) for both Muller's ratchet and adaptive evolution, and calculate analytic correction terms that compensate for inaccuracies which arise when treating discrete fitness classes as a continuum. We show that traveling wave theory makes excellent predictions for the rate of mutation accumulation in the case of Muller's ratchet, and makes good predictions for the speed of adaptation in a very broad parameter range. We predict the adaptation rate to grow logarithmically in the population size until the population size is extremely large.
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