Abstract
Evidence suggests that both the interaction of so-called Merkel cells and the epidermal stress distribution play an important role in the formation of fingerprint patterns during pregnancy. To model the formation of fingerprint patterns in a biologically meaningful way these patterns have to become stationary. For the creation of synthetic fingerprints it is also very desirable that rescaling the model parameters leads to rescaled distances between the stationary fingerprint ridges. Based on these observations, as well as the model introduced by Kücken and Champod we propose a new model for the formation of fingerprint patterns during pregnancy. In this anisotropic interaction model the interaction forces not only depend on the distance vector between the cells and the model parameters, but additionally on an underlying tensor field, representing a stress field. This dependence on the tensor field leads to complex, anisotropic patterns. We study the resulting stationary patterns both analytically and numerically. In particular, we show that fingerprint patterns can be modeled as stationary solutions by choosing the underlying tensor field appropriately.
Original language | English |
---|---|
Pages (from-to) | 2171-2206 |
Number of pages | 36 |
Journal | Journal of Mathematical Biology |
Volume | 78 |
Issue number | 7 |
DOIs | |
Publication status | Published - 1 Jun 2019 |
Bibliographical note
Funding Information:BD has been supported by the Leverhulme Trust research project Grant ?Novel discretisations for higher-order nonlinear PDE? (RPG-2015-69). SH acknowledges support from the Niedersachsen Vorab of the Volkswagen Foundation and the DFG Graduate Research School 2088. LMK was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 and the German National Academic Foundation (Studienstiftung des Deutschen Volkes). CBS acknowledges support from Leverhulme Trust project on ?Breaking the non-convexity barrier?, EPSRC Grant Nr. EP/M00483X/1, the EPSRC Centre Nr. EP/N014588/1 and the Cantab Capital Institute for the Mathematics of Information.
Funding Information:
Acknowledgements BD has been supported by the Leverhulme Trust research project Grant ‘Novel discretisations for higher-order nonlinear PDE’ (RPG-2015-69). SH acknowledges support from the Nieder-sachsen Vorab of the Volkswagen Foundation and the DFG Graduate Research School 2088. LMK was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 and the German National Academic Foundation (Studienstiftung des Deutschen Volkes). CBS acknowledges support from Leverhulme Trust project on ‘Breaking the non-convexity barrier’, EPSRC Grant Nr. EP/M00483X/1, the EPSRC Centre Nr. EP/N014588/1 and the Cantab Capital Institute for the Mathematics of Information.
Funding
BD has been supported by the Leverhulme Trust research project Grant ?Novel discretisations for higher-order nonlinear PDE? (RPG-2015-69). SH acknowledges support from the Niedersachsen Vorab of the Volkswagen Foundation and the DFG Graduate Research School 2088. LMK was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 and the German National Academic Foundation (Studienstiftung des Deutschen Volkes). CBS acknowledges support from Leverhulme Trust project on ?Breaking the non-convexity barrier?, EPSRC Grant Nr. EP/M00483X/1, the EPSRC Centre Nr. EP/N014588/1 and the Cantab Capital Institute for the Mathematics of Information. Acknowledgements BD has been supported by the Leverhulme Trust research project Grant ‘Novel discretisations for higher-order nonlinear PDE’ (RPG-2015-69). SH acknowledges support from the Nieder-sachsen Vorab of the Volkswagen Foundation and the DFG Graduate Research School 2088. LMK was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 and the German National Academic Foundation (Studienstiftung des Deutschen Volkes). CBS acknowledges support from Leverhulme Trust project on ‘Breaking the non-convexity barrier’, EPSRC Grant Nr. EP/M00483X/1, the EPSRC Centre Nr. EP/N014588/1 and the Cantab Capital Institute for the Mathematics of Information.
Keywords
- Cell movement
- Dynamical systems
- Nonlocal interactions
- Pattern formation
ASJC Scopus subject areas
- Modelling and Simulation
- Agricultural and Biological Sciences (miscellaneous)
- Applied Mathematics