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Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations
Controlled cellular automata
2565 Mc Carthy Mall, Department of Mathematics, University of Hawaii at Manoa, Honolulu, 96822, USA |
Cellular Automata have been successfully used to model evolution of complex systems based on simples rules. In this paper we introduce controlled cellular automata to depict the dynamics of systems with controls that can affect their evolution. Using theory from discrete control systems, we derive results for the control of cellular automata in specific cases. The paper is mostly oriented toward two applications: fire spreading; morphogenesis and tumor growth. In both cases, we illustrate the impact of a control on the evolution of the system. For the fire, the control is assumed to be either firelines or firebreaks to prevent spreading or dumping of water, fire retardant and chemicals (foam) on the fire to neutralize it. In the case of cellular growth, the control describes mechanisms used to regulate growth factors and morphogenic events based on the existence of extracellular matrix structures called fractones. The hypothesis is that fractone distribution may coordinate the timing and location of neural cell proliferation, thereby guiding morphogenesis, at several stages of early brain development.
References:
[1] |
T. Alarcón, H. M. Byrne and P. K. Maini,
A cellular automaton model for tumour growth in inhomogeneous environment, Journal of Theoretical Biology, 225 (2003), 257-274.
doi: 10.1016/S0022-5193(03)00244-3. |
[2] |
A. Beros, M. Chyba, A. Fronville and F. Mercier, A morphogenetic cellular automaton, in 2018 Annual American Control Conference (ACC), IEEE, 2018, 1987–1992.
doi: 10.23919/ACC.2018.8431498. |
[3] |
A. B. Bishop, Introduction to Discrete Linear Controls: Theory and Application, Elsevier, 2014. |
[4] |
H. H. Chen and G. W. Brodland,
Cell-level finite element studies of viscous cells in planar aggregates, Journal of Biomechanical Engineering, 122 (2000), 394-401.
doi: 10.1115/1.1286563. |
[5] |
V. Douet, A. Kerever, E. Arikawa-Hirasawa and F. Mercier,
Fractone-heparan sulphates mediate fgf-2 stimulation of cell proliferation in the adult subventricular zone, Cell Proliferation, 46 (2013), 137-145.
doi: 10.1111/cpr.12023. |
[6] |
S. El Yacoubi and P. Jacewicz, Cellular automata and controllability problem, in CD-Rom Proceeding of the 14th Int. Symp. on Mathematical Theory of Networks and Systems, june, 2000, 19–23. |
[7] |
S. El Yacoubi, P. Jacewicz and N. Ammor,
Analyse et contrôle par automates cellulaires, Annals of the University of Craiova-Mathematics and Computer Science Series, 30 (2003), 210-221.
|
[8] |
A. Kerever, J. Schnack, D. Vellinga, N. Ichikawa, C. Moon, E. Arikawa-Hirasawa, J. T. Efird and F. Mercier,
Novel extracellular matrix structures in the neural stem cell niche capture the neurogenic factor fibroblast growth factor 2 from the extracellular milieu, Stem Cells, 25 (2007), 2146-2157.
doi: 10.1634/stemcells.2007-0082. |
[9] |
M. Mamei, A. Roli and F. Zambonelli,
Emergence and control of macro-spatial structures in perturbed cellular automata, and implications for pervasive computing systems, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 35 (2005), 337-348.
doi: 10.1109/TSMCA.2005.846379. |
[10] |
F. Mercier,
Fractones: Extracellular matrix niche controlling stem cell fate and growth factor activity in the brain in health and disease, Cellular and Molecular Life Sciences, 73 (2016), 4661-4674.
doi: 10.1007/s00018-016-2314-y. |
[11] |
F. Mercier and V. Douet,
Bone morphogenetic protein-4 inhibits adult neurogenesis and is regulated by fractone-associated heparan sulfates in the subventricular zone, Journal of Chemical Neuroanatomy, 57 (2014), 54-61.
doi: 10.1016/j.jchemneu.2014.03.005. |
[12] |
F. Mercier, J. T. Kitasako and G. I. Hatton,
Anatomy of the brain neurogenic zones revisited: Fractones and the fibroblast/macrophage network, Journal of Comparative Neurology, 451 (2002), 170-188.
doi: 10.1002/cne.10342. |
[13] |
N. J. Popławski, M. Swat, J. S. Gens and J. A. Glazier, Adhesion between cells, diffusion of growth factors, and elasticity of the aer produce the paddle shape of the chick limb, Physica A: Statistical Mechanics and its Applications, 373 (2007), 521-532. |
[14] |
D. Walker, S. Wood, J. Southgate, M. Holcombe and R. Smallwood,
An integrated agent-mathematical model of the effect of intercellular signalling via the epidermal growth factor receptor on cell proliferation, Journal of Theoretical Biology, 242 (2006), 774-789.
doi: 10.1016/j.jtbi.2006.04.020. |
show all references
References:
[1] |
T. Alarcón, H. M. Byrne and P. K. Maini,
A cellular automaton model for tumour growth in inhomogeneous environment, Journal of Theoretical Biology, 225 (2003), 257-274.
doi: 10.1016/S0022-5193(03)00244-3. |
[2] |
A. Beros, M. Chyba, A. Fronville and F. Mercier, A morphogenetic cellular automaton, in 2018 Annual American Control Conference (ACC), IEEE, 2018, 1987–1992.
doi: 10.23919/ACC.2018.8431498. |
[3] |
A. B. Bishop, Introduction to Discrete Linear Controls: Theory and Application, Elsevier, 2014. |
[4] |
H. H. Chen and G. W. Brodland,
Cell-level finite element studies of viscous cells in planar aggregates, Journal of Biomechanical Engineering, 122 (2000), 394-401.
doi: 10.1115/1.1286563. |
[5] |
V. Douet, A. Kerever, E. Arikawa-Hirasawa and F. Mercier,
Fractone-heparan sulphates mediate fgf-2 stimulation of cell proliferation in the adult subventricular zone, Cell Proliferation, 46 (2013), 137-145.
doi: 10.1111/cpr.12023. |
[6] |
S. El Yacoubi and P. Jacewicz, Cellular automata and controllability problem, in CD-Rom Proceeding of the 14th Int. Symp. on Mathematical Theory of Networks and Systems, june, 2000, 19–23. |
[7] |
S. El Yacoubi, P. Jacewicz and N. Ammor,
Analyse et contrôle par automates cellulaires, Annals of the University of Craiova-Mathematics and Computer Science Series, 30 (2003), 210-221.
|
[8] |
A. Kerever, J. Schnack, D. Vellinga, N. Ichikawa, C. Moon, E. Arikawa-Hirasawa, J. T. Efird and F. Mercier,
Novel extracellular matrix structures in the neural stem cell niche capture the neurogenic factor fibroblast growth factor 2 from the extracellular milieu, Stem Cells, 25 (2007), 2146-2157.
doi: 10.1634/stemcells.2007-0082. |
[9] |
M. Mamei, A. Roli and F. Zambonelli,
Emergence and control of macro-spatial structures in perturbed cellular automata, and implications for pervasive computing systems, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 35 (2005), 337-348.
doi: 10.1109/TSMCA.2005.846379. |
[10] |
F. Mercier,
Fractones: Extracellular matrix niche controlling stem cell fate and growth factor activity in the brain in health and disease, Cellular and Molecular Life Sciences, 73 (2016), 4661-4674.
doi: 10.1007/s00018-016-2314-y. |
[11] |
F. Mercier and V. Douet,
Bone morphogenetic protein-4 inhibits adult neurogenesis and is regulated by fractone-associated heparan sulfates in the subventricular zone, Journal of Chemical Neuroanatomy, 57 (2014), 54-61.
doi: 10.1016/j.jchemneu.2014.03.005. |
[12] |
F. Mercier, J. T. Kitasako and G. I. Hatton,
Anatomy of the brain neurogenic zones revisited: Fractones and the fibroblast/macrophage network, Journal of Comparative Neurology, 451 (2002), 170-188.
doi: 10.1002/cne.10342. |
[13] |
N. J. Popławski, M. Swat, J. S. Gens and J. A. Glazier, Adhesion between cells, diffusion of growth factors, and elasticity of the aer produce the paddle shape of the chick limb, Physica A: Statistical Mechanics and its Applications, 373 (2007), 521-532. |
[14] |
D. Walker, S. Wood, J. Southgate, M. Holcombe and R. Smallwood,
An integrated agent-mathematical model of the effect of intercellular signalling via the epidermal growth factor receptor on cell proliferation, Journal of Theoretical Biology, 242 (2006), 774-789.
doi: 10.1016/j.jtbi.2006.04.020. |














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