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SUMMARY:Construction of quasi-potential landscapes using the Sum-of-Square
s optimisation
DTSTART;VALUE=DATE-TIME:20180709T094500Z
DTEND;VALUE=DATE-TIME:20180709T100000Z
DTSTAMP;VALUE=DATE-TIME:20200813T145317Z
UID:indico-contribution-422@conferences.maths.unsw.edu.au
DESCRIPTION:Speakers: Rowan Brackston (Imperial College London)\nThe const
ruction of effective and informative landscapes for stochastic dynamical s
ystems has proven a long-standing and complex problem\, including for biol
ogical systems. Such landscapes may refer to a true energy function for ca
ses such as protein folding or to a phenomenological metaphor in the case
of Waddingtonâ€™s epigenetic landscape. In many situations\, constructing
a landscape comes down to obtaining the quasi-potential\, a scalar functio
n that quantifies the likelihood of reaching each point in the state-space
. \n\nIn this work we provide a novel method for constructing such landsca
pes using a tool from control theory: the Sum-of-Squares method for genera
ting Lyapunov functions. Applicable to any system described by polynomials
\, this method provides an analytical polynomial expression for the potent
ial landscape\, in which the coefficients of the polynomial are obtained v
ia a convex optimisation. The resulting landscapes are based upon a decomp
osition of the vector-field of the original system\, such that the inner p
roduct of the gradient of the potential and the remaining dynamics is ever
ywhere negative. By satisfying this condition\, our derived landscapes pro
vide both upper and lower bounds on the true quasi-potential\; these bound
s becoming tight if the decomposition is orthogonal. The method is impleme
nted in the programming language Julia\, and is demonstrated to correctly
compute the quasi-potential for high-dimensional linear systems and also f
or a number of nonlinear examples. For a multi-stable stochastic system an
alogous to a developing stem cell\, we use the computed potential to evalu
ate bounds on the relative likelihood of reaching fixed points equivalent
to each of the differentiated phenotypes.\n\nhttps://conferences.maths.uns
w.edu.au/event/2/contributions/422/
LOCATION:University of Sydney Holme Building/--The Refectory
URL:https://conferences.maths.unsw.edu.au/event/2/contributions/422/
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