Critical Phenomena in a Heterogeneous Excitable System!
Rachel Sheldon (MOAC DTC, University of Warwick)!
Supervisor: Hugo van den Berg (Systems Biology, University of Warwick)!
1. Aim of the Project!
Modelling the changes in the muscular wall of the uterus in the days before labour. In par9cular, examining the excita9on of a 9ghtly coupled system of smooth muscle cells. The muscular wall of the uterus only acquires the ability to expel a fœtus in the final days before labour •
Smooth muscle cells start at a stable equilibrium. Sufficient input current is needed to excite the cell causing a voltage surge (Fig.1) •
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v
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w
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Fig.1 A sufficient perturba9on in the ini9al voltage can excite a muscle cell a) Kick = 0.3; b) kick = 0.4. •
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  Local excita9on which does not spread across the en9re 9ssue   Global excita9on where all cells are excited Threshold Value for Excitation of Cell 1
Minimum kick needed to excite highlighted cell increases with coupling (Fig.
3) Maximum amplitude is consistent for all coupled cells 2.5
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1.5
Myometrial cell synchrony is achieved by electrical conduc9on through connec9ng microfibrils1 0.5
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Fig.3 Threshold value for exci9ng the kicked cell. Maximum Amplitude
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There is a threshold coupling value for exci9ng coupled cells (Fig.4) Kicked cell 1.5
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The slow rise and fall of tension results in a contrac9on that lasts about a minute2 Threshold Value for Excited Cell
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t t
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Fig.2 The rise and fall of voltage in an excited cell. 25
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Excite middle cell Minimum kick needed to excite highlighted cell increases with coupling (Fig.5) faster than for a cell chain There is a threshold for exci9ng coupled cells (Fig.6) which is lower than for a cell chain 2
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t
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Introduce variable coupling values between cells •
Remove resistor couplings at random and inves9gate the spread of excita9on using percola9on theory 2.0
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Maximum amplitude is consistent for all coupled cells and comparable to a cell chain 1
Fig.6 Excita9on variable v over 9me for a) n length chain b) (n x n) lagce. 6. Further Work!
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vHtL
v@tD
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Coupled cells 1.0
Excited cells follow the same voltage pathway over 9me 3
4. Cell Lattices!
2.5
The phase transi9on from a locally excited state to a globally excited state has a coupling threshold and is instantaneous at this point •
1.0
Fig.4 Maximum amplitude of cells with increasing coupling. v(t) The system has two states: •
Excite a cell with a 'kick' Poten9al returns rapidly to equilibrium a\er applied current is removed (Fig.2) Cells are considered to obey Fitzhugh-­‐Nagumo dynamics3: d
1
v = (Av(1− v)(v − α ) − w − w0 )
Excita9on Variable: dt ε
Recovery Variable: d w = v − γ w − v
0
dt
•  Simula9ons were run for cells in chains and lagces of an arbitrary size •
5. Conclusions!
3. Cell Chains!
2. Background!
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Fig.5 Threshold value for exci9ng kicked cell. 7. References!
Maximum Amplitude of Coupled Cells
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Kicked cell 1.5
Coupled cells 1.0
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Fig.6 Maximum amplitude of cells with increasing coupling. Smith, R. Parturi9on. N Engl J Med. 1997, 356: 271-­‐283. Blanks, A.M. et.al. Myometrial func9on in prematurity. Best Pract Res Clin Obstet Gynaecol. 2007, 21(5): 807-­‐819. Keener, J.; Sneyd, J. Mathema=cal Physiology; Springer-­‐
Verlag: New York, 1998. Thank you to Hugo van den Berg for all his support during this project 1
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