content: 1) dynamics of beta-cells. polynomial model and gate noise. 2) the influence of noise....
TRANSCRIPT
Content:
1) Dynamics of beta-cells. Polynomial model and gate noise.
2) The influence of noise. Phenomenological.
3) The Gaussian method.
4) Wave block due to glucose gradients.
5) Summary.
The Effect of Noise on β-cell Excitation Dynamics
Mads Peter Sørensen a) and Morten Gram Pedersen b)
a) DTU Mathematics, Lyngby, Denmark, b) Dept. of Information Engineering, University of Padova, Italy
Ref.: M.G. Pedersen and M.P. Sørensen, SIAM J. Appl. Math., 67(2), pp.530-542, (2007).M.G. Pedersen and M.P. Sørensen, Jour. of Bio. Phys., Special issue on Complexity in Neurology
and Psychiatry, Vol. 34 (3-4), pp 425-432, (2008).
Coherence and Persistence in Nonlinear Waves, CPNLW09. January 6-9, 2009, Nice University, Campus Valrose, France.
The β-cell
Ion channel gates for Ca and K
B
Mathematical model for single cell dynamics Topologically equivalent and simplified models. Polynomial model
with Gaussian noise term on the gating variable.
zwufdtdu )( )()( twug
dtdw ))(( zuh
dtdz
Ref.: M. Panarowski, SIAM J. Appl. Math., 54 pp.814-832, (1994).
Voltage across the cell membrane: )(tuu Gating variable: )(tww
Gaussian gate noise term: )(t where 0)( t
)()0()( tt
Slow gate variable: )(tzz
Ref.: A. Sherman, (Eds. Othmar et al), Case studies in mathematical modelling, ecology physiology and cell biology, Prentice Hall (1996), pp.199-217.
The influence of noise on the beta-cell bursting phenomenon.
0
1.0
3.0
Ref.: M.G. Pedersen and M.P. Sørensen, SIAM J. Appl. Math., 67(2), pp.530-542, (2007).
Dynamics and bifurcations
Ref.: E.M. Izhikevich, Neural excitability spiking and bursting, Int. Jour. of Bifurcation and Chaos, p1171 (2000).
Differentiating the first equation above with respect to time leads to.
)())(()()(2
2
tzuhzuGdtdu
uFdtud
))(( zuhdtdz
Where the polynomials are given by
22ˆ)( uuauF )1(3)( 3 uuuG
)()( uuuh
Parameters: 25.0a 6.1ˆ u 4 954.0u
0025.0 7.0
Location of the left saddle-node bifurcation. The Gaussian method.
Ref.: S. Tanabe and K. Pakdaman, Phys. Rev. E. 63(3), 031911, (2001).
Mean values: uu yy
Variances: 2)()( uuuVarSu 2)()( yyyVarS y
Covariance: ))((),( yyuuyuCovC
The polynomials F(u) and G(u) are Taylor expanded aound the meanvalues of u and y. By differentiating the mean values, variances and the covariance and using the stochastic dynamical equations, we obtain:
dtdu
y
Ref.: M.G. Pedersen and M.P. Sørensen, SIAM J. Appl. Math., 67(2), pp.530-542, (2007).
ydtud
CuFSuGyuFzuGyuFdtyd
u )('))('')(''(21
)()(
Cdt
dSu 2
22 6)))(')('()((2 uyy aSCuGyuFSuF
dt
dS
CaSCuFSuGyuFSdtdC
uuy 3)())(')('(
The Fokker-Planck equation
Probability distribution function:
Fokker-Planck PDE:
),,( tyuP
LPtP
with the operator:2
22
2))()((
yzuGyuF
yy
uL
The adjoint operator is:2
22
2))()((
yyzuGyuF
uyLadj
Example
The variance:
LPdudyuududytyuPuutt
Su
22 )(),,()(
We have used the Gaussian joint variable theorem:
dudytyuPuuuuSu ),,()()( 22
Cyyuuuuu
yuuLadj 2))((2)()( 22
01 X
423143214321 XXXXXXXXXXXX
3241 XXXX
0321 XXX02 X
Analysis compared to numerical results
Mathematical model for coupled β-cells
j
jiijATPKsKCai vvgIIII
dt
dvC )()(
Coupling to nearest neighbours.
Coupling constant: ijg
Gap junctions between neighbouring cells
Ref.: A. Sherman, (Eds. Othmar et al), Case studies in mathematical modelling, ecology physiology and cell biology, Prentice Hall (1996), pp.199-217.
The gating variables
))(( CaiiCaCa vvvmgI
The gating variables obey.
Calcium current:
Potassium current: ))(( KiKK vvtngI
)()()( KiATPKATPK vvgI ATP regulated potassium current:
Slow ion current: ))(( Kiss vvtsgI
n
nvndtdn
)(
s
svsdtds
)(
)/)(exp(11
)(xxi
i svvvxx
snmx ,,
Glycose gradients through Islets of Langerhans
Ref.: J.V. Rocheleau, et al, Microfluidic glycose stimulations … , PNAS, vol 101 (35), p12899 (2004).
Coupling constant:
Glycose gradients through Islets of Langerhans. Model.
pSipSg ATPK 1)1(120)( Ni ,...,2,1
Continuous spiking for: pSg ATPK 90)(
Bursting for: pSgpS ATPK 16290 )(
Silence for: )(162 ATPKgpS
Note that 43i corresponds to pSg ATPK 162)(
Wave blocking
Units tkt tphys uku uphys msgck Cat 3.5/ mVsk mu 12
Ref.: M.G. Pedersen and M.P. Sørensen, Jour. of Bio. Phys., Special issue on Complexity in Neurology and Psychiatry, Vol. 34 (3-4), pp 425-432, (2008).
PDE model. Fisher’s equation
Continuum limit of
)2(),( 11 iiiciii vvvgsvF
dt
dv
Is approximated by the Fisher’s equation xxt uaufu );(
where )1)(();( uauuauf
2exp1
1),(
00 vtxx
txuSimple kink solution 2/)21( av
Velocity:
Ref.: O.V. Aslanidi et.al. Biophys. Jour. 80, pp 1195-1209, (2001).
Perturbed Fisher’s equation
Collective coordinate approach
)();( xguaufu xxt
),()( txuU
Insertion into the perturbed Fisher’s equation gives
)(),(),(),(),( 200100100 oaUfaaUfUaUfaUf au
Introduce:
)()( 10 xaaxa with
)(tx
)(),(''')(' xgaUfUUt
10 UUU )(')(' 10 tvt
Perturbed Fisher’s equation
Insertion into the perturbed Fisher’s equation and collecting terms of the same order of ε gives
),(''' 00000 aUfUUv
Note that
gaUfaUUaUfUvUUL au ),(''),(''' 001011001011
))((11 taa ))(( tgg
Solution condition (Fredholm’s theorem)
Adjoint operator 0WLadj )(')( 0
0 UeW v
Orthogonality condition
0,'' 011 WgUfa a and henceWU
WgWfa a
,'
,,'
0
11
Example )()( 10 xaaxa with )2/exp()(1 xxa
The integrals becomes B- and Г- functions and the final result is
1
23
)2/exp(2'0
0 av
Solution
0
2
0
0
1ln2)(vc
evc
tt
v
1
23
20a
c
with 0g
Numerical simulations and comparison to analytic result
Summary
1) Noise in the ion gates reduce the burst period.
2) Ordinary differential equations for mean values, variances and co-variances. These equations are approximate.
3) Wave blocking occurs for spatial variation of the ATP regulated potassium ion channel gate.
4) Gap junction coupling leads to enhanced excitation of otherwise silent cells
5) The homoclinic bifurcation is treated using the stochastic Melnikov function method. Shinozuka representation of Gaussian noise. Heuristic arguments.
Acknowledgements: The projet has been supported by the BioSim EU network of excelence.
Ref.: M. Shinozuka, J. Sound Vibration 25, pp.111-128, (1972). M. Shinozuka, J. Acoust. Soc. Amer.49, pp357-367, (1971).