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).