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. β-cell Excitation Dynamics Morten Gram Pedersen b) and Mads Peter Sørensen a) 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 , (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 , (2008). Rex workshop. February 18-19, 2009, Carlsberg Academy, Copenhagen.

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. Ref.: M. Panarowski, SIAM J. Appl. Math., 54 pp , (1994). Voltage across the cell membrane : Gating variable : Gaussian gate noise term : where Slow gate variable : Ref.: A. Sherman, (Eds. Othmar et al), Case studies in mathematical modelling, ecology physiology and cell biology, Prentice Hall (1996), pp

The influence of noise on the beta-cell bursting phenomenon. Ref.: M.G. Pedersen and M.P. Sørensen, SIAM J. Appl. Math., 67(2), pp , (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. Where the polynomials are given by Parameters:

Location of the left saddle-node bifurcation. The Gaussian method. Ref.: S. Tanabe and K. Pakdaman, Phys. Rev. E. 63(3), , (2001). Mean values: Variances: Covariance: The polynomials F(u) and G(u) are Taylor expanded aound the mean values of u and y. By differentiating the mean values, variances and the covariance and using the stochastic dynamical equations, we obtain:

Ref.: M.G. Pedersen and M.P. Sørensen, SIAM J. Appl. Math., 67(2), pp , (2007).

The Fokker-Planck equation Probability distribution function: Fokker-Planck PDE: with the operator: The adjoint operator is:

Example The variance: We have used the Gaussian joint variable theorem:

Analysis compared to numerical results

Mathematical model for coupled β-cells Coupling to nearest neighbours. Coupling constant: 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

The gating variables The gating variables obey. Calcium current: Potassium current: ATP regulated potassium current: Slow ion current:

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. Continuous spiking for: Bursting for: Silence for: Note thatcorresponds to

Wave blocking Units 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 , (2008).

PDE model. Fisher’s equation Continuum limit of Is approximated by the Fisher’s equation where Simple kink solution Velocity: Ref.: O.V. Aslanidi et.al. Biophys. Jour. 80, pp , (2001).

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 , (1972). M. Shinozuka, J. Acoust. Soc. Amer.49, pp , (1971).