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Stochastic and Coherent Dynamics of Single and Coupled β-cells

This workshop focuses on the dynamics of β-cells, specifically the polynomial model and gate noise, the influence of noise, and the phenomenon of wave block due to glucose gradients.

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Stochastic and Coherent Dynamics of Single and Coupled β-cells

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  1. Stochastic and Coherent Dynamics of Single and Coupled β-cells. 3rd Toyota CRDL Workshop, Mathematical Methods in Complex Systems. October 15-17, 2008, Bastide Relais de la Magdelaine, France. Mads Peter Sørensen a) and Morten Gram Pedersen b) a) DTU Mathematics, Lyngby, Denmark, b) Dept. of Information Engineering, University of Padova, Italy Content: Dynamics of beta-cells. Polynomial model and gate noise. The influence of noise. Phenomenological. The G-method. Wave block due to glucose gradients. Summary. 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, To appear in Jour. of Bio. Phys. Special issue on Complexity in Neurology and Psychiatry, (2008).

  2. The β-cell Ion channel gates for Ca and K B

  3. Mathematical model for single cell dynamics Topologically equivalent and simplified models. Polynomial model with Gaussian noise term on the gating variable. Voltage across the cell membrane: Gating variable: Slow gate variable: Gaussian gate noise term: where Ref.: M. Panarowski, SIAM J. Appl. Math., 54 pp.814-832, (1994). Ref.: A. Sherman, (Eds. Othmar et al), Case studies in mathematical modelling, ecology physiology and cell biology, Prentice Hall (1996), pp.199-217.

  4. 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.530-542, (2007).

  5. Dynamics and bifurcations

  6. Dynamics and bifurcations Ref.: E.M. Izhikevich, Neural excitability spiking and bursting, Int. Jour. of Bifurcation and Chaos, p1171 (2000).

  7. Sketch of the homoclinic bifurcation

  8. Differentiating the first equation above with respect to time leads to. Where the polynomials are given by Parameters:

  9. Location of the left saddle-node bifurcation. The G-method. 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.: S. Tanabe and K. Pakdaman, Phys. Rev. E. 63(3), 031911, (2001).

  10. Ref.: M.G. Pedersen and M.P. Sørensen, SIAM J. Appl. Math., 67(2), pp.530-542, (2007).

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

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

  13. Analysis compared to numerical results

  14. Mathematical model for coupled β-cells Gap junctions between neighbouring cells Coupling to nearest neighbours. Coupling constant: Ref.: A. Sherman, (Eds. Othmar et al), Case studies in mathematical modelling, ecology physiology and cell biology, Prentice Hall (1996), pp.199-217.

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

  16. Glycose gradients through Islets of Langerhans Ref.: J.V. Rocheleau, et al, Microfluidic glycose stimulations … , PNAS, vol 101 (35), p12899 (2004).

  17. Glycose gradients through Islets of Langerhans. Model. Continuous spiking for: Bursting for: Silence for: Coupling constant: Note that corresponds to

  18. Wave blocking Units

  19. Glycose gradients through Islets of Langerhans

  20. PDE model. Fisher’s equation Continuum limit of Is approximated by the Fisher’s equation where Velocity: Simple kink solution Ref.: O.V. Aslanidi et.al. Biophys. Jour. 80, pp 1195-1209, (2001).

  21. Perturbed Fisher’s equation with Collective coordinate approach Insertion into the perturbed Fisher’s equation gives Introduce:

  22. Perturbed Fisher’s equation Insertion into the perturbed Fisher’s equation and collecting terms of the same order of ε gives Note that Solution condition (Fredholm’s theorem) Adjoint operator

  23. Orthogonality condition and hence Example with The integrals becomes B- and Г- functions and the final result is Solution

  24. Numerical simulations and comparison to analytic result

  25. Summary Noise in the ion gates reduce the burst period. Ordinary differential equations for mean values, variances and co-variances. These equations are approximate. Wave blocking occurs for spatial variation of the ATP regulated potassium ion channel gate. Gap junction coupling leads to enhanced excitation of otherwise silent cells 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.

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