We have two sets of differential equations for neuron dynamics and astrocyte Ca2+ oscillations but with different time scales (Di Garbo 2009). Is it correct that for simulation of these sets simultaneously you must use integrators with 1000 coefficient for rescaling one set?
Not sure if I interpret your question correctly, but your equation system is probably stiff. In MATLAB, try e.g. ode15s: http://www.mathworks.se/help/matlab/ref/ode15s.html
Try to examine separately the fast and the slow subsystems (see the perturbation theory). Your integrator has to work for the fast subsystem as well.
There are dimension reduction methods for time scales systems (slow-fast system) which can be useful for this kind of problems. Try, for instance,
P. Auger, R. Bravo de la Parra, J.C. Poggiale, E. Sánchez, L. Sanz, Aggregation methods in dynamical systems variables and applications in population and community dynamics, Physics of Life Reviews, 5(2):79-105, 2008.
http://www2.uah.es/rafael_bravo/Investigacion/Publicaciones/2008-PLREV.pdf
These are all the MATLAB procedures for integrating Stiff ODE set: (1) ode15s, (2) ode23s, (3) ode23t, (4) ode23tb with the relative references:
(1) Shampine, L. F. and M. W. Reichelt, The MATLAB ODE Suite, SIAM Journal on Scientific Computing, Vol. 18, 1997, pp 1-22.
(2)Shampine, L. F., M. W. Reichelt, and J.A. Kierzenka, Solving Index-1 DAEs in MATLAB and Simulink, SIAM Review, Vol. 41, 1999, pp 538-552.
(3)Shampine, L. F. and M. E. Hosea, Analysis and Implementation of TR-BDF2, Applied Numerical Mathematics 20, 1996.
(4)Bank, R. E., W. C. Coughran, Jr., W. Fichtner, E. Grosse, D. Rose, and R. Smith, TransientSimulation of SiliconDevices and Circuits, IEEE Trans. CAD, 4 (1985), pp 436-451.
they should be helpful to solve your problem
Dear Fabio
Thanks for your answer. problem is that. all of these equations are related to each other and should be solve together by using the same method. it is impossible to solve each of them separately by using different procedures
Dera Hossein,
you don't need to separate the equations with different time scale in order to solve them. A method for stiff ordinary differential equation set will tune the integration time step differently for each equation in order to meet the accuracy required. So, in Matlab, you have just to solve a Stiff ODE set normally by selecting one of the procedures suitable to integrate stiff ODE sets.
I had the similar problem. It was necessary to have a real-time model containing aircraft motion (more inertial -slow ) and its wheels revolving (much less inertial - fast). I solved this problem with finding the analytical "semi"solution for wheels in Maple and including it into "slow" model of aircraft. That "semi solution" contained "free" members for non linear functions etc. I don't remember clearly - 10 years ago and it was not described somewhere.. But it worked.
of course, you can approximate the ODE set by decoupling the equations, but my suggestion is firstly try a MATLAB numerical procedure for stiff ODE set that could solve easily your problem
Hi, Hossein,
Did you get an answer? I have the same issue. I put a question in the Matlab forum, but no answers...
Thanks.
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