Here are some descriptions of things I am working on or have worked on in the past.
Bursting in Dynamical Systems
The square wave burster illustrated below is one of many different kinds of bursting oscillations. The bursting pattern is characterized by alternating active (oscillatory) and silent (nonoscillatory) phases.
The sawtooth pattern is that of a "slow" regulatory variable which modulates the fast bursting patterns. Square wave bursting is important since the pancreatic beta cells exhibit this type of electrical behavior. In particular, the rate at which the cells secrete insulin has been strongly correlated to the duration of the active phase. Such periodic oscillations are interesting dynamically since they represent multidimensional limit cycles which can be constructed using singular perturbation techniques (including averaging). The particular model describing the oscillation above has two "fast" variables and one "slow" variable. The periodic orbit of the associated autonomous system is therefore a closed orbit in R3. As with many systems which exhibit bursting, the differential equations contain a small parameter and the bursting solution(s) can be described asymptotically in terms of it's slow and fast subsystem (SS) and (FS), respectively. In the silent phase, the solution tracks a one dimensional slow manifold (curve) in R3. In the active phase, the solution tracks a different slow manifold (I call it the Averaged Fast Subsystem or AFS) derived using a family of period solutions of (FS). Escape from the active phase typically occurs through a homoclinic bifurcation which can be located using Melnikov theory in some models.
Classification of Bursters:
Below is a figure illustrating how bursters can be classified in terms of the fast parameters (each axis). The lines A-D separate regions describing i) where (FS) equilibria are stable, ii) what their locations are, iii) where Hopf points are, iv) their criticality and v) paremeter values for which homoclinic bifurcations of (FS) exist. Different types of bursters exist in regions bounded by these lines. The shaded region indicates parameter pairs where square-wave bursting (see above) solutions exist. For details, see the 1994 SIAM paper listed in my VITA. What is siginificant about this type of research is that not only are there many different types of bursting oscillations but they are all observed in electrophysiology experiments in nerves, endocrine system cells and muscle. That only two parameters are need to characterize all (relevant) types suggests to me that only a few genetic markers are needed for organisms to develop the correct "burster" with the right dynamics for each organism physiological subsystem. Bursters are thought to control dynamics on time scales of seconds within organisms. They are natural "clocks" and bridge the time barriers of nerve action potentials (msec) and endocrine system physiology (minutes-months).
Coupling of Bursters:
Of course most "electrically active" cells share current or are synaptically connected. Such electro-chemical communication results in different cell ensemble behavior. At left is a color-scale contour plot of numerical solutions of fast (upper) and slow (lower) variables in a PDE model for a pancreatic islet containing many diffusively coupled square-wave bursters. Coupling strength is large resulting in synchronous fast variable behavior as can be seen by the horizontal lines indicate local max of the active phase oscillations. Even though the coupling is strong, slow variables need not exhibit synchronous behavior (lower). This permits a possible mechanism for diabetic conditions (see SIAM J. Appl. Math., 58:1667-1687, 1998listed in my manuscript section for details). There are perhaps unlimited spatio-temporal dynamics possible and their functions are not well understood at present (save a few specific systems).
Below is a figure illustrating the dispersal of 9 phenotypes in a heterogeneous environment. Spatially averaged populations of phenotype "i" are plotted versus time. The dispersal rates increase in i. Thus, the diagram shows how the "slow" survive.
The model, analysis and numerics associated with the study are in "The evolution of slow dispersal rates: a reaction diffusion model", J. Dockery, V. Hutson, K. Mischaikow, M. Pernarowski. To appear in J. Math. Biol. Download Postscript File (680K)
"Recent" Progress: Below is a color contour of the same spatially averaged phenotypes in the above simulation except with the carrying capacity periodic in time t to simulate seasonal changes in the environment. In the figure, the spatial average of u_i(x,t), i=1,2..9, is located at x=x_i= 0.1*i. RED=large value, blue=small value. From the simulation, it appears that the slow disperser located at x_1 "wins" and approaches a periodic solution in the long term. Email for more details.
Glucose Diffusion in the Pancreatic Islet
Below is a figure illustrating the glucose concentration inside a pancreatic islet. The distribution represents the concentration inside beta-cells after 120 seconds in a bath concentration of 10mM glucose. For details concerning the model and numerical results, obtain the postscript version of the paper in Biophys J April 98 (above under "Papers").
Parameter Identification in Cell Models
Use experimental data and an algorithm to determine "best fit" parameters in a model of bursting in pancreatic B-cell models:Experimentally measured BEA in B-cell (10mM glucose)Perform a sliding window average:Numerically estimate derivative, then avergage then differentiate etc. The i-th row is the (i-1) rst derivative and its avergage (red).
Here's an example of how model parameters in an existing (SRK) model of beta-cell bursting activity can be altered to fit the measured potential. The predicted potential is in red, while the data is in blue data.