Research

Protein Trafficking and Synaptic Plasiticity
Behavior of solutions of nonautonomous master equations
Meanders and Noncrossing Partitions

Protein Trafficking and Synaptic Plasiticity

Dr. Paul Bressloff, of the Department of Mathematics at the University of Utah, and I have developed a number of models of AMPA receptor trafficking and its role in synaptic plasticity (LTP/LTD, STDP, homeostatic plasticity, etc.). More recently we have begun to model the trafficking of other proteins involved in plasticity, e.g. CaMKII.

Papers:

B.A. Earnshaw and P.C. Bressloff. Biophysical model of AMPA receptor trafficking and its regulation during long-term potentiation/long-term depression. J. Neurosci. 26 12362-12373 (2006). abstract
P.C. Bressloff and B.A. Earnshaw. Diffusion-trapping model of receptor trafficking in dendrites. Phys. Rev. E. 75 041915 (2007). abstract
P.C. Bressloff, B.A. Earnshaw and M.J. Ward. Diffusion of protein receptors on a cylindrical dendritic membrane with partially absorbing traps. SIAM J. Appl. Math. 68 1223-1246 (2008). abstract
B.A. Earnshaw and P.C. Bressloff. Modeling the role of lateral membrane diffusion in AMPA receptor trafficking along a spiny dendrite. J. Comput. Neurosci. 25 366-389 (2008). abstract m-file
P.C. Bressloff and B.A. Earnshaw. A dynamic corral model of receptor trafficking at a synapse. Biophys. J. 96 1786-1802 (2009). abstract
B.A. Earnshaw and P.C. Bressloff. A diffusion-activation model of CaMKII translocation waves in dendrites. J. Comput. Neurosci. To appear. abstract
Dissertation: B.A. Earnshaw. Biophysical models of alpha-amino-3-hydroxy-5-methyl-4-isoxazole-propionic acid receptor trafficking in dendrites. pdf

In the press:

Talks:

10/8/09, UBM seminar, Michigan State University: pdf
8/25/09, MBI Workshop for Young Researchers in Mathematical Biology, Columbus, OH: pdf
6/2/09, NJIT Conference on Frontiers in Applied and Computational Mathematics, Newark, NJ: pdf
5/19/09, SIAM Conference on Applications of Dynamical Systems, Snowbird, UT: pdf
3/26/09, Department of Mathematics, BYU: pdf
8/4/08, SIAM Conference on the Life Sciences, Montreal, Quebec: pdf
6/25/08, GRC on Theoretical Biology and Biomathematics, Il Ciocco, Italy: pdf
5/23/08, Neuroscience group meeting, Univ. of Utah: pdf
3/1/08, IGERT recruitment, Univ. of Utah: pdf
11/05/07, Society for Neuroscience annual meeting, San Diego, CA: pdf
10/25/07, Department of Mathematics, BYU-Idaho: pdf
6/7/07, Department of Biomathematics, UCLA: pdf
3/3/07, IGERT recruitment, Univ. of Utah: pdf
10/17/06, Graduate seminar, Univ. of Utah: pdf
10/6/06, Physiologie Cellulaire de la Synapse, Univ. Bordeaux 2: pdf
9/16/06, Department of Mathematics, BYU: pdf

Posters:

Diffusion model of AMPA receptor trafficking and expression of LTP/LTD: pdf
Model of AMPA receptor trafficking across multiple dendritic spines: pdf

Behavior of solutions of nonautonomous master equations

Dr. Jim Keener, of the Department of Mathematics at the University of Utah, and I have been studying the behavior of solutions of nonautonomous master equations, i.e., equations governing the evolution of probability density functions of jump processes.

Papers:

B.A. Earnshaw and J.P. Keener. Global asymptotic stability of solutions of nonautonmous master equations. Submitted.
B.A. Earnshaw and J.P. Keener. Invariant manifolds of binomial-like nonautonomous master equations. Submitted.

Talks:

10/17/09, AMS Central Section Meeting, Baylor University: pdf
10/8/09, Mathematical biology seminar, University of Michigan: pdf
3/21/09, MAA Intermountain Section Meeting, BYU: pdf
1/28/09, Math Biology Seminar, Univ. of Utah: pdf

Posters:

5/19/09, SIAM Conference on Applications of Dynamical Systems, Snowbird, UT: pdf

Meanders and Noncrossing Partitions

Dr. Reinhard Franz, of the Department of Engineering and Technology at BYU, and I have studied meanders (sequence id A005315), especially their partially-ordered set structure as pairs of noncrossing partitions. One of our results is an algorithm which constructs meanders recursively by rank, beginning with the supremum meander. This algorithm is novel in two ways: (1) it constructs meanders by rank, and (2) the algorithm constructs all meanders of a given order independent of lower orders.

My Masters thesis was an attempt at understanding the exterior structure of noncrossing partitions in hopes that this may provide insight into the recurrence of meanders of lower orders in higher orders (this has never come to anything though).

Papers:

R.O.W. Franz and B.A. Earnshaw. A Constructive Enumeration of Meanders. Ann. Combin. 6 7-17 (2002). abstract
Thesis: B.A. Earnshaw. Exterior Blocks and Reflexive Noncrossing Partitions. (2003). (pdf, ps)

Notes, etc.:

B.A. Earnshaw. Exterior Blocks of Noncrossing Partitions. (2003). (pdf, ps)
G. Kreweras (English translation by B.A. Earnshaw). On the Noncrossing Partitions of a Cycle. (2003). (pdf, ps)
B.A. Earnshaw. Combinatorial Aspects of Partially Ordered Sets. (2003). (pdf, ps)