Research

My research interests lie somewhere between mathematical biology and combinatorics.

AMPA Receptor Trafficking and Synaptic Plasiticity
RNA Secondary Structure and k-Noncrossing Arch Diagrams
Meanders and Noncrossing Partitions

AMPA Receptor Trafficking and Synaptic Plasiticity

Dr. Paul Bressloff, of the Department of Mathematics at the University of Utah, and I model AMPA receptor trafficking and its role in synaptic plasticity (LTP/LTD, STDP, homeostatic plasticity, etc.).

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. To appear (2008). 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 news:

Talks:

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

RNA Secondary Structure and k-Noncrossing Arch Diagrams

The combinatorics of RNA secondary structure are of particular interest to me since it involves mathematics of the "noncrossing" type (see the heading "Meanders and Noncrossing Partitions").

Talks:

3/1/08, IGERT recruitment, Univ. of Utah: pdf
2/26/08, Graduate seminar, Univ. of Utah: 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)