#### Abstract

The main objective of this project is to model the neuron activity in a sea lamprey spinal cord. Currently, the model is only forced at the ends. Eventually we hope to apply forcing at any point along the spinal cord. There are many parameters that we will change in an effort to find stable structures of the differential equations.

This project will eventually be integrated into a larger model that characterizes lamprey locomotion. This project builds on the research of Dr. Previte and Behrend student Jeremy Sopko.

#### Introduction

This project is part of a larger project that aims to understand sea lamprey locomotion. Normally brown or gray, the sea lamprey is an aquatic parasite which grows up to 90 cm long and is typically found on the Atlantic coasts of Europe and North America. This project will use coupled differential equations to model the neural activity in a sea lamprey spinal cord.

I will analyze and test generalizations of the current model. The spinal cord initiates locomotion, thus our model is a logical first step to a comprehensive model. The current model involves linear chains of neurons that are coupled together in a nearest neighbor fashion with external forcing at the ends of the chain, see:*Kopell, N. & Ermentrout, B. [1986] Symmetry and phase locking in chains of weakly coupled oscillators, Commun. Pure Appl. Math. 39, 623-660*.

This research is going to allow variable coupling among the neurons, rather than just nearest neighbor. External forcing will also be allowed to occur at any neuron along the spinal cord, as opposed to just the ends. The goal is to analyze the stable structures of the resulting differential equations and characterize them in terms of external forcing. This analysis will follow the pattern set out in the paper:*Rebecca Ellison, Virginia Gardner*, Joel Lepak, Meghan O'Malley, Joseph Paullet, Joseph P. Previte, Beth Ann Reid* and Katy Vizzard*, Pattern Formation in Small Arrays of Locally Coupled Oscillators, Int. J. of Bifurcation and Chaos 15 2283-2293. (2005), *where a similar model was analyzed. The ultimate goal is to integrate this research into a larger model that characterizes lamprey locomotion.

#### Problem Statement and Research Plan

The goal of this project is to propose and analyze a model of the neural activity in a sea lamprey spinal cord. This data will be the first step in understanding the locomotion of a sea lamprey. As previously mentioned, this model will be a generalized form of current models. The fact that the system is relatively large (50 dimensional) limits analytical analysis to a few specific cases. This leads us to a numerical approach. We have observed three possible behaviors based on initial conditions and parameters; synchrony, pseudo-synchrony, and nonsynchrony.

In order to efficiently analyze data we will write a program to automate the process. This will involve XPP, C++, and Gawk. We plan to create a computer program to determine how each parameter affects the stability of the system. To do this I will first have to understand how the current model works. The stable structures that we find should correspond to the experimental data given to us by biologists from the University of Maryland.

The collaborative effort with Maryland will be two way. As we see stable structures in our model, we will report this data to the biological researchers who will confirm our results in the actual spinal cord, and as certain behaviors are seen in the spinal cord, we will attempt to see similar behavior in he model, this process may involve model fitting.

#### Anticipated Outcome

The goal will be to analyze the data then present my findings at the Undergraduate Research Conference in the spring and at the regional MAA meeting in the spring. The larger goal is to aid in the development of a fully integrated model tat describes the motion of the sea lampey.