Date of Award

Fall 2006

Document Type


Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Farzan Nadim

Second Advisor

Amitabha Koshal Bose

Third Advisor

Jorge P. Golowasch

Fourth Advisor

Robert M. Miura

Fifth Advisor

William Corson Hunter


Projection neurons shape the activity of many neural networks. In particular, neuromodulatory substances, which are often released by projection neurons, alter the cellular and/or synaptic properties within a target network. However, neural networks in turn influence projection neuron input via synaptic feedback. This dissertation uses mathematical and biophysically-realistic modeling to investigate these issues in the gastric mill (chewing) motor network of the crab, Cancer borealis. The projection neuron MCN1 elicits a gastric mill rhythm in which the LG neuron and INTl burst in anti-phase due to their reciprocal inhibition. However, bath application of the neuromodulator PK elicits a similar gastric mill rhythm in the absence of MCN 1 participation; yet, the mechanism that underlies the PK-elicited rhythm is unknown. This dissertation develops a 2-dimensional model that is used to propose three potential mechanisms by which PK can elicit a similar gastric mill rhythm. The network dynamics of the MCN 1-elicited and PK-elicited rhythms are also compared using geometrical properties in the phase plane. Next, the two gastric mill rhythms are compared using a more biophysically-realistic model. Presynaptic inhibition of MCN 1 is necessary for coordinating network activity during the MCN 1-elicited rhythm. In contrast, the PK-elicited rhythm is shown to be coordinated by a synapse that is not functional during the MCN 1-elicited rhythm.

Next, the gastric mill rhythm that is elicited by two coactive projection neurons (MCNl and CPN2) is studied. A 2-dimensional model is used to compare the network dynamics of the MCN 1-elicited and MCN 1 /CPN2-elicited gastric mill rhythms via geometrical properties in the phase plane. While the MCN 1-elicited rhythm requires the presence of reciprocal inhibition between INTl and the LG neuron, the MCN I /CPN2-elicited rhythm persists in the absence of this reciprocal inhibition, due to an inhibitory feedback synapse from INTl to CPN2 that changes the locus of coordination in the gastric mill rhythm. Next, the effect of a second feedback synapse, from the AB neuron to MCN 1, is shown to change the motor pattern of the MCN 1- and MCN1/CPN2-elicited rhythms. Finally, a third MCNI/CPN2-elicited rhythm is studied where the AB to MCN 1 feedback synapse only affects the LG burst phase of the rhythm.

Included in

Mathematics Commons