Date of Award

Spring 2011

Document Type


Degree Name

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


Mathematical Sciences

First Advisor

Amitabha Koshal Bose

Second Advisor

Denis L. Blackmore

Third Advisor

Jorge P. Golowasch

Fourth Advisor

Farzan Nadim

Fifth Advisor

Horacio G. Rotstein


The nervous system is one of the most important organ systems in a multicellular body. Animals, including human beings perceive, learn, think and deliver motion instructions through their nervous system. The basic structural units of the nervous system are individual neurons which constitute different neuronal networks with distinct functions. In each network, constituent neurons are coupled with different connection patterns, for example, some neurons send feed-forward information to the coupling neurons while others are mutually coupled. Because it is often difficult to analyze large interconnected feedback neuronal networks, it is important to derive techniques to reduce the complexity of the analysis. My research focuses on using the information of different feed-forward neuronal networks to infer the activity of feedback networks. To accomplish this objective, I use geometric analysis combined with numerical simulations for some typical neuronal systems to determine the activity of the feedback neuronal network in the context of central pattern generating networks.

In my study, I am interested in deriving reduced methods to understand the combined effect of short-term plasticity on the phase-locked activity of networks. I consider a network of two reciprocally coupled heterogenous neurons, A and B, with synaptic depression from neuron A to neuron B. Suppose we are given two pieces of feed-forward information, the effect of neuron A on the activity of neuron B in the feed-forward network of A entraining B and vice versa. Moreover, suppose these effects are not limited to the weak coupling regime. We have developed a method to combine these pieces of feed-forward information into a 2D map that predicts the activity phase of these two neurons when they are mutually coupled. The analysis of the map is based on certain geometric constructs that arise from each of the feed-forward processes. Our analysis has two parts corresponding to different intrinsic firing patterns of these two neurons. In the first part, we assume that neuron A is oscillatory, while neuron B is not. In the second part, both neurons A and B are assumed to be oscillatory. Both sets of assumptions lead to different feedback maps.

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Mathematics Commons