Date of Award

Spring 2014

Document Type


Degree Name

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


Mathematical Sciences

First Advisor

Amitabha Koshal Bose

Second Advisor

Farzan Nadim

Third Advisor

Victor Victorovich Matveev

Fourth Advisor

Horacio G. Rotstein

Fifth Advisor

Dirk Bucher


The behavior generated by neuronal networks depends on the phase relationships of its individual neurons. Observed phases result from the combined effects of individual cells and synaptic connections whose properties change dynamically. The properties of individual cells and synapses can often be characterized by driving the cell or synapse with inputs that arrive at different phases or frequencies, thus producing a feed-forward description of these properties. In this study, a recurrent network of two oscillatory neurons that are coupled with reciprocal synapses is considered. Feed-forward descriptions of the phase response curves of the neurons and the short-term synaptic plasticity properties are used to define Poincar´e maps for the activity of the network. The fixed points of these maps correspond to the phase locked modes of the network. These maps allow analysis of the dependence of the resulting network activity on the properties of network components.

Using a combination of analysis and simulations, how various parameters of the model affect the existence and stability of phase-locked solutions is shown. It is also shown that synaptic plasticity provides flexibility and supports phase maintenance in networks. Conditions are found on the synaptic plasticity profiles and the phase response curves of the neurons for the network to be able to maintain a constant firing period, while varying the relative activity phase of the neurons or vice versa. Synaptic plasticity is shown to yield bistable phase locking modes. These results are geometrically demonstrated using a generalization to cobwebbing for two dimensional maps. Type I neurons modeled with Morris-Lecar and Quadratic Integrate-and-Fire are used to estimate the predictive power of the analytical results; however, the results hold in general.

The properties of the Negative-Leak model are also studied; a recent conductance-based model which is obtained by replacing a regenerative inward current with a negative-slope-conductance linear current. The map methods are extended to analyze networking properties of Negative-Leak neurons by including burst response curves. Finally, geometric singular perturbation techniques are applied to analyze how a hyperpolarization-activated inward current contributes to the generation of oscillations in this model.

This work introduces a general method to determine how changes in the phase response curves or synaptic dynamics affect phase locking in a recurrent network which can be generalized to study larger networks.

Included in

Mathematics Commons