Electrical activity underlies the basic functions of neural systems. This page describes my research on quantitative models for neurons and small neural networks. Most of this research has been done collaborativley with the laboratory of Ronald Harris-Warrick. The biological system we study is the
stomatogastric ganglion of a rock lobster, Panulirus interruptus.

Ions flow across the membranes of neurons through channels, creating an electrical current.
Hodgkin and Huxley developed procedures for measuring the voltage dependence of channel activation and inactivation and used these procedures to formulate differential equations that
describe how membrane potential changes with time. The Hodgkin-Huxley model for the squid giant axon has been modified and extended to account for the electrical properties of many other systems.

A comprehensive survey of models of the stomatogastric nervous system can be found in the book, Dynamic Biological Networks, edited by Harris-Warrick, Marder, Selverston and Moulins (MIT Press, 1992). The STG consists of approximately thirty neurons that coordinate movement in the crustacean foregut. Neurons within the system display complex rhythmic oscillations in which the membrane potential rises and falls, generating bursts of action potentials during part of these oscillations. Individual neurons and smaller subnetworks can be electrically isolated from the remainder of the network and manipulated in vitro. The modulatory properties of many substances that alter  rhythmic properties of the neurons and network have been tested. Gating properties of channels within the neurons have been measured and assembled into Hodgkin-Huxley like models.

The focus of my research has been to study the properties of these models and relate them to observed rhythmic properties of the system. Fitting models to data is extraordinarily difficult. The models are highly nonlinear and the effects of varying parameters on complex oscillations is difficult to predict. Moreover, the number of parameters is large. Model of a single neuron contain dozens of parameters, many of which cannot be measured directly. Simulation of the models displays their dynamics for particular sets of parameters, but interactivley "steering" the system "by hand" in response to simulation results has often been inadequate as a means of finding parameters that give even qualitative fits to multiple data sets. Consequently, we have invested effort in developing  more powerful computational tools for exploring the dynamics of the models.

We have used automated techniques for computing equilibrium point bifurcations in the models as a substrate for producing parameter space maps that show how the qualitative properties of the cell change with varying parameters. The diagram below is from the paper The Dynamics of a Conditionally Bursting Neuron, Philosophical Transactions of Royal Society, 341, 345-359, 1993 (with Shay Gueron and Ronald Harris-Warrick):
 

It shows a parameter space map for a model of the AB cell in the STG. The axes on the diagram are parameters for the maximum conductance of two potassium channels. As these parameters vary, the  model neuron enters several different regions in which there is a stable equilibrium (Quiescent), in which trajectories tend to a rapid periodic oscillation with action potentials and without bursts (Tonic action potentials), in which the neuron has slow oscillations without action potentials (Slow Oscillations) and in which there are complex oscillations with bursts of action potentials (Bursts).  Shown below are time traces of membrane potential for the last three  of these states. The first trace shows one second of data, the final two show five seconds of data.
 

In the parameter space map, the points marked with  purple "+" and blue "." were computed with algorithms that directlly locate equilibrium point bifurcations without using numerical integration. The Hopf bifurcation algorithms that we developed in the course of this work are described in the papers Computing Hopf Bifurcations I, SIAM J. Num. Anal., SIAM J. Num. Anal., 34, 1-21, 1997 (with Mark Myers and Bernd Sturmfels) and  Computing Hopf Bifurcations II, SIAM J. Sci. Comp, SIAM J. Sci. Comp, 17, 1275-1301, 1996. (with Mark Myers). The information obtained from this parameter space map was used to fit the model to data about the modulatory effects of the channel blocker 4-AP on an isolated AB cell. For details, refer to the paper cited above.

Some more recent work on neuronal models is described on the web page for multiple time scale systems, and more references are included in the publications list.