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 collaboratively with the laboratory of Ronald Harris-Warrick. 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. The models described here follow the Hodgkin-Huxley paradigm. 

Stomatogastric Ganglion

Central pattern generators are neural networks that provide inputs to muscles. The stomatogastric ganglion (STG) of crustacea is the best characterized central pattern generator. The STG consists of approximately thirty neurons that coordinate movement in the crustacean foregut. This is a small enough network that individual neurons and their synaptic connections can be identified.  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. A survey of research on the stomatogastric nervous system can be found in the book, Dynamic Biological Networks, edited by Harris-Warrick, Marder, Selverston and Moulins (MIT Press, 1992).  

My research has used numerical methods of bifurcation theory to study the properties of models for neurons in the STG and to relate these to observed rhythmic properties of the system. Fitting models to data is extraordinarily difficult and has become a focal point for our resarch. Conductance based models of neurons are highly nonlinear and the effects of varying parameters on complex oscillations is difficult to predict. Moreover, the number of parameters is large. Realistic models 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.

Subsequent experimental work on the STG has raised new mathematical issues that we have pursued. Jack Peck conducted experiments with the LP cell to identify bifurcation mechanisms that separated spiking and non-spiking regions. Two of the possible bifurcations that involve interspike intervals increasing without bound are homoclinic bifurcations and saddle-node in cycle bifurcations. Theory predicts that there will be bistability of the system in the case of homoclinic bifurcation but not it in the case of saddle-node in cycle bifurcation.  His experiments displayed spike frequency adaptation. When the cell was stimulated to begin spiking with current injection, the length of the interspike intervals increased. A slower process in the cell was activated by the spiking, and in some circumstances the cell ceased firing spontaneously during the stimulation. This prevented us from observing directly the bifurcations we sought. Instead, we viewed the dynamics as having two time scales: a fast time scale of spiking and a slow time scale in which the properties of the spiking were altered. To model this system, Allan Willms and I introduced a  non-inactivating outward current to a model cell that activated on the slow time scale. This current was sufficient to cause the model cell to display spike frequency adaptation, and in some cases to undergo the "death of periodicity." Using methods of geometric singular perturbation theory, we demonstrated that we could use measurements of interspike intervals to distinguish when the death of periodicity in the model cell was due to homoclinic or saddle-node bifurcation. We also discovered an additional dynamical mechanism that led to the death of periodicity, namely subcritical Hopf bifurcations. Analysis of the experimental data was inconclusive in associating the death of periodicity to a specific mechanism due to noise in the system. Nonetheless, the data motivated our model studies that led to new mathematical discoveries about global bifurcations.

The Harris-Warrick laboratory did extensive work correlating the expression of transient potassium A channels with their conductance and gating properties in individual cells of the STG pyloric circuit. These channels both activate and inactivate. The standard procedures for fitting voltage clamp data to channels that inactivate is to assume that the activation is complete before appreciable inactivation has occurred. Allan Willms et al. investigated the degree to which the fits depend upon this assumption. We found that with a 25:1 ratio in time constants for inactivation and activation, that the prevailing "disjoint" method for estimating channel conductance underestimates it conductance by approximately 20%. Improved methods based upon least squares estimation of residuals were applied to differential equations models for voltage clamp data of these channels. The methods were used to correlate observations of fluorescence of different classes of STG neurons labelled with monoclonal antibodies to shal protein and estimates of maximal conductance of A channels in these neurons.

I a studying neural networks in mouse spinal cord that generate rhythmic movements of hind legs in collaboration with Ronald Harris-Warrick and Ole Kiehn, Karolinska Institut, Sweden. These networks are called Central Pattern Generators or CPGs. Preparations of isolated spinal cord of rats and mice have become a model system for electrophysiological investigations of CPGs. The goal of our work is to discover principles underlying the left-right coordination of walking by developing and analyzing dynamical models of these networks. My group undertakescomputational investigations of models based upon data from experiments in the Harris-Warrick and Kiehn labs. 

Isolated spinal cords of neonatal rats and mice are the first mammalian preparation to give good access for advanced electrophysiological investigation of the neural circuits that control walking. Interneurons whose axons cross the spinal cord and synapse on motoneurons or other interneurons on the opposite side of the cord will be key elements in the models and experimental investigations.Moreover, we are using new genetic techniques that are likely to produce rapidly increasing information about the physiology and anatomy of these networks. 

The computer models we are studying are coupled cell systems of differential equations for membrane currents, whose structure incorporates what is known about the spinal cord. Connectivity of the network is now being studied in the laboratory, and we plan to conduct experiments to measure the physiological properties of neurons and their synapses that are needed to parametrize these conductance based models. This will provide the foundation for  quantitative comparison of the output of the model networks with that of the spinal cord. Our research also is 
developing new algorithms to estimate parameters that produce the best fit between rhythmic data from model and experimental observations. The parameter estimation algorithms will be used to iteratively refine the models to increase their fidelity further.