Stehanie R. Jones (2001) Rhythms in the neocortex and in CPG neurons: A dynamical systems analysis. Ph.D thesis View as Postscript File .
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Stephanie R. Jones, David J. Pinto, Tasso J. Kaper and N. Kopell (2000) Alpha Frequency Rhythms Desychronize Over Long Cortical Distances: A Modeling Study J. Comp. Neurosci. 9(3):271-291.
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Abstract Neocortical networks of excitatory and inhibitory neurons can display alpha-frequency rhythms when an animal is in a resting or unfocused state. Unlike some gamma and beta-frequency rhythms, experimental observations in cats have shown that these alpha-frequency rhythms need not synchronize over long cortical distances. Here, we develop a network model of synaptically coupled excitatory and inhibitory cells to study this asynchrony. The cells of the local circuit are modeled on the neurons found in layer V of the neocortex where alpha-frequency rhythms are thought to originate. Cortical distance is represented by a pair of local circuits coupled with a delay in synaptic propagation. Mathematical analysis of this model reveals that the h and T currents present in layer V pyramidal (excitatory) cells not only produce and regulate the alpha-frequency rhythm, but they also lead to the occurrence of spatial asynchrony. In particular, these inward currents cause excitation and inhibition to have non-intuitive effects in the network, with excitation delaying and inhibition advancing the firing time of cells; these reversed effects create the asynchrony. Moreover, increased excitatory to excitatory connections can lead to further desynchronization. However, the local rhythms have the property that, in the absence of excitatory to excitatory connections, if the participating cells are brought close to synchrony (e.g. by common input) they will remain close to synchrony for a substantial time.

Stephanie R. Jones, Tasso J. Kaper, and Nancy Kopell, Intersegmental Phase Lag and Duty Cycle in a CPG are Controlled by Intrasegmental Inhibition, in preparation.

Abstract Central Pattern Generator neurons of the crayfish swimmeret system have been experimentally observed to display rhythmic electrical activity at low frequencies (1-3 Hz) during locomotion. The oscillatory electrical recordings from neighboring segments of the 4 or 5 segment swimmeret system are 90 degrees out of phase. This phase lag, as well as the duty cycle of individual neurons, persist as the frequency of the oscillation, and hence the speed of locomotion, changes . Motivated by these experimental results, here we examine the swimmeret CPG system to understand what controls the duty cycle and inter-segmental phase lag. A modified version of an existing model that captures the dynamics of the swimmeret network provides the basis for analysis. In this model, the intrasegmental network consists of three mutually inhibitory neurons whose synaptic connections are slow and strong and whose oscillations occur with two time scales. The behavior of the full network is represented by connecting one segment to a nearest neighbor segment using intersegmental coupling that is fast and weak. Analysis reveals that the slow rate of decay of the strong inhibition within a local segment is the controlling factor of both individual cell duty cycle and intersegmental phase lag. The dynamical systems theory used to conclude this result fall into the category of singular perturbation theory, which is often used to understand the behavior of systems containing two time scales. Furthermore, the general mathematical theory of weakly coupled oscillators gives insight into why particular anatomical configurations of the unknown intersegmental coupling give appropriate phase lags, while other related configurations do not. The weak coupling theory also clarifies why other parameters, such as relative intersegmental coupling strength, can affect intersegmental phase lags alone.

David J. Pinto, Stephanie R. Jones, Tasso J. Kaper and Nancy Kopell, Neocortical Rhythms: Transitions in Frequency and Synchrony, working title in preparation.

Abstract There are many physiologically based parameters that can create transitions between rhythms of different frequencies in the cortex as behavior changes. By modifying a previously developed neocortical model, some of the possible mechanisms to switch between alpha and beta or gamma- frequency rhythms can be uncovered. It is commonly known that arousal and attention in animals is associated with an overall increase in activity in areas of cortex that are participating. Thus, to mimic changes in state, the overall level of excitation in the modeled system is varied in two ways: (1) by altering the strength of an applied constant depolarizing current and (2) by varying parameters that mimic cholinergic modulation in the system. Another modification is the addition of an AHP-current (M-current) to each of the excitatory cells, since AHP-currents are known to be important in the generation and synchronization of beta-frequency rhythms. With these modifications, the model captures the behavior of different cell assemblies that may control the neocortical rhythm at different frequencies. Moreover, alterations in the geometry of a one-dimension map which captures the dynamics of synchrony over long-distances can reveal possible mechanisms for changes in other rhythms abilities to synchronize.

Stephanie R. Jones and Tasso J. Kaper, Finding homoclinic bifurcations in the membrane potential of bursting cells using Melnikov's Method, working title in preparation.

Abstract In this work, the question of what controls the size of the plateau fraction of bursting pancreatic beta cell is addressed. The plateau fraction is defined to be the ratio of the time spent in the active phase of a cells bursting cycle to the period of the cycle. This fraction is physiologically relevant because it is known to be correlated to the rate of release of insulin in the cells, . An essential factor in determine the size of a plateau fraction is determining when the active phase of the burst cycle begins and ends. Here we construct an analytical argument for the numerical and asymptotic results of G. De Vries and R.M. Miura who use a dynamical systems theory technique to determine when the end of the active phase of the bursting activity occurs.

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