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Experimental and modeling efforts suggest that rhythms in the CA1 region of the hippocampus that are in the beta range (12-29 Hz) have a different dynamical structure than that of gamma (30-70 Hz). We use a simplified model to show that the different rhythms employ different dynamical mechanisms to synchronize, based on different ionic currents. The beta… (More)

Neuronal activity in a two-dimensional net is analyzed in the neighborhood of an instability. Bifurcation theory and group theory are used to demonstrate the existence of a variety of doubly-periodic patterns, hexagons, rolls, etc., as solutions to the field equations for the net activity. It is suggested that these simple geometric patterns are the… (More)

Hippocampal networks of excitatory and inhibitory neurons that produce gamma-frequency rhythms display behavior in which the inhibitory cells produce spike doublets when there is strong stimulation at separated sites. It has been suggested that the doublets play a key role in the ability to synchronize over a distance. Here we analyze the mechanisms by… (More)

- D G Aronson, G B Ermentrout, N Kopell, J E Marsden
- 1990

We investigate the interaction of a pair of weakly nonlinear oscillators (e.g. each near a Hopf bifurcation) when the coupling strength is comparable to the attraction of the limit cycles. Changes in amplitude cannot then be ignored, and there are new phenomena. We show that even for simple forms of these equations, there are parameter regimes in which the… (More)

Propagation of discharges in cortical and thalamic systems, which is used as a probe for examining network circuitry, is studied by constructing a one-dimensional model of integrate-and-fire neurons that are coupled by excitatory synapses with delay. Each neuron fires only one spike. The velocity and stability of propagating continuous pulses are calculated… (More)

Oscillations in many regions of the cortex have common temporal characteristics with dominant frequencies centered around the 40 Hz (gamma) frequency range and the 5-10 Hz (theta) frequency range. Experimental results also reveal spatially synchronous oscillations, which are stimulus dependent (Gray & Singer, 1987; Gray, Konig, Engel, & Singer, 1989; Engel,… (More)

We review a number of biologically motivated cellular automata (CA) that arise in models of excitable and oscillatory media, in developmental biology, in neurobiology, and in population biology. We suggest technical and theoretical arguments that permit greater speed and enhanced realism, and apply these to several classical examples of pattern formation.… (More)

- T L Williams, K A Sigvardt, N Kopell, G B Ermentrout, M P Remler
- Journal of neurophysiology
- 1990

1. This paper reports the results of an investigation of the basic mechanisms underlying intersegmental coordination in lamprey locomotion, by the use of a combined mathematical and biological approach. 2. Mathematically, the lamprey central pattern generator (CPG) is described as a chain of coupled nonlinear oscillators; experimentally, entrainment of… (More)

The problem of alignment of cells (or other objects) that interact in an angle-dependent way was described in Mogilner and Edelstein-Keshet (1995). In this sequel we consider in detail a special limiting case of nearly complete alignment. This occurs when the rotational diffusion of individual objects becomes very slow. In this case, the motion of the… (More)

- C S Peskin, G B Ermentrout, G F Oster
- 1994

Introduction Progressive enzymes are macromolecules which hydrolyze ATP while moving unidirectionally along a linear macromolecular " track ". Examples include motor molecules such as myosin, kinesin and dynein, RNA and DNA polymerases, and chaperonins. We propose a specific mechanical model for transduction of phosphate bond energy during ATP hydrolysis… (More)