To Larry Sirovich on the occasion of his 70th birthday ABSTRACT We study a class of permutation-symmetric globally-coupled, phase oscillator networks on N-dimensional tori. We focus on the effects of symmetry and of the forms of the coupling functions, derived from underlying Hodgkin-Huxley type neuron models, on the existence, stability, and degeneracy of phase-locked solutions in which subgroups of oscillators share common phases. We also estimate domains of attraction for the completely synchronized state. Implications for stochastically forced networks and ones with random natural frequencies are discussed and illustrated numerically. We indicate an application to modeling the brain structure locus coeruleus: an organ involved in cognitive control. 1 Introduction and

The Anterior Burster (AB) neuron of the lobster stomatogastric ganglion displays varied rhythmic behavior when treated with neuromodulators and channel-blocking toxins. We introduce a channelbased model for this neuron and show how bifurcation analysis can be used to investigate the response of this model to changes of its parameters. Two dimensional maps of the parameter space of the model were constructed using computational tools based on the theory of nonlinear dynamical systems. Changes in the intrinsic firing and oscillatory properties of the model AB neuron were correlated with the boundaries of Hopf and saddle-node bifurcations on these maps. Complex rhythmic patterns were observed, with a bounded region of the parameter plane producing bursting behavior of the model neuron. Experiments were performed by treating an isolated AB cell with 4-aminopyridine which selectively reduces g A , the conductance of the transient potassium channel. The model accurately predicts the qualita...

Abstract. Do cortical neurons operate as integrators or as coincidence detectors? Despite the importance of this question, no definite answer has been given yet, because each of these two views can find its own experimental support. Here we investigated this question using models of morphologically-reconstructed neocortical pyramidal neurons under in vivo like conditions. In agreement with experiments we find that the cell is capable of operating in a continuum between coincidence detection and temporal integration, depending on the characteristics of the synaptic inputs. Moreover, the presence of synaptic background activity at a level comparable to intracellular measurements in vivo can modulate the operating mode of the cell, and act as a switch between temporal integration and coincidence detection. These results suggest that background activity can be viewed as an important determinant of the integrative mode of pyramidal neurons. Thus, background activity not only sharpens cortical responses but it can also be used to tune an entire network between integration and coincidence detection modes. Keywords: cerebral cortex, synaptic background, computational model, operating mode

Pyramidal cells in the electrosensory lateral line lobe (ELL) of weakly electric fish have been observed to produce high-frequency burst discharge with constant depolarizing current (Turner et al., 1994). We present a twocompartment model of an ELL pyramidal cell that produces burst discharges similar to those seen in experiments. The burst mechanism involves a slowly changing interaction between the somatic and dendritic action potentials. Burst termination occurs when the trajectory of the system is reinjected in phase space near the "ghost" of a saddlenode bifurcation of fixed points. The burst trajectory reinjection is studied using quasi-static bifurcation theory, that shows a period doubling transition in the fast subsystem as the cause of burst termination. As the applied depolarization is increased, the model exhibits first resting, then tonic firing, and finally chaotic bursting behavior, in contrast with many other burst models. The transition between tonic firing and burst firing is due to a saddle-node bifurcation of limit cycles. Analysis of this bifurcation shows that the route to chaos in these neurons is type I intermittency, and we present experimental analysis of ELL pyramidal cell burst trains that support this model prediction. By varying parameters in a way that changes the positions of both saddle-node bifurcations in parameter space, we produce a wide gallery of burst patterns, which span a significant range of burst time scales.

We review the problem of estimating parameters and unobserved trajectory components from noisy time series measurements of continuous nonlinear dynamical systems. It is first shown that in parameter estimation techniques that do not take the measurement errors explicitly into account, like regression approaches, noisy measurements can produce inaccurate parameter estimates. Another problem is that for chaotic systems the cost functions that have to be minimized to estimate states and parameters are so complex that common optimization routines may fail. We show that the inclusion of information about the time-continuous nature of the underlying trajectories can improve parameter estimation considerably. Two approaches, which take into account both the errors-in-variables problem and the problem of complex cost functions, are described in detail: shooting approaches and recursive estimation techniques. Both are demonstrated on numerical examples.

Origin of chaos in a simple two-dimensional map model replicating the spiking and spikingbursting activity of real biological neurons is studied. The map contains one fast and one slow variable. Individual dynamics of a fast subsystem of the map is characterized by two types of possible attractors: stable fixed point (replicating silence) and superstable limit cycle (replicating spikes). Coupling this subsystem with the slow subsystem leads to the generation of periodic or chaotic spiking-bursting behavior. We study the bifurcation scenarios which reveal the dynamical mechanisms that lead to chaos at alternating silence and spiking phases.

cardiac excitation

Running head: Modeling mouse sleep-wake behavior

this paper we focus on the system of three globally coupled H--H neurons with symmetric synaptic coupling, that is, each neuron is coupled to the other two neurons bi-directionally with the same synaptic parameters. Also our main interest is on the weak coupling regime with small g syn where the phase model reduction provides a reasonable description: it is known that this occurs when g syn . 0.3 [Lee et al., 1997]. We set g syn =0.1in

The heart can be considered as a 3D mass of nonlinear excitable medium. With simple models of an excitable cell, one can probe arrhythmogenic processes with numerical experiments and possibly identify new therapeutic strategies. An important target for such experiments is the reduction in the incidence of sudden cardiac death in patients with cardiac disease. Potentially fatal cardiac rhythm disturbances are often initiated by extra endogenous heart beats. It was thought that drugs that reduced the frequency of extra heart contractions in individual cells would reduce the incidence of sudden cardiac death in patients. Two large clinical trials of drugs that exhibited significant antiarrhythmic properties in single cells were found to increase the rate of sudden cardiac death in patients by 2-3 fold over untreated patients even though the treated patients experienced an 80% reduction in extra heart beats. Presented here is a model of drug interactions with a cardiac cell that, when coupled with a nonlinear model of cardiac excitability, yields new insights into the proarrhythmic effects of supposedly antiarrhythmic drugs. With this model, a single unifying principle arises: that drug alteration of the dynamics of cardiac excitability displays both antiarrhythmic and proarrhythmic effects. In single cells, blockade of the Na channel (the action of antiarrhythmic drugs) reduces excitability and increases the interval of time that a cell remains inexcitable following a normal stimulation. When cells are coupled in 1, 2 and 3 D arrays, spatial gradients of excitability follow the activation wavefront. Reducing excitability and slowing the recovery dynamics of excitability in a multicellular medium extend the spatial gradient of the excitability recovery process. Consequently...