Intermittent synchronization in a network of bursting neurons

Synchronized oscillations in networks of inhibitory and excitatory coupled bursting neurons are common in a variety of neural systems from central pattern generators to human brain circuits. One example of the latter is the subcortical network of the basal ganglia, formed by excitatory and inhibitory bursters of the subthalamic nucleus and globus pallidus, involved in motor control and affected in Parkinson's disease. Recent experiments have demonstrated the intermittent nature of the phase-locking of neural activity in this network. Here, we explore one potential mechanism to explain the intermittent phase-locking in a network. We simplify the network to obtain a model of two inhibitory coupled elements and explore its dynamics. We used geometric analysis and singular perturbation methods for dynamical systems to reduce the full model to a simpler set of equations. Mathematical analysis was completed using three slow variables with two different time scales. Intermittently, synchronous oscillations are generated by overlapped spiking which crucially depends on the geometry of the slow phase plane and the interplay between slow variables as well as the strength of synapses. Two slow variables are responsible for the generation of activity patterns with overlapped spiking, and the other slower variable enhances the robustness of an irregular and intermittent activity pattern. While the analyzed network and the explored mechanism of intermittent synchrony appear to be quite generic, the results of this analysis can be used to trace particular values of biophysical parameters (synaptic strength and parameters of calcium dynamics), which are known to be impacted in Parkinson's disease.     

A Comparison of Two Variational Methods

A discussion is given of the relation between the maximum functionaltheorem and Schwinger's variational principle, and it is shownthat the bound given by Schwinger's principle is better thanor the same as that given by the maximum functional theorem.     

Further Gurtin-type Variational Principles for Linear Initial Value Problems

Following the variational principles for linear initial valueproblems associated with the wave and heat conduction equationsdiscussed by Gurtin and Leitmann and using time convolutionsit is shown that general variational principles exist for theseproblems with sources on the boundaries and within the regionunder consideration.     

Modified Locally One Dimensional Methods for Parabolic Partial Differential Equations in Two Space Variables

In this note we consider the locally one dimensional methodas discussed in Gourlay & Mitchell (1969, 1972) and indicatehow this scheme can be used to solve accurately a parabolicpartial differential equation in two space variables.     

Intermittent neural synchronization in Parkinson’s disease

Motor symptoms of Parkinson’s disease are related to the excessive synchronized oscillatory activity in the beta frequency band (around 20 Hz) in the basal ganglia and other parts of the brain. This review explores the dynamics and potential mechanisms of these oscillations employing ideas and methods from nonlinear dynamics. We present extensive experimental documentation of the relevance of synchronized oscillations to motor behavior in Parkinson’s disease, and we discuss the intermittent character of this synchronization. The reader is introduced to novel time-series analysis techniques aimed at the detection of the fine temporal structure of intermittent phase locking observed in the brains of Parkinsonian patients. Modeling studies of brain networks are reviewed, which may describe the observed intermittent synchrony, and we discuss what these studies reveal about brain dynamics in Parkinson’s disease. The Parkinsonian brain appears to exist on the boundary between phase-locked and nonsynchronous dynamics. Such a situation may be beneficial in the healthy state, as it may allow for easy formation and dissociation of transient patterns of synchronous activity which are required for normal motor behavior. Dopaminergic degeneration in Parkinson’s disease may shift the brain networks closer to this boundary, which would still permit some motor behavior while accounting for the associated motor deficits. Understanding the mechanisms of the intermittent synchrony in Parkinson’s disease is also important for biomedical engineering since efficient control strategies for suppression of pathological synchrony through deep brain stimulation require knowledge of the dynamics of the processes subjected to control.