Friedrich Adolph Wilhelm Diesterweg (1790-1866): Zum 200. Geburtstag

The loss of segregation of neuronal signal processing pathways is an important hypothesis for explaining the origin of functional deficits as associated with Parkinson's disease. Here we use a modeling approach which is utilized to study the influence of deep brain stimulation on the restoration of segregated activity in the target structures. Besides the spontaneous activity of the target network, the model considers a weak sensory input mimicking signal processing tasks, electrical deep brain stimulation delivered through a standard DBS electrode and synaptic plasticity. We demonstrate that the sensory input is capable of inducing a modification of the network structure which results in segregated microcircuits if the network is initialized in the healthy, desynchronized state. Depending on the strength and coverage, the sensory input is capable of restoring the functional sub-circuits even if the network is initialized in the synchronized, pathological state. Weak coordinated reset stimulation, applied to a network featuring a loss of segregation caused by global synchronization, is able to restore the segregated activity and to truncate the pathological, synchronized activity

Mathematical modeling of chemotaxis and glial scarring around implanted electrodes

It is well known that the implantation of electrodes for deep brain stimulation or microelectrode probes for the recording of neuronal activity is always accompanied by the response of the brain’s immune system leading to the formation of a glial scar around the implantation sites. The implanta- tion of electrodes causes massive release of adenosine-5′-triphosphate (ATP) and different cytokines into the extracellular space and activates the microglia. The released ATP and the products of its hydrolysis, such as ADP and adenosine, become the main elements mediating chemotactic sensitivity and motility of microglial cells via subsequent activation of P2Y2,12 as well as A3A/A2A adenosine receptors. The size and density of an insulating sheath around the electrode, formed by microglial cells, are important criteria for the optimization of the signal-to-noise ratio during microelectrode recordings or parameters of electrical current delivered to the brain tissue. Here, we study a purinergic signaling pathway underlying the chemotactic motion of microglia towards implanted electrodes as well as the possible impact of an anti-inflammatory coating consisting of the interleukin-1 receptor antagonist. We present a model describing the formation of a stable aggregate around the electrode due to the joint chemo-attractive action of ATP and ADP and the mixed influence of extracellular adenosine. The bioactive coating is modeled as a source of chemo-repellent located near the electrode surface. The obtained analytical and numerical results allowed us to reveal the dependences of size and spatial location of the insulating sheath on the amount of released ATP and estimate the impact of immune suppressive coating on the scarring process

Time scale synchronization of chaotic oscillators

This paper presents the result of the investigation of chaotic oscillator synchronization. A new approach for detecting of synchronized behaviour of chaotic oscillators has been proposed. This approach is based on the analysis of different time scales in the time series generated by the coupled chaotic oscillators. This approach has been applied for the coupled Rossler and Lorenz systems.Comment: 19 pages, 12 figure

Phase Response Curves of Coupled Oscillators

Many real oscillators are coupled to other oscillators and the coupling can affect the response of the oscillators to stimuli. We investigate phase response curves (PRCs) of coupled oscillators. The PRCs for two weakly coupled phase-locked oscillators are analytically obtained in terms of the PRC for uncoupled oscillators and the coupling function of the system. Through simulation and analytic methods, the PRCs for globally coupled oscillators are also discussed.Comment: 5 pages 4 figur

Collective dynamical response of coupled oscillators with any network structure

We formulate a reduction theory that describes the response of an oscillator network as a whole to external forcing applied nonuniformly to its constituent oscillators. The phase description of multiple oscillator networks coupled weakly is also developed. General formulae for the collective phase sensitivity and the effective phase coupling between the oscillator networks are found. Our theory is applicable to a wide variety of oscillator networks undergoing frequency synchronization. Any network structure can systematically be treated. A few examples are given to illustrate our theory.Comment: 4 pages, 2 figure

Revealing Network Connectivity From Dynamics

We present a method to infer network connectivity from collective dynamics in networks of synchronizing phase oscillators. We study the long-term stationary response to temporally constant driving. For a given driving condition, measuring the phase differences and the collective frequency reveals information about how the oscillators are interconnected. Sufficiently many repetitions for different driving conditions yield the entire network connectivity from measuring the dynamics only. For sparsely connected networks we obtain good predictions of the actual connectivity even for formally under-determined problems.Comment: 10 pages, 4 figure

Phase resetting of collective rhythm in ensembles of oscillators

Phase resetting curves characterize the way a system with a collective periodic behavior responds to perturbations. We consider globally coupled ensembles of Sakaguchi-Kuramoto oscillators, and use the Ott-Antonsen theory of ensemble evolution to derive the analytical phase resetting equations. We show the final phase reset value to be composed of two parts: an immediate phase reset directly caused by the perturbation, and the dynamical phase reset resulting from the relaxation of the perturbed system back to its dynamical equilibrium. Analytical, semi-analytical and numerical approximations of the final phase resetting curve are constructed. We support our findings with extensive numerical evidence involving identical and non-identical oscillators. The validity of our theory is discussed in the context of large ensembles approximating the thermodynamic limit.Comment: submitted to Phys. Rev.

Granger causality for circular variables

In this letter we discuss use of Granger causality to the analyze systems of coupled circular variables, by modifying a recently proposed method for multivariate analysis of causality. We show the application of the proposed approach on several Kuramoto systems, in particular one living on networks built by preferential attachment and a model for the transition from deeply to lightly anaesthetized states. Granger causalities describe the flow of information among variables.Comment: 4 pages, 5 figure

Noise-induced inhibitory suppression of malfunction neural oscillators

Motivated by the aim to find new medical strategies to suppress undesirable neural synchronization we study the control of oscillations in a system of inhibitory coupled noisy oscillators. Using dynamical properties of inhibition, we find regimes when the malfunction oscillations can be suppressed but the information signal of a certain frequency can be transmitted through the system. The mechanism of this phenomenon is a resonant interplay of noise and the transmission signal provided by certain value of inhibitory coupling. Analyzing a system of three or four oscillators representing neural clusters, we show that this suppression can be effectively controlled by coupling and noise amplitudes.Comment: 10 pages, 14 figure