A study of resonance tongues near a Chenciner bifurcation using MatcontM

MatcontM is a matlab toolbox for numerical analysis of bifurcations of fixed points and periodic orbits of maps. It computes codim 1 bifurcation curves and supports the computation of normal coefficients including branch switching from codim 2 points to secondary curves. Recently, the initialization and computation of connecting orbits was improved. Moreover, a graphical user interface was added enabling interactive control of all these computations. To further support these computations it allows to compute orbits of the map and its iterates and to represent them in 2D, 3D and numeric windows. We demonstrate the use of the toolbox in a study of Arnol'd tongues near a degenerate Neimark-Sacker (Chenciner) bifurcation. Here we illustrate the recent theory of [Baesens&Mackay,2007] how resonance tongues interact with a quasi-periodic saddle-node bifurcation of invariant curves in maps. Using normal form coefficients we find evidence for one of their cases, but not the other. Actually, we find another unfolding, i.e. a third possibility. We also find a structure that resembles a quasi-periodic cusp bifurcation of invariant curves

On phase-locking of oscillators with delay coupling

We consider two oscillators with delayed direct and velocity coupling. The oscillators have frequencies close or equal to 1:1 resonance. Due to the coupling the oscillations of the subsystems are in or out of phase. For these synchronized and anti-phase solutions, we use averaging for analytical stability results for small parameters. We also determine bifurcation curves of the delay system numerically. We identify regions in the parameter space (two coupling constants and the delay) where both solutions are stable or only one. For small parameters the averaging and numerical results are in good agreement. For larger values of the delay, we find multiple synchronized and anti-phase solutions. For small detuning we show that a minimal coupling value is needed to have almost synchronous or anti-phase behaviour

Bifurcation analysis of a model for atherosclerotic plaque evolution

We analyze two ordinary differential equation (ODE) models for atherosclerosis. The ODE models describe long time evolution of plaques in arteries. We show how the dynamics of the first atherosclerosis model (model A) can be understood using codimension-two bifurcation analysis. The Low-Density Lipoprotein (LDL) intake parameter (dd) is the first control parameter and the second control parameter is either taken to be the conversion rate of macrophages (bb) or the wall shear stress (σσ). Our analysis reveals that in both cases a Bogdanov-Takens (BT) point acts as an organizing center. The bifurcation diagrams are calculated partly analytically and to a large extent numerically using AUTO07 and MATCONT. The bifurcation curves show that the concentration of LDL in the plaque as well as the monocyte and the macrophage concentration exhibit oscillations for a certain range of values of the control parameters. Moreover, we find that there are threshold values for both the cholesterol intake rate dcritdcrit and the conversion rate of the macrophages bcritbcrit, which depend on the values of other parameters, above which the plaque volume increases with time. It is found that larger conversion rates of macrophages lower the threshold value of cholesterol intake and vice versa. We further argue that the dynamics for model A can still be discerned in the second model (model B) in which the slow evolution of the radius of the artery is coupled self-consistently to changes in the plaque volume. The very slow evolution of the radius of the artery compared to the other processes makes it possible to use a slow manifold approximation to study the dynamics in this case. We find that in this case the model predicts that the concentrations of the plaque constituents may go through a period of oscillations before the radius of the artery will start to decrease. These oscillations hence act as a precursor for the reduction of the artery radius by plaque growth

Improved homoclinic predictor for Bogdanov-Takens bifurcation

An improved homoclinic predictor at a generic codim 2 Bogdanov-Takens (BT) bifucation is derived. We use the classical "blow-up" technique to reduce the canonical smooth normal form near a generic BT bifurcation to a perturbed Hamiltonian system. With a simple perturbation method, we derive explicit rst- and second-order corrections of the unperturbed homoclinic orbit and parameter value. To obtain the normal form on the center manifold, we apply the standard parameter-dependent center manifold reduction combined with the normalization, that is based on the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we remove an ambiguity in the parameter transformation existing in the literature. The actual implementation of the improved predictor in MatCont and numerical examples illustrating its eciency are discussed

Gap junctions as modulators of synchrony in Parkinson's disease

Parkinson's disease (PD) patients show abnormal levels of synchrony and low-frequency oscillations in the basal ganglia and the motor cortex. This altered neural activity is often associated with the motor symptoms of PD, but the mechanisms for the emergence of synchrony and oscillations remain debated. We suggest that neural gap junctions in cortex and basal ganglia contribute to this transition in activity. While gap junctions between interneurons of cortex and striatum are well described, we do not know whether they appear in GPe and internal globus pallidus (GPi). Using confocal microscopy, we were able to detect the gap junction protein Cx36 in the human GPe and GPi, which was up-regulated in PD patients. Also the corresponding rat tissue showed Cx36 expression. Dopamine has already been described to modulate the conductance of gap junctions [1], especially also in the rat striatum, where dye coupling was increased after dopamine depleting 6-OHDA lesions [2]. In a conductance-based network model of the basal ganglia, we investigate the effect of gap junctional coupling in GPe and GPi on synchrony. While chemical synapses normally desynchronize the network, gap junctional coupling of sufficient strength is able to synchronize the whole basal ganglia. Also synchronized input from cortex to subthalamic nucleus has impact on synchronization, in particular in the case of numerous gap junctions in GPe. To describe the effect of gap junctional coupling between cortical interneurons on synchronized oscillations in the cortex, we introduce a diffusion term in a mean-field model. For high gap junctional coupling, large-amplitude oscillations of low frequency occur which are absent for low gap junctional coupling. Via the hyperdirect pathway, these oscillations could further synchronize the basal ganglia. We conclude that gap junctions can be a powerful trigger of synchrony in the basal ganglia. Their dependence on dopamine could explain the shifts of synchrony in PD. References 1. Li, H, Zhang, Z, Blackburn, MR, Wang, SW, Ribelayga, CP and O'Brien, J Adenosine and dopamine receptors coregulate photoreceptor coupling via gap junction phosphorylation in mouse retina. (2013) The Journal of Neuroscience, 33(7), 3135-3150. 2. Onn, SP and Grace, AA: Alterations in electrophysiological activity and dye coupling of striatal spiny and aspiny neurons in dopamine-denervated rat striatum recorded in vivo. (1999) Synapse, 33(1):1- 15

Computational modeling of Adelta-fiber-mediated nociceptive detection of electrocutaneous stimulation

Sensitization is an example of malfunctioning of the nociceptive pathway in either the peripheral or central nervous system. Using quantitative sensory testing, one can only infer sensitization, but not determine the defective subsystem. The states of the subsystems may be characterized using computational modeling together with experimental data. Here, we develop a neurophysiologically plausible model replicating experimental observations from a psychophysical human subject study. We study the effects of single temporal stimulus parameters on detection thresholds corresponding to a 0.5 detection probability. To model peripheral activation and central processing, we adapt a stochastic drift-diffusion model and a probabilistic hazard model to our experimental setting without reaction times. We retain six lumped parameters in both models characterizing peripheral and central mechanisms. Both models have similar psychophysical functions, but the hazard model is computationally more efficient. The model-based effects of temporal stimulus parameters on detection thresholds are consistent with those from human subject data

Switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in ODEs

The paper provides full algorithmic details on switching to the continuation of all possible codim 1 cycle bifurcations from generic codim 2 equilibrium bifurcation points in n-dimensional ODEs. We discuss the implementation and the performance of the algorithm in several examples, including an extended Lorenz-84 model and a laser system.Comment: 17 pages, 7 figures, submitted to Physica

Codimension 2 Bifurcations of Iterated Maps

This thesis investigates some properties of discrete-time dynamical systems, generated by iterated maps. In particular we study local bifurcations where two parameters are essential to describe the dynamical properties of the system near a fixed point or a cycle. There are 11 such cases. Knowledge of these bifurcations is important as they form boundary corners of stability regions, which is an important issue when one investigates properties of dynamical systems. In chapter two, we first review some aspects of codimension 2 bifurcations which have been studied before. After this short summary of the involved bifurcation curves and scenarios, three cases are analyzed, which are understood less, namely the fold-flip, flip-Neimark-Sacker and double Neimark-Sacker bifurcations. Parameter-dependent normal forms in the minimal possible phase dimension are given up to a certain sufficient degree. Higher order terms are neglected at first. Then local and global bifurcations of these truncated normal forms are investigated. Finally, the effect of truncation of higher order terms in the normal form is discussed. The bifurcation analysis is representative, but the dynamics of the normal form is in general not exactly the same as in the original system. The effect of certain perturbations is investigated numerically. For bifurcations of invariant curves Chenciner, Takens and Wagener, and Broer et.al. have obtained results, which we confirm, connect and extend. In chapter three we discuss codimension two bifurcations of maps in an arbitrary, but finite number of dimensions. Formulas are derived for the coefficients of the normal form with the aid of Center Manifold Reduction. These coefficients can be used to draw conclusions about the dynamics of the original systems. An implementation of this approach in the continuation package MATCONT is also described. It is capable of continuation of cycles and codimension one bifurcation curves of cycles in one and two parameters, respectively. During the continuation bifurcations can be detected and coefficients for the normal forms are computed automatically. Another new feature compared to existing software, is the possibility to switch to branches to certain bifurcation curves emanating from codimension two points. In chapter four, several examples from control, population dynamics and meteorology are discussed to illustrate the new results and the developed methods