Hil Gaétan Ellart Meijer

Numerical Normal Forms for Codim 2 Bifurcations of Fixed Points with at Most Two Critical Eigenvalues

In this paper we derive explicit formulas for the normal form coefficients to verify the nondegeneracy of eight codimension two bifurcations of fixed points with one or two critical eigenvalues. These include all strong resonances, as well as the degenerate flip and Neimark--Sacker bifurcations. Applying our results to n-dimensional maps, one avoids the computation of the center manifold, but one can directly evaluate the critical normal form coefficients in the original basis. The formulas remain valid also for n=2 and allow one to avoid the transformation of the linear part of the map into Jordan form. The developed techniques are tested on two examples: (1) a three-dimensional map appearing in adaptive control; (2) a periodically forced epidemic model. We compute numerically the critical normal form coefficients for several codim 2 bifurcations occurring in these models. To compute the required derivatives of the Poincare map for the epidemic model, the automatic differentiation package ADOL-C is used.

Numerical Methods for Two-Parameter Local Bifurcation Analysis of Maps

We discuss new and improved algorithms for the bifurcation analysis of fixed points and periodic orbits (cycles) of maps and their implementation in matcont, a MATLAB toolbox for continuation and bifurcation analysis of dynamical systems. This includes the numerical continuation of fixed points of iterates of the map with one control parameter, detecting and locating their bifurcation points (i.e., limit point, period-doubling, and Neimark-Sacker) and their continuation in two control parameters, as well as detection and location of all codimension 2 bifurcation points on the corresponding curves. For all bifurcations of codim 1 and 2, the critical normal form coefficients are computed, both numerically with finite directional differences and using symbolic derivatives of the original map. Using a parameter-dependent center manifold reduction, explicit asymptotics are derived for bifurcation curves of double and quadruple period cycles rooted at codim 2 points of cycles with arbitrary period. These asymptotics are implemented into the software and allow one to switch at codim 2 points to the continuation of the double and quadruple period bifurcations. We provide two examples illustrating the developed techniques: a generalized Henon map and a juvenile/adult competition model from mathematical biology.

Remarks on interacting Neimark–Sacker bifurcations

We study codimension-2 bifurcations of fixed points of dissipative diffeomorphisms with a pair of complex eigenvalues together with either an eigenvalue − 1 or another such a pair. In the previous studies only cubic normal forms were considered. However, in some cases the unfolding requires higher-order terms and these are investigated here. We (re)derive the normal forms and reduce them to a single amplitude map. This map is similar to the amplitude system for the double-Hopf bifurcation of vector fields. We show how the critical normal form coefficients determine the general bifurcation picture for this amplitude map. Representative nonsymmetric perturbations of the normal forms are studied numerically. Our case studies show a detailed picture near various bifurcation curves, which is richer than known theoretical predictions. For arbitrary maps with these bifurcations we give explicit formulas for critical normal form coefficients on center manifolds. We apply them to an example from robotics where we are able to demonstrate the existence of a bubble-structure, which was only observed in perturbed normal forms before.

THE FOLD-FLIP BIFURCATION

The fold-flip bifurcation occurs if a map has a fixed point with multipliers +1 and -1 simultaneously. In this paper the normal form of this singularity is calculated explicitly. Both local and global bifurcations of the unfolding are analyzed by exploring a close relationship between the derived normal form and the truncated amplitude system for the fold-Hopf bifurcation of ODEs. Two examples are presented, the generalized Henon map and an extension of the Lorenz-84 model. In the latter example the first-, second- and third-order derivatives of the Poincare map are computed using variational equations to find the normal form coefficients.

Frequency-selectivity of a thalamocortical relay neuron during parkinson's disease and deep brain stimulation: A computational study

In this computational study, we investigated (i) the functional importance of correlated basal ganglia (BG) activity associated with Parkinson’s disease (PD) motor symptoms by analysing the effects of globus pallidus internum (GPi) bursting frequency and synchrony on a thalamocortical (TC) relay neuron, which received GABAergic projections from this nucleus; (ii) the effects of subthalamic nucleus (STN) deep brain stimulation (DBS) on the response of the TC relay neuron to synchronized GPi oscillations; and (iii) the functional basis of the inverse relationship that has been reported between DBS frequency and stimulus amplitude, required to alleviate PD motor symptoms [A. L. Benabid et al. (1991)Lancet, 337, 403–406]. The TC relay neuron selectively responded to and relayed synchronized GPi inputs bursting at a frequency located in the range 2–25 Hz. Input selectivity of the TC relay neuron is dictated by low‐threshold calcium current dynamics and passive membrane properties of the neuron. STN‐DBS prevented the TC relay neuron from relaying synchronized GPi oscillations to cortex. Our model indicates that DBS alters BG output and input selectivity of the TC relay neuron, providing an explanation for the clinically observed inverse relationship between DBS frequency and stimulus amplitude.

Functional neuronal activity and connectivity within the subthalamic nucleus in Parkinson’s disease

OBJECTIVE Characterization of the functional neuronal activity and connectivity within the subthalamic nucleus (STN) in patients with Parkinsons disease (PD). METHODS Single units were extracted from micro-electrode recording (MER) of 18 PD patients who underwent STN deep brain stimulation (DBS) surgery. The firing rate and pattern of simultaneously recorded spike trains and their coherence were analyzed. To provide a precise functional assignment of position to the observed activities, for each patient we mapped its classified multichannel STN MERs to a generic atlas representation with a sensorimotor part and a remaining part. RESULTS Within the sensorimotor part we found significantly higher mean firing rate (P < 0.05) and significantly more burst-like activity (P < 0.05) than within the remaining part. The proportion of significant coherence in the beta band (13-30 Hz) is significantly higher in the sensorimotor part of the STN than elsewhere (P = 0.015). CONCLUSIONS The STN sensorimotor part distinguishes itself from the remaining part with respect to beta coherence, firing rate and burst-like activity and postoperatively was found as the preferred target area. SIGNIFICANCE Our firing behavior analysis may help to discriminate the STN sensorimotor part for the placement of the DBS electrode.

From Parkinsonian thalamic activity to restoring thalamic relay using deep brain stimulation: new insights from computational modeling.

We present a computational model of a thalamocortical relay neuron for exploring basal ganglia thalamocortical loop behavior in relation to Parkinsons disease and deep brain stimulation (DBS). Previous microelectrode, single-unit recording studies demonstrated that oscillatory interaction within and between basal ganglia nuclei is very often accompanied by synchronization at Parkinsonian rest tremor frequencies (3-10 Hz). These oscillations have a profound influence on thalamic projections and impair the thalamic relaying of cortical input by generating rebound action potentials. Our model describes convergent inhibitory input received from basal ganglia by the thalamocortical cells based on characteristics of normal activity, and/or low-frequency oscillations (activity associated with Parkinsons disease). In addition to simulated input, we also used microelectrode recordings as inputs for the model. In the resting state, and without additional sensorimotor input, pathological rebound activity is generated for even mild Parkinsonian input. We have found a specific stimulation window of amplitudes and frequencies for periodic input, which corresponds to high-frequency DBS, and which also suppresses rebound activity for mild and even more prominent Parkinsonian input. When low-frequency pathological rebound activity disables the thalamocortical cells ability to relay excitatory cortical input, a stimulation signal with parameter settings corresponding to our stimulation window can restore the thalamocortical cells relay functionality.

Generalized Henon Map and Bifurcations of Homoclinic Tangencies

We study two-parameter bifurcation diagrams of a generalized Henon map (GHM) that is known to describe dynamics of iterated maps near homoclinic and heteroclinic tangencies. We prove the nondegeneracy of codimension (codim) 2 bifurcations of fixed points of the GHM analytically and compute its various global and local bifurcation curves numerically. Special attention is given to the interpretation of the results and their application to the analysis of bifurcations of the homoclinic tangency of a neutral saddle in two-parameter families of planar diffeomorphisms. In particular, an infinite cascade of homoclinic tangencies of neutral saddle cycles is shown to exist near the homoclinic tangency of the primary neutral saddle.

Numerical Bifurcation Analysis

The theory of dynamical systems studies the behavior of solutions of systems, like nonlinear ordinary differential equations (ODEs), depending upon parameters. Using qualitative methods of bifurcation theory, the behavior of the system is characterized for various parameter combinations. In particular, the catalog of system behaviors showing qualitative differences can be identified, together with the regions in parameter space where the different behaviors occur. Bifurcations delimit such regions. Symbolic and analytical approaches are in general infeasible, but numerical bifurcation analysis is a powerful tool that aids in the understanding of a nonlinear system. When computing power became widely available, algorithms for this type of analysis matured and the first codes were developed. With the development of suitable algorithms, the advancement in the qualitative theory has found its way into several software projects evolving over time. The availability of software packages allows scientists to study and adjust their models and to draw conclusions about their dynamics.

Comparing epileptiform behavior of mesoscale detailed models and population models of neocortex.

Two models of the neocortex are developed to study normal and pathologic neuronal activity. One model contains a detailed description of a neocortical microcolumn represented by 656 neurons, including superficial and deep pyramidal cells, four types of inhibitory neurons, and realistic synaptic contacts. Simulations show that neurons of a given type exhibit similar, synchronized behavior in this detailed model. This observation is captured by a population model that describes the activity of large neuronal populations with two differential equations with two delays. Both models appear to have similar sensitivity to variations of total network excitation. Analysis of the population model reveals the presence of multistability, which was also observed in various simulations of the detailed model.