MS05.1: Slow-Fast Systems and Phenomena

The striatum brain nucleus is central for motor and cognitive functions. We propose a large-scale biophysical network for this part of the brain, using a modified Hodgkin-Huxley dynamics for neurons and a connectivity which is informed by a detailed human atlas. In this complex slow-fast system different spatio-temporal activity patterns emerge during healthy and pathological states (neurological disorder), depending on the intensity of the cortical inputs. Using equation-free methods and diffusion maps, we are able to identify a macroscopic order parameter. The equation-free approach enables a numerical analysis of the macroscopic dynamics of the striatum network. This includes a numerical bifurcation and stability analysis. Finally, the effect of deep brain stimulation on the spatiotemporal pattern formation in the network is discussed.

The upper bound response of Nonlinear Parametrically Excited (NPE) systems is investigated. Different systems including a NPE system under pure parametric excitation, a NPE system under interacting external excitation and a NPE cantilever beam with a tip mass are considered. In order to obtain the response of the system for the whole frequency range considered, the method of varying amplitudes (MVA) is used. Employing the MVA, analytical expressions for the displacement response of the system are derived. The MVA results are compared with analytical results of the Method of Multiple Scales (MMS) and the numerical results obtained from Direct Integration (DI) of the equation of motion, showing good agreement. In order to verify the MVA results in a practical application, experimental measurements of a Nonlinear Parametrically Excited Cantilever Beam (NPECB) with a tip mass are taken, demonstrating good agreement between experiments, theory and numerical findings.

Nonlinear oscillatory behavior is commonly described in terms of amplitudes, phases, and frequencies. Only in rare cases, these quantities are constant and can be readily obtained via Fourier transformation. In many measurements of, for example, ocean waves, mechanical systems, and biological systems, the amplitudes and frequencies vary slowly in time. In this presentation, the authors present a strategy to decompose measurements of nonlinear oscillations into oscillations with slowly varying amplitudes and frequencies. The amplitude and frequency variations reveal fundamental insights into the observed oscillations and can be leveraged to derive modulation equations. These equations can be used to predict the oscillatory behavior when small perturbations are added. The capabilities of the proposed methodology are demonstrated using measurements from a nonlinear mechanical oscillator experiment.