oscillations

The hearing organ of lizards -papilla- has been modelled as a chain of over-damped (inertia-less) bio-mechanical self-oscillators mutually coupled by a combination of viscous and elastic forces. In the extreme case when the elastic ones are negligible the combination of viscous coupling and overdamping leads to the study of unusual class of extended dynamical systems defined by a nonlocal spatial operator. In other words, the lack of inertia in the dynamics of the individual oscillators effectively mutates the original, locally defined coupling into one defined by a global, albeit exponentially weakening, prescription. In this talk we present a number of counterintuitive consequences of this phenomenon on the propagation of perturbations along the media, as well as on the expected synchronization behaviour of the chain. Other characteristics of papillae is tonotopy: the oscillators proper frequencies are arranged in an increasing order along the chain. The combination of different types of couplings and tonotopy, produces characteristic collective frequency spectra that one could associate with distinguishably stable spontaneous otoacoustic emissions observed in individual of certain lizards’ species like tokkai gecko for instance. We explore this phenomenon in simple settings.

Chemical and electrical synapses shape the collective dynamics of neuronal networks. Numerous theoretical studies have investigated how, separately, each of these type of synapses contributes to the generation of neuronal oscillations, but their combined effect is less understood. This limitation is further magnified by the impossibility of traditional neuronal mean field models ---often referred to as firing rate models--- to account for electrical synapses. Here we perform a comparative analysis of the dynamics of heterogeneous populations of quadratic integrate-and-fire neurons with chemical, electrical, and both chemical and electrical coupling.

In the thermodynamic limit, we show that the population's mean-field dynamics is exactly described by a system of two ordinary differential equations for the center and the width of the distribution of membrane potentials -or, equivalently, for the population-mean membrane potential and firing rate. These firing rate equations describe, in a unified framework, the collective dynamics of the ensemble of spiking neurons, and reveal that both chemical and electrical coupling are mediated by the population firing rate. Furthermore, while chemical coupling shifts the center of the distribution of membrane potentials, electrical coupling tends to reduce the width of this distribution promoting the emergence of synchronization. The analysis of the firing rate equations allows us to obtain exact formulas for all Saddle-Node and Hopf boundaries, and to construct phase diagrams characterizing the dynamics of the original network of spiking neuron. In networks with instantaneous chemical synapses the phase diagram is characterized by a codimension-two Cusp point, and by the presence of persistent states for strong excitatory coupling. In contrast, the phase diagram for electrically coupled networks is determined by a Takens-Bogdanov codimension-two point, which entails the presence of oscillations and greatly reduces the possibility of persistent states. In this case oscillations arise either via a Saddle-Node-Invariant-Circle bifurcation, or through a supercritical Hopf bifurcation -as shown using weakly nonlinear stability analysis. Finally, we show that the Takens-Bogdanov bifurcation scenario is generically present in networks with both chemical and electrical coupling.

When a ring resonator is pumped with laser light of sufficient intensity, then the refractive index -- and so the resonant frequency -- of the resonator can be modulated by the intensity of the light within it -- a phenomenon known as the Kerr nonlinearity. If the resonator is pumped with two laser beams, then this effect can give rise to spontaneous symmetry breaking in the two optical modes within the resonator. We present analytical, numerical, and experimental evidence for a rich range of exotic behaviours exhibited by this symmetry-broken light, including oscillations (implying periodic energy exchange between the modes), period-doubling, and chaos. These optical modes are described by the following coupled system of ordinary differential equations:

$$\dot{e}_{1,2}=\tilde{e}_{1,2} -[1+i(A|e_{1,2}|^{2}+B|e_{2,1}|^{2}-\Delta_{1,2})]e_{1,2},$$

where $\tilde{e}_{1,2}$ and $e_{1,2}$ are the input and coupled electric field amplitudes for each beam, respectively, and $\Delta_{1,2}$ are the frequency detunings of the laser beams, with respect to the non-Kerr-shifted cavity resonance frequency. The coefficients $A$ and $B$ denote the strengths of self- and cross-phase modulation, respectively -- i.e., the extent to which the modes interact with themselves and with each other. The physics of this dynamical system is not only of fundamental interest, but is also important for the construction of integrated all-optical circuitry and devices, such as isolators, circulators, logic gates, advanced sensors, oscillators, and scramblers.