Several neuron types have been shown to exhibit (subthreshold) membrane potential resonance (MPR), defined as the occurrence of a peak in their voltage amplitude response to oscillatory input currents at a preferred (resonant) frequency. MPR has been investigated both experimentally and theoretically. However, whether MPR is simply an epiphenomenon or it plays a functional role for the generation of neuronal network oscillations and how the latent time scales present in individual, non-oscillatory cells affect the properties of the oscillatory networks in which they are embedded are open questions. We address these issues by investigating a minimal network model consisting of (i) a non-oscillatory linear resonator (band-pass filter) with 2D dynamics, (ii) a passive cell (lowpass filter) with 1D linear dynamics, and (iii) nonlinear graded synaptic connections (excitatory or inhibitory) with instantaneous dynamics. We demonstrate that (i) the network oscillations crucially depend on the presence of MPR in the resonator, (ii) they are amplified by the network connectivity, (iii) they develop relaxation oscillations for high enough levels of mutual inhibition/excitation, and (iv) the network frequency monotonically depends on the resonators resonant frequency. We explain these phenomena using a reduced adapted version of the classical phase-plane analysis that helps uncovering the type of effective network nonlinearities that contribute to the generation of network oscillations. Our results have direct implications for network models of firing rate type and other biological oscillatory networks (e.g, biochemical, genetic). * Corresponding Investigator, CONICET, Argentina Author SummaryBiological oscillations are ubiquitous in living systems and underlie fundamental processes in healthy and diseased individuals. Understanding how the intrinsic oscillatory properties of the participating nodes interact with the network connectivity is key for the mechanistic description of biological network oscillations. In several cases these intrinsic oscillatory properties are hidden and emerge only in the presence of external oscillatory inputs in the form of preferred amplitude responses to these inputs. This phenomenon is referred to as resonance and may occur in systems that do not exhibit intrinsic oscillations. Resonance has been primarily measured in neuronal systems, but their role in the generation of neuronal network oscillations remains largely an open question. We have identified a minimal network model consisting of a resonator (a node that exhibits resonance, but not intrinsic oscillations), a low-pass filter (no resonance and no intrinsic oscillations) and nonlinear connectivity with no dynamics. This network is able to produce oscillations, even in the absence of intrinsic oscillatory components. These oscillations crucially depend on the presence of the resonator. Moreover, the resonant frequency, a dynamic property of the interaction between the resonator and oscillatory inputs, controls the network frequency in a...

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The effect of delayed feedback on the dynamics of a scalar map is studied by using a frequency-domain approach. Explicit conditions for the occurrence of period-doubling and Neimark-Sacker bifurcations in the controlled map are found analytically. The appearance of a 1:2 resonance for certain values of the delay is also formalized, revealing that this phenomenon is independent of the system parameters. A detailed study of the well-known logistic map under delayed feedback is included for illustration.

Techniques are developed to find all periodic solutions in the simple pendulum by means of the homotopy analysis method (HAM). This involves the solution of the equations of motion in two different coordinate representations. Expressions are obtained for the cycles and periods of oscillations with a high degree of accuracy in the whole range of amplitudes. Moreover, the convergence of the method is easily checked. The aim of this work is to show how the dynamics of a simple example of oscillatory systems may be studied globally with the HAM and to incentivize the interest of advanced undergraduate students in this type of techniques.