Abstract. This work arises from the purpose of relating time problems in biological systems with some known tools in dynamical systems. More precisely, how are the phase resetting curves (PRCs) around a limit cycle γ of a vector field X related to the fact that X is the infinitesimal generator of a Lie symmetry ([Y, X] = µ Y ). We show how the time variables involved in the Lie symmetry provide a natural way (a kind of normal form) to express the vector field around γ, similar to action-angle variables for integrable systems. In addition, the knowledge of the orbits of Y gives a trivial way to compute the PRC, not only on γ, but also in a neighborhood of it, thus obtaining what we call phase resetting surfaces (PRSs). However, the aim of the paper is not only to state relationships among different concepts, but also to perform the effective computation of these symmetries. The numerical scheme is based on the theoretical ground of the so-called parameterization method to compute invariant manifolds (the orbits of Y ) in a neighborhood of γ. Limit cycles in biological (more specifically, neuroscience) models encompass numerical problems that are often neglected or underestimated; we present a discussion about them and give general solutions whenever it is possible. Finally, we use all theoretical and numerical results to compute both the PRCs and PRSs and the isochronous sections of limit cycles for well-known biological models. In this part of the paper, we also explore how the PRCs evolve (in the parameter space) between different bifurcation values.

The phase response curve (PRC) is a powerful tool to study the effect of a perturbation on the phase of an oscillator, assuming that all the dynamics can be explained by the phase variable. However, factors like the rate of convergence to the oscillator, strong forcing or high stimulation frequency may invalidate the above assumption and raise the question of how is the phase variation away from an attractor. The concept of isochrons turns out to be crucial to answer this question; from it, we have built up Phase Response Functions (PRF) and, in the present paper, we complete the extension of advancement functions to the transient states by defining the Amplitude Response Function (ARF) to control changes in the transversal variables. Based on the knowledge of both the PRF and the ARF, we study the case of a pulse-train stimulus, and compare the predictions given by the PRC-approach (a 1D map) to those given by the PRF-ARF-approach (a 2D map); we observe differences up to two orders of magnitude in favor of the 2D predictions, especially when the stimulation frequency is high or the strength of the stimulus is large. We also explore the role of hyperbolicity of the limit cycle as well as geometric aspects of the isochrons. Summing up, we aim at enlightening the contribution of transient effects in predicting the phase response and showing the limits of the phase reduction approach to prevent from falling into wrong predictions in synchronization problems.List of AbbreviationsPRC phase response curve, phase resetting curve.PRF phase response function.ARF amplitude response function.

Short-term synaptic depression (STD) is a form of synaptic plasticity that has a large impact on network computations. Experimental results suggest that STD is modulated by cortical activity, decreasing with activity in the network and increasing during silent states. Here, we explored different activity-modulation protocols in a biophysical network model for which the model displayed less STD when the network was active than when it was silent, in agreement with experimental results. Furthermore, we studied how trains of synaptic potentials had lesser decay during periods of activity (UP states) than during silent periods (DOWN states), providing new experimental predictions. We next tackled the inverse question of what is the impact of modifying STD parameters on the emergent activity of the network, a question difficult to answer experimentally. We found that synaptic depression of cortical connections had a critical role to determine the regime of rhythmic cortical activity. While low STD resulted in an emergent rhythmic activity with short UP states and long DOWN states, increasing STD resulted in longer and more frequent UP states interleaved with short silent periods. A still higher synaptic depression set the network into a non-oscillatory firing regime where DOWN states no longer occurred. The speed of propagation of UP states along the network was not found to be modulated by STD during the oscillatory regime; it remained relatively stable over a range of values of STD. Overall, we found that the mutual interactions between synaptic depression and ongoing network activity are critical to determine the mechanisms that modulate cortical emergent patterns.

We report that multi-stable perception operates in a consistent, dynamical regime, balancing the conflicting goals of stability and sensitivity. When a multi-stable visual display is viewed continuously, its phenomenal appearance reverses spontaneously at irregular intervals. We characterized the perceptual dynamics of individual observers in terms of four statistical measures: the distribution of dominance times (mean and variance) and the novel, subtle dependence on prior history (correlation and time-constant). The dynamics of multi-stable perception is known to reflect several stabilizing and destabilizing factors. Phenomenologically, its main aspects are captured by a simplistic computational model with competition, adaptation, and noise. We identified small parameter volumes (~3% of the possible volume) in which the model reproduced both dominance distribution and history-dependence of each observer. For 21 of 24 data sets, the identified volumes clustered tightly (~15% of the possible volume), revealing a consistent “operating regime” of multi-stable perception. The “operating regime” turned out to be marginally stable or, equivalently, near the brink of an oscillatory instability. The chance probability of the observed clustering was <0.02. To understand the functional significance of this empirical “operating regime,” we compared it to the theoretical “sweet spot” of the model. We computed this “sweet spot” as the intersection of the parameter volumes in which the model produced stable perceptual outcomes and in which it was sensitive to input modulations. Remarkably, the empirical “operating regime” proved to be largely coextensive with the theoretical “sweet spot.” This demonstrated that perceptual dynamics was not merely consistent but also functionally optimized (in that it balances stability with sensitivity). Our results imply that multi-stable perception is not a laboratory curiosity, but reflects a functional optimization of perceptual dynamics for visual inference.

Given a centre of a planar differential system, we extend the use of the Lie bracket to the determination of the monotonicity character of the period function. As far as we know, there are no general methods to study this function, and the use of commutators and Lie bracket was restricted to prove isochronicity. We give several examples and a special method which simplifies the computations when a first integral is known. r

Canard-induced phenomena have been extensively studied in the last three decades, from both the mathematical and the application viewpoints. Canards in slow-fast systems with (at least) two slow variables, especially near folded-node singularities, give an essential generating mechanism for mixed-mode oscillations (MMOs) in the framework of smooth multiple timescale systems. There is a wealth of literature on such slow-fast dynamical systems and many models displaying canard-induced MMOs, particularly in neuroscience. In parallel, since the late 1990s several papers have shown that the canard phenomenon can be faithfully reproduced with piecewise-linear (PWL) systems in two dimensions, although very few results are available in the three-dimensional case. The present paper aims to bridge this gap by analyzing canonical PWL systems that display folded singularities, primary and secondary canards, with a similar control of the maximal winding number as in the smooth case. We also show that the singular phase portraits are compatible in both frameworks. Finally, we show using an example how to construct a (linear) global return and obtain robust PWL MMOs.Ministerio de Ciencia y Tecnología MTM2012-31821Junta de Andalucía P12-FQM-165

In this paper, we study systems in the plane having a critical point with pure imaginary eigenvalues, and we search for effective conditions to discern whether this critical point is a focus or a center; in the case of it being a center, we look for additional conditions in order to be isochronous. We stress that the essential differences between the techniques used in this work and the more usual ones are basically two: the elimination of the integration constants when we consider Ž . primitives of functions see also Remark 3.2 and the fact that we maintain the complex notation in the whole study. Thanks to these aspects, we reach with relative ease an expression of the first three Liapunov constants,¨,¨, and¨, and of the first two period ones, p and p , for a general system. As far as we 2 4

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