Antoni Guillamon

A Computational and Geometric Approach to Phase Resetting Curves and Surfaces

This work arises from the purpose of applying new tools in dynamical systems to time problems in biological systems. The main aim of this paper is to develop a numerical method to perform the effective computation of the phase advancement when we stimulate an oscillator which has not yet reached the asymptotic state (a limit cycle). That is, we want to extend the computation of the phase resetting curves (PRCs) (the classical tool to compute the phase advancement) to a neighborhood of the limit cycle, obtaining what we call the phase resetting surfaces (PRSs). To achieve this goal we first perform a careful study of the theoretical grounds (the parameterization method for invariant manifolds and another approach using Lie symmetries), which allows us to describe the isochronous sections of the limit cycle and, from them, to obtain the PRSs. In order to make this theoretical framework applicable, we use the numerical algorithms of the parameterization method and other semianalytical tools to extend invaria...

LIMIT CYCLES FOR GENERALIZED ABEL EQUATIONS

This paper deals with the problem of finding upper bounds on the number of periodic solutions of a class of one-dimensional nonautonomous differential equations: those with the right-hand sides being polynomials of degree n and whose coefficients are real smooth one-periodic functions. The case n = 3 gives the so-called Abel equations which have been thoroughly studied and are well understood. We consider two natural generalizations of Abel equations. Our results extend previous works of Lins Neto and Panov and try to step forward in the understanding of the case n > 3. They can be applied, as well, to control the number of limit cycles of some planar ordinary differential equations.

Multi-stable perception balances stability and sensitivity

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.

Estimation of synaptic conductances

In order to identify and understand mechanistically the cortical circuitry of sensory information processing estimates are needed of synaptic input fields that drive neurons. From intracellular in vivo recordings one would like to estimate net synaptic conductance time courses for excitation and inhibition, g(E)(t) and g(I)(t), during time-varying stimulus presentations. However, the intrinsic conductance transients associated with neuronal spiking can confound such estimates, and thereby jeopardize functional interpretations. Here, using a conductance-based pyramidal neuron model we illustrate errors in estimates when the influence of spike-generating conductances are not reduced or avoided. A typical estimation procedure involves approximating the current-voltage relation at each time point during repeated stimuli. The repeated presentations are done in a few sets, each with a different steady bias current. From the trial-averaged smoothed membrane potential one estimates total membrane conductance and then dissects out estimates for g(E)(t) and g(I)(t). Simulations show that estimates obtained during phases without spikes are good but those obtained from phases with spiking should be viewed with skeptism. For the simulations, we consider two different synaptic input scenarios, each corresponding to computational network models of orientation tuning in visual cortex. One input scenario mimics a push-pull arrangement for g(E)(t) and g(I)(t) and idealized as specified smooth time courses. The other is taken directly from a large-scale network simulation of stochastically spiking neurons in a slab of cortex with recurrent excitation and inhibition. For both, we show that spike-generating conductances cause serious errors in the estimates of g(E) and g(I). In some phases for the push-pull examples even the polarity of g(I) is mis-estimated, indicating significant increase when g(I) is actually decreased. Our primary message is to be cautious about forming interpretations based on estimates developed during spiking phases.

Synaptic depression and slow oscillatory activity in a biophysical network model of the cerebral cortex.

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.

The period function for second-order quadratic ODEs is monotone

AbstractVery little is known about the period function for large families of centers. In one of the pioneering works on this problem, Chicone [3] conjectured that all the centers encountered in the family of second-order differential equations

A characterization of isochronous centres in terms of symmetries

\ddot x = V(x,\dot x)

Phase portrait of Hamiltonian systems with homogeneous nonlinearities

, beingV a quadratic polynomial, should have a monotone period function. Chicone solved some of the cases but some others remain still unsolved. In this paper we fill up these gaps by using a new technique based on the existence of Lie symmetries and presented in [8]. This technique can be used as well to reprove all the cases that were already solved, providing in this way a compact proof for all the quadratic second-order differential equations. We also prove that this property on the period function is no longer true whenV is a polynomial which nonlinear part is homogeneous of degreen>2.