Gabriel B. Mindlin

Topological analysis and synthesis of chaotic time series

Abstract We have developed a topological procedure for analyzing chaotic time series which identifies the stretching and squeezing mechanisms responsible for chaotic behavior in low-dimensional dynamical systems. These mechanisms, quantitatively described by a “template” or “knot-holder”, can then be used to model the processes which generate the original chaotic data set.

Elemental gesture dynamics are encoded by song premotor cortical neurons

Quantitative biomechanical models can identify control parameters that are used during movements, and movement parameters that are encoded by premotor neurons. We fit a mathematical dynamical systems model including subsyringeal pressure, syringeal biomechanics and upper-vocal-tract filtering to the songs of zebra finches. This reduces the dimensionality of singing dynamics, described as trajectories (motor ‘gestures’) in a space of syringeal pressure and tension. Here we assess model performance by characterizing the auditory response ‘replay’ of song premotor HVC neurons to the presentation of song variants in sleeping birds, and by examining HVC activity in singing birds. HVC projection neurons were excited and interneurons were suppressed within a few milliseconds of the extreme time points of the gesture trajectories. Thus, the HVC precisely encodes vocal motor output through activity at the times of extreme points of movement trajectories. We propose that the sequential activity of HVC neurons is used as a ‘forward’ model, representing the sequence of gestures in song to make predictions on expected behaviour and evaluate feedback.

Topological analysis of chaotic time series data from the Belousov-Zhabotinskii reaction

SummaryWe have applied topological methods to analyze chaotic time series data from the Belousov-Zhabotinskii reaction. First, the periodic orbits shadowed by the data set were identified. Next, a three-dimensional embedding without self-intersections was constructed from the data set. The topological structure of that flow was visualized by constructing a branched manifold such that every periodic orbit in the flow could be held by the branched manifold. The branched manifold, or induced template, was computed using the three lowest-period orbits. The organization of the higher-period orbits predicted by this induced template was compared with the organization of the orbits reconstructed from the data set with excellent results. The consequences of the presence of certain knots found in the data are discussed.

An efficient algorithm for fast O(N∗ln(N)) box counting

Abstract A new topological ordering is defined which significantly reduces the time requirements for the fast box counting method proposed in a recent paper by Liebovitch and Toth. Only one sorting is necessary in this algorithm.

Frequency modulation during song in a suboscine does not require vocal muscles

The physiology of sound production in suboscines is poorly investigated. Suboscines are thought to develop song innately unlike the closely related oscines. Comparing phonatory mechanisms might therefore provide interesting insight into the evolution of vocal learning. Here we investigate sound production and control of sound frequency in the Great Kiskadee (Pitangus sulfuratus) by recording air sac pressure and vocalizations during spontaneously generated song. In all the songs and calls recorded, the modulations of the fundamental frequency are highly correlated to air sac pressure. To test whether this relationship reflects frequency control by changing respiratory activity or indicates synchronized vocal control, we denervated the syringeal muscles by bilateral resection of the tracheosyringeal nerve. After denervation, the strong correlation between fundamental frequency and air sac pressure patterns remained unchanged. A single linear regression relates sound frequency to air sac pressure in the intact and denervated birds. This surprising lack of control by syringeal muscles of frequency in Kiskadees, in strong contrast to songbirds, poses the question of how air sac pressure regulates sound frequency. To explore this question theoretically, we assume a nonlinear restitution force for the oscillating membrane folds in a two mass model of sound production. This nonlinear restitution force is essential to reproduce the frequency modulations of the observed vocalizations.

Temperature Induced Syllable Breaking Unveils Nonlinearly Interacting Timescales in Birdsong Motor Pathway

The nature of telencephalic control over premotor and motor circuits is debated. Hypotheses range from complete usurping of downstream circuitry to highly interactive mechanisms of control. We show theoretically and experimentally, that telencephalic song motor control in canaries is consistent with a highly interactive strategy. As predicted from a theoretical model of respiratory control, mild cooling of a forebrain nucleus (HVC) led to song stretching, but further cooling caused progressive restructuring of song, consistent with the hypothesis that respiratory gestures are subharmonic responses to a timescale present in the output of HVC. This interaction between a life-sustaining motor function (respiration) and telencephalic song motor control suggests a more general mechanism of how nonlinear integration of evolutionarily new brain structures into existing circuitry gives rise to diverse, new behavior.

Neurophysiological bases of exponential sensory decay and top-down memory retrieval: a model

Behavioral observations suggest that multiple sensory elements can be maintained for a short time, forming a perceptual buffer which fades after a few hundred milliseconds. Only a subset of this perceptual buffer can be accessed under top-down control and broadcasted to working memory and consciousness. In turn, single-cell studies in awake-behaving monkeys have identified two distinct waves of response to a sensory stimulus: a first transient response largely determined by stimulus properties and a second wave dependent on behavioral relevance, context and learning. Here we propose a simple biophysical scheme which bridges these observations and establishes concrete predictions for neurophsyiological experiments in which the temporal interval between stimulus presentation and top-down allocation is controlled experimentally. Inspired in single-cell observations, the model involves a first transient response and a second stage of amplification and retrieval, which are implemented biophysically by distinct operational modes of the same circuit, regulated by external currents. We explicitly investigated the neuronal dynamics, the memory trace of a presented stimulus and the probability of correct retrieval, when these two stages were bracketed by a temporal gap. The model predicts correctly the dependence of performance with response times in interference experiments suggesting that sensory buffering does not require a specific dedicated mechanism and establishing a direct link between biophysical manipulations and behavioral observations leading to concrete predictions.

Simple Model of a Stochastically Excited Solar Dynamo

The aim of this paper is to investigate the dynamical nature of the complexity observed in the time evolution of the sunspot number. We report a detailed analysis of the sunspot number time series, and use the daily records to build the phase space of the underlying dynamical system. The observed features of the phase space prompted us to describe the global behavior of the solar cycle in terms of a noise-driven relaxation oscillator. We find the equations whose solutions best fit the observed series, which adequately describe the shape of the peaks and the oscillations of the system. The system of equations obtained from this fitting procedure is shown to be equivalent to a truncation of the dynamo equations. A linear transformation maps the phase space of these equations into the phase space reconstructed from the observations. The irregularities of the solar cycle were modeled through the introduction of a stochastic parameter in the equations to simulate the randomness arising in the process of eruption of magnetic flow to the solar surface. The mean values and deviations obtained for the periods, rise times and peak values, are in good agreement with the values obtained from the sunspot time series.

Beyond harmonic sounds in a simple model for birdsong production.

In this work we present an analysis of the dynamics displayed by a simple bidimensional model of labial oscillations during birdsong production. We show that the same model capable of generating tonal sounds can present, for a wide range of parameters, solutions which are spectrally rich. The role of physiologically sensible parameters is discussed in each oscillatory regime, allowing us to interpret previously reported data.

A circular model for song motor control in Serinus canaria.

Song production in songbirds is controlled by a network of nuclei distributed across several brain regions, which drives respiratory and vocal motor systems to generate sound. We built a model for birdsong production, whose variables are the average activities of different neural populations within these nuclei of the song system. We focus on the predictions of respiratory patterns of song, because these can be easily measured and therefore provide a validation for the model. We test the hypothesis that it is possible to construct a model in which (1) the activity of an expiratory related (ER) neural population fits the observed pressure patterns used by canaries during singing, and (2) a higher forebrain neural population, HVC, is sparsely active, simultaneously with significant motor instances of the pressure patterns. We show that in order to achieve these two requirements, the ER neural population needs to receive two inputs: a direct one, and its copy after being processed by other areas of the song system. The model is capable of reproducing the measured respiratory patterns and makes specific predictions on the timing of HVC activity during their production. These results suggest that vocal production is controlled by a circular network rather than by a simple top-down architecture.