Talks

Nonlinear dynamics aims to elucidate the basic mechanisms necessary to reflect the temporal behavior of a natural system. The data analysis and modeling techniques proposed by artificial intelligence (deep networks, computational reservoirs, recurrent networks, as examples), on the other hand, ostensibly resign the mechanistic vision for a data-oriented modeling paradigm.  In these lectures, these apparently antagonistic approaches will be analyzed in parallel, using as examples my work on the physics of birdsong production and vocal learning.

How insects find and feed on flowers is crucial for plant pollination. Plants provide several chemical and visual cues to attract insects. However, even as insects find their host flowers, they have the additional challenge of targeting a tiny nectary opening on floral surfaces to reach the sugary reward. This becomes especially challenging for moths and butterflies because they use a long and flexible mouthpart called proboscis to draw the nectary. Additionally, nocturnal hawkmoths feed while hovering in front of flowers, often at very low lights levels during nighttime. How do moths target the tiny nectary hole in flowers, despite the low visual resolution of the nectary opening? We tracked the motion of hawkmoth proboscis as they fed from artificial, 3D-printed flowers and show that hawkmoths systematically explore floral surfaces to detect tactile features such as curvatures to target the nectary. We found that over repeated visits, they preferentially explore in ways that increases the efficiency of finding the nectary. Systematic exploration and targeting objects in the environment require expert control over appendage movements. How do hawkmoths sense and control the proboscis motion to achieve such precise movements? Pilifers are paired bristled organs at the proboscis base and are well placed to provide proprioceptive feedback about relative movements of the proboscis. To study the role of pilifers as proprioceptive organs, we drove lateral motions of the proboscis in anaesthetized head-fixed moths while simultaneously recording neural responses from the pilifer nerve. Our recordings reveal that pilifer mechanosensory neurons are sensitive to lateral motions, either to the left or to the right. Like other mechanosensory organs, they respond extremely rapidly, often within a few milliseconds. We build neural models which reveal that the neural filtering properties such as the stimulus feature and selectivity function of the pilifer mechosensors are strikingly like other insect mechanosensors, including the strain sensors on wings and halteres, abdominal mechanosensors etc. suggesting an important role for sensor mechanics and motion in encoding relevant information.

Grid cells in the medial entorhinal cortex (MEC) fire when an animal is located at the vertices of a hexagonal grid that extends across the environment. The population activity of grid cells serves as an allocentric representation of the current location of the animal. Recent studies have identified a class of grid cells that represent locations ahead of the animal. How do these predictive representations emerge from the wetware of the MEC? To address this question, we developed a detailed conductance-based model of the MEC network, constrained by existing data on the biophysical properties of stellate cells and the topology of the MEC network. Our model revealed two mechanisms underlying the emergence of a predictive code in the MEC. The first relied on a time scale associated with the HCN conductance. The other depended on the degree of asymmetry in the topology of the MEC network. The former mechanism was sufficient to explain predictive coding in layer II grid cells that represented locations shifted ahead of the current location. The shift was equivalent to ~5% of the diameter of a grid field. The latter mechanism was required to model predictive representations in layer III grid cells that were shifted forward by a distance of ~25% of the diameter of a grid field. A corollary of our model, that the extent of the predictive code changes monotonically along the dorsoventral axis of the MEC following observed changes in the properties of the HCN conductance, is borne out by recent experiments.

Oscillatory phenomena are ubiquitous in the brain. Although there are oscillator-based models of brain dynamics, they do not seem to enjoy the universal computational properties of rate-coded and spiking neuron network models. Use of oscillator-based models is often limited to special phenomena like locomotor rhythms and oscillatory attractor-based memories. If neuronal ensembles are taken to be the basic functional units of brain dynamics, it is desirable to develop oscillator-based models that can explain a wide variety of neural phenomena. To this end, we aim to develop a generalized network of oscillatory neurons. Specifically we propose a novel neural network architecture consisting of Hopf oscillators described in the complex domain. The oscillators can adapt their intrinsic frequencies by tracking the frequency components of the input signals. The oscillators are also laterally connected with each other through a special form of coupling we labeled as “power coupling”. Power coupling allows two oscillators with arbitrarily different intrinsic frequencies to interact at a constant normalized phase difference. The network can be operated in two phases. In the encoding phase the oscillators comprising the network perform a Fourier-like decomposition of the input signal(s). In the reconstruction phase, outputs the trained oscillators are combined to reconstruct the training signals. As a salient example, the network can be trained to reconstruct Electroencephalogram (EEG) signals, paving the way to an exciting class of large scale brain models.

Birdsong is a complex motor activity that arises from the interaction between the central and peripheral nervous systems, the body, and the environment. Its striking similarities to human speech, both in production and learning, make songbirds powerful animal models for studying learned motor skills. In this talk, I will present an interdisciplinary approach to understanding the emergence of behavior. Specifically, I will show neuronal recordings from a telencephalic region involved in sensorimotor integration, revealing well-defined oscillations in local field potentials synchronized with the rhythmic structure of canary (Serinus canaria) song. I will introduce a low-dimensional mathematical model of a neural network that replicates the neural dynamics observed in the experiments. This work highlights the value of low-dimensional models as tools for exploring the neuroscience of perception and generation of complex motor behaviors.

Nonlinear dynamics aims to elucidate the basic mechanisms necessary to reflect the temporal behavior of a natural system. The data analysis and modeling techniques proposed by artificial intelligence (deep networks, computational reservoirs, recurrent networks, as examples), on the other hand, ostensibly resign the mechanistic vision for a data-oriented modeling paradigm.  In these lectures, these apparently antagonistic approaches will be analyzed in parallel, using as examples my work on the physics of birdsong production and vocal learning.

In 2010, Kontsevich and Soibelman defined Cohomological Hall Algebras for quivers and potential as a mathematical construction of the algebra of BPS states. These algebras are modeled on the cohomology of vanishing cycles, which makes these algebras particularly hard to study but often result in interesting algebraic structures. A deformation of a particular case of them gives rise to a positive half of Maulik-Okounkov Yangians. The goal of my talk is to give an introduction to these ideas and explain how for the case of tripled cyclic quiver with canonical cubic potential, this algebra turns out to be one-half of the universal enveloping algebra of the Lie algebra of matrix differential operators on the torus, while its deformation turn out be one half of an explicit integral form of the Affine Yangian of gl(n).