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Kadakia, N, Rey D, Ye J, Abarbanel HDI.  2017.  Symplectic structure of statistical variational data assimilation. Quarterly Journal of the Royal Meteorological Society. 143:756-771.   10.1002/qj.2962   Abstract

Data assimilation variational principles (4D-Var) exhibit a natural symplectic structure among the state variables x(t) and. x(t). We explore the implications of this structure in both Lagrangian coordinates {x(t), x(t)} andHamiltonian canonical coordinates {x(t), p(t)} through a numerical examination of the chaotic Lorenz 1996 model in ten dimensions. We find that there are a number of subtleties associated with discretization, boundary conditions, and symplecticity, suggesting differing approaches when working in the the Lagrangian versus the Hamiltonian description. We investigate these differences in detail, and accordingly develop a protocol for searching for optimal trajectories in a Hamiltonian space. We find that casting the problem into canonical coordinates can, in some situations, considerably improve the quality of predictions.

Kadakia, N, Armstrong E, Breen D, Morone U, Daou A, Margoliash D, Abarbanel HDI.  2016.  Nonlinear statistical data assimilation for HVCRA neurons in the avian song system. Biological Cybernetics. 110:417-434.   10.1007/s00422-016-0697-3   AbstractWebsite

With the goal of building a model of the HVC nucleus in the avian song system, we discuss in detail a model of HVCRA projection neurons comprised of a somatic compartment with fast Na+ and K+ currents and a dendritic compartment with slower Ca2+ dynamics. We show this model qualitatively exhibits many observed electrophysiological behaviors. We then show in numerical procedures how one can design and analyze feasible laboratory experiments that allow the estimation of all of the many parameters and unmeasured dynamical variables, given observations of the somatic voltage V-s(t) alone. A key to this procedure is to initially estimate the slow dynamics associated with Ca, blocking the fast Na and K variations, and then with the Ca parameters fixed estimate the fast Na and K dynamics. This separation of time scales provides a numerically robust method for completing the full neuron model, and the efficacy of the method is tested by prediction when observations are complete. The simulation provides a framework for the slice preparation experiments and illustrates the use of data assimilation methods for the design of those experiments.

Armstrong, E, Abarbanel HDI.  2016.  Model of the songbird nucleus HVC as a network of central pattern generators. Journal of Neurophysiology. 116:2405-2419.   10.1152/jn.00438.2016   AbstractWebsite

We propose a functional architecture of the adult songbird nucleus HVC in which the core element is a "functional syllable unit" (FSU). In this model, HVC is organized into FSUs, each of which provides the basis for the production of one syllable in vocalization. Within each FSU, the inhibitory neuron population takes one of two operational states: 1) simultaneous firing wherein all inhibitory neurons fire simultaneously, and 2) competitive firing of the inhibitory neurons. Switching between these basic modes of activity is accomplished via changes in the synaptic strengths among the inhibitory neurons. The inhibitory neurons connect to excitatory projection neurons such that during state 1 the activity of projection neurons is suppressed, while during state 2 patterns of sequential firing of projection neurons can occur. The latter state is stabilized by feedback from the projection to the inhibitory neurons. Song composition for specific species is distinguished by the manner in which different FSUs are functionally connected to each other. Ours is a computational model built with biophysically based neurons. We illustrate that many observations of HVC activity are explained by the dynamics of the proposed population of FSUs, and we identify aspects of the model that are currently testable experimentally. In addition, and standing apart from the core features of an FSU, we propose that the transition between modes may be governed by the biophysical mechanism of neuromodulation.

Laurent, G, Stopfer M, Friedrich RW, Rabinovich MI, Volkovskii A, Abarbanel HDI.  2001.  Odor encoding as an active, dynamical process: Experiments, computation, and theory. Annual Review of Neuroscience. 24:263-297.   10.1146/annurev.neuro.24.1.263   AbstractWebsite

We examine early olfactory processing in the vertebrate and insect olfactory systems, using a computational perspective. What transformations occur between the first and second olfactory processing stages? What are the causes and consequences of these transformations? To answer these questions, we focus on the functions of olfactory circuit structure and on the role of time in odor-evoked integrative processes. We argue that early olfactory relays are active and dynamical networks, whose actions change the format of odor-related information in very specific ways, so as to refine stimulus identification. Finally, we introduce a new theoretical framework ("winnerless competition") for the interpretation of these data.