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Wang, J, Breen D, Akinin A, Broccard F, Abarbanel HDI, Cauwenberghs G.  2017.  Assimilation of biophysical neuronal dynamics in neuromorphic VLSI. Ieee Transactions on Biomedical Circuits and Systems. 11:1258-1270.   10.1109/tbcas.2017.2776198   AbstractWebsite

Representing the biophysics of neuronal dynamics and behavior offers a principled analysis-by-synthesis approach toward understanding mechanisms of nervous system functions. We report on a set of procedures assimilating and emulating neurobiological data on a neuromorphic very large scale integrated (VLSI) circuit. The analog VLSI chip, NeuroDyn, features 384 digitally programmable parameters specifying for 4 generalized Hodgkin-Huxley neurons coupled through 12 conductance-based chemical synapses. The parameters also describe reversal potentials, maximal conductances, and spline regressed kinetic functions for ion channel gating variables. In one set of experiments, we assimilated membrane potential recorded from one of the neurons on the chip to the model structure upon which NeuroDyn was designed using the known current input sequence. We arrived at the programmed parameters except for model errors due to analog imperfections in the chip fabrication. In a related set of experiments, we replicated songbird individual neuron dynamics on NeuroDyn by estimating and configuring parameters extracted using data assimilation from intracellular neural recordings. Faithful emulation of detailed biophysical neural dynamics will enable the use of NeuroDyn as a tool to probe electrical and molecular properties of functional neural circuits. Neuroscience applications include studying the relationship between molecular properties of neurons and the emergence of different spike patterns or different brain behaviors. Clinical applications include studying and predicting effects of neuromodulators or neurodegenerative diseases on ion channel kinetics.

An, Z, Rey D, Ye JX, Abarbanel HDI.  2017.  Estimating the state of a geophysical system with sparse observations: time delay methods to achieve accurate initial states for prediction. Nonlinear Processes in Geophysics. 24:9-22.   10.5194/npg-24-9-2017   AbstractWebsite

The problem of forecasting the behavior of a complex dynamical system through analysis of observational time-series data becomes difficult when the system expresses chaotic behavior and the measurements are sparse, in both space and/or time. Despite the fact that this situation is quite typical across many fields, including numerical weather prediction, the issue of whether the available observations are "sufficient" for generating successful forecasts is still not well understood. An analysis by Whartenby et al. (2013) found that in the context of the nonlinear shallow water equations on a beta plane, standard nudging techniques require observing approximately 70% of the full set of state variables. Here we examine the same system using a method introduced by Rey et al. (2014a), which generalizes standard nudging methods to utilize time delayed measurements. We show that in certain circumstances, it provides a sizable reduction in the number of observations required to construct accurate estimates and high-quality predictions. In particular, we find that this estimate of 70% can be reduced to about 33% using time delays, and even further if Lagrangian drifter locations are also used as measurements.

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.

Whartenby, WG, Quinn JC, Abarbanel HDI.  2013.  The number of required observations in data assimilation for a shallow-water flow. Monthly Weather Review. 141:2502-2518.   10.1175/mwr-d-12-00103.1   AbstractWebsite

The authors consider statistical ensemble data assimilation for a one-layer shallow-water equation in a twin experiment: data are generated by an N x N enstrophy-conserving grid integration scheme along with an Ekman vertical velocity at the bottom of an Ekman layer driving the flow and Rayleigh and eddy viscosity dissipation damping the flow. Data are generated for N = 16 and the chaotic flow that results is analyzed. This analysis is performed in a path-integral formulation of the data assimilation problem. These path integrals are estimated by a Monte Carlo method using a Metropolis Hastings algorithm. The authors' concentration is on the number of measurements L-c that must be assimilated by the model to allow accurate estimation of the full state of the model at the end of an observation window. It is found that for this shallow-water flow approximately 70% of the full set of state variables must be observed to accomplish either goal. The number of required observations is determined by examining the number needed to synchronize the observed data L-c and the model output when L data streams are assimilated by the model. Synchronization occurs when L >= L-c and the correct selection of which L-c data are observed is made. If the number of observations is too small, so synchronization does not occur, or the selection of observations does not lead to synchronization of the data with the model output, state estimates during and at the end of the observation window and predictions beyond the observation window are inaccurate.

Haas, JS, Kreuz T, Torcini A, Politi A, Abarbanel HDI.  2010.  Rate maintenance and resonance in the entorhinal cortex. European Journal of Neuroscience. 32:1930-1939.   10.1111/j.1460-9568.2010.07455.x   AbstractWebsite

Throughout the brain, neurons encode information in fundamental units of spikes. Each spike represents the combined thresholding of synaptic inputs and intrinsic neuronal dynamics. Here, we address a basic question of spike train formation: how do perithreshold synaptic inputs perturb the output of a spiking neuron? We recorded from single entorhinal principal cells in vitro and drove them to spike steadily at similar to 5 Hz (theta range) with direct current injection, then used a dynamic-clamp to superimpose strong excitatory conductance inputs at varying rates. Neurons spiked most reliably when the input rate matched the intrinsic neuronal firing rate. We also found a striking tendency of neurons to preserve their rates and coefficients of variation, independently of input rates. As mechanisms for this rate maintenance, we show that the efficacy of the conductance inputs varied with the relationship of input rate to neuronal firing rate, and with the arrival time of the input within the natural period. Using a novel method of spike classification, we developed a minimal Markov model that reproduced the measured statistics of the output spike trains and thus allowed us to identify and compare contributions to the rate maintenance and resonance. We suggest that the strength of rate maintenance may be used as a new categorization scheme for neuronal response and note that individual intrinsic spiking mechanisms may play a significant role in forming the rhythmic spike trains of activated neurons; in the entorhinal cortex, individual pacemakers may dominate production of the regional theta rhythm.

Abarbanel, HDI.  2009.  Effective actions for statistical data assimilation. Physics Letters A. 373:4044-4048.   10.1016/j.physleta.2009.08.072   AbstractWebsite

Data assimilation is a problem in estimating the fixed parameters and state of a model of an observed dynamical system as it receives inputs from measurements passing information to the model. Using methods developed in statistical physics, we present effective actions and equations of motion for the mean orbits associated with the temporal development of a dynamical model when it has errors, there is uncertainty in its initial state, and it receives information from noisy measurements. If there are statistical dependences among errors in the measurements they can be included in this approach. (C) 2009 Elsevier B.V. All rights reserved.

Abarbanel, HDI, Kennel MB, Illing L, Tang S, Chen HF, Liu JM.  2001.  Synchronization and communication using semiconductor lasers with optoelectronic feedback. IEEE Journal of Quantum Electronics. 37:1301-1311.   10.1109/3.952542   AbstractWebsite

Semiconductor lasers provide an excellent opportunity for communication using chaotic waveforms. We discuss the characteristics and the synchronization of two semiconductor lasers with optoelectronic feedback. The systems exhibit broadband chaotic intensity oscillations whose dynamical dimension generally increases with the time delay in the feedback loop. We explore the robustness of this synchronization with parameter mismatch in the lasers, with mismatch in the optoelectronic feedback delay, and with the strength of the coupling between the systems. Synchronization is robust to mismatches between the intrinsic parameters of the lasers, but it is sensitive to mismatches of the time delay in the transmitter and receiver feedback loops. An open-loop receiver configuration Is suggested, eliminating feedback delay mismatch issues. Communication strategies for arbitrary amplitude of modulation onto the chaotic signals are discussed, and the bit-error rate for one such scheme is evaluated as a function of noise in the optical channel.

Lewis, CT, Abarbanel HDI, Kennel MB, Buhl M, Illing L.  2001.  Synchronization of chaotic oscillations in doped fiber ring lasers. Physical Review E. 63 AbstractWebsite

The synchronization of chaotic rare-earth-doped fiber ring lasers is analyzed. The lasers are first coupled by transmitting a fraction c of the circulating electric field in the transmitter and injecting it into the optical cavity of the receiver. A coupling strategy which relies on modulation of the intensity of the light alone is also examined. Synchronization is studied as a function of the coupling strength, and we see excellent synchronization, even with very small c. We prove that in an open loop configuration (c=1) synchronization is guaranteed due to the particular structure of our equations and of the injection method we use. The generalized synchronization of these model lasers is examined when there is parameter mismatch between the transmitter and receiver lasers. The synchronization is found to be insensitive to a wide range of mismatch in laser parameters, but it is sensitive to other parameters, in particular those associated with the phase and the polarization of the circulating electric field. Communicating information between the transmitter and receiver lasers is also addressed. We investigate a scheme for modulating information onto the chaotic electric field and then demodulating and detecting the information embedded in the chaotic signal passed down the communications channel. We show full recovery with very low error for a wide range of coupling strengths.

Szucs, A, Elson RC, Rabinovich MI, Abarbanel HDI, Selverston AI.  2001.  Nonlinear behavior of sinusoidally forced pyloric pacemaker neurons. Journal of Neurophysiology. 85:1623-1638. AbstractWebsite

Periodic current forcing was used to investigate the intrinsic dynamics of a small group of electrically coupled neurons in the pyloric central pattern generator (CPG) of the lobster. This group contains three neurons, namely the two pyloric dilator (PD) motoneurons and the anterior burster (AB) interneuron. Intracellular current injection, using sinusoidal waveforms of varying amplitude and frequency, was applied in three configurations of the pacemaker neurons: 1) the complete pacemaker group, 2) the two PDs without the AB, and 3) the AB neuron isolated from the PDs. Depending on the frequency and amplitude of the injected current, the intact pacemaker group exhibited a wide variety of nonlinear behaviors, including synchronization to the forcing, quasiperiodicity, and complex dynamics. In contrast, a single, broad 1:1 entrainment zone characterized the response of the PD neurons when isolated from the main pacemaker neuron AB. The isolated AB responded to periodic forcing in a manner similar to the complete pacemaker group, but with wider zones of synchronization. We have built an analog electronic circuit as an implementation of a modified Hindmarsh-Rose model for simulating the membrane potential activity of pyloric neurons. We subjected this electronic model neuron to the same periodic forcing as used in the biological experiments. This four-dimensional electronic model neuron reproduced the autonomous oscillatory firing patterns of biological pyloric pacemaker neurons, and it expressed the same stationary nonlinear responses to periodic forcing as its biological counterparts. This adds to our confidence in the model. These results strongly support the idea that the intact pyloric pacemaker group acts as a uniform low-dimensional deterministic nonlinear oscillator, and the regular pyloric oscillation is the outcome of cooperative behavior of strongly coupled neurons, having different dynamical and biophysical properties when isolated.

Pinto, RD, Varona P, Volkovskii AR, Szucs A, Abarbanel HDI, Rabinovich MI.  2000.  Synchronous behavior of two coupled electronic neurons. Physical Review E. 62:2644-2656.   10.1103/PhysRevE.62.2644   AbstractWebsite

We report on experimental studies of synchronization phenomena in a pair of analog electronic neurons (ENs). The ENs were designed to reproduce the observed membrane voltage oscillations of isolated biological neurons from the stomatogastric ganglion of the California spiny lobster Panulirus interruptus. The ENs are simple analog circuits which integrate four-dimensional differential equations representing fast and slow subcellular mechanisms that produce the characteristic regular/chaotic spiking-bursting behavior of these cells. In this paper we study their dynamical behavior as we couple them in the same configurations as we have done for their counterpart biological neurons. The interconnections we use for these neural oscillators are both direct electrical connections and excitatory and inhibitory chemical connections: each realized by analog circuitry and suggested by biological examples. We provide here quantitative evidence that the ENs and the biological neurons behave similarly when coupled in the same manner. They each display well defined bifurcations in their mutual synchronization and regularization. We report briefly on an experiment on coupled biological neurons and four-dimensional ENs, which provides further ground for testing the validity of our numerical and electronic models of individual neural behavior. Our experiments as a whole present interesting new examples of regularization and synchronization in coupled nonlinear oscillators.

Sarnthein, J, Abarbanel HDI, Pockberger H.  1998.  Nonlinear analysis of epileptic activity in rabbit neocortex. Biological Cybernetics. 78:37-44.   10.1007/s004220050410   AbstractWebsite

We report on the nonlinear analysis of electroencephalogram (EEG) recordings in the rabbit visual cortex. Epileptic seizures were induced by local penicillin application and triggered by visual stimulation. The analysis procedures for nonlinear signals have been developed over the past few years and applied primarily to physical systems. This is an early application to biological systems and the first to EEG data. We find that during epileptic activity, both global and local embedding dimensions are reduced with respect to nonepileptic activity. Interestingly, these values are very low (d(E) approximate to 3) and do not change between preictal and tonic stages of epileptic activity, also the Lyapunov dimension remains constant. However, between these two stages the manifestations of the local dynamics change quite drastically, as can be seen, e.g., from the shape of the attractors. Furthermore, the largest Lyapunov exponent is reduced by a factor of about two in the second stage and characterizes the difference in dynamics. Thus, the occurrence of clinical symptoms associated with the tonic seizure activity seems to be mainly related to the local dynamics of the nonlinear system. These results thus seem to give a strong indication that the dynamics remains much the same in these stages of behavior, and changes are due to alterations in model parameters and consequent bifurcations of the observed orbits.

Lam, BC, Sushchik MM, Abarbanel HDI.  1995.  Relaxation-oscillation-induced chaotic instabilities in modulated external-cavity semiconductor lasers. Journal of the Optical Society of America B-Optical Physics. 12:1150-1156.   10.1364/josab.12.001150   AbstractWebsite

We theoretically examine the consequences of modulating an external-cavity semiconductor laser around its mode-locking resonant frequency. When the modulation frequency is below resonance, the laser exhibits a three-frequency route to chaos. When the modulation frequency is above resonance, the laser oscillates in two- and three-frequency states. The chaotic instability is a result of the nonlinear interaction of three periodic modes of the laser system. These modes are dynamical manifestations of the composite cavity mode-locking resonance, the applied field that is due to the modulation, and the laser relaxation oscillation.