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Drawing on ergodic theory, we introduce a novel training method for machine learning based forecasting methods for chaotic dynamical systems. The training enforces dynamical invariants--such as the Lyapunov exponent spectrum and fractal dimension--in the systems of interest, enabling longer and more stable forecasts when operating with limited data. The technique is demonstrated in detail using the recurrent neural network architecture of reservoir computing. Results are given for the Lorenz 1996 chaotic dynamical system and a spectral quasi-geostrophic model, both typical test cases for numerical weather prediction.

A reservoir computer (RC) is a type of simplified recurrent neural network architecture that has demonstrated success in the prediction of spatiotemporally chaotic dynamical systems. A further advantage of RC is that it reproduces intrinsic dynamical quantities essential for its incorporation into numerical forecasting routines such as the ensemble Kalman filter -- used in numerical weather prediction to compensate for sparse and noisy data. We explore here the architecture and design choices for a "best in class" RC for a number of characteristic dynamical systems, and then show the application of these choices in scaling up to larger models using localization. Our analysis points to the importance of large scale parameter optimization. We also note in particular the importance of including input bias in the RC design, which has a significant impact on the forecast skill of the trained RC model. In our tests, the the use of a nonlinear readout operator does not affect the forecast time or the stability of the forecast. The effects of the reservoir dimension, spinup time, amount of training data, normalization, noise, and the RC time step are also investigated. While we are not aware of a generally accepted best reported mean forecast time for different models in the literature, we report over a factor of 2 increase in the mean forecast time compared to the best performing RC model of Vlachas et.al (2020) for the 40 dimensional spatiotemporally chaotic Lorenz 1996 dynamics, and we are able to accomplish this using a smaller reservoir size.

Data assimilation (DA) is integrated with machine learning in order to perform entirely data-driven online state estimation. To achieve this, recurrent neural networks (RNNs) are implemented as surrogate models to replace key components of the DA cycle in numerical weather prediction (NWP), including the conventional numerical forecast model, the forecast error covariance matrix, and the tangent linear and adjoint models. It is shown how these RNNs can be initialized using DA methods to directly update the hidden/reservoir state with observations of the target system. The results indicate that these techniques can be applied to estimate the state of a system for the repeated initialization of short-term forecasts, even in the absence of a traditional numerical forecast model. Further, it is demonstrated how these integrated RNN-DA methods can scale to higher dimensions by applying domain localization and parallelization, providing a path for practical applications in NWP., Comment: 22 pages, 16 figures

Reservoir computers (RC) are a form of recurrent neural network (RNN) used for forecasting timeseries data. As with all RNNs, selecting the hyperparameters presents a challenge when training onnew inputs. We present a method based on generalized synchronization (GS) that gives direction in designing and evaluating the architecture and hyperparameters of an RC. The 'auxiliary method' for detecting GS provides a computationally efficient pre-training test that guides hyperparameterselection. Furthermore, we provide a metric for RC using the reproduction of the input system's Lyapunov exponentsthat demonstrates robustness in prediction., Comment: Full Version of arXiv:2102.08930

Reservoir computers (RC) are a form of recurrent neural network (RNN) used for forecasting time series data. As with all RNNs, selecting the hyperparameters presents a challenge when training on new inputs. We present a method based on generalized synchronization (GS) that gives direction in designing and evaluating the architecture and hyperparameters of a RC. The 'auxiliary method' for detecting GS provides a pre-training test that guides hyperparameter selection. Furthermore, we provide a metric for a "well trained" RC using the reproduction of the input system's Lyapunov exponents., Comment: This is the Shortened Version of the Paper, the longer paper, Robust Forecasting through Generalized Synchronization in Reservoir Computing, can be found at arXiv:2103.00362

We present a novel machine learning architecture for classification suggested by experiments on olfactory systems. The network separates input stimuli, represented as spatially distinct currents, via winnerless competition---a process based on the intrinsic sequential dynamics of the neural system---then uses a support vector machine (SVM) to provide precision to the space-time separation of the output. The combined network uses biophysical models of neurons and shows high discrimination among inputs and robustness to noise. While using the SVM alone does not permit determination of the components of mixtures of classified inputs, the combined network is able to tell the precise concentrations of the constituent parts.

In statistical data assimilation (SDA) and supervised machine learning (ML), we wish to transfer information from observations to a model of the processes underlying those observations. For SDA, the model consists of a set of differential equations that describe the dynamics of a physical system. For ML, the model is usually constructed using other strategies. In this paper, we develop a systematic formulation based on Monte Carlo sampling to achieve such information transfer. Following the derivation of an appropriate target distribution, we present the formulation based on the standard Metropolis-Hasting (MH) procedure and the Hamiltonian Monte Carlo (HMC) method for performing the high dimensional integrals that appear. To the extensive literature on MH and HMC, we add (1) an annealing method using a hyperparameter that governs the precision of the model to identify and explore the highest probability regions of phase space dominating those integrals, and (2) a strategy for initializing the state space search. The efficacy of the proposed formulation is demonstrated using a nonlinear dynamical model with chaotic solutions widely used in geophysics.

Tasking machine learning to predict segments of a time series requires estimating the parameters of a ML model with input/output pairs from the time series. Using the equivalence between statistical data assimilation and supervised machine learning, we revisit this task. The training method for the machine utilizes a precision annealing approach to identifying the global minimum of the action (-log[P]). In this way we are able to identify the number of training pairs required to produce good generalizations (predictions) for the time series. We proceed from a scalar time series $s(t_n); t_n = t_0 + n \Delta t$ and using methods of nonlinear time series analysis show how to produce a $D_E > 1$ dimensional time delay embedding space in which the time series has no false neighbors as does the observed $s(t_n)$ time series. In that $D_E$-dimensional space we explore the use of feed forward multi-layer perceptrons as network models operating on $D_E$-dimensional input and producing $D_E$-dimensional outputs.

Statistical Data Assimilation (SDA) is the transfer of information from field or laboratory observations to a user selected model of the dynamical system producing those observations. The data is noisy and the model has errors; the information transfer addresses properties of the conditional probability distribution of the states of the model conditioned on the observations. The quantities of interest in SDA are the conditional expected values of functions of the model state, and these require the approximate evaluation of high dimensional integrals. We introduce a conditional probability distribution and use the Laplace method with annealing to identify the maxima of the conditional probability distribution. The annealing method slowly increases the precision term of the model as it enters the Laplace method. In this paper, we extend the idea of precision annealing (PA) to Monte Carlo calculations of conditional expected values using Metropolis-Hastings methods.

In statistical data assimilation one seeks the largest maximum of the conditional probability distribution $P(\mathbf{X},\mathbf{p}|\mathbf{Y})$ of model states, $\mathbf{X}$, and parameters,$\mathbf{p}$, conditioned on observations $\mathbf{Y}$ through minimizing the `action', $A(\mathbf{X}) = -\log P(\mathbf{X},\mathbf{p}|\mathbf{Y})$. This determines the dominant contribution to the expected values of functions of $\mathbf{X}$ but does not give information about the structure of $P(\mathbf{X},\mathbf{p}|\mathbf{Y})$ away from the maximum. We introduce a Monte Carlo sampling method, called Strategic Monte Carlo (SMC) sampling, for estimating $P(\mathbf{X}, \mathbf{p}|\mathbf{Y})$ in the neighborhood of its largest maximum to remedy this limitation. SMC begins with a systematic variational annealing (VA) procedure for finding the smallest minimum of $A(\mathbf{X})$. SMC generates accurate estimates for the mean, standard deviation and other higher moments of $P(\mathbf{X},\mathbf{p}|\mathbf{Y})$. Additionally, the random search allows for an understanding of any multimodal structure that may underly the dynamics of the problem. SMC generates a gaussian probability control term based on the paths determined by VA to minimize a cost function $A(\mathbf{X},\mathbf{p})$. This probability is sampled during the Monte Carlo search of the cost function to constrain the search to high probability regions of the surface thus substantially reducing the time necessary to sufficiently explore the space.

We assess the utility of an optimization-based data assimilation (D.A.) technique for treating the problem of nonlinear neutrino flavor transformation in core collapse supernovae. D.A. uses measurements obtained from a physical system to estimate the state variable evolution and parameter values of the associated model. Formulated as an optimization procedure, D.A. can offer an integration-blind approach to predicting model evolution, which offers an advantage for models that thwart solution via traditional numerical integration techniques. Further, D.A. performs most optimally for models whose equations of motion are nonlinearly coupled. In this exploratory work, we consider a simple steady-state model with two mono-energetic neutrino beams coherently interacting with each other and a background medium. As this model can be solved via numerical integration, we have an independent consistency check for D.A. solutions. We find that the procedure can capture key features of flavor evolution over the entire trajectory, even given measurements of neutrino flavor only at the endpoint, and with an assumed known initial flavor distribution. Further, the procedure permits an examination of the sensitivity of flavor evolution to estimates of unknown model parameters, locates degeneracies in parameter space, and can identify the specific measurements required to break those degeneracies., Comment: 26 pages, 4 figures

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: (A) simultaneous firing wherein all inhibitory neurons fire simultaneously, and (B) 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 (A) the activity of projection neurons is suppressed, while during state (B) 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., Comment: 20 pages, 9 figures. Submitted to the Journal of Neurophysiology

Most data based state and parameter estimation methods require suitable initial values or guesses to achieve convergence to the desired solution, which typically is a global minimum of some cost function. Unfortunately, however, other stable solutions (e.g., local minima) may exist and provide suboptimal or even wrong estimates. Here we demonstrate for a 9-dimensional Lorenz-96 model how to characterize the basin size of the global minimum when applying some particular optimization based estimation algorithm. We compare three different strategies for generating suitable initial guesses and we investigate the dependence of the solution on the given trajectory segment (underlying the measured time series). To address the question of how many state variables have to be measured for optimal performance, different types of multivariate time series are considered consisting of 1, 2, or 3 variables. Based on these time series the local observability of state variables and parameters of the Lorenz-96 model is investigated and confirmed using delay coordinates. This result is in good agreement with the observation that correct state and parameter estimation results are obtained if the optimization algorithm is initialized with initial guesses close to the true solution. In contrast, initialization with other exact solutions of the model equations (different from the true solution used to generate the time series) typically fails, i.e. the optimization procedure ends up in local minima different from the true solution. Initialization using random values in a box around the attractor exhibits success rates depending on the number of observables and the available time series (trajectory segment)., Comment: 15 pages, 2 figures

Utilizing the information in observations of a complex system to make accurate predictions through a quantitative model when observations are completed at time $T$, requires an accurate estimate of the full state of the model at time $T$. When the number of measurements $L$ at each observation time within the observation window is larger than a sufficient minimum value $L_s$, the impediments in the estimation procedure are removed. As the number of available observations is typically such that $L \ll L_s$, additional information from the observations must be presented to the model. We show how, using the time delays of the measurements at each observation time, one can augment the information transferred from the data to the model, removing the impediments to accurate estimation and permitting dependable prediction. We do this in a core geophysical fluid dynamics model, the shallow water equations, at the heart of numerical weather prediction. The method is quite general, however, and can be utilized in the analysis of a broad spectrum of complex systems where measurements are sparse. When the model of the complex system has errors, the method still enables accurate estimation of the state of the model and thus evaluation of the model errors in a manner separated from uncertainties in the data assimilation procedure.

The answers to data assimilation questions can be expressed as path integrals over all possible state and parameter histories. We show how these path integrals can be evaluated numerically using a Markov Chain Monte Carlo method designed to run in parallel on a Graphics Processing Unit (GPU). We demonstrate the application of the method to an example with a transmembrane voltage time series of a simulated neuron as an input, and using a Hodgkin-Huxley neuron model. By taking advantage of GPU computing, we gain a parallel speedup factor of up to about 300, compared to an equivalent serial computation on a CPU, with performance increasing as the length of the observation time used for data assimilation increases., Comment: 5 figures, submitted to Journal of Computational Physics

In using data assimilation to import information from observations to estimate parameters and state variables of a model, one must assume a distribution for the noise in the measurements and in the model errors. Using the path integral formulation of data assimilation~ cite{abar2009}, we introduce the idea of self consistency of the distribution of stochastic model errors: the distribution of model errors from the path integral with observed data should be consistent with the assumption made in formulating the the path integral. The path integral setting for data assimilation is discussed to provide the setting for the consistency test. Using two examples drawn from the 1996 Lorenz model, for $D = 100$ and for $D = 20$ we show how one can test for this inconsistency with essential no additional effort than that expended in extracting answers to interesting questions from data assimilation itself. \end{abstract}

Transferring information from observations of a dynamical system to estimate the fixed parameters and unobserved states of a system model can be formulated as the evaluation of a discrete time path integral in model state space. The observations serve as a guiding potential working with the dynamical rules of the model to direct system orbits in state space. The path integral representation permits direct numerical evaluation of the conditional mean path through the state space as well as conditional moments about this mean. Using a Monte Carlo method for selecting paths through state space we show how these moments can be evaluated and demonstrate in an interesting model system the explicit influence of the role of transfer of information from the observations. We address the question of how many observations are required to estimate the unobserved state variables, and we examine the assumptions of Gaussianity of the underlying conditional probability., Comment: Submitted to the Quarterly Journal of the Royal Meteorological Society, 19 pages, 5 figures

Ensemble data assimilation is a problem in determining the most likely phase space trajectory of a model of an observed dynamical sys- tem 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 associ- ated with the temporal development of a dynamical model when it has errors, there is uncertainty in its initial state, and it receives informa- tion from measurements. If there are correlations among errors in the measurements they are naturally included in this approach., Comment: 10 pages

We examine the use of synchronization as a mechanism for extracting parameter and state information from experimental systems. We focus on important aspects of this problem that have received little attention previously, and we explore them using experiments and simulations with the chaotic Colpitts oscillator as an example system. We explore the impact of model imperfection on the ability to extract valid information from an experimental system. We compare two optimization methods: an initial value method and a constrained method. Each of these involve coupling the model equations to the experimental data in order to regularize the chaotic motions on the synchronization manifold. We explore both time dependent and time independent coupling. We also examine both optimized and fixed (or manually adjusted) coupling. For the case of an optimized time dependent coupling function u(t) we find a robust structure which includes sharp peaks and intervals where it is zero. This structure shows a strong correlation with the location in phase space and appears to depend on noise, imperfections of the model, and the Lyapunov direction vectors. Comparison of this result with that obtained using simulated data may provide one measure of model imperfection. The constrained method with time dependent coupling appears to have benefits in synchronizing long datasets with minimal impact, while the initial value method with time independent coupling tends to be substantially faster, more flexible and easier to use. We also describe a new method of coupling which is useful for sparse experimental data sets. Our use of the Colpitts oscillator allows us to explore in detail the case of a system with one positive Lyapunov exponent. The methods we explored are easily extended to driven systems such as neurons with time dependent injected current., Comment: submitted to Physical Review E

Estimating the degree of synchrony or reliability between two or more spike trains is a frequent task in both experimental and computational neuroscience. In recent years, many different methods have been proposed that typically compare the timing of spikes on a certain time scale to be fixed beforehand. Here, we propose the ISI-distance, a simple complementary approach that extracts information from the interspike intervals by evaluating the ratio of the instantaneous frequencies. The method is parameter free, time scale independent and easy to visualize as illustrated by an application to real neuronal spike trains obtained in vitro from rat slices. In a comparison with existing approaches on spike trains extracted from a simulated Hindemarsh-Rose network, the ISI-distance performs as well as the best time-scale-optimized measure based on spike timing., Comment: 11 pages, 13 figures; v2: minor modifications; v3: minor modifications, added link to webpage that includes the Matlab Source Code for the method (http://inls.ucsd.edu/~kreuz/Source-Code/Spike-Sync.html)

Sensory systems pass information about an animal's environment to higher nervous system units through sequences of action potentials. When these action potentials have essentially equivalent waveforms, all information is contained in the interspike intervals (ISIs) of the spike sequence. We address the question: How do neural circuits recognize and read these ISI sequences? Our answer is given in terms of a biologically inspired neural circuit that we construct using biologically realistic neurons. The essential ingredients of the ISI Reading Unit (IRU) are (i) a tunable time delay circuit modelled after one found in the anterior forebrain pathway of the birdsong system and (ii) a recently observed rule for inhibitory synaptic plasticity. We present a circuit that can both learn the ISIs of a training sequence using inhibitory synaptic plasticity and then recognize the same ISI sequence when it is presented on subsequent occasions. We investigate the ability of this IRU to learn in the presence of two kinds of noise: jitter in the time of each spike and random spikes occurring in the ideal spike sequence. We also discuss how the circuit can be detuned by removing the selected ISI sequence and replacing it by an ISI sequence with ISIs drawn from a probability distribution. We have investigated realizations of the time delay circuit using Hodgkin-Huxley conductance based neurons connected by realistic excitatory and inhibitory synapses. Our models for the time delay circuit are tunable from about 10 ms to 100 ms allowing one to learn and recognize ISI sequences within that range of ISIs. ISIs down to a few ms and longer than 100 ms are possible with other intrinsic and synaptic currents in the component neurons.

Environmental signals sensed by nervous systems are often represented in spike trains carried from sensory neurons to higher neural functions where decisions and functional actions occur. Information about the environmental stimulus is contained (encoded) in the train of spikes. We show how to "read" the encoding using state space methods of nonlinear dynamics. We create a mapping from spike signals which are output from the neural processing system back to an estimate of the analog input signal. This mapping is realized locally in a reconstructed state space embodying both the dynamics of the source of the sensory signal and the dynamics of the neural circuit doing the processing. We explore this idea using a Hodgkin-Huxley conductance based neuron model and input from a low dimensional dynamical system, the Lorenz system. We show that one may accurately learn the dynamical input/output connection and estimate with high precision the details of the input signals from spike timing output alone. This form of "reading the neural code" has a focus on the neural circuitry as a dynamical system and emphasizes how one interprets the dynamical degrees of freedom in the neural circuit as they transform analog environmental information into spike trains., Comment: Published in Springer Applied Mathematical Sciences Series Celebratory Volume for the Occasion of the 70th birthday of Larry Sirovich

We study the synchronization of two model neurons coupled through a synapse having an activity-dependent strength. Our synapse follows the rules of Spike-Timing Dependent Plasticity (STDP). We show that this plasticity of the coupling between neurons produces enlarged frequency locking zones and results in synchronization that is more rapid and much more robust against noise than classical synchronization arising from connections with constant strength. We also present a simple discrete map model that demonstrates the generality of the phenomenon., Comment: 4 pages, accepted for publication in PRE

We suggest a mechanism based on spike time dependent plasticity (STDP) of synapses to store, retrieve and predict temporal sequences. The mechanism is demonstrated in a model system of simplified integrate-and-fire type neurons densely connected by STDP synapses. All synapses are modified according to the so-called normal STDP rule observed in various real biological synapses. After conditioning through repeated input of a limited number of of temporal sequences the system is able to complete the temporal sequence upon receiving the input of a fraction of them. This is an example of effective unsupervised learning in an biologically realistic system. We investigate the dependence of learning success on entrainment time, system size and presence of noise. Possible applications include learning of motor sequences, recognition and prediction of temporal sensory information in the visual as well as the auditory system and late processing in the olfactory system of insects., Comment: 13 pages, 14 figures, completely revised and augmented version

In the analysis of time series from nonlinear sources, mutual information (MI) is used as a nonlinear statistical criterion for the selection of an appropriate time delay in time delay reconstruction of the state space. MI is a statistic over the sets of sequences associated with the dynamical source, and we examine here the distribution of MI, thus going beyond the familiar analysis of its average alone. We give for the first time the distribution of MI for a standard, classical communications channel with Gaussian, additive white noise. For time series analysis of a dynamical system, we show how to determine the distribution of MI and discuss the implications for the use of average mutual information (AMI) in selecting time delays in phase space reconstruction., Comment: 4 pages, 2 figures, RevTeX V4b4, submitted to Phys. Rev. Lett

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., Comment: 26 pages, 10 figures

Central Pattern Generators (CPGs) in invertebrates are comprised of networks of neurons in which every neuron has reciprocal connections to other members of the CPG. This is a ``closed'' network topology. An ``open'' topology, where one or more neurons receives input but does not send output to other member neurons, is not found in these CPGs. In this paper we investigate a possible reason for this topological structure using the ability to perform a biological functional task as a measure of the efficacy of the network. When the CPG is composed of model neurons which exhibit regular membrane voltage oscillations, open topologies are essentially as able to maximize this functionality as closed topologies. When we replace these models by neurons which exhibit chaotic membrane voltage oscillations, the functional criterion selects closed topologies when the demands of the task are increased, and these are the topologies observed in known CPG networks. As isolated neurons from invertebrate CPGs are known in some cases to undergo chaotic oscillations (Abarbanel et al 1996, Hayashi Ishuzuka 1992) this provides a biological basis for understanding the class of closed network topologies we observe., Comment: 16 pages, 6 figures

We investigate synchronization and subsequently communication using chaotic rare-earth-doped fiber ring lasers, represented by a physically realistic model. The lasers are coupled by transmitting a fraction c of the circulating electric field in the transmitter and injecting it into the optical cavity of the receiver. We then analyze a coupling strategy which relies on modulation of the intensity of the light alone. This avoids problems associated with the polarization and phase of the laser light. We study synchronization as a function of the coupling strength and see excellent convergence, even with small coupling constants. 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 for these coupled laser systems. We also analyze the generalized synchronization of these model lasers when there is parameter mismatch between the transmitter and the receiver. We then address communicating information between the transmitter and receiver lasers by modulation of the chaotic optical waveform. Finally we comment on the cryptographic setting of our algorithms, especially the open-loop strategy at c=1, and hope this may lead others to perform the cryptographic analyses to determine which, if any, of the communications strategies we investigate are secure., Comment: 37 pages, 25 encapsulated postscript figures, 1 table. To be submitted to Phys. Rev. E

Studies of insect olfactory processing indicate that odors are represented by rich spatio-temporal patterns of neural activity. These patterns are very difficult to predict a priori, yet they are stimulus specific and reliable upon repeated stimulation with the same input. We formulate here a theoretical framework in which we can interpret these experimental results. We propose a paradigm of ``dynamic competition'' in which inputs (odors) are represented by internally competing neural assemblies. Each pattern is the result of dynamical motion within the network and does not involve a ``winner'' among competing possibilities. The model produces spatio-temporal patterns with strong resemblance to those observed experimentally and possesses many of the general features one desires for pattern classifiers: large information capacity, reliability, specific responses to specific inputs, and reduced sensitivity to initial conditions or influence of noise. This form of neural processing may thus describe the organizational principles of neural information processing in sensory systems and go well beyond the observations on insect olfactory processing which motivated its development., Comment: 19 pages, 16 figures. Originally submitted to the neuro-sys archive which was never publicly announced (was 9905002)

We report experimental studies of synchronization phenomena in a pair of biological neurons that interact through naturally occurring, electrical coupling. When these neurons generate irregular bursts of spikes, the natural coupling synchronizes slow oscillations of membrane potential, but not the fast spikes. By adding artificial electrical coupling we studied transitions between synchrony and asynchrony in both slow oscillations and fast spikes. We discuss the dynamics of bursting and synchronization in living neurons with distributed functional morphology., Comment: 4 pages, 6 figures, to be published in PRL

It is frequently asserted that in a chaotic system two initially close points will separate at an exponential rate governed by the largest global Lyapunov exponent. Local Lyapunov exponents, however, are more directly relevant to predictability. The difference between the local and global Lyapunov exponents, the large variations of local exponents over an attractor, and the saturation of error growth near the size of the attractor---all result in non-exponential scalings in errors at both short and long prediction times, sometimes even obscuring evidence of exponential growth. Failure to observe exponential error scaling cannot rule out deterministic chaos as an explanation. We demonstrate a simple model that quantitatively predicts observed error scaling from the local Lyapunov exponents, for both short and surprisingly long times. We comment on the relevance to atmospheric predictability as studied in the meteorological literature., Comment: In REVTeX format, followed by uuencoded, compressed ".tar.Z" file which includes Postscript version of manuscript and 5 Postscript figures. PACS:{05.45.+b,92.60.Wc}, email:[mbk,hdia,sid]@inls1.ucsd.edu