Henry D. I. Abarbanel

Synchronous Behavior of Two Coupled Biological Neurons

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. [S0031-9007(98)08008-9] The dynamics of many neural ensembles such as central pattern generators (CPGs) or thalamo-cortical circuits pose questions related to cooperative behavior of neurons. Individual neurons may show irregular behavior [1], while ensembles of different neurons can synchronize in order to process biological information [2] or to produce regular, rhythmical activity [3]. How do the irregular neurons synchronize? How do they inhibit noise and intrinsic fluctuations? What parameters of the ensemble are responsible for such synchronization and regularization? Answers to these and similar questions may be found through experiments that enable one to follow qualitatively the cooperative dynamics of neurons as intrinsic and synaptic parameters are varied. Despite their interest, these problems have not received extensive study. Results of such an experiment for a minimal ensemble of two coupled, living neurons are reported in this communication. The experiment was carried out on two electrically coupled neurons (the pyloric dilators, PD) from the pyloric CPG of the lobster stomatogastric ganglion [3]. Individually, these neurons can generate spiking-bursting activity that is irregular and seemingly chaotic. This activity pattern can be altered by injecting dc current (I1 and I2) into the neurons; see Fig. 1. In parallel to their natural coupling, we added artificial coupling by a dynamic current clamp device [7]. Varying these control parameters (offset current and artificial coupling), we found the following regimes of cooperative behavior. Natural coupling produces state-dependent synchronization; see Fig. 2. (i) When depolarized by positive dc current, both neurons fire a continuous pattern of synchronized spikes (Fig. 2d). (ii) With little or no applied current, the neurons fire spikes in irregular bursts: now the slow oscillations are well synchronized while spikes are not (Fig. 2a). Changing the magnitude and sign of electrical coupling restructures the cooperative dynamics. (iii) Increasing the strength of coupling produces complete synchronization of both irregular slow oscillations and fast spikes (see below). (iv) Compensating the natural coupling leads to the onset

Model of Transient Oscillatory Synchronization in the Locust Antennal Lobe

Transient pairwise synchronization of locust antennal lobe (AL) projection neurons (PNs) occurs during odor responses. In a Hodgkin-Huxley-type model of the AL, interactions between excitatory PNs and inhibitory local neurons (LNs) created coherent network oscillations during odor stimulation. GABAergic interconnections between LNs led to competition among them such that different groups of LNs oscillated with periodic Ca(2+) spikes during different 50-250 ms temporal epochs, similar to those recorded in vivo. During these epochs, LN-evoked IPSPs caused phase-locked, population oscillations in sets of postsynaptic PNs. The model shows how alternations of the inhibitory drive can temporally encode sensory information in networks of neurons without precisely tuned intrinsic oscillatory properties.

Measuring spike train synchrony

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 optimized by the analyst. Here, we propose the ISI-distance, a simple complementary approach that extracts information from the interspike intervals by evaluating the ratio of the instantaneous firing rates. 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.

Model of Cellular and Network Mechanisms for Odor-Evoked Temporal Patterning in the Locust Antennal Lobe

Locust antennal lobe (AL) projection neurons (PNs) respond to olfactory stimuli with sequences of depolarizing and hyperpolarizing epochs, each lasting hundreds of milliseconds. A computer simulation of an AL network was used to test the hypothesis that slow inhibitory connections between local neurons (LNs) and PNs are responsible for temporal patterning. Activation of slow inhibitory receptors on PNs by the same GABAergic synapses that underlie fast oscillatory synchronization of PNs was sufficient to shape slow response modulations. This slow stimulus- and neuron-specific patterning of AL activity was resistant to blockade of fast inhibition. Fast and slow inhibitory mechanisms at synapses between LNs and PNs can thus form dynamical PN assemblies whose elements synchronize transiently and oscillate collectively, as observed not only in the locust AL, but also in the vertebrate olfactory bulb.

Chaotic pulse position modulation: a robust method of communicating with chaos

In this letter we investigate a communication strategy for digital ultra-wide bandwidth impulse radio, where the separation between the adjacent pulses is chaotic arising from a dynamical system with irregular behavior. A pulse position method is used to modulate binary information onto the carrier. The receiver is synchronized to the chaotic pulse train, thus providing the time reference for information extraction. We characterize the performance of this scheme in terms of error probability versus E/sub b//N/sub 0/ by numerically simulating its operation in the presence of noise and filtering.

LYAPUNOV EXPONENTS IN CHAOTIC SYSTEMS: THEIR IMPORTANCE AND THEIR EVALUATION USING OBSERVED DATA

We review the idea of Lyapunov exponents for chaotic systems and discuss their evaluation from observed data alone. These exponents govern the growth or decrease of small perturbations to orbits of a dynamical system. They are critical to the predictability of models made from observations as well as known analytic models. The Lyapunov exponents are invariants of the dynamical system and are connected with the dimension of the system attractor and to the idea of information generation by the system dynamics. Lyapunov exponents are among the many ways we can classify observed nonlinear systems, and their appeal to physicists remains their clear interpretation in terms of system stability and predictability. We discuss the familiar global Lyapunov exponents which govern the evolution of perturbations for long times and local Lyapunov exponents which determine the predictability over a finite number of time steps.

Synchronous behavior of two coupled electronic neurons

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.

Variation of Lyapunov exponents on a strange attractor

SummaryWe introduce the idea of local Lyapunov exponents which govern the way small perturbations to the orbit of a dynamical system grow or contract after afinite number of steps,L, along the orbit. The distributions of these exponents over the attractor is an invariant of the dynamical system; namely, they are independent of the orbit or initial conditions. They tell us the variation of predictability over the attractor. They allow the estimation of extreme excursions of perturbations to an orbit once we know the mean and moments about the mean of these distributions. We show that the variations about the mean of the Lyapunov exponents approach zero asL → ∞ and argue from our numerical work on several chaotic systems that this approach is asL−v. In our examplesv ≈ 0.5–1.0. The exponents themselves approach the familiar Lyapunov spectrum in this same fashion.

Nonlinear Dynamics of the Great Salt Lake: Dimension Estimation

We study the possibility that variations in the volume of the Great Salt Lake (GSL), a large, closed basin lake, may be described as a low-dimensional nonlinear dynamical system. There is growing evidence for structure in the recurrence patterns of climatic fluctuations that drive western United States hydrology. Moreover, the time behavior of such lakes is generally more regular than that of the climatic forcing. This suggests the possibility that an analysis of the 144-year, biweekly time series of the GSL volume may shed some light on the underlying dynamics of lake variations. Three methods (correlation dimension, nearest neighbor dimension, and false neighbor dimension) of estimating attractor dimension are applied and compared. The analysis suggests that the GSL dynamics may be described by a dimension of about four. Implications of such analyses relative to low-frequency variations and colored noise and limitations of such analyses are discussed.

Synchronized action of synaptically coupled chaotic model neurons

Experimental observations of the intracellular recorded electrical activity in individual neurons show that the temporal behavior is often chaotic. We discuss both our own observations on a cell from the stom-atogastric central pattern generator of lobster and earlier observations in other cells. In this paper we work with models of chaotic neurons, building on models by Hindmarsh and Rose for bursting, spiking activity in neurons. The key feature of these simplified models of neurons is the presence of coupled slow and fast subsystems. We analyze the model neurons using the same tools employed in the analysis of our experimental data. We couple two model neurons both electrotonically and electrochemically in inhibitory and excitatory fashions. In each of these cases, we demonstrate that the model neurons can synchronize in phase and out of phase depending on the strength of the coupling. For normal synaptic coupling, we have a time delay between the action of one neuron and the response of the other. We also analyze how the synchronization depends on this delay. A rich spectrum of synchronized behaviors is possible for electrically coupled neurons and for inhibitory coupling between neurons. In synchronous neurons one typically sees chaotic motion of the coupled neurons. Excitatory coupling produces essentially periodic voltage trajectories, which are also synchronized. We display and discuss these synchronized behaviors using two distance measures of the synchronization.