The field of chaotic synchronization has grown considerably since its advent in 1990. Several subdisciplines and "cottage industries" have emerged that have taken on bona fide lives of their own. Our purpose in this paper is to collect results from these various areas in a review article format with a tutorial emphasis. Fundamentals of chaotic synchronization are reviewed first with emphases on the geometry of synchronization and stability criteria. Several widely used coupling configurations are examined and, when available, experimental demonstrations of their success (generally with chaotic circuit systems) are described. Particular focus is given to the recent notion of synchronous substitution-a method to synchronize chaotic systems using a larger class of scalar chaotic coupling signals than previously thought possible. Connections between this technique and well-known control theory results are also outlined. Extensions of the technique are presented that allow so-called hyperchaotic systems (systems with more than one positive Lyapunov exponent) to be synchronized. Several proposals for "secure" communication schemes have been advanced; major ones are reviewed and their strengths and weaknesses are touched upon. Arrays of coupled chaotic systems have received a great deal of attention lately and have spawned a host of interesting and, in some cases, counterintuitive phenomena including bursting above synchronization thresholds, destabilizing transitions as coupling increases (short-wavelength bifurcations), and riddled basins. In addition, a general mathematical framework for analyzing the stability of arrays with arbitrary coupling configurations is outlined. Finally, the topic of generalized synchronization is discussed, along with data analysis techniques that can be used to decide whether two systems satisfy the mathematical requirements of generalized synchronization. (c) 1997 American Institute of Physics.

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Noise can assist neurons in the detection of weak signals via a mechanism known as stochastic resonance (SR). We demonstrate experimentally that SR-type effects can be obtained in rat sensory neurons with white noise, 1͞f noise, or 1͞f 2 noise. For low-frequency input noise, we show that the optimal noise intensity is the lowest and the output signal-to-noise ratio the highest for conventional white noise. We also show that under certain circumstances, 1͞f noise can be better than white noise for enhancing the response of a neuron to a weak signal. We present a theory to account for these results and discuss the biological implications of 1͞f noise. [S0031-9007(99)08727-X]

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Biological information-processing systems, such as populations of sensory and motor neurons, may use correlations between the firings of individual elements to obtain lower noise levels and a systemwide performance improvement in the dynamic range or the signal-to-noise ratio. Here, we implement such correlations in networks of coupled integrate-and-fire neurons using inhibitory coupling and demonstrate that this can improve the system dynamic range and the signal-to-noise ratio in a population rate code. The improvement can surpass that expected for simple averaging of uncorrelated elements. A theory that predicts the resulting power spectrum is developed in terms of a stochastic point-process model in which the instantaneous population firing rate is modulated by the coupling between elements.An important issue in neuroscience is how neurons encode information (1-8). Here, we consider the problem of encoding an analog signal in the firing times of a population of neurons. Several factors must be accounted for in addressing this issue. First, the firing records of neurons are often noisy and irregular (8, 9). Second, cortical neurons sometimes fire relatively slowly compared to many of the signals they may need to encode (10, 11).We explore a method for population rate coding by which a network of coupled noisy neurons can encode relatively highfrequency signals. We consider a system of N neurons that receive the same analog input. The relevant output is the population firing rate F N (t), the number of neuronal firings per unit time summed across the population. This quantity does not require averaging over a significant time window, and it can respond quickly to rapidly changing inputs (12-15). For neurons firing asynchronously, F N is approximately N times the single-neuron rate. The input signal is encoded in the modulation of the firing times of the neurons in the network. When the analog input is converted to a train of discrete spike events, ''quantization'' noise from the errors made in digitization is unavoidable and limits the fidelity with which the output signal can be decoded. For N uncoupled, independent neurons, the quantization noise power grows as N, and the coherent signal power grows as N 2 . The system signal-to-noise ratio (SNR), defined as the ratio of the output signal power to the noise power, grows as N. The SNR is improved if either the output signal is increased or the noise is reduced. Some systems are limited in the maximal signal power that they can process. In such cases, another important figure of merit is the system dynamic range (DR), which we take to be the ratio between the maximum signal power the system can tolerate and the noise power. For a fixed maximum output signal power, improving the DR is equivalent to reducing the noise. An example of a system requiring high DR is the human auditory system, which processes signals ranging from a soft whisper to a loud jet engine.We propose a method to improve the DR and SNR for a population rate code beyond simple averagin...