Partners

Abstract:
Even in the absence of external stimuli there is ongoing activity in the cerebral cortex as a result of recurrent connectivity. This paper attempts to characterize one aspect of this ongoing activity by examining how the information content carried by specific neurons varies as a function of brain state. We recorded from rats chronically implanted with tetrodes in the primary visual cortex during awake and sleep periods. Electro-encephalogram and spike trains were recorded during 30-min periods, and 2–4 neuronal spikes were isolated per tetrode off-line. All the activity included in the analysis was spontaneous, being recorded from the visual cortex in the absence of visual stimuli. The brain state was determined through a combination of behavior evaluation, electroencephalogram and electromyogram analysis. Information in the spike trains was determined by using Lempel–Ziv Complexity. Complexity was used to estimate the entropy of neural discharges and thus the information content (Amigóet al. Neural Comput., 2004, 16: 717–736). The information content in spike trains (range 4–70 bits s−1) was evaluated during different brain states and particularly during the transition periods. Transitions toward states of deeper sleep coincided with a decrease of information, while transitions to the awake state resulted in an increase in information. Changes in both directions were of the same magnitude, about 30%. Information in spike trains showed a high temporal correlation between neurons, reinforcing the idea of the impact of the brain state in the information content of spike trains.

Keywords:
awake, brain states, entropy, firing rate, information, sleep, spike train

Abstract:
An important question in neural information processing is how neurons cooperate to transmit information. To study this question, we resort to the concept of redundancy in the information transmitted by a group of neurons and, at the same time, we introduce a novel concept for measuring cooperation between pairs of neurons called relative mutual information (RMI). Specifically, we studied these two parameters for spike trains generated by neighboring neurons from the primary visual cortex in the awake, freely moving rat. The spike trains studied here were spontaneously generated in the cortical network, in the absence of visual stimulation. Under these conditions, our analysis revealed that while the value of RMI oscillated slightly around an average value, the redundancy exhibited a behavior characterized by a higher variability. We conjecture that this combination of approximately constant RMI and greater variable redundancy makes information transmission more resistant to noise disturbances. Furthermore, the redundancy values suggest that neurons can cooperate in a flexible way during information transmission. This mostly occurs via a leading neuron with higher transmission rate or, less frequently, through the information rate of the whole group being higher than the sum of the individual information rates—in other words in a synergetic manner. The proposed method applies not only to the stationary, but also to locally stationary neural signals.

Keywords:
Neurons, Shannon information, Entropy, Mutual information, Redundancy, Visual cortex, Spikes train, Spontaneous activity

Abstract:
One of the possible by-products of discrete chaos is the application of its tools, in particular of the discrete Lyapunov exponent, to cryptography. In this Letter we explore this question in a very general setting.

Abstract:
In this Letter we address some basic questions about chaotic cryptography, not least the very definition of chaos in discrete systems. We propose a conceptual framework and illustrate it with different examples from private and public key cryptography. We elaborate also on possible limits of chaotic cryptography.

Abstract:
In a recent paper, Jakimoski and Subbalakshmi provided a nice connection between the so-called discrete Lyapunov exponent of a permutation F defined on a finite lattice and its maximal differential probability, a parameter that measures the complexity of a differential cryptanalysis attack on the substitution defined by F. In this brief, we take a second look at their result to find some practical shortcomings. We also discuss more general aspects.

Keywords:
Differential cryptanalysis, discrete Lyapunov exponent (DLE), maximum differential probability (DP)

Abstract:
Chaotic maps can mimic random behavior in a quite impressive way. In particular, those possessing a generating partition can produce any symbolic sequence by properly choosing the initial state. We study in this Letter the ability of chaotic maps to generate order patterns and come to the conclusion that their performance in this respect falls short of expectations. This result reveals some basic limitation of a deterministic dynamic as compared to a random one. This being the case, we propose a non-statistical test based on ‘forbidden’ order patterns to discriminate chaotic from truly random time series with, in principle, arbitrarily high probability. Some relations with discrete chaos and chaotic cryptography are also discussed.

Keywords:
Chaotic maps, Order patterns, Permutation entropy, Discrete Lyapunov exponent, Chaotic cryptography

Abstract:
We propose a definition of the discrete Lyapunov exponent for an arbitrary permutation of a finite lattice. For discrete-time dynamical systems, it measures the local (between neighboring points) average spreading of the system. We justify our definition by proving that, for large classes of chaotic maps, the corresponding discrete Lyapunov exponent approaches the largest Lyapunov exponent of a chaotic map when Mrarrinfin, where M is the cardinality of the discrete phase space. In analogy with continuous systems, we say the system has discrete chaos if its discrete Lyapunov exponent tends to a positive number, when Mrarrinfin. We present several examples to illustrate the concepts being introduced.

Keywords:
Chaos, discrete chaos, Lyapunov components

Abstract:
We show that chaotic maps may be used for designing so-called substitution boxes for ciphers resistant to linear and differential cryptanalysis, providing an alternative to the algebraic methods. Our approach is based on the approximation of mixing maps by periodic transformations. The expectation behind is, of course, that the nice chaotic properties of such maps will be inherited by their approximations, at least if the convergence rate is appropriate and the associated partitions are sufficiently fine. We show that this is indeed the case and that, in principle, substitutions with close-to-optimal immunity to linear and differential cryptanalysis can be designed along these guidelines. We provide also practical examples and numerical evidence for this approximation philosophy

Keywords:
Chaotic maps, Periodic approximations, Bit permutations, Cryptanalysis

Abstract:
In this paper, we explore, following Shannon’s suggestion that diffusion should be one of the ingredients of resistant block ciphers, the feasibility of designing cryptographically secure substitutions (think of S-boxes, say) via approximation of mixing maps by periodic transformations. The expectation behind this approach is, of course, that the nice diffusion properties of such maps will be inherited by their approximations, at least if the convergence rate is appropriate and the associated partitions are sufficiently fine. Our results show that this is indeed the case and that, in principle, block ciphers with close-to-optimal immunity to linear and differential cryptanalysis (as measured by the linear and differential approximation probabilities) can be designed along these guidelines. We provide also practical examples and numerical evidence for this approximation philosophy.

Keywords:
Black cipher, differential cryptanalysis, linear cryptanalysis, mixing dynamical system, periodic approximation, S box

Abstract:
Up to now biometric methods have been used in cryptography for authentication purposes. In this paper we propose to use biological data for generating sequences of random bits. We point out that this new approach could be particularly useful to generate seeds for pseudo-random number generators and so-called “key sessions”. Our method is very simple and is based on the observation that, for typical biometric readings, the last binary digits fluctuate “randomly”. We apply our method to two data sets, the first based on animal neurophysiological brain responses and the second on human galvanic skin response. For comparison we also test our approach on numerical samplings of the Ornstein–Uhlenbeck stochastic process. To verify the randomness of the sequences generated, we apply the standard suite of statistical tests (FIPS 140-2) recommended by the National Institute of Standard and Technology for studying the quality of the physical random number generators, especially those implemented in cryptographic modules. Additionally, to confirm the high cryptographic quality of the biometric generators, we also use the often recommended Maurer's universal test and the Lempel–Ziv complexity test, which estimate the entropy of the source. The results of all these verifications show that, after appropriate choice of encoding and experimental parameters, the sequences obtained exhibit excellent statistical properties, which opens the possibility of a new design technology for true random number generators. It remains a challenge to find appropriate biological phenomena characterized by easy accessibility, fast sampling rate, high accuracy of measurement and variability of sampling rate.

Abstract:
We review several applications of Lempel–Ziv complexity to the characterization of neural responses. In particular, Lempel–Ziv complexity allows to estimate the entropy of binned spike trains in an alternative way to the usual method based on the relative frequencies of words, with the definitive advantage of no requiring very long registers. We also use complexity to discriminate neural responses to different kinds of stimuli and to evaluate the number of states of neuronal sources.

Keywords:
Lempel–Ziv complexity, Entropy, Spike trains, Neuronal sources

Abstract:
Normalized Lempel-Ziv complexity, which measures the generation rate of new patterns along a digital sequence, is closely related to such important source properties as entropy and compression ratio, but, in contrast to these, it is a property of individual sequences. In this article, we propose to exploit this concept to estimate (or, at least, to bound from below) the entropy of neural discharges (spike trains). The main advantages of this method include fast convergence of the estimator (as supported by numerical simulation) and the fact that there is no need to know the probability law of the process generating the signal. Furthermore, we present numerical and experimental comparisons of the new method against the standard method based on word frequencies, providing evidence that this new approach is an alternative entropy estimator for binned spike trains.

Abstract:
In a previous paper (Proceedings of the World Congress on Neuroinformatics (2001)) the authors applied the so-called Lempel–Ziv complexity to study neural discharges (spike trains) from an information-theoretical point of view. Along with other results, it is shown there that this concept of complexity allows to characterize the responses of primary visual cortical neurons to both random and periodic stimuli. To this aim we modeled the neurons as information sources and the spike trains as messages generated by them. In this paper, we study further consequences of this mathematical approach, this time concerning the number of states of such neuronal information sources. In this context, the state of an information source means an internal degree of freedom (or parameter) which allows outputs with more general stochastic properties, since symbol generation probabilities at every time step may additionally depend on the value of the current state of the neuron. Furthermore, if the source is ergodic and Markovian, the number of states is directly related to the stochastic dependence lag of the source and provides a measure of the autocorrelation of its messages. Here, we find that the number of states of the neurons depends on the kind of stimulus and the type of preparation ( in vivo versus in vitro recordings), thus providing another way of differentiating neuronal responses. In particular, we observed that (for the encoding methods considered) in vitro sources have a higher lag than in vivo sources for periodic stimuli. This supports the conclusion put forward in the paper mentioned above that, for the same kind of stimulus, in vivo responses are more random (hence, more difficult to compress) than in vitro responses and, consequently, the former transmit more information than the latter.

Keywords:
Spike trains, Encoding, Lempel–Ziv complexity, Entropy, Internal states, Numerical invariants for neuronal responses

Abstract:
During the last years a new approach to construct safe block and stream ciphers has been developed using the theory of dynamical systems. Since a block cryptosystem is generally, from the mathematical point of view, a family (parametrized by the keys) of permutations of n-bit numbers, one of the main problems of this approach is to adapt the dynamics defined by a map f to the block structure of the cryptosystem. In this paper we propose a method based on the approximation of f by periodic maps Tn (v.g. some interval exchange transformations). The approximation of automorphisms of measure spaces by periodic automorphisms was introduced by Halmos and Rohlin. One important aspect studied in our paper is the relation between the dynamical properties of the map f (say, ergodicity or mixing) and the immunity of the resulting cipher to cryptolinear attacks, which is currently one of the standard benchmarks for cryptosystems to be considered secure. Linear cryptanalysis, first proposed by M. Matsui, exploits some statistical inhomogeneities of expressions called linear approximations for a given cipher. Our paper quantifies immunity to cryptolinear attacks in terms of the approximation speed of the map f by the periodic Tn. We show that the most resistant block ciphers are expected when the approximated dynamical system is mixing.

Abstract:
Pattern matching is a simple method for studying the properties of information sources based on individual sequences (Wyner et al 1998 IEEE Trans. Inf. Theory 44 2045–56). In particular, the normalized Lempel–Ziv complexity (Lempel and Ziv 1976 IEEE Trans. Inf. Theory 22 75–88), which measures the rate of generation of new patterns along a sequence, is closely related to such important source properties as entropy and information compression ratio. We make use of this concept to characterize the responses of neurons of the primary visual cortex to different kinds of stimulus, including visual stimulation (sinusoidal drifting gratings) and intracellular current injections (sinusoidal and random currents), under two conditions (in vivo and in vitro preparations). Specifically, we digitize the neuronal discharges with several encoding techniques and employ the complexity curves of the resulting discrete signals as fingerprints of the stimuli ensembles. Our results show, for example, that if the neural discharges are encoded with a particular one-parameter method (‘interspike time coding’), the normalized complexity remains constant within some classes of stimuli for a wide range of the parameter. Such constant values of the normalized complexity allow then the differentiation of the stimuli classes. With other encodings (e.g. ‘bin coding’), the whole complexity curve is needed to achieve this goal. In any case, it turns out that the normalized complexity of the neural discharges in vivo are higher (and hence carry more information in the sense of Shannon) than in vitro for the same kind of stimulus.