Deterministic prediction and chaos in squid axon response

Mees, A. I.; Aihara, Kazuyuki; Adachi, Masaharu; Judd, Kevin; Ikeguchi, Tohru; Matsumoto, Gen

doi:10.1016/0375-9601(92)90802-s

Cited by 48 publications

(24 citation statements)

References 14 publications

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“…Future problems are to apply these techniques not only to real physiological data (Mees et al 1992;Aihara 1995) but also to neuron models incorporated in neural networks (Aihara et al 1990;Aihara 1993, 2000). (lower lines) using inclusion (solid), exclusion (dotted), and a typical orbit of length 10 5 (dash-dot) Fig.…”

Section: Discussion Of Resultsmentioning

confidence: 99%

“…For example, a one-dimensional map to be introduced in Sect. 4 (Ichinose et al 1998) can well represent characteristics observed by electrophysiological experiments with squid giant axons (Mees et al 1992;Aihara 1995). These studies imply that variability and irregularity of neuronal ®ring may be explained to some extent from the viewpoint of dynamical system theory.…”

Section: Introductionmentioning

confidence: 85%

“…On the other hand, well-controlled electrophysiological experiments with real neurons have shown that their dynamical behavior can be approximately described by deterministic nonlinear maps on appropriate PoincareÂ sections, especially by one-dimensional nonlinear maps with possibly chaotic dynamics (Aihara et al 1986;Segundo et al 1991;Hayashi and Ishizuka 1992;Mees et al 1992;Aihara 1995). On the basis of such neuronal properties, simple mathematical models of one-dimensional nonlinear maps on neuronal response have been proposed (Aihara et al 1990;Ichinose et al 1998) as models of neurons in neural networks to aid our understanding of complex neurodynamics.…”

Section: Introductionmentioning

confidence: 99%

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Estimating statistics of neuronal dynamics via Markov chains

Froyland

Aihara

2001

Biol Cybern

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We present an efficient computational method for estimating the mean and variance of interspike intervals defined by the timing of spikes in typical orbits of one-dimensional neuronal maps. This is equivalent to finding the mean and variance of return times of orbits to particular regions of phase space. Rather than computing estimates directly from time series, the system is modelled as a finite state Markov chain to extract stationary behaviour in the form of invariant measures and average absorption times. Ergodic-theoretic formulae are then applied to produce the estimates without the need to generate orbits directly. The approach may be applied to both deterministic and randomly forced systems.

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Section: Discussion Of Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

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Estimating statistics of neuronal dynamics via Markov chains

Froyland

Aihara

2001

Biol Cybern

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“…A similar approach has been employed, for example, by Mees et al (1992) and Kennel and Isabelle (1992), although other methods for estimating the time delay and embedding dimension also exist (Buzug and Pfister 1992).…”

Section: Methods For the Analysis Of Time Seriesmentioning

confidence: 96%

The presence of both stochastic and deterministic dynamics in electroretinograms measured under identical experimental conditions

Andrushchenko

Samoilenko

Yatsenko

1997

Biological Cybernetics

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We have studied whether living things investigated under the same measuring conditions can generate signals with different types of dynamics. We also wanted to detect the possible effects of low-intensity microwaves using parameters of deterministic chaos. For this purpose, two sets of electroretinograms were analysed by methods aimed at recognizing different types of dynamics. Both sets included the time series recorded from objects exposed to low-intensity microwaves and those that were not exposed. The analytical methods are based on nonlinear forecasting and a "surrogate data" technique. Although the experimental conditions were identical for the two sets, we have shown that both have time series with deterministic and stochastic dynamics. We also found that the use of parameters of deterministic dynamics is insufficient to distinguish between the sets.

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“…Figure 4 shows the corresponding periodic responses of the membrane potential in a squid giant axon that is in the resting state and stimulated by periodic pulses with a constant amplitude. The model of equations (4.4) and (4.5) and its variants (Aihara et al 1990) reproduce these responses of squid giant axons well, including even chaotic responses (Aihara & Matsumoto 1986;Matsumoto et al 1987;Mees et al 1992;Aihara 2002;Suzuki & Aihara 2008).…”

Section: Complex Behaviour In a Simple Neuron Model And Its Variantsmentioning

confidence: 99%

Theory of hybrid dynamical systems and its applications to biological and medical systems

Aihara

Suzuki

2010

Phil. Trans. R. Soc. A.

105

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In this introductory article, we survey the contents of this Theme Issue. This Theme Issue deals with a fertile region of hybrid dynamical systems that are characterized by the coexistence of continuous and discrete dynamics. It is now well known that there exist many hybrid dynamical systems with discontinuities such as impact, switching, friction and sliding. The first aim of this Issue is to discuss recent developments in understanding nonlinear dynamics of hybrid dynamical systems in the two main theoretical fields of dynamical systems theory and control systems theory. A combined study of the hybrid systems dynamics in the two theoretical fields might contribute to a more comprehensive understanding of hybrid dynamical systems. In addition, mathematical modelling by hybrid dynamical systems is particularly important for understanding the nonlinear dynamics of biological and medical systems as they have many discontinuities such as threshold-triggered firing in neurons, on-off switching of gene expression by a transcription factor, division in cells and certain types of chronotherapy for prostate cancer. Hence, the second aim is to discuss recent applications of hybrid dynamical systems in biology and medicine. Thus, this Issue is not only general to serve as a survey of recent progress in hybrid systems theory but also specific to introduce interesting and stimulating applications of hybrid systems in biology and medicine. As the introduction to the topics in this Theme Issue, we provide a brief history of nonlinear dynamics and mathematical modelling, different mathematical models of hybrid dynamical systems, the relationship between dynamical systems theory and control systems theory, examples of complex behaviour in a simple neuron model and its variants, applications of hybrid dynamical systems in biology and medicine as a road map of articles in this Theme Issue and future directions of hybrid systems modelling.Keywords: hybrid dynamical systems; mathematical modelling; biological systems; medical systems; nonlinear dynamics; control Nonlinear dynamics and mathematical modellingThe history of natural science can be traced back to Isaac Newton in the seventeenth century. He formulated equations relating force and motion and co-invented calculus. This seminal work inaugurated the field of dynamical systems. His laws of force and motion are also the origin of mathematical modelling of dynamical phenomena in this world. Because calculus generally hates discontinuities and instabilities, most classical studies of dynamical systems and mathematical modelling concentrated on solutions that were smooth and stable. The universe was thus conceived of as a precision machine with smoothness and stability; based on the initial conditions, one could accurately predict future behaviour such as that of planets orbiting the Sun or that of a pendulum clock. However, in the 1960s and 1970s, 'catastrophe theory' (Thom 1975) and 'chaos theory' (Li & Yorke 1975;May 1976) were formulated as special branches of dynamical syst...

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Deterministic prediction and chaos in squid axon response

Cited by 48 publications

References 14 publications

Estimating statistics of neuronal dynamics via Markov chains

Estimating statistics of neuronal dynamics via Markov chains

The presence of both stochastic and deterministic dynamics in electroretinograms measured under identical experimental conditions

Theory of hybrid dynamical systems and its applications to biological and medical systems

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