Chaotic behavior in probabilistic cellular neural networks

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 1997

The results of neurobiological studies in both vertebrates and invertebrates lead to the general question: How is a population of neurons, whose individual activity is chaotic and uncorrelated able to form functional circuits with regular and stable behavior? What are the circumstances which support these regular oscillations? What are the mechanisms that promote this transition? We address these questions using our experimental and modeling studies describing the behavior of groups of spiking-bursting neurons. We show that the role of inhibitory synaptic coupling between neurons is crucial in the self-control of chaos.

International Journal of Neuroscience, 2003

We have shortly reviewed the occurrence of the post-synaptic potentials between neurons, the relation between EEG and neuron dynamics, as well as methods of signal analysis. We supposed a simple stochastic model representing electrical activity of neuronal systems. The model is constructed using the Monte Carlo simulation technique. The results yielded EEG- * 1 like signals with their phase portraits in three-dimensional space. The Lyapunov exponent was positive, indicating a chaotic behavior. The correlation dimension of the EEG-like signals was found to be 0.92, which was smaller than those reported by others. It was concluded that this neuron model may provide valuable clues about the dynamic behavior of neural systems.

Physical Review Letters, 2002

A neural network model that exhibits stochastic population bursting is studied by simulation. First return maps of inter-burst intervals exhibit recurrent unstable periodic orbit (UPO)-like trajectories similar to those found in experiments on hippocampal slices. Applications of various control methods and surrogate analysis for UPO-detection also yield results similar to those of experiments. Our results question the interpretation of the experimental data as evidence for deterministic chaos and suggest caution in the use of UPO-based methods for detecting determinism in time-series data. 05.45.Gg, 05.45.Tp, 87.18.Sn With recent advances in the theory of dynamical systems [1] and nonlinear time-series analysis [2], many recent studies have looked for signatures of deterministic chaos in brain dynamics . This is expected to help in understanding brain functions in terms of the collective dynamics of neurons and in achieving short-term prediction and control with potential medical applications.

IEEE Transactions on Systems, Man, and Cybernetics, 1983

Deterministic mathematical models of neural systems can give rise to complex aperiodic ("chaotic") dynamics in the absence of stochastic fluctuations ("noise") in the variables or parameters of the model or in the inputs to the system. We show that chaotic dynamics are expected in nonlinear feedback systems possessing time delays such as are found in recurrent inhibition and from the periodic forcing of neural oscillators. The implications of the possible occurrence of chaotic dynamics for experimental work and mathematical modeling of normal and abnormal function in neurophysiology are mentioned. I.

Experimental Chaos, 2004

We compare the recorded activity of cultured neuronal networks with hybridized model simulations, in which the model neurons are driven by the recorded activity of special neurons. The latter, named 'spiker' neurons, that exhibit fast firing with homoclinic chaos like characteristics, are expected to play an important role in the networks' self regulation. The cultured networks are grown from dissociated mixtures of cortical neurons and glia cells. Despite the artificial manner of their construction, the spontaneous activity of these networks exhibits rich dynamical behavior, marked by the formation of temporal sequences of synchronized bursting events (SBEs), and additional features which seemingly reflect the action of underlying regulating mechanism, rather than arbitrary causes and effects. Our model neurons are composed of soma described by the two Morris-Lecar dynamical variables (voltage and fraction of open potassium channels), with dynamical synapses described by the Tsodyks-Markram three variables dynamics. To study the recorded and simulated activities we evaluated the inter-neuron correlation matrices, and analyzed them utilizing the functional holography approach: the correlations are re-normalized by the correlation distances -Euclidean distances between the matrix columns. Then, we project the N-dimensional (for N channels) space spanned by the matrix of re-normalized correlations, or correlation affinities, onto a corresponding 3-D causal manifold (3-D Cartesian space constructed by the 3 leading principal vectors of the N-dimensional space. The neurons are located by their principal eigenvalues and linked by their original (not-normalized) correlations. This reveals hidden causal motifs: the neuron locations and their links form simple structures. Similar causal motifs are exhibited by the model simulations when feeded by the recorded activity of the spiker neurons. We illustrate that the homoclinic chaotic behavior of the spiker neurons can be generated by glia-regulated self-synapses. Since in real networks the glia are regulated back by the networks' neuronal activity, our findings hint that the structure of the causal manifolds in the affinity space is self-regulated via collective regulation of the homoclinic behavior of the spiker neurons.

Journal of Neurophysiology, 2009

There is great interest in the role of coherent oscillations in the brain. In some cases, high-frequency oscillations (HFOs) are integral to normal brain function, whereas at other times they are implicated as markers of epileptic tissue. Mechanisms underlying HFO generation, especially in abnormal tissue, are not well understood. Using a physiological computer model of hippocampus, we investigate random synaptic activity (noise) as a potential initiator of HFOs. We explore parameters necessary to produce these oscillations and quantify the response using the tools of stochastic resonance (SR) and coherence resonance (CR). As predicted by SR, when noise was added to the network the model was able to detect a subthreshold periodic signal. Addition of basket cell interneurons produced two novel SR effects: 1) improved signal detection at low noise levels and 2) formation of coherent oscillations at high noise that were entrained to harmonics of the signal frequency. The periodic signal was then removed to study oscillations generated only by noise. The combined effects of network coupling and synaptic noise produced coherent, periodic oscillations within the network, an example of CR. Our results show that, under normal coupling conditions, synaptic noise was able to produce gamma (30-100 Hz) frequency oscillations. Synaptic noise generated HFOs in the ripple range (100-200 Hz) when the network had parameters similar to pathological findings in epilepsy: increased gap junctions or recurrent synaptic connections, loss of inhibitory interneurons such as basket cells, and increased synaptic noise. The model parameters that generated these effects are comparable with published experimental data. We propose that increased synaptic noise and physiological coupling mechanisms are sufficient to generate gamma oscillations and that pathologic changes in noise and coupling similar to those in epilepsy can produce abnormal ripples.

Chaotic dynamics of neural oscillations has been shown at the single neuron and network levels, both in experimental data and numerical simulations. Theoretical studies over the last twenty years have demonstrated an underlying role of chaos in neural systems. Nevertheless, whether chaotic neural oscillators make a significant contribution to relevant network behavior and whether the dynamical richness of neural networks are sensitive to the dynamics of isolated neurons, still remain open questions. We investigated transition dynamics of a medium-sized heterogeneous neural network of neurons connected by electrical coupling in a small world topology. We make use of an oscillatory neuron model (HB+Ih) that exhibits either chaotic or non-chaotic behavior at different combinations of conductance parameters. Measuring order parameter as a measure of synchrony, we find that the heterogeneity of firing rate and types of firing patterns make a greater contribution than chaos to the steepne...

2017

The role of chaos in biological information processing has been established as an important breakthrough of nonlinear dynamics, after the early pioneering work of J.S. Nicolis and notably in neuroscience by the work of Walter J. Freeman and co-workers spanning more than three decades. In this work we revisit the subject and we further focus on novel results that reveal its underlying logical structure when faced with the cognition of ambiguous stimuli. We demonstrate, by utilizing a minimal model for apprehension and judgement related to Bayesian updating, that the fundamental characteristics of a biological processor obey in this case an extended, non-Boolean, logic which is characterized as a quantum logic. And we realize that in its essence the role of chaos in biological information processing accounts for, and is fully compatible with, the logic of “quantum cognition” in psychology and neuroscience.

Behavioral and Brain Sciences, 2001

Using the concepts of chaotic dynamical systems, we present an interpretation of dynamic neural activity found in cortical and subcortical areas. The discovery of chaotic itinerancy in high-dimensional dynamical systems with and without a noise term has motivated a new interpretation of this dynamic neural activity, cast in terms of the high-dimensional transitory dynamics among “exotic” attractors. This interpretation is quite different from the conventional one, cast in terms of simple behavior on low-dimensional attractors. Skarda and Freeman (1987) presented evidence in support of the conclusion that animals cannot memorize odor without chaotic activity of neuron populations. Following their work, we study the role of chaotic dynamics in biological information processing, perception, and memory. We propose a new coding scheme of information in chaos-driven contracting systems we refer to as Cantor coding. Since these systems are found in the hippocampal formation and also in the...

Psycoloquy, 1994

This critique of Wright et al.'s (1993) macroscopic EEG model is based on recent findings by Kaneko (1990) concerning dynamic behaviors in a globally coupled map. A coupled chaotic system does not always follow linear and equilibrium statistical mechanics. Some aspects of attractive neural networks and chaotic learning are also discussed.

2008

We have shortly reviewed the occurrence of the post-synaptic potentials between neurons, the relation between EEG and neuron dynamics, as well as methods of signal analysis. We supposed a simple stochastic model representing electrical activity of neuronal systems. The model is constructed using the Monte Carlo simulation technique. The results yielded EEG-Address correspondence to Dr. Ekrem Aydiner, Department of Physics, Faculty of Arts and Sciences,

Physical Review E, 2010

We propose a stochastic dynamical model of noisy neural networks with complex architectures and discuss activation of neural networks by a stimulus, pacemakers and spontaneous activity. This model has a complex phase diagram with self-organized active neural states, hybrid phase transitions, and a rich array of behavior. We show that if spontaneous activity (noise) reaches a threshold level then global neural oscillations emerge. Stochastic resonance is a precursor of this dynamical phase transition. These oscillations are an intrinsic property of even small groups of 50 neurons.

Neural networks, 2007

We present a neurobiologically-inspired stochastic cellular automaton whose state jumps with time between the attractors corresponding to a series of stored patterns. The jumping varies from regular to chaotic as the model parameters are modified. The resulting irregular behavior, which mimics the state of attention in which a systems shows a great adaptability to changing stimulus, is a consequence in the model of shorttime presynaptic noise which induces synaptic depression. We discuss results from both a mean-field analysis and Monte Carlo simulations.

Mathematical Modelling of Natural Phenomena

We demonstrate that unidirectional electrical coupling between two periodically spiking Hindmarsh-Rose neurons induces bistability in the system. We find that for certain values of intermediate coupling, the slave neuron exhibits coexistence of two attractors. One of them is the periodic orbit similar to the original attractor without coupling, and the other one is a chaotic attractor or a periodic orbit with higher periodicity, depending on the coupling strength. For strong coupling, the slave neuron is monostable at a periodic orbit similar to the attractor of the master neuron. When the master and slave neurons are in a similar attractor they are completely synchronized, whereas being in different states they are in generalized synchronization. We also present the experimental evidence of this behavior with electronic circuits based on the Hindmarsh-Rose model.

Proceedings of the National Academy of Sciences, 1989

Self-organization of frequencies is studied by using model neurons called VCONs (voltage-controlled oscillator neuron models). These models give direct access to frequency information, in contrast to all-or-none neuron models, and they generate voltage spikes that phase-lock to oscillatory stimulation, similar to phase-locking of action potentials to oscillatory voltage stimulation observed in Hodgkin-Huxley preparations of squid axons. The rotation vector method is described and used to study how networks synchronize, even in the presence of noise or when damaged; the entropy of ratios of phases is used to construct an energy function that characterizes organized behavior. Computer simulations show that rotation numbers (output frequency/input frequency) describe both chaotic and nonchaotic behavior. Learning occurs when synaptic connections strengthen in response to stimulation that is synchronous with cell activity. It is shown that intermittent chaotic firing is suppressed and s...