Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator

Neuron-like spiking and bursting in Josephson junctions: A review

The superconducting Josephson junction shows spiking and bursting behaviors, which have similarities with neuronal spiking and bursting. This phenomenon had been observed long ago by some researchers; however, they overlooked the biological similarity of this particular dynamical feature and never attempted to interpret it from the perspective of neuronal dynamics. In recent times, the origin of such a strange property of the superconducting junction has been explained and such neuronal functional behavior has also been observed in superconducting nanowires. The history of this research is briefly reviewed here with illustrations from studies of two junction models and their dynamical interpretation in the sense of biological bursting.

Toward New Modalities in VEP-Based BCI Applications Using Dynamical Stimuli: Introducing Quasi-Periodic and Chaotic VEP-Based BCI

Visual evoked potentials (VEPs) to periodic stimuli are commonly used in brain computer interfaces for their favorable properties such as high target identification accuracy, less training time, and low surrounding target interference. Conventional periodic stimuli can lead to subjective visual fatigue due to continuous and high contrast stimulation. In this study, we compared quasi-periodic and chaotic complex stimuli to common periodic stimuli for use with VEP-based brain computer interfaces (BCIs). Canonical correlation analysis (CCA) and coherence methods were used to evaluate the performance of the three stimulus groups. Subjective fatigue caused by the presented stimuli was evaluated by the Visual Analogue Scale (VAS). Using CCA with the M2 template approach, target identification accuracy was highest for the chaotic stimuli (M = 86.8, SE = 1.8) compared to the quasi-periodic (M = 78.1, SE = 2.6, p = 0.008) and periodic (M = 64.3, SE = 1.9, p = 0.0001) stimulus groups. The evaluation of fatigue rates revealed that the chaotic stimuli caused less fatigue compared to the quasi-periodic (p = 0.001) and periodic (p = 0.0001) stimulus groups. In addition, the quasi-periodic stimuli led to lower fatigue rates compared to the periodic stimuli (p = 0.011). We conclude that the target identification results were better for the chaotic group compared to the other two stimulus groups with CCA. In addition, the chaotic stimuli led to a less subjective visual fatigue compared to the periodic and quasi-periodic stimuli and can be suitable for designing new comfortable VEP-based BCIs.

Analytic solutions throughout a period doubling route to chaos

We show examples of dynamical systems that can be solved analytically at any point along a period doubling route to chaos. Each system consists of a linear part oscillating about a set point and a nonlinear rule for regularly updating that set point. Previously it has been shown that such systems can be solved analytically even when the oscillations are chaotic. However, these solvable systems show few bifurcations, transitioning directly from a steady state to chaos. Here we show that a simple change to the rule for updating the set point allows for a greater variety of nonlinear dynamical phenomena, such as period doubling, while maintaining solvability. Two specific examples are given. The first is an oscillator whose set points are determined by a logistic map. We present analytic solutions describing an entire period doubling route to chaos. The second example is an electronic circuit. We show experimental data confirming both solvability and a period doubling route to chaos. These results suggest that analytic solutions may be a more useful tool in studying nonlinear dynamics than was previously recognized.

Hyperpolarization-Activated Current Induces Period-Doubling Cascades and Chaos in a Cold Thermoreceptor Model

In this article, we describe and analyze the chaotic behavior of a conductance-based neuronal bursting model. This is a model with a reduced number of variables, yet it retains biophysical plausibility. Inspired by the activity of cold thermoreceptors, the model contains a persistent Sodium current, a Calcium-activated Potassium current and a hyperpolarization-activated current (I_h) that drive a slow subthreshold oscillation. Driven by this oscillation, a fast subsystem (fast Sodium and Potassium currents) fires action potentials in a periodic fashion. Depending on the parameters, this model can generate a variety of firing patterns that includes bursting, regular tonic and polymodal firing. Here we show that the transitions between different firing patterns are often accompanied by a range of chaotic firing, as suggested by an irregular, non-periodic firing pattern. To confirm this, we measure the maximum Lyapunov exponent of the voltage trajectories, and the Lyapunov exponent and Lempel-Ziv's complexity of the ISI time series. The four-variable slow system (without spiking) also generates chaotic behavior, and bifurcation analysis shows that this is often originated by period doubling cascades. Either with or without spikes, chaos is no longer generated when the I_h is removed from the system. As the model is biologically plausible with biophysically meaningful parameters, we propose it as a useful tool to understand chaotic dynamics in neurons.

A basic bifurcation structure from bursting to spiking of injured nerve fibers in a two-dimensional parameter space

Two different bifurcation scenarios of firing patterns with decreasing extracellular calcium concentrations were observed in identical sciatic nerve fibers of a chronic constriction injury (CCI) model when the extracellular 4-aminopyridine concentrations were fixed at two different levels. Both processes proceeded from period-1 bursting to period-1 spiking via complex or simple processes. Multiple typical experimental examples manifested dynamics closely matching those simulated in a recently proposed 4-dimensional model to describe the nonlinear dynamics of the CCI model, which included most cases of the bifurcation scenarios. As the extracellular 4-aminopyridine concentrations is increased, the structure of the bifurcation scenario becomes more complex. The results provide a basic framework for identifying the relationships between different neural firing patterns and different bifurcation scenarios and for revealing the complex nonlinear dynamics of neural firing patterns. The potential roles of the basic bifurcation structures in identifying the information process mechanism are discussed.

Chaos and Hyperchaos in a Model of Ribosome Autocatalytic Synthesis

Any vital activities of the cell are based on the ribosomes, which not only provide the basic machinery for the synthesis of all proteins necessary for cell functioning during growth and division, but for biogenesis itself. From this point of view, ribosomes are self-replicating and autocatalytic structures. In current work we present an elementary model in which the autocatalytic synthesis of ribosomal RNA and proteins, as well as enzymes ensuring their degradation are described with two monotonically increasing functions. For certain parameter values, the model, consisting of one differential equation with delayed argument, demonstrates both stationary and oscillatory dynamics of the ribosomal protein synthesis, which can be chaotic and hyperchaotic dependent on the value of the delayed argument. The biological interpretation of the modeling results and parameter estimation suggest the feasibility of chaotic dynamics in molecular genetic systems of eukaryotes, which depends only on the internal characteristics of functioning of the translation system.

Dynamics of on-off neural firing patterns and stochastic effects near a sub-critical Hopf bifurcation

On-off firing patterns, in which repetition of clusters of spikes are interspersed with epochs of subthreshold oscillations or quiescent states, have been observed in various nervous systems, but the dynamics of this event remain unclear. Here, we report that on-off firing patterns observed in three experimental models (rat sciatic nerve subject to chronic constrictive injury, rat CA1 pyramidal neuron, and rabbit blood pressure baroreceptor) appeared as an alternation between quiescent state and burst containing multiple period-1 spikes over time. Burst and quiescent state had various durations. The interspike interval (ISI) series of on-off firing pattern was suggested as stochastic using nonlinear prediction and autocorrelation function. The resting state was changed to a period-1 firing pattern via on-off firing pattern as the potassium concentration, static pressure, or depolarization current was changed. During the changing process, the burst duration of on-off firing pattern increased and the duration of the quiescent state decreased. Bistability of a limit cycle corresponding to period-1 firing and a focus corresponding to resting state was simulated near a sub-critical Hopf bifurcation point in the deterministic Morris-Lecar (ML) model. In the stochastic ML model, noise-induced transitions between the coexisting regimes formed an on-off firing pattern, which closely matched that observed in the experiment. In addition, noise-induced exponential change in the escape rate from the focus, and noise-induced coherence resonance were identified. The distinctions between the on-off firing pattern and stochastic firing patterns generated near three other types of bifurcations of equilibrium points, as well as other viewpoints on the dynamics of on-off firing pattern, are discussed. The results not only identify the on-off firing pattern as noise-induced stochastic firing pattern near a sub-critical Hopf bifurcation point, but also offer practical indicators to discriminate bifurcation types and neural excitability types.

Experimental dynamical characterization of five autonomous chaotic oscillators with tunable series resistance

In this paper, an experimental characterization of the dynamical properties of five autonomous chaotic oscillators, based on bipolar-junction transistors and obtained de-novo through a genetic algorithm in a previous study, is presented. In these circuits, a variable resistor connected in series to the DC voltage source acts as control parameter, for a range of which the largest Lyapunov exponent, correlation dimension, approximate entropy, and amplitude variance asymmetry are calculated, alongside bifurcation diagrams and spectrograms. Numerical simulations are compared to experimental measurements. The oscillators can generate a considerable variety of regular and chaotic sine-like and spike-like signals.

Membrane tension influences the spike propagation between voltage-gated ion channel clusters of excitable membranes

Ion channels of excitable membranes are known to be sensitive to various kinds of stimuli, but the case of simultaneous occurrence of different stimuli is poorly understood. Here, we theoretically analyze the influence of membrane tension on the dynamics of voltage-gated ion channels of excitable membranes. To do so, we develop a modification of the well-known Hodgkin-Huxley model to study numerically the spike generation and propagation in a single and two coupled excitable cells. We find that these cells can use membrane tension to trigger sub-threshold spike propagation, to suppress spike propagation and to alter the intensity of the signal transmission. These effects indicate that cells could use membrane tension to regulate cell-to-cell communication.

Biological experimental observations of an unnoticed chaos as simulated by the Hindmarsh-Rose model

An unnoticed chaotic firing pattern, lying between period-1 and period-2 firing patterns, has received little attention over the past 20 years since it was first simulated in the Hindmarsh-Rose (HR) model. In the present study, the rat sciatic nerve model of chronic constriction injury (CCI) was used as an experimental neural pacemaker to investigate the transition regularities of spontaneous firing patterns. Chaotic firing lying between period-1 and period-2 firings was observed located in four bifurcation scenarios in different, isolated neural pacemakers. These bifurcation scenarios were induced by decreasing extracellular calcium concentrations. The behaviors after period-2 firing pattern in the four scenarios were period-doubling bifurcation not to chaos, period-doubling bifurcation to chaos, period-adding sequences with chaotic firings, and period-adding sequences with stochastic firings. The deterministic structure of the chaotic firing pattern was identified by the first return map of interspike intervals and a short-term prediction using nonlinear prediction. The experimental observations closely match those simulated in a two-dimensional parameter space using the HR model, providing strong evidences of the existence of chaotic firing lying between period-1 and period-2 firing patterns in the actual nervous system. The results also present relationships in the parameter space between this chaotic firing and other firing patterns, such as the chaotic firings that appear after period-2 firing pattern located within the well-known comb-shaped region, periodic firing patterns and stochastic firing patterns, as predicted by the HR model. We hope that this study can focus attention on and help to further the understanding of the unnoticed chaotic neural firing pattern.

Deformation of attractor landscape via cholinergic presynaptic modulations: a computational study using a phase neuron model

Corticopetal acetylcholine (ACh) is released transiently from the nucleus basalis of Meynert (NBM) into the cortical layers and is associated with top-down attention. Recent experimental data suggest that this release of ACh disinhibits layer 2/3 pyramidal neurons (PYRs) via muscarinic presynaptic effects on inhibitory synapses. Together with other possible presynaptic cholinergic effects on excitatory synapses, this may result in dynamic and temporal modifications of synapses associated with top-down attention. However, the system-level consequences and cognitive relevance of such disinhibitions are poorly understood. Herein, we propose a theoretical possibility that such transient modifications of connectivity associated with ACh release, in addition to top-down glutamatergic input, may provide a neural mechanism for the temporal reactivation of attractors as neural correlates of memories. With baseline levels of ACh, the brain returns to quasi-attractor states, exhibiting transitive dynamics between several intrinsic internal states. This suggests that top-down attention may cause the attention-induced deformations between two types of attractor landscapes: the quasi-attractor landscape (Q-landscape, present under low-ACh, non-attentional conditions) and the attractor landscape (A-landscape, present under high-ACh, top-down attentional conditions). We present a conceptual computational model based on experimental knowledge of the structure of PYRs and interneurons (INs) in cortical layers 1 and 2/3 and discuss the possible physiological implications of our results.

Chaos in neurons and its application: perspective of chaos engineering

We review our recent work on chaos in neurons and its application to neural networks from perspective of chaos engineering. Especially, we analyze a dataset of a squid giant axon by newly combining our previous work of identifying Devaney's chaos with surrogate data analysis, and show that an axon can behave chaotically. Based on this knowledge, we use a chaotic neuron model to investigate possible information processing in the brain.

Neurodynamics, tonality, and the auditory brainstem response

Tonal relationships are foundational in music, providing the basis upon which musical structures, such as melodies, are constructed and perceived. A recent dynamic theory of musical tonality predicts that networks of auditory neurons resonate nonlinearly to musical stimuli. Nonlinear resonance leads to stability and attraction relationships among neural frequencies, and these neural dynamics give rise to the perception of relationships among tones that we collectively refer to as tonal cognition. Because this model describes the dynamics of neural populations, it makes specific predictions about human auditory neurophysiology. Here, we show how predictions about the auditory brainstem response (ABR) are derived from the model. To illustrate, we derive a prediction about population responses to musical intervals that has been observed in the human brainstem. Our modeled ABR shows qualitative agreement with important features of the human ABR. This provides a source of evidence that fundamental principles of auditory neurodynamics might underlie the perception of tonal relationships, and forces reevaluation of the role of learning and enculturation in tonal cognition.

Modeling brain resonance phenomena using a neural mass model

Stimulation with rhythmic light flicker (photic driving) plays an important role in the diagnosis of schizophrenia, mood disorder, migraine, and epilepsy. In particular, the adjustment of spontaneous brain rhythms to the stimulus frequency (entrainment) is used to assess the functional flexibility of the brain. We aim to gain deeper understanding of the mechanisms underlying this technique and to predict the effects of stimulus frequency and intensity. For this purpose, a modified Jansen and Rit neural mass model (NMM) of a cortical circuit is used. This mean field model has been designed to strike a balance between mathematical simplicity and biological plausibility. We reproduced the entrainment phenomenon observed in EEG during a photic driving experiment. More generally, we demonstrate that such a single area model can already yield very complex dynamics, including chaos, for biologically plausible parameter ranges. We chart the entire parameter space by means of characteristic Lyapunov spectra and Kaplan-Yorke dimension as well as time series and power spectra. Rhythmic and chaotic brain states were found virtually next to each other, such that small parameter changes can give rise to switching from one to another. Strikingly, this characteristic pattern of unpredictability generated by the model was matched to the experimental data with reasonable accuracy. These findings confirm that the NMM is a useful model of brain dynamics during photic driving. In this context, it can be used to study the mechanisms of, for example, perception and epileptic seizure generation. In particular, it enabled us to make predictions regarding the stimulus amplitude in further experiments for improving the entrainment effect.

Neuroscience aims to understand the enormously complex function of the normal and diseased brain. This, in turn, is the key to explaining human behavior and to developing novel diagnostic and therapeutic procedures. We develop and use models of mean activity in a single brain area, which provide a balance between tractability and plausibility. We use such a model to explain the resonance phenomenon in a photic driving experiment, which is routinely applied in the diagnosis of various diseases including epilepsy, migraine, schizophrenia and depression. Based on the model, we make predictions on the outcome of similar resonance experiments with periodic stimulation of the patients or participants. Our results are important for researchers and clinicians analyzing brain or behavioral data following periodic input.

Rewiring-induced chaos in pulse-coupled neural networks

The dependence of the dynamics of pulse-coupled neural networks on random rewiring of excitatory and inhibitory connections is examined. When both excitatory and inhibitory connections are rewired, periodic synchronization emerges with a Hopf-like bifurcation and a subsequent period-doubling bifurcation; chaotic synchronization is also observed. When only excitatory connections are rewired, periodic synchronization emerges with a saddle node-like bifurcation, and chaotic synchronization is also observed. This result suggests that randomness in the system does not necessarily contaminate the system, and sometimes it even introduces rich dynamics to the system such as chaos.

Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns

Period-doubling bifurcation to chaos were discovered in spontaneous firings of Onchidium pacemaker neurons. In this paper, we provide three cases of bifurcation processes related to period-doubling bifurcation cascades to chaos observed in the spontaneous firing patterns recorded from an injured site of rat sciatic nerve as a pacemaker. Period-doubling bifurcation cascades to period-4 (π(2,2)) firstly, and then to chaos, at last to a periodicity, which can be period-5, period-4 (π(4)) and period-3, respectively, in different pacemakers. The three bifurcation processes are labeled as case I, II and III, respectively, manifesting procedures different to those of period-adding bifurcation. Higher-dimensional unstable periodic orbits (UPOs) can be detected in the chaos, built close relationships to the periodic firing patterns. Case III bifurcation process is similar to that discovered in the Onchidium pacemaker neurons and simulated in theoretical model-Chay model. The extra-large Feigenbaum constant manifesting in the period-doubling bifurcation process, induced by quasi-discontinuous characteristics exhibited in the first return maps of both ISI series and slow variable of Chay model, shows that higher-dimensional periodic behaviors appeared difficult within the period-doubling bifurcation cascades. The results not only provide examples of period-doubling bifurcation to chaos and chaos with higher-dimensional UPOs, but also reveal the dynamical features of the period-doubling bifurcation cascades to chaos.

Mode-locked spike trains in responses of ventral cochlear nucleus chopper and onset neurons to periodic stimuli

We report evidence of mode-locking to the envelope of a periodic stimulus in chopper units of the ventral cochlear nucleus (VCN). Mode-locking is a generalized description of how responses in periodically forced nonlinear systems can be closely linked to the input envelope, while showing temporal patterns of higher order than seen during pure phase-locking. Re-analyzing a previously unpublished dataset in response to amplitude modulated tones, we find that of 55% of cells (6/11) demonstrated stochastic mode-locking in response to sinusoidally amplitude modulated (SAM) pure tones at 50% modulation depth. At 100% modulation depth SAM, most units (3/4) showed mode-locking. We use interspike interval (ISI) scattergrams to unravel the temporal structure present in chopper mode-locked responses. These responses compared well to a leaky integrate-and-fire model (LIF) model of chopper units. Thus the timing of spikes in chopper unit responses to periodic stimuli can be understood in terms of the complex dynamics of periodically forced nonlinear systems. A larger set of onset (33) and chopper units (24) of the VCN also shows mode-locked responses to steady-state vowels and cosine-phase harmonic complexes. However, while 80% of chopper responses to complex stimuli meet our criterion for the presence of mode-locking, only 40% of onset cells show similar complex-modes of spike patterns. We found a correlation between a unit's regularity and its tendency to display mode-locked spike trains as well as a correlation in the number of spikes per cycle and the presence of complex-modes of spike patterns. These spiking patterns are sensitive to the envelope as well as the fundamental frequency of complex sounds, suggesting that complex cell dynamics may play a role in encoding periodic stimuli and envelopes in the VCN.

Mode locking in a periodically forced resonate-and-fire neuron model

The resonate-and-fire (RF) model is a spiking neuron model which from a dynamical systems perspective is a piecewise smooth system (impact oscillator). We analyze the response of the RF neuron oscillator to periodic stimuli by expressing the firing events in terms of an implicit one-dimensional time map. Based on such a firing map, we describe mode-locked solutions and their stability, leading to the so-called Arnol'd tongues. The boundaries of these tongues correspond to either local bifurcations of the firing time map or grazing bifurcations of the discontinuity of the flow. Despite the fact that the periodically driven RF system shows periodic firing, its behavior may become chaotic when the forcing frequency is near the resonant frequency. We compare these results to numerical simulations of the model undergoing sinusoidal forcing. Furthermore, upon varying a system parameter, the RF system can be reduced to the integrate-and-fire system and in this case we show the consistency of the results on mode-locked solutions.

Dynamical foundations of the neural circuit for bayesian decision making

On the basis of accumulating behavioral and neural evidences, it has recently been proposed that the brain neural circuits of humans and animals are equipped with several specific properties, which ensure that perceptual decision making implemented by the circuits can be nearly optimal in terms of Bayesian inference. Here, I introduce the basic ideas of such a proposal and discuss its implications from the standpoint of biophysical modeling developed in the framework of dynamical systems.

Dynamical instability determines the effect of ongoing noise on neural firing

At low stimulation rates, electrically stimulated auditory nerve fibers typically fire regularly, in lock-step to the applied stimulus. At high stimulation rates, however, these same fibers fire irregularly. Firing irregularity has been attributed to the random opening and closing of voltage-gated sodium channels at the spike generation site. We demonstrate, however, that the nonlinear dynamics of neural excitation and refractoriness embodied in the FitzHugh-Nagumo (FN) model produce realistic firing irregularity at high stimulus rates, even in the complete absence of ongoing physiological noise. Indeed, we show that ongoing noise can actually regularize the response at low discharge rates. The degree of stimulus-dependent irregularity is determined not so much by the level of ongoing physiological noise as by the dynamical instability. Our work suggests that the dynamical instability, quantified by the Lyapunov exponent, controls neural sensitivity to input signals and to physiological noise, as well the amount of mutual desynchronization between similarly stimulated fibers. This instability, quantified by the value of the Lyapunov exponent, may play a critical role in determining modulation sensitivity and dynamic range in cochlear implants.

Noise-assisted spike propagation in myelinated neurons

We consider noise-assisted spike propagation in myelinated axons within a multicompartment stochastic Hodgkin-Huxley model. The noise originates from a finite number of ion channels in each node of Ranvier. For the subthreshold internodal electric coupling, we show that (i) intrinsic noise removes the sharply defined threshold for spike propagation from node to node and (ii) there exists an optimum number of ion channels which allows for the most efficient signal propagation and it corresponds to the actual physiological values.

Chaotic solutions in the quadratic integrate-and-fire neuron with adaptation

The quadratic integrate-and-fire (QIF) model with adaptation is commonly used as an elementary neuronal model that reproduces the main characteristics of real neurons. In this paper, we introduce a QIF neuron with a nonlinear adaptive current. This model reproduces the neuron-computational features of real neurons and is analytically tractable. It is shown that under a constant current input chaotic firing is possible. In contrast to previous study the neuron is not sinusoidally forced. We show that the spike-triggered adaptation is a key parameter to understand how chaos is generated.

Synchronization and firing death in the dynamics of two interacting excitable units with heterogeneous signals

We study the response of two coupled FitzHugh-Nagumo systems to heterogeneous external inputs. The latter, modeled by periodic parametric stimuli, force the uncoupled excitable systems into a regime of chaotic firing. Due to parameter dispersion involved in randomly distributed amplitudes and/or phases of the external forces the units are nonidentical and their firing events will be asynchronous. Interest is focused on mutually synchronized spikings arising through the coupling. It is demonstrated that the phase difference of the two external forces crucially affects the onset of spike synchronization as well as the resulting degree of synchrony. For large phase differences the degree of spike synchrony is constricted to a maximal possible value and cannot be enhanced upon increasing the coupling strength. We even found that overcritically strong couplings lead to suppression of firing so that the units perform synchronous subthreshold oscillations. This effect, which we call "firing death," is due to a coupling-induced modification of the excitation threshold impeding spiking of the units. In clear contrast, when only the amplitudes of the forces are distributed perfect spike synchrony is achieved for sufficiently strong coupling.

Intrinsic coherence resonance in excitable membrane patches

The influence of intrinsic channel noise on the spiking activity of excitable membrane patches is studied by use of a stochastic generalization of the Hodgkin-Huxley model. Internal noise stemming from the stochastic dynamics of individual ion channels does affect the electric properties of the cell-membrane patches. There exists an optimal size of the membrane patch for which the internal noise alone can cause a nearly regular spontaneous generation of action potentials. We consider the influence of intrinsic channel noise in presence of a constant and an oscillatory current driving for both, the mean interspike interval and the phenomenon of coherence resonance for neuronal spiking. Given small membrane patches, implying that channel noise dominates the excitable dynamics, we find the phenomenon of intrinsic coherence resonance. In this case, the relatively regular spiking behavior becomes essentially independent of an applied stimulus. We observed, however, the occurrence of a skipping of supra-threshold input events due to channel noise for intermediate patch sizes. This effect consequently reduces the overall coherence of the spiking.

Periodic forcing of a mathematical model of the eukaryotic cell cycle

In a differential equation model of the molecular network governing cell growth and division, cell cycle phases and transitions through checkpoints are associated with certain bifurcations of the underlying vector field. If the cell cycle is driven by another rhythmic process, interactions between forcing and bifurcations lead to emergent orbits and oscillations. In this paper, by varying the amplitude and frequency of forcing of the synthesis rates of regulatory proteins and the mass growth rate in a minimal model of the eukaryotic cell cycle, we study changes of the probability distributions of interdivision time and mass at division. By computing numerically the Lyapunov exponent of the model, we show that the splitting of probability distributions is associated with mode-locked solutions. We also introduce a simple, integrate-and-fire model to analyze mode locking in the cell cycle.

Bidirectional modulation of neuronal responses by depolarizing GABAergic inputs

The reversal potential of GABAA receptor channels is known to be less negative than the resting membrane potential under some cases. Recent electrophysiological experiments revealed that a GABAergic unitary conductance with such a depolarized reversal potential could not only prevent but also facilitate action potential generation depending on the timing of its application relative to the excitatory unitary conductance. Using a two-dimensional point neuron model, we simulate the experiments regarding the integration of unitary conductances, and execute bifurcation analysis. Then we extend our analysis to the case in which the neuron receives two kinds of periodic input trains-an excitatory one and a GABAergic one. We show that the periodic depolarizing GABAergic input train can modulate the output time-averaged firing rate bidirectionally, namely as an increase or a decrease, in a devil's-staircase-like manner depending on the phase difference with the excitatory input train. Bifurcation analysis reveals the existence of a wide variety of phase-locked solutions underlying such a graded response of the neuron. We examine how the input time-width and the value of the GABAA reversal potential affect the response. Moreover, considering a neuronal population, we show that depolarizing GABAergic inputs bidirectionally modulate the amplitude of the oscillatory population activity.

A MOSFET-based model of a Class 2 nerve membrane

We have constructed a nerve membrane using MOSFET circuitry, which can be a basic element of an FET-based neural system. Its mechanism of action potentials generation is designed to reproduce that of the Hodgkin-Huxley equations. The responses to singlet, doublet, repetitive pulse, and sustained stimuli are analyzed to show that it exhibits similar properties to the Hodgkin-Huxley equations; namely, 1) excitable dynamics with generation of action potentials, 2) the existence of a chaotic response to periodic stimuli, and 3) Class 2 excitability. It is known that Class 2 excitability is generated by an inverted Hopf bifurcation. We have applied Hopf bifurcation theory to our nerve membrane's system equations and have shown a routine for ascertaining whether a certain parameter set generates an inverted Hopf bifurcation.

A Network of Electronic Neural Oscillators Reproduces the Dynamics of the Periodically Forced Pyloric Pacemaker Group

Low-dimensional oscillators are a valuable model for the neuronal activity of isolated neurons. When coupled, the self-sustained oscillations of individual free oscillators are replaced by a collective network dynamics. Here, dynamical features of such a network, consisting of three electronic implementations of the Hindmarsh-Rose mathematical model of bursting neurons, are compared to those of a biological neural motor system, specifically the pyloric CPG of the crustacean stomatogastric nervous system. We demonstrate that the network of electronic neurons exhibits realistic synchronized bursting behavior comparable to the biological system. Dynamical properties were analyzed by injecting sinusoidal currents into one of the oscillators. The temporal bursting structure of the electronic neurons in response to periodic stimulation is shown to bear a remarkable resemblance to that observed in the corresponding biological network. These findings provide strong evidence that coupled nonlinear oscillators realistically reproduce the network dynamics experimentally observed in assemblies of several neurons.

BDNF boosts spike fidelity in chaotic neural oscillations

Oscillatory activity and its nonlinear dynamics are of fundamental importance for information processing in the central nervous system. Here we show that in aperiodic oscillations, brain-derived neurotrophic factor (BDNF), a member of the neurotrophin family, enhances the accuracy of action potentials in terms of spike reliability and temporal precision. Cultured hippocampal neurons displayed irregular oscillations of membrane potential in response to sinusoidal 20-Hz somatic current injection, yielding wobbly orbits in the phase space, i.e., a strange attractor. Brief application of BDNF suppressed this unpredictable dynamics and stabilized membrane potential fluctuations, leading to rhythmical firing. Even in complex oscillations induced by external stimuli of 40 Hz (gamma) on a 5-Hz (theta) carrier, BDNF-treated neurons generated more precisely timed spikes, i.e., phase-locked firing, coupled with theta-phase precession. These phenomena were sensitive to K252a, an inhibitor of tyrosine receptor kinases and appeared attributable to BDNF-evoked Na(+) current. The data are the first indication of pharmacological control of endogenous chaos. BDNF diminishes the ambiguity of spike time jitter and thereby might assure neural encoding, such as spike timing-dependent synaptic plasticity.

CHAOTIC SPIKING IN THE HODGKIN-HUXLEY NERVE MODEL WITH SLOW INACTIVATION OF THE SODIUM CURRENT

The Hodgkin-Huxley (HH) equations with a modification in which the inactivation process (h variable) of sodium channels is slightly slowed down are investigated. It is shown that this slight modification changes the HH dynamics to a completely different one, with chaotic spiking and very long interspike intervals appearing in a generic manner, although the initiation mechanism of repetitive firing is a simple Hopf bifurcation.

Phase locking in integrate-and-fire models with refractory periods and modulation

It is known [8, 11, 16, 26] that phase locking can entrain frequency information when the leaky integrate-and-fire (IF) model of a neuron is forced by a periodic function. We show that this is still the case when the IF model is made more biologically realistic. We incorporate into our model spike dependent threshold modulation and refractory periods. Consecutive firing times from this model and their respective interspike intervals are related by an annulus map. We prove a general theorem concerning orientation reversing annulus twist homeomorphisms, which shows that our map admits a unique rotation number. This implies, in particular, that chaotic behaviour is not possible in our model and phase locking is predicted.

Is there chaos in the brain? II. Experimental evidence and related models

The search for chaotic patterns has occupied numerous investigators in neuroscience, as in many other fields of science. Their results and main conclusions are reviewed in the light of the most recent criteria that need to be satisfied since the first descriptions of the surrogate strategy. The methods used in each of these studies have almost invariably combined the analysis of experimental data with simulations using formal models, often based on modified Huxley and Hodgkin equations and/or of the Hindmarsh and Rose models of bursting neurons. Due to technical limitations, the results of these simulations have prevailed over experimental ones in studies on the nonlinear properties of large cortical networks and higher brain functions. Yet, and although a convincing proof of chaos (as defined mathematically) has only been obtained at the level of axons, of single and coupled cells, convergent results can be interpreted as compatible with the notion that signals in the brain are distributed according to chaotic patterns at all levels of its various forms of hierarchy. This chronological account of the main landmarks of nonlinear neurosciences follows an earlier publication [Faure, Korn, C. R. Acad. Sci. Paris, Ser. III 324 (2001) 773-793] that was focused on the basic concepts of nonlinear dynamics and methods of investigations which allow chaotic processes to be distinguished from stochastic ones and on the rationale for envisioning their control using external perturbations. Here we present the data and main arguments that support the existence of chaos at all levels from the simplest to the most complex forms of organization of the nervous system. We first provide a short mathematical description of the models of excitable cells and of the different modes of firing of bursting neurons (Section 1). The deterministic behavior reported in giant axons (principally squid), in pacemaker cells, in isolated or in paired neurons of Invertebrates acting as coupled oscillators is then described (Section 2). We also consider chaotic processes exhibited by coupled Vertebrate neurons and of several components of Central Pattern Generators (Section 3). It is then shown that as indicated by studies of synaptic noise, deterministic patterns of firing in presynaptic interneurons are reliably transmitted, to their postsynaptic targets, via probabilistic synapses (Section 4). This raises the more general issue of chaos as a possible neuronal code and of the emerging concept of stochastic resonance Considerations on cortical dynamics and of EEGs are divided in two parts. The first concerns the early attempts by several pioneer authors to demonstrate chaos in experimental material such as the olfactory system or in human recordings during various forms of epilepsies, and the belief in 'dynamical diseases' (Section 5). The second part explores the more recent period during which surrogate-testing, definition of unstable periodic orbits and period-doubling bifurcations have been used to establish more firmly the nonlinear features of retinal and cortical activities and to define predictors of epileptic seizures (Section 6). Finally studies of multidimensional systems have founded radical hypothesis on the role of neuronal attractors in information processing, perception and memory and two elaborate models of the internal states of the brain (i.e. 'winnerless competition' and 'chaotic itinerancy'). Their modifications during cognitive functions are given special attention due to their functional and adaptive capabilities (Section 7) and despite the difficulties that still exist in the practical use of topological profiles in a state space to identify the physical underlying correlates. The reality of 'neurochaos' and its relations with information theory are discussed in the conclusion (Section 8) where are also emphasized the similarities between the theory of chaos and that of dynamical systems. Both theories strongly challenge computationalism and suggest that new models are needed to describe how the external world is represented in the brain.

Array-enhanced coherence resonance and forced dynamics in coupled FitzHugh-Nagumo neurons with noise

Nonlinear dynamics of coupled FitzHugh-Nagumo neurons subject to independent noise is analyzed. A kind of self-sustained global oscillation with almost synchronous firing is generated by array-enhanced coherence resonance. Further, forced dynamics of the self-sustained global oscillation stimulated by sinusoidal input is analyzed and classified as synchronized, quasiperiodic, and chaotic responses just like the forced oscillations in nerve membranes observed by in vitro experiments with squid giant axons. Possible physiological importance of such forced oscillations is also discussed.

Construction of an associative memory using unstable periodic orbits of a chaotic attractor

Unstable periodic orbits are the skeleton of a chaotic attractor. We constructed an associative memory based on the chaotic attractor of an artificial neural network, which associates input patterns to unstable periodic orbits. By processing an input, the system is driven out of the ground state to one of the pre-defined disjunctive areas of the attractor. Each of these areas is associated with a different unstable periodic orbit. We call an input pattern learned if the control mechanism keeps the system on the unstable periodic orbit during the response. Otherwise, the system relaxes back to the ground state on a chaotic trajectory. The major benefits of this memory device are its high capacity and low-energy consumption. In addition, new information can be simply added by linking a new input to a new unstable periodic orbit.

Birhythmicity induced by perturbing an oscillating electrochemical system

We describe the generation of new limit cycles in electrochemical systems under the influence of external periodic perturbations. For certain specific parameters of a nonharmonic forcing function, two coexisting periodic orbits can be generated from a single limit cycle observed in the unperturbed dynamics. This inception of birhythmicity (bistability) is observed in both simulations and actual experiments involving potentiostatic electrodissolution of copper in an acetate buffer.

Coherence resonance and discharge time reliability in neurons and neuronal models

Neurons are subject to internal and external noise that have been known to modify the way they process incoming signals. Recent studies have suggested that such alterations have functional roles and can also be used in biomedical applications. The present work goes over experimental and theoretical descriptions of the response of neurons to white noise stimulation. It examines various forms of noise related behavior in a standard neuronal model, namely the leaky integrate and fire. This clarifies the conditions under which specific noise induced changes occur in neurons, and consequently can help in determining whether nervous systems operate under similar circumstances.

Periodically forced leaky integrate-and-fire model

The discharge pattern of periodically forced leaky integrate-and-fire models is studied. While previous analyses have been mainly concerned with the response of this model to sinusoidal stimulation, our results hold for arbitrary periodic inputs. It is shown that, for any periodic input, the map representing the relation between input phases at consecutive discharge times can be restricted to a piecewise continuous, orientation preserving circle map. This implies that (i) the rotation number is well defined and independent of the initial condition, and (ii) in the same way as for sinusoidal forcing, other forms of periodic stimuli can evoke only one of four types of response, namely, phase locking, quasiperiodic discharges, nonchaotic aperiodic firing, and termination of the discharge after a finite number of firings.

Nonlinear behavior of sinusoidally forced pyloric pacemaker neurons

Periodic current forcing was used to investigate the intrinsic dynamics of a small group of electrically coupled neurons in the pyloric central pattern generator (CPG) of the lobster. This group contains three neurons, namely the two pyloric dilator (PD) motoneurons and the anterior burster (AB) interneuron. Intracellular current injection, using sinusoidal waveforms of varying amplitude and frequency, was applied in three configurations of the pacemaker neurons: 1) the complete pacemaker group, 2) the two PDs without the AB, and 3) the AB neuron isolated from the PDs. Depending on the frequency and amplitude of the injected current, the intact pacemaker group exhibited a wide variety of nonlinear behaviors, including synchronization to the forcing, quasiperiodicity, and complex dynamics. In contrast, a single, broad 1:1 entrainment zone characterized the response of the PD neurons when isolated from the main pacemaker neuron AB. The isolated AB responded to periodic forcing in a manner similar to the complete pacemaker group, but with wider zones of synchronization. We have built an analog electronic circuit as an implementation of a modified Hindmarsh-Rose model for simulating the membrane potential activity of pyloric neurons. We subjected this electronic model neuron to the same periodic forcing as used in the biological experiments. This four-dimensional electronic model neuron reproduced the autonomous oscillatory firing patterns of biological pyloric pacemaker neurons, and it expressed the same stationary nonlinear responses to periodic forcing as its biological counterparts. This adds to our confidence in the model. These results strongly support the idea that the intact pyloric pacemaker group acts as a uniform low-dimensional deterministic nonlinear oscillator, and the regular pyloric oscillation is the outcome of cooperative behavior of strongly coupled neurons, having different dynamical and biophysical properties when isolated.

Responses of a Hodgkin-Huxley neuron to various types of spike-train inputs

Numerical investigations have been made of responses of a Hodgkin-Huxley (HH) neuron to spike-train inputs whose interspike interval (ISI) is modulated by deterministic, semi-deterministic (chaotic), and stochastic signals. As deterministic one, we adopt inputs with the time-independent ISI and with time-dependent ISI modulated by sinusoidal signal. The Rössler and Lorentz models are adopted for chaotic modulations of ISI. Stochastic ISI inputs with the gamma distribution are employed. It is shown that distribution of output ISI data depends not only on the mean of ISIs of spike-train inputs but also on their fluctuations. The distinction of responses to the three kinds of inputs can be made by return maps of input and output ISIs, but not by their histograms. The relation between the variations of input and output ISIs is shown to be different from that of the integrate and fire (IF) model because of the refractory period in the HH neuron.

The origins of aperiodicities in sensory neuron entrainment

Aperiodic entrainment to rhythmic sensory input was obtained with either a single neuron or an excitatory network model, without addition of a stochastic or "noisy" element. The entrainment properties of primary sensory neurons were well captured by the dynamics of the Hodgkin-Huxley ordinary differential equations with a quiescent resting state or threshold for spike output. The frequency-amplitude parameter space was compressed and aperiodic regimes were small in comparison to those of periodically activated pacemaker-like neurons. Transitions between phase-locked and aperiodic entrainment patterns were predictable and determined by the equation dynamics, supporting the contention that some aperiodicities observed in situ arise from the inherent membrane properties of neurons. When the rhythmically activated neuron was embedded in an excitatory network of Hodgkin-Huxley neurons with heterogeneous synaptic delays, aperiodic entrainment patterns were more frequently encountered and these were associated with asynchronous output from the network. Embedding the rhythmically activated neuron in a network with synaptic delays greatly reduced the range of entrained spike frequencies and increased the variability in the neuronal firing. The temporal coding of sensory stimuli may be dependent on these findings. Sensory stimuli are signaled in the periphery by a mixture of periodic and irregular interspike intervals. Most models of such temporal codes assume intrinsic rhythmicity arising from the ionic currents, with variations attributed to membrane or synaptic noise. In contrast, we demonstrate irregular neural codes that arise completely in the absence of noise. In the proposed model, the sources of these irregular sensory patterns are the extensive cross-connections and resultant interactions between neurons. The balance between the regular and irregular entrainment of a neuron in situ could uniquely identify a stimulus. Other biological mechanisms of modifying the entrainment properties and promoting aperiodic entrainment are discussed.

On the complex dynamics of intracellular ganglion cell light responses in the cat retina

We recorded intracellular responses from cat retinal ganglion cells to sinusoidal flickering lights, and compared the response dynamics with a theoretical model based on coupled nonlinear oscillators. Flicker responses for several different spot sizes were separated in a "smooth" generator (G) potential and corresponding spike trains. We have previously shown that the G-potential reveals complex, stimulus-dependent, oscillatory behavior in response to sinusoidally flickering lights. Such behavior could be simulated by a modified van der Pol oscillator. In this paper, we extend the model to account for spike generation as well, by including extended Hodgkin-Huxley equations describing local membrane properties. We quantified spike responses by several parameters describing the mean and standard deviation of spike burst duration, timing (phase shift) of bursts, and the number of spikes in a burst. The dependence of these response parameters on stimulus frequency and spot size could be reproduced in great detail by coupling the van der Pol oscillator and Hodgkin-Huxley equations. The model mimics many experimentally observed response patterns, including non-phase-locked irregular oscillations. Our findings suggest that the information in the ganglion cell spike train reflects both intraretinal processing, simulated by the van der Pol oscillator, and local membrane properties described by Hodgkin-Huxley equations. The interplay between these complex processes can be simulated by changing the coupling coefficients between the two oscillators. Our simulations therefore show that irregularities in spike trains, which normally are considered to be noise, may be interpreted as complex oscillations that might carry information.

Chaotic responses of the hippocampal CA3 region to a mossy fiber stimulation in vitro

Undoubted evidence of chaotic activity of a biological neural network are given. Spontaneous epileptiform bursts of a neuron population in the CA3 region of rat hippocampal slices were caused in a perfusing medium with 2 mM penicillin and 8 mM K+ ions, and responses of field potential in the CA3 region to a periodic mossy fiber stimulation were investigated. Phase-locked and chaotic responses occur depending on stimulus parameters; for example, when the frequency of the stimulation increases, 1:1 phase-locking bifurcates to chaos through 1:2 phase-locking. The chaotic responses show a broad-band spectrum, and their trajectories in the three-dimensional phase space (V(t), V(t + tau), V(t + 2 tau)) reconstruct a strange attractor. Lyapunov exponents of the strange attractors estimated by the Wolf's algorithm are positive. Moreover, one-dimensional strobomaps obtained from the chaotic responses show a non-invertible function. Since the slope of each strobomap at their fixed point is more negative than -1, the fixed points are unstable. These are undoubted evidences for chaotic responses of the CA3 region in hippocampal slices maintained in vitro. Cross-correlation functions between field potential responses which were simultaneously observed at different sites show that the responses are spatially coherent throughout the CA3 region even when the responses are chaotic.

Discrete-time stimulation of the oscillatory and excitable forms of a FitzHugh-Nagumo model applied to the pulsatile release of luteinizing hormone releasing hormone

A model for the pulsatile release of luteinizing hormone releasing hormone (LHRH) can be reduced to a FitzHugh-Nagumo model subject to regular and quasiregular (i.e., with slight random variation in the interstimulus interval), discrete-time stimulation. The relationship of output pulse frequency (OPF) to stimulus frequency is compared between the excitable and oscillatory forms of the model and discussed in the context of results from other pulse-driven model systems. Some examples of the changes in OPF caused by quasiregular and purely Poissonian stimuli are given for the excitable case. The unstimulated system frequently interacts with the stimulation in such a complex manner that the OPF bears little resemblance to the frequency of stimulation or of the unstimulated system. Furthermore, the inability of the oscillatory form of the model to allow complete suppression of output pulses for moderate stimulation frequencies suggests that the LHRH system can be more appropriately described by the excitable form of the model. (c) 1995 American Institute of Physics.