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.To the memory of Prof. Otto-Joachim Grusser     

Chaos and synchrony in a model of a hypercolumn in visual cortex

Neurons in cortical slices emit spikes or bursts of spikes regularly in response to a suprathreshold current injection. This behavior is in marked contrast to the behavior of cortical neurons in vivo, whose response to electrical or sensory input displays a strong degree of irregularity. Correlation measurements show a significant degree of synchrony in the temporal fluctuations of neuronal activities in cortex. We explore the hypothesis that these phenomena are the result of the synchronized chaos generated by the deterministic dynamics of local cortical networks. A model of a hypercolumn in the visual cortex is studied. It consists of two populations of neurons, one inhibitory and one excitatory. The dynamics of the neurons is based on a Hodgkin-Huxley type model of excitable voltage-clamped cells with several cellular and synaptic conductances. A slow potassium current is included in the dynamics of the excitatory population to reproduce the observed adaptation of the spike trains emitted by these neurons. The pattern of connectivity has a spatial structure which is correlated with the internal organization of hypercolumns in orientation columns. Numerical simulations of the model show that in an appropriate parameter range, the network settles in a synchronous chaotic state, characterized by a strong temporal variability of the neural activity which is correlated across the hypercolumn. Strong inhibitory feedback is essential for the stabilization of this state. These results show that the cooperative dynamics of large neuronal networks are capable of generating variability and synchrony similar to those observed in cortex. Auto-correlation and cross-correlation functions of neuronal spike trains are computed, and their temporal and spatial features are analyzed. In other parameter regimes, the network exhibits two additional states: synchronized oscillations and an asynchronous state. We use our model to study cortical mechanisms for orientation selectivity. It is shown that in a suitable parameter regime, when the input is not oriented, the network has a continuum of states, each representing an inhomogeneous population activity which is peaked at one of the orientation columns. As a result, when a weakly oriented input stimulates the network, it yields a sharp orientation tuning. The properties of the network in this regime, including the appearance of virtual rotations and broad stimulus-dependent cross-correlations, are investigated. The results agree with the predictions of the mean field theory which was previously derived for a simplified model of stochastic, two-state neurons. The relation between the results of the model and experiments in visual cortex are discussed.     

Six types of multistability in a neuronal model based on slow calcium current

Background

Multistability of oscillatory and silent regimes is a ubiquitous phenomenon exhibited by excitable systems such as neurons and cardiac cells. Multistability can play functional roles in short-term memory and maintaining posture. It seems to pose an evolutionary advantage for neurons which are part of multifunctional Central Pattern Generators to possess multistability. The mechanisms supporting multistability of bursting regimes are not well understood or classified.

Methodology/Principal Findings

Our study is focused on determining the bio-physical mechanisms underlying different types of co-existence of the oscillatory and silent regimes observed in a neuronal model. We develop a low-dimensional model typifying the dynamics of a single leech heart interneuron. We carry out a bifurcation analysis of the model and show that it possesses six different types of multistability of dynamical regimes. These types are the co-existence of 1) bursting and silence, 2) tonic spiking and silence, 3) tonic spiking and subthreshold oscillations, 4) bursting and subthreshold oscillations, 5) bursting, subthreshold oscillations and silence, and 6) bursting and tonic spiking. These first five types of multistability occur due to the presence of a separating regime that is either a saddle periodic orbit or a saddle equilibrium. We found that the parameter range wherein multistability is observed is limited by the parameter values at which the separating regimes emerge and terminate.

Conclusions

We developed a neuronal model which exhibits a rich variety of different types of multistability. We described a novel mechanism supporting the bistability of bursting and silence. This neuronal model provides a unique opportunity to study the dynamics of networks with neurons possessing different types of multistability.     

A model for the characterization of the spatial properties in vestibular neurons

Quantitative study of the static and dynamic response properties of some otolith-sensitive neurons has been difficult in the past partly because their responses to different linear acceleration vectors exhibited no null plane and a dependence of phase on stimulus orientation. The theoretical formulation of the response ellipse provides a quantitative way to estimate the spatio-temporal properties of such neurons. Its semi-major axis gives the direction of the polarization vector (i.e., direction of maximal sensitivity) and it estimates the neuronal response for stimulation along that direction. In addition, the semi-minor axis of the ellipse provides an estimate of the neuron's maximal sensitivity in the null plane. In this paper, extracellular recordings from otolith-sensitive vestibular nuclei neurons in decerebrate rats were used to demonstrate the practical application of the method. The experimentally observed gain and phase dependence on the orientation angle of the acceleration vector in a head-horizontal plane was described and satisfactorily fit by the response ellipse model. In addition, the model satisfactorily fits neuronal responses in three-dimensions and unequivocally demonstrates that the response ellipse formulation is the general approach to describe quantitatively the spatial properties of vestibular neurons.     

Surprising Effects of Synaptic Excitation

Typically, excitatory synaptic coupling is thought of as an influence that accelerates and propagates firing in neuronal networks. This paper reviews recent results explaining how, contrary to these expectations, the presence of excitatory synaptic coupling can drastically slow oscillations in a network and how localized, sustained activity can arise in a network with purely excitatory coupling, without sustained inputs. These two effects stem from interactions of excitatory coupling with two different forms of intrinsic neuronal dynamics, and both serve to highlight the fact that the influence of synaptic coupling in a network depends strongly on the intrinsic properties of cells in the network.This work was partially supported by the National Science Foundation, under award DMS-0414023     

Modelling of the active reaction of stick insects by a network of neuromimes

Depending on the activity status of the animal, the control system of the femur-tibia joint in stick insects exhibits either a resistance reflex, or the active reaction, a totally different action pattern (Bässler 1988a). Using analog electronic neuron models, several different neuronal circuits are explored that model the active reaction with all its features. The models differ in complexity, redundance and the robustness against small variations of network parameters (e.g. coupling strengths). The circuit with the highest robustness and redundancy requires interneurons with special features, such as found in real animals. When inserted into a closed loop modeling movement and sensory feedback from the periphery, this circuit produces oscillations similar to searching movements found in the real animal. In addition to intracellular recording methods, the authors propose modeling with realistic neuromimes as a complementary method in the investigation of neuronal networks which have well documented input-output relationships.     

On the neuronal basis of figure-ground discrimination by relative motion in the visual system of the fly

A fly can discriminate an object (figure) from its background on the basis of motion information alone. This information processing task has been analysed, so far, mainly in behavioural studies but also in electrophysiological experiments (Reichardt et al., 1983). The present study represents a further attempt to bridge the gap between the behavioural and the neuronal level. It is based on behavioural and electrophysiological experiments as well as on computer simulations. The characteristic properties of figureground discrimination behaviour impose specific constraints on the spatial integration properties of the output cells of the underlying neuronal network, the heterolateral interactions in their input circuitry, as well as on the range of variability of their response. These constraints are derived partly from previous behavioural studies (Reichardt et al., 1983), partly, however, from behavioural response characteristics which have not been addressed explicitly so far. They are interpreted in terms of one of the alternative model circuits shown by Reichardt et al. (1983) to be sufficient to account for figure-ground discrimination. It will be demonstrated, however, that this can be done equally well by means of a further alternative model circuit. These constraints are used in the electrophysiological analysis for establishing visual interneurones as output elements of the neuronal network underlying figure-ground discrimination.In the behavioural experiments on figure-ground discrimination as well as on the optomotor course control the yaw torque generated by the tethered flying fly under visual stimulation was used as a measure for the strength and time course of the reaction. Therefore, it has initially been proposed that the three Horizontal Cells, which are regarded as the output elements of the neuronal network underlying the optomotor reaction (e.g. Hausen, 1981), might also control yaw torque generation in figure-ground discrimination (Reichardt et al., 1983). New behavioural data show, however, that the Horizontal Cells do not meet all the constraints imposed on the presumed output cells of the figure-ground discrimination network: (1) The Horizontal Cells are not sensitive enough to motion of small objects. (2) The heterolateral interactions within their input circuitry are not in accordance with the behavioural data (see also Reichardt et al., 1983). (3) The variability found in the time course of certain components of the yaw torque response to relative motion of figure and ground cannot be explained by their response characteristics. Hence, the Horizontal Cells cannot account for figure-ground discrimination on their own and additional output cells of the optic lobes with different functional properties are required to accomplish this task.     

History-dependent excitability as a single-cell substrate of transient memory for information discrimination

Neurons react differently to incoming stimuli depending upon their previous history of stimulation. This property can be considered as a single-cell substrate for transient memory, or context-dependent information processing: depending upon the current context that the neuron "sees" through the subset of the network impinging on it in the immediate past, the same synaptic event can evoke a postsynaptic spike or just a subthreshold depolarization. We propose a formal definition of History-Dependent Excitability (HDE) as a measure of the propensity to firing in any moment in time, linking the subthreshold history-dependent dynamics with spike generation. This definition allows the quantitative assessment of the intrinsic memory for different single-neuron dynamics and input statistics. We illustrate the concept of HDE by considering two general dynamical mechanisms: the passive behavior of an Integrate and Fire (IF) neuron, and the inductive behavior of a Generalized Integrate and Fire (GIF) neuron with subthreshold damped oscillations. This framework allows us to characterize the sensitivity of different model neurons to the detailed temporal structure of incoming stimuli. While a neuron with intrinsic oscillations discriminates equally well between input trains with the same or different frequency, a passive neuron discriminates better between inputs with different frequencies. This suggests that passive neurons are better suited to rate-based computation, while neurons with subthreshold oscillations are advantageous in a temporal coding scheme. We also address the influence of intrinsic properties in single-cell processing as a function of input statistics, and show that intrinsic oscillations enhance discrimination sensitivity at high input rates. Finally, we discuss how the recognition of these cell-specific discrimination properties might further our understanding of neuronal network computations and their relationships to the distribution and functional connectivity of different neuronal types.     

Phase-Coherence Transitions and Communication in the Gamma Range between Delay-Coupled Neuronal Populations

Synchronization between neuronal populations plays an important role in information transmission between brain areas. In particular, collective oscillations emerging from the synchronized activity of thousands of neurons can increase the functional connectivity between neural assemblies by coherently coordinating their phases. This synchrony of neuronal activity can take place within a cortical patch or between different cortical regions. While short-range interactions between neurons involve just a few milliseconds, communication through long-range projections between different regions could take up to tens of milliseconds. How these heterogeneous transmission delays affect communication between neuronal populations is not well known. To address this question, we have studied the dynamics of two bidirectionally delayed-coupled neuronal populations using conductance-based spiking models, examining how different synaptic delays give rise to in-phase/anti-phase transitions at particular frequencies within the gamma range, and how this behavior is related to the phase coherence between the two populations at different frequencies. We have used spectral analysis and information theory to quantify the information exchanged between the two networks. For different transmission delays between the two coupled populations, we analyze how the local field potential and multi-unit activity calculated from one population convey information in response to a set of external inputs applied to the other population. The results confirm that zero-lag synchronization maximizes information transmission, although out-of-phase synchronization allows for efficient communication provided the coupling delay, the phase lag between the populations, and the frequency of the oscillations are properly matched.     

Effectiveness factor calculation for immobilised growing cell systems

Summary Gel immobilised living cell systems represent a special form of heterogeneous catalysis. Due to mass transfer limitations on substrate delivery and product removal, time-dependent spatial variations in growth rate and biomass densities are created. A dynamic mathematical model has been developed which was used to simulate the start-up dynamics of growing yeast cells in calcium alginate beads. By coupling the transient model with the effectiveness factor calculation, the dynamic effectiveness factor could be calculated. The influence of the bead radius and the initial biomass concentration on the dynamic effectiveness factor have been investigated.     

Spatial and temporal coding in single neurons

Convergence between cells which differ in both spatial and temporal properties create higher order neurons with response properties that are distinctly different from those of the input neurons. The spatial properties of target neurons are not necessarily cosinetuned. In addition, unlike the independence between spatial and temporal properties in cosine-tuned afferent neurons, higher-order target cells generally exhibit a dependence of temporal dynamics on spatial properties. The response properties of target neurons receiving spatio-temporal convergence (STC) from tonic and phasic-tonic or phasic afferents is investigated here by considering a general case where the dynamic input is represented by a fractional, leaky, derivative transfer function. It is shown that, at frequencies below the corner frequency of the dynamic input, the temporal properties of target neurons can be described by leaky differentiators having time constants that are a function of spatial direction. Thus, STC target neurons exhibit tonic temporal response properties during stimulation along some spatial directions (having small time constants) and phasic properties along other directions (having large time constants). Specifically, target neurons encode the complete derivative of the stimulus along certain spatial directions. Thus, STC acts as a directionally specific high-pass filter and produces complete derivatives from fractional, leaky derivative afferent signals. In addition, spatio-temporal transformations can generate novel temporal dynamics in the central nervous system. These observations suggest that spatio-temporal computations might constitute an alternative to parallel, independent spatial and temporal channels.     

Innate synchronous oscillations in freely-organized small neuronal circuits

Background

Information processing in neuronal networks relies on the network''s ability to generate temporal patterns of action potentials. Although the nature of neuronal network activity has been intensively investigated in the past several decades at the individual neuron level, the underlying principles of the collective network activity, such as the synchronization and coordination between neurons, are largely unknown. Here we focus on isolated neuronal clusters in culture and address the following simple, yet fundamental questions: What is the minimal number of cells needed to exhibit collective dynamics? What are the internal temporal characteristics of such dynamics and how do the temporal features of network activity alternate upon crossover from minimal networks to large networks?

Methodology/Principal Findings

We used network engineering techniques to induce self-organization of cultured networks into neuronal clusters of different sizes. We found that small clusters made of as few as 40 cells already exhibit spontaneous collective events characterized by innate synchronous network oscillations in the range of 25 to 100 Hz. The oscillation frequency of each network appeared to be independent of cluster size. The duration and rate of the network events scale with cluster size but converge to that of large uniform networks. Finally, the investigation of two coupled clusters revealed clear activity propagation with master/slave asymmetry.

Conclusions/Significance

The nature of the activity patterns observed in small networks, namely the consistent emergence of similar activity across networks of different size and morphology, suggests that neuronal clusters self-regulate their activity to sustain network bursts with internal oscillatory features. We therefore suggest that clusters of as few as tens of cells can serve as a minimal but sufficient functional network, capable of sustaining oscillatory activity. Interestingly, the frequencies of these oscillations are similar those observed in vivo.     

Model of brain rhythmic activity

1. A model of a neuronal network has been set up in a digital computer based on histological and biophysical data experimentally obtained from the thalamus; the model includes two populations of neurons interconnected by means of negative feedback; in the model allowance is also made for other sort of interactions.
2. To test the hypothesis that the alpha-rhythm (8–13 Hz rhythmic activity characteristic of the EEG) is a filtered noise signal the simulated neuronal network was stimulated by random trains of pulses with a Poisson distribution. The density of pulses fired by the simulated neurons was computed as well as the oscillations of the mean membrane potential of the population of simulated neurons. The latter was found to be equivalent to the experimentally obtained alpha rhythms.
3. In order to test the hypothesis that several noise sources are responsible for thalamo-cortical coherences three simulated neuronal networks were coupled together using several noise sources as secondary inputs. It was shown that although all the networks produced simulated alpha signals with identical spectra they could have significantly different values of coherence depending on the relation between correlated and uncorrelated input signals.
4. The model was analysed by means of linear systems analysis after introducing the necessary simplifications and approximations. In this way it was possible to evaluate the influence of different physiological or histological parameters upon the statistical properties of the resulting rhythmic activity in an analytical form.
5. By changing the model parameters it was shown that a family of spectral curves could be obtained which simulated the development of the EEG as function of age from a predominantly low frequency to a clearly rhythmic type of signal. This was shown to depend mainly on the feedback coupling parameters.

     

Systems-level modeling of neuronal circuits for leech swimming

This paper describes a mathematical model of the neuronal central pattern generator (CPG) that controls the rhythmic body motion of the swimming leech. The systems approach is employed to capture the neuronal dynamics essential for generating coordinated oscillations of cell membrane potentials by a simple CPG architecture with a minimal number of parameters. Based on input/output data from physiological experiments, dynamical components (neurons and synaptic interactions) are first modeled individually and then integrated into a chain of nonlinear oscillators to form a CPG. We show through numerical simulations that the values of a few parameters can be estimated within physiologically reasonable ranges to achieve good fit of the data with respect to the phase, amplitude, and period. This parameter estimation leads to predictions regarding the synaptic coupling strength and intrinsic period gradient along the nerve cord, the latter of which agrees qualitatively with experimental observations.     

The origin of glucocorticoid hormone oscillations

Oscillating levels of adrenal glucocorticoid hormones are essential for optimal gene expression, and for maintaining physiological and behavioural responsiveness to stress. The biological basis for these oscillations is not known, but a neuronal "pulse generator" within the hypothalamus has remained a popular hypothesis. We demonstrate that pulsatile hypothalamic activity is not required for generating ultradian glucocorticoid oscillations. We show that a constant level of corticotrophin-releasing hormone (CRH) can activate a dynamic pituitary-adrenal peripheral network to produce ultradian adrenocorticotrophic hormone and glucocorticoid oscillations with a physiological frequency. This oscillatory response to CRH is dose dependent and becomes disrupted for higher levels of CRH. These data suggest that glucocorticoid oscillations result from a sub-hypothalamic pituitary-adrenal system, which functions as a deterministic peripheral hormone oscillator with a characteristic ultradian frequency. This constitutes a novel mechanism by which the level, rather than the pattern, of CRH determines the dynamics of glucocorticoid hormone secretion.     

Impact of adaptation currents on synchronization of coupled exponential integrate-and-fire neurons

The ability of spiking neurons to synchronize their activity in a network depends on the response behavior of these neurons as quantified by the phase response curve (PRC) and on coupling properties. The PRC characterizes the effects of transient inputs on spike timing and can be measured experimentally. Here we use the adaptive exponential integrate-and-fire (aEIF) neuron model to determine how subthreshold and spike-triggered slow adaptation currents shape the PRC. Based on that, we predict how synchrony and phase locked states of coupled neurons change in presence of synaptic delays and unequal coupling strengths. We find that increased subthreshold adaptation currents cause a transition of the PRC from only phase advances to phase advances and delays in response to excitatory perturbations. Increased spike-triggered adaptation currents on the other hand predominantly skew the PRC to the right. Both adaptation induced changes of the PRC are modulated by spike frequency, being more prominent at lower frequencies. Applying phase reduction theory, we show that subthreshold adaptation stabilizes synchrony for pairs of coupled excitatory neurons, while spike-triggered adaptation causes locking with a small phase difference, as long as synaptic heterogeneities are negligible. For inhibitory pairs synchrony is stable and robust against conduction delays, and adaptation can mediate bistability of in-phase and anti-phase locking. We further demonstrate that stable synchrony and bistable in/anti-phase locking of pairs carry over to synchronization and clustering of larger networks. The effects of adaptation in aEIF neurons on PRCs and network dynamics qualitatively reflect those of biophysical adaptation currents in detailed Hodgkin-Huxley-based neurons, which underscores the utility of the aEIF model for investigating the dynamical behavior of networks. Our results suggest neuronal spike frequency adaptation as a mechanism synchronizing low frequency oscillations in local excitatory networks, but indicate that inhibition rather than excitation generates coherent rhythms at higher frequencies.     

Neuronal Spike Initiation Modulated by Extracellular Electric Fields

Based on a reduced two-compartment model, the dynamical and biophysical mechanism underlying the spike initiation of the neuron to extracellular electric fields is investigated in this paper. With stability and phase plane analysis, we first investigate in detail the dynamical properties of neuronal spike initiation induced by geometric parameter and internal coupling conductance. The geometric parameter is the ratio between soma area and total membrane area, which describes the proportion of area occupied by somatic chamber. It is found that varying it could qualitatively alter the bifurcation structures of equilibrium as well as neuronal phase portraits, which remain unchanged when varying internal coupling conductance. By analyzing the activating properties of somatic membrane currents at subthreshold potentials, we explore the relevant biophysical basis of spike initiation dynamics induced by these two parameters. It is observed that increasing geometric parameter could greatly decrease the intensity of the internal current flowing from soma to dendrite, which switches spike initiation dynamics from Hopf bifurcation to SNIC bifurcation; increasing internal coupling conductance could lead to the increase of this outward internal current, whereas the increasing range is so small that it could not qualitatively alter the spike initiation dynamics. These results highlight that neuronal geometric parameter is a crucial factor in determining the spike initiation dynamics to electric fields. The finding is useful to interpret the functional significance of neuronal biophysical properties in their encoding dynamics, which could contribute to uncovering how neuron encodes electric field signals.     

Transmission of stimulus-locked responses in two oscillators with bistable coupling

Numerous electroencephalography (EEG) and magnetoencephalography (MEG) studies aim at identifying the chronological order of activation of brain areas. This paper demonstrates that the timing sequence obtained with the gold standard for EEG/MEG analysis (averaging across trials) may not correlate at all with the actual transmission of a stimulus effect within a pathway formed by connected brain areas. This is shown by studying transmission of stimulus-locked responses in a model that shares basic features with stimulated neuronal rhythms: in two phase oscillators with bistable coupling and noise one oscillator is stimulated. The model presents a mechanism that causes a response clustering, i.e., a switching between two different responses across trials, without extinction of the averaged response (calculated over all trials). Transmission times are calculated for all trials as well as for the two clusters separately with standard averaged responses and with a stochastic phase resetting analysis. The stochastic phase resetting analysis provides reliable estimates of the transmission time. In contrast, transmission times calculated by averaging across trials correspond to the phase difference in the different stable synchronized states (when calculated for the two clusters separately) or their weighted superposition (when calculated over all trials). The standard method does not detect the time elapsing during the transmission of the stimulus action. The results presented here call into question many findings reported in the evoked response literature.     

Multiple Stable Periodic Oscillations in a Mathematical Model of CTL Response to HTLV-I Infection

Stable periodic oscillations have been shown to exist in mathematical models for the CTL response to HTLV-I infection. These periodic oscillations can be the result of mitosis of infected target CD4⁺ cells, of a general form of response function, or of time delays in the CTL response. In this study, we show through a simple mathematical model that time delays in the CTL response process to HTLV-I infection can lead to the coexistence of multiple stable periodic solutions, which differ in amplitude and period, with their own basins of attraction. Our results imply that the dynamic interactions between the CTL immune response and HTLV-I infection are very complex, and that multi-stability in CTL response dynamics can exist in the form of coexisting stable oscillations instead of stable equilibria. Biologically, our findings imply that different routes or initial dosages of the viral infection may lead to quantitatively and qualitatively different outcomes.