Pulse coupled oscillators and the phase resetting curve

Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators. In these oscillators, the interactions take the form of brief pulses such that the effect of one input dies out before the next is received. A phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. PRCs can be used to predict phase locking in networks of pulse coupled oscillators. In some studies of pulse coupled oscillators, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is generated using a perturbation that approximates the input received in the real biological network. In general, this approach requires that circuit architecture and a specific firing pattern be assumed. This allows the construction of discrete maps from one event to the next. The fixed points of these maps correspond to periodic firing modes and are easier to locate and analyze for stability compared to locating and analyzing periodic modes in the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks.     

Slow Noise in the Period of a Biological Oscillator Underlies Gradual Trends and Abrupt Transitions in Phasic Relationships in Hybrid Neural Networks

In order to study the ability of coupled neural oscillators to synchronize in the presence of intrinsic as opposed to synaptic noise, we constructed hybrid circuits consisting of one biological and one computational model neuron with reciprocal synaptic inhibition using the dynamic clamp. Uncoupled, both neurons fired periodic trains of action potentials. Most coupled circuits exhibited qualitative changes between one-to-one phase-locking with fairly constant phasic relationships and phase slipping with a constant progression in the phasic relationships across cycles. The phase resetting curve (PRC) and intrinsic periods were measured for both neurons, and used to construct a map of the firing intervals for both the coupled and externally forced (PRC measurement) conditions. For the coupled network, a stable fixed point of the map predicted phase locking, and its absence produced phase slipping. Repetitive application of the map was used to calibrate different noise models to simultaneously fit the noise level in the measurement of the PRC and the dynamics of the hybrid circuit experiments. Only a noise model that added history-dependent variability to the intrinsic period could fit both data sets with the same parameter values, as well as capture bifurcations in the fixed points of the map that cause switching between slipping and locking. We conclude that the biological neurons in our study have slowly-fluctuating stochastic dynamics that confer history dependence on the period. Theoretical results to date on the behavior of ensembles of noisy biological oscillators may require re-evaluation to account for transitions induced by slow noise dynamics.     

Multi-Stability and Pattern-Selection in Oscillatory Networks with Fast Inhibition and Electrical Synapses

A model or hybrid network consisting of oscillatory cells interconnected by inhibitory and electrical synapses may express different stable activity patterns without any change of network topology or parameters, and switching between the patterns can be induced by specific transient signals. However, little is known of properties of such signals. In the present study, we employ numerical simulations of neural networks of different size composed of relaxation oscillators, to investigate switching between in-phase (IP) and anti-phase (AP) activity patterns. We show that the time windows of susceptibility to switching between the patterns are similar in 2-, 4- and 6-cell fully-connected networks. Moreover, in a network (N = 4, 6) expressing a given AP pattern, a stimulus with a given profile consisting of depolarizing and hyperpolarizing signals sent to different subpopulations of cells can evoke switching to another AP pattern. Interestingly, the resulting pattern encodes the profile of the switching stimulus. These results can be extended to different network architectures. Indeed, relaxation oscillators are not only models of cellular pacemakers, bursting or spiking, but are also analogous to firing-rate models of neural activity. We show that rules of switching similar to those found for relaxation oscillators apply to oscillating circuits of excitatory cells interconnected by electrical synapses and cross-inhibition. Our results suggest that incoming information, arriving in a proper time window, may be stored in an oscillatory network in the form of a specific spatio-temporal activity pattern which is expressed until new pertinent information arrives.     

An oscillatory correlation model of auditory streaming

We present a neurocomputational model for auditory streaming, which is a prominent phenomenon of auditory scene analysis. The proposed model represents auditory scene analysis by oscillatory correlation, where a perceptual stream corresponds to a synchronized assembly of neural oscillators and different streams correspond to desynchronized oscillator assemblies. The underlying neural architecture is a two-dimensional network of relaxation oscillators with lateral excitation and global inhibition, where one dimension represents time and another dimension frequency. By employing dynamic connections along the frequency dimension and a random element in global inhibition, the proposed model produces a temporal coherence boundary and a fissure boundary that closely match those from the psychophysical data of auditory streaming. Several issues are discussed, including how to represent physical time and how to relate shifting synchronization to auditory attention.     

动态神经网络中的同步振荡

目前有一种假设认为同一视觉对象是由一群神经元的同步振荡活动来表征的。这一神经元发放活动的时间特性,是解决视觉信息处理中“结合问题（Ｂｉｎｄｉｎｇｐｒｏｂｌｅｍ）”的可能机制。本文用我们所提出的一种简化现实性神经网络模型［１］所构造的时滞非线性振子网络［２］,模拟生物神经网络的同步振荡活动。并考虑了振子各参数的设置与振荡活动的关系,以及网络振子间耦联对同步活动的影响．     

Synaptic mechanisms in invertebrate pattern generation

Although individual neurons can be intrinsically oscillatory and can be network pacemakers, motor patterns are often generated in a more distributed manner. Synaptic connections with other neurons are important because they either modify the rhythm of the pacemaker cell or are essential for pattern generation in the first place. Computational studies of half-center oscillators have made much progress in describing how neurons make transitions between active and inactive phases in these simple networks. In addition to characterizing phase transitions, recent studies have described the synaptic mechanisms that are important for the initiation and maintenance of activity in half-center oscillators.     

Control of multistability in ring circuits of oscillators

The essential dynamics of some biological central pattern generators (CPGs) can be captured by a model consisting of N neurons connected in a ring. These circuits, like many oscillatory nonlinear circuits of sufficient complexity, are capable of multistability, that is, of generating different firing patterns distinguished by the phasic relationships between the firing in each circuit element (neuron). Moreover, a shift in firing pattern can be induced by a transient perturbation. A systematic approach, based on phase-response curve (PRC) theory, was used to determine the optimum timing for perturbations that induce a shift in the firing pattern. The first step was to visualize the solution space of the ring circuit, including the attractive basins for each stable firing pattern; this was possible using the relative phase of N−1 oscillators, with respect to an arbitrarily selected reference oscillator, as coordinate axes. The trajectories in this phase space were determined using an iterative mapping based only on the PRCs of the uncoupled component oscillators; this algorithm was called a circuit emulator. For an accurate mapping of the attractive basin of each pattern exhibited by the ring circuit, the emulator had to take into account the effect of a perturbation or input on the timing of two bursts following the onset of the perturbation, rather than just one. The visualization of the attractive basins for rings of two, three, and four oscillators enabled the accurate prediction of the amounts of phase resetting applied to up to N−1 oscillators within a cycle that would induce a transition from any pattern to any another pattern. Finally, the timing and synaptic characterization of an input called the switch signal was adjusted to produce the desired amount of phase resetting. Received: 29 May 1998 / Accepted in revised form: 18 September 1998     

Timing as an intrinsic property of neural networks: evidence from in vivo and in vitro experiments

The discrimination and production of temporal patterns on the scale of hundreds of milliseconds are critical to sensory and motor processing. Indeed, most complex behaviours, such as speech comprehension and production, would be impossible in the absence of sophisticated timing mechanisms. Despite the importance of timing to human learning and cognition, little is known about the underlying mechanisms, in particular whether timing relies on specialized dedicated circuits and mechanisms or on general and intrinsic properties of neurons and neural circuits. Here, we review experimental data describing timing and interval-selective neurons in vivo and in vitro. We also review theoretical models of timing, focusing primarily on the state-dependent network model, which proposes that timing in the subsecond range relies on the inherent time-dependent properties of neurons and the active neural dynamics within recurrent circuits. Within this framework, time is naturally encoded in populations of neurons whose pattern of activity is dynamically changing in time. Together, we argue that current experimental and theoretical studies provide sufficient evidence to conclude that at least some forms of temporal processing reflect intrinsic computations based on local neural network dynamics.     

Symmetry-breaking bifurcation: A possible mechanism for 2:1 frequency-locking in animal locomotion

The generation and control of animal locomotion is believed to involve central pattern generators — networks of neurons which are capable of producing oscillatory behavior. In the present work, the quadrupedal locomotor central pattern generator is modelled as four distinct but symmetrically coupled nonlinear oscillators. We show that the typical patterns for two such networks of oscillators include 2:1 frequency-locked oscillations. These patterns, which arise through symmetry-breaking Hopf bifurcation, correspond in part to observed patterns of 2:1 frequency-locking of limb movements during electrically elicited locomotion of decerebrate and spinal quadrupeds. We briefly describe how our theoretical predictions could be tested experimentally.     

Mesoscopic Patterns of Neural Activity Support Songbird Cortical Sequences

Time-locked sequences of neural activity can be found throughout the vertebrate forebrain in various species and behavioral contexts. From “time cells” in the hippocampus of rodents to cortical activity controlling movement, temporal sequence generation is integral to many forms of learned behavior. However, the mechanisms underlying sequence generation are not well known. Here, we describe a spatial and temporal organization of the songbird premotor cortical microcircuit that supports sparse sequences of neural activity. Multi-channel electrophysiology and calcium imaging reveal that neural activity in premotor cortex is correlated with a length scale of 100 µm. Within this length scale, basal-ganglia–projecting excitatory neurons, on average, fire at a specific phase of a local 30 Hz network rhythm. These results show that premotor cortical activity is inhomogeneous in time and space, and that a mesoscopic dynamical pattern underlies the generation of the neural sequences controlling song.     

An oscillator network exhibiting a long-lasting response to an external signal

This study proposes an oscillator network to model the long-lasting responses observed in neural circuits. The responses of the proposed network model are represented by the temporal synchronization of the oscillators. The response duration does not depend on the natural frequency of the oscillators, which allows the responses to last much longer than the oscillation period of the oscillators. We can control the response duration by tuning the connection strengths between the oscillators and the external signal that triggers the responses. It is possible to break and restart the responses regardless of the way in which the oscillators are connected.     

A Simple State-Determined Model Reproduces Entrainment and Phase-Locking of Human Walking

Theoretical studies and robotic experiments have shown that asymptotically stable periodic walking may emerge from nonlinear limit-cycle oscillators in the neuro-mechanical periphery. We recently reported entrainment of human gait to periodic mechanical perturbations with two essential features: 1) entrainment occurred only when the perturbation period was close to the original (preferred) walking period, and 2) entrainment was always accompanied by phase locking so that the perturbation occurred at the end of the double-stance phase. In this study, we show that a highly-simplified state-determined walking model can reproduce several salient nonlinear limit-cycle behaviors of human walking: 1) periodic gait that is 2) asymptotically stable; 3) entrainment to periodic mechanical perturbations only when the perturbation period is close to the model''s unperturbed period; and 4) phase-locking to locate the perturbation at the end of double stance. Importantly, this model requires neither supra-spinal control nor an intrinsic self-sustaining neural oscillator such as a rhythmic central pattern generator. Our results suggest that several prominent limit-cycle features of human walking may stem from simple afferent feedback processes without significant involvement of supra-spinal control or a self-sustaining oscillatory neural network.     

Effect of the topology and delayed interactions in neuronal networks synchronization

As important as the intrinsic properties of an individual nervous cell stands the network of neurons in which it is embedded and by virtue of which it acquires great part of its responsiveness and functionality. In this study we have explored how the topological properties and conduction delays of several classes of neural networks affect the capacity of their constituent cells to establish well-defined temporal relations among firing of their action potentials. This ability of a population of neurons to produce and maintain a millisecond-precise coordinated firing (either evoked by external stimuli or internally generated) is central to neural codes exploiting precise spike timing for the representation and communication of information. Our results, based on extensive simulations of conductance-based type of neurons in an oscillatory regime, indicate that only certain topologies of networks allow for a coordinated firing at a local and long-range scale simultaneously. Besides network architecture, axonal conduction delays are also observed to be another important factor in the generation of coherent spiking. We report that such communication latencies not only set the phase difference between the oscillatory activity of remote neural populations but determine whether the interconnected cells can set in any coherent firing at all. In this context, we have also investigated how the balance between the network synchronizing effects and the dispersive drift caused by inhomogeneities in natural firing frequencies across neurons is resolved. Finally, we show that the observed roles of conduction delays and frequency dispersion are not particular to canonical networks but experimentally measured anatomical networks such as the macaque cortical network can display the same type of behavior.     

Deterministic characterization of phase noise in biomolecular oscillators

On top of the many external perturbations, cellular oscillators also face intrinsic perturbations due the randomness of chemical kinetics. Biomolecular oscillators, distinct in their parameter sets or distinct in their architecture, show different resilience with respect to such intrinsic perturbations. Assessing this resilience can be done by ensemble stochastic simulations. These are computationally costly and do not permit further insights into the mechanistic cause of the observed resilience. For reaction systems operating at a steady state, the linear noise approximation (LNA) can be used to determine the effect of molecular noise. Here we show that methods based on LNA fail for oscillatory systems and we propose an alternative ansatz. It yields an asymptotic expression for the phase diffusion coefficient of stochastic oscillators. Moreover, it allows us to single out the noise contribution of every reaction in an oscillatory system. We test the approach on the one-loop model of the Drosophila circadian clock. Our results are consistent with those obtained through stochastic simulations with a gain in computational efficiency of about three orders of magnitude.     

一端受外力的交联振荡器链中的内部传输

本文研究了二类一端受外力的交联振荡器链：最邻近多相位交联振荡器链,以及多重交联振荡器链,讨论了它们产生内部传输,即各振荡器与外力具有相同频率的现象。文中近似相位差方程、指数二分性理论和中心流形理论被应用于系统的渐近近似。研究。本文得到了更符合于实际情况的神经网络CPG链动态特性分析结论。     

Phase angle difference alters coupling relations of functionally distinct circadian oscillators revealed by rhythm splitting

The interactions (i.e., coupling) between multiple oscillators of a circadian system determine basic properties of the integrated pacemaker. Unfortunately, there are few experimental models to investigate the putative interactions of functionally defined oscillators comprising the mammalian circadian pacemaker. Here the authors induce in hamsters a novel circadian entrainment pattern that is characterized by the daily expression of robust wheel-running activity in each scotophase of a 24-h light:dark:light:dark cycle. The daily activity bouts are mediated by 2 circadian oscillators, here designated "daytime" and "nighttime," that have been temporally dissociated under this light regime. To assess the phase dependence of interactions between oscillatory components, the phase relationship of the 2 daily scotophases was manipulated over a 4-h range, and the timing of activity of the daytime and nighttime components under entrained and probe conditions was examined. The average phase angle of entrainment and the day-to-day variability of activity onset of each activity component depended on the phase relationship of the respective scotophases and not on whether the component occurred in the daytime or the nighttime. Short-term denial of wheel access subsequently influenced amount and duration of wheel running but not timing of its onset, suggesting that only the former measures depend on a homeostatic mechanism sensitive to the time elapsed since prior intense running. Replacement of individual photophases with darkness revealed phase attraction between oscillators that was not dependent on the phase relationship of component oscillators but differed for daytime versus nighttime activity components. Entrainment patterns shown here cannot be accounted for by only nonparametric actions of light. Instead, the phase-dependent interactions of oscillators strongly influence entrainment properties, whereas intrinsic functional differences in dissociated oscillators apparently influence their attraction in darkness. This model system may be ideal for identifying genomic and physiological factors that mediate these interactions and thus contribute importantly to system properties of the mammalian circadian clock.     

Stability of two cluster solutions in pulse coupled networks of neural oscillators

Phase response curves (PRCs) have been widely used to study synchronization in neural circuits comprised of pacemaking neurons. They describe how the timing of the next spike in a given spontaneously firing neuron is affected by the phase at which an input from another neuron is received. Here we study two reciprocally coupled clusters of pulse coupled oscillatory neurons. The neurons within each cluster are presumed to be identical and identically pulse coupled, but not necessarily identical to those in the other cluster. We investigate a two cluster solution in which all oscillators are synchronized within each cluster, but in which the two clusters are phase locked at nonzero phase with each other. Intuitively, one might expect this solution to be stable only when synchrony within each isolated cluster is stable, but this is not the case. We prove rigorously the stability of the two cluster solution and show how reciprocal coupling can stabilize synchrony within clusters that cannot synchronize in isolation. These stability results for the two cluster solution suggest a mechanism by which reciprocal coupling between brain regions can induce local synchronization via the network feedback loop.     

Saccadic Reaction Times to Audiovisual Stimuli Show Effects of Oscillatory Phase Reset

Initiating an eye movement towards a suddenly appearing visual target is faster when an accessory auditory stimulus occurs in close spatiotemporal vicinity. Such facilitation of saccadic reaction time (SRT) is well-documented, but the exact neural mechanisms underlying the crossmodal effect remain to be elucidated. From EEG/MEG studies it has been hypothesized that coupled oscillatory activity in primary sensory cortices regulates multisensory processing. Specifically, it is assumed that the phase of an ongoing neural oscillation is shifted due to the occurrence of a sensory stimulus so that, across trials, phase values become highly consistent (phase reset). If one can identify the phase an oscillation is reset to, it is possible to predict when temporal windows of high and low excitability will occur. However, in behavioral experiments the pre-stimulus phase will be different on successive repetitions of the experimental trial, and average performance over many trials will show no signs of the modulation. Here we circumvent this problem by repeatedly presenting an auditory accessory stimulus followed by a visual target stimulus with a temporal delay varied in steps of 2 ms. Performing a discrete time series analysis on SRT as a function of the delay, we provide statistical evidence for the existence of distinct peak spectral components in the power spectrum. These frequencies, although varying across participants, fall within the beta and gamma range (20 to 40 Hz) of neural oscillatory activity observed in neurophysiological studies of multisensory integration. Some evidence for high-theta/alpha activity was found as well. Our results are consistent with the phase reset hypothesis and demonstrate that it is amenable to testing by purely psychophysical methods. Thus, any theory of multisensory processes that connects specific brain states with patterns of saccadic responses should be able to account for traces of oscillatory activity in observable behavior.     

Rapid synchronization through fast threshold modulation

Synchronization properties of locally coupled neural oscillators were investigated analytically and by computer simulation. When coupled in a manner that mimics excitatory chemical synapses, oscillators having more than one time scale (relaxation oscillators) are shown to approach synchrony using mechanisms very different from that of oscillators with a more sinusoidal waveform. The relaxation oscillators make critical use of fast modulations of their thresholds, leading to a rate of synchronization relatively independent of coupling strength within some basin of attraction; this rate is faster for oscillators that have conductance-based features than for neural caricatures such as the FitzHugh-Nagumo equations that lack such features. Computer simulations of one-dimensional arrays show that oscillators in the relaxation regime synchronize much more rapidly than oscillators with the same equations whose parameters have been modulated to yield a more sinusoidal waveform. We present a heuristic explanation of this effect based on properties of the coupling mechanisms that can affect the way the synchronization scales with array length. These results suggest that the emergent synchronization behavior of oscillating neural networks can be dramatically influenced by the intrinsic properties of the network components. Possible implications for perceptual feature binding and attention are discussed.Supported in part by NASA (NGT-50497)Supported in part by NSF (DMS-8901913), and NIMH-47150 Present address and address for correspondence: Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, E25-618, Cambridge, MA 02139, USA     

Temporal and spatial oscillations in bacteria

Oscillations pervade biological systems at all scales. In bacteria, oscillations control fundamental processes, including gene expression, cell cycle progression, cell division, DNA segregation and cell polarity. Oscillations are generated by biochemical oscillators that incorporate the periodic variation in a parameter over time to generate an oscillatory output. Temporal oscillators incorporate the periodic accumulation or activity of a protein to drive temporal cycles such as the cell and circadian cycles. Spatial oscillators incorporate the periodic variation in the localization of a protein to define subcellular positions such as the site of cell division and the localization of DNA. In this Review, we focus on the mechanisms of oscillators and discuss the design principles of temporal and spatial oscillatory systems.