The Sleep Cycle Modelled as a Cortical Phase Transition

We present a mean-field model of the cortex that attempts to describe the gross changes in brain electrical activity for the cycles of natural sleep. We incorporate within the model two major sleep modulatory effects: slow changes in both synaptic efficiency and in neuron resting voltage caused by the ∼90-min cycling in acetylcholine, together with even slower changes in resting voltage caused by gradual elimination during sleep of somnogens (fatigue agents) such as adenosine. We argue that the change from slow-wave sleep (SWS) to rapid-eye-movement (REM) sleep can be understood as a first-order phase transition from a low-firing, coherent state to a high-firing, desychronized cortical state. We show that the model predictions for changes in EEG power, spectral distribution, and correlation time at the SWS-to-REM transition are consistent not only with those observed in clinical recordings of a sleeping human subject, but also with the on-cortex EEG patterns recently reported by Destexhe et al. [J. Neurosci. 19(11), (1999) 4595–4608] for the sleeping cat.     

An analysis of the transitions between down and up states of the cortical slow oscillation under urethane anaesthesia

We study the dynamics of the transition between the low- and high-firing states of the cortical slow oscillation by using intracellular recordings of the membrane potential from cortical neurons of rats. We investigate the evidence for a bistability in assemblies of cortical neurons playing a major role in the maintenance of this oscillation. We show that the trajectory of a typical transition takes an approximately exponential form, equivalent to the response of a resistor–capacitor circuit to a step-change in input. The time constant for the transition is negatively correlated with the membrane potential of the low-firing state, and values are broadly equivalent to neural time constants measured elsewhere. Overall, the results do not strongly support the hypothesis of a bistability in cortical neurons; rather, they suggest the cortical manifestation of the oscillation is a result of a step-change in input to the cortical neurons. Since there is evidence from previous work that a phase transition exists, we speculate that the step-change may be a result of a bistability within other brain areas, such as the thalamus, or a bistability among only a small subset of cortical neurons, or as a result of more complicated brain dynamics.     

The phase response of the cortical slow oscillation

Cortical slow oscillations occur in the mammalian brain during deep sleep and have been shown to contribute to memory consolidation, an effect that can be enhanced by electrical stimulation. As the precise underlying working mechanisms are not known it is desired to develop and analyze computational models of slow oscillations and to study the response to electrical stimuli. In this paper we employ the conductance based model of Compte et al. (J Neurophysiol 89:2707–2725, 2003) to study the effect of electrical stimulation. The population response to electrical stimulation depends on the timing of the stimulus with respect to the state of the slow oscillation. First, we reproduce the experimental results of electrical stimulation in ferret brain slices by Shu et al. (Nature 423:288–293, 2003) from the conductance based model. We then numerically obtain the phase response curve for the conductance based network model to quantify the network’s response to weak stimuli. Our results agree with experiments in vivo and in vitro that show that sensitivity to stimulation is weaker in the up than in the down state. However, we also find that within the up state stimulation leads to a shortening of the up state, or phase advance, whereas during the up–down transition a prolongation of up states is possible, resulting in a phase delay. Finally, we compute the phase response curve for the simple mean-field model by Ngo et al. (EPL Europhys Lett 89:68002, 2010) and find that the qualitative shape of the PRC is preserved, despite its different mechanism for the generation of slow oscillations.     

Thalamic T-type Ca2+ channels and NREM sleep

T-type Ca2+ channels play a number of different and pivotal roles in almost every type of neuronal oscillation expressed by thalamic neurones during non-rapid eye movement (NREM) sleep, including those underlying sleep theta waves, the K-complex and the slow (<1 Hz) sleep rhythm, sleep spindles and delta waves. In particular, the transient opening of T channels not only gives rise to the 'classical' low threshold Ca2+ potentials, and associated high frequency burst of action potentials, that are characteristically present during sleep spindles and delta waves, but also contributes to the high threshold bursts that underlie the thalamic generation of sleep theta rhythms. The persistent opening of a small fraction of T channels, i.e. I(Twindow), is responsible for the large amplitude and long lasting depolarization, or UP state, of the slow (<1 Hz) sleep oscillation in thalamic neurones. These cellular findings are in part matched by the wake-sleep phenotype of global and thalamic-selective CaV3.1 knockout mice that show a decreased amount of total NREM sleep time. T-type Ca2+ channels, therefore, constitute the single most crucial voltage-dependent conductance that permeates all activities of thalamic neurones during NREM sleep. Since I(Twindow) and high threshold bursts are not restricted to thalamic neurones, the cellular neurophysiology of T channels should now move away from the simplistic, though historically significant, view of these channels as being responsible only for low threshold Ca2+ potentials.     

睡眠2期脑电信号产生的生理机制模型仿真研究

目前慢波睡眠生理机制研究已有的理论及动物实验结果显示,慢波睡眠中,皮层-丘脑系统神经元存在三种不同节律的振荡:慢振荡(<1 HZ)、δ振荡(1-4Hz)和纺锤振荡(7-14Hz)。这些神经元电活动在皮层水平广泛同步化,产生慢波睡眠脑电。提出了能分别产生这三种节律的三种神经元环路模型,并将之组合简化成一个七细胞环路模型。由这样的大量环路组成的网络模型在合适的突触连接参数范围内,能在皮层处产生人类慢波睡眠脑电2期的完整波形。这一结果说明了如何将动物实验观察到的睡眠生理机制的结果与人自然睡眠活动的脑电结果联系起来,并提示脑信息处理中由微观神经元群放电特征整合为脑的宏观功能状态的主要环节。     

Working memory dynamics and spontaneous activity in a flip-flop oscillations network model with a Milnor attractor

Many cognitive tasks require the ability to maintain and manipulate simultaneously several chunks of information. Numerous neurobiological observations have reported that this ability, known as the working memory, is associated with both a slow oscillation (leading to the up and down states) and the presence of the theta rhythm. Furthermore, during resting state, the spontaneous activity of the cortex exhibits exquisite spatiotemporal patterns sharing similar features with the ones observed during specific memory tasks. Here to enlighten neural implication of working memory under these complicated dynamics, we propose a phenomenological network model with biologically plausible neural dynamics and recurrent connections. Each unit embeds an internal oscillation at the theta rhythm which can be triggered during up-state of the membrane potential. As a result, the resting state of a single unit is no longer a classical fixed point attractor but rather the Milnor attractor, and multiple oscillations appear in the dynamics of a coupled system. In conclusion, the interplay between the up and down states and theta rhythm endows high potential in working memory operation associated with complexity in spontaneous activities.

入睡K-综合波产生的生理机制模型仿真研究

在确定人类睡眠脑电客观分期的国际标准中,有两类脑电特征波可以用来确定入睡状态（睡眠第二期）,即纺锤波和K－综合波。在前文中已提出了产生纺锤波的生理机制模型。按照1998年后对K－综合波形成的生理机制的看法,建立了微观神经元环路模型,其放电节律与实验中入睡时神经元放电的振荡节律相一致。而由大量这种相同环路组成的网络模型则在皮层处可产生符合K－综合波的波形。这一结果再次启示了脑信息处理中如何由微观神经元群放电特征整合为脑的宏观功能状态的过程。     

Origins and suppression of oscillations in a computational model of Parkinson’s disease

Efficacy of deep brain stimulation (DBS) for motor signs of Parkinson’s disease (PD) depends in part on post-operative programming of stimulus parameters. There is a need for a systematic approach to tuning parameters based on patient physiology. We used a physiologically realistic computational model of the basal ganglia network to investigate the emergence of a 34 Hz oscillation in the PD state and its optimal suppression with DBS. Discrete time transfer functions were fit to post-stimulus time histograms (PSTHs) collected in open-loop, by simulating the pharmacological block of synaptic connections, to describe the behavior of the basal ganglia nuclei. These functions were then connected to create a mean-field model of the closed-loop system, which was analyzed to determine the origin of the emergent 34 Hz pathological oscillation. This analysis determined that the oscillation could emerge from the coupling between the globus pallidus external (GPe) and subthalamic nucleus (STN). When coupled, the two resonate with each other in the PD state but not in the healthy state. By characterizing how this oscillation is affected by subthreshold DBS pulses, we hypothesize that it is possible to predict stimulus frequencies capable of suppressing this oscillation. To characterize the response to the stimulus, we developed a new method for estimating phase response curves (PRCs) from population data. Using the population PRC we were able to predict frequencies that enhance and suppress the 34 Hz pathological oscillation. This provides a systematic approach to tuning DBS frequencies and could enable closed-loop tuning of stimulation parameters.     

The Relationship between Respiration-Related Membrane Potential Slow Oscillations and Discharge Patterns in Mitral/Tufted Cells: What Are the Rules?

Background

A slow respiration-related rhythm strongly shapes the activity of the olfactory bulb. This rhythm appears as a slow oscillation that is detectable in the membrane potential, the respiration-related spike discharge of the mitral/tufted cells and the bulbar local field potential. Here, we investigated the rules that govern the manifestation of membrane potential slow oscillations (MPSOs) and respiration-related discharge activities under various afferent input conditions and cellular excitability states.

Methodology and Principal Findings

We recorded the intracellular membrane potential signals in the mitral/tufted cells of freely breathing anesthetized rats. We first demonstrated the existence of multiple types of MPSOs, which were influenced by odor stimulation and discharge activity patterns. Complementary studies using changes in the intracellular excitability state and a computational model of the mitral cell demonstrated that slow oscillations in the mitral/tufted cell membrane potential were also modulated by the intracellular excitability state, whereas the respiration-related spike activity primarily reflected the afferent input. Based on our data regarding MPSOs and spike patterns, we found that cells exhibiting an unsynchronized discharge pattern never exhibited an MPSO. In contrast, cells with a respiration-synchronized discharge pattern always exhibited an MPSO. In addition, we demonstrated that the association between spike patterns and MPSO types appeared complex.

Conclusion

We propose that both the intracellular excitability state and input strength underlie specific MPSOs, which, in turn, constrain the types of spike patterns exhibited.     

Regional slow waves and spindles in human sleep

The most prominent EEG events in sleep are slow waves, reflecting a slow (<1 Hz) oscillation between up and down states in cortical neurons. It is unknown whether slow oscillations are synchronous across the majority or the minority of brain regions--are they a global or local phenomenon? To examine this, we recorded simultaneously scalp EEG, intracerebral EEG, and unit firing in multiple brain regions of neurosurgical patients. We find that most sleep slow waves and the underlying active and inactive neuronal states occur locally. Thus, especially in late sleep, some regions can be active while others are silent. We also find that slow waves can propagate, usually from medial prefrontal cortex to the medial temporal lobe and hippocampus. Sleep spindles, the other hallmark of NREM sleep EEG, are likewise predominantly local. Thus, intracerebral communication during sleep is constrained because slow and spindle oscillations often occur out-of-phase in different brain regions.     

Linearized oscillations in population dynamics

A linearized oscillation theorem due to Kulenović, Ladas and Meimaridou (1987,Quart. appl. Math. XLV, 155–164) and an extension of it are applied to obtain the oscillation of solutions of several equations which have appeared in population dynamics. They include the logistic equation with several delays, Nicholson's blowflies model as described by Gurney, Blythe and Nisbet (1980,Nature, Lond. 287, 17–21) and the Lasota-Wazewska model of the red blood cell supply in an animal. We also developed a linearized oscillation result for difference equations and applied it to several equations taken from the biological literature.     

Simple models for complex systems: exploiting the relationship between local and global densities

Simple temporal models that ignore the spatial nature of interactions and track only changes in mean quantities, such as global densities, are typically used under the unrealistic assumption that individuals are well mixed. These so-called mean-field models are often considered overly simplified, given the ample evidence for distributed interactions and spatial heterogeneity over broad ranges of scales. Here, we present one reason why such simple population models may work even when mass-action assumptions do not hold: spatial structure is present but it relates to global densities in a special way. With an individual-based predator–prey model that is spatial and stochastic, and whose mean-field counterpart is the classic Lotka–Volterra model, we show that the global densities and densities of pairs (or spatial covariances) establish a bi-power law at the stationary state and also in their transient approach to this state. This relationship implies that the dynamics of global densities can be written simply as a function of those densities alone without invoking pairs (or higher order moments). The exponents of the bi-power law for the predation rate exhibit a remarkable robustness to changes in model parameters. Evidence is presented for a connection of our findings to the existence of a critical phase transition in the dynamics of the spatial system. We discuss the application of similar modified mean-field equations to other ecological systems for which similar transitions have been described, both in models and empirical data.     

The theoretical mechanism of Parkinson’s oscillation frequency bands: a computational model study

Excessive synchronous oscillation activities appear in the brain is a key pathological feature of Parkinson’s disease, the mechanism of which is still unclear. Although some previous studies indicated that β oscillation (13–30 Hz) may directly originate in the network composed of the subthalamic nucleus (STN) and external globus pallidus (GPe) neurons, specific onset mechanisms of which are unclear, especially theoretical evidences in numerical simulation are still little. In this paper, we employ a STN–GPe mean-field model to explore the onset mechanism of Parkinson’s oscillation. In addition to β oscillation, we find that some other common oscillation frequency bands can appear in this network, such as the α oscillation band (8–12 Hz), the θ oscillation band (4–7 Hz) and δ oscillation band (1–3 Hz). In addition to the coupling weight between the STN and GPe, the delay is also a critical factor to affect oscillatory activities, which can not be neglected compared to other parameters. Through simulation and analysis, we propose two possible theories may induce the system to transfer from the stable state to the oscillatory state in this model: (1). The oscillation activity can be induced when the firing activation level of the population increases to large enough; (2). In some special cases, the population may stay in the high firing rate stable state and the mean discharge rate of which is too large to induce oscillations, then oscillation activities may be induced as the MD decreases to moderate value. In most situations, the change trends of the MD and oscillation dominant frequency are contrary, which may be an important physiological phenomenon shown in this model. The delays and firing rates were obtained by simulating, which may be verified in the experiment in the future.     

In vivo electrophysiological evidences for cortical neuron-glia interactions during slow (<1 Hz) and paroxysmal sleep oscillations.

The cortical activity results from complex interactions within networks of neurons and glial cells. The dialogue signals consist of neurotransmitters and various ions, which cross through the extracellular space. Slow (<1 Hz) sleep oscillations were first disclosed and investigated at the neuronal level where they consist of an alternation of the membrane potential between a depolarized and a hyperpolarized state. However, neuronal properties alone could not account for the mechanisms underlying the oscillatory nature of the sleeping cortex. Here I will show the behavior of glial cells during the slow sleep oscillation and its relationship with the variation of the neuronal membrane potential (pairs of neurons and glia recorded simultaneously and intracellularly) suggesting that, in contrast with previous assumptions, glial cells are not idle followers of neuronal activity. I will equally present measurements of the extracellular concentration of K(+) and Ca(2+), ions known to modulate the neuronal excitability. They are also part of the ionic flux that is spatially buffered by glial cells. The timing of the spatial buffering during the slow oscillation suggests that, during normal oscillatory activity, K(+) ions are cleared from active spots and released in the near vicinity, where they modulate the excitability of the neuronal membrane and contribute to maintain the depolarizing phase of the oscillation. Ca(2+) ions undergo a periodic variation of their extracellular concentration, which modulates the synaptic efficacy. The depolarizing phase of the slow oscillation is associated with a gradual depletion of the extracellular Ca(2+) promoting a progressive disfacilitation in the network. This functional synaptic neuronal disconnection is responsible for the ending of the depolarizing phase of the slow oscillation and the onset of a phasic hyperpolarization during which the neuronal network is silent and the intra- and extracellular ionic concentrations return to normal values. Spike-wave seizures often develop during sleep from the slow oscillation. Here I will show how the increased gap junction communication substantiates the facility of the glial syncytium to spatially buffer K(+) ions that were uptaken during the spike-wave seizures, and therefore contributing to the long-range recruitment of cortical territories. Similar mechanisms as those described during the slow oscillation promote the periodic (2-3 Hz) recurrence of spike-wave complexes.     

Going Beyond a Mean-field Model for the Learning Cortex: Second-Order Statistics

Mean-field models of the cortex have been used successfully to interpret the origin of features on the electroencephalogram under situations such as sleep, anesthesia, and seizures. In a mean-field scheme, dynamic changes in synaptic weights can be considered through fluctuation-based Hebbian learning rules. However, because such implementations deal with population-averaged properties, they are not well suited to memory and learning applications where individual synaptic weights can be important. We demonstrate that, through an extended system of equations, the mean-field models can be developed further to look at higher-order statistics, in particular, the distribution of synaptic weights within a cortical column. This allows us to make some general conclusions on memory through a mean-field scheme. Specifically, we expect large changes in the standard deviation of the distribution of synaptic weights when fluctuation in the mean soma potentials are large, such as during the transitions between the “up” and “down” states of slow-wave sleep. Moreover, a cortex that has low structure in its neuronal connections is most likely to decrease its standard deviation in the weights of excitatory to excitatory synapses, relative to the square of the mean, whereas a cortex with strongly patterned connections is most likely to increase this measure. This suggests that fluctuations are used to condense the coding of strong (presumably useful) memories into fewer, but dynamic, neuron connections, while at the same time removing weaker (less useful) memories.     

Spatial patterns in coupled biochemical oscillators

Summary The effects of diffusion on the dynamics of biochemical oscillators are investigated for general kinetic mechanisms and for a simplified model of glycolysis. When diffusion is sufficiently rapid a population of oscillators relaxes to a globally-synchronized oscillation, but when diffusion of one or more species is slow enough, the synchronized oscillation can be unstable and a nonuniform steady state or an asynchronous oscillation can arise. The significance of these results vis-a-vis models of contact inhibition and zonation patterns is discussed.     

Inter-Network Interactions: Impact of Connections between Oscillatory Neuronal Networks on Oscillation Frequency and Pattern

Oscillations in electrical activity are a characteristic feature of many brain networks and display a wide variety of temporal patterns. A network may express a single oscillation frequency, alternate between two or more distinct frequencies, or continually express multiple frequencies. In addition, oscillation amplitude may fluctuate over time. The origin of this complex repertoire of activity remains unclear. Different cortical layers often produce distinct oscillation frequencies. To investigate whether interactions between different networks could contribute to the variety of oscillation patterns, we created two model networks, one generating on its own a relatively slow frequency (20 Hz; slow network) and one generating a fast frequency (32 Hz; fast network). Taking either the slow or the fast network as source network projecting connections to the other, or target, network, we systematically investigated how type and strength of inter-network connections affected target network activity. For high inter-network connection strengths, we found that the slow network was more effective at completely imposing its rhythm on the fast network than the other way around. The strongest entrainment occurred when excitatory cells of the slow network projected to excitatory or inhibitory cells of the fast network. The fast network most strongly imposed its rhythm on the slow network when its excitatory cells projected to excitatory cells of the slow network. Interestingly, for lower inter-network connection strengths, multiple frequencies coexisted in the target network. Just as observed in rat prefrontal cortex, the target network could express multiple frequencies at the same time, alternate between two distinct oscillation frequencies, or express a single frequency with alternating episodes of high and low amplitude. Together, our results suggest that input from other oscillating networks may markedly alter a network''s frequency spectrum and may partly be responsible for the rich repertoire of temporal oscillation patterns observed in the brain.     

Dissection of a model for neuronal parabolic bursting

We have obtained new insight into the mechanisms for bursting in a class of theoretical models. We study Plant's model [24] for Aplysia R-15 to illustrate our view of these so-called parabolic bursters, which are characterized by low spike frequency at the beginning and end of a burst. By identifying and analyzing the fast and slow processes we show how they interact mutually to generate spike activity and the slow wave which underlies the burst pattern. Our treatment is essentially the first step of a singular perturbation approach presented from a geometrical viewpoint and carried out numerically with AUTO [12]. We determine the solution sets (steady state and oscillatory) of the fast subsystem with the slow variables treated as parameters. These solutions form the slow manifold over which the slow dynamics then define a burst trajectory. During the silent phase of a burst, the solution trajectory lies approximately on the steady state branch of the slow manifold and during the active phase of spiking, the trajectory sweeps through the oscillation branch. The parabolic nature of bursting arises from the (degenerate) homoclinic transition between the oscillatory branch and the steady state branch. We show that, for some parameter values, the trajectory remains strictly on the steady state branch (to produce a resting steady state or a pure slow wave without spike activity) or strictly in the oscillatory branch (continuous spike activity without silent phases). Plant's model has two slow variables: a calcium conductance and the intracellular free calcium concentration, which activates a potassium conductance. We also show how bursting arises from an alternative mechanism in which calcium inactivates the calcium current and the potassium conductance is insensitive to calcium. These and other biophysical interpretations are discussed.