Metastable states and quasicycles in a stochastic Wilson-Cowan\ud model of neuronal population dynamics

We analyze a stochastic model of neuronal population dynamics with intrinsic noise. In the thermodynamic limit N -> infinity, where N determines the size of each population, the dynamics is described by deterministic Wilson–Cowan equations. On the other hand, for finite N the dynamics is described by a master equation that determines the probability of spiking activity within each population. We first consider a single excitatory population that exhibits bistability in the deterministic limit. The steady–state probability distribution of the stochastic network has maxima at points corresponding to the stable fixed points of the deterministic network; the relative weighting of the two maxima depends on the system size. For large but finite N, we calculate the exponentially small rate of noise–induced transitions between the resulting metastable states using a Wentzel–Kramers–Brillouin (WKB) approximation and matched asymptotic expansions. We then consider a two-population excitatory/inhibitory network that supports limit cycle oscillations. Using a diffusion approximation, we reduce the dynamics to a neural Langevin equation, and show how the intrinsic noise amplifies subthreshold oscillations (quasicycles)

Spatially structured oscillations in a two-dimensional excitatory neuronal network with synaptic depression

We study the spatiotemporal dynamics of a two-dimensional excitatory neuronal network with synaptic depression. Coupling between populations of neurons is taken to be nonlocal, while depression is taken to be local and presynaptic. We show that the network supports a wide range of spatially structured oscillations, which are suggestive of phenomena seen in cortical slice experiments and in vivo. The particular form of the oscillations depends on initial conditions and the level of background noise. Given an initial, spatially localized stimulus, activity evolves to a spatially localized oscillating core that periodically emits target waves. Low levels of noise can spontaneously generate several pockets of oscillatory activity that interact via their target patterns. Periodic activity in space can also organize into spiral waves, provided that there is some source of rotational symmetry breaking due to external stimuli or noise. In the high gain limit, no oscillatory behavior exists, but a transient stimulus can lead to a single, outward propagating target wave

Short term synaptic depression improves information transfer in perceptual multistability

Competitive neural networks are often used to model the dynamics of perceptual bistability. Switching between percepts can occur through fluctuations and/or a slow adaptive process. Here, we analyze switching statistics in competitive networks with short term synaptic depression and noise. We start by analyzing a ring model that yields spatially structured solutions and complement this with a study of a space-free network whose populations are coupled with mutual inhibition. Dominance times arising from depression driven switching can be approximated using a separation of timescales in the ring and space-free model. For purely noise-driven switching, we use energy arguments to justify how dominance times are exponentially related to input strength. We also show that a combination of depression and noise generates realistic distributions of dominance times. Unimodal functions of dominance times are more easily differentiated from one another using Bayesian sampling, suggesting synaptic depression induced switching transfers more information about stimuli than noise-driven switching. Finally, we analyze a competitive network model of perceptual tristability, showing depression generates a memory of previous percepts based on the ordering of percepts.Comment: 26 pages, 15 figure

Intrinsic and synaptic membrane properties of neurons in the thalamic reticular nucleus

Tableau d’honneur de la Faculté des études supérieures et postdoctorales, 2004-2005Le noyau réticulaire thalamique (RE) est une structure qui engendre des fuseaux, une oscillation bioélectrique de marque pendant les stades précoces du sommeil. De multiples propriétés neuronales, intrinsèques et synaptiques, sont impliquées dans la génération, la propagation, le maintien et la terminaison des ondes en fuseaux. D’un autre côté, ce rythme constitue un état spécial de l’activité du réseau qui est généré par le réseau lui-même et affecte les propriétés cellulaires du noyau RE. Cette étude se concentre sur ces sujets: comment les propriétés cellulaires et les propriétés du réseau sont inter-reliées et interagissent pour engendrer les ondes fuseaux dans les neurones du RE et leurs cibles, les neurones thalamocorticaux. La présente thèse fournit de nouvelles évidences montrant le rôle fondamental joué par les neurones du noyau RE dans la genèse des ondes en fuseaux, dû aux synapses chimiques établies par ces neurones. La propagation et la synchronisation de l’activité sont modulées par les synapses électriques entre les neurones réticulaires thalamiques, mais aussi par les composantes dépolarisantes secondaires des réponses synaptiques évoquées par le cortex. De plus, la forme générale et la terminaison des oscillations thalamiques sont probablement contrôlées en grande partie par les neurones du RE, lesquels expriment une conductance intrinsèque leurs procurant une membrane avec un comportement bistable. Finalement, les oscillations thalamiques en fuseaux sont aussi capables de moduler les propriétés membranaires et l’activité des neurones individuels du RE.The thalamic reticular nucleus (RE) is a key structure related to spindles, a hallmark bioelectrical oscillation during early stages of sleep. Multiple neuronal properties, both intrinsic and synaptic, are implicated in the generation, propagation, maintenance and termination of spindle waves. On the other hand, this rhythm constitutes a special state of network activity, which is generated within, and affects single-cell properties of the RE nucleus. This study is focused on these topics: how cellular and network properties are interrelated and interact to generate spindle waves in the pacemaking RE neurons and their targets, thalamocortical neurons. The present thesis provides new evidence showing the fundamental role played by the RE nucleus in the generation of spindle waves, due to chemical synapses established by its neurons. The propagation and synchronization of activity is modulated by electrical synapses between thalamic reticular neurons, but also by the secondary depolarizing component of cortically-evoked synaptic responses. Additionally, the general shaping and probably the termination of thalamic oscillations could be controlled to a great extent by RE neurons, which express an intrinsic conductance endowing them with membrane bistable behaviour. Finally, thalamic spindle oscillations are also able to modulate the membrane properties and activities of individual RE neurons

Dynamics of phase locking in neuronal networks in the presence of synaptic plasticity

The behavior generated by neuronal networks depends on the phase relationships of its individual neurons. Observed phases result from the combined effects of individual cells and synaptic connections whose properties change dynamically. The properties of individual cells and synapses can often be characterized by driving the cell or synapse with inputs that arrive at different phases or frequencies, thus producing a feed-forward description of these properties. In this study, a recurrent network of two oscillatory neurons that are coupled with reciprocal synapses is considered. Feed-forward descriptions of the phase response curves of the neurons and the short-term synaptic plasticity properties are used to define Poincar´e maps for the activity of the network. The fixed points of these maps correspond to the phase locked modes of the network. These maps allow analysis of the dependence of the resulting network activity on the properties of network components. Using a combination of analysis and simulations, how various parameters of the model affect the existence and stability of phase-locked solutions is shown. It is also shown that synaptic plasticity provides flexibility and supports phase maintenance in networks. Conditions are found on the synaptic plasticity profiles and the phase response curves of the neurons for the network to be able to maintain a constant firing period, while varying the relative activity phase of the neurons or vice versa. Synaptic plasticity is shown to yield bistable phase locking modes. These results are geometrically demonstrated using a generalization to cobwebbing for two dimensional maps. Type I neurons modeled with Morris-Lecar and Quadratic Integrate-and-Fire are used to estimate the predictive power of the analytical results; however, the results hold in general. The properties of the Negative-Leak model are also studied; a recent conductance-based model which is obtained by replacing a regenerative inward current with a negative-slope-conductance linear current. The map methods are extended to analyze networking properties of Negative-Leak neurons by including burst response curves. Finally, geometric singular perturbation techniques are applied to analyze how a hyperpolarization-activated inward current contributes to the generation of oscillations in this model. This work introduces a general method to determine how changes in the phase response curves or synaptic dynamics affect phase locking in a recurrent network which can be generalized to study larger networks

Stationary bumps in a piecewise smooth neural field model with synaptic depression

We analyze the existence and stability of stationary pulses or bumps in a one–dimensional piecewise smooth neural field model with synaptic depression. The continuum dynamics is described in terms of a nonlocal integrodifferential equation, in which the integral kernel represents the spatial distribution of synaptic weights between populations of neurons whose mean firing rate is taken to be a Heaviside function of local activity. Synaptic depression dynamically reduces the strength of synaptic weights in response to increases in activity. We show that in the case of a Mexican hat weight distribution, there exists a stable bump for sufficiently weak synaptic depression. However, as synaptic depression becomes stronger, the bump became unstable with respect to perturbations that shift the boundary of the bump, leading to the formation of a traveling pulse. The local stability of a bump is determined by the spectrum of a piecewise linear operator that keeps track of the sign of perturbations of the bump boundary. This results in a number of differences from previous studies of neural field models with Heaviside firing rate functions, where any discontinuities appear inside convolutions so that the resulting dynamical system is smooth. We also extend our results to the case of radially symmetric bumps in two–dimensional neural field models

Enhancement of synchronization in a hybrid neural circuit by spike timing dependent plasticity

Synchronization of neural activity is fundamental for many functions of the brain. We demonstrate that spike-timing dependent plasticity (STDP) enhances synchronization (entrainment) in a hybrid circuit composed of a spike generator, a dynamic clamp emulating an excitatory plastic synapse, and a chemically isolated neuron from the Aplysia abdominal ganglion. Fixed-phase entrainment of the Aplysia neuron to the spike generator is possible for a much wider range of frequency ratios and is more precise and more robust with the plastic synapse than with a nonplastic synapse of comparable strength. Further analysis in a computational model of HodgkinHuxley-type neurons reveals the mechanism behind this significant enhancement in synchronization. The experimentally observed STDP plasticity curve appears to be designed to adjust synaptic strength to a value suitable for stable entrainment of the postsynaptic neuron. One functional role of STDP might therefore be to facilitate synchronization or entrainment of nonidentical neurons

Mesoscale Systems, Finite Size Effects, and Balanced Neural Networks

Cortical populations are typically in an asynchronous state, sporadically interrupted by brief epochs of coordinated population activity. Current cortical models are at a loss to explain this combination of states. At one extreme are network models where recurrent in- hibition dynamically stabilizes an asynchronous low activity state. While these networks are widely used they cannot produce the coherent population-wide activity that is reported in a variety of datasets. At the other extreme are models where short term synaptic depression between excitatory neurons can generate the epochs of population-wide activity. However, in these networks inhibition plays only a perfunctory role in network stability, which is at odds with many reports across cortex. In this study we analyze spontaneously active in vitro preparations of primary auditory cortex that show dynamics that are emblematic of this mix- ture of states. To capture this complex population activity we consider models where large excitation is balanced by recurrent inhibition yet we include short term synaptic depression dynamics of the excitatory connections. This model gives very rich nonlinear behavior that mimics the core features of the in vitro data, including the possibility of low frequency (2- 12 Hz) rhythmic dynamics within population events. Our study extends balanced network models to account for nonlinear, population-wide correlated activity, thereby providing a critical step in a mechanistic theory of realistic cortical activity. We further investigate an extension of this model that l exhibits clearly non-Arrhenius behavior, whereby lower noise systems may exhibit faster escape from a stable state. We show that this behavior is due to the system size dependent vector field, intrinsically linking noise and dynamics

Characterization of K-Complexes and Slow Wave Activity in a Neural Mass Model

NREM sleep is characterized by two hallmarks, namely K-complexes (KCs) during sleep stage N2 and cortical slow oscillations (SOs) during sleep stage N3. While the underlying dynamics on the neuronal level is well known and can be easily measured, the resulting behavior on the macroscopic population level remains unclear. On the basis of an extended neural mass model of the cortex, we suggest a new interpretation of the mechanisms responsible for the generation of KCs and SOs. As the cortex transitions from wake to deep sleep, in our model it approaches an oscillatory regime via a Hopf bifurcation. Importantly, there is a canard phenomenon arising from a homoclinic bifurcation, whose orbit determines the shape of large amplitude SOs. A KC corresponds to a single excursion along the homoclinic orbit, while SOs are noise-driven oscillations around a stable focus. The model generates both time series and spectra that strikingly resemble real electroencephalogram data and points out possible differences between the different stages of natural sleep