Critical behaviour of the stochastic Wilson-Cowan model

Spontaneous brain activity is characterized by bursts and avalanche-like dynamics, with scale-free features typical of critical behaviour. The stochastic version of the celebrated Wilson-Cowan model has been widely studied as a system of spiking neurons reproducing non-trivial features of the neural activity, from avalanche dynamics to oscillatory behaviours. However, to what extent such phenomena are related to the presence of a genuine critical point remains elusive. Here we address this central issue, providing analytical results in the linear approximation and extensive numerical analysis. In particular, we present results supporting the existence of a bona fide critical point, where a second-order-like phase transition occurs, characterized by scale-free avalanche dynamics, scaling with the system size and a diverging relaxation time-scale. Moreover, our study shows that the observed critical behaviour falls within the universality class of the mean-field branching process, where the exponents of the avalanche size and duration distributions are, respectively, 3/2 and 2. We also provide an accurate analysis of the system behaviour as a function of the total number of neurons, focusing on the time correlation functions of the firing rate in a wide range of the parameter space.     

A quantitative population model of whisker barrels: Re-examining the Wilson-Cowan equations

Beginning from a biologically based integrate and fire model of a rat whisker barrel, we employ semirigorous techniques to reduce the system to a simple set of equations, similar to the Wilson-Cowan equations, while retaining the ability for both qualitative and quantitative comparisons with the biological system. This is made possible through the clarification of three distinct measures of population activity: voltage, firing rate, and a new term called synaptic drive. The model is activated by prerecorded neural activity obtained from thalamic barreloid neurons in response to whisker stimuli. Output is produced in the form of population PSTHs, one each corresponding to activity of spiny (excitatory) and smooth (inhibitory) barrel neurons, which is quantitatively comparable to PSTHs from electrophysiologically studied regular-spike and fast-spike neurons. Through further analysis, the model yields novel physiological predictions not readily apparent from the full model or from experimental studies.Abbreviations PW Principal Whisker - AW Adjacent Whisker - RSU Regular-Spike Unit (excitatory/spiny) - FSU Fast-Spike Unit (inhibitory/smooth) - TCU Thalamocortical Unit - VPM Ventral Posterior Medical Nucleus - PSTH Peristimulus Histogram - PSP Post-Synaptic Potential - CTR Conditioned-Test Ratio     

A discrete dynamic model of a neuronal network with space-time organization]

A dynamical model of the neuronal network based on the principles of spatial-temporal organization of information processing was constructed. The additivity law for the formation of input connection factors in this model provides its versatility for a variety of logical functions. The model takes into account real properties of the systemic organization of brain neurones.     

Inhibition modifies the effects of slow calcium-activated potassium channels on epileptiform activity in a neuronal network model

Generation of epileptiform activity typically results from a change in the balance between network excitation and inhibition. Experimental evidence indicates that alterations of either synaptic activity or intrinsic membrane properties can produce increased network excitation. The slow Ca²⁺-activated K⁺ currents (sI_AHP) are important modulators of neuronal firing rate and excitability and have important established and potential roles in epileptogenesis. While the effects of changes in sI_AHP on individual neuronal excitability are readily studied and well established, the effects of such changes on network behavior are less well known. The experiments here utilize a defined small network model of multicompartment pyramidal cells and an inhibitory interneuron to study the effects of changes in sI_AHP on network behavior. The benefits of this model system include the ability to observe activity in all cells in a network and the effects of interactions of multiple simultaneous influences. In the model with no inhibitory interneuron, increasing sI_AHP results in progressively decreasing burst activity. Adding an inhibitory interneuron changes the observed effects; at modest inhibitory strengths, increasing sI_AHP in all network neurons actually results in increased network bursting (except at very high values). The duration of the burst activity is influenced by the length of delay in a feedback loop, with longer loops resulting in more prolonged bursting. These observations illustrate that the study of potential antiepileptogenic membrane effects must be extended to realistic networks. Network inhibition can dramatically alter the observations seen in pure excitatory networks.     

Spiking regularity in a noisy small-world neuronal network

The regularity of spiking oscillations is studied in the networks with different topological structures. The network is composed of coupled Fitz-Hugh-Nagumo neurons driven by colored noise. The investigation illustrates that the spike train in both the regular and the Watts-Strogatz small-world neuronal networks can show the best regularity at a moderate noise intensity, indicating the existence of coherence resonance. Moreover, the temporal coherence of the spike train in the small-world network is superior to that in a regular network due to the increase of the randomness of the network topology. Besides the noise intensity, the spiking regularity can be optimized by tuning the randomness of the network topological structure or by tuning the correlation time of the colored noise. In particular, under the cooperation of the small-world topology and the correlation time, the spike train with good regularity could sustain a large magnitude of the local noise.     

Bifurcations of emergent bursting in a neuronal network

Complex neuronal networks are an important tool to help explain paradoxical phenomena observed in biological recordings. Here we present a general approach to mathematically tackle a complex neuronal network so that we can fully understand the underlying mechanisms. Using a previously developed network model of the milk-ejection reflex in oxytocin cells, we show how we can reduce a complex model with many variables and complex network topologies to a tractable model with two variables, while retaining all key qualitative features of the original model. The approach enables us to uncover how emergent synchronous bursting can arise from a neuronal network which embodies known biological features. Surprisingly, the bursting mechanisms are similar to those found in other systems reported in the literature, and illustrate a generic way to exhibit emergent and multiple time scale oscillations at the membrane potential level and the firing rate level.     

Neuromimetic model of a neuronal filter

Experimental work in cats has shown that a series of afferent impulses from muscle receptors activated during contractions of an ankle extensor elicit declining inhibitory potentials in homonymous and synergic motoneurones. Inhibitory potentials were ascribed to the action of Ib afferents from Golgi tendon organs that are specific contraction-sensitive mechano-receptors. The decline of inhibition was, at least partly, due to presynaptic inhibition acting as a filter of tendon organ information in the spinal cord. In the present work, a computer model of the simplest spinal pathways from Ib fibres to motoneurones was designed. In order to make the model as realistic as possible, the most pertinent of the known functional properties of the neuronal elements were incorporated. Functions simulating primary afferent depolarizations of Ib terminals, i.e. the electrophysiological correlate of presynaptic inhibition, were introduced in the network. Simulations showed that declining inhibitory potentials were computed in the output stage of the network that represented the motoneurone-like element. These results support the assumption that the filtering out of Ib inputs is to a great extent due to presynaptic inhibition. The model behaved as expected, suggesting that predictions of the behaviour of neural components in the biological network should be possible upon introduction in the model of other, more complex, spinal pathways from Ib fibres to motoneurones.     

PCDH12 loss results in premature neuronal differentiation and impeded migration in a cortical organoid model

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A neuronal network model of primary visual cortex explains spatial frequency selectivity

We address how spatial frequency selectivity arises in Macaque primary visual cortex (V1) by simulating V1 with a large-scale network model consisting of O(10⁴) excitatory and inhibitory integrate-and-fire neurons with realistic synaptic conductances. The new model introduces variability of the widths of subregions in V1 neuron receptive fields. As a consequence different model V1 neurons prefer different spatial frequencies. The model cortex has distributions of spatial frequency selectivity and of preference that resemble experimental findings from the real V1. Two main sources of spatial frequency selectivity in the model are the spatial arrangement of feedforward excitation, and cortical nonlinear suppression, a result of cortical inhibition. Action Editor: Jonathan D. Victor     

A model of the neuronal network using eye micromovements for distinguishing contrasts on an image

A monochromatic model of a neuron network of foveal retina with the straight way of information stream of the system "cone-diminutive bipolar-diminutive ganglion cell" is proposed. The network can distinguish contrasts on an image. It is shown that cells of the output level have concentric receptive fields formed due to eye micromovements. The paper discusses the functioning of the model in the case when eye movements contain resetting microsaccades as well as drift. A special attention is paid to the compensation of transient processes.     

Bifurcations of lurching waves in a thalamic neuronal network

We consider a two-layer, one-dimensional lattice of neurons; one layer consists of excitatory thalamocortical neurons, while the other is comprised of inhibitory reticular thalamic neurons. Such networks are known to support “lurching” waves, for which propagation does not appear smooth, but rather progresses in a saltatory fashion; these waves can be characterized by different spatial widths (different numbers of neurons active at the same time). We show that these lurching waves are fixed points of appropriately defined Poincaré maps, and follow these fixed points as parameters are varied. In this way, we are able to explain observed transitions in behavior, and, in particular, to show how branches with different spatial widths are linked with each other. Our computer-assisted analysis is quite general and could be applied to other spatially extended systems which exhibit this non-trivial form of wave propagation.     

双层Hodgkin-Huxley神经元网络中的随机共振

随机共振是一种非零噪声优化系统响应的现象。运用信噪比的评价方式,研究单个Hodgkin-Huxley神经元及其所构建的双层神经元网络中的随机共振,来模拟生物感觉系统中检测微弱信号的随机共振现象。结果表明,单个神经元在阈值下存在噪声优化系统检测性能的随机共振现象,但是最优的噪声强度却随外部信号性质的改变而变化;双层神经元网络不但可以在固定的噪声强度上对一定幅度范围内的阈下信号进行优化检测,而且噪声的存在并没有降低网络对阈上信号的检测能力。     

Parameters for a model of an oscillating neuronal network in the cochlear nucleus defined by genetic algorithms

Chopper neurons in the cochlear nucleus are characterized by intrinsic oscillations with short average interspike intervals (ISIs) and relative level independence of their response (Pfeiffer, Exp Brain Res 1:220–235, 1966; Blackburn and Sachs, J Neurophysiol 62:1303–1329, 1989), properties which are unattained by models of single chopper neurons (e.g., Rothman and Manis, J Neurophysiol 89:3070–3082, 2003a). In order to achieve short ISIs, we optimized the time constants of Rothman and Manis single neuron model with genetic algorithms. Some parameters in the optimization, such as the temperature and the capacity of the cell, turned out to be crucial for the required acceleration of their response. In order to achieve the relative level independence, we have simulated an interconnected network consisting of Rothman and Manis neurons. The results indicate that by stabilization of intrinsic oscillations, it is possible to simulate the physiologically observed level independence of ISIs. As previously reviewed and demonstrated (Bahmer and Langner, Biol Cybern 95:371–379, 2006a), chopper neurons show a preference for ISIs which are multiples of 0.4 ms. It was also demonstrated that the network consisting of two optimized Rothman and Manis neurons which activate each other with synaptic delays of 0.4 ms shows a preference for ISIs of 0.8 ms. Oscillations with various multiples of 0.4 ms as ISIs may be derived from neurons in a more complex network that is activated by simultaneous input of an onset neuron and several auditory nerve fibers.     

Arginine vasopressin neuronal loss results from autophagy-associated cell death in a mouse model for familial neurohypophysial diabetes insipidus

Familial neurohypophysial diabetes insipidus (FNDI) characterized by progressive polyuria is mostly caused by mutations in the gene encoding neurophysin II (NPII), which is the carrier protein of the antidiuretic hormone, arginine vasopressin (AVP). Although accumulation of mutant NPII in the endoplasmic reticulum (ER) could be toxic for AVP neurons, the precise mechanisms of cell death of AVP neurons, reported in autopsy studies, remain unclear. Here, we subjected FNDI model mice to intermittent water deprivation (WD) in order to promote the phenotypes. Electron microscopic analyses demonstrated that, while aggregates are confined to a certain compartment of the ER in the AVP neurons of FNDI mice with water access ad libitum, they were scattered throughout the dilated ER lumen in the FNDI mice subjected to WD for 4 weeks. It is also demonstrated that phagophores, the autophagosome precursors, emerged in the vicinity of aggregates and engulfed the ER containing scattered aggregates. Immunohistochemical analyses revealed that expression of p62, an adapter protein between ubiquitin and autophagosome, was elicited on autophagosomal membranes in the AVP neurons, suggesting selective autophagy induction at this time point. Treatment of hypothalamic explants of green fluorescent protein (GFP)-microtubule-associated protein 1 light chain 3 (LC3) transgenic mice with an ER stressor thapsigargin increased the number of GFP-LC3 puncta, suggesting that ER stress could induce autophagosome formation in the hypothalamus of wild-type mice as well. The cytoplasm of AVP neurons in FNDI mice was occupied with vacuoles in the mice subjected to WD for 12 weeks, when 30–40% of AVP neurons are lost. Our data thus demonstrated that autophagy was induced in the AVP neurons subjected to ER stress in FNDI mice. Although autophagy should primarily be protective for neurons, it is suggested that the organelles including ER were lost over time through autophagy, leading to autophagy-associated cell death of AVP neurons.     

A stochastic model for the spread of a sexually transmitted disease which results in a scale-free network

A stochastic model for the spread of a sexually transmitted disease (STD) is presented. To reflect varying degrees of promiscuity among individuals it is assumed that the infectivity of any infected individual is proportional to the number of previous contacts the individual has had with other infected individuals. In both the simple single-sex model and in the more complex two-sex model, the tree graphs of the infection exhibit scale-free network behaviour (i.e. power-law behaviour in the upper tail of the degree distribution). The distributions of the size of the infection and of the ring number (distance from the original source of the infection) are determined.     

Effect of spike-timing-dependent plasticity on stochastic burst synchronization in a scale-free neuronal network

We consider an excitatory population of subthreshold Izhikevich neurons which cannot fire spontaneously without noise. As the coupling strength passes a threshold, individual neurons exhibit noise-induced burstings. This neuronal population has adaptive dynamic synaptic strengths governed by the spike-timing-dependent plasticity (STDP). However, STDP was not considered in previous works on stochastic burst synchronization (SBS) between noise-induced burstings of sub-threshold neurons. Here, we study the effect of additive STDP on SBS by varying the noise intensity D in the Barabási–Albert scale-free network (SFN). One of our main findings is a Matthew effect in synaptic plasticity which occurs due to a positive feedback process. Good burst synchronization (with higher bursting measure) gets better via long-term potentiation (LTP) of synaptic strengths, while bad burst synchronization (with lower bursting measure) gets worse via long-term depression (LTD). Consequently, a step-like rapid transition to SBS occurs by changing D, in contrast to a relatively smooth transition in the absence of STDP. We also investigate the effects of network architecture on SBS by varying the symmetric attachment degree \(l^*\) and the asymmetry parameter \(\Delta l\) in the SFN, and Matthew effects are also found to occur by varying \(l^*\) and \(\Delta l\). Furthermore, emergences of LTP and LTD of synaptic strengths are investigated in details via our own microscopic methods based on both the distributions of time delays between the burst onset times of the pre- and the post-synaptic neurons and the pair-correlations between the pre- and the post-synaptic instantaneous individual burst rates (IIBRs). Finally, a multiplicative STDP case (depending on states) with soft bounds is also investigated in comparison with the additive STDP case (independent of states) with hard bounds. Due to the soft bounds, a Matthew effect with some quantitative differences is also found to occur for the case of multiplicative STDP.     

Dual oscillator model of the respiratory neuronal network generating quantal slowing of respiratory rhythm

We developed a dual oscillator model to facilitate the understanding of dynamic interactions between the parafacial respiratory group (pFRG) and the preBötzinger complex (preBötC) neurons in the respiratory rhythm generation. Both neuronal groups were modeled as groups of 81 interconnected pacemaker neurons; the bursting cell model described by Butera and others [model 1 in Butera et al. (J Neurophysiol 81:382–397, 1999a)] were used to model the pacemaker neurons. We assumed (1) both pFRG and preBötC networks are rhythm generators, (2) preBötC receives excitatory inputs from pFRG, and pFRG receives inhibitory inputs from preBötC, and (3) persistent Na⁺ current conductance and synaptic current conductances are randomly distributed within each population. Our model could reproduce 1:1 coupling of bursting rhythms between pFRG and preBötC with the characteristic biphasic firing pattern of pFRG neurons, i.e., firings during pre-inspiratory and post-inspiratory phases. Compatible with experimental results, the model predicted the changes in firing pattern of pFRG neurons from biphasic expiratory to monophasic inspiratory, synchronous with preBötC neurons. Quantal slowing, a phenomena of prolonged respiratory period that jumps non-deterministically to integer multiples of the control period, was observed when the excitability of preBötC network decreased while strengths of synaptic connections between the two groups remained unchanged, suggesting that, in contrast to the earlier suggestions (Mellen et al., Neuron 37:821–826, 2003; Wittmeier et al., Proc Natl Acad Sci USA 105(46):18000–18005, 2008), quantal slowing could occur without suppressed or stochastic excitatory synaptic transmission. With a reduced excitability of preBötC network, the breakdown of synchronous bursting of preBötC neurons was predicted by simulation. We suggest that quantal slowing could result from a breakdown of synchronized bursting within the preBötC.     

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.     

一种基于Wilson-Cowan双节点耦合振子模型的图像增强方法

针对模拟大脑节律现象的Wilson-Cowan双节点耦合振子模型,选取满足该模型产生极限环振荡条件的参数,采用连续灰度阶图像块作为输入,兴奋性亚群节点响应作为图像增强的输出,做出刺激响应曲线,发现该曲线与图像处理领域中用于图像增强的Gamma校正曲线相似,说明Wilson-Cowan双节点耦合振子模型可以作为一种新的图像增强方法,同时也为基于Gamma校正曲线的图像增强提供了一种神经机制。将新方法与Retinex算法做对比,表现出了更好的图像增强性能。