Coarse-grained event tree analysis for quantifying Hodgkin-Huxley neuronal network dynamics

We present an event tree analysis of studying the dynamics of the Hodgkin-Huxley (HH) neuronal networks. Our study relies on a coarse-grained projection to event trees and to the event chains that comprise these trees by using a statistical collection of spatial-temporal sequences of relevant physiological observables (such as sequences of spiking multiple neurons). This projection can retain information about network dynamics that covers multiple features, swiftly and robustly. We demonstrate that for even small differences in inputs, some dynamical regimes of HH networks contain sufficiently higher order statistics as reflected in event chains within the event tree analysis. Therefore, this analysis is effective in discriminating small differences in inputs. Moreover, we use event trees to analyze the results computed from an efficient library-based numerical method proposed in our previous work, where a pre-computed high resolution data library of typical neuronal trajectories during the interval of an action potential (spike) allows us to avoid resolving the spikes in detail. In this way, we can evolve the HH networks using time steps one order of magnitude larger than the typical time steps used for resolving the trajectories without the library, while achieving comparable statistical accuracy in terms of average firing rate and power spectra of voltage traces. Our numerical simulation results show that the library method is efficient in the sense that the results generated by using this numerical method with much larger time steps contain sufficiently high order statistical structure of firing events that are similar to the ones obtained using a regular HH solver. We use our event tree analysis to demonstrate these statistical similarities.     

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。     

神经元网络动力学的统计物理方法

本文回顾了利用统计物理的方法研究神经元网络动力学的数学降维描述.以一个全兴奋性的“整合-发放”神经元网络为出发点,导出了描写神经元群体活动的概率分布函数的(2+1)-维对流-扩散方程.在没有引入任何新参数的情况下,讨论了如何利用moment closure scheme得到(1+1)-维的动力学方程.我们将此方程的预测...     

Emergence and fragmentation of the alpha-band driven by neuronal network dynamics

Rhythmic neuronal network activity underlies brain oscillations. To investigate how connected neuronal networks contribute to the emergence of the α-band and to the regulation of Up and Down states, we study a model based on synaptic short-term depression-facilitation with afterhyperpolarization (AHP). We found that the α-band is generated by the network behavior near the attractor of the Up-state. Coupling inhibitory and excitatory networks by reciprocal connections leads to the emergence of a stable α-band during the Up states, as reflected in the spectrogram. To better characterize the emergence and stability of thalamocortical oscillations containing α and δ rhythms during anesthesia, we model the interaction of two excitatory networks with one inhibitory network, showing that this minimal topology underlies the generation of a persistent α-band in the neuronal voltage characterized by dominant Up over Down states. Finally, we show that the emergence of the α-band appears when external inputs are suppressed, while fragmentation occurs at small synaptic noise or with increasing inhibitory inputs. To conclude, α-oscillations could result from the synaptic dynamics of interacting excitatory neuronal networks with and without AHP, a principle that could apply to other rhythms.     

Population dynamics: limits of predictability

Problems pertaining to the complex character of ecological system dynamics are discussed. Examples of the complex dynamics of plankton populations in a heterogeneous environment and agricultural ecosystems under invasion of pests resistant to Bt toxins produced by genetically modified insecticidal crops are given.     

Spike latency and jitter of neuronal membrane patches with stochastic Hodgkin-Huxley channels

Ion channel stochasticity can influence the voltage dynamics of neuronal membrane, with stronger effects for smaller patches of membrane because of the correspondingly smaller number of channels. We examine this question with respect to first spike statistics in response to a periodic input of membrane patches including stochastic Hodgkin-Huxley channels, comparing these responses to spontaneous firing. Without noise, firing threshold of the model depends on frequency—a sinusoidal stimulus is subthreshold for low and high frequencies and suprathreshold for intermediate frequencies. When channel noise is added, a stimulus in the lower range of subthreshold frequencies can influence spike output, while high subthreshold frequencies remain subthreshold. Both input frequency and channel noise strength influence spike timing. Specifically, spike latency and jitter have distinct minima as a function of input frequency, showing a resonance like behavior. With either no input, or low frequency subthreshold input, or input in the low or high suprathreshold frequency range, channel noise reduces latency and jitter, with the strongest impact for the lowest input frequencies. In contrast, for an intermediate range of suprathreshold frequencies, where an optimal input gives a minimum latency, the noise effect reverses, and spike latency and jitter increase with channel noise. Thus, a resonant minimum of the spike response as a function of frequency becomes more pronounced with less noise. Spike latency and jitter also depend on the initial phase of the input, resulting in minimal latencies at an optimal phase, and depend on the membrane time constant, with a longer time constant broadening frequency tuning for minimal latency and jitter. Taken together, these results suggest how stochasticity of ion channels may influence spike timing and thus coding for neurons with functionally localized concentrations of channels, such as in “hot spots” of dendrites, spines or axons.     

Population dynamics: Limits of predictability

Problems related to the complex pattern of ecosystem dynamics are discussed. Examples of studies on the complex population dynamics are considered, including those of plankton populations in a spatially heterogeneous environment and of an agroecosystem invaded by pests resistant to Bt toxins produced by genetically modified insecticidal crops.     

Intraseasonal predictability of natural phytoplankton population dynamics

It is difficult to make skillful predictions about the future dynamics of marine phytoplankton populations. Here, we use a 22‐year time series of monthly average abundances for 198 phytoplankton taxa from Station L4 in the Western English Channel (1992–2014) to test whether and how aggregating phytoplankton into multi‐species assemblages can improve predictability of their temporal dynamics. Using a non‐parametric framework to assess predictability, we demonstrate that the prediction skill is significantly affected by how species data are grouped into assemblages, the presence of noise, and stochastic behavior within species. Overall, we find that predictability one month into the future increases when species are aggregated together into assemblages with more species, compared with the predictability of individual taxa. However, predictability within dinoflagellates and larger phytoplankton (>12 μm cell radius) is low overall and does not increase by aggregating similar species together. High variability in the data, due to observational error (noise) or stochasticity in population growth rates, reduces the predictability of individual species more than the predictability of assemblages. These findings show that there is greater potential for univariate prediction of species assemblages or whole‐community metrics, such as total chlorophyll or biomass, than for the individual dynamics of phytoplankton species.     

Behavior-related neuronal reactions and dynamics of neuronal activity

Functionally, behavior-related discharges of associative neurons are an efferent flow of pulses continuously generated over the course of each behavioral act of an animal. However, predominant research approaches are based on the "stimulus - reaction" principle. Analysis of the dynamics of unit activity in monkeys during performance of a multi-step behavioral complex showed that differences related to different behavioral acts consist in composition changes in the active neurons (or their recombination) rather than in a number of responsive cells or involvement of action-specific neurons. Each combination of active neurons ensures the distribution of efferent signals characteristic of the given combination. These findings suggest the addressing coding of the efferent neuronal signals.     

Complex nonlinear dynamics of the Hodgkin-Huxley equations induced by time scale changes

 The Hodgkin–Huxley equations with a slight modification are investigated, in which the inactivation process (h) of sodium channels or the activation process of potassium channels (n) is slowed down. We show that the equations produce a variety of action potential waveforms ranging from a plateau potential, such as in heart muscle cells, to chaotic bursting firings. When h is slowed down – differently from the case of n variable being slow – chaotic bursting oscillations are observed for a wide range of parameter values although both variables cause a decrease in the membrane potential. The underlying nonlinear dynamics of various action potentials are analyzed using bifurcation theory and a so-called slow–fast decomposition analysis. It is shown that a simple topological property of the equilibrium curves of slow and fast subsystems is essential to the production of chaotic oscillations, and this is the cause of the large difference in global firing characteristics between the h-slow and n-slow cases. Received: 9 August 2000 / Accepted in revised form: 10 January 2001     

Age- and sex-dependent alterations in primary somatosensory cortex neuronal calcium network dynamics during locomotion

Over the past 30 years, the calcium (Ca²⁺) hypothesis of brain aging has provided clear evidence that hippocampal neuronal Ca²⁺ dysregulation is a key biomarker of aging. Age-dependent Ca²⁺-mediated changes in intrinsic excitability, synaptic plasticity, and activity have helped identify some of the mechanisms engaged in memory and cognitive decline based on work done mostly at the single-cell level and in the slice preparation. Recently, our lab identified age- and Ca²⁺-related neuronal network dysregulation in the cortex of the anesthetized animal. Still, investigations in the awake animal are needed to test the generalizability of the Ca²⁺ hypothesis of brain aging. Here, we used in vigilo two-photon imaging in ambulating mice, to image GCaMP8f in the primary somatosensory cortex (S1), during ambulation and at rest. We investigated aging- and sex-related changes in neuronal networks in the C56BL/6J mouse. Following imaging, gait behavior was characterized to test for changes in locomotor stability. During ambulation, in both young adult and aged mice, an increase in network connectivity and synchronicity was noted. An age-dependent increase in synchronicity was seen in ambulating aged males only. Additionally, females displayed increases in the number of active neurons, Ca²⁺ transients, and neuronal activity compared to males, particularly during ambulation. These results suggest S1 Ca²⁺ dynamics and network synchronicity are likely contributors of locomotor stability. We believe this work raises awareness of age- and sex-dependent alterations in S1 neuronal networks, perhaps underlying the increase in falls with age.     

Modeling the contribution of lamina 5 neuronal and network dynamics to low frequency EEG phenomena

The Electroencephalogram (EEG) is an important clinical and research tool in neurophysiology. With the advent of recording techniques, new evidence is emerging on the neuronal populations and wiring in the neocortex. A main challenge is to relate the EEG generation mechanisms to the underlying circuitry of the neocortex. In this paper, we look at the principal intrinsic properties of neocortical cells in layer 5 and their network behavior in simplified simulation models to explain the emergence of several important EEG phenomena such as the alpha rhythms, slow-wave sleep oscillations, and a form of cortical seizure. The models also predict the ability of layer 5 cells to produce a resonance-like neuronal recruitment known as the augmenting response. While previous models point to deeper brain structures, such as the thalamus, as the origin of many EEG rhythms (spindles), the current model suggests that the cortical circuitry itself has intrinsic oscillatory dynamics which could account for a wide variety of EEG phenomena.Electronic Supplementary Material Supplementary material is available for this article at Sensorimotor Control Project- MIT Harvard NeuroEngineering Research Collaborative.     

Stochastic models of neuronal dynamics

Cortical activity is the product of interactions among neuronal populations. Macroscopic electrophysiological phenomena are generated by these interactions. In principle, the mechanisms of these interactions afford constraints on biologically plausible models of electrophysiological responses. In other words, the macroscopic features of cortical activity can be modelled in terms of the microscopic behaviour of neurons. An evoked response potential (ERP) is the mean electrical potential measured from an electrode on the scalp, in response to some event. The purpose of this paper is to outline a population density approach to modelling ERPs.We propose a biologically plausible model of neuronal activity that enables the estimation of physiologically meaningful parameters from electrophysiological data. The model encompasses four basic characteristics of neuronal activity and organization: (i) neurons are dynamic units, (ii) driven by stochastic forces, (iii) organized into populations with similar biophysical properties and response characteristics and (iv) multiple populations interact to form functional networks. This leads to a formulation of population dynamics in terms of the Fokker-Planck equation. The solution of this equation is the temporal evolution of a probability density over state-space, representing the distribution of an ensemble of trajectories. Each trajectory corresponds to the changing state of a neuron. Measurements can be modelled by taking expectations over this density, e.g. mean membrane potential, firing rate or energy consumption per neuron. The key motivation behind our approach is that ERPs represent an average response over many neurons. This means it is sufficient to model the probability density over neurons, because this implicitly models their average state. Although the dynamics of each neuron can be highly stochastic, the dynamics of the density is not. This means we can use Bayesian inference and estimation tools that have already been established for deterministic systems. The potential importance of modelling density dynamics (as opposed to more conventional neural mass models) is that they include interactions among the moments of neuronal states (e.g. the mean depolarization may depend on the variance of synaptic currents through nonlinear mechanisms).Here, we formulate a population model, based on biologically informed model-neurons with spike-rate adaptation and synaptic dynamics. Neuronal sub-populations are coupled to form an observation model, with the aim of estimating and making inferences about coupling among sub-populations using real data. We approximate the time-dependent solution of the system using a bi-orthogonal set and first-order perturbation expansion. For didactic purposes, the model is developed first in the context of deterministic input, and then extended to include stochastic effects. The approach is demonstrated using synthetic data, where model parameters are identified using a Bayesian estimation scheme we have described previously.     

Evaluation of neuronal coupling dynamics

Temporary correlated activity of neuron assemblies is believed to play a substantial role for the brain's pattern recognition ability. To study the underlying principles of such mechanisms, a method is proposed for the characterization of the interneuronal and stimulus-response coupling changes of two periodically driven and simultaneously recorded units. The coupling measure is derived from the cross correlation function by calculating the actual correlation contributions without performing the subsequent time-average (which would give the cross correlation function). Examples are given for simultaneously recorded spike trains from visual cortical units, but the method can be applied equally well to evoked potentials or intracellular recordings.     

Nonlinear dynamics of the blood flow studied by Lyapunov exponents

In order to gain an insight into the dynamics of the cardiovascular system throughout which the blood circulates, the signals measured from peripheral blood flow in humans were analyzed by calculating the Lyapunov exponents. Over a wide range of algorithm parameters, paired values of both the global and the local Lyapunov exponents were obtained, and at least one exponent equaled zero within the calculation error. This may be an indication of the deterministic nature and finite number of degrees of freedom of the cardiovascular system governing the blood-flow dynamics on a time scale of minutes. A difference was observed in the Lyapunov dimension of controls and athletes.     

Microtubule dynamics in neuronal morphogenesis

Microtubules (MTs) are essential for neuronal morphogenesis in the developing brain. The MT cytoskeleton provides physical support to shape the fine structure of neuronal processes. MT-based motors play important roles in nucleokinesis, process formation and retraction. Regulation of MT stability downstream of extracellular cues is proposed to be critical for axonogenesis. Axons and dendrites exhibit different patterns of MT organization, underlying the divergent functions of these processes. Centrosomal positioning has drawn the attention of researchers because it is a major clue to understanding neuronal MT organization. In this review, we focus on how recent advances in live imaging have revealed the dynamics of MT organization and centrosome positioning during neural development.     

Complexities and uncertainties of neuronal network function

The nervous system generates behaviours through the activity in groups of neurons assembled into networks. Understanding these networks is thus essential to our understanding of nervous system function. Understanding a network requires information on its component cells, their interactions and their functional properties. Few networks come close to providing complete information on these aspects. However, even if complete information were available it would still only provide limited insight into network function. This is because the functional and structural properties of a network are not fixed but are plastic and can change over time. The number of interacting network components, their (variable) functional properties, and various plasticity mechanisms endows networks with considerable flexibility, but these features inevitably complicate network analyses. This review will initially discuss the general approaches and problems of network analyses. It will then examine the success of these analyses in a model spinal cord locomotor network in the lamprey, to determine to what extent in this relatively simple vertebrate system it is possible to claim detailed understanding of network function and plasticity.