Fractal Stochastic Modeling of Spiking Activity in Suprachiasmatic Nucleus Neurons

Individual neurons in the suprachiasmatic nucleus (SCN), the master biological clock in mammals, autonomously produce highly complex patterns of spikes. We have shown that most (~90%) SCN neurons exhibit truly stochastic interspike interval (ISI) patterns. The aim of this study was to understand the stochastic nature of the firing patterns in SCN neurons by analyzing the ISI sequences of 150 SCN neurons in hypothalamic slices. Fractal analysis, using the periodogram, Fano factor, and Allan factor, revealed the presence of a 1/f-type power-law (fractal) behavior in the ISI sequences. This fractal nature was persistent after the application of the GABA_A receptor antagonist bicuculline, suggesting that the fractal stochastic activity is an intrinsic property of individual SCN neurons. Based on these physiological findings, we developed a computational model for the stochastic SCN neurons to find that their stochastic spiking activity was best described by a gamma point process whose mean firing rate was modulated by a fractal binomial noise. Taken together, we suggest that SCN neurons generate temporal spiking patterns using the fractal stochastic point process.Action Editor: Carson C. Chow     

Gain adjustment of inhibitory synapses in the auditory system

A group of central auditory neurons residing in the lateral superior olivary nucleus (LSO) responds selectively to interaural level differences and may contribute to sound localization. In this simple circuit, ipsilateral sound increases firing of LSO neurons, whereas contralateral sound inhibits the firing rate via activation of the medial nucleus of the trapezoid body (MNTB). During development, individual MNTB fibers arborize within the LSO, but they undergo a restriction of their boutons that ultimately leads to mature topography. A critical issue is whether a distinct form of inhibitory synaptic plasticity contributes to MNTB synapse elimination within LSO. Whole-cell recording from LSO neurons in brain slices from developing gerbils show robust long-term depression (LTD) of the MNTB-evoked IPSP/Cs when the MNTB was activated at a low frequency (1 Hz). These inhibitory synapses also display mixed GABA/glycinergic transmission during development, as assessed physiologically and immunohistochemically (Kotak et al. 1998). While either glycine or GABA_A receptors could independently display inhibitory LTD, focal delivery of GABA, but not glycine, at the postsynaptic-locus induces depression. Furthermore, the GABA_B receptor antagonist, SCH-50911, prevents GABA or synaptically induced depression. Preliminary evidence also indicated strengthening of inhibitory transmission (LTP) by a distinct pattern of inhibitory activity. These data support the idea that GABA is crucial for the expression inhibitory LTD and that this plasticity may underlie the early refinement of inhibitory synaptic connections in the LSO.     

Tuning bat LSO neurons to interaural intensity differences through spike-timing dependent plasticity

Bats, like other mammals, are known to use interaural intensity differences (IID) to determine azimuthal position. In the lateral superior olive (LSO) neurons have firing behaviors which vary systematically with IID. Those neurons receive excitatory inputs from the ipsilateral ear and inhibitory inputs from the contralateral one. The IID sensitivity of a LSO neuron is thought to be due to delay differences between the signals coming from both ears, differences due to different synaptic delays and to intensity-dependent delays. In this paper we model the auditory pathway until the LSO. We propose a learning scheme where inputs to LSO neurons start out numerous with different relative delays. Spike timing-dependent plasticity (STDP) is then used to prune those connections. We compare the pruned neuron responses with physiological data and analyse the relationship between IID’s of teacher stimuli and IID sensitivities of trained LSO neurons.     

Responses of neurons in primary and inferior temporal visual cortices to natural scenes.

The primary visual cortex (V1) is the first cortical area to receive visual input, and inferior temporal (IT) areas are among the last along the ventral visual pathway. We recorded, in area V1 of anaesthetized cats and area IT of awake macaque monkeys, responses of neurons to videos of natural scenes. Responses were analysed to test various hypotheses concerning the nature of neural coding in these two regions. A variety of spike-train statistics were measured including spike-count distributions, interspike interval distributions, coefficients of variation, power spectra, Fano factors and different sparseness measures. All statistics showed non-Poisson characteristics and several revealed self-similarity of the spike trains. Spike-count distributions were approximately exponential in both visual areas for eight different videos and for counting windows ranging from 50 ms to 5 seconds. The results suggest that the neurons maximize their information carrying capacity while maintaining a fixed long-term-average firing rate, or equivalently, minimize their average firing rate for a fixed information carrying capacity.     

On the existence of spontaneous neuronal bursting activity at the periphery of the amphibian auditory pathway

The firing activity of 104 neurons from the medullar dorsal nucleus (homologous to the mammalian cochlear nucleus complex) of immobilized common frogs (Rana t. temporaria) was recorded extracellularly in the absence of external auditory stimuli. This background activity was analyzed by using stochastic point process parameters such as the coefficient of variation, local coefficient of variation, interdependence of neighboring intervals, interpulse interval distribution, autocorrelation, and hazard function. All these parameters were put in correspondence with bursting rate, i.e., relative number of interpulse intervals grouped in clusters, or bursts, determined by the proposed method. Neurons with pronounced refractoriness, as a rule, exhibited lower bursting rates whereas neurons with a distinct maximum of the correlation function showed higher values than expected from a random Poisson process. However, even in the latter group of cells, bursting rates usually did not decrease after random shuffling of all the interpulse intervals. As a result, we must admit the absence of clustered spontaneous activity in second-order auditory neurons in the amphibian species studied. These data are compared with the specificities of spontaneous firing activity in different parts of the mammalian auditory pathway.     

Role of hyperpolarization-activated conductances in the lateral superior olive: A modeling study

This modeling study examines the possible functional roles of two hyperpolarization-activated conductances in lateral superior olive (LSO) principal neurons. Inputs of these LSO neurons are transformed into an output, which provides a firing-rate code for a certain interaural sound intensity difference (IID) range. Recent experimental studies have found pharmacological evidence for the presence of both the Gh conductance as well as the inwardly rectifying outward GKIR conductance in the LSO. We addressed the question of how these conductances influence the dynamic range (IID versus firing rate). We used computer simulations of both a point-neuron model and a two-compartmental model to investigate this issue, and to determine the role of these conductances in setting the dynamic range of these neurons. The width of the dynamic regime, the frequency-current (f-I) function, first-spike latency, subthreshold oscillations and the interplay between the two hyperpolarization activated conductances are discussed in detail. The in vivo non-monotonic IID-firing rate function in a subpopulation of LSO neurons is in good correspondence with our simulation predictions. Two compartmental model simulation results suggest segregation of Gh and GKIR conductances on different compartments, as this spatial configuration could explain certain experimental results.     

A fractional calculus approach to self-similar protein dynamics.

Relaxation processes and reaction kinetics of proteins deviate from exponential behavior because of their large amount of conformational substrates. The dynamics are governed by many time scales and, therefore, the decay of the relaxation function or reactant concentration is slower than exponential. Applying the idea of self-similar dynamics, we derive a fractal scaling model that results in an equation in which the time derivative is replaced by a differentiation (d/dt)beta of non-integer order beta. The fractional order differential equation is solved by a Mittag-Leffler function. It depends on two parameters, a fundamental time scale tau 0 and a fractional order beta that can be interpreted as a self-similarity dimension of the dynamics. Application of the fractal model to ligand rebinding and pressure release measurements of myoglobin is demonstrated, and the connection of the model to considerations of energy barrier height distributions is shown.     

L-type Calcium Channel Cav1.2 Is Required for Maintenance of Auditory Brainstem Nuclei

Ca_v1.2 and Ca_v1.3 are the major L-type voltage-gated Ca²⁺ channels in the CNS. Yet, their individual in vivo functions are largely unknown. Both channel subunits are expressed in the auditory brainstem, where Ca_v1.3 is essential for proper maturation. Here, we investigated the role of Ca_v1.2 by targeted deletion in the mouse embryonic auditory brainstem. Similar to Ca_v1.3, loss of Ca_v1.2 resulted in a significant decrease in the volume and cell number of auditory nuclei. Contrary to the deletion of Ca_v1.3, the action potentials of lateral superior olive (LSO) neurons were narrower compared with controls, whereas the firing behavior and neurotransmission appeared unchanged. Furthermore, auditory brainstem responses were nearly normal in mice lacking Ca_v1.2. Perineuronal nets were also unaffected. The medial nucleus of the trapezoid body underwent a rapid cell loss between postnatal days P0 and P4, shortly after circuit formation. Phosphorylated cAMP response element-binding protein (CREB), nuclear NFATc4, and the expression levels of p75NTR, Fas, and FasL did not correlate with cell death. These data demonstrate for the first time that both Ca_v1.2 and Ca_v1.3 are necessary for neuronal survival but are differentially required for the biophysical properties of neurons. Thus, they perform common as well as distinct functions in the same tissue.     

Deficiency of neural recognition molecule NB-2 affects the development of glutamatergic auditory pathways from the ventral cochlear nucleus to the superior olivary complex in mouse

Neural recognition molecule NB-2/contactin 5 is expressed transiently during the first postnatal week in glutamatergic neurons of the central auditory system. Here, we investigated the effect of NB-2 deficiency on the auditory brainstem in mouse. While almost all principal neurons are wrapped with the calyces of Held in the medial nucleus of the trapezoid body (MNTB) in wild type, 8% of principal neurons in NB-2 knockout (KO) mice lack the calyces of Held at postnatal day (P) 6. At P10 and P15, apoptotic principal neurons were detected in NB-2 KO mice, but not in wild type. Apoptotic cells were also increased in the ventral cochlear nucleus (VCN) of NB-2 KO mice, which contains bushy neurons projecting to the MNTB and the lateral superior olive (LSO). At the age of 1 month, the number of principal neurons in the MNTB and of glutamatergic synapses in the LSO was reduced in NB-2 KO mice. Finally, interpeak latencies for auditory brainstem response waves II-III and III-IV were significantly increased in NB-2 KO mice. Together, these findings suggest that NB-2 deficiency causes a deficit in synapse formation and then induces apoptosis in MNTB and VCN neurons, affecting auditory brainstem function.     

听觉电生理实时分析系统及其在大鼠下丘信息编码研究中的应用

基于TDT神经电生理软硬件平台和Matlab软件环境,开发了专用于听觉电生理研究的实时分析软件。通过对神经元胞外记录信号的在线处理和分析,可以在实验过程中得到刺激后放电活动时间直方图、平均发放率、首次发放潜伏期等定量分析结果,以及刺激参数变化时神经元发放率的变化曲线,如发放率-刺激强度曲线等。此分析软件被用于大鼠下丘神经元听觉信息编码的研究中,观察到下丘神经元对于纯音和噪声刺激不同的时间响应模式,以及神经元发放率和首次发放潜伏期对声音刺激强度的编码。     

Robustness of neuronal tuning to binaural sound localization cues against age-related loss of inhibitory synaptic inputs

Sound localization relies on minute differences in the timing and intensity of sound arriving at both ears. Neurons of the lateral superior olive (LSO) in the brainstem process these interaural disparities by precisely detecting excitatory and inhibitory synaptic inputs. Aging generally induces selective loss of inhibitory synaptic transmission along the entire auditory pathways, including the reduction of inhibitory afferents to LSO. Electrophysiological recordings in animals, however, reported only minor functional changes in aged LSO. The perplexing discrepancy between anatomical and physiological observations suggests a role for activity-dependent plasticity that would help neurons retain their binaural tuning function despite loss of inhibitory inputs. To explore this hypothesis, we use a computational model of LSO to investigate mechanisms underlying the observed functional robustness against age-related loss of inhibitory inputs. The LSO model is an integrate-and-fire type enhanced with a small amount of low-voltage activated potassium conductance and driven with (in)homogeneous Poissonian inputs. Without synaptic input loss, model spike rates varied smoothly with interaural time and level differences, replicating empirical tuning properties of LSO. By reducing the number of inhibitory afferents to mimic age-related loss of inhibition, overall spike rates increased, which negatively impacted binaural tuning performance, measured as modulation depth and neuronal discriminability. To simulate a recovery process compensating for the loss of inhibitory fibers, the strength of remaining inhibitory inputs was increased. By this modification, effects of inhibition loss on binaural tuning were considerably weakened, leading to an improvement of functional performance. These neuron-level observations were further confirmed by population modeling, in which binaural tuning properties of multiple LSO neurons were varied according to empirical measurements. These results demonstrate the plausibility that homeostatic plasticity could effectively counteract known age-dependent loss of inhibitory fibers in LSO and suggest that behavioral degradation of sound localization might originate from changes occurring more centrally.     

From spiking neuron models to linear-nonlinear models

Neurons transform time-varying inputs into action potentials emitted stochastically at a time dependent rate. The mapping from current input to output firing rate is often represented with the help of phenomenological models such as the linear-nonlinear (LN) cascade, in which the output firing rate is estimated by applying to the input successively a linear temporal filter and a static non-linear transformation. These simplified models leave out the biophysical details of action potential generation. It is not a priori clear to which extent the input-output mapping of biophysically more realistic, spiking neuron models can be reduced to a simple linear-nonlinear cascade. Here we investigate this question for the leaky integrate-and-fire (LIF), exponential integrate-and-fire (EIF) and conductance-based Wang-Buzsáki models in presence of background synaptic activity. We exploit available analytic results for these models to determine the corresponding linear filter and static non-linearity in a parameter-free form. We show that the obtained functions are identical to the linear filter and static non-linearity determined using standard reverse correlation analysis. We then quantitatively compare the output of the corresponding linear-nonlinear cascade with numerical simulations of spiking neurons, systematically varying the parameters of input signal and background noise. We find that the LN cascade provides accurate estimates of the firing rates of spiking neurons in most of parameter space. For the EIF and Wang-Buzsáki models, we show that the LN cascade can be reduced to a firing rate model, the timescale of which we determine analytically. Finally we introduce an adaptive timescale rate model in which the timescale of the linear filter depends on the instantaneous firing rate. This model leads to highly accurate estimates of instantaneous firing rates.     

Modeling the electrophysiology of suprachiasmatic nucleus neurons

Neurons in the SCN act as the central circadian (approximately 24-h) pacemaker in mammals. Using measurements of the ionic currents in SCN neurons, the authors fit a Hodgkin-Huxley-type model that accurately reproduces slow (approximately 28 Hz) neural firing as well as the contributions of ionic currents during an action potential. When inputs of other SCN neurons are considered, the model accurately predicts the fractal nature of firing rates and the appearance of random bursting. In agreement with experimental data, the molecular clock within these neurons modulates the firing rate through small changes in the concentration of internal calcium, calcium channels, or potassium channels. Predictions are made on how signals from other neurons can start, stop, speed up, or slow down firing. Only a slow sodium inactivation variable and voltage do not reach equilibrium during the interval between action potentials, and based on this finding, a reduced model is formulated.     

自发和诱发神经放电序列的内在时序特性

运用Fano因子分析法,考察豚鼠听神经单纤维的自发放电序列、小鼠海马CAl区神经元的自发放电序列以及蟾蜍缝匠肌肌梭传入神经的诱发放电序列的时序特性,结果显示自发和诱发放电时间序列均存在Fano因子随计算窗口时间的增大而持续增长的特点,而原始数据的随机重排替代数据则没有这一特性,说明这些神经放电时间序列与一般的随机点过程不同,存在长时程相关性,在时序上具有某种结构特征。进一步的研究表明,这一时序结构特征可以通过将随机产生的一维正态分布序列数据,与神经放电时间序列数据进行跟随排序后而体现,提示这一特征与放电间隔的分布特点无关。     

Sparse representation of sounds in the unanesthetized auditory cortex

How do neuronal populations in the auditory cortex represent acoustic stimuli? Although sound-evoked neural responses in the anesthetized auditory cortex are mainly transient, recent experiments in the unanesthetized preparation have emphasized subpopulations with other response properties. To quantify the relative contributions of these different subpopulations in the awake preparation, we have estimated the representation of sounds across the neuronal population using a representative ensemble of stimuli. We used cell-attached recording with a glass electrode, a method for which single-unit isolation does not depend on neuronal activity, to quantify the fraction of neurons engaged by acoustic stimuli (tones, frequency modulated sweeps, white-noise bursts, and natural stimuli) in the primary auditory cortex of awake head-fixed rats. We find that the population response is sparse, with stimuli typically eliciting high firing rates (>20 spikes/second) in less than 5% of neurons at any instant. Some neurons had very low spontaneous firing rates (<0.01 spikes/second). At the other extreme, some neurons had driven rates in excess of 50 spikes/second. Interestingly, the overall population response was well described by a lognormal distribution, rather than the exponential distribution that is often reported. Our results represent, to our knowledge, the first quantitative evidence for sparse representations of sounds in the unanesthetized auditory cortex. Our results are compatible with a model in which most neurons are silent much of the time, and in which representations are composed of small dynamic subsets of highly active neurons.     

Sensitivity of firing rate to input fluctuations depends on time scale separation between fast and slow variables in single neurons

Neuronal responses are often characterized by the firing rate as a function of the stimulus mean, or the f–I curve. We introduce a novel classification of neurons into Types A, B−, and B+ according to how f–I curves are modulated by input fluctuations. In Type A neurons, the f–I curves display little sensitivity to input fluctuations when the mean current is large. In contrast, Type B neurons display sensitivity to fluctuations throughout the entire range of input means. Type B− neurons do not fire repetitively for any constant input, whereas Type B+ neurons do. We show that Type B+ behavior results from a separation of time scales between a slow and fast variable. A voltage-dependent time constant for the recovery variable can facilitate sensitivity to input fluctuations. Type B+ firing rates can be approximated using a simple “energy barrier” model.     

Fractal fluctuations in human respiration.

The present study was designed to characterize respiratory fluctuations in awake, healthy adult humans under resting conditions. For this purpose, we recorded respiratory movements with a strain-gauge pneumograph in 20 subjects. We then used Allan factor, Fano factor, and dispersional analysis to test whether the fluctuations in the number of breaths, respiratory period, and breath amplitude were fractal (i.e., time-scale-invariant) or random in occurrence. Specifically, we measured the slopes of the power laws in the Allan factor, Fano factor, and dispersional analysis curves for original time series and compared these with the slopes of the curves for surrogates (randomized data sets). In addition, the Hurst exponent was calculated from the slope of the power law in the Allan factor curve to determine whether the long-range correlations among the fluctuations in breath number were positively or negatively correlated. The results can be summarized as follows. Fluctuations in all three parameters were fractal in nine subjects. There were four subjects in whom only the fluctuations in number of breaths and breath amplitude were fractal, three subjects in whom only the fluctuations in number of breaths were fractal, and two subjects in whom only fluctuations in breath number and respiratory period were fractal. Time-scale-invariant behavior was absent in the two remaining subjects. The results indicate that, in most cases, apparently random fluctuations in respiratory pattern are, in fact, correlated over more than one time scale. Moreover, the data suggest that fractal fluctuations in breath number, respiratory period, and breath amplitude are controlled by separate processes.     

A Markov model for interspike interval distributions of auditory cortical neurons that do not show periodic firings

Spontaneous firing properties of individual auditory cortical neurons are interpreted in terms of local and global order present in functioning brain networks, such as alternating “up” and “down” states. A four-state modulated Markov process is used to model neuronal firings. The system alternates between a bound and an unbound state, both with Poisson-distributed lifetimes. During the unbound state, active and closed states alternate with Poisson-distributed lifetimes. Inside the active state, spikes are generated as a realization of a Poisson process. This combination of processes constitutes a four-state modulated Markov process, determined by five independent parameters. Analytical expressions for the probability density functions (pdfs) that describe the interspike interval (ISI) distribution and autocorrelation function are derived. The pdf for the ISI distribution is shown to be a linear combination of three exponential functions and is expressed through the five system parameters. Through fitting experimental ISI histograms by the theoretical ones, numerical values of the system parameters are obtained for the individual neurons. Both Monte Carlo simulations and goodness-of-fit tests are used to validate the fitting procedure. The values of the estimated system parameters related to the active-closed and bound–unbound processes and their independence on the neurons’ mean firing rate suggest that the underlying quasi-periodic processes reflect properties of the network in which the neurons are embedded. The characteristic times of autocorrelations, determined by the bound–unbound and active-closed processes, are also independent of the neuron’s firing rate. The agreement between experimental and theoretical ISI histograms and autocorrelation functions allows interpretation of the system parameters of the individual neurons in terms of slow and delta waves, and high-frequency oscillations observed in cortical networks. This procedure can identify and track the influence of changing brain states on the single-unit firing patterns in experimental animals.     

Electrophysiological properties and modeling of murine vomeronasal sensory neurons in acute slice preparations

The vomeronasal system is involved in the detection of pheromones in many mammals. Vomeronasal sensory neurons encode the behaviorally relevant information into action potentials that are directly transmitted to the accessory olfactory bulb. We developed a model of the electrical activity of mouse basal vomeronasal sensory neurons, which mimics both the voltage-gated current properties and the firing behavior of these neurons in their near-native state, using a minimal number of parameters. Data were obtained by recordings with the whole-cell voltage-clamp or current-clamp techniques from mouse basal vomeronasal sensory neurons in acute slice preparations. The resting potential ranged from -50 to -70 mV, and current injections of less than 2-10 pA induced tonic firing in most neurons. The experimentally determined firing frequency as a function of injected current was well described by a Michaelis-Menten equation and was exactly reproduced by the model, which could be used in combination with future models that will include details of the mouse vomeronasal transduction cascade.