Inference of intrinsic spiking irregularity based on the Kullback-Leibler information

We have recently established an empirical Bayes method that extracts both the intrinsic irregularity and the time-dependent rate from a spike sequence [Koyama, S., Shinomoto, S., 2005. Empirical Bayes interpretations of random point events. J. Phys. A: Math. Gen. 38, L531-L537]. In the present paper, we examine an alternative method based on the more fundamental principle of minimizing the Kullback-Leibler information from the original distribution of spike sequences to a model distribution. Not only the empirical Bayes method but also the Kullback-Leibler information method exhibits a switch of the most plausible interpretation of the spikes between (I) being derived irregularly from a nearly constant rate, and (II) being derived rather regularly from a significantly fluctuating rate.The model distributions selected by both methods are similar for the same spike sequences derived from a given rate-fluctuating gamma process.     

Coefficient of variation of interspike intervals greater than 0.5. How and when?

Using Stein's model with and without reversal potentials, we investigated the mechanism of production of spike trains with a CV (ISI) (standard deviation/mean interspike interval) greater than 0.5, as observed in the visual cortex. When the attractor of the deterministic part of the dynamics is below the firing threshold, spike generation results primarily from random fluctuations. Using computer simulation for a range of membrane decay times and with other model parameters set to values appropriate for the visual cortex, we demonstrate that CV (ISI) is then usually greater than 0.5; if the attractor is above the threshold, spike generation is mainly due to deterministic forces, and CV (ISI) is then usually lower than 0.5. The critical value of the inhibitory postsynaptic potential (IPSP) rate at which CV (ISI) becomes greater than 0.5 is determined, resulting in specifications of how neurones might adjust their synaptic inputs to elicit irregular spike trains. Received: 25 June 1998/Accepted in revised form: 16 December 1998     

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     

Computation of spiking activity for a stochastic spatial neuron model: effects of spatial distribution of input on bimodality and CV of the ISI distribution

We obtain computational results for a new extended spatial neuron model in which the neuronal electrical depolarization from resting level satisfies a cable partial differential equation and the synaptic input current is also a function of space and time, obeying a first order linear partial differential equation driven by a two-parameter random process. The model is first described explicitly with the inclusion of all biophysical parameters. Simplified equations are obtained with dimensionless space and time variables. A standard parameter set is described, based mainly on values appropriate for cortical pyramidal cells. When the noise is small and the mean voltage crosses threshold, a formula is derived for the expected time to spike. A simulation algorithm, involving one-dimensional random processes is given and used to obtain moments and distributions of the interspike interval (ISI). The parameters used are those for a near balanced state and there is great sensitivity of the firing rate around the balance point. This sensitivity may be related to genetically induced pathological brain properties (Rett's syndrome). The simulation procedure is employed to find the ISI distribution for some simple patterns of synaptic input with various relative strengths for excitation and inhibition. With excitation only, the ISI distribution is unimodal of exponential type and with a large coefficient of variation. As inhibition near the soma grows, two striking effects emerge. The ISI distribution shifts first to bimodal and then to unimodal with an approximately Gaussian shape with a concentration at large intervals. At the same time the coefficient of variation of the ISI drops dramatically to less than 1/5 of its value without inhibition.     

See globally, spike locally: oscillations in a retinal model encode large visual features

We show that coherent oscillations among neighboring ganglion cells in a retinal model encode global topological properties, such as size, that cannot be deduced unambiguously from their local, time-averaged firing rates. Whereas ganglion cells may fire similar numbers of spikes in response to both small and large spots, only large spots evoke coherent high frequency oscillations, potentially allowing downstream neurons to infer global stimulus properties from their local afferents. To determine whether such information might be extracted over physiologically realistic spatial and temporal scales, we analyzed artificial spike trains whose oscillatory correlations were similar to those measured experimentally. Oscillatory power in the upper gamma band, extracted on single-trials from multi-unit spike trains, supported good to excellent size discrimination between small and large spots, with performance improving as the number of cells and/or duration of the analysis window was increased. By using Poisson distributed spikes to normalize the firing rate across stimulus conditions, we further found that coincidence detection, or synchrony, yielded substantially poorer performance on identical size discrimination tasks. To determine whether size encoding depended on contiguity independent of object shape, we examined the total oscillatory activity across the entire model retina in response to random binary images. As the ON-pixel probability crossed the percolation threshold, which marks the sudden emergence of large connected clusters, the total gamma-band activity exhibited a sharp transition, a phenomena that may be experimentally observable. Finally, a reanalysis of previously published oscillatory responses from cat ganglion cells revealed size encoding consistent with that predicted by the retinal model.     

Statistical characteristics of climbing fiber spikes necessary for efficient cerebellar learning

 Mean firing rates (MFRs), with analogue values, have thus far been used as information carriers of neurons in most brain theories of learning. However, the neurons transmit the signal by spikes, which are discrete events. The climbing fibers (CFs), which are known to be essential for cerebellar motor learning, fire at the ultra-low firing rates (around 1 Hz), and it is not yet understood theoretically how high-frequency information can be conveyed and how learning of smooth and fast movements can be achieved. Here we address whether cerebellar learning can be achieved by CF spikes instead of conventional MFR in an eye movement task, such as the ocular following response (OFR), and an arm movement task. There are two major afferents into cerebellar Purkinje cells: parallel fiber (PF) and CF, and the synaptic weights between PFs and Purkinje cells have been shown to be modulated by the stimulation of both types of fiber. The modulation of the synaptic weights is regulated by the cerebellar synaptic plasticity. In this study we simulated cerebellar learning using CF signals as spikes instead of conventional MFR. To generate the spikes we used the following four spike generation models: (1) a Poisson model in which the spike interval probability follows a Poisson distribution, (2) a gamma model in which the spike interval probability follows the gamma distribution, (3) a max model in which a spike is generated when a synaptic input reaches maximum, and (4) a threshold model in which a spike is generated when the input crosses a certain small threshold. We found that, in an OFR task with a constant visual velocity, learning was successful with stochastic models, such as Poisson and gamma models, but not in the deterministic models, such as max and threshold models. In an OFR with a stepwise velocity change and an arm movement task, learning could be achieved only in the Poisson model. In addition, for efficient cerebellar learning, the distribution of CF spike-occurrence time after stimulus onset must capture at least the first, second and third moments of the temporal distribution of error signals. Received: 28 January 2000 / Accepted in revised form: 2 August 2000     

Noise in attractor networks in the brain produced by graded firing rate representations

Representations in the cortex are often distributed with graded firing rates in the neuronal populations. The firing rate probability distribution of each neuron to a set of stimuli is often exponential or gamma. In processes in the brain, such as decision-making, that are influenced by the noise produced by the close to random spike timings of each neuron for a given mean rate, the noise with this graded type of representation may be larger than with the binary firing rate distribution that is usually investigated. In integrate-and-fire simulations of an attractor decision-making network, we show that the noise is indeed greater for a given sparseness of the representation for graded, exponential, than for binary firing rate distributions. The greater noise was measured by faster escaping times from the spontaneous firing rate state when the decision cues are applied, and this corresponds to faster decision or reaction times. The greater noise was also evident as less stability of the spontaneous firing state before the decision cues are applied. The implication is that spiking-related noise will continue to be a factor that influences processes such as decision-making, signal detection, short-term memory, and memory recall even with the quite large networks found in the cerebral cortex. In these networks there are several thousand recurrent collateral synapses onto each neuron. The greater noise with graded firing rate distributions has the advantage that it can increase the speed of operation of cortical circuitry.     

急性大鼠癫痫模型海马神经元原发性单位后放电脉冲间隔特征

受到刺激后即刻出现的海马（hippocampus,HPC）原发性单位后放电是癫痫相关性细胞电活动的重要形式之一,其放电脉冲间隔（interspike interval,ISI）和串内平均频率（Hz）特征及其在网络癫痫形成中的作用值得探讨。实验用急性强直电刺激（60Hz,2S,0．4-0．6mA）大鼠右侧后背HPC（acute tetanization of the fight posterior dorsal hippocampus,以后简称ATPDH）或右侧尾壳核（acute tetanization of the fight caudate putamen nucleus,以后简称ATRC）诱导HPC或皮层网络癫痫,重点观察HPC神经元原发性单位后放电模式和上述的瞬时时间编码特征。结果表明：（1）HPC原发性单位后放电表现为两种不同的放电模式,即先易化后抑制或先抑制后易化,其ISI序列分别表现为先小后大的“头尾”式分布或先大后小的“尾头”式分布。（2）ATFDH主要引起“尾头”式（35／57串）、而ATRC主要引起“头尾”式（12／22串）ISI点分布的原发性单位后放电,串内“头”、“尾”平均持续时间均具有明显差异（P〈0．05）。（3）ATRC可以诱导双侧HPC单位后放电出现交互的“头尾”、‘呢头”式ISI点分布特征。（4）多串电刺激可以诱导HPC原发性单位后放电特征性ISI点分布重复显现。（5）特征性HPC原发性单位后放电伴随出现网络癫痫发作样高频电振荡。这提示：强直电刺激诱导的HPC神经元原发性单位后放电“头尾”或呢头”式ISI序列分布规律,可以较准确地反映所记录神经元的诱发性易化或抑制活动的程度,用于网络癫痫形成中单个成员细胞癫痫相关性电活动机制的分析。     

A cross-interval spike train analysis: the correlation between spike generation and temporal integration of doublets

A stochastic spike train analysis technique is introduced to reveal the correlation between the firing of the next spike and the temporal integration period of two consecutive spikes (i.e., a doublet). Statistics of spike firing times between neurons are established to obtain the conditional probability of spike firing in relation to the integration period. The existence of a temporal integration period is deduced from the time interval between two consecutive spikes fired in a reference neuron as a precondition to the generation of the next spike in a compared neuron. This analysis can show whether the coupled spike firing in the compared neuron is correlated with the last or the second-to-last spike in the reference neuron. Analysis of simulated and experimentally recorded biological spike trains shows that the effects of excitatory and inhibitory temporal integration are extracted by this method without relying on any subthreshold potential recordings. The analysis also shows that, with temporal integration, a neuron driven by random firing patterns can produce fairly regular firing patterns under appropriate conditions. This regularity in firing can be enhanced by temporal integration of spikes in a chain of polysynaptically connected neurons. The bandpass filtering of spike firings by temporal integration is discussed. The results also reveal that signal transmission delays may be attributed not just to conduction and synaptic delays, but also to the delay time needed for temporal integration. Received: 3 March 1997 / Accepted in revised form: 6 November 1997     

体感皮层神经元放电间隔的概率密度函数与分布参数

本文建立了估计ISI概率密度函数的标准化ISI直方图和分布参数拟合方法,对34例猫体感皮层神经元自发和诱发放电活动进行了统计分析.     

Structure of spike activity of cold thermoreceptors at its various levels

In acute experiments on anesthetized rabbits we investigated the spike activity of cold fibers of the infraorbital nerve during a steady decrease in skin temperature from 39 to 7°C at a rate of 0.8 ± 0.05°/min. Analysis of interspike intervals (ISI) in the firing of receptors demonstrated that in the investigated range of skin temperatures the ISI histograms changed significantly several times, reflecting a shift in the pattern of firing. In addition, the reactions of each cold thermoreceptor had individual aspects, which lays the foundation for discussion of the perception of various characteristics of the temperature stimulus of the set of thermoreceptors.I. P. Pavlov Physiology Institute, Russian Academy of Sciences, Saint Petersburg. Translated from Neirofiziologiya, Vol. 24, No. 5, pp. 559–566, September–October, 1992.     

Variability and randomness in stationary neuronal activity

The patterns of neuronal activity can be different even if the mean firing rate is fixed. Investigating the variability of the firing may not be sufficient and we suggest to take into account the notion of randomness. The randomness is related to the entropy of the firing, which is bounded from above by the entropy of the Poisson process (given the mean interspike interval). Thus, we propose the Kullback-Leibler distance with respect to the Poisson process as a measure of randomness in a stationary neuronal activity. Under the condition of equal mean values the KL distance does not depend on the time scale and therefore can be compared to the coefficient of variation employed to measure the variability. Furthermore, this measure can be extended to account for correlated neuronal firing. Finally, we analyze the variability and randomness for three common ISI distributions in detail: gamma, lognormal and inverse Gaussian.     

Inhibitory synchrony as a mechanism for attentional gain modulation

Recordings from area V4 of monkeys have revealed that when the focus of attention is on a visual stimulus within the receptive field of a cortical neuron, two distinct changes can occur: The firing rate of the neuron can change and there can be an increase in the coherence between spikes and the local field potential (LFP) in the gamma-frequency range (30-50 Hz). The hypothesis explored here is that these observed effects of attention could be a consequence of changes in the synchrony of local interneuron networks. We performed computer simulations of a Hodgkin-Huxley type neuron driven by a constant depolarizing current, I, representing visual stimulation and a modulatory inhibitory input representing the effects of attention via local interneuron networks. We observed that the neuron's firing rate and the coherence of its output spike train with the synaptic inputs was modulated by the degree of synchrony of the inhibitory inputs. When inhibitory synchrony increased, the coherence of spiking model neurons with the synaptic input increased, but the firing rate either increased or remained the same. The mean number of synchronous inhibitory inputs was a key determinant of the shape of the firing rate versus current (f-I) curves. For a large number of inhibitory inputs (approximately 50), the f-I curve saturated for large I and an increase in input synchrony resulted in a shift of sensitivity-the model neuron responded to weaker inputs I. For a small number (approximately 10), the f-I curves were non-saturating and an increase in input synchrony led to an increase in the gain of the response-the firing rate in response to the same input was multiplied by an approximately constant factor. The firing rate modulation with inhibitory synchrony was highest when the input network oscillated in the gamma frequency range. Thus, the observed changes in firing rate and coherence of neurons in the visual cortex could be controlled by top-down inputs that regulated the coherence in the activity of a local inhibitory network discharging at gamma frequencies.     

Bistable,Irregular Firing and Population Oscillations in a Modular Attractor Memory Network

Attractor neural networks are thought to underlie working memory functions in the cerebral cortex. Several such models have been proposed that successfully reproduce firing properties of neurons recorded from monkeys performing working memory tasks. However, the regular temporal structure of spike trains in these models is often incompatible with experimental data. Here, we show that the in vivo observations of bistable activity with irregular firing at the single cell level can be achieved in a large-scale network model with a modular structure in terms of several connected hypercolumns. Despite high irregularity of individual spike trains, the model shows population oscillations in the beta and gamma band in ground and active states, respectively. Irregular firing typically emerges in a high-conductance regime of balanced excitation and inhibition. Population oscillations can produce such a regime, but in previous models only a non-coding ground state was oscillatory. Due to the modular structure of our network, the oscillatory and irregular firing was maintained also in the active state without fine-tuning. Our model provides a novel mechanistic view of how irregular firing emerges in cortical populations as they go from beta to gamma oscillations during memory retrieval.     

Predicting the Responses of Repetitively Firing Neurons to Current Noise

We used phase resetting methods to predict firing patterns of rat subthalamic nucleus (STN) neurons when their rhythmic firing was densely perturbed by noise. We applied sequences of contiguous brief (0.5–2 ms) current pulses with amplitudes drawn from a Gaussian distribution (10–100 pA standard deviation) to autonomously firing STN neurons in slices. Current noise sequences increased the variability of spike times with little or no effect on the average firing rate. We measured the infinitesimal phase resetting curve (PRC) for each neuron using a noise-based method. A phase model consisting of only a firing rate and PRC was very accurate at predicting spike timing, accounting for more than 80% of spike time variance and reliably reproducing the spike-to-spike pattern of irregular firing. An approximation for the evolution of phase was used to predict the effect of firing rate and noise parameters on spike timing variability. It quantitatively predicted changes in variability of interspike intervals with variation in noise amplitude, pulse duration and firing rate over the normal range of STN spontaneous rates. When constant current was used to drive the cells to higher rates, the PRC was altered in size and shape and accurate predictions of the effects of noise relied on incorporating these changes into the prediction. Application of rate-neutral changes in conductance showed that changes in PRC shape arise from conductance changes known to accompany rate increases in STN neurons, rather than the rate increases themselves. Our results show that firing patterns of densely perturbed oscillators cannot readily be distinguished from those of neurons randomly excited to fire from the rest state. The spike timing of repetitively firing neurons may be quantitatively predicted from the input and their PRCs, even when they are so densely perturbed that they no longer fire rhythmically.     

α-Calcium calmodulin kinase II modulates the temporal structure of hippocampal bursting patterns

The alpha calcium calmodulin kinase II (α-CaMKII) is known to play a key role in CA1/CA3 synaptic plasticity, hippocampal place cell stability and spatial learning. Additionally, there is evidence from hippocampal electrophysiological slice studies that this kinase has a role in regulating ion channels that control neuronal excitability. Here, we report in vivo single unit studies, with α-CaMKII mutant mice, in which threonine 305 was replaced with an aspartate (α-CaMKII(T305D) mutants), that indicate that this kinase modulates spike patterns in hippocampal pyramidal neurons. Previous studies showed that α-CaMKII(T305D) mutants have abnormalities in both hippocampal LTP and hippocampal-dependent learning. We found that besides decreased place cell stability, which could be caused by their LTP impairments, the hippocampal CA1 spike patterns of α-CaMKII(T305D) mutants were profoundly abnormal. Although overall firing rate, and overall burst frequency were not significantly altered in these mutants, inter-burst intervals, mean number of intra-burst spikes, ratio of intra-burst spikes to total spikes, and mean intra-burst intervals were significantly altered. In particular, the intra burst intervals of place cells in α-CaMKII(T305D) mutants showed higher variability than controls. These results provide in vivo evidence that besides its well-known function in synaptic plasticity, α-CaMKII, and in particular its inhibitory phosphorylation at threonine 305, also have a role in shaping the temporal structure of hippocampal burst patterns. These results suggest that some of the molecular processes involved in acquiring information may also shape the patterns used to encode this information.     

Alterations in the Intrinsic Burst Activity of Purkinje Neurons in Offspring Maternally Exposed to the CB1 Cannabinoid Agonist WIN 55212-2

Burst firing plays an important role in normal neuronal function and dysfunction. In Purkinje neurons, where the firing rate and discharge pattern encode the timing signals necessary for motor function, any alteration in firing properties, including burst activity, may affect the motor output. Therefore, we examined whether maternal exposure to the cannabinoid receptor agonist WIN 55212-2 (WIN) may affect the burst firing properties of cerebellar Purkinje cells in offspring. Whole-cell somatic patch-clamp recordings were made from cerebellar slices of adult male rats that were exposed to WIN prenatally. WIN exposure during pregnancy induced long-term alterations in the burst firing behavior of Purkinje neurons in rat offspring as evidenced by a significant increase in the mean number of spikes per burst (p < 0.05) and the prolongation of burst firing activity (p < 0.01). The postburst afterhyperpolarization potential (p < 0.001), the mean intraburst interspike intervals (p < 0.001) and the mean intraburst firing frequency (p < 0.001) were also significantly increased in the WIN-treated group. Prenatal exposure to WIN enhanced the firing irregularity as reflected by a significant decrease in the coefficient of variation of the intraburst interspike interval (p < 0.05). Furthermore, whole-cell voltage-clamp recordings revealed that prenatal WIN exposure significantly enhanced Ca²⁺ channel current amplitude in offspring Purkinje neurons compared to control cells. Overall, the data presented here strongly suggest that maternal exposure to cannabinoids can induce long-term changes in complex spike burst activity, which in turn may lead to alterations in neuronal output.     

Phase-of-firing coding of natural visual stimuli in primary visual cortex

We investigated the hypothesis that neurons encode rich naturalistic stimuli in terms of their spike times relative to the phase of ongoing network fluctuations rather than only in terms of their spike count. We recorded local field potentials (LFPs) and multiunit spikes from the primary visual cortex of anaesthetized macaques while binocularly presenting a color movie. We found that both the spike counts and the low-frequency LFP phase were reliably modulated by the movie and thus conveyed information about it. Moreover, movie periods eliciting higher firing rates also elicited a higher reliability of LFP phase across trials. To establish whether the LFP phase at which spikes were emitted conveyed visual information that could not be extracted by spike rates alone, we compared the Shannon information about the movie carried by spike counts to that carried by the phase of firing. We found that at low LFP frequencies, the phase of firing conveyed 54% additional information beyond that conveyed by spike counts. The extra information available in the phase of firing was crucial for the disambiguation between stimuli eliciting high spike rates of similar magnitude. Thus, phase coding may allow primary cortical neurons to represent several effective stimuli in an easily decodable format.     

Quantitative and statistical analysis of the shape of amperometric spikes recorded from two populations of cells

Previously used methods of comparing amperometric spike characteristics from two separate groups of cells have entailed pooling all the values for a spike characteristic from each group of cells and then statistically comparing the two samples. Although this approach has indicated that there are significant differences between the spike characteristics from coloboma and control mouse chromaffin cells, the results are not consistent between experiments. We have reexamined the assumptions of the statistical tests used as well as the variability inherent in amperometric data measured from two groups of cells. Our findings indicate that when comparing amperometric spike characteristics between groups of cells, it is more appropriate to compare samples of mean spike values. This method consistently indicates that there is no difference between coloboma and control amperometric spikes. These results have been validated by using samples of mean spike characteristics to detect changes in the shape of amperometric spikes from both mouse chromaffin cells at 37 degrees C and PC12 cells previously exposed to 50 microM L-3,4-dihydroxyphenylalanine and by the use of an additional analysis method, the nested ANOVA. Together, these results indicate that pooled samples of amperometric spike characteristics can give results that may confound the interpretation of amperometric data.     

A joint interspike interval difference stochastic spike train analysis: detecting local trends in the temporal firing patterns of single neurons

We introduce a stochastic spike train analysis method called joint interspike interval difference (JISID) analysis. By design, this method detects changes in firing interspike intervals (ISIs), called local trends, within a 4-spike pattern in a spike train. This analysis classifies 4-spike patterns that have similar incremental changes. It characterizes the higher-order serial dependence in spike firing relative to changes in the firing history. Mathematically, this spike train analysis describes the statistical joint distribution of consecutive changes in ISIs, from which the serial dependence of the changes in higher-order intervals can be determined. It is similar to the joint interspike interval (JISI) analysis, except that the joint distribution of consecutive ISI differences (ISIDs) is quantified. The graphical location of points in the JISID scatter plot reveals the local trends in firing (i.e., monotonically increasing, monotonically decreasing, or transitional firing). The trajectory of these points in the serial-JISID plot traces the time evolution of these trends represented by a 5-spike pattern, while points in the JISID scatter plot represent trends of a 4-spike pattern. We provide complete theoretical interpretations of the JISID analysis. We also demonstrate that this method indeed identifies firing trends in both simulated spike trains and spike trains recorded from cultured neurons. Received: 13 May 1997 / Accepted in revised form: 9 December 1998