Shinsuke Koyama

Empirical Bayes interpretations of random point events

Given a sequence of apparently random point events, such as neuronal spikes, one may interpret them as being derived either irregularly in time from a constant rate or regularly from a fluctuating rate. To determine which interpretation is more plausible in any given case, we employ the empirical Bayes method. For a sequence of point events derived from a rate-fluctuating gamma process, the marginal likelihood function can possess two local maxima and the system exhibits a first-order phase transition representing the switch of the most plausible interpretation from one to the other.

Histogram bin width selection for time-dependent Poisson processes

In constructing a time histogram of the event sequences derived from a nonstationary point process, we wish to determine the bin width such that the mean squared error of the histogram from the underlying rate of occurrence is minimized. We find that the optimal bin widths obtained for a doubly stochastic Poisson process and a sinusoidally regulated Poisson process exhibit different scaling relations with respect to the number of sequences, time scale and amplitude of rate modulation, but both diverge under similar parametric conditions. This implies that under these conditions, no determination of the time-dependent rate can be made. We also apply the kernel method to these point processes, and find that the optimal kernels do not exhibit any critical phenomena, unlike the time histogram method.

The effect of interspike interval statistics on the information gain under the rate coding hypothesis.

The question, how much information can be theoretically gained from variable neuronal firing rate with respect to constant average firing rate is investigated. We employ the statistical concept of information based on the Kullback-Leibler divergence, and assume rate-modulated renewal processes as a model of spike trains. We show that if the firing rate variation is sufficiently small and slow (with respect to the mean interspike interval), the information gain can be expressed by the Fisher information. Furthermore, under certain assumptions, the smallest possible information gain is provided by gamma-distributed interspike intervals. The methodology is illustrated and discussed on several different statistical models of neuronal activity.

On the spike train variability characterized by variance-to-mean power relationship

We propose a statistical method for modeling the non-Poisson variability of spike trains observed in a wide range of brain regions. Central to our approach is the assumption that the variance and the mean of interspike intervals are related by a power function characterized by two parameters: the scale factor and exponent. It is shown that this single assumption allows the variability of spike trains to have an arbitrary scale and various dependencies on the firing rate in the spike count statistics, as well as in the interval statistics, depending on the two parameters of the power function. We also propose a statistical model for spike trains that exhibits the variance-to-mean power relationship. Based on this, a maximum likelihood method is developed for inferring the parameters from rate-modulated spike trains. The proposed method is illustrated on simulated and experimental spike trains.

Information transmission using non-poisson regular firing

In many cortical areas, neural spike trains do not follow a Poisson process. In this study, we investigate a possible benefit of non-Poisson spiking for information transmission by studying the minimal rate fluctuation that can be detected by a Bayesian estimator. The idea is that an inhomogeneous Poisson process may make it difficult for downstream decoders to resolve subtle changes in rate fluctuation, but by using a more regular non-Poisson process, the nervous system can make rate fluctuations easier to detect. We evaluate the degree to which regular firing reduces the rate fluctuation detection threshold. We find that the threshold for detection is reduced in proportion to the coefficient of variation of interspike intervals.

Fluctuation scaling in neural spike trains.

Fluctuation scaling has been observed universally in a wide variety of phenomena. In time series that describe sequences of events, fluctuation scaling is expressed as power function relationships between the mean and variance of either inter-event intervals or counting statistics, depending on measurement variables. In this article, fluctuation scaling has been formulated for a series of events in which scaling laws in the inter-event intervals and counting statistics were related. We have considered the first-passage time of an Ornstein-Uhlenbeck process and used a conductance-based neuron model with excitatory and inhibitory synaptic inputs to demonstrate the emergence of fluctuation scaling with various exponents, depending on the input regimes and the ratio between excitation and inhibition. Furthermore, we have discussed the possible implication of these results in the context of neural coding.

On the relation between encoding and decoding of neuronal spikes

Sensory information is represented in neuronal responses. Determining which code is used by the neurons is important for understanding how the brain processes the information [1]. n nCoding schemes used by neurons can be divided approximately into two categories. In rate coding, information about the stimulus depends solely on the firing rate, which is the average number of spikes per unit time. In temporal coding, on the other hand, there is significant correlation between the stimulus and any moments in the spike pattern having higher order than the mean [2]. n nWhile neural codes are characterized in terms of these encoding schemes, i.e., how the neurons encode the stimulus into the features of spike responses, experimentalists can access the neural codes only through decoding. From the decoding viewpoint, rate coding is operationally defined by counting the number of spikes over a period of time, without taking into account any correlation structure among spikes. Any scheme based on such an operation is equivalent to decoding under the Poisson assumption, because the number of spikes over a period of time, or the sample mean, is a sufficient statistic of the rate parameter of a Poisson process. Similarly, temporal coding can be defined by decoding the stimulus using a statistical model with a correlation structure between spikes (such as the m-IMI model, introduced below). If such a decoder improves on the performance of the rate decoder, it indicates that significant information about the stimulus is carried on the temporal aspect of spike trains [3]. n n nWe introduce a simple statistical model that has a correlation structure, taking the intensity function of a point process to be a product of two factors: n n n n n(1) n n n nwhere s*(t) represents the last spike time preceding t. The statistical model with the intensity function (1) has been called the multiplicative inhomogeneous Markov interval (m-IMI) model [4]. ϕ(t) is the free firing rate, which depends only on the stimulus, and g(t- s*(t)) is the recovery function, which describes the dependency of the last spike time preceding t and hence allows the m-IMI model to have a correlation structure between spikes. Note that (1) becomes the intensity function of a Poisson process if the recovery function is constant in time. It has been reported that the m-IMI model enhances decoding performance in real data analysis [3], which encourages use of the m-IMI model to test temporal codes. n nAlthough neural codes can be defined in terms of either encoding or decoding, the resulting codes are generally different from one another. Here, we investigate the relation between the two viewpoints of neural coding in terms of rate and temporal coding schemes. Specifically, we investigate the extent to which decoders of each scheme decode rate and temporal codes that are defined in terms of encoding. Our main claim is that temporal decoding does not necessarily mean decoding a temporal code that the rate decoder fails to read, but also decoding certain rate codes with greater efficiency than the rate decoder.

Information gain on variable neuronal firing rate

The question of how much information can be theoretically gained from variable neuronal firing rate with respect to constant mean firing rate is investigated. For this purpose, we employ the Kullback-Leibler divergence as a measure of information gain. We first give a statistical interpretation of this information in terms of detectability of rate variation: the lower bound of detectable rate variation, below which the temporal variation of firing rate is undetectable with a Bayesian decoder, is entirely determined by this information [1]. We derive a formula for the lower bound, which tells how much information is necessary for the rate variation to be detected from spike trains. For instance, if a spike, on average, is expected to be observed in the characteristic timescale of the rate variation, it is necessary for the spike train to carry more than 0.36 bits per spike information so that the underlying rate variation is detectable. We show that the information depends not only of the variation of firing rates (i.e., signals), but also significantly on the dispersion properties of neuronal firing described by the shape of interspike interval (ISI) distribution (i.e., noise properties). It is shown that under certain condition, the gamma distribution attains the theoretical lower bound of the information among all ISI distributions when the coefficient of variation of ISIs is given [2]. We also give a useful formula for the (approximate) information, which provides a theoretical prediction for the range of the information gain. With the basis of the theoretical investigations, we propose a practical procedure for estimating the information from spike trains, and apply this method to biological spike data recorded from a cortical area. The estimated information ranges up to 0.8 bits/spike, which roughly matches the theoretical prediction. Acknowledgements The author was supported by JSPS KAKENHI Grant Number 24700287.