Estimating Spiking Irregularities Under Changing Environments

Abstract We considered a gammadistribution of interspike intervals as a statistical model for neuronal spike generation. A gamma distribution is a natural extension of the Poisson process taking the effect of a refractory period into account. The model is specified by two parameters: a time-dependent firing rate and a shape parameter that characterizes spiking irregularities of individual neurons. Because the environment changes over time, observed data are generated from a model with a time-dependent firing rate, which is an unknown function. A statistical model with an unknown function is called a semiparametric model and is generally very difficult to solve. We used a novel method of estimating functions in information geometry to estimate the shape parameter without estimating the unknown function. We obtained an optimal estimating function analytically for the shape parameter independent of the functional form of the firing rate. This estimation is efficient without Fisher information loss and better than maximum likelihood estimation. We suggest a measure of spiking irregularity based on the estimating function, which may be useful for characterizing individual neurons in changing environments.

[1]  J. Neyman,et al.  Consistent Estimates Based on Partially Consistent Observations , 1948 .

[2]  V. P. Godambe.  An Optimum Property of Regular Maximum Likelihood Estimation , 1960 .

[3]  D. Cox,et al.  The statistical analysis of series of events , 1966 .

[4]  D. Cox,et al.  The statistical analysis of series of events , 1966 .

[5]  V. P. Godambe.  Conditional likelihood and unconditional optimum estimating equations , 1976 .

[6]  S. Amari.  Differential Geometry of Curved Exponential Families-Curvatures and Information Loss , 1982 .

[7]  Shun-ichi Amari,et al.  Differential-geometrical methods in statistics , 1985 .

[8]  S. Amari,et al.  Estimation in the Presence of Infinitely many Nuisance Parameters--Geometry of Estimating Functions , 1988 .

[9]  J. Pfanzagl.  Estimation in semiparametric models , 1990 .

[10]  P. Bickel,et al.  Achieving Information Bounds in Non and Semiparametric Models , 1990 .

[11]  Shun-ichi Amari,et al.  Information geometry of Boltzmann machines , 1992, IEEE Trans. Neural Networks.

[12]  J. Wellner,et al.  Information Bounds and Nonparametric Maximum Likelihood Estimation , 1992 .

[13]  P. Bickel.  Efficient and Adaptive Estimation for Semiparametric Models , 1993 .

[14]  M. Murray,et al.  Differential Geometry and Statistics , 1993 .

[15]  K. Do,et al.  Efficient and Adaptive Estimation for Semiparametric Models. , 1994 .

[16]  William R. Softky,et al.  Comparison of discharge variability in vitro and in vivo in cat visual cortex neurons. , 1996, Journal of neurophysiology.

[17]  S. Amari,et al.  Information geometry of estimating functions in semi-parametric statistical models , 1997 .

[18]  B. Knight,et al.  The Power Ratio and the Interval Map: Spiking Models and Extracellular Recordings , 1998, The Journal of Neuroscience.

[19]  Shun-ichi Amari,et al.  Natural Gradient Works Efficiently in Learning , 1998, Neural Computation.

[20]  Yutaka Sakai,et al.  Temporally correlated inputs to leaky integrate-and-fire models can reproduce spiking statistics of cortical neurons , 1999, Neural Networks.

[21]  Shun-ichi Amari,et al.  Methods of information geometry , 2000 .

[22]  R N Lemon,et al.  Precise spatiotemporal repeating patterns in monkey primary and supplementary motor areas occur at chance levels. , 2000, Journal of neurophysiology.

[23]  Emery N. Brown,et al.  The Time-Rescaling Theorem and Its Application to Neural Spike Train Data Analysis , 2002, Neural Computation.

[24]  Julian Havil.  Gamma: Exploring Euler's Constant , 2003 .

[25]  Shigeru Shinomoto,et al.  Differences in Spiking Patterns Among Cortical Neurons , 2003, Neural Computation.

[26]  Parimal Mukhopadhyay,et al.  An Introduction to Estimating Functions , 2004 .

[27]  Kazushi Ikeda.  Information Geometry of Interspike Intervals in Spiking Neurons , 2005, Neural Computation.

[28]  H. Tamura,et al.  Regional and laminar differences in in vivo firing patterns of primate cortical neurons. , 2005, Journal of neurophysiology.

[29]  S. Shinomoto,et al.  A measure of local variation of inter-spike intervals. , 2005, Bio Systems.

[30]  Shigeru Shinomoto,et al.  A solution to the controversy between rate and temporal coding , 2007, Statistics in medicine.