Spike-density Estimation

Histogram Method

Kernel Method

Binwidth optimization

Bandwidth optimization

Shimazaki & Shinomoto
Neural Computation, 2007

Shimazaki & Shinomoto
J. Comput. Neurosci. 2010

Abstract
Histogram and kernel method have been used as standard tools for
capturing the instantaneous rate of neuronal spike discharges in
Neurophysiological community.
These methods are left with one free parameter that determines thesmoothness of the estimated rate, namely a binwidth or a bandwidth.
In most of the neurophysiological literature, however, the binwidth or thebandwidth that critically determines the goodness of the fit of the estimatedrate to the underlying rate has been selected by individual researchers in an
unsystematic manner.
Here we established a method for optimizing the histogram binwidth as wellas the kernel bandwidth, with which the estimated rate best approximates theunknown underlying rate.

Histogram Optimization
Unknown

Underlying Rate
Data

Histogram
Estimate

4

2

1

4

3

1
Shimazaki & Shinomoto Neural Computation, 2007

Extrapolation Method
We have n=30 Sequneces

Original:

Optimal bin size diverges

Extrapolated:
Finite optimal bin sizeOptimal bin size v.s. m

Required # of trials

Required # ofsequences(Estimation)

Data Size Used

# of sequences used

Extrapolation Method
Optimized Histogram

Too few to makea Histogram !
Extrapolation

Estimation:
At least 12 trials are required.
Data : Britten et al. (2004) neural signal archive

Spike Rate [spike/s]

Kernel Optimization
λt

Underlying rate

20012

Spike Sequences

Cn (w )

Cost function−100

Theoretical cost function

Empirical cost function

Too small w
0λˆ t

12−140Too small w

40Too large w

Spike Rate [spike/s]

20−180λˆ tOptimized w *Optimized w *2000.1

0.2

0.3

0.4

Bandwidth w [s]λˆ t

Too large w

2001Time [s]2Shimazaki & Shinomoto, J. Comp. Neurosci under review

Kernel Optimization Formula
For a Gauss kernel function,w
find w that minimizes the formula,
Total number of spikes

Bandwidth

Spike Time
Shimazaki & Shinomoto, J. Comp. Neurosci

Variable Kernel Method40

Spike Rate [spike/s]

Underlying Rate
Optimized variable kernel method (γ∗ = 0.8)30

Variable kernel method (γ = 0.1)

20Cost function
Cn ( γ )10Empirical cost funtion
Theoretical cost funtion

0012Bandwidth [s]Variable bandwidth
0.2

Optimized γ∗
Too small γ

0.1−185Too small γ

0012

Optimized γ∗

Spike Sequences

−190
00.2

0.4

0.6

0.8

1Stiffness constant γ

012Time [s]Our method automatically adjusts the stiffness of bandwidth variability
Shimazaki & Shinomoto, J. Comp. Neurosci.

Performance Comparison
40Sinusoid
40040

Sawtooth

2001Time [s]2HISTOGRAMFIXED KERNELVARIABLE KERNELABRAMSON’SLOCFITBARS300

ABRAMSON’SLOCFIT
BARS

BadHistogram
Our Methods

200Kernel

Good100

Underlying RateHISTOGRAM
FIXED KERNELVARIABLE KERNEL

Integrated Squared Error (Sawtooth)

Spike Rate [spikes/s]

20050

Adaptive Kernel
100150200250Integrated Squared Error (Sinusoid)

Kernel method is comparable to, or even better than modern methods.
Shimazaki & Shinomoto, J. Comp. Neurosci.

Appendix

Histogram construction
Time-Varying Rate

The mean underlying rate in an interval [0, Δ]:

Spike Sequences

The spike count in the bin obeys the Poisson
distribution*:

Time Histogram
A histogram bar-height is an estimator of θ :*When the spikes are obtained by repeating an independent trial, the accumulated
data obeys the Poisson point process due to a general limit theorem.

Theory for Histogram Optimization

Expectation by the Poisson
statistics, given the rate.

Sampling Error

Average oversegmented bins.

Systematic Error

Decomposition of the Systematic Error

Systematic Error

Variance of the rate
Independent of Δ

Variance ofan ideal histogram

Introduction of the cost function:

Sampling error

Unknown: Variance of an ideal histogram

The variance decomposition:
Variance of a histogram

Sampling error

The Poisson statistics obeys:

Mean of a Histogram

Variance of a Histogram