Spike-density Estimation

Histogram Method Kernel Method

Shimazaki & Shinomoto Neural Computation, 2007

Shimazaki & Shinomoto J. Comput. Neurosci. 2010

Binwidth optimization Bandwidth optimization

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 the smoothness of the estimated rate, namely a binwidth or a bandwidth. In most of the neurophysiological literature, however, the binwidth or the bandwidth that critically determines the goodness of the fit of the estimated rate 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 well as the kernel bandwidth, with which the estimated rate best approximates the unknown underlying rate.

Unknown

Histogram Underlying Rate

Estimate Data

Histogram Optimization

4 2 1 4 3 1

Shimazaki & Shinomoto Neural Computation, 2007

Extrapolation Method

Extrapolated: Finite optimal bin size

Original: Optimal bin size diverges

Required # of sequences (Estimation)

Optimal bin size v.s. m

Data Size Used # of sequences used

Required # of trials

We have n=30 Sequneces

Data : Britten et al. (2004) neural signal archive

Extrapolation Method

Too few to make a Histogram !

Extrapolation

Estimation: At least 12 trials are required.

Optimized Histogram

Shimazaki & Shinomoto, J. Comp. Neurosci under review

Kernel Optimization

0 0.1 0.2 0.3 0.4

−180

−140

−100

0 1 2

20

20

40

20

0 1 2

20

0 1 2

Optimized

Too large w

( )nC w

Bandwidth w [s]

Cost function

Underlying rate

Too small w

Too large w

ˆ tλ

ˆ tλ

ˆ tλ

tλ

Empirical cost functionTheoretical cost function

w *Optimized w*

Too small w

Time [s]

Spike SequencesS

pike

Rat

e [s

pike

/s]

Spi

ke R

ate

[spi

ke/s

]

w

Kernel Optimization Formula

Bandwidth Spike Time

Total number of spikes

Shimazaki & Shinomoto, J. Comp. Neurosci

For a Gauss kernel function,

find w that minimizes the formula,

Variable Kernel Method

Shimazaki & Shinomoto, J. Comp. Neurosci.

0 10

10

20

30

40

0

0.1

0.2

0 1 2

0 0.2 0.4 0.6 0.8 1−190

−185

2

0 1 2

Variable bandwidth

( )nC γ

Stiffness constant γ

Optimized variable kernel method (γ∗ = 0.8)

Cost function

Variable kernel method (γ = 0.1)

Optimized γ∗ Too small γ

Underlying Rate

Time [s]

Theoretical cost funtionEmpirical cost funtion

Ban

dwid

th [s

]S

pike

Rat

e [s

pike

/s]

Optimized γ∗

Too small γ

Spike Sequences

Our method automatically adjusts the stiffness of bandwidth variability

Performance Comparison

20

40

20

40

0 21

0 50 100 150 200 250100

200

300

400

Time [s]

Underlying RateHISTOGRAM

LOCFITABRAMSON’S

BARSFIXED KERNELVARIABLE KERNEL

Sinusoid

Sawtooth

Spi

ke R

ate

[spi

kes/

s]

HISTOGRAMFIXED KERNELVARIABLE KERNELABRAMSON’S

BARSLOCFIT

Integrated Squared Error (Sinusoid)In

tegr

ated

Squ

ared

Err

or (S

awto

oth)

Shimazaki & Shinomoto, J. Comp. Neurosci.

Kernel method is comparable to, or even better than modern methods.

Good

Bad

Our Methods

Histogram

Kernel

Adaptive Kernel

Appendix

Time-Varying Rate

Spike Sequences

Time Histogram

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

A histogram bar-height is an estimator of θ :

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

*When the spikes are obtained by repeating an independent trial, the accumulated data obeys the Poisson point process due to a general limit theorem.

Histogram construction

Expectation by the Poisson statistics, given the rate.

Variance of the rate Variance of an ideal histogram

Decomposition of the Systematic Error

Systematic Error Sampling Error

Average over segmented bins.

Independent of Δ Systematic Error

Theory for Histogram Optimization

Variance of a Histogram

Introduction of the cost function:

The variance decomposition:

Mean of a Histogram

The Poisson statistics obeys:

Unknown: Variance of an ideal histogram Sampling error

Variance of a histogram Sampling error