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