Transcript Document

Tahereh Toosi
IPM
Brief Review of Spike Train Analysis
10:30-11:30
12-13
Thursday, 31 Jan
Estimating the Firing Rate of Spike
Trains
Tahereh Toosi
Introduction to Parameter
Estimation
HaDi MaBouDi
Thursday, 7 Feb
Spike-Train Statistics
Ehsan Sabri
Entropy and Information Theory
Tahereh Toosi
Thursday, 14 Feb
Spike-Train Encoding and Decoding
Safura Rashid Shomali
Statistical models of neural data
HaDi MaBouDi
Thursday, 21 Feb
An Introductory to Information
Geometry of Spike Trains
HaDi MaBouDi
Population coding: Ising model and
GLM
Safura Rashid Shomali
Thursday, 28 Feb,
Works on Real Data!!!
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Outline
• Extracting Spikes
• Spike Sorting
• Neuronal coding types
• Estimating the firing rate
• Optimizing the rate estimation
• Summary
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Neural recordings
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Spike Sorting
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[NeuroQuest]
Neural Coding
• Three Coding schemes
• Rate coding
• Spike-count rate
• Time-dependent firing rate
• Temporal coding
• Phase-of-firing code
• Spike Latency codes
• Population coding
• Position coding
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Estimating the firing rate
• Methods for estimating the firing rate
• PSTH
• The Kernel Density Estimation
• Methods for optimizing the rate
estimation
• Minimizing Mean Squared error (MISE)
• Maximizing likelihood
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“Analysis of Parallel Spike Trains”, Chapter 2
Estimating the Firing Rate, S. Shinomoto
Methods for estimating instantaneous
rate
Challenges to rate estimation
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PSTH Method
Methods for estimating the firing rate
• Peristimulus time histogram
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PSTH Method
Methods for estimating the firing rate
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Methods for estimating the firing rate
Kernel Density Function Method
Kernel features:
1. the normalization to unit area, f (t)dt = 1.
2. nonnegative,f (t) ≥ 0,
3. have a finite bandwidth defined by the variance that
is normally finite, 2 = t2f (t)dt <∞,
4. symmetric, f (t) = f (−t).
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Methods for estimating the firing rate
Kernel Density Function Method
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Methods for Optimizing the Rate
Estimation
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Methods for optimizing rate estimation
MISE
• Minimizing Mean Integrated Squared Error
• Assumption on r(t) :
• spikes are drawn from nonhomogeneous Possion process
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[Shimazaki and Shinomoto, 2007]
Methods for optimizing rate estimation
MISE for PSTH
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Methods for optimizing rate estimation
MISE : results
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Methods for optimizing rate estimation
MISE for Kernel Density Function
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Methods for optimizing rate estimation
MISE : Comparison of the optimized PSTH and
optimized kernel density estimator
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Methods for optimizing rate estimation
Maximum likelihood
• Time- dependent Poisson process
The rate-modulated Poisson process.
The probability for a spike to occur in each short interval δt is r(t)δt<< 1,
and the probability of having no spike is 1− r(t)δt ≈ exp(−r(t)δt)
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Methods for optimizing rate estimation
probability of having no spike from the time t1 to t2 :
Bayes rule gives the “inverse probability”:
flatness
the estimated rate becomes
in/sensitive to individual spike
occurrences as β is large/small
The probability of having spikes at {ti} ≡ {t1, t2, . . . , tNs} is given by the “marginal
likelihood function”:
obtaining the maximum a posteriori (MAP) estimate of the rate ˆr(t), so that their
posterior distribution
maximized.
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Methods for optimizing rate estimation
Comparison of the optimized KD
and Empirical Bayes rate estimators
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Methods for optimizing rate estimation
Empirical Bayes method
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Summary
• Neuronal activity is measured by the number of spikes
• Challenges to grasping the time-varying rate of spike firing
• Binsize of the time histogram
• Bandwidth of the kernel smoother
• Standard rate estimation tools, such as
• the peri-stimulus time histogram (PSTH)
• kernel density estimation
• Optimization of rate estimation
• MISE
• Maximum likelihood
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