Estimating the firing rate

Tahereh ToosiIPM

Brief Review of Spike Train Analysis

10:30-11:30 12-13

Thursday, 31 JanEstimating the Firing Rate of Spike

TrainsTahereh Toosi

Introduction to Parameter Estimation

HaDi MaBouDi

Thursday, 7 Feb Spike-Train StatisticsEhsan Sabri

Entropy and Information TheoryTahereh Toosi

Thursday, 14 Feb Spike-Train Encoding and DecodingSafura Rashid Shomali

Statistical models of neural dataHaDi MaBouDi

Thursday, 21 FebAn Introductory to Information

Geometry of Spike TrainsHaDi 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]

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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

8“Analysis of Parallel Spike Trains”, Chapter 2 Estimating the Firing Rate, S. Shinomoto

Methods for estimating instantaneous rate

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Challenges to rate estimation

PSTH Method• Peristimulus time histogram

Methods for estimating the firing rate

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PSTH Method

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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).

Kernel Density Function Method

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Methods for estimating the firing rate

Methods for Optimizing the Rate Estimation

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MISE• Minimizing Mean Integrated Squared Error

• Assumption on r(t) : • spikes are drawn from nonhomogeneous Possion process

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Methods for optimizing rate estimation

[Shimazaki and Shinomoto, 2007]

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

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Methods for optimizing rate estimation

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)

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

probability of having no spike from the time t1 to t2 :

Bayes rule gives the “inverse probability”:

flatness

the estimated rate becomesin/sensitive to individual spike occurrences as β is large/small

The probability of having spikes at {ti} ≡ {t1, t2, . . . , tNs} is given by the “marginallikelihood function”:

Comparison of the optimized KDand Empirical Bayes rate estimators

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Methods for optimizing rate estimation

Empirical Bayes methodMethods for optimizing rate estimation

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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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