Local Field Potentials, Spikes and Modeling Strategies for Both

Local Field Potentials, Spikes and Modeling Strategies for

Both

Local Field Potentials, Spikes and Modeling Strategies for

BothRobert HaslingerRobert Haslinger

Dept. of Brain and Cog. Sciences: MITDept. of Brain and Cog. Sciences: MITMartinos Center for Biomedical Imaging: MGHMartinos Center for Biomedical Imaging: MGH

Robert HaslingerRobert HaslingerDept. of Brain and Cog. Sciences: MITDept. of Brain and Cog. Sciences: MIT

Martinos Center for Biomedical Imaging: MGHMartinos Center for Biomedical Imaging: MGH

OutlineOutline

1.1. Extra-cellular electric potentials, their Extra-cellular electric potentials, their origin, and their filtering origin, and their filtering

2.2. Local Field Potential and Continuous Local Field Potential and Continuous Models Models

3.3. Spikes and Generalized Linear ModelsSpikes and Generalized Linear Models4.4. Example of GLM modeling in rat barrel Example of GLM modeling in rat barrel

cortexcortex

1.1. Extra-cellular electric potentials, their Extra-cellular electric potentials, their origin, and their filtering origin, and their filtering

2.2. Local Field Potential and Continuous Local Field Potential and Continuous Models Models

3.3. Spikes and Generalized Linear ModelsSpikes and Generalized Linear Models4.4. Example of GLM modeling in rat barrel Example of GLM modeling in rat barrel

cortexcortex

The Brain is a Complex Structured Network of NeuronsThe Brain is a Complex Structured Network of Neurons

Neural ActivityNeural Activity Neurons have a ~ - 70 mV potential gradient across their membranesNeurons have a ~ - 70 mV potential gradient across their membranes Synaptic activity can depolarize the membraneSynaptic activity can depolarize the membrane Enough depolarization leads to a sharp (1msec 80 mV) change in Enough depolarization leads to a sharp (1msec 80 mV) change in

potential (spike) which propagates down axons to other neurons potential (spike) which propagates down axons to other neurons

Neurons have a ~ - 70 mV potential gradient across their membranesNeurons have a ~ - 70 mV potential gradient across their membranes Synaptic activity can depolarize the membraneSynaptic activity can depolarize the membrane Enough depolarization leads to a sharp (1msec 80 mV) change in Enough depolarization leads to a sharp (1msec 80 mV) change in

potential (spike) which propagates down axons to other neurons potential (spike) which propagates down axons to other neurons

synapses, leak currents, capacitive currents, spikes etc. all move charge across synapses, leak currents, capacitive currents, spikes etc. all move charge across the neural membrane: the neural membrane: GENERATE ELECTRIC POTENTIALGENERATE ELECTRIC POTENTIAL

synapses, leak currents, capacitive currents, spikes etc. all move charge across synapses, leak currents, capacitive currents, spikes etc. all move charge across the neural membrane: the neural membrane: GENERATE ELECTRIC POTENTIALGENERATE ELECTRIC POTENTIAL

What do we actually measure ?What do we actually measure ?


Experiments record Experiments record extra-cellular voltage extra-cellular voltage changeschanges

Voltage changes Voltage changes generated by generated by movement of charge movement of charge (Na, K, Ca, Cl) across (Na, K, Ca, Cl) across neuronal membranesneuronal membranes






Generally extra-cellular Generally extra-cellular voltage is filtered into voltage is filtered into two types of signals: two types of signals: spikes and LFPspikes and LFP



Generally extra-cellular Generally extra-cellular voltage is filtered into voltage is filtered into two types of signals: two types of signals: spikes and LFPspikes and LFP

Local Field Potential

Local Field Potential (LFP)Local Field Potential (LFP)

Low pass filtered (0.1 -Low pass filtered (0.1 -250 Hz) signal250 Hz) signal

““slower” processes, slower” processes, synapses, leak synapses, leak currents, capacitive currents, capacitive currents etc.currents etc.



Haslinger & Neuenschwander

Local Field Potential (LFP)Local Field Potential (LFP)





Haslinger & Devor

Local Field PotentialLocal Field Potential

LFP generated through LFP generated through current “sinks” and current “sinks” and

“sources”“sources”

LFP generated through LFP generated through current “sinks” and current “sinks” and

“sources”“sources”


LFP generated through LFP generated through current “sinks” and current “sinks” and “sources”“sources”

LFP generated through LFP generated through current “sinks” and current “sinks” and “sources”“sources”


LFP generated through LFP generated through current “sinks” and current “sinks” and “sources” “sources”







Can think of charge Can think of charge imbalances creating imbalances creating extracellular voltageextracellular voltage






Or can think in terms of Or can think in terms of voltage drops due to voltage drops due to current loops , e.g. current loops , e.g. Ohm’s Law (V=IR)Ohm’s Law (V=IR)



Or can think in terms of Or can think in terms of voltage drops due to voltage drops due to current loops , e.g. current loops , e.g. Ohm’s Law (V=IR)Ohm’s Law (V=IR)

current loop

LFP results from the superposition of potentials from ALL sinks and sourcesLFP results from the superposition of potentials from ALL sinks and sources

We only see sinks and We only see sinks and source pairs that don’t source pairs that don’t overlap with each other, or overlap with each other, or with sinks and sources from with sinks and sources from other neurons.other neurons.

Elongated pyramidal Elongated pyramidal neurons, YES.neurons, YES.

Compact interneurons or Compact interneurons or layer IV stellate cells : NOlayer IV stellate cells : NO

Synchronous (across cells) Synchronous (across cells) events: YESevents: YES

Sinks not always excitatory, Sinks not always excitatory, sources not always inhibitorysources not always inhibitory

We only see sinks and We only see sinks and source pairs that don’t source pairs that don’t overlap with each other, or overlap with each other, or with sinks and sources from with sinks and sources from other neurons.other neurons.

Elongated pyramidal Elongated pyramidal neurons, YES.neurons, YES.

Compact interneurons or Compact interneurons or layer IV stellate cells : NOlayer IV stellate cells : NO

Synchronous (across cells) Synchronous (across cells) events: YESevents: YES

Sinks not always excitatory, Sinks not always excitatory, sources not always inhibitorysources not always inhibitory

LFP Changes With PositionLFP Changes With Position

LFP Is Not Well Localized SpatiallyLFP Is Not Well Localized Spatially LFP localized within .5 - LFP localized within .5 -

3 mm (at best)3 mm (at best) V~1/r … but in cortex its V~1/r … but in cortex its

worse !!!!worse !!!! Caused by dendritic and Caused by dendritic and

cortical geometrycortical geometry

LFP localized within .5 - LFP localized within .5 - 3 mm (at best)3 mm (at best)

V~1/r … but in cortex its V~1/r … but in cortex its worse !!!!worse !!!!

Caused by dendritic and Caused by dendritic and cortical geometrycortical geometry

Pseudo 1D GeometryPseudo 1D Geometry

Cortex is a thin (2-3 mm thick) sheetCortex is a thin (2-3 mm thick) sheet Greatest anatomical variation perpendicular to the Greatest anatomical variation perpendicular to the

sheetsheet Essentially a Essentially a ONE DIMENSIONALONE DIMENSIONAL geometry geometry In 1D , V ~ z , not 1/r In 1D , V ~ z , not 1/r

Cortex is a thin (2-3 mm thick) sheetCortex is a thin (2-3 mm thick) sheet Greatest anatomical variation perpendicular to the Greatest anatomical variation perpendicular to the

sheetsheet Essentially a Essentially a ONE DIMENSIONALONE DIMENSIONAL geometry geometry In 1D , V ~ z , not 1/r In 1D , V ~ z , not 1/r

LFP is Non-LocalLFP is Non-Local



Characterizing LFPCharacterizing LFP

Highly complex continuous time signalHighly complex continuous time signal Intuition about phenomena at different Intuition about phenomena at different

time scalestime scalesCan apply all sorts of signal processing Can apply all sorts of signal processing

techniquestechniquesHelps to have some idea of what you’re Helps to have some idea of what you’re

looking forlooking for

Highly complex continuous time signalHighly complex continuous time signal Intuition about phenomena at different Intuition about phenomena at different

time scalestime scalesCan apply all sorts of signal processing Can apply all sorts of signal processing

techniquestechniquesHelps to have some idea of what you’re Helps to have some idea of what you’re

looking forlooking for

Signal Processing TechniquesSignal Processing Techniques

Fourier and Windowed Fourier Transform (multi-Fourier and Windowed Fourier Transform (multi-taper) (Chronux)taper) (Chronux)

Wavelets and Multi-Resolution AnalysisWavelets and Multi-Resolution Analysis Time - Frequency representationTime - Frequency representation Hilbert TransformHilbert Transform Power, PhasePower, Phase Empirical Mode DecompositionEmpirical Mode Decomposition Autoregressive ModelingAutoregressive Modeling Coherency Analysis (Chronux)Coherency Analysis (Chronux) Granger CausalityGranger Causality Information TheoryInformation Theory

Fourier and Windowed Fourier Transform (multi-Fourier and Windowed Fourier Transform (multi-taper) (Chronux)taper) (Chronux)

Wavelets and Multi-Resolution AnalysisWavelets and Multi-Resolution Analysis Time - Frequency representationTime - Frequency representation Hilbert TransformHilbert Transform Power, PhasePower, Phase Empirical Mode DecompositionEmpirical Mode Decomposition Autoregressive ModelingAutoregressive Modeling Coherency Analysis (Chronux)Coherency Analysis (Chronux) Granger CausalityGranger Causality Information TheoryInformation Theory

Frequency - ologyFrequency - ology

Alpha (8-12 Hz) attention Alpha (8-12 Hz) attention Beta (12-20 Hz) Beta (12-20 Hz) Gamma (40-80 Hz) complex Gamma (40-80 Hz) complex

processing, mediated by inhibitionprocessing, mediated by inhibitionDelta (1-4 Hz) slow wave sleepDelta (1-4 Hz) slow wave sleepMu (8-12 Hz) but in motor cortexMu (8-12 Hz) but in motor cortexTheta (4-8 Hz) HippocampusTheta (4-8 Hz) Hippocampus

Alpha (8-12 Hz) attention Alpha (8-12 Hz) attention Beta (12-20 Hz) Beta (12-20 Hz) Gamma (40-80 Hz) complex Gamma (40-80 Hz) complex

processing, mediated by inhibitionprocessing, mediated by inhibitionDelta (1-4 Hz) slow wave sleepDelta (1-4 Hz) slow wave sleepMu (8-12 Hz) but in motor cortexMu (8-12 Hz) but in motor cortexTheta (4-8 Hz) HippocampusTheta (4-8 Hz) Hippocampus

Statistical Modeling of LFPStatistical Modeling of LFP

SystemEEG, LFPcovariates

( | )p neural activity covariates

Y X β ε= +Linear Regression(Gaussian Model of Variability)

Many standard methods for regression, model selection, goodness of fit and so forth

Spikes

Spikes : “high pass filtered”Spikes : “high pass filtered” Extra-cellular voltage is high pass filtered and Extra-cellular voltage is high pass filtered and

discrete spikes are identified through spike sortingdiscrete spikes are identified through spike sorting Extra-cellular voltage is high pass filtered and Extra-cellular voltage is high pass filtered and

discrete spikes are identified through spike sortingdiscrete spikes are identified through spike sorting

Neurons generating spikes Neurons generating spikes are located near the are located near the electrodeelectrode

See spikes from all types of See spikes from all types of neurons (pyramids, neurons (pyramids, interneurons etc.) interneurons etc.)

Functional distinctions Functional distinctions based on spike shape (FS = based on spike shape (FS = inhibitory, RS = excitatory)inhibitory, RS = excitatory)

Neurons generating spikes Neurons generating spikes are located near the are located near the electrodeelectrode

See spikes from all types of See spikes from all types of neurons (pyramids, neurons (pyramids, interneurons etc.) interneurons etc.)

Functional distinctions Functional distinctions based on spike shape (FS = based on spike shape (FS = inhibitory, RS = excitatory)inhibitory, RS = excitatory)

Spikes are discrete events Spikes are discrete events

Smooth into spike rate - continuous processSmooth into spike rate - continuous process Interspike interval distribution (ISI)Interspike interval distribution (ISI) Spectral techniques (multi-taper)Spectral techniques (multi-taper)

Smooth into spike rate - continuous processSmooth into spike rate - continuous process Interspike interval distribution (ISI)Interspike interval distribution (ISI) Spectral techniques (multi-taper)Spectral techniques (multi-taper)

Spikes are discrete events Spikes are discrete events

Smooth into spike rate - continuous processSmooth into spike rate - continuous process Interspike interval distribution (ISI)Interspike interval distribution (ISI) Spectral techniques (multi-taper)Spectral techniques (multi-taper) Point Process Statistical ModelingPoint Process Statistical Modeling

Smooth into spike rate - continuous processSmooth into spike rate - continuous process Interspike interval distribution (ISI)Interspike interval distribution (ISI) Spectral techniques (multi-taper)Spectral techniques (multi-taper) Point Process Statistical ModelingPoint Process Statistical Modeling

Introducing Generalized Linear ModelsIntroducing Generalized Linear Models

SystemEEG, LFPcovariates

( | )p neural activity covariates

Y X β ε= +Linear Regression(Gaussian Model of Variability)

Systemspikescovariates

Conditional Intensity FunctionConditional Intensity Function

Spikes depend upon both external covariates (stimuli) and the previous history of the spiking process

(t) = ( x(t) | Ht )

(t) dt is the probability of a spike conditioned on the past spiking history Ht and a function of the external covariates (stimuli) x(t)

Conditional Intensity FunctionConditional Intensity Function

Spikes depend upon both external covariates (stimuli) and the previous history of the spiking process

(t) = ( x(t) | Ht )

(t) dt is the probability of a spike conditioned on the past spiking history Ht and a function of the external covariates (stimuli) x(t)

Our goal in statistical modeling is to get (t). Once we know that, we know “everything” (probability of any spike sequence for example)

Regression for Event-Like DataRegression for Event-Like Data

““Standard” regression (linear or non-linear) Standard” regression (linear or non-linear) assumes continuous data and Gaussian assumes continuous data and Gaussian noisenoise

Spikes are localized events, we should Spikes are localized events, we should respect the nature of the datarespect the nature of the data

A statistical model can be used for A statistical model can be used for inference, inference, prediction, decoding and simulationprediction, decoding and simulation

There are standard techniques for modeling There are standard techniques for modeling point process data, e.g. point process data, e.g. logistic regressionlogistic regression and other and other Generalized Linear ModelsGeneralized Linear Models

““Standard” regression (linear or non-linear) Standard” regression (linear or non-linear) assumes continuous data and Gaussian assumes continuous data and Gaussian noisenoise

Spikes are localized events, we should Spikes are localized events, we should respect the nature of the datarespect the nature of the data

A statistical model can be used for A statistical model can be used for inference, inference, prediction, decoding and simulationprediction, decoding and simulation

There are standard techniques for modeling There are standard techniques for modeling point process data, e.g. point process data, e.g. logistic regressionlogistic regression and other and other Generalized Linear ModelsGeneralized Linear Models

Linear vs. Logistic RegressionLinear vs. Logistic Regression

(t) = i i xi(t)

restricted only by range of {xi}


(t) = i i xi(t)


log[ (t) / (1 - (t) ] = i i xi(t)

is restricted between 0 and 1

is a PROBABILITY


(t) = i i xi(t)


log[ (t) / (1 - (t) ] = i i xi(t)

is restricted between 0 and 1

is a PROBABILITY

LINK FUNCTION

Generalized Linear ModelsGeneralized Linear Models

Logistic regression is one example of a Logistic regression is one example of a Generalized Linear Model (GLM)Generalized Linear Model (GLM)

Can be solved by Can be solved by maximum likelihood maximum likelihood estimationestimation (log-concave problem) (log-concave problem)

There exist efficient estimation techniques There exist efficient estimation techniques (iterative re-weighted least squares)(iterative re-weighted least squares)

They can be solved in Matlab (glmfit.m) and They can be solved in Matlab (glmfit.m) and almost all statistical packagesalmost all statistical packages

Logistic regression is one example of a Logistic regression is one example of a Generalized Linear Model (GLM)Generalized Linear Model (GLM)

Can be solved by Can be solved by maximum likelihood maximum likelihood estimationestimation (log-concave problem) (log-concave problem)

There exist efficient estimation techniques There exist efficient estimation techniques (iterative re-weighted least squares)(iterative re-weighted least squares)

They can be solved in Matlab (glmfit.m) and They can be solved in Matlab (glmfit.m) and almost all statistical packagesalmost all statistical packages

Possible Covariates to IncludePossible Covariates to Include

log[ (t) / (1 - (t)) ] = i i fi (stimulus)



j j gj (spiking history)




k k hk (ensemble spiking)





p p rp (LFP)





p p rp (LFP)

Fitted parameters give the importance of different contributions

Goodness-of-FitGoodness-of-FitGoodness-of-FitGoodness-of-Fit

1

( | )i

i

t

i utz u H duλ+=∫

Time Rescalenttt ..., 21 nzzz ..., 21

Time-Rescaling Theorem: zi’s are i.i.d. exponential rate 1

Kolmogorov-Smirnov (KS) Plot:EC

DF(z

i)

CDF(exp(1))

GLM Example : Rat Barrel Cortex

M. Andermann

Inclusion of Different Covariates

Time since stimulus i i Bi (t)

spline basis functions


Time since stimulus

Deflection Angle

i i Bi (t)


cos ( 0) =

1 cos( ) - 2 sin( )


Time since stimulus

Deflection Angle

Spike History

i i Bi (t)


cos ( 0) =

1 cos( ) - 2 sin( )

j j gj ( t - tj )

g(t) = 0 (no spike at t)

= 1 (spike at t)


log[ (t) / (1 - (t) ) ] = i i Bi (t)

+ 1 cos( ) - 2 sin( )

j j gj ( t - tj )

FINAL MODEL

log[ (t) / (1 - (t) ) ] = i i Bi (t)

+ 1 cos( ) - 2 sin( )

j j gj ( t - tj )

FINAL MODEL

History Term

Often spike history effects account for most of the statistics !!!!!!!

log[ (t) / (1 - (t) ) ] = i i Bi (t)

+ 1 cos( ) - 2 sin( )

j j gj ( t - tj )

FINAL MODEL

History Term

Often spike history effects account for most of the statistics !!!!!!!

refractory

bursting

ConclusionsConclusions

Important to understand the physical origins of what we record Important to understand the physical origins of what we record and modeland model

LFP and Spikes are fundamentally different types of data and LFP and Spikes are fundamentally different types of data and require different modeling strategiesrequire different modeling strategies

LFP requires some thought about what to model, but techniques LFP requires some thought about what to model, but techniques are standardare standard

Spikes effectively described by probability but are point Spikes effectively described by probability but are point processes and require different techniquesprocesses and require different techniques

Logistic Regression (and other GLMs) for spikes. Kolmogorov Logistic Regression (and other GLMs) for spikes. Kolmogorov Smirnov test for goodness of fitSmirnov test for goodness of fit

Rigorous model identification is important to determine the Rigorous model identification is important to determine the importance of different covariates.importance of different covariates.

This can be a prelude to developing more effective BMIsThis can be a prelude to developing more effective BMIs

Important to understand the physical origins of what we record Important to understand the physical origins of what we record and modeland model

LFP and Spikes are fundamentally different types of data and LFP and Spikes are fundamentally different types of data and require different modeling strategiesrequire different modeling strategies

LFP requires some thought about what to model, but techniques LFP requires some thought about what to model, but techniques are standardare standard

Spikes effectively described by probability but are point Spikes effectively described by probability but are point processes and require different techniquesprocesses and require different techniques

Logistic Regression (and other GLMs) for spikes. Kolmogorov Logistic Regression (and other GLMs) for spikes. Kolmogorov Smirnov test for goodness of fitSmirnov test for goodness of fit

Rigorous model identification is important to determine the Rigorous model identification is important to determine the importance of different covariates.importance of different covariates.

This can be a prelude to developing more effective BMIsThis can be a prelude to developing more effective BMIs

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Local Field Potentials, Spikes and Modeling Strategies for Both