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Howdoyoudetermine thestimuliencodedbyaneuron?

• Getlucky/genious (lec 1– Hubel+Wiesel,Hollywood)• Havereallysimplesensor (lec 1- winddirecton)• Tryeverything• But…thereasmany16x16black/whiteimagesthanatomsintheuniverse

• Tryrandomthings,seewhat ‘lightsup’theneuron,andgeneralize!…Reverseengineeringthebrainviaspiketriggeredaverages

MultivariateStatistics• Probability:

• Say:probabilityofx• Mean:whatarethechancesofeventxhappening?• Example:whenyouroll ad6,whatistheprobability oflandinga5?

• ConditionalProbability:• Say:probabilityofxgiveny• Mean:giventheknowledgeofyhavinghappened, howprobable isx?• Example:whatistheprobability oflandinga5giventheroll wasover3?

• BayesInversion• Conditional probabilities canbe‘inverted’:

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Intuition• Considerasimpleexample,a‘colordetector’neuron

Stimuli:

Spikes:

Intuition:Orangedetector?

Time:

Goal:

Intuition• Considerasimpleexample,a‘colordetector’neuron

Stimuli:

Spikes:

Intuition:Orangedetector?

Time:

Goal: P (stimt|spiket)P (spiket)

P (stimt)

=~

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Intuition• Considerasimpleexample,a‘colordetector’neuron

Stimuli:

Spikes:

Intuition:Orangedetector?

Time:

…lots

Goal: P (stimt|spiket)P (spiket)

P (stimt)

=~

More complicatedsituation,2DGrayscale (independent, uniform)

Spiketriggeredaverage(STA)• Assumestimulusiszero-meanandcompletelyrandom(independent)

• Ifapixel‘drives’aneuron, itwill likelybepresentinstimuli evokingspikesThiswillresultinabiasofthatpixel inallstimuli thatevokedaspike

• Ifapixelisirrelevanttoneuron’s response, itmay/maynotbeinspikingstimuliSincepixelvaluesareindependent andzeromean,averagevalueis0

• Taketheexpectedvalueofeachpixelacrossspike-triggeredensemble• Spiketriggeredensemble:thesetofallstimuli thatevokedaspike

• Notetheconceptualsimilaritytotheprobability• TheSTAthengivesusanideaaboutneural activity

SpikingStimuli:

AverageStimuli:

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Spiketriggeredaverage(STA)• Let’s doanexamplefora1-D,temporal stimulus

Spiketriggeredaverage(STA)• Let’s doanexamplefora1-D,temporal stimulus

generate_spiketrain_from_linear_filter.m

T=100 * 10^3; %total duration of spike train, in millisecondsdeltat=1; %in ms

time_list=deltat*(1:length(stim_list)); %list of times

spike_train %list of 0/1 spike/or not each timestepstim_list %list of stimulus values at each timestep…

figure;subplot(211)plot(time_list,stim_list);title('stimulus','FontSize',18)subplot(212)stem(time_list,spike_train,'.')xlabel('time (ms)','FontSize',15)title('spike raster plot','FontSize',15)

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Spiketriggeredaverage(STA)• Let’s doanexamplefora1-D,temporal stimulus

Your turn!

First run generate_spiketrain_from_linear_filter.mto make the vectors spike_train and stim_list

Then write a code that computes STA for this stimulus and spike train.

Discuss its form, and what it means intuitively for what stimuli drive the neuron to fire.

Note, you might need to increase T to get an interpretable result!

Predictingresponsestonewstimuli

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Spiketriggeredaverage(STA)• Ideaoftheoptimalfiltertopredictneuralfiring:

• Takea(brandnew)stimulusstim(x,y)• Compute“dotproduct”

• Lcangivethebest(linear)estimateofp(spike|stim(x,y))…forthisNEWstimulus:i.e.,that’sthefiringrate!(SeeCh.2forconditions)

L =P

x,y

stim(x, y)⇥ STA(x, y)

Spiketriggeredaverage(STA)• Ideaoftheoptimalfiltertopredictneuralfiring:

• Takea(brandnew)stimulusstim(x,y)• Compute“dotproduct”

• UseLas(linear)estimateofp(spike|stim(x,y))…forthisNEWstimulus:i.e.,that’sthefiringrate!

Literally,asin:p=L*deltatspike=round(rand + (p-1/2))

L =P

x,y

stim(x, y)⇥ STA(x, y)

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Spiketriggeredaverage(STA)

STA(x, t, ⌧) = average stim preceding spike by ⌧

(2.24,AbbottandDayan)

L(t) =P

x,y,⌧

stim(x, y, t� ⌧)⇥ STA(x, y, ⌧)

Extension totemporalstimuli:

p=L(t)*deltatspike(t)=round(rand + (p-1/2))

Spiketriggeredaverage(STA)• Ideaoftheoptimalfiltertopredictneuralfiring:

• Takea(brandnew)stimulusstim(x,y)• Compute“dotproduct”

• UseLas(linear)estimateofp(spike|stim(x,y))…forthisNEWstimulus:i.e.,that’sthefiringrate!

Makes INTUITIVEsense…similaritytothe“average”stimulusthatcreatedaspike.

WhencanweshowthatthismakesMATHEMATICAL sense? Lcangivethebest(linear)estimateofp(spike|stim(x,y))(See Ch.2forconditions)

L =P

x,y

stim(x, y)⇥ STA(x, y)

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Usinglinearfiltertopredictresponses

Linear-Nonlinear-PoissonModel(LNP)• Incorporatingnonlinearitiesinto• LNCascade

Stimulus STA

dot(inner)product

Spikerateatt

F (L(t)) = max(0, L(t))

= r(t)

AfterdetermineSTA,nonlinearity(F)maybe fitbasedonsamplesofLandsamplesofr

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STAsusedtoModel“ReceptiveFields”• Gaussian

• Gainalphameanmuvariancesigmasquared

• 2DGaussianProductofGaussiansineachdimension

• GaborProductofsinusoidandaGaussian

Rod/Ganglionreceptive field

Ganglion/LGNreceptive field

%make 2D arrays of X and Y positions[DX, DY] = meshgrid(staData.X, staData.Y);

%inline function definitiong = @(D,mu,sigma,alpha) alpha*exp(-(D-mu).^2./(2*sigma 2));

%make 2d RFrf = g(DX, 0, 2, 1).*g(DY, 5, 4, 1);

%display RFimagesc(staData.X, staData.Y, rf); rf = g(DX, 0, 2, 1.5).*g(DY, 0, 2, 1.5) –

g(DX, 0, 1, 2).*g(DY, 0, 1, 2);

DifferenceofGaussians:

V1simplecellreceptive field

• AbhishekDe’s V1Cells• CourtesyHorwitz Lab• Brainsarenoisy

• The‘complexity’ofcomputationsbetweenastimulusandtheneuron’s spikerateeffecttheabilityofSTAtoestimatetheRF

STA’susedtorecoverV1RFsCell1 Cell2 Cell3

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Linear-Nonlinear-PoissonModel(LNP)• WehaveaRF,howdoyoumodelspikes?• LNCascade

Stimulus RF

dot(inner)product

Lambdaoft:likelihoodofspikingatt

LimitsofSTA• Spike-triggeredaveragesseemlikemagicWhyhaven’twesolvedthebrainandvision?• LetslookatsomedatarecordedfromV4

• Wouldthesestimulidrivethecell?

BasisShapes Rotations Single-unitV4responses

0 Spk/Sec 37

• Howlongwouldittakebeforeyourandomlysampledashape?

• STAonlyguaranteed toworkinthelimitofinfinitestimuli(notpracticalforexperimentation)

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Decodingneuronsprobabilistically• STArequiresuncorrelatedstimuli

• GoodforRetina,LGN,V1

• “GLM”andpointprocessmethods provideimportant alliedapproaches• [Gerstner,Paninski etal,Book“NeuronalDynamics”]

• Deeperregionsofventralcortexrespondtocomplexstructureandform

• MaximallyInformativeDimensions• AnalyzingNeuralResponsestoNaturalSignals:MaximallyInformativeDimensions.TatyanaSharpee,NicoleC.Rust,andWilliamBialek,NeuralComputation200416:2,223-250

• Givenamodelofstimulitospikeoutput,maximizethedifferencebetween:

• Agnostictostimuliandcomputation,hardtofit

HaxSerre etal.,2005