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1/22/18
1
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
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