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fMRI Methods Lecture3 – Modeling the neurovascular coupling Hemodynamic changes Neurons Synaptic transmission Majority of the synapses in the cortex are excitatory glutamate synapses Synaptic transmission Neurotransmitter vesicles Majority of the synapses in the cortex are excitatory glutamate synapses Synaptic transmission Neurotransmitter releases into the synaptic cleft and binds to receptors Synaptic transmission Post-synaptic influx of Sodium Local depolarization of membrane Na+ Neural activity input from >5000 neurons Only once the soma is depolarized above threshold the neuron fires output onto 5000 neurons Input and output Synaptic integration Neural selectivity Cortical architecture Lot’s of local reciprocal connections in the cortex. 80% of synapses are onto neighboring neurons within 1mm. What’s the input and what’s the output? Lot’s of correlated activity among neighbors (columns) Resolution With an electrode we can isolate the activity of a few neighboring neurons. Do single neurons represent what’s happening in the network? You can describe brain activity at different levels of resolution. Perspective Each voxel = 3 mm3 ~3,000,000 neurons Typical cortical area 300 voxels ~1,000,000,000 neurons We’re measuring the summed activity of a huge neural network. Neurovascular coupling Relationship between neural activity and hemodynamics Birth of the HRF How do we characterize the hemodynamic response associated with a particular neural response? We look at primary visual cortex Birth of the HRF Boynton et. al. 1996 Birth of the HRF The HRFs above have a characteristic shape that can be approximated by a gamma function. This function has two free parameters: Tau - time to peak n - time shift (amount of delay) Boynton et. al. 1996 Birth of the HRF By playing with the parameters we can model different HRFs Boynton et. al. 1996 Birth of the HRF So we can either measure our subject’s specific HRF or use a “canonical” HRF from the literature We assume that the same relationship found in primary visual cortex applies everywhere. Might be reasonable for the cortex where the neural architecture is more or less the same. Subcortical areas? Linear shift invariant system Stimulus HRF Time Invariance Very simple coupling between neural activity and hemodynamics Linear shift invariant system Stimulus HRF Time Invariance Scaling Linear shift invariant system Stimulus HRF Time Invariance Scaling Measured Response: Additivity Linear shift invariant system The linear transformation step is simply a convolution with a hemodynamic impulse response function Convolution Multiply each timepoint of the neural response by a copy of the HRF Temporal summation When convolving with an HRF we are actually “smoothing” our data. We loose temporal resolution because we create a lot of correlation between neighboring timepoints. Original audio: Smoothed audio: Neural temporal resolution On the order of milliseconds The challenges Spatial: we’re sampling the average activity of millions of neurons distributed across space. Temporally: we’re sampling the average activity of these neurons across several seconds in time. But, we don’t need to cut anybody’s head open… Estimating neural activity So far we’ve been estimating hemodynamic responses from neural activity. We actually want to go the other way around. Experimental design Because of the sluggishness of the hemodynamic responses we want to build slow experiments. Need to consider our signal to noise: How clean are our measurements? Should we repeat them many times and average? Block design Present long “blocks” of stimulation (a few seconds) interleaved with blank sections and see how the brain responds. Turn the visual system “on and off” Repeat and average to get rid of noise Block design We assume that the stimulus is generating prolonged sustained neural activity for the entire length of stimulus presentation and with equal amplitude on consecutive blocks. Model of expected neural activity Block design We can build a model of the expected hemodynamic changes. Correlation How can we relate the model with the actual data we measured in the scanner? One option is to correlate… Correlation A measure of similarity Covariance Correlation is based on covariance – a measure that reflects the degree to which two variables vary together. Similar signals will have large positive covariance Covariance Correlation is based on covariance – a measure that reflects the degree to which two variables vary together. Opposite signals will have large negative covariance Covariance Correlation is based on covariance – a measure that reflects the degree to which two variables vary together. Different signals will have small positive or negative covariance Correlation Correlation is the covariance divided (normalized) by the variance of the two signals This last bit ensures that correlation coefficients have values between -1 and 1. It also means that the scaling/amplitude/variance of the signal doesn’t matter when computing correlation! Correlation maps Paint the voxels according to the correlation level How big are the correlation values? Is there a chance we would get strong correlations from random hemodynamic fluctuations? Activity localization Estimating response amplitude But we also want to estimate response strength How much do we need to scale the model so that it best fits the data? General linear model Explain the recorded data with a model composed from a combination of linear predictors. data = a0 + a1x1 + a2x2 + … + a3x3 + error data: voxel time-course a: parameter weights (often called beta weights) x: model factor/predictor e: error (what’s left over in the data that is not explained by the model) General linear model In our example so far we had a model with only one predictor. data = a0 + a1x1 + error Our predictor described the hemodynamic activity expected based on our experiment structure. x1 = General linear model We can describe the previous equation as: design matrix beta data residuals = * a1 + error We want to find a1 that will minimize the error term (best fit). Regression In our example the beta weight minimizing the error term will equal the slope of the regression line when regressing the predictor (x) onto the data (y): regression line slope intercept y’ = a*x + c predictor What happened to the ‘error’ (residuals)? Regression The slope (a) is the covariance divided/normalized only by the variance of the predictor. This makes the slope dependant on the variance in the data (y)… Correlation: regression slope intercept line y’ = a*x + c predictor Regression Least squares optimization Find the beta weight (a) that will minimize the squared error: data design matrix In our example the solution is to find the projection of the single predictor onto the data (it’s their dot product). Open least squares handout. Multiple predictors In most experiments we have more than one predictor. We’ll have different experimental conditions and we’ll want to compare the responses to each. Multiple predictors We will have a separate column for each predictor in our design matrix and a separate associated beta weight. design matrix data = * a1 a2 beta residuals + error Least squares optimization We’ll generalize the previous solution to: Matrix inversion As long as the predictors are linearly independent (perpendicular vectors), we can solve separately for each. Basis set: vectors that are independent (dot product = 0). A space is defined by its basis set. Beta value maps Paint the voxels by their beta value (response amplitude): Does not take model fit (noise) into account. Statistical parameter maps Paint the voxels by the statistical significance of the betas: p values from a ttest. Takes model fit (noise) into account. Break Lab #3 Open a folder for your code on the local computer. Try to keep the path name simple (e.g. “C:\Your_name”). Download code and MRI data from: http://www.weizmann.ac.il/neurobiology/labs/malach/ilan/lecture_notes.html Save Lab3.zip in the folder you’ve created and unzip. Open Matlab. Change the “current directory” to the directory you’ve created. Open: “Lab3_SimulatingData.m” When done move on to: “Lab3_AnalyzingData.m” Homework! Read Chapters 6-8 of Huettel et. al. Go over least squares handout Matlab exercise: email me the report as a word document. The report should include answers, figures, and the actual Matlab code used to generate them (copy it into word).