Generalized Linear Models (GLMs) II Jonathan Pillow 1 <latexit - PowerPoint PPT Presentation

Statistical modeling and analysis of neural data NEU 560, Spring 2018 Lecture 10 Generalized Linear Models (GLMs) II Jonathan Pillow 1

<latexit sha1_base64="miR0nT8tJiZuVKiaUfnxELw/TEo=">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</latexit> <latexit sha1_base64="UNt4lEq3y3m8KLk2pIT2tsWp4c=">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</latexit> <latexit sha1_base64="HP+VfMA0E+XGBPwmKdZYtaDvzk=">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</latexit> Summary: 1. “Linear-Gaussian” GLM: Y | X, ~ k ∼ N ( X ~ k, � 2 I ) k = ( X T X ) − 1 X T Y ˆ 2. Bernoulli GLM: x t , ~ x t · ~ y t | ~ k ∼ Ber( f ( ~ k )) 3. Poisson GLM: 2

Linear-Nonlinear-Poisson exponential Poisson stimulus filter nonlinearity spiking stimulus λ ( t ) k f conditional intensity (spike rate) 3

<latexit sha1_base64="IqiPS5IzX5l+5dK3I8KUFXVZ9E=">AB7HicbVBNT8JAEJ3iF+IX6tHLRmLibRe1BvRi0dMLJBAQ7bLFla2Z3akIa/oMXD2q8+oO8+W9coAcFXzLJy3szmZkXplIYdN1vp7S2vrG5Vd6u7Ozu7R9UD49aJsk04z5LZKI7ITVcCsV9FCh5J9WcxqHk7XB8O/PbT1wbkagHnKQ8iOlQiUgwilZq9UYUybhfrbl1dw6ySryC1KBAs1/96g0SlsVcIZPUmK7nphjkVKNgk8rvczwlLIxHfKupYrG3AT5/NopObPKgESJtqWQzNXfEzmNjZnEoe2MKY7MsjcT/O6GUZXQS5UmiFXbLEoyiTBhMxeJwOhOUM5sYQyLeythI2opgxtQBUbgrf8irxL+rXde/erTVuijTKcAKncA4eXEID7qAJPjB4hGd4hTcncV6cd+dj0Vpyiplj+APn8weYqo6q</latexit> Fitting the nonlinearity Filter k specifies a direction in stimulus space 
 (i.e. a 1D subspace) 4

<latexit sha1_base64="flj8ApT0XAd06blhKZPNAqW50=">ACDHicbZC9TsMwFIUdfkv5CzCyWBQkpiphAQakChbGIhFaqYkix3VaK4T2U6lKskLsPAqLAyAWHkANt4Gt80ALUey9Once3V9T5AyKpVlfRtLyura+u1jfrm1vbOrm3/yCTGDi4IQlohsgSRjlxFUMdJNBUFxwEgniG4m9c6ICEkTfq/GKfFiNOA0pBgpbfnmsTsiGZ+BK+gGwqEc3eIFIzKvChmVBSlbzaspjUVXAS7gao1PbNL7ef4CwmXGpOzZVq8HAlFMSNl3c0kSRGO0ID0NHIUE+nl02tKeKdPgwToR9XcOr+nshRLOU4DnRnjNRQztcm5n+1XqbCy+nPM0U4Xi2KMwYVAmcRAP7VBCs2FgDwoLqv0I8RDoTpQOs6xDs+ZMXwTlrXjbtO6vRuq7SqIFDcAROgQ3OQvcgjZwAaP4Bm8gjfjyXgx3o2PWeuSUc0cgD8yPn8AlRabeQ=</latexit> <latexit sha1_base64="O9GuRVuYx2v+9x/1eESUfYxnZko=">AB/XicbVC7TsMwFHXKq5RXADGxWFRITFXCAmwVLIxFIrRSE0WO47RWHTuynYoqsSvsDAYuU/2Pgb3DQDtBzpSsfn3Cvfe6KMUaUd59uqrayurW/UNxtb2zu7e/b+wYMSucTEw4IJ2YuQIoxy4mqGelkqA0YqQbjW5mfndMpKC3+tJRoIUDThNKEbaSKF95I8Jho/Qx7HQsHzk4Si0m07LKQGXiVuRJqjQCe0vPxY4TwnXmCGl+q6T6aBAUlPMyLTh54pkCI/QgPQN5SglKijK9afw1CgxTIQ0xTUs1d8TBUqVmqSR6UyRHqpFbyb+5/VznVwGBeVZrgnH84+SnEt4CwLGFNJsGYTQxCW1OwK8RBJhLVJrGFCcBdPXibeuq5d45zfZ1lUYdHIMTcAZcAHa4BZ0gAcwKMAzeAVv1pP1Yr1bH/PWmlXNHI/sD5/AIQHlL4=</latexit> Fitting the nonlinearity 1) Project onto subspace spanned by k 5

<latexit sha1_base64="flj8ApT0XAd06blhKZPNAqW50=">ACDHicbZC9TsMwFIUdfkv5CzCyWBQkpiphAQakChbGIhFaqYkix3VaK4T2U6lKskLsPAqLAyAWHkANt4Gt80ALUey9Once3V9T5AyKpVlfRtLyura+u1jfrm1vbOrm3/yCTGDi4IQlohsgSRjlxFUMdJNBUFxwEgniG4m9c6ICEkTfq/GKfFiNOA0pBgpbfnmsTsiGZ+BK+gGwqEc3eIFIzKvChmVBSlbzaspjUVXAS7gao1PbNL7ef4CwmXGpOzZVq8HAlFMSNl3c0kSRGO0ID0NHIUE+nl02tKeKdPgwToR9XcOr+nshRLOU4DnRnjNRQztcm5n+1XqbCy+nPM0U4Xi2KMwYVAmcRAP7VBCs2FgDwoLqv0I8RDoTpQOs6xDs+ZMXwTlrXjbtO6vRuq7SqIFDcAROgQ3OQvcgjZwAaP4Bm8gjfjyXgx3o2PWeuSUc0cgD8yPn8AlRabeQ=</latexit> <latexit sha1_base64="O9GuRVuYx2v+9x/1eESUfYxnZko=">AB/XicbVC7TsMwFHXKq5RXADGxWFRITFXCAmwVLIxFIrRSE0WO47RWHTuynYoqsSvsDAYuU/2Pgb3DQDtBzpSsfn3Cvfe6KMUaUd59uqrayurW/UNxtb2zu7e/b+wYMSucTEw4IJ2YuQIoxy4mqGelkqA0YqQbjW5mfndMpKC3+tJRoIUDThNKEbaSKF95I8Jho/Qx7HQsHzk4Si0m07LKQGXiVuRJqjQCe0vPxY4TwnXmCGl+q6T6aBAUlPMyLTh54pkCI/QgPQN5SglKijK9afw1CgxTIQ0xTUs1d8TBUqVmqSR6UyRHqpFbyb+5/VznVwGBeVZrgnH84+SnEt4CwLGFNJsGYTQxCW1OwK8RBJhLVJrGFCcBdPXibeuq5d45zfZ1lUYdHIMTcAZcAHa4BZ0gAcwKMAzeAVv1pP1Yr1bH/PWmlXNHI/sD5/AIQHlL4=</latexit> Fitting the nonlinearity 1) Project onto subspace spanned by k 2) take histogram of projected stimuli projection onto u k 6

<latexit sha1_base64="flj8ApT0XAd06blhKZPNAqW50=">ACDHicbZC9TsMwFIUdfkv5CzCyWBQkpiphAQakChbGIhFaqYkix3VaK4T2U6lKskLsPAqLAyAWHkANt4Gt80ALUey9Once3V9T5AyKpVlfRtLyura+u1jfrm1vbOrm3/yCTGDi4IQlohsgSRjlxFUMdJNBUFxwEgniG4m9c6ICEkTfq/GKfFiNOA0pBgpbfnmsTsiGZ+BK+gGwqEc3eIFIzKvChmVBSlbzaspjUVXAS7gao1PbNL7ef4CwmXGpOzZVq8HAlFMSNl3c0kSRGO0ID0NHIUE+nl02tKeKdPgwToR9XcOr+nshRLOU4DnRnjNRQztcm5n+1XqbCy+nPM0U4Xi2KMwYVAmcRAP7VBCs2FgDwoLqv0I8RDoTpQOs6xDs+ZMXwTlrXjbtO6vRuq7SqIFDcAROgQ3OQvcgjZwAaP4Bm8gjfjyXgx3o2PWeuSUc0cgD8yPn8AlRabeQ=</latexit> <latexit sha1_base64="O9GuRVuYx2v+9x/1eESUfYxnZko=">AB/XicbVC7TsMwFHXKq5RXADGxWFRITFXCAmwVLIxFIrRSE0WO47RWHTuynYoqsSvsDAYuU/2Pgb3DQDtBzpSsfn3Cvfe6KMUaUd59uqrayurW/UNxtb2zu7e/b+wYMSucTEw4IJ2YuQIoxy4mqGelkqA0YqQbjW5mfndMpKC3+tJRoIUDThNKEbaSKF95I8Jho/Qx7HQsHzk4Si0m07LKQGXiVuRJqjQCe0vPxY4TwnXmCGl+q6T6aBAUlPMyLTh54pkCI/QgPQN5SglKijK9afw1CgxTIQ0xTUs1d8TBUqVmqSR6UyRHqpFbyb+5/VznVwGBeVZrgnH84+SnEt4CwLGFNJsGYTQxCW1OwK8RBJhLVJrGFCcBdPXibeuq5d45zfZ1lUYdHIMTcAZcAHa4BZ0gAcwKMAzeAVv1pP1Yr1bH/PWmlXNHI/sD5/AIQHlL4=</latexit> Fitting the nonlinearity 1) Project onto subspace spanned by k 2) take histogram of projected stimuli STA response 3) ML estimate of Poisson rate in each bin is # spikes / # stimuli projection onto u k 7