Computational Models of Neural Intent x Two different levels of neurophysiology realism Black Box models -no realism. function relation between input desired response Generative Models -minimal realism state space models using neuroscience elements
Computational Models of Neural Intent Two different levels of neurophysiology realism Black Box models – no realism, function relation between input desired response Generative Models – minimal realism, state space models using neuroscience elements
Signal Processing Approaches with Black Box Modeling Accessing 2 types of signals(cortical activity and behavior) leads us to a general class of 1O models Data for these models are rate codes obtained by binning spikes on 100 msec windows x Optimal FIR Filter-linear, feedforward *K TDNN-nonlinear feedforward x Multiple FIR filters-mixture of experts xK RMLP-nonlinear, dynamic
Signal Processing Approaches with Black Box Modeling Accessing 2 types of signals (cortical activity and behavior) leads us to a general class of I/O models. Data for these models are rate codes obtained by binning spikes on 100 msec windows. Optimal FIR Filter – linear, feedforward TDNN – nonlinear, feedforward Multiple FIR filters – mixture of experts RMLP – nonlinear, dynamic
Linear Model (Wiener-Hopf solution) Consider a set of spike counts from M neurons, and a hand position vector dEc(c is the output dimension, C= 2 or 3). The spike count of each neuron is embedded by an L-tap discrete time-delay line. Then, the input vector for a linear model at a given time instance n is composed as x(n=X,n), X, (n-1) X,(n-L+1), X2(n).XM(n-L+1)]T, XE M, where x( n-n denotes the spike count of neuron i at a time instance n-1 A linear model estimating hand position at time instance n from the embedded spike counts can be described as y=∑∑x(n-)m+b i=0j=1 where y is the c-coordinate of the estimated hand position by the model, Wi, is a weight on the connection from X(n-)to yc, and bc is a bias for the c-coordinate
Linear Model (Wiener-Hopf solution) Consider a set of spike counts from M neurons, and a hand position vector dC (C is the output dimension, C = 2 or 3). The spike count of each neuron is embedded by an L-tap discrete time-delay line. Then, the input vector for a linear model at a given time instance n is composed as x(n) = [x1 (n), x1 (n-1) … x1 (n-L+1), x2 (n) … xM(n-L+1)]T, xL M, where xi (n-j) denotes the spike count of neuron i at a time instance n-j. A linear model estimating hand position at time instance n from the embedded spike counts can be described as where yc is the c-coordinate of the estimated hand position by the model, wji is a weight on the connection from xi (n-j) to yc , and bc is a bias for the c-coordinate. c L i M j c i ji c y =x n − j w + b − = = 1 0 1 ( )
Linear Model (Wiener-Hopf solution) In a matrix form, we can rewrite the previous equation as w where y is a C-dimensional output vector, and w is a weight matrix of dimension(LM+1)XC. Each column of w consists of [W1o, Wi1, W12 ., Wil °,W21°.,WM0,…,wM1 x1() y(n) xun (m) y(n)
Linear Model (Wiener-Hopf solution) In a matrix form, we can rewrite the previous equation as where y is a C-dimensional output vector, and W is a weight matrix of dimension (LM+1)C. Each column of W consists of [w10 c , w11 c , w12 c…, w1L- 1 c , w20 c , w21 c…, wM0 c , …, wML-1 c ] T. y W x T = x1(n) xM(n) z -1 z -1 … z -1 z -1 … … y x (n) y y (n) y z (n)
Linear Model ( Wiener-Hopf solution) For the MIMO case, the weight matrix in the Wiener filter system is estimated by =RP Wiener R is the correlation matrix of neural spike inputs with the dimension of (LAM)×(LA, R= P=/p2 p2c TMI IM2 PMI pMe」 fi ls the LxL autocorrelation matrix of neuron/ h neurons i and j(i+), and where ri is the LxL cross-correlation matrix betwee P is the(L M)xc cross-correlation matrix between the neuronal bin count and hand position, where pic is the cross-correlation vector between neuron i and the c-coordinate of hand position. The estimated weights Wiener are optimal based on the assumption that the error is drawn from white Gaussian distribution and the data are stationary
Linear Model (Wiener-Hopf solution) For the MIMO case, the weight matrix in the Wiener filter system is estimated by R is the correlation matrix of neural spike inputs with the dimension of (LM)(LM), where rij is the LL cross-correlation matrix between neurons i and j (i ≠ j), and rii is the LL autocorrelation matrix of neuron i. P is the (LM)C cross-correlation matrix between the neuronal bin count and hand position, where pic is the cross-correlation vector between neuron i and the c-coordinate of hand position. The estimated weights WWiener are optimal based on the assumption that the error is drawn from white Gaussian distribution and the data are stationary. W R P −1 Wiener = = M M MM M M r r r r r r r r r R 1 2 21 22 2 11 12 1 = M MC C C p p p p p p P 1 21 2 11 1