Using Embedding Algorithms to Find a Low Dimensional Representation of Neural Activity During Motor Planning

Neural Activity During Motor Planning CS229 Final Project – Fall 2005 Afsheen “the Plumber” Afshar and John “the Whale” Cunningham Introduction Patterns of neural activity in certain brain areas are understood to drive motor behavior. In the time immediately preceding a movement, there is a period of preparatory neural activity, called the "plan period." This activity can be measured as cell firing patterns, where we record as data, for each neuron, the number of observed action potentials in small time bins. Thus, the recorded data is equivalent to a high dimensional firing rate vector as a function of time, with one dimension for each neuron. Prior work has supported the hypothesis that neurons in the dorsal premotor cortex (PMd) region of the brain modulate their activity depending on the direction, distance, and speed of an upcoming movement [Churchland and Shenoy]. Numerous theories have attempted to explain what these neurons are representing by their firing patterns, but no single encoding scheme has proven universally successful in explaining the observed data. Recently, various researchers have begun to propose models of plan period activity in PMd that do not restrict neurons to solely representing a small subset of the dimensions of the upcoming movement (e.g. parameters in a kinematic model of the arm) [Yu et al]. Specifically, they have proposed that, while planning, the population’s firing rate vector is in a low dimensional manifold of the high dimensional firing rate space. Further, it posits that when the operation of planning involves altering firing rates so that they move to a subspace of that manifold. This is called the ‘optimal subspace hypothesis’. Various experimental results, such as the finding that firing rate variance across similar trials decreases as a function of time, agree with this hypothesis [Churchland et al]. If firing rates during planning occupy a low dimensional manifold, there should be a way to represent these data using many fewer dimensions than the number of neurons. This low dimensional representation could reveal the fundamental neural signatures of motor planning as well as correlate with behavioral features of the impending movement. Such a representation could be very useful for neural prosthetics in addition to basic neuroscience. Current work involves using expectation maximization to do exactly this [Yu et al]. This work investigates how well the simpler algorithms of Principle Component Analysis (PCA), Isomap, Local Linear Embedding (LLE), and Sensible PCA (SPCA) perform on this problem. Background Linear Methods – PCA and SPCA [Roweis] PCA is a linear projection method that works on the simple principle of ordering the axes in terms of their variability (an eigenvector decomposition) and selecting the dimensions of most variability (highest eigenvalues). This well known analysis has the benefit of simple implementation, quick runtime, and optimal mean squared error over the class of linear methods. While it is very useful in assessing important dimensions in the data, it lacks a proper probabilistic model under which one can evaluate test data; that is, given a n dimensional (n=number of neural units) training set{ } m i x i ,..., 1 ; ) ( = , PCA does not learn a generative model for x. Hence, one can not calculate the likelihood of any training or test data. SPCA addresses this problem by adding a proper probabilistic framework to PCA via a Factor Analysis approach. Calling the data x and the latent variables (iid) y, we assume the model: v y x + = C ) , ( ~ I 0 y N ) , ( ~ I 0 v ε N ) , ( ~ I 0 x ε + ⇒ T CC N We can calculate the likelihood of any data under this model using ) | ( θ x p . To learn this model, we will employ the EM algorithm. Using Bayes’s Rule and the rules for conditioning on Gaussian Random Vectors, we can write: