146 CHAPTER 5 Graphical Models the problem as one of graph partitioning in which a minimum cut algorithm is applied to identify clusters with strongly similar objects. Specifically, if a ic denotes the probability of the i th neuron being in the c th cluster, the minimum cut problem is solved by finding the set of probabilities {a ic } that maximize the following objective function (Jin et al., 2005): N C N a ic a jc w ij i=1 j=1 N N O = c=1 (5.13) a ic w ij i=1 j=1 The final cluster memberships are derived from c j = arg max a ic . Refer- c[1,...,C] ring to the example in Figure 5.6, only edges of small weights are removed with this approach. The probabilistic spectral clustering algorithm overcomes many limitations when clustering data with arbitrary structures, since it is nonparametric and therefore can track a wide variety of cluster shapes. The probabilistic nature of the algorithm is useful in quantifying the degree of uncertainty in the membership of a given neuron in each cluster. The clustering algorithm is useful in reducing the dimension of the neural space to a set of subspaces where statistical dependency is significant between neurons within a cluster and insignificant across clusters. Therefore, the two- step approach illustrated in Figure 5.7 constitutes an integrated framework for dealing with large-scale systems. 5.4 RESULTS We tested the methods just discussed in analyzing both synthetic spike data and experimental spike data collected from awake, behaving rats in working memory tasks. Here we detail our most relevant findings. 5.4.1 Spiking Neural Model We used a general formulation of the GLM to generate synthetic spike data. Specifically, the output of neuron i, i (t), was expressed as i (t|H i (t), (t), i ) = f i b , h i h , H i (t) , g i , (t) , (5.14) This formulation includes the following set of parameters: i = { i b , i h , i } is a vector in which i b expresses the neuron's background activity level, i h