Latent Variable Models for Uncovering Motor Cortical Ensemble Dynamics Zhe Chen 1,* , Shizhao Liu 1,2 , Jose Iriate-Diaz 3 , Nicholas G. Hatsopoulos 4 , Callum F. Ross 4 and Kazutaka Takahashi 4,* 1 Department of Psychiatry, School of Medicine, New York University, New York, NY, USA 2 Department of Biomedical Engineering, Tsinghua University, Beijing, China 3 Department of Oral Biology, University of Illinois at Chicago, Chicago, IL, USA 4 Department of Organismal Biology and Anatomy, University of Chicago, Chicago, IL, USA * Correspondence email: zhe.chen3@nyumc.org, kazutaka@uchicago.edu Abstract— Neural activity is dynamic at various spatiotempo- ral scales. We consider a general class of latent variable models for analyses of neuronal ensemble spikes. Specifically, the latent variable follows a Markovian dynamics, and the observations are modulated by the latent state variable through an assumed or unknown mapping function. The inference of latent variable models lead to novel solutions for unsupervised decoding analy- sis, data visualization, variable and model selection. We propose two unsupervised analysis paradigms and assess the validity of the proposed approaches by computer simulations and experimental population spike data recorded from monkey’s primary motor cortices during orofacial movement. I. I NTRODUCTION Single-neuron responses in motor cortex are complex, and there is marked disagreement regarding which movement pa- rameters are represented [9]. Therefore, it is more important to discover latent structure of motor population dynamics [1], [11]. One popular approach is to use supervised learning for establishing encoding models for individual motor neurons, and then use the assumed encoding model in population decoding. However, this approach has several drawbacks: First, we need to make strong statistical assumptions for the neuronal encoding model. Furthermore, the measured move- ment behavior is high-dimensional, which may induce over- fitting. Second, there is often strong heterogeneity among neuronal populations. Without prior information, it is unwise to assume that a common encoding model across all neurons. In contrast, the alternative approach is to use unsupervised learning for unbiased assessment of neuronal population codes. Given the measured motor population spike activity, unsupervised learning is aimed to unfold the latent state trajectory that drives the motor population dynamics, where the relationship between the inferred state trajectory and measured behavior can be established a posteriori. The latent variable approach can be viewed as a subclass of “neural trajectory” methods, with the aim to uncover the low-dimensional neural trajectory. Methods are either based on trial averaging—such as principal component analysis (PCA) or other subspace methods [2], [9], or based on single- trial dynamics—-such as the Gaussian process factor analysis (GPFA) [25], [26], linear and nonlinear dynamical systems [12], [14], [17], [23]. The orofacial primary motor cortex (MIo) plays critical roles in processing and controlling oral motor functions, such as tongue protrusion, chewing, and swallowing [20]. The MIo receives sensory inputs from teeth, bone, mucosa and joints that provide feedforward and feedback information crucial for modulating jaw and tongue movements. However, compared with current knowledge of neural processes in the limb MI and their role in modulating reach-and-grasp behaviors, little is known of the neural processes in the MIo for their role in modulating feeding behaviors. Previously, we have used latent variable models, such as Poisson linear dynamical system (PLDS) and hidden Markov models (HMMs) discovering latent structures of hippocampal-neocoritcal population codes [4]–[6], [8]. Here we extend these two unsupervised approaches to monkey MIo data. In the first approach, we use PLDS with con- tinuous latent states. In the second approach, we use a hierarchical Dirichlet process (HDP)-HMM with discrete latent states [15]. In both cases, the dimensionality of latent states is unknown, and we don’t need to explicitly define the latent state a priori. II. METHODS A. PLDS Let y t =[y 1,t ,...,y C,t ] ⊤ denote a population vector of C neurons, with each element consisting of the neuronal spike count at the t-th time bin (bin size Δ). We assume that the latent univariate variable z t ∈ R m represents an unobserved common input that drives neuronal ensemble spiking activity, as specified by the following state space model: z t = Az t−1 + ǫ t (1) y t ∼ Poisson ( exp(Cz t + d)Δ ) (2) where the state equation (1) describes a first-order autore- gressive (AR) model driven by a zero-mean Gaussian noise process ǫ t ∈N (0, Σ ǫ ). Inference of unknown state variable {z t } and parameters Θ = {A, C, d, Σ ǫ } from observations y 1:T can be tackled by the expectation-maximization (EM) algorithm using different optimization schemes [3], [16]. B. HDP-HMM First, we assume that the latent state process follows a first-order finite m-state Markov chain {S t }∈{1, 2,...,m}. We also assume that the spike counts of individual place cells at discrete time index t, conditional on the latent state S t , follow a Poisson probability distribution with associated tuning curve functions Λ = {λ c,i }: p(y1:T ,S1:T |θ)= p(S1|π) T  t=2 p(St |St-1, P ) T  t=1 p(yt |St , Λ) 331 978-1-5386-1823-3/17/$31.00 ©2017 IEEE Asilomar 2017 Authorized licensed use limited to: UNIV OF CHICAGO LIBRARY. Downloaded on April 05,2021 at 06:18:57 UTC from IEEE Xplore. Restrictions apply.