The title of my PhD thesis is "Statistical Methods for Neural Data: Cointegration Analysis of Coupled Neurons & Generalized Linear Models for Spike Train Data". This basically summarizes my overall interests in neuroscience models.
Using cointegration analysis to infer network connectivity is a new approach in neuroscience and I am currently working on applications for high-dimensional networks. I am working with Susanne Ditlevsen and Anders Rahbek on these matters.
With respect to spike train modeling, I started working in this area in 2016 during a research stay at Boston University. I am working on bridging the intuition behind well-known models for neurons and the statistical model class of Generalized Linear Models. My collaborators in this area are Uri Eden and Mark Kramer.
Abstract: We present cointegration analysis as a method to infer the network struc- ture of a linearly phase coupled oscillating system. By defining a class of oscillating systems with interacting phases, we derive a data generating process where we can specify the coupling structure of a network that resembles biological processes. In particular we study a network of Winfree oscillators, for which we present a statistical analysis of various simulated networks, where we conclude on the coupling structure: the direction of feedback in the phase processes and proportional coupling strength between individual components of the system. We show that we can correctly classify the network structure for such a system by cointegration analysis, for various types of coupling, including uni-/bi-directional and all-to-all coupling. Finally, we analyze a set of EEG recordings and discuss the current applicability of cointegration analysis in the field of neuroscience.
Abstract: To understand neural activity, two broad categories of models exist: statistical and dynamical. While statistical models possess rigorous methods for parameter estimation and goodness-of-fit assessment, dynamical models provide mechanistic insight. In general, these two categories of models are separately applied; understanding the relationships between these modeling approaches remains an area of active research. In this letter, we examine this relationship using simulation. To do so, we first generate spike train data from a well-known dynamical model, the Izhikevich neuron, with a noisy input current. We then fit these spike train data with a statistical model (a generalized linear model, GLM, with multiplicative influences of past spiking). For different levels of noise, we show how the GLM captures both the deterministic features of the Izhikevich neuron and the variability driven by the noise. We conclude that the GLM captures essential features of the simulated spike trains, but for near-deterministic spike trains, goodness-of-fit analyses reveal that the model does not fit very well in a statistical sense; the essential random part of the GLM is not captured.
Abstract: Bursting is a complex behavior observed in neural spike dynamics that is characterized by a cluster of rapid successive spikes followed by a longer period of quiescence. It is well known that some classical neuron models such as the FitzHugh-Nagumo model and the Morris-Lecar model among others, cannot produce this behavior. The duality of this behavior requires a model that can account for dual time scales to handle a fast and slow subsystem such as the Izhikevich model. However, while the Izhikevich model is capable of producing bursting behavior in simulations, it is cumbersome to fit model parameters to observed data. In contrast, a statistical model class such as the Generalized Linear Model (GLM) is capable of capturing the features of a variety of spike train behaviors. The flexibility of the GLM has been illustrated previously by the broad applications of this model class. However, the interpretation of a GLM is not always intuitive and this is indeed the case for a bursting neuron. Although the GLM is capable of capturing bursting, evident from the estimated intensity, the model itself have a more complex interpretation. In this paper, we show how the GLM can be extended to a State Space GLM (SSGLM) to explicitly account for the dual behavior observed for a bursting neuron. We demonstrate how the model can be fitted to an observed spike train by utilizing a marginalized particle filter to simultaneously decode the state of the neuron (bursting/resting) and estimate history dependent kernels that modulate the baseline firing rate, dependent on the behavior. This leads to a simple statistical model that captures the bursting behavior very well when evaluated by a goodness-of-fit analysis based on the Kolmogorov-Smirnov statistic.
Upcoming and past presentations (click on link to view slides/posters):