Latent variable models for neural data analysis

The brain is perhaps the most complex system to have ever been subjected to rigorous scientific investigation. The scale is staggering: over 10^11 neurons, each making an average of 10^3 synapses, with computation occurring on scales ranging from a single dendritic spine, to an entire cortical area. Slowly, we are beginning to acquire experimental tools that can gather the massive amounts of data needed to characterize this system. However, to understand and interpret these data will also require substantial strides in inferential and statistical techniques. This dissertation attempts to meet this need, extending and applying the modern tools of latent variable modeling to problems in neural data analysis.

It is divided into two parts. The first begins with an exposition of the general techniques of latent variable modeling. A new, extremely general, optimization algorithm is proposed - called Relaxation Expectation Maximization (REM) - that may be used to learn the optimal parameter values of arbitrary latent variable models. This algorithm appears to alleviate the common problem of convergence to local, sub-optimal, likelihood maxima. REM leads to a natural framework for model size selection; in combination with standard model selection techniques the quality of fits may be further improved, while the appropriate model size is automatically and efficiently determined. Next, a new latent variable model, the mixture of sparse hidden Markov models, is introduced, and approximate inference and learning algorithms are derived for it. This model is applied in the second part of the thesis.

The second part brings the technology of part I to bear on two important problems in experimental neuroscience. The first is known as spike sorting; this is the problem of separating the spikes from different neurons embedded within an extracellular recording. The dissertation offers the first thorough statistical analysis of this problem, which then yields the first powerful probabilistic solution. The second problem addressed is that of characterizing the distribution of spike trains recorded from the same neuron under identical experimental conditions. A latent variable model is proposed. Inference and learning in this model leads to new principled algorithms for smoothing and clustering of spike data.

Biophysical and network mechanisms of high frequency extracellular potentials in the rat hippocampus

A fundamental question in neuroscience is how distributed networks of neurons communicate and coordinate dynamically and specifically. Several models propose that oscillating local networks can transiently couple to each other through phase-locked firing. Coherent local field potentials (LFP) between synaptically connected regions is often presented as evidence for such coupling. The physiological correlates of LFP signals depend on many anatomical and physiological factors, however, and how the underlying neural processes collectively generate features of different spatiotemporal scales is poorly understood. High frequency oscillations in the hippocampus, including gamma rhythms (30-100 Hz) that are organized by the theta oscillations (5-10 Hz) during active exploration and REM sleep, as well as sharp wave-ripples (SWRs, 140-200 Hz) during immobility or slow wave sleep, have each been associated with various aspects of learning and memory. Deciphering their physiology and functional consequences is crucial to understanding the operation of the hippocampal network.

We investigated the origins and coordination of high frequency LFPs in the hippocampo-entorhinal network using both biophysical models and analyses of large-scale recordings in behaving and sleeping rats. We found that the synchronization of pyramidal cell spikes substantially shapes, or even dominates, the electrical signature of SWRs in area CA1 of the hippocampus. The precise mechanisms coordinating this synchrony are still unresolved, but they appear to also affect CA1 activity during theta oscillations. The input to CA1, which often arrives in the form of gamma-frequency waves of activity from area CA3 and layer 3 of entorhinal cortex (EC3), did not strongly influence the timing of CA1 pyramidal cells. Rather, our data are more consistent with local network interactions governing pyramidal cells' spike timing during the integration of their inputs. Furthermore, the relative timing of input from EC3 and CA3 during the theta cycle matched that found in previous work to engage mechanisms for synapse modification and active dendritic processes. Our work demonstrates how local networks interact with upstream inputs to generate a coordinated hippocampal output during behavior and sleep, in the form of theta-gamma coupling and SWRs.

Firing Patterns of Cerebellar Purkinje Cells During Locomotion and Sleep

The cerebellum is a major supraspinal center involved in the coordination of movement. The principal neurons of the cerebellar cortex, Purkinje cells, receive excitatory synaptic input from two sources: the parallel and climbing fibers. These pathways have markedly different effects: the parallel fibers control the rate of simple sodium spikes, while the climbing fibers induce characteristic complex spike bursts, which are accompanied by dendritic calcium transients and play a key role in regulating synaptic plasticity. While many studies using a variety of species, behaviors, and cerebellar regions have documented modulation in Purkinje cell activity during movement, few have attempted to record from these neurons in unrestrained rodents. In this dissertation, we use chronic, multi-tetrode recording in freely-behaving rats to study simple and complex spike firing patterns during locomotion and sleep. Purkinje cells discharge rhythmically during stepping, but this activity is highly variable across steps. We show that behavioral variables systematically influence the step-locked firing rate in a step-phase-dependent way, revealing a functional clustering of Purkinje cells. Furthermore, we find a pronounced disassociation between patterns of variability driven by the parallel and climbing fibers, as well as functional differences between cerebellar lobules. These results suggest that Purkinje cell activity not only represents step phase within each cycle, but is also shaped by behavior across steps, facilitating control of movement under dynamic conditions. During sleep, we observe an attenuation of both simple and complex spiking, relative to awake behavior. Although firing rates during slow wave sleep (SWS) and rapid eye movement sleep (REM) are similar, simple spike activity is highly regular in SWS, while REM is characterized by phasic increases and pauses in simple spiking. This phasic activity in REM is associated with pontine waves, which propagate into the cerebellar cortex and modulate both simple and complex spiking. Such a temporal coincidence between parallel and climbing fiber activity is known to drive plasticity at parallel fiber synapses; consequently, pontocerebellar waves may provide a mechanism for tuning synaptic weights in the cerebellum during active sleep.

The Riesz space structure of an Abelian W*-algebra

Let M be an Abelian W*-algebra of operators on a Hilbert space H. Let M₀ be the set of all linear, closed, densely defined transformations in H which commute with every unitary operator in the commutant M’ of M. A well known result of R. Pallu de Barriere states that if ɸ is a normal positive linear functional on M, then ɸ is of the form T → (Tx, x) for some x in H, where T is in M. An elementary proof of this result is given, using only those properties which are consequences of the fact that ReM is a Dedekind complete Riesz space with plenty of normal integrals. The techniques used lead to a natural construction of the class M₀, and an elementary proof is given of the fact that a positive self-adjoint transformation in M₀ has a unique positive square root in M₀. It is then shown that when the algebraic operations are suitably defined, then M₀ becomes a commutative algebra. If ReM₀ denotes the set of all self-adjoint elements of M₀, then it is proved that ReM₀ is Dedekind complete, universally complete Riesz spaces which contains ReM as an order dense ideal. A generalization of the result of R. Pallu de la Barriere is obtained for the Riesz space ReM₀ which characterizes the normal integrals on the order dense ideals of ReM₀. It is then shown that ReM₀ may be identified with the extended order dual of ReM, and that ReM₀ is perfect in the extended sense.

Some secondary questions related to the Riesz space ReM are also studied. In particular it is shown that ReM is a perfect Riesz space, and that every integral is normal under the assumption that every decomposition of the identity operator has non-measurable cardinal. The presence of atoms in ReM is examined briefly, and it is shown that ReM is finite dimensional if and only if every order bounded linear functional on ReM is a normal integral.