Introduction to Focus Issue: Rhythms and Dynamic Transitions in Neurological Disease: Modeling, Computation, and Experiment

Abstract

Rhythmic neuronal oscillations across a broad range of frequencies, as well as spatiotemporal phenomena, such as waves and bumps, have been observed in various areas of the brain and proposed as critical to brain function. While there is a long and distinguished history of studying rhythms in nerve cells and neuronal networks in healthy organisms, the association and analysis of rhythms to diseases are more recent developments. Indeed, it is now thought that certain aspects of diseases of the nervous system, such as epilepsy, schizophrenia, Parkinson's, and sleep disorders, are associated with transitions or disruptions of neurological rhythms. This focus issue brings together articles presenting modeling, computational, analytical, and experimental perspectives about rhythms and dynamic transitions between them that are associated to various diseases.

To have and to hold—and to oscillate—in sickness and in health.

I. INTRODUCTION

Dynamic phenomena, such as rhythmic oscillations and waves, are ubiquitous in the nervous system and have been associated with cognition and motor behavior in both health and disease.^1–5 Oscillations in different frequency ranges have been associated with healthy brain function, including learning, memory, spatial navigation, attention, sleep, and motor behavior.^2,6–14 Wave-like phenomena have been associated with sensory processing¹⁵ and working memory.¹⁶

Rhythmic neuronal activity—and transitions between rhythms—are also central phenomena in neurological disease.^5,17 Perhaps the most famous manifestations occur in epilepsy, which is characterized as a paroxysmal cerebral dysrhythmia (see, e.g., Ref. 18). More recently, similar phenomena have been proposed in other neurological disorders, including schizophrenia (see, e.g., Refs. 19–22), Alzheimer's disease (see, e.g., Ref. 23), Parkinson's disease (see, e.g., Refs. 24 and 25), and sleep disorders (see, e.g., Ref. 26).

A broad range of rhythms, and transitions between different rhythmic regimes, have been observed for these diseases. In epilepsy, the rhythms range from extremely fast (e.g., hundreds of Hertz^27–30) to slow (e.g., a few Hertz^31,32) and are bracketed by abrupt transitions at the onset and termination of seizure events.^33–36 Schizophrenia—characterized as a failure of cognitive integration—manifests in brain rhythms as altered beta and gamma band (15–70 Hz) synchronization.^22,37 In Alzheimer's disease, brain rhythms shift to power at lower frequencies, and the coherence of fast rhythms decreases.^23,38,39 In Parkinson's disease, pathologically exaggerated beta oscillations characterize the abnormal rhythms.^25,40–44 Transitions between dynamic regimes have been observed during sleep, and abnormal transitions have been associated with sleep disorders.^26,45–51 Finally, the slow waves of cortical spreading depression⁵² underlie the reduction of excitability in neuronal tissue associated with migraine,^53,54 in particular, migraine with aura.⁵²

On the three fronts of modeling, computation, and experiment, there have been a series of important recent advances, and many new avenues of research have emerged. These include: (1) new techniques to record high-density brain activity;^55–57 (2) theoretical advances and clinical application of deep brain stimulation methods;⁵⁸ (3) sophisticated biophysical models of rhythms and disease;^59,60 (4) novel optogenetic techniques that permit interrogation of neural circuits in vivo;⁶¹ and, (5) increasingly-sophisticated data-analysis techniques.⁶² The modeling and computational work—both deterministic and stochastic—have focused in part on reproducing experimental results, and on the investigation of the underlying biophysical and dynamical mechanisms, especially when experiments are not possible or very difficult to perform.

The dynamic transitions between healthy and diseased regimes may occur in many ways, including through alterations in the intrinsic properties of the participating neurons, changes in the network connectivity between neurons, modulation in the extracellular environment, or combinations of these and other mechanisms. Dynamic transitions may involve bifurcations or abrupt transitions, which do not involve local bifurcations but rather result from the multiplicity of time scales present in neurons and neuronal networks. The basic ingredients of dynamic transitions may exist at the individual cell level or, alternatively, may result from circuit interactions (i.e., at the network level). The complexity of these interactions spanning spatial and temporal scales often leads to counterintuitive results. Identifying the mechanisms that govern these transitions and where these transitions originate is one of the challenges of systems neuroscience that may shed light on the cellular and network substrate of diseases of the nervous system.

II. THIS FOCUS ISSUE

This focus issue brings together scientists from widely disparate fields, who collectively apply a variety of techniques to assess the features of rhythmic neuronal activity and the transitions between rhythms. Despite the different objects of study and terminologies, common underlying mechanisms and similarities exist. We think that both experimental and theoretical scientists will benefit from exploring the mechanisms and ideas described in the presented articles. Also, we hope that this focus issue will further foster cross-fertilization of techniques and ideas between different fields, and promote continued collaboration between these fields.

Rhythms and dynamical changes of states are of critical importance in the development of migraines, as has been known for over 70 years. Significant current research is devoted to determining the upstream mechanisms, including cortical spreading depression, responsible for the onset of various types of migraines. Miura et al.⁶³ present a thorough review of eight of the mathematical models that have been used to study the physiological and dynamical mechanisms underlying the development of cortical spreading depression, and its role in migraine with aura. These models incorporate ion diffusion, membrane currents and pumps, spatial buffering, osmosis, and cell swelling. The authors also identify some important open questions. Dahlem⁶⁴ develops a novel mathematical model of the migraine generator network that takes into account the chemical imbalances associated with the generation of migraine. This network model incorporates the trigeminal nerve, which innervates the cranial circulation, an associated network of brainstem nuclei, and parasympathetic vasomotor efferents. It also includes the important physiological process of cortical spreading depression, as a spatio-temporal perturbation of homeostasis. The author shows how this extended network model may be used to understand the nonlinear interactions between the key physiological processes and to develop new experiments involving noninvasive and minimally-invasive neuromodulation techniques.

The intrinsic properties of individual cells, and synaptic connections between cells, typically affect network behavior in complicated ways. Proddutur et al.⁶⁵ use network models of fast spiking basket cells to examine how biophysical changes occurring in inhibitory synaptic interactions in epileptic-like (pilocarpine-treated) animals modulate network activity in the gamma frequency range (30–100 Hz). These synaptic changes consist mostly in the enhancement of extra synaptic gamma-amino butyric acid (GABA) currents and the depolarization of the GABA reversal potential. Cabral et al.⁶⁶ examine the effects of alterations in the connectivity properties that underlie the development of schizophrenia. These authors examine whether changes in the functional connectivity necessarily result from changes in the structural connectivity or whether other dynamic mechanisms are involved. Rotstein⁶⁷ investigates the conditions under which the interaction between E1-Amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid excitation and the intrinsic properties of neurons leads to abrupt transitions between low and high firing frequencies. This has been associated to the phenomenon of hyperexcitability in medial entorhinal cortex layer II stellate cells in pilocarpine treated rats.

How synaptic connections support robust neuronal dynamics remains an active research area. Jalil et al.⁶⁸ consider a mathematical model of a central pattern generator, a small network of synaptically-coupled interneurons, governing various motor behaviors in animals. The authors develop a four-cell network of biophysically-motivated model neurons, and examine the observed phase-locked activity states using Poincare return maps. Through variations of the model parameters, the authors match the model dynamics with observed electrophysiology, and thereby investigate the minimal wiring of synaptic connections supporting robust neuronal dynamics observed in experimental studies.

While brain disease is often characterized by intervals of strongly correlated activity, healthy brain activity is typically irregular and neuronal activity uncorrelated. Terman et al.⁶⁹ use conductance-based models to describe a novel mechanism by which irregular neuronal activity emerges in a two-cell network reciprocally connected by synaptic inhibition.

In addition to the computational modeling of neuronal activity, a fundamental issue in understanding the brain's rhythmic dynamics is the development of statistical tools to assess the data associated to these rhythms and their changes. To this end, Deng et al.⁷⁰ develop a point process modeling framework to characterize rhythmic spiking dynamics, test for significant changes in those dynamics, and track the temporal evolution of those changes. The authors apply this approach to spike train data recorded from patients with Parkinson's disease.

In a related vein, it is important to understand the relationships between results obtained from computational models and experiments. In particular, when models are built to address specific experimental questions, but the subsequent construction of hypotheses and the testing of predictions involve details not included in the original models, techniques are required to modify the original models. Skinner and Ferguson⁷¹ propose a multi-level integrative approach to address the cyclic interactions between modeling and experiments. Using whole hippocampus in vitro preparations as a prototypical system, they show how to incorporate experimental measurements made at the cellular- and network-levels into a computational model and simultaneously how model output can be used to control the experiments. Their proposed cycling dynamically in real-time between experiment and model may help with resolving the accuracy of the models and for hypothesis testing regarding the transitions observed between rhythms in healthy organisms and those observed in neurological disease.

More broadly, these latter two research topics fall in the broader category of “data assimilation,” which is also playing an increasingly-important role in other fields of science and engineering, including Lagrangian mixing in geophysical fluid mechanics, complex turbulent flows, among many others.

The two final articles focus on complicated dynamic activities that emerge in single cells. Osinga and Tsaneva-Atanasova⁷² study spike-adding as a transient response in fast-slow models of transmembrane voltage. Based on the classical Hindmarsh-Rose system, their model consists of one slow and two fast variables with a polynomial vector field, and its underlying structure is similar to that of models of pyramidal CA1 and CA3 excitable cells. They demonstrate how an applied current can take the cell out of the globally-attracting quiescent state and lead to the formation of spikes. They also analyze how the number of spikes in the transient response is determined by the amplitude of the applied current and by the geometry of the system's slow manifold. Desroches et al.⁷³ analyze a novel mechanism that governs the abrupt transitions between two oscillatory modes: subthreshold oscillations (STOs) and bursting. A key ingredient is the slow passage through a spike-adding bifurcation. These results extend previous work on mixed-mode oscillatory behavior between sub- and supra-threshold activity, which has almost exclusively focused on abrupt transitions between STOs and spikes.