Shared inputs, entrainment, and desynchrony in elliptic bursters: from slow passage to discontinuous circle maps

What input signals will lead to synchrony vs. desynchrony in a group of biological oscillators? This question connects with both classical dynamical systems analyses of entrainment and phase locking and with emerging studies of stimulation patterns for controlling neural network activity. Here, we focus on the response of a population of uncoupled, elliptically bursting neurons to a common pulsatile input. We extend a phase reduction from the literature to capture inputs of varied strength, leading to a circle map with discontinuities of various orders. In a combined analytical and numerical approach, we apply our results to both a normal form model for elliptic bursting and to a biophysically-based neuron model from the basal ganglia. We find that, depending on the period and amplitude of inputs, the response can either appear chaotic (with provably positive Lyaponov exponent for the associated circle maps), or periodic with a broad range of phase-locked periods. Throughout, we discuss the critical underlying mechanisms, including slow-passage effects through Hopf bifurcation, the role and origin of discontinuities, and the impact of noiseComment: 17 figures, 40 page

Analyses at microscopic, mesoscopic, and mean-field scales

Die Aktivität des Hippocampus im Tiefschlaf ist geprägt durch sharp wave-ripple Komplexe (SPW-R): kurze (50–100 ms) Phasen mit erhöhter neuronaler Aktivität, moduliert durch eine schnelle “Ripple”-Oszillation (140–220 Hz). SPW-R werden mit Gedächtniskonsolidierung in Verbindung gebracht, aber ihr Ursprung ist unklar. Sowohl exzitatorische als auch inhibitorische Neuronpopulationen könnten die Oszillation generieren. Diese Arbeit analysiert Ripple-Oszillationen in inhibitorischen Netzwerkmodellen auf mikro-, meso- und makroskopischer Ebene und zeigt auf, wie die Ripple-Dynamik von exzitatorischem Input, inhibitorischer Kopplungsstärke und dem Rauschmodell abhängt. Zuerst wird ein stark getriebenes Interneuron-Netzwerk mit starker, verzögerter Kopplung analysiert. Es wird eine Theorie entwickelt, die die Drift-bedingte Feuerdynamik im Mean-field Grenzfall beschreibt. Die Ripple-Frequenz und die Dynamik der Membranpotentiale werden analytisch als Funktion des Inputs und der Netzwerkparameter angenähert. Die Theorie erklärt, warum die Ripple-Frequenz im Verlauf eines SPW-R-Ereignisses sinkt (intra-ripple frequency accommodation, IFA). Weiterhin zeigt eine numerische Analyse, dass ein alternatives Modell, basierend auf einem transienten Störungseffekt in einer schwach gekoppelten Interneuron-Population, unter biologisch plausiblen Annahmen keine IFA erzeugen kann. IFA kann somit zur Modellauswahl beitragen und deutet auf starke, verzögerte inhibitorische Kopplung als plausiblen Mechanismus hin. Schließlich wird die Anwendbarkeit eines kürzlich entwickelten mesoskopischen Ansatzes für die effiziente Simulation von Ripples in endlich großen Netzwerken geprüft. Dabei wird das Rauschen nicht im Input der Neurone beschrieben, sondern als stochastisches Feuern entsprechend einer Hazard-Rate. Es wird untersucht, wie die Wahl des Hazards die dynamische Suszeptibilität einzelner Neurone, und damit die Ripple-Dynamik in rekurrenten Interneuron-Netzwerken beeinflusst.Hippocampal activity during sleep or rest is characterized by sharp wave-ripples (SPW-Rs): transient (50–100 ms) periods of elevated neuronal activity modulated by a fast oscillation — the ripple (140–220 Hz). SPW-Rs have been linked to memory consolidation, but their generation mechanism remains unclear. Multiple potential mechanisms have been proposed, relying on excitation and/or inhibition as the main pacemaker. This thesis analyzes ripple oscillations in inhibitory network models at micro-, meso-, and macroscopic scales and elucidates how the ripple dynamics depends on the excitatory drive, inhibitory coupling strength, and the noise model. First, an interneuron network under strong drive and strong coupling with delay is analyzed. A theory is developed that captures the drift-mediated spiking dynamics in the mean-field limit. The ripple frequency as well as the underlying dynamics of the membrane potential distribution are approximated analytically as a function of the external drive and network parameters. The theory explains why the ripple frequency decreases over the course of an event (intra-ripple frequency accommodation, IFA). Furthermore, numerical analysis shows that an alternative inhibitory ripple model, based on a transient ringing effect in a weakly coupled interneuron population, cannot account for IFA under biologically realistic assumptions. IFA can thus guide model selection and provides new support for strong, delayed inhibitory coupling as a mechanism for ripple generation. Finally, a recently proposed mesoscopic integration scheme is tested as a potential tool for the efficient numerical simulation of ripple dynamics in networks of finite size. This approach requires a switch of the noise model, from noisy input to stochastic output spiking mediated by a hazard function. It is demonstrated how the choice of a hazard function affects the linear response of single neurons and therefore the ripple dynamics in a recurrent interneuron network

Multiscale Modeling of Coupled Oscillators with Applications to the Mammalian Circadian Clock.

Many biological systems function based on two essential motifs: network interactions between cells and integration across timescales. Both of these are ubiquitous in coupled biological oscillators, such as in heart and brain tissues, which require communication and coordination between cellular oscillators across timescales in order to generate tissue-level rhythms. Mathematical modeling can provide invaluable insight in order to explain this complexity. In this dissertation, we develop a detailed model of the suprachiasmatic nucleus (SCN), the central circadian (daily) pacemaker in mammals, and numerical methods for simulating it. This new multiscale model resolves the intracellular molecular events that generate circadian rhythms as well as the cellular electrical activity important for relaying timing information to the rest of the brain. The model not only reproduces experimental findings, but also makes several new predictions about the role of intercellular signaling in the SCN, some of which we experimentally validate. First, the model explains how intercellular signaling in the SCN increases robustness of tissue-level rhythms. Second, it proposes a new mechanism by which the SCN can differentially regulate intra-SCN synchrony and SCN output signals through a single neurotransmitter signaling on disparate timescales. Third, it shows how the response polarity of cells to this neurotransmitter changes depending on the particular daylength an animal has been entrained to. By modulating the response and the intrinsic period of subsets of SCN neurons, phase-locked populations of cells are used to encode the daylength of the entraining signal. Finally, the model predicts that a kinase can be used to modulate the firing rate of cells to control SCN output. On the whole, the model answers many open questions about signaling within the SCN and control of SCN output to the rest of the brain. It presents a holistic picture of the SCN as a robustly oscillating network with its synchrony and output signals modulated on two different timescales in response to the entraining light signal. The computational framework developed herein with parallelization using GPUs provides an important tool for circadian research, and a model computational system for the many multiscale projects currently studying brain function.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/113644/1/dewoskin_1.pd

Stochastic and deterministic dynamics of intrinsically irregular firing in cortical inhibitory interneurons

Most cortical neurons fire regularly when excited by a constant stimulus. In contrast, irregular-spiking (IS) interneurons are remarkable for the intrinsic variability of their spike timing, which can synchronize amongst IS cells via specific gap junctions. Here, we have studied the biophysical mechanisms of this irregular spiking in mice, and how IS cells fire in the context of synchronous network oscillations. Using patch-clamp recordings, artificial dynamic conductance injection, pharmacological analysis and computational modeling, we show that spike time irregularity is generated by a nonlinear dynamical interaction of voltage-dependent sodium and fast-inactivating potassium channels just below spike threshold, amplifying channel noise. This $\textit{active irregularity}$ may help IS cells synchronize with each other at gamma range frequencies, while resisting synchronization to lower input frequencies.Biotechnology and Biological Sciences Research Council, Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Cambridge Overseas Trus

Intrinsic Neuronal Properties Switch the Mode of Information Transmission in Networks

Diverse ion channels and their dynamics endow single neurons with complex biophysical properties. These properties determine the heterogeneity of cell types that make up the brain, as constituents of neural circuits tuned to perform highly specific computations. How do biophysical properties of single neurons impact network function? We study a set of biophysical properties that emerge in cortical neurons during the first week of development, eventually allowing these neurons to adaptively scale the gain of their response to the amplitude of the fluctuations they encounter. During the same time period, these same neurons participate in large-scale waves of spontaneously generated electrical activity. We investigate the potential role of experimentally observed changes in intrinsic neuronal properties in determining the ability of cortical networks to propagate waves of activity. We show that such changes can strongly affect the ability of multi-layered feedforward networks to represent and transmit information on multiple timescales. With properties modeled on those observed at early stages of development, neurons are relatively insensitive to rapid fluctuations and tend to fire synchronously in response to wave-like events of large amplitude. Following developmental changes in voltage-dependent conductances, these same neurons become efficient encoders of fast input fluctuations over few layers, but lose the ability to transmit slower, population-wide input variations across many layers. Depending on the neurons' intrinsic properties, noise plays different roles in modulating neuronal input-output curves, which can dramatically impact network transmission. The developmental change in intrinsic properties supports a transformation of a networks function from the propagation of network-wide information to one in which computations are scaled to local activity. This work underscores the significance of simple changes in conductance parameters in governing how neurons represent and propagate information, and suggests a role for background synaptic noise in switching the mode of information transmission

Controllability analysis and design for underactuated stochastic neurocontrol

Neuroengineering has advanced tremendously over the past decade, but for sensory prosthetics and similar applications, it remains an extraordinary challenge to access neurons at the single cell resolution of most sensory encoding theories. In particular, if each neuron is “tuned” to particular stimulus features, then eliciting a target percept requires activating only neurons tuned to that percept and not others. However, most available technology is underactuated, with orders of magnitude fewer independent control inputs than neural degrees of freedom, possibly limiting its effectiveness given the inherent trade-off of resolution with network size. Here I analyze controllability for pairs of neurons receiving a common input. In particular, I extend previous work on the deterministic control problem to include stochastic membrane dynamics, treating both cases as a bifurcation problem in the noise parameter. I determine controllable regions in parameter space using a combination of mathematical analysis and numerical solution of stochastic differential and Fokker-Planck equations. I explain how boundaries between these regions change with noise level, and connect to the dynamical mechanisms by which controllability is lost. I show that in stochastic systems, in contrast to deterministic systems, expanding the allowable input space to include exponential ramps expands the parameter range over which neuron pairs are controllable. I also describe an alternative controllability definition using only mean spike times, as compared to the probability distribution of spiking within prespecified time intervals. These results could guide future control strategies in the development of sensory neuroprosthetics and other neurocontrol application

Multi-Scale Modeling of the Neural Control of Respiration

The generation of respiration in mammals begins in the lower brainstem where groups of neurons, that together comprise the respiratory central pattern generator (CPG), interact to produce a motor output that controls breathing. The pre-Bötzinger complex (pre-BötC) in the ventrolateral respiratory column (VRC) is believed to be a major contributor to rhythmic inspiratory activity that interacts with other neural compartments within the VRC as well as with other brainstem areas, including the pons. Though there has been a substantial push to understand the exact cellular and network mechanisms operating within the pre-BötC, as well as the way it is incorporated into the larger respiratory network, there is still much to be resolved. The overarching goal of the work presented in this dissertation is to contribute to our understanding of the neural control of respiration at several hierarchical levels. It is my hope that better insight into the complexities of these multiscale neural control mechanisms will provide a more complete framework for understanding various respiratory pathologies, and ultimately guide the development of novel therapies that will improve patient outcomes. I applied techniques from the fields of mathematics and computer science to develop computational models that reproduced results from electrophysiological recordings (done by our collaborators) and generated verifiable predictions. The scale of my modeling work encompasses the interaction of neurons in a single population, several interconnected populations of neurons that encompass the core of the mammalian respiratory network, and an integration of the respiratory network into a larger control system that includes afferent feedback loops. At each level I address specific, but related, topics that add to the general understanding of the neural control of respiration. The aims of my thesis address specific issues at each of the scales mentioned above. These issues may be summarized as follows: (i) the characteristic rhythmic bursting behavior observed in the pre-BötC, which was studied at the cellular levels with a particular interest in how this behavior impacts respiratory rhythmogenesis; (ii) a respiratory network connectome that defines interactions between several populations of neurons that together form the VRC, which produces an alternating pattern of inspiration, post-inspiration and expiration, and, how such a pattern may be affected by changes in chemical environment, e.g. elevated carbon dioxide or diminished oxygen concentrations; and (iii) the role of afferent feedback to the VRC, from the pons and lungs, which was studied in the context of respiratory phase switching mechanisms.Ph.D., Biomedical Engineering -- Drexel University, 201

Circadian clocks optimally adapt to sunlight for reliable synchronization

Circadian oscillation provides selection advantages through synchronization to the daylight cycle. However, a reliable clock must be designed through two conflicting properties: entrainability to synchronize internal time with periodic stimuli such as sunlight, and regularity to oscillate with a precise period. These two aspects do not easily coexist because better entrainability favors higher sensitivity, which may sacrifice the regularity. To investigate conditions for satisfying the two properties, we analytically calculated the optimal phase-response curve with a variational method. Our result indicates an existence of a dead zone, i.e., a time period during which input stimuli neither advance nor delay the clock. A dead zone appears only when input stimuli obey the time course of actual solar radiation but a simple sine curve cannot yield a dead zone. Our calculation demonstrates that every circadian clock with a dead zone is optimally adapted to the daylight cycle.Comment: 24 pages, 7 figure

Modulation of Long-Range Connectivity Patterns via Frequency-Specific Stimulation of Human Cortex

There is increasing interest in how the phase of local oscillatory activity within a brain area determines the long-range functional connectivity of that area. For example, increasing convergent evidence from a range of methodologies suggests that beta (20 Hz) oscillations may play a vital role in the function of the motor system [1-5]. The "communication through coherence" hypothesis posits that the precise phase of coherent oscillations in network nodes is a determinant of successful communication between them [6, 7]. Here we set out to determine whether oscillatory activity in the beta band serves to support this theory within the cortical motor network in vivo. We combined non-invasive transcranial alternating-current stimulation (tACS) [8-12] with resting-state functional MRI (fMRI) [13] to follow both changes in local activity and long-range connectivity, determined by inter-areal blood-oxygen-level-dependent (BOLD) signal correlation, as a proxy for communication in the human cortex. Twelve healthy subjects participated in three fMRI scans with 20 Hz, 5 Hz, or sham tACS applied separately on each scan. Transcranial magnetic stimulation (TMS) at beta frequency has previously been shown to increase local activity in the beta band [14] and to modulate long-range connectivity within the default mode network [15]. We demonstrated that beta-frequency tACS significantly changed the connectivity pattern of the stimulated primary motor cortex (M1), without changing overall local activity or network connectivity. This finding is supported by a simple phase-precession model, which demonstrates the plausibility of the results and provides emergent predictions that are consistent with our empirical findings. These findings therefore inform our understanding of how local oscillatory activity may underpin network connectivity