Jonathan E. Rubin

High Frequency Stimulation of the Subthalamic Nucleus Eliminates Pathological Thalamic Rhythmicity in a Computational Model

Deep brain stimulation (DBS) of the subthalamic nucleus (STN) or the internal segment of the globus pallidus (GPi) has recently been recognized as an important form of intervention for alleviating motor symptoms associated with Parkinsons disease, but the mechanism underlying its effectiveness remains unknown. Using a computational model, this paper considers the hypothesis that DBS works by replacing pathologically rhythmic basal ganglia output with tonic, high frequency firing. In our simulations of parkinsonian conditions, rhythmic inhibition from GPi to the thalamus compromises the ability of thalamocortical relay (TC) cells to respond to depolarizing inputs, such as sensorimotor signals. High frequency stimulation of STN regularizes GPi firing, and this restores TC responsiveness, despite the increased frequency and amplitude of GPi inhibition to thalamus that result. We provide a mathematical phase plane analysis of the mechanisms that determine TC relay capabilities in normal, parkinsonian, and DBS states in a reduced model. This analysis highlights the differences in deinactivation of the low-threshold calcium T-current that we observe in TC cells in these different conditions. Alternative scenarios involving convergence of thalamic signals in the cortex are also discussed, and predictions associated with these results, including the occurrence of rhythmic rebound bursts in certain TC cells in parkinsonian states and their drastic reduction by DBS, are stated. These results demonstrate how DBS could work by increasing firing rates of target cells, rather than shutting them down.

Equilibrium Properties of Temporally Asymmetric Hebbian Plasticity

A theory of temporally asymmetric Hebb rules, which depress or potentiate synapses depending upon whether the postsynaptic cell fires before or after the presynaptic one, is presented. Using the Fokker-Planck formalism, we show that the equilibrium synaptic distribution induced by such rules is highly sensitive to the manner in which bounds on the allowed range of synaptic values are imposed. In a biologically plausible multiplicative model, the synapses in asynchronous networks reach a distribution that is invariant to the firing rates of either the presynaptic or postsynaptic cells. When these cells are temporally correlated, the synaptic strength varies smoothly with the degree and phase of their synchrony.

Calcium-activated nonspecific cation current and synaptic depression promote network-dependent burst oscillations

Central pattern generators (CPGs) produce neural-motor rhythms that often depend on specialized cellular or synaptic properties such as pacemaker neurons or alternating phases of synaptic inhibition. Motivated by experimental evidence suggesting that activity in the mammalian respiratory CPG, the preBötzinger complex, does not require either of these components, we present and analyze a mathematical model demonstrating an unconventional mechanism of rhythm generation in which glutamatergic synapses and the short-term depression of excitatory transmission play key rhythmogenic roles. Recurrent synaptic excitation triggers postsynaptic Ca2+-activated nonspecific cation current (ICAN) to initiate a network-wide burst. Robust depolarization due to ICAN also causes voltage-dependent spike inactivation, which diminishes recurrent excitation and thus attenuates postsynaptic Ca2+ accumulation. Consequently, activity-dependent outward currents—produced by Na/K ATPase pumps or other ionic mechanisms—can terminate the burst and cause a transient quiescent state in the network. The recovery of sporadic spiking activity rekindles excitatory interactions and initiates a new cycle. Because synaptic inputs gate postsynaptic burst-generating conductances, this rhythm-generating mechanism represents a new paradigm that can be dubbed a ‘group pacemaker’ in which the basic rhythmogenic unit encompasses a fully interdependent ensemble of synaptic and intrinsic components. This conceptual framework should be considered as an alternative to traditional models when analyzing CPGs for which mechanistic details have not yet been elucidated.

Thalamocortical Relay Fidelity Varies Across Subthalamic Nucleus Deep Brain Stimulation Protocols in a Data-Driven Computational Model

The therapeutic effectiveness of deep brain stimulation (DBS) of the subthalamic nucleus (STN) may arise through its effects on inhibitory basal ganglia outputs, including those from the internal segment of the globus pallidus (GPi). Changes in GPi activity will impact its thalamic targets, representing a possible pathway for STN-DBS to modulate basal ganglia-thalamocortical processing. To study the effect of STN-DBS on thalamic activity, we examined thalamocortical (TC) relay cell responses to an excitatory input train under a variety of inhibitory signals, using a computational model. The inhibitory signals were obtained from single-unit GPi recordings from normal monkeys and from monkeys rendered parkinsonian through arterial 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine injection. The parkinsonian GPi data were collected in the absence of STN-DBS, under sub-therapeutic STN-DBS, and under therapeutic STN-DBS. Our simulations show that inhibition from parkinsonian GPi activity recorded without DBS-compromised TC relay of excitatory inputs compared with the normal case, whereas TC relay fidelity improved significantly under inhibition from therapeutic, but not sub-therapeutic, STN-DBS GPi activity. In a heterogeneous model TC cell population, response failures to the same input occurred across multiple TC cells significantly more often without DBS than in the therapeutic DBS case and in the normal case. Inhibitory signals preceding successful TC relay were relatively constant, whereas those before failures changed more rapidly. Computationally generated inhibitory inputs yielded similar effects on TC relay. These results support the hypothesis that STN-DBS alters parkinsonian GPi activity in a way that may improve TC relay fidelity.

Mechanistic simulations of inflammation: current state and future prospects.

Inflammation is a normal, robust physiological process. It can also be viewed as a complex system that senses and attempts to resolve homeostatic perturbations initiated from within the body (for example, in autoimmune disease) or from the outside (for example, in infections). Virtually all acute and chronic diseases are either driven or modulated by inflammation. The complex interplay between beneficial and harmful arms of the inflammatory response may underlie the lack of fully effective therapies for many diseases. Mathematical modeling is emerging as a frontline tool for understanding the complexity of the inflammatory response. A series of articles in this issue highlights various modeling approaches to inflammation in the larger context of health and disease, from intracellular signaling to whole-animal physiology. Here we discuss the state of this emerging field. We note several common features of inflammation models, as well as challenges and prospects for future studies.

IN SILICO MODELS OF ACUTE INFLAMMATION IN ANIMALS

ABSTRACT Trauma and hemorrhagic shock elicit an acute inflammatory response, predisposing patients to sepsis, organ dysfunction, and death. Few approved therapies exist for these acute inflammatory states, mainly due to the complex interplay of interacting inflammatory and physiological elements working at multiple levels. Various animal models have been used to simulate these phenomena, but these models often do not replicate the clinical setting of multiple overlapping insults. Mathematical modeling of complex systems is an approach for understanding the interplay among biological interactions. We constructed a mathematical model using ordinary differential equations that encompass the dynamics of cells and cytokines of the acute inflammatory response, as well as global tissue dysfunction. The model was calibrated in C57Bl/6 mice subjected to (1) various doses of lipopolysaccharide (LPS) alone, (2) surgical trauma, and (3) surgery + hemorrhagic shock. We tested the models predictive ability in scenarios on which it had not been trained, namely, (1) surgery ± hemorrhagic shock + LPS given at times after the beginning of surgical instrumentation, and (2) surgery + hemorrhagic shock + bilateral femoral fracture. Software was created that facilitated fitting of the mathematical model to experimental data, as well as for simulation of experiments with various inflammatory challenges and associated variations (gene knockouts, inhibition of specific cytokines, etc.). Using this software, the C57Bl/6-specific model was recalibrated for inflammatory analyte data in CD14−/− mice and was used to elucidate altered features of inflammation in these animals. In other experiments, rats were subjected to surgical trauma ± LPS or to bacterial infection via fibrin clots impregnated with various inocula of Escherichia coli. Mathematical modeling may provide insights into the complex dynamics of acute inflammation in a manner that can be tested in vivo using many fewer animals than has been possible previously.

Geometric Singular Perturbation Analysis of Neuronal Dynamics

Abstract In this chapter, we consider recent models for neuronal activity. We review the sorts of oscillatory behavior which may arise from the models and then discuss how geometric singular perturbation methods have been used to analyze these rhythms. We begin by discussing models for single cells which display bursting oscillations. There are, in fact, several different classes of bursting solutions; these have been classified by the geometric properties of how solutions evolve in phase space. We describe several of the bursting classes and then review related rigorous mathematical analysis. We then discuss the dynamics of small networks of neurons. We are primarily interested in whether excitatory or inhibitory synaptic coupling leads to either synchronous or desynchronous rhythms. We demonstrate that all four combinations are possible, depending on the details of the intrinsic and synaptic properties of the cells. Finally, we discuss larger networks of neuronal oscillators involving two distinct cell populations. In particular, we demonstrate how dynamical systems methods can be used to analyze recent models for sleep rhythms and other oscillations generated in the thalamus. The analysis helps to explain the generation of the different thalamic rhythms and the transitions between them, in both of which inhibition plays a crucial role.

Multiple Rhythmic States in a Model of the Respiratory Central Pattern Generator

The three-phase respiratory pattern observed during normal breathing changes with alterations in metabolic or physiological conditions. A recent study using in situ perfused rat brain preparations demonstrated a reorganization of the respiratory pattern with sequential reduction of the brain stem respiratory network. Specifically, with removal of the pons, the normal three-phase pattern transformed to a two-phase inspiratory-expiratory pattern and, with more caudal transections, to one-phase, intrinsically generated inspiratory oscillations. A minimal neural network proposed to reproduce these transformations includes 1) a ringlike mutually inhibitory network composed of the postinspiratory, augmenting expiratory, and early-inspiratory neurons and 2) an excitatory preinspiratory neuron, with persistent sodium current (I(NaP))-dependent intrinsic bursting properties, that dynamically participates in the expiratory-inspiratory phase transition and inspiratory phase generation. We used activity-based single-neuron models and applied numerical simulations, bifurcation methods, and fast-slow decomposition to describe the behavior of this network in the functional states corresponding to the three-, two-, and one-phase oscillatory regimes, as well as to analyze the transitions between states and between respiratory phases within each state. We demonstrate that, although I(NaP) is not necessary for the generation of three- and two-phase oscillations, it contributes to control of the oscillation period in each state. We also show that the transitions between states can be produced by progressive changes of drives to particular neurons and proceed through intermediate regimes, featuring high-amplitude late-expiratory and biphasic-expiratory activities or ectopic burst generation. Our results provide important insights for understanding the state-dependent mechanisms for respiratory rhythm generation and control.

Timing in synaptic plasticity: from detection to integration

Timing of cellular and subcellular events contributes to spiking-induced modification of synapses in a variety of ways. Initially, the timing of presynaptic and postsynaptic action potentials must be translated into signals that can initiate intracellular processes. Recent experimental and computational findings suggest that the spatiotemporal details of such signals, in particular the time courses and locations of postsynaptic Ca(2+) transients, might themselves be crucial for driving potentiation and depression modules that interact in a time-dependent way to determine plasticity outcomes. On longer timescales, the effects of multiple spikes are integrated in a nonlinear manner, yielding non-intuitive plasticity results that are likely to be sensitive to local conditions and, finally, additional elements must be called into action to stabilize changes in synaptic strengths. This review is part of the TINS Synaptic Connectivity series.

Control of oscillation periods and phase durations in half-center central pattern generators: a comparative mechanistic analysis

Central pattern generators (CPGs) consisting of interacting groups of neurons drive a variety of repetitive, rhythmic behaviors in invertebrates and vertebrates, such as arise in locomotion, respiration, mastication, scratching, and so on. These CPGs are able to generate rhythmic activity in the absence of afferent feedback or rhythmic inputs. However, functionally relevant CPGs must adaptively respond to changing demands, manifested as changes in oscillation period or in relative phase durations in response to variations in non-patterned inputs or drives. Although many half-center CPG models, composed of symmetric units linked by reciprocal inhibition yet varying in their intrinsic cellular properties, have been proposed, the precise oscillatory mechanisms operating in most biological CPGs remain unknown. Using numerical simulations and phase-plane analysis, we comparatively investigated how the intrinsic cellular features incorporated in different CPG models, such as subthreshold activation based on a slowly inactivating persistent sodium current, adaptation based on slowly activating calcium-dependent potassium current, or post-inhibitory rebound excitation, can contribute to the control of oscillation period and phase durations in response to changes in excitatory external drive to one or both half-centers. Our analysis shows that both the sensitivity of oscillation period to alterations of excitatory drive and the degree to which the duration of each phase can be separately controlled depend strongly on the intrinsic cellular mechanisms involved in rhythm generation and phase transitions. In particular, the CPG formed from units incorporating a slowly inactivating persistent sodium current shows the greatest range of oscillation periods and the greatest degree of independence in phase duration control by asymmetric inputs. These results are explained based on geometric analysis of the phase plane structures corresponding to the dynamics for each CPG type, which in particular helps pinpoint the roles of escape and release from synaptic inhibition in the effects we find.