Phase and amplitude responses for delay equations using harmonic balance.

Robust delay induced oscillations, common in nature, are often modeled by delay-differential equations (DDEs). Motivated by the success of phase-amplitude reductions for ordinary differential equations with limit cycle oscillations, there is now a growing interest in the development of analogous approaches for DDEs to understand their response to external forcing. When combined with Floquet theory, the fundamental quantities for this reduction are phase and amplitude response functions. Here, we develop a framework for their construction that utilizes the method of harmonic balance.

Insights into oscillator network dynamics using a phase-isostable framework.

Networks of coupled nonlinear oscillators can display a wide range of emergent behaviors under the variation of the strength of the coupling. Network equations for pairs of coupled oscillators where the dynamics of each node is described by the evolution of its phase and slowest decaying isostable coordinate have previously been shown to capture bifurcations and dynamics of the network, which cannot be explained through standard phase reduction. An alternative framework using isostable coordinates to obtain higher-order phase reductions has also demonstrated a similar descriptive ability for two oscillators. In this work, we consider the phase-isostable network equations for an arbitrary but finite number of identical coupled oscillators, obtaining conditions required for the stability of phase-locked states including synchrony. For the mean-field complex Ginzburg-Landau equation where the solutions of the full system are known, we compare the accuracy of the phase-isostable network equations and higher-order phase reductions in capturing bifurcations of phase-locked states. We find the former to be the more accurate and, therefore, employ this to investigate the dynamics of globally linearly coupled networks of Morris-Lecar neuron models (both two and many nodes). We observe qualitative correspondence between results from numerical simulations of the full system and the phase-isostable description demonstrating that in both small and large networks, the phase-isostable framework is able to capture dynamics that the first-order phase description cannot.

Cell-specific effects of Dyt1 knock-out on sensory processing, network-level connectivity, and motor deficits.

DYT1 dystonia is a debilitating movement disorder characterized by repetitive, unintentional movements and postures. The disorder has been linked to mutation of the TOR1A/DYT1 gene encoding torsinA. Convergent evidence from studies in humans and animal models suggest that striatal medium spiny neurons and cholinergic neurons are important in DYT1 dystonia. What is not known is how torsinA dysfunction in these specific cell types contributes to the pathophysiology of DYT1 dystonia. In this study we sought to determine whether torsinA dysfunction in cholinergic neurons alone is sufficient to generate the sensorimotor dysfunction and brain changes associated with dystonia, or if torsinA dysfunction in a broader subset of cell types is needed. We generated two genetically modified mouse models, one with selective Dyt1 knock-out from dopamine-2 receptor expressing neurons (D2KO) and one where only cholinergic neurons are impacted (Ch2KO). We assessed motor deficits and performed in vivo 11.1 T functional MRI to assess sensory-evoked brain activation and connectivity, along with diffusion MRI to assess brain microstructure. We found that D2KO mice showed greater impairment than Ch2KO mice, including reduced sensory-evoked brain activity in key regions of the sensorimotor network, and altered functional connectivity of the striatum that correlated with motor deficits. These findings suggest that (1) the added impact of torsinA dysfunction in medium spiny and dopaminergic neurons of the basal ganglia generate more profound deficits than the dysfunction of cholinergic neurons alone, and (2) that sensory network impairments are linked to motor deficits in DYT1 dystonia.

Standing and travelling waves in a spherical brain model: The Nunez model revisited.

The Nunez model for the generation of electroencephalogram (EEG) signals is naturally described as a neural field model on a sphere with space-dependent delays. For simplicity, dynamical realisations of this model either as a damped wave equation or an integro-differential equation, have typically been studied in idealised one dimensional or planar settings. Here we revisit the original Nunez model to specifically address the role of spherical topology on spatio-temporal pattern generation. We do this using a mixture of Turing instability analysis, symmetric bifurcation theory, centre manifold reduction and direct simulations with a bespoke numerical scheme. In particular we examine standing and travelling wave solutions using normal form computation of primary and secondary bifurcations from a steady state. Interestingly, we observe spatio-temporal patterns which have counterparts seen in the EEG patterns of both epileptic and schizophrenic brain conditions.

Increasing the quantitative bandwidth of NMR measurements.

The frequency range of quantitative NMR is increased from tens to hundreds of kHz by a new pulse sequence, CHORUS. It uses chirp pulses to excite uniformly over very large bandwidths, yielding accurate integrals even for nuclei such as (19)F that have very wide spectra.

Unifying principles of calcium wave propagation - Insights from a three-dimensional model for atrial myocytes.

Atrial myocytes in a number of species lack transverse tubules. As a consequence the intracellular calcium signals occurring during each heartbeat exhibit complex spatio-temporal dynamics. These calcium patterns arise from saltatory calcium waves that propagate via successive rounds of diffusion and calcium-induced calcium release. The many parameters that impinge on calcium-induced calcium release and calcium signal propagation make it difficult to know a priori whether calcium waves will successfully travel, or be extinguished. In this study, we describe in detail a mathematical model of calcium signalling that allows the effect of such parameters to be independently assessed. A key aspect of the model is to follow the triggering and evolution of calcium signals within a realistic three-dimensional cellular volume of an atrial myocyte, but with low computational costs. This is achieved by solving the linear transport equation for calcium analytically between calcium release events and by expressing the onset of calcium liberation as a threshold process. The model makes non-intuitive predictions about calcium signal propagation. For example, our modelling illustrates that the boundary of a cell produces a wave-guiding effect that enables calcium ions to propagate further and for longer, and can subtly alter the pattern of calcium wave movement. The high spatial resolution of the modelling framework allows the study of any arrangement of calcium release sites. We demonstrate that even small variations in randomly positioned release sites cause highly heterogeneous cellular responses. This article is part of a Special Issue entitled: 13th European Symposium on Calcium.

The impact of beef cattle temperament assessed using flight speed on muscle glycogen, muscle lactate and plasma lactate concentrations at slaughter.

This study evaluated the effect of animal temperament measured using flight speed (FS) on plasma lactate, muscle glycogen and lactate concentrations at slaughter plus ultimate pH in 648 lot finished cattle of mixed breed and sex. Muscle samples were collected at slaughter from the m. semimembranosus, m. semitendinosus and m. longissimus thoracis (LT) for analysis of glycogen and lactate concentration. Blood was collected after exsanguination and analysed for plasma lactate concentration and ultimate pH of the LT was measured. FS had no effect on muscle glycogen concentration in any muscle or ultimate pH of the LT (P>0.05). As FS increased from 1 to 5m/s, plasma and muscle lactate concentration increased by 54% and 11.4% (P<0.01). The mechanisms through which temperament contributes to variation in glycogen metabolism remain unclear. The risk of dark cutting was not impacted by temperament, indicating that other production and genetic factors have a greater impact on the incidence of dark cutting.

Pulsating fronts in periodically modulated neural field models.

We consider a coarse-grained neural field model for synaptic activity in spatially extended cortical tissue that possesses an underlying periodicity in its microstructure. The model is written as an integrodifferential equation with periodic modulation of a translationally invariant spatial kernel. This modulation can have a strong effect on wave propagation through the tissue, including the creation of pulsating fronts with widely varying speeds and wave-propagation failure. Here we develop a new analysis for the study of such phenomena, using two complementary techniques. The first uses linearized information from the leading edge of a traveling periodic wave to obtain wave speed estimates for pulsating fronts, and the second develops an interface description for waves in the full nonlinear model. For weak modulation and a Heaviside firing rate function the interface dynamics can be analyzed exactly and gives predictions that are in excellent agreement with direct numerical simulations. Importantly, the interface dynamics description improves on the standard homogenization calculation, which is restricted to modulation that is both fast and weak.

Neuronal spike-train responses in the presence of threshold noise.

The variability of neuronal firing has been an intense topic of study for many years. From a modelling perspective it has often been studied in conductance based spiking models with the use of additive or multiplicative noise terms to represent channel fluctuations or the stochastic nature of neurotransmitter release. Here we propose an alternative approach using a simple leaky integrate-and-fire model with a noisy threshold. Initially, we develop a mathematical treatment of the neuronal response to periodic forcing using tools from linear response theory and use this to highlight how a noisy threshold can enhance downstream signal reconstruction. We further develop a more general framework for understanding the responses to large amplitude forcing based on a calculation of first passage times. This is ideally suited to understanding stochastic mode-locking, for which we numerically determine the Arnol'd tongue structure. An examination of data from regularly firing stellate neurons within the ventral cochlear nucleus, responding to sinusoidally amplitude modulated pure tones, shows tongue structures consistent with these predictions and highlights that stochastic, as opposed to deterministic, mode-locking is utilised at the level of the single stellate cell to faithfully encode periodic stimuli.

Calcium window currents, periodic forcing, and chaos: understanding single neuron response with a discontinuous one-dimensional map.

Thalamocortical (TC) neurones are known to express the low-voltage activated, inactivating Ca2+ current I(T). The triggering of this current underlies the generation of low threshold Ca2+ potentials that may evoke single or bursts of action potentials. Moreover, this current can contribute to an intrinsic slow (<1 Hz) oscillation whose rhythm is partly determined by the steady state component of I(T) and its interaction with a leak current. This steady state, or window current as it is so often called, has received relatively little theoretical attention despite its importance in determining the electroresponsiveness and input-output relationship of TC neurones. In this paper, we introduce an integrate-and-fire spiking neuron model that includes a biophysically realistic model of I(T). We briefly review the subthreshold bifurcation diagram of this model with constant current injection before moving on to consider its response to periodic forcing. Direct numerical simulations show that as well as the expected mode-locked responses there are regions of parameter space that support chaotic behavior. To reveal the mechanism by which the window current generates a chaotic response to periodic forcing we consider a piecewise linear caricature of the dynamics for the gating variables in the model of I(T). This model can be analyzed in closed form and is shown to support an unstable set of periodic orbits. Trajectories are repelled from these organizing centers until they reach the threshold for firing. By determining the condition for a grazing bifurcation (at the border between a spiking and nonspiking event) we show how knowledge of the unstable periodic orbits (existence and stability) can be combined with the grazing condition to determine an effective one-dimensional map that captures the essentials of the chaotic behavior. This map is discontinuous and has strong similarities with the universal limit mapping in grazing bifurcations derived in the context of impacting mechanical systems [A. B. Nordmark, Phys. Rev. E 55, 266 (1997)].

Depolarization induced suppression of excitation and the emergence of ultraslow rhythms in neural networks.

Ultraslow fluctuations (0.01-0.1 Hz) are a feature of intrinsic brain activity of as yet unclear origin. We propose a candidate mechanism based on retrograde endocannabinoid signaling in a synaptically coupled network of excitatory neurons. This is known to cause depolarization-induced suppression of excitation (DISE), which we model phenomenologically. We construct emergent network oscillations in a globally coupled network and show that for strong synaptic coupling DISE can lead to a synchronized population burst at the frequencies of resting brain rhythms.

Large-scale neural dynamics: simple and complex.

We review the use of neural field models for modelling the brain at the large scales necessary for interpreting EEG, fMRI, MEG and optical imaging data. Albeit a framework that is limited to coarse-grained or mean-field activity, neural field models provide a framework for unifying data from different imaging modalities. Starting with a description of neural mass models, we build to spatially extend cortical models of layered two-dimensional sheets with long range axonal connections mediating synaptic interactions. Reformulations of the fundamental non-local mathematical model in terms of more familiar local differential (brain wave) equations are described. Techniques for the analysis of such models, including how to determine the onset of spatio-temporal pattern forming instabilities, are reviewed. Extensions of the basic formalism to treat refractoriness, adaptive feedback and inhomogeneous connectivity are described along with open challenges for the development of multi-scale models that can integrate macroscopic models at large spatial scales with models at the microscopic scale.

Google governance: increasing the effectiveness of critical care physicians through the use of an online usergroup.

AIMS: The aim of this study was to describe the use of an online user group to enhance communication and productivity by critical care specialists. METHODS: In this article, we provide a description of the first 6 months of use of an online user group by senior retrieval physicians. RESULTS: Initially developed as a communication and online discussion tool, our online user group evolved to include a number of other utilities that support clinical governance. These included a repository for useful files, educational presentations, online rostering and "portfolio pages", updating aspects of an individual specialist's non-clinical activity. Its applications continue to evolve in number and utility. Participating physicians perceive an increase in organisational efficiency. CONCLUSIONS: An online user group such as Google Groups may provide powerful support to an organisation's clinical governance. We recommend this tool to other services with limited administrative personnel.

Understanding cardiac alternans: a piecewise linear modeling framework.

Cardiac alternans is a beat-to-beat alternation in action potential duration (APD) and intracellular calcium (Ca(2+)) cycling seen in cardiac myocytes under rapid pacing that is believed to be a precursor to fibrillation. The cellular mechanisms of these rhythms and the coupling between cellular Ca(2+) and voltage dynamics have been extensively studied leading to the development of a class of physiologically detailed models. These have been shown numerically to reproduce many of the features of myocyte response to pacing, including alternans, and have been analyzed mathematically using various approximation techniques that allow for the formulation of a low dimensional map to describe the evolution of APDs. The seminal work by Shiferaw and Karma is of particular interest in this regard [Shiferaw, Y. and Karma, A., "Turing instability mediated by voltage and calcium diffusion in paced cardiac cells," Proc. Natl. Acad. Sci. U.S.A. 103, 5670-5675 (2006)]. Here, we establish that the key dynamical behaviors of the Shiferaw-Karma model are arranged around a set of switches. These are shown to be the main elements for organizing the nonlinear behavior of the model. Exploiting this observation, we show that a piecewise linear caricature of the Shiferaw-Karma model, with a set of appropriate switching manifolds, can be constructed that preserves the physiological interpretation of the original model while being amenable to a systematic mathematical analysis. In illustration of this point, we formulate the dynamics of Ca(2+) cycling (in response to pacing) and compute the properties of periodic orbits in terms of a stroboscopic map that can be constructed without approximation. Using this, we show that alternans emerge via a period-doubling instability and track this bifurcation in terms of physiologically important parameters. We also show that when coupled to a spatially extended model for Ca(2+) transport, the model supports spatially varying patterns of alternans. We analyze the onset of this instability with a generalization of the master stability approach to accommodate the nonsmooth nature of our system.

Democratization in a passive dendritic tree: an analytical investigation.

One way to achieve amplification of distal synaptic inputs on a dendritic tree is to scale the amplitude and/or duration of the synaptic conductance with its distance from the soma. This is an example of what is often referred to as "dendritic democracy". Although well studied experimentally, to date this phenomenon has not been thoroughly explored from a mathematical perspective. In this paper we adopt a passive model of a dendritic tree with distributed excitatory synaptic conductances and analyze a number of key measures of democracy. In particular, via moment methods we derive laws for the transport, from synapse to soma, of strength, characteristic time, and dispersion. These laws lead immediately to synaptic scalings that overcome attenuation with distance. We follow this with a Neumann approximation of Green's representation that readily produces the synaptic scaling that democratizes the peak somatic voltage response. Results are obtained for both idealized geometries and for the more realistic geometry of a rat CA1 pyramidal cell. For each measure of democratization we produce and contrast the synaptic scaling associated with treating the synapse as either a conductance change or a current injection. We find that our respective scalings agree up to a critical distance from the soma and we reveal how this critical distance decreases with decreasing branch radius.

A bidomain threshold model of propagating calcium waves.

We present a bidomain fire-diffuse-fire model that facilitates mathematical analysis of propagating waves of elevated intracellular calcium (Ca(2+)) in living cells. Modeling Ca(2+) release as a threshold process allows the explicit construction of traveling wave solutions to probe the dependence of Ca(2+) wave speed on physiologically important parameters such as the threshold for Ca(2+) release from the endoplasmic reticulum (ER) to the cytosol, the rate of Ca(2+) resequestration from the cytosol to the ER, and the total [Ca(2+)] (cytosolic plus ER). Interestingly, linear stability analysis of the bidomain fire-diffuse-fire model predicts the onset of dynamic wave instabilities leading to the emergence of Ca(2+) waves that propagate in a back-and-forth manner. Numerical simulations are used to confirm the presence of these so-called 'tango waves' and the dependence of Ca(2+) wave speed on the total [Ca(2+)].

Branching dendrites with resonant membrane: a "sum-over-trips" approach.

Dendrites form the major components of neurons. They are complex branching structures that receive and process thousands of synaptic inputs from other neurons. It is well known that dendritic morphology plays an important role in the function of dendrites. Another important contribution to the response characteristics of a single neuron comes from the intrinsic resonant properties of dendritic membrane. In this paper we combine the effects of dendritic branching and resonant membrane dynamics by generalising the "sum-over-trips" approach (Abbott et al. in Biol Cybernetics 66, 49-60 1991). To illustrate how this formalism can shed light on the role of architecture and resonances in determining neuronal output we consider dual recording and reconstruction data from a rat CA1 hippocampal pyramidal cell. Specifically we explore the way in which an Ih current contributes to a voltage overshoot at the soma.

Modeling electrocortical activity through improved local approximations of integral neural field equations.

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular, we are able to treat "patchy" connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a "lattice-directed" traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs.

Waves, bumps, and patterns in neural field theories.

Neural field models of firing rate activity have had a major impact in helping to develop an understanding of the dynamics seen in brain slice preparations. These models typically take the form of integro-differential equations. Their non-local nature has led to the development of a set of analytical and numerical tools for the study of waves, bumps and patterns, based around natural extensions of those used for local differential equation models. In this paper we present a review of such techniques and show how recent advances have opened the way for future studies of neural fields in both one and two dimensions that can incorporate realistic forms of axo-dendritic interactions and the slow intrinsic currents that underlie bursting behaviour in single neurons.