Chaotic dynamics in spatially distributed neuronal networks generate population-wide shared variability.

Neural activity in the cortex is highly variable in response to repeated stimuli. Population recordings across the cortex demonstrate that the variability of neuronal responses is shared among large groups of neurons and concentrates in a low dimensional space. However, the source of the population-wide shared variability is unknown. In this work, we analyzed the dynamical regimes of spatially distributed networks of excitatory and inhibitory neurons. We found chaotic spatiotemporal dynamics in networks with similar excitatory and inhibitory projection widths, an anatomical feature of the cortex. The chaotic solutions contain broadband frequency power in rate variability and have distance-dependent and low-dimensional correlations, in agreement with experimental findings. In addition, rate chaos can be induced by globally correlated noisy inputs. These results suggest that spatiotemporal chaos in cortical networks can explain the shared variability observed in neuronal population responses.

The inhibitory control of traveling waves in cortical networks.

Propagating waves of activity can be evoked and can occur spontaneously in vivo and in vitro in cerebral cortex. These waves are thought to be instrumental in the propagation of information across cortical regions and as a means to modulate the sensitivity of neurons to subsequent stimuli. In normal tissue, the waves are sparse and tightly controlled by inhibition and other negative feedback processes. However, alterations of this balance between excitation and inhibition can lead to pathological behavior such as seizure-type dynamics (with low inhibition) or failure to propagate (with high inhibition). We develop a spiking one-dimensional network of neurons to explore the reliability and control of evoked waves and compare this to a cortical slice preparation where the excitability can be pharmacologically manipulated. We show that the waves enhance sensitivity of the cortical network to stimuli in specific spatial and temporal ways. To gain further insight into the mechanisms of propagation and transitions to pathological behavior, we derive a mean-field model for the synaptic activity. We analyze the mean-field model and a piece-wise constant approximation of it and study the stability of the propagating waves as spatial and temporal properties of the inhibition are altered. We show that that the transition to seizure-like activity is gradual but that the loss of propagation is abrupt and can occur via either the loss of existence of the wave or through a loss of stability leading to complex patterns of propagation.

Mechanisms regulating the properties of inhibition-based gamma oscillations in primate prefrontal and parietal cortices.

In primates, the dorsolateral prefrontal (DLPFC) and posterior parietal (PPC) cortices are key nodes in the working memory network. The working memory-related gamma oscillations induced in these areas, predominantly in layer 3, exhibit higher frequency in DLPFC. Although these regional differences in oscillation frequency are likely essential for information transfer between DLPFC and PPC, the mechanisms underlying these differences remain poorly understood. We investigated, in rhesus monkey, the DLPFC and PPC layer 3 pyramidal neuron (L3PN) properties that might regulate oscillation frequency and assessed the effects of these properties simulating oscillations in computational models. We found that GABAAR-mediated synaptic inhibition synchronizes L3PNs in both areas, but analysis of GABAAR mRNA levels and inhibitory synaptic currents suggested similar mechanisms of inhibition-mediated synchrony in DLPFC and PPC. Basal dendrite spine density and AMPAR/NMDAR mRNA levels were higher in DLPFC L3PNs, whereas excitatory synaptic currents were similar between areas. Therefore, synaptically evoked excitation might be stronger in DLPFC L3PNs due to a greater quantity of synapses in basal dendrites, a main target of recurrent excitation. Simulations in computational networks showed that oscillation frequency and power increased with increasing recurrent excitation, suggesting a mechanism by which the DLPFC-PPC differences in oscillation properties are generated.

Age-Related Changes to the Immune System Exacerbate the Inflammatory Response to Pandemic H1N1 Infection.

Age-induced dysregulation of the immune response is a major contributor to the morbidity and mortality related to influenza a virus infections. Experimental data have shown substantial changes to the activation and maintenance of the immune response will occur with age, but it remains unclear which of these many interrelated changes are most critical to controlling the survival of the host during infection. To ascertain which mechanisms are predominantly responsible for the increased morbidity in elderly hosts, we developed an ordinary differential equation model to simulate the immune response to pandemic H1N1 infection. We fit this model to experimental data measured in young and old macaques. We determined that the severity of the infection in the elderly hosts is caused by a dysregulation in the innate immune response. We also simulated CD8+ T cell exhaustion, a common consequence of chronic and extensive infections. Our simulations indicate that while T cell exhaustion is possible in both age groups, its effects are more severe in the elderly population, as their dysregulated immune response cannot easily compensate for the exhausted T cells. Finally, we explore a therapeutic approach to reversing T cell exhaustion through an inflammatory stimulus. A controlled increase in inflammatory signals can lead to a higher chance of surviving the infection, but excess inflammation will likely lead to septic death. These results indicate that our model captures distinctions in the predominant mechanisms controlling the immune response in younger and older hosts and allows for simulations of clinically relevant therapeutic strategies post-infection.

Traveling waves in non-local pulse-coupled networks.

Traveling phase waves are commonly observed in recordings of the cerebral cortex and are believed to organize behavior across different areas of the brain. We use this as motivation to analyze a one-dimensional network of phase oscillators that are nonlocally coupled via the phase response curve (PRC) and the Dirac delta function. Existence of waves is proven and the dispersion relation is computed. Using the theory of distributions enables us to write and solve an associated stability problem. First and second order perturbation theory is applied to get analytic insight and we show that long waves are stable while short waves are unstable. We apply the results to PRCs that come from mitral neurons. We extend the results to smooth pulse-like coupling by reducing the nonlocal equation to a local one and solving the associated boundary value problem.

Synchronization and locking in oscillators with flexible periods.

Entrainment of a nonlinear oscillator by a periodic external force is a much studied problem in nonlinear dynamics and characterized by the well-known Arnold tongues. The circle map is the simplest such system allowing for stable N:M entrainment where the oscillator produces N cycles for every M stimulus cycles. There are a number of experiments that suggest that entrainment to external stimuli can involve both a shift in the phase and an adjustment of the intrinsic period of the oscillator. Motivated by a recent model of Loehr et al. [J. Exp. Psychol.: Hum. Percept. Perform. 37, 1292 (2011)], we explore a two-dimensional map in which the phase and the period are allowed to update as a function of the phase of the stimulus. We characterize the number and stability of fixed points for different N:M-locking regions, specifically, 1:1, 1:2, 2:3, and their reciprocals, as a function of the sensitivities of the phase and period to the stimulus as well as the degree that the oscillator has a preferred period. We find that even in the limited number of locking regimes explored, there is a great deal of multi-stability of locking modes, and the basins of attraction can be complex and riddled. We also show that when the forcing period changes between a starting and final period, the rate of this change determines, in a complex way, the final locking pattern.

Phase Models Beyond Weak Coupling.

We use the theory of isostable reduction to incorporate higher order effects that are lost in the first order phase reduction of coupled oscillators. We apply this theory to weakly coupled complex Ginzburg-Landau equations, a pair of conductance-based neural models, and finally to a short derivation of the Kuramoto-Sivashinsky equations. Numerical and analytical examples illustrate bifurcations occurring in coupled oscillator networks that can cause standard phase-reduction methods to fail.

Oscillations in working memory and neural binding: A mechanism for multiple memories and their interactions.

Neural oscillations have been recorded and implicated in many different basic brain and cognitive processes. For example, oscillatory neural activity has been suggested to play a role in binding and in the maintenance of information in working memory. With respect to the latter, the majority of work has focused primarily on oscillations in terms of providing a "code" in working memory. However, oscillations may additionally play a fundamental role by enabling or facilitating essential properties and behaviors that neuronal networks must exhibit in order to produce functional working memory and the processes it supports, such as combining items in memory into bound objects or separating bound objects into distinct items. In the present work, we present a biologically plausible working memory model and demonstrate that specific types of stable oscillatory dynamics that arise may play critical roles in providing mechanisms for working memory and the cognitive functions that it supports. Specifically, these roles include (1) enabling a range of different types of binding, (2) both enabling and limiting capacities of bound and distinct items held active in working memory, and (3) facilitating transitions between active working memory states as required in cognitive function. Several key results arise within the examinations, such as the occurrence of different network capacities for working memory and binding, differences in processing times for transitions in working memory states, and the emergence of a combinatorially rich and complex range of oscillatory states that are sufficient to map onto a wide range of cognitive operations supported by working memory, such as variable binding, reasoning, and language. In particular, we show that these oscillatory states and their transitions can provide a specific instantiation of current established connectionist models in representing these functions. Finally, we further characterize the dependence of the relevant oscillatory solutions on certain critical parameters, including mutual inhibition and synaptic timescales.

Recent advances in coupled oscillator theory.

We review the theory of weakly coupled oscillators for smooth systems. We then examine situations where application of the standard theory falls short and illustrate how it can be extended. Specific examples are given to non-smooth systems with applications to the Izhikevich neuron. We then introduce the idea of isostable reduction to explore behaviours that the weak coupling paradigm cannot explain. In an additional example, we show how bifurcations that change the stability of phase-locked solutions in a pair of identical coupled neurons can be understood using the notion of isostable reduction. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Optogenetic Stimulation Shifts the Excitability of Cerebral Cortex from Type I to Type II: Oscillation Onset and Wave Propagation.

Constant optogenetic stimulation targeting both pyramidal cells and inhibitory interneurons has recently been shown to elicit propagating waves of gamma-band (40-80 Hz) oscillations in the local field potential of non-human primate motor cortex. The oscillations emerge with non-zero frequency and small amplitude-the hallmark of a type II excitable medium-yet they also propagate far beyond the stimulation site in the manner of a type I excitable medium. How can neural tissue exhibit both type I and type II excitability? We investigated the apparent contradiction by modeling the cortex as a Wilson-Cowan neural field in which optogenetic stimulation was represented by an external current source. In the absence of any external current, the model operated as a type I excitable medium that supported propagating waves of gamma oscillations similar to those observed in vivo. Applying an external current to the population of inhibitory neurons transformed the model into a type II excitable medium. The findings suggest that cortical tissue normally operates as a type I excitable medium but it is locally transformed into a type II medium by optogenetic stimulation which predominantly targets inhibitory neurons. The proposed mechanism accounts for the graded emergence of gamma oscillations at the stimulation site while retaining propagating waves of gamma oscillations in the non-stimulated tissue. It also predicts that gamma waves can be emitted on every second cycle of a 100 Hz oscillation. That prediction was subsequently confirmed by re-analysis of the neurophysiological data. The model thus offers a theoretical account of how optogenetic stimulation alters the excitability of cortical neural fields.

Greater accuracy and broadened applicability of phase reduction using isostable coordinates.

The applicability of phase models is generally limited by the constraint that the dynamics of a perturbed oscillator must stay near its underlying periodic orbit. Consequently, external perturbations must be sufficiently weak so that these assumptions remain valid. Using the notion of isostables of periodic orbits to provide a simplified coordinate system from which to understand the dynamics transverse to a periodic orbit, we devise a strategy to correct for changing phase dynamics for locations away from the limit cycle. Consequently, these corrected phase dynamics allow for perturbations of larger magnitude without invalidating the underlying assumptions of the reduction. The proposed reduction strategy yields a closed set of equations and can be applied to periodic orbits embedded in arbitrarily high dimensional spaces. We illustrate the utility of this strategy in two models with biological relevance. In the first application, we find that an optimal control strategy for modifying the period of oscillation can be improved with the corrected phase reduction. In the second, the corrected phase reduced dynamics are used to understand adaptation and memory effects resulting from past perturbations.

Stochastic Pacing Inhibits Spatially Discordant Cardiac Alternans.

Depressed heart rate variability is a well-established risk factor for sudden cardiac death in survivors of acute myocardial infarction and for those with congestive heart failure. Although measurements of heart rate variability provide a valuable prognostic tool, it is unclear whether reduced heart rate variability itself is proarrhythmic or if it simply correlates with the severity of autonomic nervous system dysfunction. In this work, we investigate a possible mechanism by which heart rate variability could protect against cardiac arrhythmia. Specifically, in numerical simulations, we observe an inverse relationship between the variance of stochastic pacing and the occurrence of spatially discordant alternans, an arrhythmia that is widely believed to facilitate the development of cardiac fibrillation. By analyzing the effects of conduction velocity restitution, cellular dynamics, electrotonic coupling, and stochastic pacing on the nodal dynamics of spatially discordant alternans, we provide intuition for this observed behavior and propose control strategies to inhibit discordant alternans.

A model of cardiac ryanodine receptor gating predicts experimental Ca2+-dynamics and Ca2+-triggered arrhythmia in the long QT syndrome.

Abnormal Ca2+ handling is well-established as the trigger of cardiac arrhythmia in catecholaminergic polymorphic ventricular tachycardia and digoxin toxicity, but its role remains controversial in Torsade de Pointes (TdP), the arrhythmia associated with the long QT syndrome (LQTS). Recent experimental results show that early afterdepolarizations (EADs) that initiate TdP are caused by spontaneous (non-voltage-triggered) Ca2+ release from Ca2+-overloaded sarcoplasmic reticulum (SR) rather than the activation of the L-type Ca2+-channel window current. In bradycardia and long QT type 2 (LQT2), a second, non-voltage triggered cytosolic Ca2+ elevation increases gradually in amplitude, occurs before overt voltage instability, and then precedes the rise of EADs. Here, we used a modified Shannon-Puglisi-Bers model of rabbit ventricular myocytes to reproduce experimental Ca2+ dynamics in bradycardia and LQT2. Abnormal systolic Ca2+-oscillations and EADs caused by SR Ca2+-release are reproduced in a modified 0-dimensional model, where 3 gates in series control the ryanodine receptor (RyR2) conductance. Two gates control RyR2 activation and inactivation and sense cytosolic Ca2+ while a third gate senses luminal junctional SR Ca2+. The model predicts EADs in bradycardia and low extracellular [K+] and cessation of SR Ca2+-release terminate salvos of EADs. Ca2+-waves, systolic cell-synchronous Ca2+-release, and multifocal diastolic Ca2+ release seen in subcellular Ca2+-mapping experiments are observed in the 2-dimensional version of the model. These results support the role of SR Ca2+-overload, abnormal SR Ca2+-release, and the subsequent activation of the electrogenic Na+/Ca2+-exchanger as the mechanism of TdP. The model offers new insights into the genesis of cardiac arrhythmia and new therapeutic strategies.

Weakly coupled oscillators in a slowly varying world.

We extend the theory of weakly coupled oscillators to incorporate slowly varying inputs and parameters. We employ a combination of regular perturbation and an adiabatic approximation to derive equations for the phase-difference between a pair of oscillators. We apply this to the simple Hopf oscillator and then to a biophysical model. The latter represents the behavior of a neuron that is subject to slow modulation of a muscarinic current such as would occur during transient attention through cholinergic activation. Our method extends and simplifies the recent work of Kurebayashi (Physical Review Letters, 111, 214101, 2013) to include coupling. We apply the method to an all-to-all network and show that there is a waxing and waning of synchrony of modulated neurons.

Phase-locked patterns of the Kuramoto model on 3-regular graphs.

We consider the existence of non-synchronized fixed points to the Kuramoto model defined on sparse networks: specifically, networks where each vertex has degree exactly three. We show that "most" such networks support multiple attracting phase-locked solutions that are not synchronized and study the depth and width of the basins of attraction of these phase-locked solutions. We also show that it is common in "large enough" graphs to find phase-locked solutions where one or more of the links have angle difference greater than π/2.

Stochastic representations of ion channel kinetics and exact stochastic simulation of neuronal dynamics.

In this paper we provide two representations for stochastic ion channel kinetics, and compare the performance of exact simulation with a commonly used numerical approximation strategy. The first representation we present is a random time change representation, popularized by Thomas Kurtz, with the second being analogous to a "Gillespie" representation. Exact stochastic algorithms are provided for the different representations, which are preferable to either (a) fixed time step or (b) piecewise constant propensity algorithms, which still appear in the literature. As examples, we provide versions of the exact algorithms for the Morris-Lecar conductance based model, and detail the error induced, both in a weak and a strong sense, by the use of approximate algorithms on this model. We include ready-to-use implementations of the random time change algorithm in both XPP and Matlab. Finally, through the consideration of parametric sensitivity analysis, we show how the representations presented here are useful in the development of further computational methods. The general representations and simulation strategies provided here are known in other parts of the sciences, but less so in the present setting.

A Boundary Value Approach to Optimization with an Application to Salmonella Competition.

We develop a novel optimization framework to study strategies in ecological competition processes. The optimization method uses theory from dynamical systems describing the asymptotic behavior of a bistable system based on initial conditions, which we implement using a numerical boundary value problem. As an application of our method, we develop a model of the competition between Salmonella Typhimurium and the host's native microflora, which constantly and densely inhabit the intestinal lining of most mammals. S. Typhimurium invades the gut in two distinct phenotypic populations, one virulent and one avirulent, though the avirulent bacteria have the ability to activate a virulence factor and thereby "switch" into the virulent population. Counterintuitively, some studies have found that the combined population of S. Typhimurium gains an environmental advantage over the commensal microbiota after the virulent subpopulation provokes the body's inflammatory defenses. Our model represents the competition between the commensal microbiota, the avirulent salmonella, and the virulent salmonella populations and incorporates a simple representation of the immune response. We use our model to predict optimal strategies that would favor salmonella in its competition with the commensal bacteria.

Evolution of patterns on Conus shells.

The pigmentation patterns of shells in the genus Conus can be generated by a neural-network model of the mantle. We fit model parameters to the shell pigmentation patterns of 19 living Conus species for which a well resolved phylogeny is available. We infer the evolutionary history of these parameters and use these results to infer the pigmentation patterns of ancestral species. The methods we use allow us to characterize the evolutionary history of a neural network, an organ that cannot be preserved in the fossil record. These results are also notable because the inferred patterns of ancestral species sometimes lie outside the range of patterns of their living descendants, and illustrate how development imposes constraints on the evolution of complex phenotypes.

Optimizing working memory with heterogeneity of recurrent cortical excitation.

A neural correlate of parametric working memory is a stimulus-specific rise in neuron firing rate that persists long after the stimulus is removed. Network models with local excitation and broad inhibition support persistent neural activity, linking network architecture and parametric working memory. Cortical neurons receive noisy input fluctuations that cause persistent activity to diffusively wander about the network, degrading memory over time. We explore how cortical architecture that supports parametric working memory affects the diffusion of persistent neural activity. Studying both a spiking network and a simplified potential well model, we show that spatially heterogeneous excitatory coupling stabilizes a discrete number of persistent states, reducing the diffusion of persistent activity over the network. However, heterogeneous coupling also coarse-grains the stimulus representation space, limiting the storage capacity of parametric working memory. The storage errors due to coarse-graining and diffusion trade off so that information transfer between the initial and recalled stimulus is optimized at a fixed network heterogeneity. For sufficiently long delay times, the optimal number of attractors is less than the number of possible stimuli, suggesting that memory networks can under-represent stimulus space to optimize performance. Our results clearly demonstrate the combined effects of network architecture and stochastic fluctuations on parametric memory storage.