Understanding the fundamental mechanisms governing fluctuating oscillations in large-scale cortical circuits is a crucial prelude to a proper knowledge of their role in both adaptive and pathological cortical processes. Neuroscience research in this area has much to gain from understanding the Kuramoto model, a mathematical model that speaks to the very nature of coupled oscillating processes, and which has elucidated the core mechanisms of a range of biological and physical phenomena. In this paper, we provide a brief introduction to the Kuramoto model in its original, rather abstract, form and then focus on modifications that increase its neurobiological plausibility by incorporating topological properties of local cortical connectivity. The extended model elicits elaborate spatial patterns of synchronous oscillations that exhibit persistent dynamical instabilities reminiscent of cortical activity. We review how the Kuramoto model may be recast from an ordinary differential equation to a population level description using the nonlinear Fokker–Planck equation. We argue that such formulations are able to provide a mechanistic and unifying explanation of oscillatory phenomena in the human cortex, such as fluctuating beta oscillations, and their relationship to basic computational processes including multistability, criticality, and information capacity.

Traveling waves of neuronal oscillations have been observed in many cortical regions, including the motor and sensory cortex. Such waves are often modulated in a task-dependent fashion although their precise functional role remains a matter of debate. Here we conjecture that the cortex can utilize the direction and wavelength of traveling waves to encode information. We present a novel neural mechanism by which such information may be decoded by the spatial arrangement of receptors within the dendritic receptor field. In particular, we show how the density distributions of excitatory and inhibitory receptors can combine to act as a spatial filter of wave patterns. The proposed dendritic mechanism ensures that the neuron selectively responds to specific wave patterns, thus constituting a neural basis of pattern decoding. We validate this proposal in the descending motor system, where we model the large receptor fields of the pyramidal tract neurons — the principle outputs of the motor cortex — decoding motor commands encoded in the direction of traveling wave patterns in motor cortex. We use an existing model of field oscillations in motor cortex to investigate how the topology of the pyramidal cell receptor field acts to tune the cells responses to specific oscillatory wave patterns, even when those patterns are highly degraded. The model replicates key findings of the descending motor system during simple motor tasks, including variable interspike intervals and weak corticospinal coherence. By additionally showing how the nature of the wave patterns can be controlled by modulating the topology of local intra-cortical connections, we hence propose a novel integrated neuronal model of encoding and decoding motor commands.

Adaptive changes in behavior require rapid changes in brain states yet the brain must also remain stable. We investigated two neural mechanisms for evoking rapid transitions between spatiotemporal synchronization patterns of beta oscillations (13–30 Hz) in motor cortex. Cortex was modeled as a sheet of neural oscillators that were spatially coupled using a center-surround connection topology. Manipulating the inhibitory surround was found to evoke reliable transitions between synchronous oscillation patterns and traveling waves. These transitions modulated the simulated local field potential in agreement with physiological observations in humans. Intermediate levels of surround inhibition were also found to produce bistable coupling topologies that supported both waves and synchrony. State-dependent perturbation between bistable states produced very rapid transitions but were less reliable. We surmise that motor cortex may thus employ state-dependent computation to achieve very rapid changes between bistable motor states when the demand for speed exceeds the demand for accuracy.

Computational models of neuromotor control require forward models of limb movement that can replicate the natural relationships between muscle activation and joint dynamics without the burdens of excessive anatomical detail. We present a model of a three-link biomechanical limb that emphasizes the dynamics of limb movement within a simplified two-dimensional framework. Muscle co-contraction effects were incorporated into the model by flanking each joint with a pair of antagonist muscles that may be activated independently. Muscle co-contraction is known to alter the damping and stiffness of limb joints without altering net joint torque. Idealized muscle actuators were implemented using the Voigt muscle model which incorporates the parallel elasticity of muscle and tendon but omits series elasticity. The natural force-length-velocity relationships of contractile muscle tissue were incorporated into the actuators using ideal mathematical forms. Numerical stability analysis confirmed that co-contraction of these simplified actuators increased damping in the biomechanical limb consistent with observations of human motor control. Dynamic changes in joint stiffness were excluded by the omission of series elasticity. The analysis also revealed the unexpected finding that distinct stable (bistable) equilibrium positions can co-exist under identical levels of muscle co-contraction. We map the conditions under which bistability arises and prove analytically that monostability (equifinality) is guaranteed when the antagonist muscles are identical. Lastly we verify these analytic findings in the full biomechanical limb model.

The study of fluctuations in time-resolved functional connectivity is a topic of substantial current interest. As the term “dynamic functional connectivity” implies, such fluctuations are believed to arise from dynamics in the neuronal systems generating these signals. While considerable activity currently attends to methodological and statistical issues regarding dynamic functional connectivity, less attention has been paid toward its candidate causes. Here, we review candidate scenarios for dynamic (functional) connectivity that arise in dynamical systems with two or more subsystems; generalized synchronization, itinerancy (a form of metastability), and multistability. Each of these scenarios arises under different configurations of local dynamics and intersystem coupling: We show how they generate time series data with nonlinear and/or nonstationary multivariate statistics. The key issue is that time series generated by coupled nonlinear systems contain a richer temporal structure than matched multivariate (linear) stochastic processes. In turn, this temporal structure yields many of the phenomena proposed as important to large-scale communication and computation in the brain, such as phase-amplitude coupling, complexity, and flexibility. The code for simulating these dynamics is available in a freeware software platform, the Brain Dynamics Toolbox.

Aims Cardiac electrical activity is extraordinarily robust. However, when it goes wrong it can have fatal consequences. Electrical activity in the heart is controlled by the carefully orchestrated activity of more than a dozen different ion conductances. Whilst there is considerable variability in cardiac ion channel expression levels between individuals, studies in rodents have indicated that there are modules of ion channels whose expression co-vary. The aim of this study was to investigate whether meta-analytic co-expression analysis of large-scale gene expression data sets could identify modules of co-expressed cardiac ion channel genes in human hearts that are of functional importance. Methods and Results Meta-analysis of 3653 public human RNA-seq datasets identified a strong correlation between expression of CACNA1C (L-type calcium current, ICaL) and KCNH2 (rapid delayed rectifier K+ current, IKr), which was also observed in human adult heart tissue samples. In silico modeling suggested that co-expression of CACNA1C and KCNH2 would limit the variability in action potential duration seen with variations in expression of ion channel genes and reduce susceptibility to early afterdepolarizations, a surrogate marker for pro-arrhythmia. We also found that levels of KCNH2 and CACNA1C expression are correlated in human induced pluripotent stem cell derived cardiac myocytes and the levels of CACNA1C and KCNH2 expression were inversely correlated with the magnitude of changes in repolarization duration following inhibition of IKr. Conclusions Meta-analytic approaches of multiple independent human gene expression datasets can be used to identify gene modules that are important for regulating heart function. Specifically, we have verified that there is co-expression of CACNA1C and KCNH2 ion channel genes in human heart tissue, and in silico analyses suggest that CACNA1C-KCNH2 co-expression increases the robustness of cardiac electrical activity. Translational Perspective Here, we show, using meta-analysis of multiple independent human gene expression datasets, that there is co-expression of KCNH2-CACNA1C in human heart tissue which was then confirmed in human cardiac myocytes derived from induced pluripotent stem cells. Both in silico and functional studies show that the co-expression of CACNA1C and KCNH2 increases the robustness of cardiac electrical signalling. Our data also suggest that those patients who express higher levels of CACNA1C and KCNH2 are likely to be more susceptible to arrhythmias when exposed to drugs that block IKr, the major cause of drug-induced cardiac arrhythmias.

Nonlinear dynamical systems are increasingly informing both theoretical and empirical branches of neuroscience. The Brain Dynamics Toolbox provides an interactive simulation platform for exploring such systems in Matlab. It supports the major classes of differential equations that arise in computational neuroscience: Ordinary Differential Equations, Delay Differential Equations and Stochastic Differential Equations. The design of the graphical interface fosters intuitive exploration of the dynamics while still supporting scripted parameter explorations and large-scale simulations. Although the toolbox is intended for dynamical models in computational neuroscience, it can be applied to dynamical systems from any domain.Computational neuroscience relies heavily on numerical methods for sim-2 ulating non-linear models of brain dynamics. Software toolkits are the man-3 ifestation of those endeavors. Each one represents an attempt to balance 4 mathematical flexibility with computational convenience. Toolkits such as 5 Genesis [1], Neuron [2] and Brian[3] provide convenient methods to sim-6 ulate conductance-based models of single neurons and networks thereof. The 7 Virtual Brain [4] scales up that approach to the macroscopic dynamics of 8 the whole brain by combining neural field models [5] with anatomical con-9 nectivity datasets [6]. Mathematical toolkits such as Auto [7], Xppaut 10 [8], Matcont [9], PyDSTool [10] and CoCo [11] are useful for analyzing 11 non-linear dynamics but assume advanced mathematical theory. 12 In our experience, the existing computational toolkits often present tech-14 nical barriers to broader audiences in cognitive neuroscience, systems neuro-15 science and neuroimaging. For example, Genesis [1], Neuron [2], Brian [3] 16 and Xppaut [8] each use idiosyncratic languages for defining the differential 17 equations. The Virtual Brain [4], CoCo [11] and PyDSTool [10] use conven-18 tional programming languages (Python and Matlab) but assume advanced 19 object-oriented programming techniques that broader audiences often find 20 confusing. Of all of the existing toolkits, only Xppaut [8] and the Virtual 21 Brain [4] are capable of supporting Ordinary Differential Equations (ODEs), 22 Delay Differential Equations (DDEs) and Stochastic Differential Equations 23 (SDEs). Our Brain Dynamics Toolbox aims to bridge these technical barri-24 ers by allowing those with diverse backgrounds to explore neuronal dynamics 25 through phase space analysis, time series exploration and other methods with 26 minimal programming burden. A custom system of ODEs, DDEs or SDEs 27 can typically be implemented in fewer than 100 lines of standard Matlab 28 code. Object-oriented programming techniques are not required. Once the 29 model is implemented, it can be run interactively in the graphical interface 30 (Figure 1) where a variety of different plotting panels and numerical solvers 31 can be applied with no additional programming effort. The internal states 32 of the graphical interface are accessible to the user's worksp...

Hallucinations occur in both normal and clinical populations. Due to their unpredictability and complexity, the mechanisms underlying hallucinations remain largely untested. Here we show that visual hallucinations can be induced in the normal population by visual flicker, limited to an annulus that constricts content complexity to simple moving grey blobs, allowing objective mechanistic investigation. Hallucination strength peaked at ~11 Hz flicker and was dependent on cortical processing. Hallucinated motion speed increased with flicker rate, when mapped onto visual cortex it was independent of eccentricity, underwent local sensory adaptation and showed the same bistable and mnemonic dynamics as sensory perception. A neural field model with motion selectivity provides a mechanism for both hallucinations and perception. Our results demonstrate that hallucinations can be studied objectively, and they share multiple mechanisms with sensory perception. We anticipate that this assay will be critical to test theories of human consciousness and clinical models of hallucination.DOI: http://dx.doi.org/10.7554/eLife.17072.001