Gap junctions mediate large-scale Turing structures in a mean-field cortex driven by subcortical noise

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

The Anterolateral Barrel Subfield Differs from the Posteromedial Barrel Subfield in the Morphology and Cell Density of Parvalbumin-Positive GABAergic Interneurons

Layer 4 of the rodent somatosensory cortex has unitary structures called barrels that receive tactile information from individual vibrissae. Barrels in the anterolateral barrel subfield (ALBSF) are much smaller and have gained less attention than larger barrels in the posteromedial barrel subfield (PMBSF), though the former outnumber the latter. We compared the morphological features of barrels between the ALBSF and PMBSF in male mice using deformation-free tangential sections and confocal optical slice-based, precise reconstructions of barrels. The average volume of a single barrel in the ALBSF was 34.7% of that in the PMBSF, but the numerical density of parvalbumin (PV)-positive interneurons in the former was 1.49 times higher than that in the latter. Moreover, PV neuron density in septa was 2.08 times higher in the ALBSF than that in the PMBSF. The proportions of PV neuron number to both all neuron number and all GABAergic neuron number in the ALBSF were also higher than those in the PMBSF. Somata of PV neurons in barrels and septa in the ALBSF received 1.64 and 1.50 times more vesicular glutamate transporter Type 2-labeled boutons than those in the PMBSF, suggesting more potent feedforward inhibitory circuits in the ALBSF. The mode of connectivity through dendritic gap junctions among PV neurons also differed between the ALBSF and PMBSF. Clusters of smaller unitary structures containing a higher density of representative GABAergic interneurons with differential morphological features in the ALBSF suggest a division of functional roles in the two vibrissa-barrel systems, as has been demonstrated by behavioral studies.

Complexity of cortical wave patterns of the wake mouse cortex

Rich spatiotemporal dynamics of cortical activity, including complex and diverse wave patterns, have been identified during unconscious and conscious brain states. Yet, how these activity patterns emerge across different levels of wakefulness remain unclear. Here we study the evolution of wave patterns utilizing data from high spatiotemporal resolution optical voltage imaging of mice transitioning from barbiturate-induced anesthesia to wakefulness (N = 5) and awake mice (N = 4). We find that, as the brain transitions into wakefulness, there is a reduction in hemisphere-scale voltage waves, and an increase in local wave events and complexity. A neural mass model recapitulates the essential cellular-level features and shows how the dynamical competition between global and local spatiotemporal patterns and long-range connections can explain the experimental observations. These mechanisms possibly endow the awake cortex with enhanced integrative processing capabilities.

Mean-Field Models for EEG/MEG: From Oscillations to Waves

Neural mass models have been used since the 1970s to model the coarse-grained activity of large populations of neurons. They have proven especially fruitful for understanding brain rhythms. However, although motivated by neurobiological considerations they are phenomenological in nature, and cannot hope to recreate some of the rich repertoire of responses seen in real neuronal tissue. Here we consider a simple spiking neuron network model that has recently been shown to admit an exact mean-field description for both synaptic and gap-junction interactions. The mean-field model takes a similar form to a standard neural mass model, with an additional dynamical equation to describe the evolution of within-population synchrony. As well as reviewing the origins of this next generation mass model we discuss its extension to describe an idealised spatially extended planar cortex. To emphasise the usefulness of this model for EEG/MEG modelling we show how it can be used to uncover the role of local gap-junction coupling in shaping large scale synaptic waves.

Investigating the role of gap junctions in seizure wave propagation

The effect of gap junctions as well as the biological mechanisms behind seizure wave propagation is not completely understood. In this work, we use a simple neural field model to study the possible influence of gap junctions specifically on cortical wave propagation that has been observed in vivo preceding seizure termination. We consider a voltage-based neural field model consisting of an excitatory and an inhibitory population as well as both chemical and gap junction-like synapses. We are able to approximate important properties of cortical wave propagation previously observed in vivo before seizure termination. This model adds support to existing evidence from models and clinical data suggesting a key role of gap junctions in seizure wave propagation. In particular, we found that in this model gap junction-like connectivity determines the propagation of one-bump or two-bump traveling wave solutions with features consistent with the clinical data. For sufficiently increased gap junction connectivity, wave solutions cease to exist. Moreover, gap junction connectivity needs to be sufficiently low or moderate to permit the existence of linearly stable solutions of interest.

Spinodal decomposition in a mean-field model of the cortex: Emergence of hexagonally symmetric activation patterns

Spinodal decomposition is a well-known pattern-forming mechanism in metallurgic alloys, semiconductor crystals, and colloidal gels. In metallurgy, if a heated sample of a homogeneous Zn-Al alloy is suddenly quenched below a critical temperature, then the sample can spontaneously precipitate into inhomogenous textures of Zn- and Al-rich regions with significantly altered material properties such as ductility and hardness. Here we report on our recent discovery that a two-dimensional model of the human cortex with inhibitory diffusion can, under particular homogeneous initial conditions, exhibit a form of nonconserved spinodal decomposition in which regions of the cortex self-organize into hexagonally distributed binary patches of activity and inactivity. Fine-scale patterns precipitate rapidly, and then the dynamics slows to render coarser-scale shapes which can ripen into a range of slowly evolving patterns including mazelike labyrinths, hexagonal islands and continents, nucleating "mitotic cells" which grow to a critical size then subdivide, and inverse nucleations in which quiescent islands are surrounded by a sea of activity. One interesting class of activity coalesces into a soliton-like narrow ribbon of depolarization that traverses the cortex at ∼4cm/s. We speculate that this may correspond to the thus far unexplained interictal waves of cortical activation that precede grand-mal seizure in an epileptic event. We note that spinodal decomposition is quite distinct from the Turing mechanism for symmetry breaking in cortex investigated in earlier work by the authors [Steyn-Ross et al., Phys. Rev. E 76, 011916 (2007)PLEEE81539-375510.1103/PhysRevE.76.011916].

Anesthesia modifies subthreshold critical slowing down in a stochastic Hodgkin-Huxley-like model with inhibitory synaptic input

The dynamics of a stochastic type-I Hodgkin-Huxley-like point neuron model exposed to inhibitory synaptic noise are investigated as a function of distance from spiking threshold and the inhibitory influence of the general anesthetic agent propofol. The model is biologically motivated and includes the effects of intrinsic ion-channel noise via a stochastic differential equation description as well as inhibitory synaptic noise modeled as multiple Poisson-distributed impulse trains with saturating response functions. The effect of propofol on these synapses is incorporated through this drug's principal influence on fast inhibitory neurotransmission mediated by γ-aminobutyric acid (GABA) type-A receptors via reduction of the synaptic response decay rate. As the neuron model approaches spiking threshold from below, we track membrane voltage fluctuation statistics of numerically simulated stochastic trajectories. We find that for a given distance from spiking threshold, increasing the magnitude of anesthetic-induced inhibition is associated with augmented signatures of critical slowing: fluctuation amplitudes and correlation times grow as spectral power is increasingly focused at 0 Hz. Furthermore, as a function of distance from threshold, anesthesia significantly modifies the power-law exponents for variance and correlation time divergences observable in stochastic trajectories. Compared to the inverse square root power-law scaling of these quantities anticipated for the saddle-node bifurcation of type-I neurons in the absence of anesthesia, increasing anesthetic-induced inhibition results in an observable exponent <-0.5 for variance and >-0.5 for correlation time divergences. However, these behaviors eventually break down as distance from threshold goes to zero with both the variance and correlation time converging to common values independent of anesthesia. Compared to the case of no synaptic input, linearization of an approximating multivariate Ornstein-Uhlenbeck model reveals these effects to be the consequence of an additional slow eigenvalue associated with synaptic activity that competes with those of the underlying point neuron in a manner that depends on distance from spiking threshold.

Gap junctional signaling in pattern regulation: Physiological network connectivity instructs growth and form

Gap junctions (GJs) are aqueous channels that allow cells to communicate via physiological signals directly. The role of gap junctional connectivity in determining single-cell functions has long been recognized. However, GJs have another important role: the regulation of large-scale anatomical pattern. GJs are not only versatile computational elements that allow cells to control which small molecule signals they receive and emit, but also establish connectivity patterns within large groups of cells. By dynamically regulating the topology of bioelectric networks in vivo, GJs underlie the ability of many tissues to implement complex morphogenesis. Here, a review of recent data on patterning roles of GJs in growth of the zebrafish fin, the establishment of left-right patterning, the developmental dysregulation known as cancer, and the control of large-scale head-tail polarity, and head shape in planarian regeneration has been reported. A perspective in which GJs are not only molecular features functioning in single cells, but also enable global neural-like dynamics in non-neural somatic tissues has been proposed. This view suggests a rich program of future work which capitalizes on the rapid advances in the biophysics of GJs to exploit GJ-mediated global dynamics for applications in birth defects, regenerative medicine, and morphogenetic bioengineering. © 2016 Wiley Periodicals, Inc. Develop Neurobiol 77: 643-673, 2017.

Weakly coupled map lattice models for multicellular patterning and collective normalization of abnormal single-cell states

We present a weakly coupled map lattice model for patterning that explores the effects exerted by weakening the local dynamic rules on model biological and artificial networks composed of two-state building blocks (cells). To this end, we use two cellular automata models based on (i) a smooth majority rule (model I) and (ii) a set of rules similar to those of Conway's Game of Life (model II). The normal and abnormal cell states evolve according to local rules that are modulated by a parameter κ. This parameter quantifies the effective weakening of the prescribed rules due to the limited coupling of each cell to its neighborhood and can be experimentally controlled by appropriate external agents. The emergent spatiotemporal maps of single-cell states should be of significance for positional information processes as well as for intercellular communication in tumorigenesis, where the collective normalization of abnormal single-cell states by a predominantly normal neighborhood may be crucial.

From individual spiking neurons to population behavior: Systematic elimination of short-wavelength spatial modes

Mean-field models of the brain approximate spiking dynamics by assuming that each neuron responds to its neighbors via a naive spatial average that neglects local fluctuations and correlations in firing activity. In this paper we address this issue by introducing a rigorous formalism to enable spatial coarse-graining of spiking dynamics, scaling from the microscopic level of a single type 1 (integrator) neuron to a macroscopic assembly of spiking neurons that are interconnected by chemical synapses and nearest-neighbor gap junctions. Spiking behavior at the single-neuron scale ℓ≈10μm is described by Wilson's two-variable conductance-based equations [H. R. Wilson, J. Theor. Biol. 200, 375 (1999)], driven by fields of incoming neural activity from neighboring neurons. We map these equations to a coarser spatial resolution of grid length Bℓ, with B≫1 being the blocking ratio linking micro and macro scales. Our method systematically eliminates high-frequency (short-wavelength) spatial modes q(->) in favor of low-frequency spatial modes Q(->) using an adiabatic elimination procedure that has been shown to be equivalent to the path-integral coarse graining applied to renormalization group theory of critical phenomena. This bottom-up neural regridding allows us to track the percolation of synaptic and ion-channel noise from the single neuron up to the scale of macroscopic population-average variables. Anticipated applications of neural regridding include extraction of the current-to-firing-rate transfer function, investigation of fluctuation criticality near phase-transition tipping points, determination of spatial scaling laws for avalanche events, and prediction of the spatial extent of self-organized macrocolumnar structures. As a first-order exemplar of the method, we recover nonlinear corrections for a coarse-grained Wilson spiking neuron embedded in a network of identical diffusively coupled neurons whose chemical synapses have been disabled. Intriguingly, we find that reblocking transforms the original type 1 Wilson integrator into a type 2 resonator whose spike-rate transfer function exhibits abrupt spiking onset with near-vertical takeoff and chaotic dynamics just above threshold.

Closed-loop feedback control and bifurcation analysis of epileptiform activity via optogenetic stimulation in a mathematical model of human cortex

Optogenetics provides a method of neuron stimulation that has high spatial, temporal, and cell-type specificity. Here we present a model of optogenetic feedback control that targets the inhibitory population, which expresses light-sensitive channelrhodopsin-2 channels, in a mean-field model of undifferentiated cortex that is driven to seizures. The inhibitory population is illuminated with an intensity that is a function of electrode measurements obtained via the cortical model. We test the efficacy of this control method on seizurelike activity observed in two parameter spaces of the cortical model that most closely correspond to seizures observed in patients. We also compare the effect of closed-loop and open-loop control on seizurelike activity using a less-complicated ordinary differential equation model of the undifferentiated cortex in parameter space. Seizurelike activity is successfully suppressed in both parameter planes using optimal illumination intensities less likely to have adverse effects on cortical tissue.

EEG slow-wave coherence changes in propofol-induced general anesthesia: experiment and theory

The electroencephalogram (EEG) patterns recorded during general anesthetic-induced coma are closely similar to those seen during slow-wave sleep, the deepest stage of natural sleep; both states show patterns dominated by large amplitude slow waves. Slow oscillations are believed to be important for memory consolidation during natural sleep. Tracking the emergence of slow-wave oscillations during transition to unconsciousness may help us to identify drug-induced alterations of the underlying brain state, and provide insight into the mechanisms of general anesthesia. Although cellular-based mechanisms have been proposed, the origin of the slow oscillation has not yet been unambiguously established. A recent theoretical study by Steyn-Ross et al. (2013) proposes that the slow oscillation is a network, rather than cellular phenomenon. Modeling anesthesia as a moderate reduction in gap-junction interneuronal coupling, they predict an unconscious state signposted by emergent low-frequency oscillations with chaotic dynamics in space and time. They suggest that anesthetic slow-waves arise from a competitive interaction between symmetry-breaking instabilities in space (Turing) and time (Hopf), modulated by gap-junction coupling strength. A significant prediction of their model is that EEG phase coherence will decrease as the cortex transits from Turing-Hopf balance (wake) to Hopf-dominated chaotic slow-waves (unconsciousness). Here, we investigate changes in phase coherence during induction of general anesthesia. After examining 128-channel EEG traces recorded from five volunteers undergoing propofol anesthesia, we report a significant drop in sub-delta band (0.05-1.5 Hz) slow-wave coherence between frontal, occipital, and frontal-occipital electrode pairs, with the most pronounced wake-vs.-unconscious coherence changes occurring at the frontal cortex.

Simulations of pattern dynamics for reaction-diffusion systems via SIMULINK

BACKGROUND

Investigation of the nonlinear pattern dynamics of a reaction-diffusion system almost always requires numerical solution of the system's set of defining differential equations. Traditionally, this would be done by selecting an appropriate differential equation solver from a library of such solvers, then writing computer codes (in a programming language such as C or Matlab) to access the selected solver and display the integrated results as a function of space and time. This "code-based" approach is flexible and powerful, but requires a certain level of programming sophistication. A modern alternative is to use a graphical programming interface such as Simulink to construct a data-flow diagram by assembling and linking appropriate code blocks drawn from a library. The result is a visual representation of the inter-relationships between the state variables whose output can be made completely equivalent to the code-based solution.

RESULTS

As a tutorial introduction, we first demonstrate application of the Simulink data-flow technique to the classical van der Pol nonlinear oscillator, and compare Matlab and Simulink coding approaches to solving the van der Pol ordinary differential equations. We then show how to introduce space (in one and two dimensions) by solving numerically the partial differential equations for two different reaction-diffusion systems: the well-known Brusselator chemical reactor, and a continuum model for a two-dimensional sheet of human cortex whose neurons are linked by both chemical and electrical (diffusive) synapses. We compare the relative performances of the Matlab and Simulink implementations.

CONCLUSIONS

The pattern simulations by Simulink are in good agreement with theoretical predictions. Compared with traditional coding approaches, the Simulink block-diagram paradigm reduces the time and programming burden required to implement a solution for reaction-diffusion systems of equations. Construction of the block-diagram does not require high-level programming skills, and the graphical interface lends itself to easy modification and use by non-experts.

Open loop optogenetic control of simulated cortical epileptiform activity

We present a model for the use of open loop optogenetic control to inhibit epileptiform activity in a meso scale model of the human cortex. The meso scale cortical model first developed by Liley et al. (2001) is extended to two dimensions and the nature of the seizure waves is studied. We adapt to the meso scale a 4 state functional model of Channelrhodopsin-2 (ChR2) ion channels. The effects of pulsed and constant illumination on the conductance of these ion channels is presented. The inhibitory cell population is targeted for the application of open loop control. Seizure waves are successfully suppressed and the inherent properties of the optogenetic channels ensures charge balance in the cortex, protecting it from damage.

In vitro electrical conductivity of seizing and non-seizing mouse brain slices at 10 kHz

The electrical conductivity of small samples of mouse cortex (in vitro) has been measured at 10 kHz through the four-electrode method of van der Pauw. Brain slices from three mice were prepared under seizing and non-seizing conditions by changing the concentration of magnesium in the artificial cerebrospinal fluid used to maintain the tissue. These slices provided 121 square samples of cortical tissue; the conductivity of these samples was measured with an Agilent E4980A four-point impedance monitor. Of these, 73 samples were considered acceptable on the grounds of having good electrical contact between electrodes and tissue excluding outlier measurements. Results show that there is a significant difference (p = 0.03) in the conductivities of the samples under the two conditions. The seizing and non-seizing samples have mean conductivities of 0.33 and 0.36 S m(-1), respectively; however, these quantitative values should be used with caution as they are both subject to similar systematic uncertainties due to non-ideal temperature conditions and non-ideal placement of electrodes. We hypothesize that the difference between them, which is more robust to uncertainty, is due to the changing gap junction connectivity during seizures.

Gap junctions modulate seizures in a mean-field model of general anesthesia for the cortex

During slow-wave sleep, general anesthesia, and generalized seizures, there is an absence of consciousness. These states are characterized by low-frequency large-amplitude traveling waves in scalp electroencephalogram. Therefore the oscillatory state might be an indication of failure to form coherent neuronal assemblies necessary for consciousness. A generalized seizure event is a pathological brain state that is the clearest manifestation of waves of synchronized neuronal activity. Since gap junctions provide a direct electrical connection between adjoining neurons, thus enhancing synchronous behavior, reducing gap-junction conductance should suppress seizures; however there is no clear experimental evidence for this. Here we report theoretical predictions for a physiologically-based cortical model that describes the general anesthetic phase transition from consciousness to coma, and includes both chemical synaptic and direct electrotonic synapses. The model dynamics exhibits both Hopf (temporal) and Turing (spatial) instabilities; the Hopf instability corresponds to the slow (≲8 Hz) oscillatory states similar to those seen in slow-wave sleep, general anesthesia, and seizures. We argue that a delicately balanced interplay between Hopf and Turing modes provides a canonical mechanism for the default non-cognitive rest state of the brain. We show that the Turing mode, set by gap-junction diffusion, is generally protective against entering oscillatory modes; and that weakening the Turing mode by reducing gap conduction can release an uncontrolled Hopf oscillation and hence an increased propensity for seizure and simultaneously an increased sensitivity to GABAergic anesthesia.

The "conscious pilot"-dendritic synchrony moves through the brain to mediate consciousness

Cognitive brain functions including sensory processing and control of behavior are understood as "neurocomputation" in axonal-dendritic synaptic networks of "integrate-and-fire" neurons. Cognitive neurocomputation with consciousness is accompanied by 30- to 90-Hz gamma synchrony electroencephalography (EEG), and non-conscious neurocomputation is not. Gamma synchrony EEG derives largely from neuronal groups linked by dendritic-dendritic gap junctions, forming transient syncytia ("dendritic webs") in input/integration layers oriented sideways to axonal-dendritic neurocomputational flow. As gap junctions open and close, a gamma-synchronized dendritic web can rapidly change topology and move through the brain as a spatiotemporal envelope performing collective integration and volitional choices correlating with consciousness. The "conscious pilot" is a metaphorical description for a mobile gamma-synchronized dendritic web as vehicle for a conscious agent/pilot which experiences and assumes control of otherwise non-conscious auto-pilot neurocomputation.

Transient Turing patterns in a neural field model

We investigate Turing bifurcations in a neural field model with one spatial dimension. For some parameter values the resulting Turing patterns are stable, while for others the patterns appear transiently. We show that this difference is due to the relative position in parameter space of the saddle-node bifurcation of a spatially periodic pattern and the Turing bifurcation point. By varying parameters we are able to observe transient patterns whose duration scales in the same way as type-I intermittency. Similar behavior occurs in two spatial dimensions.

Modeling brain activation patterns for the default and cognitive states

We argue that spatial patterns of cortical activation observed with EEG, MEG and fMRI might arise from spontaneous self-organisation of interacting populations of excitatory and inhibitory neurons. We examine the dynamical behavior of a mean-field cortical model that includes chemical and electrical (gap-junction) synapses, focusing on two limiting cases: the "slow-soma" limit with slow voltage feedback from soma to dendrite, and the "fast-soma" limit in which the feedback action of soma voltage onto dendrite reversal potentials is instantaneous. For slow soma-dendrite feedback, we find a low-frequency (approximately 1 Hz) dynamic Hopf instability, and a stationary Turing instability that catalyzes formation of patterned distributions of cortical firing-rate activity with pattern wavelength approximately 2 cm. Turing instability can only be triggered when gap-junction diffusion between inhibitory neurons is strong, but patterning is destroyed if the tonic level of subcortical excitation is raised sufficiently. Interaction between the Hopf and Turing instabilities may describe the non-cognitive background or "default" state of the brain, as observed by BOLD imaging. In the fast-soma limit, the model predicts a high-frequency Hopf (approximately 35 Hz) instability, and a traveling-wave gamma-band instability that manifests as a 2-D standing-wave pattern oscillating in place at approximately 30 Hz. Small levels of inhibitory diffusion enhance and broaden the definition of the gamma antinodal regions by suppressing higher-frequency spatial modes, but gamma emergence is not contingent on the presence of inhibitory gap junctions; higher levels of diffusion suppress gamma activity. Fast-soma instabilities are enhanced by increased subcortical stimulation. Prompt soma-dendrite feedback may be an essential component of the genesis and large-scale cortical synchrony of gamma activity observed at the point of cognition.