Population based models of cortical drug response: insights from anaesthesia

A great explanatory gap lies between the molecular pharmacology of psychoactive agents and the neurophysiological changes they induce, as recorded by neuroimaging modalities. Causally relating the cellular actions of psychoactive compounds to their influence on population activity is experimentally challenging. Recent developments in the dynamical modelling of neural tissue have attempted to span this explanatory gap between microscopic targets and their macroscopic neurophysiological effects via a range of biologically plausible dynamical models of cortical tissue. Such theoretical models allow exploration of neural dynamics, in particular their modification by drug action. The ability to theoretically bridge scales is due to a biologically plausible averaging of cortical tissue properties. In the resulting macroscopic neural field, individual neurons need not be explicitly represented (as in neural networks). The following paper aims to provide a non-technical introduction to the mean field population modelling of drug action and its recent successes in modelling anaesthesia

The complexity of dynamics in small neural circuits

Mean-field theory is a powerful tool for studying large neural networks. However, when the system is composed of a few neurons, macroscopic differences between the mean-field approximation and the real behavior of the network can arise. Here we introduce a study of the dynamics of a small firing-rate network with excitatory and inhibitory populations, in terms of local and global bifurcations of the neural activity. Our approach is analytically tractable in many respects, and sheds new light on the finite-size effects of the system. In particular, we focus on the formation of multiple branching solutions of the neural equations through spontaneous symmetry-breaking, since this phenomenon increases considerably the complexity of the dynamical behavior of the network. For these reasons, branching points may reveal important mechanisms through which neurons interact and process information, which are not accounted for by the mean-field approximation.Comment: 34 pages, 11 figures. Supplementary materials added, colors of figures 8 and 9 fixed, results unchange

Work Toward a Theory of Brain Function

This dissertation reports research from 1971 to the present, performed in three parts. The first part arose from unilateral electrical stimulation of motivational/reward pathways in the lateral hypothalamus and brain stem of “split-brain” cats, in which the great cerebral commissures were surgically divided. This showed that motivation systems in split-brain animals exert joint influence upon learning in both of the divided cerebral hemispheres, in contrast to the separation of cognitive functions produced by commissurotomy. However, attempts to identify separate signatures of electrocortical activity associated with the diffuse motivational/alerting effects and those of the cortically lateralised processes failed to achieve this goal, and showed that an adequate model of cerebral information processing was lacking. The second part describes how this recognition of inadequacy led into computer simulations of large populations of cortical neurons – work which slowly led my colleagues and me to successful explanations of mechanisms for cortical synchrony and oscillation, and of evoked potentials and the global EEG. These results complemented the work of overseas groups led by Nunez, by Freeman, by Lopes da Silva and others, but also differed from the directions taken by these workers in certain important respects. It became possible to conceive of information transfer in the active cortex as a series of punctuated synchronous equilibria of signal exchange among cortical neurons – equilibria reached repeatedly, with sequential perturbations of the neural activity away from equilibrium caused by exogenous inputs and endogenous pulse-bursting, thus forming a basis for cognitive sequences. The third part reports how the explanation of synchrony gave rise to a new theory of the regulation of embryonic cortical growth and the emergence of mature functional connections. This work was based upon very different assumptions, and reaches very different conclusions, to that of pioneers of the field such as Hubel and Wiesel, whose ideas have dominated cortical physiology for more than fifty years. In conclusion, findings from all the stages of this research are linked together, to show they provide a sketch of the working brain, fitting within and helping to unify wider contemporary concepts of brain function

Metabifurcation analysis of a mean field model of the cortex

Mean field models (MFMs) of cortical tissue incorporate salient features of neural masses to model activity at the population level. One of the common aspects of MFM descriptions is the presence of a high dimensional parameter space capturing neurobiological attributes relevant to brain dynamics. We study the physiological parameter space of a MFM of electrocortical activity and discover robust correlations between physiological attributes of the model cortex and its dynamical features. These correlations are revealed by the study of bifurcation plots, which show that the model responses to changes in inhibition belong to two families. After investigating and characterizing these, we discuss their essential differences in terms of four important aspects: power responses with respect to the modeled action of anesthetics, reaction to exogenous stimuli, distribution of model parameters and oscillatory repertoires when inhibition is enhanced. Furthermore, while the complexity of sustained periodic orbits differs significantly between families, we are able to show how metamorphoses between the families can be brought about by exogenous stimuli. We unveil links between measurable physiological attributes of the brain and dynamical patterns that are not accessible by linear methods. They emerge when the parameter space is partitioned according to bifurcation responses. This partitioning cannot be achieved by the investigation of only a small number of parameter sets, but is the result of an automated bifurcation analysis of a representative sample of 73,454 physiologically admissible sets. Our approach generalizes straightforwardly and is well suited to probing the dynamics of other models with large and complex parameter spaces

Investigation and Modelling of Fetal Sheep Maturation

In this thesis, I study the maturational changes of the fetal sheep ECoG (electrocorticogram) in its third-trimester of gestation (95-140 days of gestation), investigate three continuum models for electrical behaviour of the cortex, and tune the parameters in one of these models to generate the discontinuous EEG waves in the immature cortex. Visual inspection of the ECoG time-series shows that the third-trimester of fetal sheep is comprised of two stages: early third-trimester characterised by bursting activity separated by silent intervals, and late third-trimester with well-defined SWS (slow wave sleep) and REM (rapid eye movement) sleep states. For the late third-trimester, the results of power, correlation time, and SVD (singular value decomposition) entropy analysis demonstrate that the sleep state change is a cortical phase transition—with SWS-to-REM transition being a first-order transition, and REM-to-SWS second-order. Further analyses by correlation time, SVD entropy, and spectral edge frequency display that the differentiation of the two distinct SWS and REM sleep states occurs at about 125 dGA (day gestational age). Spectral analysis divides the third-trimester into four stages in terms of the frequency and amplitude variations of the major resonances. Spindle-like resonances only occur in the first stage. A power surge is observed immediately prior to the emergence of the two sleep states. Most significant changes of the spectrum occur during the fourth stage for both SWS (in amplitude) and REM (in frequency) sleep states. For the modelling of the immature cortex, different theoretical descriptions of cortical behaviour are investigated, including the ccf (cortical column field) model of J. J. Wright, and the Waikato cortical model. For the ccf model at centimetric scale, the time-series, fluctuation power, power law relation, gamma oscillation, phase relation between excitatory and inhibitory elements, power spectral density, and spatial Fourier spectrum are quantified from numerical simulations. From these simulations, I determined that the physiologically sophisticated ccf model is too large and unwieldy for easy tuning to match the electrical response of the immature cortex. The Waikato near-far fast-soma model is constructed by incorporating the back-propagation effect of the action potential into the Waikato fast-soma model, state equations are listed and stability prediction are performed by varying the gap junction diffusion strength, subcortical drive, and the rate constants of the near- and far-dendritic tree. In the end, I selected the classic and simpler Waikato slow-soma mean-field model to use for my immature cortex simulations. Model parameters are customised based on the physiology of the immature cortex, including GABA (an inhibitory neurotransmitter in adult) excitatory effect, number of synaptic connections, and rate constants of the IPSPs (inhibitory postsynaptic potential). After hyperpolarising the neuron resting voltage sufficiently to cause the immature inhibitory neuron to act as an excitatory agent, I alter the rate constant of the IPSP, and study the stability of the immature cortex. The bursting activity and quiet states of the discontinuous EEG are simulated and the gap junction diffusion effect in the immature cortex is also examined. For a rate constant of 18.6 s-1, slow oscillations in the quiet states are generated, and for rate constant of 25 s-1, a possible cortical network oscillation emerges. As far as I know, this is the first time that the GABA excitatory effect has been integrated into a mean-field cortical model and the discontinuous EEG wave successfully simulated in a qualitative way

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. © 2010 Elsevier Inc

Dynamics of biologically informed neural mass models of the brain

This book contributes to the development and analysis of computational models that help brain function to be understood. The mean activity of a brain area is mathematically modeled in such a way as to strike a balance between tractability and biological plausibility. Neural mass models (NMM) are used to describe switching between qualitatively different regimes (such as those due to pharmacological interventions, epilepsy, sleep, or context-induced state changes), and to explain resonance phenomena in a photic driving experiment. The description of varying states in an ordered sequence gives a principle scheme for the modeling of complex phenomena on multiple time scales. The NMM is matched to the photic driving experiment routinely applied in the diagnosis of such diseases as epilepsy, migraine, schizophrenia and depression. The model reproduces the clinically relevant entrainment effect and predictions are made for improving the experimental setting.Die vorliegende Arbeit stellt einen Beitrag zur Entwicklung und Analyse von Computermodellen zum Verständnis von Hirnfunktionen dar. Es wird die mittlere Aktivität eines Hirnareals analytisch einfach und dabei biologisch plausibel modelliert. Auf Grundlage eines Neuronalen Massenmodells (NMM) werden die Wechsel zwischen Oszillationsregimen (z.B. durch pharmakologisch, epilepsie-, schlaf- oder kontextbedingte Zustandsänderungen) als geordnete Folge beschrieben und Resonanzphänomene in einem Photic-Driving-Experiment erklärt. Dieses NMM kann sehr komplexe Dynamiken (z.B. Chaos) innerhalb biologisch plausibler Parameterbereiche hervorbringen. Um das Verhalten abzuschätzen, wird das NMM als Funktion konstanter Eingangsgrößen und charakteristischer Zeitenkonstanten vollständig auf Bifurkationen untersucht und klassifiziert. Dies ermöglicht die Beschreibung wechselnder Regime als geordnete Folge durch spezifische Eingangstrajektorien. Es wird ein Prinzip vorgestellt, um komplexe Phänomene durch Prozesse verschiedener Zeitskalen darzustellen. Da aufgrund rhythmischer Stimuli und der intrinsischen Rhythmen von Neuronenverbänden die Eingangsgrößen häufig periodisch sind, wird das Verhalten des NMM als Funktion der Intensität und Frequenz einer periodischen Stimulation mittels der zugehörigen Lyapunov-Spektren und der Zeitreihen charakterisiert. Auf der Basis der größten Lyapunov-Exponenten wird das NMM mit dem Photic-Driving-Experiment überein gebracht. Dieses Experiment findet routinemäßige Anwendung in der Diagnostik verschiedener Erkrankungen wie Epilepsie, Migräne, Schizophrenie und Depression. Durch die Anwendung des vorgestellten NMM wird der für die Diagnostik entscheidende Mitnahmeeffekt reproduziert und es werden Vorhersagen für eine Verbesserung der Indikation getroffen

Conditions for wave trains in spiking neural networks

Spatiotemporal patterns such as traveling waves are frequently observed in recordings of neural activity. The mechanisms underlying the generation of such patterns are largely unknown. Previous studies have investigated the existence and uniqueness of different types of waves or bumps of activity using neural-field models, phenomenological coarse-grained descriptions of neural-network dynamics. But it remains unclear how these insights can be transferred to more biologically realistic networks of spiking neurons, where individual neurons fire irregularly. Here, we employ mean-field theory to reduce a microscopic model of leaky integrate-and-fire (LIF) neurons with distance-dependent connectivity to an effective neural-field model. In contrast to existing phenomenological descriptions, the dynamics in this neural-field model depends on the mean and the variance in the synaptic input, both determining the amplitude and the temporal structure of the resulting effective coupling kernel. For the neural-field model we employ liner stability analysis to derive conditions for the existence of spatial and temporal oscillations and wave trains, that is, temporally and spatially periodic traveling waves. We first prove that wave trains cannot occur in a single homogeneous population of neurons, irrespective of the form of distance dependence of the connection probability. Compatible with the architecture of cortical neural networks, wave trains emerge in two-population networks of excitatory and inhibitory neurons as a combination of delay-induced temporal oscillations and spatial oscillations due to distance-dependent connectivity profiles. Finally, we demonstrate quantitative agreement between predictions of the analytically tractable neural-field model and numerical simulations of both networks of nonlinear rate-based units and networks of LIF neurons.Comment: 36 pages, 8 figures, 4 table

Neural field models with a dendritic dimension

Neural field models (NFMs) describe the spatio-temporal evolution of neuronal populations as a continuous excitable medium. The resulting tissue-level description can be employed to fit data from macroscopic recordings of electrocortical brain ac-tivity like the electroencephalogram (EEG) and local field potentials (LFPs). The standard neural field approach models the cortex as a two-dimensional sheet, ne-glecting the actual cortical depth. Although a small number of studies have con-sidered the anatomical cortical layers to model different connectivity patterns, their mathematical description does not commonly use the cortical depth to determine the model dynamics. Therefore, within the framework of neural field theory, the impact of dendrites on brain activity remains far from being exhaustively explored. In the present work, we extend the geometry of a two-dimensional (2D) NFM to incorporate a dendritic dimension for the excitatory neural populations, repre-senting the cortical depth. Dendritic trees are modelled as linear cables, spatially discretized in multiple subsections (compartments). Spatio-temporal patterns of the new cortical model are studied for systems consisting of either a single or multiple microcolumns. A powerful approximation, extended from the one for the 2D NFM, is introduced to predict the power spectral density of the mean membrane potential from the Jacobian matrix of the linearized system evaluated at a singular point. Our numerical analysis reveals a variety of dynamics, ranging from those characterized by "flat" power spectra without alpha rhythmicity due to signal loss over the tree, up to sharp alpha resonances corresponding to proximity to a Hopf bifurcation. The research focuses on the identification of plausible EEG dynamics, e.g., those exhibit-ing a dominant alpha activity, conceived as the central rhythm of spontaneous EEG. Crucial to this endeavour has been the careful tuning of key dendritic parameters introduced with the three-dimensional (3D) geometry, such as the "synaptic factor" (i.e. synaptic conductance) and the membrane length constant, and wider parameter sweeps using the Particle Swarm Optimization (PSO) technique. The dynamics are mainly studied for a single microcolumn systems with different dendritic configurations (e.g. varying conductance and length constant) during synchronous and asynchronous synaptic activation in either a single or multiple dendritic domains. Our results explain the impact of key dendritic parameters on the 3D NFM dynamics. Heuristics characterizing these effects can be regarded as representative of the well-known phenomenon of "dendritic democracy", classically indicating the normalisation of post-synaptic somatic potentials compensating for dendritic filtering activity. While several experimental studies have investigated the genesis of this compensation, to date this phenomenon has not be explored concerning a potential interplay with the alpha rhythm. Our findings suggest that physiological conditions enhancing the onset of action potentials in active models also promote alphoid dynamics in our passive neural field models including the dendritic dimension. In particular, synaptic strength has to increase with distance from the soma. We found several parameter configurations giving rise to alpha rhythmicity in the 3D geometry, Dynamical analysis highlights the impact of the key dendritic parameters at different cortical depths on the genesis of alpha rhythm, providing a clearer insight into the dendritic mechanisms and cortical dynamics. Indeed, the model can be used as a valid starting point for NF studies aiming to encompass further dendritic properties, implement more detailed connctivity schemes and incorporate data from depth electrode recordings