Mean field modelling of human EEG: application to epilepsy

Aggregated electrical activity from brain regions recorded via an electroencephalogram (EEG), reveal that the brain is never at rest, producing a spectrum of ongoing oscillations that change as a result of different behavioural states and neurological conditions. In particular, this thesis focusses on pathological oscillations associated with absence seizures that typically affect 2–16 year old children. Investigation of the cellular and network mechanisms for absence seizures studies have implicated an abnormality in the cortical and thalamic activity in the generation of absence seizures, which have provided much insight to the potential cause of this disease. A number of competing hypotheses have been suggested, however the precise cause has yet to be determined. This work attempts to provide an explanation of these abnormal rhythms by considering a physiologically based, macroscopic continuum mean-field model of the brain's electrical activity. The methodology taken in this thesis is to assume that many of the physiological details of the involved brain structures can be aggregated into continuum state variables and parameters. The methodology has the advantage to indirectly encapsulate into state variables and parameters, many known physiological mechanisms underlying the genesis of epilepsy, which permits a reduction of the complexity of the problem. That is, a macroscopic description of the involved brain structures involved in epilepsy is taken and then by scanning the parameters of the model, identification of state changes in the system are made possible. Thus, this work demonstrates how changes in brain state as determined in EEG can be understood via dynamical state changes in the model providing an explanation of absence seizures. Furthermore, key observations from both the model and EEG data motivates a number of model reductions. These reductions provide approximate solutions of seizure oscillations and a better understanding of periodic oscillations arising from the involved brain regions. Local analysis of oscillations are performed by employing dynamical systems theory which provide necessary and sufficient conditions for their appearance. Finally local and global stability is then proved for the reduced model, for a reduced region in the parameter space. The results obtained in this thesis can be extended and suggestions are provided for future progress in this area

On the genesis of spike-wave activity in a mean-field model of human brain activity

In this letter, the genesis of spike-wave activity - a hallmark of many generalized epileptic seizures - is investigated in a reduced mean-field model of human neural activity. Drawing upon brain modeling and dynamical systems theory, we demonstrate that the thalamic circuitry of the system is crucial for the generation of these abnormal rhythms, observing that the combination of inhibition from reticular nuclei and excitation from the external signal, interplay to generate the spike-wave oscillation. We demonstarte that this is a nonlinear phenomena and that linear stability analysis is not appropriate to explain such solutions

A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis

The aim of this paper is to explain critical features of the human primary generalized epilepsies by investigating the dynamical bifurcations of a nonlinear model of the brain’s mean field dynamics. The model treats the cortex as a medium for the propagation of waves of electrical activity, incorporating key physiological processes such as propagation delays, membrane physiology and corticothalamic feedback. Previous analyses have demonstrated its descriptive validity in a wide range of healthy states and yielded specific predictions with regards to seizure phenomena. We show that mapping the structure of the nonlinear bifurcation set predicts a number of crucial dynamic processes, including the onset of periodic and chaotic dynamics as well as multistability. Quantitative study of electrophysiological data supports the validity of these predictions and reveals processes unique to the global bifurcation set. Specifically, we argue that the core electrophysiological and cognitive differences between tonic-clonic and absence seizures are predicted by the global bifurcation diagram of the model’s dynamics. The present study is the first to present a unifying explanation of these generalized seizures using the bifurcation analysis of a dynamical model of the brain

Elemental biochemical processes involved in the energy status of cells.

<p>The synthesis sources of ATP are coupled to energy-consumption processes through a network of enzymatic reactions which, interconverting ATP, ADP and AMP, shapes a permanent cycle of synthesis-degradation for the adenine nucleotides. This dynamic functional structure defines the elemental processes of the adenylate energy network, a thermodynamically open system able to accept, store, and supply energy to cells.</p

Values of the kinetic parameters used to simulate some of the dynamics of the adenylate energy system.

<p>Values of the kinetic parameters used to simulate some of the dynamics of the adenylate energy system.</p

Numerical analysis for the model of the adenylate energy system.

<p>a–c: (cf. Scenario I in text) In y-axis we are plotting the max and the min of the different variables α, β and γ. For situations with no oscillations (stable fixed point colored in solid black lines) the max and the min are coincident. For situations with oscillations, the max and the min of the oscillations are plotted separately; in blue we are coloring the max of the oscillation, in red, its minimum value. is the control parameter. The numerical integration shows simple solutions. For small values () the adenine nucleotide concentrations present different stable steady states which lose stability at a Hopf bifurcation at ∼1. For , the attractor is a stable limit cycle. d–f: (Scenario II) The delay r₂ is the control parameter. The numerical bifurcation analysis reveals that the temporal structure is complex, emerging 5 Hopf bifurcations as well as a secondary bifurcation of Neimark-Sacker type. Two pairs of Hopf bifurcations are connected in the parameter space. A third supercritical Hopf bifurcation occurs at r₂∼71.94, rapidly followed by another Hopf bifurcation, subcritical, at r₂∼72.83. This marks the beginning of the region where the system is multi-stable. The last Hopf bifurcation, born at r₂∼72.83, which is subcritical exhibiting the presence of several Torus bifurcations, occurs on a branch of limit cycles when a pair of complex-conjugated Floquet multipliers, leave the unit circle. Branches of stable (resp. unstable) steady states are represented by solid (resp. dashed) black lines; branches of stable (resp. unstable) limit cycles are represented by the max of the oscillation in blue and the minimum in red and by solid (resp. dashed). Hopf bifurcation points are black dots labeled H; Torus bifurcation points are blue dots labeled TR. The bifurcation parameters (Scenario I) and r₂ (Scenario II) are represented on the horizontal axis. The max and min values of each variable are represented on the vertical axis.</p

Emergence of oscillations in the AEC (Scenario II).

<p>Different oscillatory behavior appears when varying r₂, controlling the ADP time delays. (a) For r₂ = 37 s the AEC periodically oscillates with a very low relative amplitude of 0.045. (b–c) Existence of complex AEC oscillatory patterns for: (b) r₂ = 72 s and (c) r2 = 94 s. (d–e) AEC transitions between different oscillatory behavior and steady state patterns for several r₂ values. (d) 50 s, 27 s, 30 s, 32 s, 33 s, 72 s, 52 s. (e) 50 s, 27 s, 30 s, 32 s, 34 s, 36 s, 33 s, 36 s, 38 s, 40 s.</p

AEC dynamics under low production of ATP.

<p>AEC values as a function of time. At very small values (), which represents a strong reduction of the ATP synthesis due to low substrate intake, the dynamic of the adenylate energy system shows a steady state behavior that slowly starts to descend, in a monotone way, up to reach the lowest energy values (AEC ∼0.59) at which the steady state loses stability and oscillatory patterns emerge with a decreasing trend. Finally, when the maximum of the energy charge oscillations reaches a very small value (AEC ∼0.28) the adenylate system suddenly collapses after 12,000 seconds of temporal evolution.</p

Dynamical solutions of Scenario I.

<p>For  = 1.02 (normal activity for the ATP synthesis), periodic oscillations emerge. (a) ATP concentrations. (b) ADP concentrations. (c) AMP concentrations. (d) The Gibbs free energy change for ATP hydrolysis to ADP. (e) The total adenine nucleotide (TAN) pool. It can be observed that ATP and ADP oscillate in anti-phase (the ATP maximum concentration corresponds to the ADP minimum concentration). Likewise, it is noted that the total adenine nucleotide pool shows very small amplitude of only 0.27 and a period around 65 s. (f) ATP transitions between different periodic oscillations and a steady state pattern for several values of (0.97, 1.08, 1.02, 0.97). Maxima and minima values per oscillation are shown in y-axis.</p