Section: New Results

From the mesoscopic to the macroscopic scale

Participants: Laurent Bougrain, Axel Hutt, Pedro Garcia-Rodriguez, Eric Nichols, Guillaume Serrière, Tamara Tosic, Nicole Voges, Mariia Fedotenkova, Meysam Hashemi, Cecilia Lindig-Leon, Kevin Green, Sébastian Rimbert, Thomas Tassone.

To understand the action of anaesthetic drugs on the EEG-signal observed experimentally, Meysam Hashemi has developed and studied several neural mass models [18] , [15] , [16] , [3] . He has identified the thalamo-cortical loop (TCL) as a possible origin of $\delta -$activity. Since loss of consciousness is accompanied by emerging $\delta -$activity, this work relates the TCL to the loss of consciousness.

Increasing the anaesthetic concentration beyond the point of loss of consciousness, EEG-signals exhibit alternating patterns of high and low activity. This activity is called burst suppression. Since these alternations resemble stochastic jumps between low and high activity resting states, Pedro Garcia-Rodriguez and colleagues are working on a stochastic theory based on neural mass models to describe and reproduce these experimental results. Since the minimum mathematical model for such an effect is two-dimensional and does not exhibit potential dynamics, whereas the majority of literature up to date considers one-dimensional stochastic models obeying potential dynamics, Pedro and colleagues had to develop a new stochastic theory. They can show that the two-dimensional dynamics of the neural mass model can be mapped to a one-dimensional stochastic potential model [14] , [13] . This reduction allows to apply standard stochastic theory to describe burst suppression as stochastic transistions. This finding indicates the presence of multiple resting states in the brain and supports a heavily discussed hypothesis on the loss of consciousness.

Biological neural networks are subject to random fluctuations, originating from intrinsic random fluctuations of ions or from external stimulus. The latter neural mass models take into account these fluctuations by assuming additive random input fluctuations. For many decades, these additive fluctuations have been assumed to not affect the stability of the system. However, previous own work has revealed that additive fluctuations tune the stability of nonlinear high-dimensional systems. Since random fluctuations play an important role in the description of neural population dynamics and realistic models consider , it is necessary to study in detail how random fluctuations affect the stability of neural mass models and, hence, how our mathematical model analyses have to be modified. To this end, Axel Hutt and colleagues have performed a stochastic center manifold analysis in a delayed stochastic neural mass model [5] and have found conditions for the stability shift. A first application to delayed stochastic neural fields has revealed how additive random fluctuations may affect EEG-signals [19] , [6] , however additional detailed mathematical studies and the comparison to experimental data are necessary to affirm the importance of the stochastic effect. Essentially, this work emphasizes to take into account nonlinear noise effects in neural mass and neural field models.

Neural mass models do not consider the spatial extension of neural populations and consequently neglect transmission or interaction delay between neurons at different spatial locations. Taking into account the spatial extension and axonal transmission delay, Axel Hutt and colleagues have shown mathematically [7] how travelling activity fronts propagate through neural tissue and how the fronts properties, such as speed, depend on the neural field properties.

The latter neural field model is embedded in a one-dimensional space. Since biological neural populations in the neocortex are organized in two-dimensional layers or sheets, it is necessary to employ neural field models in two spatial dimensions. This causes both theoretically and numerically problems in the presence of axonal transmission delay. Eric Nichols and colleagues has implemented a recent numerical integration algorithm [8] in the visualization software NeuralFieldSimulator, cf. section 5.1 . This software is the basis of numerical bifurcation studies of two-dimensional neural field models [12] , [20] . First analytical results [10] show good accordance to numerical results obtained by the NeuralFieldSimulator.

The latter neural field models assume homogeneous spatial interactions, i.e., neural interactions whose strength just depends on the distance between the two neurons. This assumption is strong and not biologically realistic in certain brain areas. In addition, this assumption constrains the model description of recurrent sequences of EEG patterns, which have been found experimentally, e.g., during the emergence from general anaesthesia. Consequently to be able to describe such recurrent EEG-pattern sequences, it is necessary to improve the mathematical description of EEG-patterns. A promising new model has been derived by Axel Hutt and collagues based on heterogeneous neural fields [1] . In order to extract the recurrence EEG-patterns from data, we have extended a recent recurrence analysis technique [2] . The next step will consist in the combination of the heterogeneous neural field model and the results from the recurrence analysis.

Recurrence analysis extracts temporally reccurrent time windows in multi-dimensional datasets. Typical EEG-signals obtained durin surgery under anaesthesia include one electrode and hence a single time series only. To extract recurrence structures of such one-dimensional signals, Mariia Fedotenkova computes the multi-dimensional time-frequency representation of the signal and has worked out the best analysis technique for this step [9] . In the next step she will compute the recurrence plot for a large dataset of 110 patients under surgery (data obtained from University of Auckland).

In order to understand immobility during anaesthesia and how to supervise unconscious patients automatically in hospital emergency rooms, Cecilia Lindig-Leon studies motor imagery and its detection by BCI techniques. Limb movement execution or imagination induce sensorimotor rhythms that can be detected in EEG recordings. Her recent work considers signal power changes in two frequency bands to detect the elicited EEG rebound, i.e. the increasing of synchronization, at the end of motor imageries. The analysis is based on the database 2a of the BCI competition IV and shows that rebound can be stronger over the alpha frequency band (8-12Hz) than the beta frequency band (12-20Hz). She can demonstrate that the analysis of the alpha frequency band improves the detection of the end of motor imageries. In this context, Cecilia has compared intrinsic multi-class classifiers (i.e., one-step methods) with ensembles of two-class classifiers on dataset 2a of the BCI competition IV for motor imagery. Subsequently, she has compared the classical Common Spatial Pattern (CSP) approach and the CSP by Joint Approximate Diagonalization in order to identify whether the latter method represents an outperforming alternative.

Sleep is strongly related to anaesthesia and we have started working on the improvement of sleep monitors. The basic idea is to consider not only EEG-signals but multiple different physiological signals (e.g. heart pulses, electrocardiogram, EEG, respiration cycle, body movements) to classify sleep stages. By virtue of the different signal natures of different physiological signals, it is challenging to put together these so-called multi-modal signals in a single analysis method. To this end, Tamara Tosic and colleagues employ recurrence analysis techniques which allow to estimate time windows exhibiting temporal synchronization between physiological signals [11] . They have developed a method that is based on artificial data sets and Local Field Potentials measured under anaesthesia. In the next step, applications to sleep data (obtained from CHU Nancy) will allow to extract sleep stages and will evaluate the method.