The human brain remains one of the most complex and enigmatic systems, driving research in both neuroscience and computing. Understanding its macroscopic dynamics is essential for uncovering and intervening in cognitive processes. As part of the Dutch Brain Interface Initiative, we develop mathematical models to understand and influence neural processes via closed-loop neurostimulation. In this thesis we study memory consolidation, a process in which temporary memories in the Hippocampus are transformed into long-term, stable memories in the Neocortex. This relies on bidirectional communication between the two brain regions occurring via slowand fast oscillatory signals. The main goal of the current work is to obtain a data-driven identification method yielding linear models of these oscillatory signals, and of their interaction.
Current neuroscience modeling efforts often result in nonlinear and complex models, that are hard to fit to measurement data and to employ for control purposes. In the present work, we have developed a method that can reliably identify a linear model of memory consolidation dynamics. Inspired by the Hankel Alternative View of Koopman (HAVOK) Analysis, the HAVOK-Informed AR Model (HAVOK-AR) and HAVOK-Informed ARX Model (HAVOK-ARX) are developed for identification of autonomousand forced nonlinear systems, respectively, finding a finite approximation of the Koopman operator. Our models are shown to perform well on numerical simulation examples and nonlinear benchmark datasets, as well as experimental data of the mouse brain.
Cross-validation indicates that internal brain dynamics are comparable across animal subjects.Lastly, the HAVOK-AR(X) approach confirms the findings of our collaborators at the Donders Institute for Brain, Cognition, and Behaviour. That is, the causal relation between cortical transients and slow-gamma oscillations is stronger than between rippleband oscillations and cortical transients.

In this project, a data-driven modelling algorithm for learning new dynamical models from experimental magnetoencephalography (MEG) data is introduced. This algorithm provides a contrast to existing hypothesis-driven modelling techniques in neuronal dynamics, and is useful for generating new insights from data when hypotheses for the neural mechanisms underlying a process are not readily available. The algorithm utilises universal differential equations (UDEs), combining white-box modelling with machine learning techniques. The algorithm is applied to a single-subject human MEG dataset to produce an oscillator network model. The model captures the frequency-domain behaviour of and interaction between several brain regions of interest during completion of a working memory (WM) task. The machine learning techniques are used to identify the role of attention mechanisms in these interaction dynamics, providing neuroscientists with data-driven insights into the brain dynamics underlying the attention-WM interplay.

This thesis reports the results of research into the stability of the all-to-all coupled discrete time Kuramoto model under constant, matched input disturbances. The discrete time Kuramoto model can be used as a dynamic, decentralized multi-agent orientation coordination system: once initialized, the agents will communicate their orientations to all other agents and calculate their own step-update based on the received data. A properly controlled and undisturbed Kuramoto model can direct agents to two final, stable sets of orientations: either all agents align to the same orientation or they form a balanced set of orientations with the characteristic that the centre of gravity of all orientations on the unit circle is at the origin.

These final states will not be reached when at least one of the agents is influenced by a disturbance. Not only will this agent be affected, but because of the networked system, the disturbance in one agent will influence other agents as well. Understanding the Kuramoto model enables the design of controllers that can attenuate the influence of disturbances. All controllers are designed with the assumption of constant matched input disturbances.
The first controller is an error feedback controller. For this strategy, the original Kuramoto model had to be modified. The resulting controller can attenuate the effects of matched input disturbances in a system of agents, but individual agents with matched input disturbances will not reach a steady state.

The second controller is based on predictor-error feedback and the Kuramoto characteristic that the average orientation in a Kuramoto model is constant. The controller is augmented with an algorithm that generates a one-step ahead prediction based on the known states and inputs. Since the disturbance is assumed to be constant, its effects can be calculated and attenuated in the next time step. This controller succeeds in directing the system to the same aligned set as the undisturbed system, although via a different trajectory. The controller also succeeds in directing the system to a balanced set, but for systems with N ≥ 4 agents that balanced set is different from the undisturbed set.

Since the second controller showed that a deviation from the undisturbed trajectory leads to a different balanced set, the third controller is designed with reference trajectories that do not use the actual states and inputs, but are generated fully autonomously. The difference between reference and actual state is processed by a proportional-integral algorithm to ensure zero steady state error. This controller however has the possibility of destabilizing the system, when not properly tuned.

All controllers have their merits: the first controller decouples the agents, thereby preventing that a disturbed agent affects others. Under constant disturbance, the second controller guarantees stability, but will let the agents follow different trajectories than the undisturbed system, leading to different balanced sets. The third controller can direct all agents to their undisturbed trajectory, but can negatively impact the stability properties of the Kuramoto controller when improperly tuned.

The central nervous system is the key controller in the human body. All aspects of live - from basic functions such as breathing to higher cognitive processes as memory building or complex decision making - have a network of neural cells at its core. While neuronal cells have been subject to intensive research for more than a century, the role of glial cells - the second major group of cell types in the brain - including astrocytes is still relatively unknown or contradictory. The focus of this thesis lies on memory processes and the astrocytic impact on them.
Working memory is a memory module which describes the short-term storage and processing of input data. A classic example is remembering a phone number before noting it down or typing it. Working memory is the fundamental building block for many cognitive functions such as prediction, goal-directed behavior and decision-making. Its impairment is tragically visible in various forms of dementia which can be caused by Alzheimer’s Disease or Parkinson’s Disease. Understanding the role of an astrocytic pathway as a possible underlying mechanism of working memory, could help strategies to prevent and treat these diseases and improve patients’ quality of life. The goal of this thesis is to provide support for astrocytic gliotransmission as a fundamental pathway in regard to Working memory. To this end, stability analysis of a tripartite model
with novel modifications shows the ability of extended firing in the presence of gliotransmission and hence, the principal ability for memory storage. Building upon this cell model, a complex network consisting of an established neuron-astrocyte structure is investigated. Different simulation scenarios show the possibility of astrocytes to store neuronal information during the absence of input signal and thus, showing a possible method for working memory.
Simultaneously, two competing theories of persistent and sparse delay activity are explained by varying strength of astrocytic gliotransmission while showing working memory functionality for both cases. Classical working memory experiments on primates are used as guiding points for the simulation sequence in order to provide comparability and authenticity.

Researchers have been interested in studying the connection between emotion and memory for decades but much remains unknown due to the elusive nature of the human brain. Furthering our understanding of the phenomenon is crucial for improving the treatment of neurological disorders associated with emotion dysregulation, as well as for enhancing autonomous systems that interact with humans, such as robotic prostheses and artificial intelligence. A promising approach to studying cognitive processes is developing computational models of brain activity, which have the potential to uncover the underlying neural mechanisms.
This master's thesis presents a novel dynamical model of the neural activity underpinning the emotional associative memory encoding process. This type of memory is especially complex and poorly understood as it requires memorizing the relationships between various environmental stimuli that may elicit distinct emotions. To model this phenomenon, the thesis proposes a network model using the Dynamic Causal Modeling (DCM) framework, focusing on the Amygdala (Amy), the Hippocampus (Hip), and the Orbitofrontal Cortex (OFC) due to their central roles in emotional associative memory. Furthermore, the thesis provides an analysis of network state coordination and state error stability, deriving analytical bounds for state error dynamics that illustrate how the model's properties are linked to improved memory encoding.
The dataset used to estimate the DCM model comes from a functional Magnetic Resonance Imaging (fMRI) study conducted by Zhu et al. at Donders Institute for Brain, Cognition and Behaviour, where participants were asked to memorize pairs of two emotionally potent images (group EE), two neutral images (group NN), or one neutral and one emotional image (group NE). The DCM model developed in this thesis reflects how the neural dynamics differ among the groups and the modeled brain regions, corroborating many of the behavioral results of the experiment and bringing novel insights. Based on stability and error dynamics analysis, the model of group NE is found to exhibit the best overall information flow, which is consistent with the experiment, where this group achieved the best memorization results. Furthermore, the coupling between Amy-Hip is shown to play a key role in improving memory encoding.
Another important contribution of this thesis is the extensive overview of DCM in Chapter 2, which discusses the mathematical foundations, the system identification algorithm, criticism raised against the method, studies of its statistical validity, and advice on implementation. Additionally, Appendix B presents the first guide for DCM script development in the SPM12 toolbox in MATLAB. Furthermore, this thesis compares the performance of four different variations of DCM, proving that Stochastic DCM achieves superior results in experimental paradigms with brief emotional stimuli. The success of this approach is believed to stem from its capacity to handle model misspecification and neuronal noise. Overall, the Stochastic DCM developed in this thesis is considered to provide the most comprehensive representation of neural dynamics governing the encoding of emotional associative memory compared to other existing dynamical models.

The human brain, with its intricate web of billions of neurons and trillions of synaptic connections, is a remarkable organ responsible for performing complex cognitive processes. While brain imaging techniques like fMRI and EEG provide insights into neural activity, there is no broadly accepted mathematical framework for the collaborative activity of neuronal populations and their communication. In the interdisciplinary fields of neuroscience and computational modelling, this research introduces a data-driven mathematical framework for modelling the brain's cortical network, derived from EEG data, while incorporating an exogenous input.

This framework captures brain dynamics, forming a state-space model using subspace identification, and evaluates statistical dependencies among brain sources, known as functional connectivity. Employed on a dataset mimicking a sensorimotor task, it proves effective in characterizing brain dynamics, with the PO-MOESP method outperforming N4SID. Additionally, the framework is competent to assess functional connectivity between brain sources, resulting in a network connectivity diagram that offers valuable insights into the statistical relationships between distinct brain regions. Altogether, it is concluded the designed algorithm can determine the intricate interactions within the brain during a simple passively performed task stimulating the brain. Consequently, this achievement forms a robust foundation for advancing Brain-Computer Interfaces (BCIs) and enhancing the diagnosis of neurological disorders.

This thesis considers the problem of nonlinear output regulation in a Koopman operator framework. The goal of output regulation is to asymptotically track a reference and/or simultaneously reject a disturbance signal, both generated by some external autonomous system called the exosystem. The nonlinear output regulation problem is solvable if and only if a set of partial differential equations (PDE) are satisfied. From the solution, a feedback law can be obtained that achieves output regulation. However, solving the PDE is difficult. In this thesis, we instead aim to construct a feedback law by utilizing the Koopman operator instead.
The Koopman operator associated with a state-space model of a (nonlinear) dynamical system describes the evolution of functions of the states, called observable functions, by propagating the state forward in time according to the flow of the system, and evaluating this at each possible observable function. The space of observables is an infinite-dimensional vector field. Therefore, the Koopman operator is infinite-dimensional and linear. The Koopman operator of an autonomous system associated with a nonlinear control system provides a bilinear description of the control system instead. The use of the Koopman operator to tackle the output regulation problem has not been done before in the literature. We identify conditions under which the Koopman operator can be used to rephrase the nonlinear output regulation problem as a bilinear output regulation problem. We then show when the bilinear output regulation problem is solved using linear dynamic error feedback. In particular, a Lyapunov-based approach is used to characterize a set of initial conditions for which the output is regulated. Finally, to verify the results, a numerical example is presented.

This thesis proposes a novel algorithm, Wi-Closure, to improve computational efficiency and robustness of map matching in multi-robot SLAM. Current state-of-the-art techniques connect maps with inter-robot loop closures, that are usually found through place recognition. Wi-Closure decreases the computational overhead of these approaches by pruning the search space of potential loop closures, prior to evaluation by a typical place recognition algorithm. Wi-Closure achieves this by identifying where trajectories are close to each other through sensing spatial information directly from the wireless communication signal. Then, place recognition is only performed on scans taken at locations close to each other. Wireless sensing provides information even when operating in non-line-of-sight or without existing communication infrastructure. The validity of Wi-Closure is demonstrated in simulation and hardware experiments. Results show that using Wi-closure greatly reduces computation time, by 54% in simulation and by 77% in hardware, compared with a multi-robot SLAM baseline. Importantly, this is achieved without sacrificing accuracy. Using Wi-Closure reduces absolute trajectory estimation error by 99% in simulation and 89% in hardware experiments. This improvement is due in part to Wi-Closure’s ability to avoid catastrophic optimization failure that typically occurs with classical approaches in challenging repetitive environments.

An intracranial aneurysm is a bulge in the cerebral vasculature. The rupture of an aneurysm results in a brain bleed. As a consequence, most patients become severely handicapped or may even die. Preventive treatment with endovascular coiling is controversial due to the high risk of complications. These complications are partly caused by the current uncontrolled delivery of coils to the aneurysm. While recent developments in microcatheter design allow for improved positioning, no method has been devised to model and control the tension applied by the coil on the aneurysm wall throughout a coiling procedure. This thesis aims to come up with a method to improve the safety of aneurysm treatment.

This thesis presents a new way to dynamically model an endovascular coil and its interaction with a microcatheter and the aneurysm wall throughout a coil deployment procedure. The main advantage of this model is its low computational complexity allowing real-time control computation. A control architecture is presented that enables regulation of the contact force between the coil and the aneurysm wall while obeying the constraints imposed by the equipment and the environment. The presented architecture comprises an augmented energy-shaping controller working in parallel with a constraint preservation controller.

This thesis shows that this control architecture asymptotically stabilizes the aneurysm wall tension at the desired value throughout the coil deployment procedure.

This work provides a basis for modelling and control in future experimental validations. Therefore, this work is a promising first step in the modelling and control of robotic systems for neurovascular interventions and a step forward in the preventive treatment of intracranial aneurysms.

This thesis contains two contributions to the stabilization of visually guided robotic lampreys: the head stabilization method and the head-led target tracking design. Both approach the problem that camera inputs, attached to the head segment, are disturbed due to the participation of the head in the locomotion gait of the robot. Head stabilization is designed to stabilize the head segment itself, and head-led target tracking is designed to stabilize the target in the field of view of the cameras. The head stabilization and head-led target tracking designs are build upon the Ijspeert model. The Ijspeert model is an biologically inspired oscillator-based central pattern generator, capable of producing locomotion signals to achieve a lateral undulation gait. Analysis of this Ijspeert model is done by rewriting the model as a network of Kuramoto oscillators. This analysis concludes with a proof for the convergence of the phase differences of the Ijspeert oscillators. Although methods that mitigate the head stabilization problem for robotic lampreys have been designed before, the head stabilization method in this thesis approaches the head stabilization problem as a control problem for the fist time, to the best of our knowledge. The head stabilization method is designed to align the head segment with the average body direction, by providing head stabilizing parameters to the Ijspeert model. Perfect head stabilization is achieved, under the assumption that the motor dynamics are instant. Even though head stabilization is not perfectly achieved in reality with non-instant motor dynamics, we have verified that the head stabilized Ijspeert model is an improvement in terms of head stability, compared to the Ijspeert model without head stabilization. The designed head stabilization method is applied to a novel head-led target tracking design. This method combines a forward locomotion gait with a turning controller to perform target tracking, and is designed to increase the accuracy of visual information by directing the head segment towards the target. The head-led target tracking design is verified by placing the robotic lamprey in a virtual fluid environment with a target, which showed that the target is reached by the robot. Furthermore, the head-led target tracking design is compared to a design from the literature that does not direct the head segment towards the target. The head-led target tracking design shows improvements to the design from the literature in terms of the used performance measures.