A Computational Approach to Verbal Width for Engel Words in Alternating Groups

It is known that every element in the alternating group A n , with n ≥ 5 , can be written as a product of at most two Engel words of arbitrary length. However, it is still unknown if every element in an alternating group is an Engel word of Arbitrary length. In this paper, a different approach to this problem is presented, getting new results for small alternating groups

Algebraic properties of profinite groups

Recently there has been a lot of research and progress in profinite groups. We survey some of the new results and discuss open problems. A central theme is decompositions of finite groups into bounded products of subsets of various kinds which give rise to algebraic properties of topological groups.Comment: This version has some references update

On the effects of transcranial alternating stimulation (tACS) on neuronal dynamics and cognition.

A few minutes at a busy square in London allow one to appreciate the wide array of actions that humans are capable of expressing— walking, reading a book, tapping touch screen of smartphone, eating, shaking hands and crossing the street. Response inhibition is an essential mechanism of action control and is one of the most studied processes. For example, crossing the street when a fast motorcycle is approaching might necessitate inhibition of stepping forward to avoid being hurt. This ability to quickly suppress a response in a dynamic environment has traditionally been associated with conscious control. Crucially, recent experimental evidence has challenged the view that inhibitory control is restricted to conditions where stimuli are accessible to conscious awareness. Such an unconscious and automatic activation of the motor response system does not necessarily require stimuli to be consciously perceived and is deemed essential to act in a constantly changing environment. This has been interpreted as a basic motor process allowing preparatory mechanisms to automatically suppress an activated movement without the need of conscious cognitive processes. Thus, while there may be differences between automatic and voluntary processes, they might not have entirely distinct neural representations. Indeed, automatic control appears to rely on the corticobasal ganglia network that has been associated with voluntary control. Contemporary research has shown that an up-regulation of neural beta oscillations in the cortico-basal ganglia dynamics can be functionally relevant for inhibition of movement. Consequently, beta oscillations have been proposed as an essential mechanism that allows the motor network to communicate in a dynamic and flexible manner. Present research has demonstrated that it is possible to interact with the neuronal activity by non invasive brain stimulation (NIBS) techniques such as transcranial Direct Current Stimulation (tDCS), transcranial Alternating Current Stimulation (tACS). Specifically, tACS allows delivery of alternating current at different frequencies and it has been used to manipulate ongoing brain oscillations in a controllable way. This concept is still in the very early stages of research, and much needs to be done in order to fully grasp the underlying mechanisms. Building upon these discoveries, the research presented in this thesis aimed to demonstrate a causal role of beta frequency oscillations on unconscious and automatic inhibition adopting tACS over the primary motor cortex and supplementary motor area. Furthermore combining tACS with TMS and EEG allowed me to characterise the underlying basic mechanisms of its action on corticospinal excitability and neuronal dynamics. Overall, this work contributes to our understanding of the human motor system while offering new insights into the combined approach of tACS and EEG in the characterization of a causal role of neuronal oscillatory dynamics on behaviour

Change blindness: eradication of gestalt strategies

Arrays of eight, texture-defined rectangles were used as stimuli in a one-shot change blindness (CB) task where there was a 50% chance that one rectangle would change orientation between two successive presentations separated by an interval. CB was eliminated by cueing the target rectangle in the first stimulus, reduced by cueing in the interval and unaffected by cueing in the second presentation. This supports the idea that a representation was formed that persisted through the interval before being 'overwritten' by the second presentation (Landman et al, 2003 Vision Research 43149–164]. Another possibility is that participants used some kind of grouping or Gestalt strategy. To test this we changed the spatial position of the rectangles in the second presentation by shifting them along imaginary spokes (by ±1 degree) emanating from the central fixation point. There was no significant difference seen in performance between this and the standard task [F(1,4)=2.565, p=0.185]. This may suggest two things: (i) Gestalt grouping is not used as a strategy in these tasks, and (ii) it gives further weight to the argument that objects may be stored and retrieved from a pre-attentional store during this task

Neuronal oscillations, information dynamics, and behaviour: an evolutionary robotics study

Oscillatory neural activity is closely related to cognition and behaviour, with synchronisation mechanisms playing a key role in the integration and functional organization of different cortical areas. Nevertheless, its informational content and relationship with behaviour - and hence cognition - are still to be fully understood. This thesis is concerned with better understanding the role of neuronal oscillations and information dynamics towards the generation of embodied cognitive behaviours and with investigating the efficacy of such systems as practical robot controllers. To this end, we develop a novel model based on the Kuramoto model of coupled phase oscillators and perform three minimally cognitive evolutionary robotics experiments. The analyses focus both on a behavioural level description, investigating the robot’s trajectories, and on a mechanism level description, exploring the variables’ dynamics and the information transfer properties within and between the agent’s body and the environment. The first experiment demonstrates that in an active categorical perception task under normal and inverted vision, networks with a definite, but not too strong, propensity for synchronisation are more able to reconfigure, to organise themselves functionally, and to adapt to different behavioural conditions. The second experiment relates assembly constitution and phase reorganisation dynamics to performance in supervised and unsupervised learning tasks. We demonstrate that assembly dynamics facilitate the evolutionary process, can account for varying degrees of stimuli modulation of the sensorimotor interactions, and can contribute to solving different tasks leaving aside other plasticity mechanisms. The third experiment explores an associative learning task considering a more realistic connectivity pattern between neurons. We demonstrate that networks with travelling waves as a default solution perform poorly compared to networks that are normally synchronised in the absence of stimuli. Overall, this thesis shows that neural synchronisation dynamics, when suitably flexible and reconfigurable, produce an asymmetric flow of information and can generate minimally cognitive embodied behaviours

State-Dependent Cortical Network Dynamics

Neuropsychiatric illness represents a major health burden in the United States with a paucity of effective treatment. Many neuropsychiatric illnesses are network disorders, exhibiting aberrant organization of coordinated activity within and between brain areas. Cortical oscillations, arising from the synchronized activity of groups of neurons, are important in mediating both local and long-range communication in the brain and are particularly affected in neuropsychiatric diseases. A promising treatment approach for such network disorders entails ‘correcting’ abnormal oscillatory activity through non-invasive brain stimulation. However, we lack a clear understanding of the functional role of oscillatory activity in both health and disease. Thus, basic science and translational work is needed to elucidate the role of oscillatory activity and other network dynamics in neuronal processing and behavior. Organized activity in the brain occurs at many spatial and temporal scales, ranging from the millisecond duration of individual action potentials to the daily circadian rhythm. The studies comprising this dissertation focused on organization in cortex at the time scale of milliseconds, assessing local field potential and spiking activity, and contribute to understanding (1) the effects of non-invasive brain stimulation on behavioral responses, (2) network dynamics within and across cortical areas during different states, and (3) how oscillatory activity organizes spiking activity locally and long-range during sustained attention. Taken together, this work provides insight into the physiological organization of network dynamics and can provide the basis for future rational design of non-invasive brain stimulation treatments.Doctor of Philosoph