title: Going beyond a mean-field model for the learning cortex: second-order statistics

authors: Wilson, Marcus T.

subjects: mean-field

abstract: Mean-field models of the cortex have been used successfully to interpret the origin of features on the electroencephalogram under situations such as sleep, anesthesia, and seizures. In a mean-field scheme, dynamic changes in synaptic weights can be considered through fluctuation-based Hebbian learning rules. However, because such implementations deal with population-averaged properties, they are not well suited to memory and learning applications where individual synaptic weights can be important. We demonstrate that, through an extended system of equations, the mean-field models can be developed further to look at higher-order statistics, in particular, the distribution of synaptic weights within a cortical column. This allows us to make some general conclusions on memory through a mean-field scheme. Specifically, we expect large changes in the standard deviation of the distribution of synaptic weights when fluctuation in the mean soma potentials are large, such as during the transitions between the “up” and “down” states of slow-wave sleep. Moreover, a cortex that has low structure in its neuronal connections is most likely to decrease its standard deviation in the weights of excitatory to excitatory synapses, relative to the square of the mean, whereas a cortex with strongly patterned connections is most likely to increase this measure. This suggests that fluctuations are used to condense the coding of strong (presumably useful) memories into fewer, but dynamic, neuron connections, while at the same time removing weaker (less useful) memories.

text: Going beyond a mean-field model for the learning cortex: second-order statistics Mean-field models of the cortex have been used successfully to interpret the origin of features on the electroencephalogram under situations such as sleep, anesthesia, and seizures. In a mean-field scheme, dynamic changes in synaptic weights can be considered through fluctuation-based Hebbian learning rules. However, because such implementations deal with population-averaged properties, they are not well suited to memory and learning applications where individual synaptic weights can be important. We demonstrate that, through an extended system of equations, the mean-field models can be developed further to look at higher-order statistics, in particular, the distribution of synaptic weights within a cortical column. This allows us to make some general conclusions on memory through a mean-field scheme. Specifically, we expect large changes in the standard deviation of the distribution of synaptic weights when fluctuation in the mean soma potentials are large, such as during the transitions between the “up” and “down” states of slow-wave sleep. Moreover, a cortex that has low structure in its neuronal connections is most likely to decrease its standard deviation in the weights of excitatory to excitatory synapses, relative to the square of the mean, whereas a cortex with strongly patterned connections is most likely to increase this measure. This suggests that fluctuations are used to condense the coding of strong (presumably useful) memories into fewer, but dynamic, neuron connections, while at the same time removing weaker (less useful) memories.

year: 2007

title: Decomposing complete equipartite graphs into short odd cycles

authors: Smith, Benjamin R.

subjects: mathematics

abstract: In this paper we examine the problem of decomposing the lexicographic product of a cycle with an empty graph into cycles of uniform length. We determine necessary and sufficient conditions for a solution to this problem when the cycles are of odd length. We apply this result to find necessary and sufficient conditions to decompose a complete equipartite graph into cycles of uniform length, in the case that the length is both odd and short relative to the number of parts.

text: Decomposing complete equipartite graphs into short odd cycles In this paper we examine the problem of decomposing the lexicographic product of a cycle with an empty graph into cycles of uniform length. We determine necessary and sufficient conditions for a solution to this problem when the cycles are of odd length. We apply this result to find necessary and sufficient conditions to decompose a complete equipartite graph into cycles of uniform length, in the case that the length is both odd and short relative to the number of parts.

year: 2010

title: Ngā wāhine ariki tapairu o Te Arawa: Te Kuraimonoa, Kearoa, Whakaotirangi

authors: Retikaukau, Georgia-Louise

subjects: Te Arawa

abstract: Similar to many other iwi (tribes) in Aotearoa, the Arawa people have a rich ancestral heritage that traces back to our origins in Hawaiki through our esteemed canoe traditions. However, within these narratives, the prominence of our heroic male ancestors tends to overshadow the stories of their wives, who are often mentioned merely as secondary characters. Te Kuraimonoa, Kearoa, and Whakaotirangi represent the first three women to be named in Te Arawa traditions, yet their narratives remain vague, existing only from a male perspective. Although their names can be found scattered throughout European and Māori literature, their stories are fragmented and incomplete, varying from tribe to tribe.

The objective of this research is to delve into these three ancestresses' lives from a Te Arawa female perspective, aiming to provide a more comprehensive and nuanced understanding. Employing a mixed-method research approach, this study will investigate Māori and European literature, supplemented by thematical analysis to study and compare data gathered from interviews with female Te Arawa descendants. The insights gained from this endeavour will contribute to the creation of substantial knowledge that can benefit both present and future generations.

Comprising of seven chapters, this thesis unfolds as follows: Chapter one introduces the research purpose and provides an overview of the subsequent chapters. Chapter two outlines the research methodology employed. Chapter three presents an in-depth exploration of Te Arawa iwi, elucidating their significance and cultural context. Chapter four encompasses a comprehensive literature review, while chapter five is solely dedicated to the insights derived from the interview dialogues. Chapter six delves into the findings and engages in detailed discussions, offering a comprehensive analysis. Lastly, chapter seven serves as the conclusion, summarizing the key findings and their implications.

text: Ngā wāhine ariki tapairu o Te Arawa: Te Kuraimonoa, Kearoa, Whakaotirangi Similar to many other iwi (tribes) in Aotearoa, the Arawa people have a rich ancestral heritage that traces back to our origins in Hawaiki through our esteemed canoe traditions. However, within these narratives, the prominence of our heroic male ancestors tends to overshadow the stories of their wives, who are often mentioned merely as secondary characters. Te Kuraimonoa, Kearoa, and Whakaotirangi represent the first three women to be named in Te Arawa traditions, yet their narratives remain vague, existing only from a male perspective. Although their names can be found scattered throughout European and Māori literature, their stories are fragmented and incomplete, varying from tribe to tribe.

The objective of this research is to delve into these three ancestresses' lives from a Te Arawa female perspective, aiming to provide a more comprehensive and nuanced understanding. Employing a mixed-method research approach, this study will investigate Māori and European literature, supplemented by thematical analysis to study and compare data gathered from interviews with female Te Arawa descendants. The insights gained from this endeavour will contribute to the creation of substantial knowledge that can benefit both present and future generations.

Comprising of seven chapters, this thesis unfolds as follows: Chapter one introduces the research purpose and provides an overview of the subsequent chapters. Chapter two outlines the research methodology employed. Chapter three presents an in-depth exploration of Te Arawa iwi, elucidating their significance and cultural context. Chapter four encompasses a comprehensive literature review, while chapter five is solely dedicated to the insights derived from the interview dialogues. Chapter six delves into the findings and engages in detailed discussions, offering a comprehensive analysis. Lastly, chapter seven serves as the conclusion, summarizing the key findings and their implications.

year: 2023