title: Describing the stochastic dynamics of neurons using Hamilton’s equations of classical mechanics

authors: Wilson, Marcus T.

subjects: neurons

abstract: We consider the most likely behaviour of neuron models by formulating them in terms of Hamilton’s equations. Starting from a Lagrangian for a stochastic system, we describe how Hamilton’s equations of classical mechanics can be used to derive an equivalent description in terms of canonical co-ordinates and momenta. We give physical meaning to these generalized momenta; speciﬁcally they are linear combinations of the noise terms in the stochastic model. Pseudo-kinetic energy and potential energy terms are also derived. The conjugate momenta can be considered as growing modes, and by implication the most likely noise input to a system will grow exponentially at large times; this surprising prediction agrees with existing experimental work on a single neuron. For many-neuron models, multiple growing modes will exist, and the numerical analysis of these is more complicated; however, the approach may still provide insight on the more detailed dynamics of these systems.

text: Describing the stochastic dynamics of neurons using Hamilton’s equations of classical mechanics We consider the most likely behaviour of neuron models by formulating them in terms of Hamilton’s equations. Starting from a Lagrangian for a stochastic system, we describe how Hamilton’s equations of classical mechanics can be used to derive an equivalent description in terms of canonical co-ordinates and momenta. We give physical meaning to these generalized momenta; speciﬁcally they are linear combinations of the noise terms in the stochastic model. Pseudo-kinetic energy and potential energy terms are also derived. The conjugate momenta can be considered as growing modes, and by implication the most likely noise input to a system will grow exponentially at large times; this surprising prediction agrees with existing experimental work on a single neuron. For many-neuron models, multiple growing modes will exist, and the numerical analysis of these is more complicated; however, the approach may still provide insight on the more detailed dynamics of these systems.

year: 2008

title: The matching law

authors: Poling, Alan

subjects: concurrent schedule of reinforcement

abstract: This article introduces the quantitative analysis of choice behavior by describing a number of equations developed over the years to describe the relation between the allocation of behavior under concurrent schedules of reinforcement and the consequences received for alternative responses. Direct proportionality between rate of responding and rate of reinforcement was observed in early studies, suggesting that behavioral output matched environmental input in a mathematical sense. This relation is termed "strict matching," and the equation that describes it is referred to as "the matching law." Later data showed systematic departures from strict matching, and a generalized version of the matching equation is now used to describe such data. This equation, referred to as "the generalized matching equation," also describes data that follow strict matching. It has become convention to refer to either of these equations as "the matching law." Empirical support for the matching law is briefly summarized, as is the applied and practical significance of matching analyses.

text: The matching law This article introduces the quantitative analysis of choice behavior by describing a number of equations developed over the years to describe the relation between the allocation of behavior under concurrent schedules of reinforcement and the consequences received for alternative responses. Direct proportionality between rate of responding and rate of reinforcement was observed in early studies, suggesting that behavioral output matched environmental input in a mathematical sense. This relation is termed "strict matching," and the equation that describes it is referred to as "the matching law." Later data showed systematic departures from strict matching, and a generalized version of the matching equation is now used to describe such data. This equation, referred to as "the generalized matching equation," also describes data that follow strict matching. It has become convention to refer to either of these equations as "the matching law." Empirical support for the matching law is briefly summarized, as is the applied and practical significance of matching analyses.

year: 2011

title: Evaluation of iwi and hapū participation in the resource consents processes of six district councils

authors: Backhurst, Michael

subjects: New Zealand

abstract: This working paper analyses the processes adopted by councils for involving hapū/iwi in plan implementation, including the resource consents process. Three topic issues were investigated to assess plan implementation — urban amenity, storm water, and issues of importance to iwi. Questions were asked about the capacity of hapū/iwi to engage in the resource consent process, which resource issues were of concern to them, their relationship with council and consent applicants, and their perception of the consent process. Most resources listed in the questionnaire were of concern to hapū/iwi, with water quality, wāhi tapu and heritage the most commonly cited. In conclusion, we found a general dissatisfaction on the part of hapū/iwi with councils’ performance with respect to both Treaty relationships and consent processing under the RMA. A further contributing factor to the poor relationships found between hapū/iwi and councils, was the lack of clarity over the role of hapū and iwi in resource management. In several districts, diverging responses from hapū/iwi and councils to questions about level of understanding and commitment suggests there is a need for more effective communication. These problems are compounded by the generally low capacity of hapū/iwi to participate in resource consent processes. These findings suggest that there is much to be done to improve relationships and behaviour of these key stakeholder groups in the plan implementation process if key provisions in the RMA related to hapū/iwi interests are to be fulfilled. The differences shown in reciprocal perceptions have serious implications for establishing a sound working partnership between councils and hapū/iwi in their areas. Making clear these discrepancies is a first step towards taking the measures needed for building a better partnership. Further, the capacity of hapū/iwi to participate could be better utilised if there was greater integration between regional and district councils on issues of significance and processes for iwi involvement.

text: Evaluation of iwi and hapū participation in the resource consents processes of six district councils This working paper analyses the processes adopted by councils for involving hapū/iwi in plan implementation, including the resource consents process. Three topic issues were investigated to assess plan implementation — urban amenity, storm water, and issues of importance to iwi. Questions were asked about the capacity of hapū/iwi to engage in the resource consent process, which resource issues were of concern to them, their relationship with council and consent applicants, and their perception of the consent process. Most resources listed in the questionnaire were of concern to hapū/iwi, with water quality, wāhi tapu and heritage the most commonly cited. In conclusion, we found a general dissatisfaction on the part of hapū/iwi with councils’ performance with respect to both Treaty relationships and consent processing under the RMA. A further contributing factor to the poor relationships found between hapū/iwi and councils, was the lack of clarity over the role of hapū and iwi in resource management. In several districts, diverging responses from hapū/iwi and councils to questions about level of understanding and commitment suggests there is a need for more effective communication. These problems are compounded by the generally low capacity of hapū/iwi to participate in resource consent processes. These findings suggest that there is much to be done to improve relationships and behaviour of these key stakeholder groups in the plan implementation process if key provisions in the RMA related to hapū/iwi interests are to be fulfilled. The differences shown in reciprocal perceptions have serious implications for establishing a sound working partnership between councils and hapū/iwi in their areas. Making clear these discrepancies is a first step towards taking the measures needed for building a better partnership. Further, the capacity of hapū/iwi to participate could be better utilised if there was greater integration between regional and district councils on issues of significance and processes for iwi involvement.

year: 2004-03