The investigation of variable nernst equilibria on isolated neurons and coupled neurons forming discrete and continuous networks

Since the introduction of the Hodgkin-Huxley equations, used to describe the excitation of neurons, the Nernst equilibria for individual ion channels have assumed to be constant in time. Recent biological recordings call into question the validity of this assumption. Very little theoretical work has been done to address the issue of accounting for these non-static Nernst equilibria using the Hodgkin-Huxley formalism. This body of work incorporates non-static Nernst equilibria into the generalized Hodgkin-Huxley formalism by considering the first-order effects of the Nernst equation. It is further demonstrated that these effects are likely dominate in neurons with diameters much smaller than that of the squid giant axon and permeate important information processing regions of the brain such as the hippocampus. Particular results of interest include single-cell bursting due to the interplay of spatially separated neurons, pattern formation via spiral waves within a soliton-like regime, and quantifiable shifts in the multifractality of hippocampal neurons under the administration of various drugs at varying dosages. This work provides a new perspective on the variability of Nernst equilibria and demonstrates its utility in areas such as pharmacology and information processing

Understanding spiking and bursting electrical activity through piece-wise linear systems

In recent years there has been an increased interest in working with piece-wise linear caricatures of nonlinear models. Such models are often preferred over more detailed conductance based models for their small number of parameters and low computational overhead. Moreover, their piece-wise linear (PWL) form, allow the construction of action potential shapes in closed form as well as the calculation of phase response curves (PRC). With the inclusion of PWL adaptive currents they can also support bursting behaviour, though remain amenable to mathematical analysis at both the single neuron and network level. In fact, PWL models caricaturing conductance based models such as that of Morris-Lecar or McKean have also been studied for some time now and are known to be mathematically tractable at the network level. In this work we proceed to analyse PWL neuron models of conductance type. In particular we focus on PWL models of the FitzHugh-Nagumo type and describe in detail the mechanism for a canard explosion. This model is further explored at the network level in the presence of gap junction coupling. The study moves to a different area where excitable cells (pancreatic beta-cells) are used to explain insulin secretion phenomena. Here, Ca2+ signals obtained from pancreatic beta-cells of mice are extracted from image data and analysed using signal processing techniques. Both synchrony and functional connectivity analyses are performed. As regards to PWL bursting models we focus on a variant of the adaptive absolute IF model that can support bursting. We investigate the bursting electrical activity of such models with an emphasis on pancreatic beta-cells

Models of self-organization in biological development

Bibliography: p. 297-320.In this thesis we thus wish to consider the concept of self-organization as an overall paradigm within which various theoretical approaches to the study of development may be described and evaluated. In the process, an attempt is made to give a fair and reasonably comprehensive overview of leading modelling approaches in developmental biology, with particular reference to self-organization. The work proceeds from a physical or mathematical perspective, but not unduly so - the major mathematical derivations and results are relegated to appendices - and attempts to fill a perceived gap in the extant review literature, in its breadth and attempted impartiality of scope. A characteristic of the present account is its markedly interdisciplinary approach: it seeks to place self-organization models that have been proposed for biological pattern formation and morphogenesis both within the necessary experimentally-derived biological framework, and in the wider physical context of self-organization and the mathematical techniques that may be employed in its study. Hence the thesis begins with appropriate introductory chapters to provide the necessary background, before proceeding to a discussion of the models themselves. It should be noted that the work is structured so as to be read sequentially, from beginning to end; and that the chapters in the main text were designed to be understood essentially independently of the appendices, although frequent references to the latter are given. In view of the vastness of the available information and literature on developmental biology, a working knowledge of embryological principles must be assumed. Consequently, rather than attempting a comprehensive introduction to experimental embryology, chapter 2 presents just a few biological preliminaries, to 'set the scene', outlining some of the major issues that we are dealing with, and sketching an indication of the current status of knowledge and research on development. The chapter is aimed at furnishing the necessary biological, experimental background, in the light of which the rest of the thesis should be read, and which should indeed underpin and motivate any theoretical discussions. We encounter the different hierarchical levels of description in this chapter, as well as some of the model systems whose experimental study has proved most fruitful, some of the concepts of experimental embryology, and a brief reference to some questions that will not be addressed in this work. With chapter 3, we temporarily move away from developmental biology, and consider the wider physical and mathematical concepts related to the study of self-organization. Here we encounter physical and chemical examples of spontaneous structure formation, thermodynamic considerations, and different approaches to the description of complexity. Mathematical approaches to the dynamical study of self-organization are also introduced, with specific reference to reaction-diffusion equations, and we consider some possible chemical and biochemical realizations of self-organizing kinetics. The chapter may be read in conjunction with appendix A, which gives a somewhat more in-depth study of reaction-diffusion equations, their analysis and properties, as an example of the approach to the analysis of self-organizing dynamical systems and mathematically-formulated models. Appendix B contains a more detailed discussion of the Belousov-Zhabotinskii reaction, which provides a vivid chemical paradigm for the concepts of symmetry-breaking and self-organization. Chapter 3 concludes with a brief discussion of a model biological system, the cellular slime mould, which displays rudimentary development and has thus proved amenable to detailed study and modelling. The following two chapters form the core of the thesis, as they contain discussions of the detailed application of theoretical concepts and models, largely based on self-organization, to various developmental situations. We encounter a diversity of models which has arisen largely in the last quarter century, each of which attempts to account for some aspect of biological pattern formation and morphogenesis; an aim of the discussion is to assess the extent of the underlying unity of these models in terms of the self-organization paradigm. In chapter 4 chemical pre-patterns and positional information are considered, without the overt involvement of cells in the patterning. In chapter 5, on the other hand, cellular interactions and activities are explicitly taken into account; this chapter should be read together with appendix C, which contains a brief introduction to the mathematical formulation and analysis of some of the models discussed. The penultimate chapter, 6, considers two other approaches to the study of development; one of these has faded away, while the other is still apparently in the ascendant. The assumptions underlying catastrophe theory, the value of its applications to developmental biology and the reasons for its decline in popularity, are considered. Lastly, discrete approaches, including the recently fashionable cellular automata, are dealt with, and the possible roles of rule-based interactions, such as of the so-called L-systems, and of fractals and chaos are evaluated. Chapter 7 then concludes the thesis with a brief assessment of the value of the self-organization concept to the study of biological development

Locomotor patterns and persistent activity in self-organizing neural models

The thesis investigates principles of self-organization that may account for the observed structure and behaviour of neural networks that generate locomotor behaviour and complex spatiotemporal patterns such as spiral waves, metastable states and persistent activity. This relates to the general neuroscience problem of finding the correspondence between the structure of neural networks and their function. This question is both extremely important and difficult to answer because the structure of a neural network defines a specific type of neural dynamics which underpins some function of the neural system and also influences the structure and parameters of the network including connection strengths. This loop of influences results in a stable and reliable neural dynamics that realises a neural function. In order to study the relationship between neural network structure and spatiotemporal dynamics, several computational models of plastic neural networks with different architectures are developed. Plasticity includes both modification of synaptic connection strengths and adaptation of neuronal thresholds. This approach is based on a consideration of general modelling concepts and focuses on a relatively simple neural network which is still complex enough to generate a broad spectrum of spatio-temporal patterns of neural activity such as spiral waves, persistent activity, metastability and phase transitions. Having considered the dynamics of networks with fixed architectures, we go on to consider the question of how a neural circuit which realizes some particular function establishes its architecture of connections. The approach adopted here is to model the developmental process which results in a particular neural network structure which is relevant to some particular functionality; specifically we develop a biologically realistic model of the tadpole spinal cord. This model describes the self-organized process through which the anatomical structure of the full spinal cord of the tadpole develops. Electrophysiological modelling shows that this architecture can generate electrical activity corresponding to the experimentally observed swimming behaviour