The two variable Fitzhugh Nagumo model behaves qualitatively like the four variable Hodgkin-Huxley space clamped system and is more mathematically tractable than the Hodgkin Huxley model, thus allowing the action potential and other properties of the Hodgkin Huxley system to be more readily be visualized. In this thesis, it is shown that the Carleman Embedding Technique can be applied to both the Fitzhugh Nagumo model and to Van der Pol's model of nonlinear oscillation, which are both finite nonlinear systems of differential equations. The Carleman technique can thus be used to obtain approximate solutions of the Fitzhugh Nagumo model and to study neural activity such as excitability.

Neurons are specialized cells that communicate via electrical and chemical signaling. It is well-known that homeostatic mechanisms exist to potentiate neuronal output when activity falls. Likewise, while neurons rely on excitable states to function, these same excitable states must be kept in check for stable function. However, the identity of molecular factors and pathways regulating these pathways remain elusive. Chapter 2 of this thesis reports the findings from an RNA interference- and electrophysiology-based screen to identify factors necessary for the long-term maintenance of homeostatic synaptic potentiation. Data is reported to resolve a long-standing question as to the role of presynaptic Cav2-type channels in homeostatic synaptic potentiation at the Drosophila NMJ. It is shown that reduction in Cav2 channel expression and resultant activity is not sufficient to occlude homeostatic potentiation. Thus, the homeostatic block of a amino-acid substituted Cav2-type calcium channel (cacS) channel is presumed to be due to loss of a specific signaling or binding activity, but not due to overall diminishment in channel function. It is also reported that both Drosophila homologs of phospholipase Cβ (PLCβ) and its putative activator Gαq were found to be necessary for a scaling up of neurotransmitter release upon genetic ablation of glutamate receptors. These factors are canonically involved in the activation of intracellular calcium stores through the inositol trisphosphate receptor (IP3R) and the closely related ryanodine receptor (RyR). Likewise, the Drosophila homolog of Cysteine String Protein (Csp) is identified as important for long-term homeostatic potentiation. CSP has also been reported to be involved in regulation of intracellular calcium. PLCβ, Gαq, and CSP are also known to regulate Cav2-type channels directly, and this possibility, as well as others, are discussed as mechanisms underlying their roles in homeostatic potentiation. Chapter 3 of this thesis reports the extended findings from expression of a gain-of-function Cav2-type channel. The Cav2.1 channel in humans is known to cause a dominant, heritable form of migraine called familial hemiplegic migraine (FHM). Two amino-acid substitutions causative for migraine were cloned into their analogous residues of the Drosophila Cav2 homolog. Expression of these migraine-modeled channels gave rise to several forms of hyperexcitability. Hyperexcitability defects included abnormal evoked waveforms, generation of spontaneous action potential-like events, and multi-quantal release. It is shown that these forms of hyperexcitability can be mitigated through targeted down-regulation of the PLCβ-IP3R-RyR intracellular signaling pathway. Chapter 4 presents an extended discussion as to the roles for presynaptic calcium channels, PLCβ, and CSP in homeostatic synaptic potentiation, and the mechanism underlying hyperexcitability downstream of gain-of-function Cav2-type channels. The proposed model aims to bridge the involvement of the PLCβ pathway in both homeostatic potentiation and neuronal excitability. Last, the implications for these findings on human disease conditions are elucidated.

The human central nervous system (CNS) is capable of significant architectural and physiological reorganization in response to environmental stimuli. Novel sensorimotor experiences stimulate neuronal networks to modify their intrinsic excitability and spatial connectivity within and between CNS structures. Early learning-induced adaptations in the primary motor cortex are thought to serve as a priming stimulus for long term CNS reorganization underlying long-lasting changes in motor skill. Recent animal and human studies suggest that whole body exercise and core temperature elevation as systemic stressors also recruit activity-dependent processes that prime the motor cortex, cerebellum, and hippocampus to process sensorimotor stimuli from the environment, enhancing overall CNS learning and performance. A primary goal of rehabilitation specialists is to evaluate and design activity-based intervention strategies that induce or enhance beneficial neuroplastic processes across the lifespan. As such, an investgation of the influence of physical, non-pharmacological interventions on cortical excitability, motor learning, and cognitive function provide the central theme of this dissertation. The first study investigated the effects of a visually-guided motor learning task on motor cortex excitability at rest and during voluntary activation measured via transcranical magnetic stimulation (TMS). Motor learning significantly increased resting cortical excitability that was not accompanied by changes in excitability as a function of voluntary muscle activation. The cortical silent period, a measure of inhibition, increased after learning and was associated with the magnitude of learning at low activation. These findings suggest that separate excitatory and inhibitory mechanisms may influence motor output as a function of learning success. The following studies investigated the influence of systemic whole-body thermal stress on motor cortex excitability, motor learning and cognitive performance. We established the reliability of a novel TMS cortical mapping procedure to study neurophysiological responses after whole-body heat stress. Heat stress significantly potentiated motor cortex excitability, though acute motor learning and cognitive test performance did not differ between subjects receiving heat stress and control subjects. Future research is needed to delineate the potential of whole body heat stress as a therapeutic modality to influence central nervous system plasticity and performance.

Mechanical loading can modulate tissue plasticity and has potential applications in rehabilitation science and regenerative medicine. To safely and effectively introduce mechanical loads to human cells, tissues, and the entire body, we need to understand the optimal loading environment to promote growth and health. The purpose of this research was 1) to validate a limb vibration and compression system; 2) to determine the effect of limb vibration on neural excitability measured by sub-threshold TMS-conditioned H-reflexes and supra-threshold TMS; 3) to determine changes in center of pressure, muscle activity, and kinematics during a postural task following limb vibration; 4) to determine the effect of vibration on accuracy and long latency responses during a weight bearing visuomotor task. The major findings of this research are 1) the mechanical system presented in the manuscript can deliver limb vibration and compression reliably, accurate, and safely to human tissue; 2) sub-threshold cortical stimulation reduces the vibration-induced presynaptic inhibition of the H-reflex. This reduction cannot be attributed to an increase in cortical excitability during limb vibration because the MEP remains unchanged with limb vibration; 3) limb vibration altered the soleus and tibialis EMG activity during a postural control task. The vibration-induced increase in muscle activity was associated with unchanged center of pressure variability and reduced center of pressure complexity; 4) healthy individuals were able to accommodate extraneous afferent information due to the vibration interventions They maintained similar levels of accuracy of a visuomotor tracking task and unchanged long latency responses during an unexpected perturbation.

Pendant des décennies, la plasticité synaptique a été considérée comme le substrat principal de la plasticité fonctionnelle cérébrale. Récemment, plusieurs études expérimentales indiquent que des régulations à long terme de l’excitabilité intrinsèque participent à la plasticité dépendante de l’activité. En effet, la modulation des canaux ioniques dépendants du potentiel, lesquels régulent fortement l’excitabilité intrinsèque et l’intégration des entrées synaptiques, a été démontrée essentielle dans les processus d’apprentissage. Cependant, la régulation, dépendante de l’activité, du courant ionique activé par l’hyperpolarisation (Ih) et ses conséquences sur l’induction de futures plasticités reste à éclaircir, tout comme la présence d’une régulation de conductances dépendantes du potentiel dans les neurones inhibiteurs. Dans la première partie de ma thèse, nous caractérisons les mécanismes d’induction et d’expression de la plasticité à long terme de l’excitabilité (LTP-IE) dans les interneurons en panier de la région CA1 exprimant la parvalbumine. Dans une seconde partie, le rôle de Ih dans la régulation homéostatique de l’excitabilité neuronale induite par des manipulations de l’activité neuronale dans sa globalité a été étudié. Dans la troisième étude, nous montrons que la magnitude de la Dépression à Long Terme (LTD) détermine le sens de la régulation de Ih dans les neurones pyramidaux de CA1. En conclusion, cette thèse montre qu’à la fois dans les neurones excitateurs et inhibiteurs, les régulations des conductances dépendantes du potentiel aident à maintenir une relative stabilité dans l’activité du réseau. / Synaptic plasticity has been considered for decades as the main substrate of functional plasticity in the brain. Recently, experimental evidences suggest that long-lasting regulation of intrinsic neuronal excitability may also account for activity-dependent plasticity. Indeed, voltage-dependent ionic channels strongly regulate intrinsic excitability and inputs integration and their regulation was found to be essential in learning process. However, activity-dependent regulation of the hyperpolarization-activated ionic current (Ih) and its consequences for future plasticity remain unclear, so as the presence of any voltage-dependent conductances regulation in inhibitory neurons. In the first part of this thesis, we report the characterization of the induction and expression mechanisms of Long-Term Potentiation of Intrinsic Excitability (LTP-IE) in CA1 parvalbumin-positive basket interneurons. In a second part, the role of Ih in the homeostatic regulation of intrinsic neuronal excitability induced by global manipulations of neuronal activity was reported. In the third experimental study, we showed that the magnitude of Long-term Depression (LTD) determines the sign of Ih regulation in CA1 pyramidal neurons. In conclusion, this thesis shows that in both excitatory and inhibitory neurons, activity-dependent regulations of voltage-dependent conductances help to maintain a relative stability in the network activity.

This thesis consists in the study of a periodically forced slow-fast system in both its excitable and oscillatory regimes. The slow-fast system under consideration is the FitzHugh-Nagumo model, and the periodic forcing consists of a train of Gaussian-shaped pulses, the width of which is much shorter than the action potential duration. This system is a qualitative model for both an excitable cell and a spontaneously beating cell submitted to periodic electrical stimulation. Such a configuration has often been studied in cardiac electrophysiology, due to the fact that it constitutes a simplified model of the situation of a cardiac cell in the intact heart, and might therefore contribute to the understanding of cardiac arrhythmias. Using continuation methods (AUTO software), we compute periodic-solution branches for the periodically forced system, taking the stimulation period as bifurcation parameter. We then study the evolution of the resulting bifurcation diagram as the stimulation amplitude is raised. In both the excitable and the oscillatory regimes, we find that a critical amplitude of stimulation exists below which the behaviour of the system is trivial: in the excitable case, the bifurcation diagram is restricted to a stable subthreshold period-1 branch, and in the oscillatory case, all the stable periodic solutions belong to isolated loops (i.e., to distinct closed solution branches). Due to the slow-fast nature of the system, the changes that take place in the bifurcation diagram as the critical amplitude is crossed are drastic, while the way the bifurcation diagram re-simplifies above some second amplitude is much more gentle. In the oscillatory case, we show that the critical amplitude is also the amplitude at which the topology of phase-resetting changes type. We explain the origin of this coincidence by considering a one-dimensional discrete map of the circle derived from the phase-resetting curve of the oscillator (the phase-resetting map), map which constitutes a good approximation of the original differential equations under certain conditions. We show that the bifurcation diagram of any such circle map, where the bifurcation parameter appears only in an additive fashion, is always characterized by the period-1 solutions belonging to isolated loops when the topological degree of the map is one, while these period-1 solutions belong to a unique branch when the topological degree of the map is zero.

The biophysical properties and distribution of voltage gated ion channels shape the spatio-temporal pattern of synaptic inputs and determine the input-output properties of the neuron. Of the various voltage-gated ion channels, persistent Na⁺ current (INaP) is of interest because of its activation near rest, slow inactivation kinetics, and consequent effects on excitability. Overshadowed by transient Na⁺ current (INaT) of large amplitude and fast inactivation, various quantitative characterizations of INaP have yet to provide a clear understanding of their role in neuronal excitability. We addressed this question using quantitative electrophysiology to compare somatic INaP and INaT in 4–7 week old Sprague-Dawley rat hippocampal CA1 pyramidal neurons. INaP was evoked with 0.4 mV/ms ramp voltage commands and INaT with step commands in hippocampal neurons from in vitro brain slices utilizing nucleated patch-clamp recording. INaP was found to have a density of 1.4 ± 0.7 pA/pF in the soma. Compared to INaT, it has a much smaller amplitude (2.38% of INaT) and distinct voltage dependence of activation (16.7 mV lower half maximal activation voltage and 41.3% smaller slope factor than those of INaT). The quantitative measurement of INaT gave the activation time constant ([tau]m) of 22.2 ± 2.3 [mu]s at 40 mV. Hexanol, which has anesthetic effects, was shown to preferentially block INaP compared to INaT with a significant voltage threshold elevation (4.6 ± 0.7 mV) and delayed 1st spike latency (221 ± 54.6 ms) suggesting reduced neuronal excitability. The number of spikes evoked by either given step current injections or [alpha]-EPSP integration was also significantly decreased. The differential blocking of INaP by halothane, a popularly used volatile anesthetic, further supports the critical role of INaP in setting voltage threshold. Taken together, the presence of INaP in the soma demonstrates an intrinsic mechanism utilized by hippocampal CA1 pyramidal neurons to regulate axonal spike initiation through different biophysical properties of the Na⁺ channel. Furthermore, INaP becomes an interesting target of intrinsic plasticity because of its profound effect on the input-output function of the neuron. / text

The main goal of this dissertation was to study the bifurcation structure underlying families of low dimensional dynamical systems that model cellular excitability. One of the main contributions of this work is a mathematical characterization of profiles of electrophysiological activity in excitable cells of the same identified type, and across cell types, as a function of the relative levels of expression of ion channels coded by specific genes. In doing so, a generic formulation for transmembrane transport was derived from first principles in two different ways, expanding previous work by other researchers. The relationship between the expression of specific membrane proteins mediating transmembrane transport and the electrophysiological profile of excitable cells is well reproduced by electrodiffusion models of membrane potential involving as few as 2 state variables and as little as 2 transmembrane currents. Different forms of the generic electrodiffusion model presented here can be used to study the geometry underlying different forms of excitability in cardiocytes, neurons, and other excitable cells, and to simulate different patterns of response to constant, time-dependent, and (stochastic) time- and voltage-dependent stimuli. In all cases, an initial analysis performed on a deterministic, autonoumous version of the system of interest is presented to develop basic intuition that can be used to guide analyses of non-autonomous or stochastic versions of the model. Modifications of the biophysical models presented here can be used to study complex physiological systems involving single cells with specific membrane proteins, possibly linking different levels of biological organization and spatio-temporal scales.

The plasma membrane electrical activities of neurons that secrete gonadotropin releasing hormone (GnRH), referred to as GnRH neurons hereafter, have been studied extensively. A couple of mathematical models have been developed previously to explain different aspects of these activities including spontaneous spiking and responses to stimuli such as current injections, GnRH, thapsigargin (Tg) and apamin. The goal of this paper is to develop one single, minimal model that accounts for the experimental results reproduced by previously existing models and results that were not accounted for by these models. The latter includes two types of membrane potential bursting mechanisms and the associated calcium oscillations in the cytosol. One of them has not been reported in experimental literatures on GnRH neurons and is thus regarded as a model prediction. Other improvements achieved in this model include the incorporation of a more detailed description of calcium dynamics in a three dimensional cell body with the ion channels evenly distributed on the cell surface. Although the model is mainly based on data collected in cultured GnRH cell lines, we show that it is capable of explaining some properties of GnRH neurons observed in several of other preparations including mature GnRH neurons in hypothalamic slices. One potential explanation is suggested. A phenomenological reduction of this model into a simplified form is presented. The simplified model will facilitate the study of the roles of plasma membrane electrical activities on the pulsatile release of GnRH by these neurons when it is coupled with a model of pulsatile GnRH release based on the autoregulation mechanism.

Impulse control, an executive process that restrains inappropriate actions, is impaired in numerous psychiatric conditions. This thesis reports three experiments that utilized a novel animal model of impulse control, the response inhibition (RI) task, to examine the substrates that underlie learning this task. In the first experiment, rats were trained to withhold responding on the RI task, and then euthanized for electrophysiological testing. Training in the RI task increased the AMPA/NMDA ratio at the synapses of pyramidal neurons in the prelimbic, but not infralimbic, region of the medial prefrontal cortex. This enhancement paralleled performance as subjects underwent acquisition and extinction of the inhibitory response. AMPA/NMDA was elevated only in neurons that project to the ventral striatum. Thus, this experiment identified a synaptic correlate of impulse control. In the second experiment, a separate group of rats were trained in the RI task prior to electrophysiological testing. Training in the RI task produced a decrease in membrane excitability in prelimbic, but not infralimbic, neurons as measured by maximal spiking evoked in response to increasing current injection. Importantly, this decrease was strongly correlated with successful inhibition in the task. Fortuitously, subjects trained in an operant control condition showed elevated infralimbic, but not prelimbic, excitability, which was produced by learning an anticipatory signal that predicted imminent reward availability. These experiments revealed two cellular correlates of performance, corresponding to learning two different associations under distinct task conditions. In the final experiment, rats were trained on the RI task under three conditions: Short (4-s), long (60-s), or unpredictable (1-s to 60-s) premature phases. These conditions produced distinct errors on the RI task. Interestingly, amphetamine increased premature responding in the short and long conditions, but decreased premature responding in the unpredictable condition. This dissociation may arise from interactions between amphetamine and underlying cognitive processes, such as attention, timing, and conditioned avoidance. In summary, this thesis showed that learning to inhibit a response produces distinct synaptic, cellular, and pharmacological changes. It is hoped that these advances will provide a starting point for future therapeutic interventions of disorders of impulse control. / Thesis (Ph.D, Neuroscience Studies) -- Queen's University, 2011-07-11 09:44:54.815