Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Rajaei, Arman
Master of Applied Science - MASc
University of British Columbia
This thesis explores the analysis of neuronal action potentials (AP) within sympathetic nerve activity (SNA) from rats' splanchnic nerves. Chosen for their physiological resemblance to humans, rats serve as a viable model for this research. We introduce an advanced AP detection algorithm employing wavelet transforms (WT), addressing limitations observed in previous methodologies. A primary focus is the identification of the noise floor in the recorded data, a pivotal step for distinguishing neural activity from background noise. Analyzing rats’ SNA, allows for the integration of pharmacological interventions to induce ganglionic blockades, which lead to the significant reduction or elimination of APs. By comparing pre- and post-blockade recordings, we establish a baseline threshold, enhancing our capability to identify true neural signals. Our new algorithm consists of two steps. In the first step, we have used the maximal overlap discrete wavelet transform (MODWT) to identify the timestamps of APs in the raw data. Since MODWT employs multiresolution analysis and each level acts as a bandpass filter, choosing the appropriate level of decomposition is important. Therefore, by incorporating frequency and spectrum analysis of APs, we have determined the appropriate decomposition level that overlaps with the frequency components of the APs. In the next step, we constructed a mother wavelet from the average of extracted APs for use in continuous wavelet transform (CWT). This approach likely yields more AP extractions compared to using predefined wavelets from the Symlet and Daubechies families. By iterating this process with the APs extracted from CWT, we refine our template, enabling its adaptation as a mother wavelet and aiding in the accurate determination of the shape and length of rat APs. Although Shafer et al., (2022) also refined their template, they initiated it from large APs, potentially introducing a bias towards larger spikes in their analysis. Additionally, through burst analysis and the examination of AP synchronicity within bursts, we validated our AP identification approach by demonstrating the correlation between APs and burst intervals.
Enhanced Biological Signal Processing: Integrating Advanced Frequency Analysis and Wavelet Techniques for Action Potential Detection in Rats’ Sympathetic Nerve Recordings by Arman Rajaei B.A., Shiraz University, 2015 MASc Shiraz University, 2018 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE COLLEGE OF GRADUATE STUDIES (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) April 2024 © Arman Rajaei, 2024 ii The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis entitled: Enhanced Biological Signal Processing: Integrating Advanced Frequency Analysis and Wavelet Techniques for Action Potential Detection in Rats’ Sympathetic Nerve Recordings submitted by Arman Rajaei in partial fulfillment of the requirements of the degree of Master of Applied Science Dr. Rudolf Seethaler, School of Engineering, Supervisor Dr. Klaske van Heusden, School of Engineering Co-Supervisor Dr. Christopher R West, University of British Columbia-Vancouver Supervisory Committee Member Dr. Guy Dumont, University of British Columbia-Vancouver Supervisory Committee Member Dr. Christopher Collier, School of Engineering University Examiner iii Abstract This thesis explores the analysis of neuronal action potentials (AP) within sympathetic nerve activity (SNA) from rats' splanchnic nerves. Chosen for their physiological resemblance to humans, rats serve as a viable model for this research. We introduce an advanced AP detection algorithm employing wavelet transforms (WT), addressing limitations observed in previous methodologies. A primary focus is the identification of the noise floor in the recorded data, a pivotal step for distinguishing neural activity from background noise. Analyzing rats’ SNA, allows for the integration of pharmacological interventions to induce ganglionic blockades, which lead to the significant reduction or elimination of APs. By comparing pre- and post-blockade recordings, we establish a baseline threshold, enhancing our capability to identify true neural signals. Our new algorithm consists of two steps. In the first step, we have used the maximal overlap discrete wavelet transform (MODWT) to identify the timestamps of APs in the raw data. Since MODWT employs multiresolution analysis and each level acts as a bandpass filter, choosing the appropriate level of decomposition is important. Therefore, by incorporating frequency and spectrum analysis of APs, we have determined the appropriate decomposition level that overlaps with the frequency components of the APs. In the next step, we constructed a mother wavelet from the average of extracted APs for use in continuous wavelet transform (CWT). This approach likely yields more AP extractions compared to using predefined wavelets from the Symlet and Daubechies families. By iterating this process with the APs extracted from CWT, we refine our template, enabling its adaptation as a mother wavelet and aiding in the accurate determination of the shape and length of rat APs. Although Shafer et al., (2022) also refined their template, they initiated it from large APs, potentially iv introducing a bias towards larger spikes in their analysis. Additionally, through burst analysis and the examination of AP synchronicity within bursts, we validated our AP identification approach by demonstrating the correlation between APs and burst intervals. v Lay Summary This research focuses on studying the electrical spikes in nerve activity from rats, which helps us understand similar processes in humans. The study introduces a new method for detecting these spikes using WTs, a mathematical technique, improving upon older methods. We identified irrelevant background noise and used drug interventions to better distinguish the true nerve signals. We have applied a maximal overlap approach to analyze the data more accurately. We also created a new template for analyzing rat nerve signals, based on a larger dataset than previous studies, which could lead to more reliable results. Additionally, we examined how these spikes fire synchronously together, helping validate their detection method. This work could significantly enhance our understanding of nerve signal patterns and their implications for human health. vi Preface The data that form the backbone of our research were meticulously recorded in Dr. West's laboratory. I owe a debt of gratitude to Dr. West for providing such a rich dataset for examination. The opportunities for research provided by these data have been invaluable, allowing for a detailed and thorough exploration of neural signal processing. As of the writing of this thesis, we have not yet published our findings in academic journals. However, it is important to note that the groundwork laid out in the following pages is being prepared for publication. We are in the process of compiling our research, findings, and methodologies into a cohesive paper. This upcoming publication is intended to encapsulate all the materials and methods discussed throughout this thesis. vii Table of Contents Abstract ......................................................................................................................................... iii Lay Summary .................................................................................................................................v Preface ........................................................................................................................................... vi Table of Contents ........................................................................................................................ vii List of Tables ..................................................................................................................................x List of Figures ............................................................................................................................... xi List of Abbreviations ................................................................................................................. xiv Acknowledgements ......................................................................................................................xv Dedication ................................................................................................................................... xvi Chapter 1: Introduction ................................................................................................................1 1.1 General Background ....................................................................................................... 1 1.2 Objectives and Aims ....................................................................................................... 2 1.2.1 Defining the Noise Floor in AP Extraction ............................................................. 2 1.2.2 Adapting Algorithms for Enhanced AP Detection in Neuroscience ...................... 4 1.2.3 Determine the Length and Shape of APs in Rats. ................................................... 5 1.2.4 Validation of the new AP extraction algorithms ..................................................... 5 1.3 Organization of the thesis ............................................................................................... 6 Chapter 2: Literature Review .......................................................................................................8 2.1 General Physiology of Nervous System ......................................................................... 8 2.1.1 Functions of the Nervous System ........................................................................... 8 viii 2.1.2 Exploring the Components of the Nervous System: Structure, Function, and Connectivity ............................................................................................................................ 9 2.1.3 The Vital Role of APs in Neural Communication ................................................ 10 2.2 The Importance of Measuring AP ................................................................................. 12 2.2.1 The Central Role of APs in Cardiovascular Regulation ....................................... 12 2.2.2 The Pivotal Role of APs in Regulating Blood Pressure ....................................... 13 2.3 Current and potential Methods for Measuring AP ........................................................ 14 2.3.1 Integrated SNA: Burst Analysis............................................................................ 15 2.3.2 Wavelet Transform ............................................................................................... 18 2.3.2.1 General Principles of Wavelet Transform ........................................................ 18 2.3.2.2 Integrating the Uncertainty Principle with FT and STFT to Advance Wavelet Analysis............................................................................................................................. 20 2.3.2.3 Discrete Wavelet Transformations ................................................................... 24 2.3.2.4 Continuous Wavelet Transform ........................................................................ 29 2.4 Challenges in AP Extraction from Rat Neural Recordings .......................................... 34 Chapter 3: Recording and Data Analysis of SNA Signals .......................................................37 3.1 Data Collection ............................................................................................................. 37 3.2 Experimental Plan ......................................................................................................... 41 3.3 Algorithm Development ............................................................................................... 43 3.3.1 Burst Analysis ....................................................................................................... 43 3.3.2 Analytical Techniques for Extraction of APs from Noisy SNA ........................... 46 3.3.2.1 AP extraction using Maximal Overlap Discrete Wavelet Transform (MODWT)… .................................................................................................................... 46 ix 3.3.2.2 Refining AP extraction using an iterative CWT approach. .............................. 55 Chapter 4: Results and Discussion .............................................................................................63 4.1 Neural Analysis Outcomes: Burst Detection ................................................................ 63 4.2 AP Extraction and Template Analysis .......................................................................... 66 4.2.1.1 AP Detection Through MODWT Analysis....................................................... 67 4.2.1.2 Synthesis of the First Template and Mother Wavelet Construction ................. 69 4.2.1.3 Enhanced Detection through CWT and AP Template Refinement Using the Constructed Mother Wavelet ............................................................................................ 70 4.3 Temporal Correlation of APs with Burst Intervals in Neuronal Activity ..................... 74 Chapter 5: Conclusion .................................................................................................................77 5.1 Summary of Major Findings ......................................................................................... 77 5.2 Challenges and Limitations........................................................................................... 80 5.3 Future Directions .......................................................................................................... 80 Bibliography .................................................................................................................................82 x List of Tables Table 4.1: Comparison of the percentage of APs occurring during burst intervals as detected by our algorithm versus that of Shafer et al. (2022), with the total number of detected APs for each rat indicated in parentheses. .......................................................................................................... 76 xi List of Figures Figure 2.1: Stages of iSNA Burst Analysis - Transition from Raw Nerve Recordings to Rectified and Time-Integrated Signals ......................................................................................................... 17 Figure 2.2: Time-Frequency Localization. (a) High time uncertainty and low frequency uncertainty. (b) Low time uncertainty and high frequency uncertainty (Ghaderpour et al., 2021)....................................................................................................................................................... 21 Figure 2.3: Time-frequency resolution trade-off in WT: Stretched wavelets for detailed low-frequency analysis and compressed wavelets for precise high-frequency timing. (Ghaderpour et al., 2021) ....................................................................................................................................... 23 Figure 2.4: Three-level DWT decomposition process. (Chowdhury et al., 2019)........................ 24 Figure 2.5: Selection of commonly used mother wavelets in WT analysis. ................................. 26 Figure 2.6: Schematic overview of the AP extraction procedure using DWT denoising techniques........................................................................................................................................................ 28 Figure 2.7: Main steps of the CWT. ............................................................................................. 31 Figure 2.8: Overview of Salmanpour et al.'s procedure for AP detection using CWT with a custom-designed wavelet. .......................................................................................................................... 32 Figure 2.9: Shafer et al's iterative refinement of the AP detection algorithm using CWT-based template creation. .......................................................................................................................... 33 Figure 3.1: Anatomical Diagram of the Sympathetic Nervous System Highlighting the Left Greater Splanchnic Nerve (Haroun, 2018) ................................................................................................ 38 Figure 3.2: Comprehensive surgical setup. (a) Diagrammatic representation. (b) Actual laboratory procedure ....................................................................................................................................... 39 xii Figure 3.3: Surgical exposure of the rat's splanchnic sympathetic nerve (SSN) with real-time recording of SNA .......................................................................................................................... 40 Figure 3.4: Electrophysiological setup for nerve recording with active and reference electrodes 41 Figure 3.5: iSNA with burst detection based on threshold crossing ............................................. 45 Figure 3.6: (a) Averaged AP waveform extracted from neural data, and (b) the corresponding magnitude spectrum, revealing the dominant frequency at 444.44 Hz. ........................................ 48 Figure 3.7: Overlay of a 444.44 Hz sine wave on the central portion of the averaged AP waveform, demonstrating the alignment with the dominant frequency. ......................................................... 49 Figure 3.8: (a) Profile of the Symlet 6 wavelet, illustrating its wave-like structure (Shan et al., 2009) and (b) Comparative overlay of the averaged manually extracted APs with the Symlet 6 wavelet. ......................................................................................................................................... 51 Figure 3.9: The distribution of wavelet coefficients of noise with the selected threshold indicated by the red dashed lines. ................................................................................................................. 53 Figure 3.10: Half-second segment of thresholded approximation coefficients highlighting peaks corresponding to APs. ................................................................................................................... 54 Figure 3.11: Overview of the presented procedure for AP detection utilizing MODWT. ........... 55 Figure 3.12: (a) Averaged AP template, and (b) Corresponding magnitude spectrum with a peak at 444.44 Hz. ................................................................................................................................. 56 Figure 3.13: Comparison of the constructed wavelet with the averaged AP template. ................ 58 Figure 3.14: Comparative illustration of the original and updated AP templates. ....................... 60 Figure 3.15: Comparison of the second and third template refinements indicating near convergence. ................................................................................................................................. 61 xiii Figure 3.16: Overview of applying CWT for AP detection utilizing the constructed mother wavelet and its refinement. ......................................................................................................................... 61 Figure 4.1: One-Second Burst Analysis Snapshot for Rats 1-9 .................................................... 66 Figure 4.2: (a) Half-second segment of thresholded MODWT approximation coefficients and (b) overlay of extracted APs from Phase I recordings. ....................................................................... 68 Figure 4.3: Comparison of the averaged AP template and the custom-constructed mother wavelet....................................................................................................................................................... 69 Figure 4.4: (a) Half-second segment of Phase I recordings' CWT coefficients using the custom mother wavelet and (b) aligned and overlapped extracted APs. ................................................... 71 Figure 4.5: Comparative overlay of the first and second AP templates. ...................................... 72 Figure 4.6: Comparative analysis of first and second AP templates for Rats 2-9, highlighting improved extraction accuracy and AP morphology. ..................................................................... 74 Figure 4.7: iSNA with marked burst intervals (top) and the corresponding raw signal with APs detected by wavelet analysis (bottom) for Rat 1. .......................................................................... 75 xiv List of Abbreviations AG Adrenal gland AP Action potential Ao Aorta CG Celiac ganglion CMAP Compound muscle action potential CNS Central nervous system DWT Discrete wavelet transform FFT Fast Fourier Transform IVC Inferior vena cava MODWT Maximal overlap discrete wavelet transform MSNA Muscular sympathetic nerve activity MUAP Motor unit action potential nAChRs Nicotinic acetylcholine receptors PNS Peripheral nervous system SNS Sympathetic nervous system WT Wavelet transform xv Acknowledgements I would like to extend my deepest gratitude to several individuals whose guidance and support were invaluable throughout the course of this research. Firstly, I would like to thank Dr. Seethaler for his extensive help and mentorship. His insights and suggestions were crucial in shaping the entire research project, and his encouragement kept me motivated during challenging times. I am also immensely grateful to Dr. Van Heusden for her technical advice and expertise, which greatly contributed to the development of this thesis. Her meticulous attention to detail and profound knowledge in the field were instrumental in refining my work. Special thanks are also owed to Dr. West, whose innovative ideas and assistance with the biological aspects of this research were indispensable. Additionally, I would like to acknowledge the members of Dr. West’s lab, who played a significant role in recording the data essential for this study. Last but not least, I am thankful to Dr. Foster for sharing his vast experience and knowledge in this field of study. His wisdom and guidance were greatly appreciated and have left a lasting impact on my work. This thesis would not have been possible without the contributions and support of these individuals. I am truly grateful for their involvement and dedication. xvi Dedication This work is dedicated to my loving family: my father, mother, and little sister, whose unwavering support, love, and encouragement have been the bedrock of my existence. Their endless sacrifices and steadfast belief in me have been the cornerstone of my strength and inspiration, guiding me through life's myriad challenges. I would also like to extend my deepest appreciation to Mr. Reza Golshan, who has been like a brother to me since my arrival in Canada. His companionship and guidance have made my journey here less daunting and more hopeful. Furthermore, I am deeply grateful to Dr. Ehsan Pashanejad for his support and guidance during my initial days in Kelowna. His kindness and wisdom were crucial in helping me adapt to new beginnings, providing me with comfort and direction when I needed it most. Last but not least, my sincerest gratitude goes to Dr. Milad Goodarzi, who stood by me during my most challenging times. His friendship and unwavering support have been a beacon of light, illuminating the darkest of times with hope and solidarity.1 Chapter 1: Introduction 1.1 General Background Neuroscience is at the forefront of deciphering the intricacies of the brain, with electrophysiological signal recording serving as a cornerstone of this investigation. Neuronal electrical activity, represented as AP or spikes, is important in understanding brain function. This information is not only academic; it is used in practical applications such as brain-computer interfaces and provides important insights into a variety of neurological illnesses. Despite considerable advances, the field confronts challenges such as consistently capturing APs in live tissue recordings among an overwhelming amount of background noise and interference from neighboring cells. This challenge highlights a gap in existing technology, where correct detection and interpretation of brain signals is essential for both advancing neuroscience and improving therapeutic outcomes. Recent breakthroughs in this area, such as tetrode electrode layouts (Ferguson et al., 2009, Scholvin et al., 2016) and spectrum transformations (Liang et al., 2013, Biswal et al., 2015), represent considerable leaps. Notably, WTs have been useful in neuroscientific research due to their de-noising and compression characteristics (Quotb et al., 2011). Their characteristic strength in multi-resolution time-frequency analysis enables researchers to spot frequency fluctuations that are often missed in traditional time-domain investigations. Despite these technical advancements, important gaps remain. Numerous spike detection methods have been developed, but their performance in noisy extracellular settings remains poor, as Brychta et al. (2007a) note. In this context, the appropriate selection and deployment of mother wavelets, which are important for WT effectiveness, are under research. Furthermore, precisely quantifying the duration of APs 2 and determining the noise floor in AP measurements is essential. These factors are paramount not only for enhancing the accuracy of AP detection but also for improving the overall functionality and reliability of diagnostic tools and brain-computer interfaces. Addressing these challenges is not just about refining existing methodologies; it marks a significant step towards more precise, dependable, and therapeutically useful understandings of brain activity. In this context, rats have been utilized as subjects in this study due to their physiological similarities to humans, providing a relevant model for extrapolating findings to human conditions. To the best of our knowledge, there have been no previously published characterizations of rat APs. Therefore, besides addressing the challenges in previous studies we have tried to analyze rat’s APs characteristics. This makes our research pioneering in its field. Such advancements extend beyond theoretical significance; they pave the way for tangible progress in detecting and treating neurological illnesses, refining brain-computer interface technology, deepening our comprehension of the brain's core workings, and advancing our investigation of the blood pressure control loop, including its actuators and sensors. 1.2 Objectives and Aims 1.2.1 Defining the Noise Floor in AP Extraction The noise floor notion is of great importance in signal processing and pattern detection, particularly when denoising data using WT methods. The noise floor is the amount of background noise in a signal or the lowest level of a signal observable from this noise. Accurately identifying the noise floor is essential in wavelet-based denoising, which is a popular approach due to its ability to minimize in-band noise while keeping signal properties (Baldazzi et al., 2020). This is especially important for measuring APs in neural signals. APs are characterized by fast voltage 3 changes across cellular membranes and are frequently interfered with noise, hindering their exact detection. As a result, precisely determining the noise floor is more than a technical need; it is utilized for assuring the quality and reliability of neural signal processing. Previous investigators have employed a modified version of a formula originally proposed by Donoho (1995) to determine the noise floor in the extraction of AP. Their methodology is inherently subjective, as it involves selecting an appropriate correction factor through a trial-and-error process, which may yield different results across recordings with varying signal-to-noise ratios. This approach introduces variability and potential bias in the determination of the noise floor. A major element in our strategy was to determine the noise floor utilizing pharmacological intervention. This approach aims to provide a more standardized and objective measure by directly influencing the noise components within the recordings, thereby reducing the subjectivity associated with manual correction factor adjustments. The experimental framework is methodically split into two distinct phases. Phase I, referred to as 'Normal State Recording,' records the subject's baseline physiological activity, providing the data set with the APs of interest. Phase II, titled 'Ganglionic Blockade and Threshold Analysis,' offers a paradigm change by injecting hexamethonium bromide, which causes a ganglionic blockade. This results in a significant decrease or entire absence of APs, providing a baseline threshold for the noise floor. The noise threshold from Phase II is then applied to the Phase I recordings. The use of drug-induced neural inhibition to estimate the noise is a unique addition to neuroscience research. For the first time, it allows for a direct measurement of the noise floor in nerve recordings, eliminating tuning parameters common in traditional approaches. However, the 4 two-phase recording approach, while successfully tested in rat models, has limits in its direct application to human subjects. 1.2.2 Adapting Algorithms for Enhanced AP Detection in Neuroscience New advancements in neuroscience require the adaptation of AP detection algorithms. The rising complexity and variety of brain recordings, as noted in the works of Quian Quiroga & Panzeri (2009), need a dynamic approach to algorithm development. This requirement for continuous adaptation is driven by developments in recording technology and computational methodologies, which have greatly extended the breadth and depth of brain data accessible for analysis (Buzsáki et al., 2012). In our work, we proposed a two-step algorithm that utilizes the MODWT in Step 1 and the CWT in Step 2 to extract APs from the raw data. In the first step, we applied multiresolution analysis using MODWT, where each level functions as a bandpass filter. This setup necessitates performing frequency and spectrum analysis to effectively extract patterns from the signals. Consequently, we manually extracted a limited number of APs and conducted spectrum analysis on their average, enabling us to identify the decomposition level in a manner that enhances the robustness of our algorithm. The average of all extracted APs from the raw data using MODWTs provides a good estimation of the rat’s AP shape. The construction of a mother wavelet that best fits this average template and meets the CWT criteria marks the starting point for the second step of our algorithm. In Step 2, CWT coefficients are computed using the constructed mother wavelet. Iteratively applying the CWT refines the AP’s shape. Although this step resembles the approach used by Shafer et al. (2022), who began their analysis with MATLAB's built-in 'sym7' wavelet spanning 14 samples—5 thus limiting their initial extractions to larger APs—our method begins with templates of APs extracted using MODWT. This ensures that our starting template is more comprehensive and less biased towards larger APs, potentially yielding better results. 1.2.3 Determine the Length and Shape of APs in Rats. The study of AP duration and shape is of great importance in neuroscience for understanding the frequency and speed of neural transmission. The length and shape of an AP are essentially related to the rate at which neurons may fire, which directly affects the speed of information processing inside the brain. By looking into these characteristics, we obtain valuable insights into the nervous system's signaling pathways while also improving our ability to precisely detect APs. The final template that we have developed after several rounds of refinements in the proposed AP extraction algorithm, can be regarded as representative of the AP shapes in rats. 1.2.4 Validation of the new AP extraction algorithms Validating the accuracy of AP extraction is difficult since a ground truth for comparison is not readily available. While our recordings were conducted under normal conditions for the rats, APs’ shapes, sizes, and frequencies change due to various subject conditions, such as animal physiology, and variations in breathing or blood pressure. There is also a rhythmic pattern of clusters with a high frequency of AP followed by intervals with low AP activity. This interplay of few APs and concentrated bursts is being studied by neuroscientists since it may provide important insights into the intricate mechanics of neural coding and transmission. Research shows that about 75% of all APs in humans are expected to occur during burst. We were able to show that our algorithms applied to rats provide the same results, providing a sanity check of our approach in the absence of a ground truth to compare to. 6 By doing so, we also aimed to provide a tool for future research to uncover the underlying temporal structure of neuronal communication, with a particular emphasis on determining the fraction of APs that emerge during intervals between bursts. 1.3 Organization of the thesis The thesis is structured to ensure a comprehensive and systematic evaluation of AP detection and analysis in the field of neuroscience. Chapter 2 provides a Literature Review. This chapter explains the general physiology of the nervous system and emphasizes the importance of measuring APs. It evaluates existing approaches for measuring APs. This part also addresses the inherent difficulties in AP identification and examines various methodologies used in the area, laying the way for a broader knowledge of the issue. Chapter 3 focuses on the research's practical sections, describing the Recording Procedure and Data Analysis of Sympathetic Nerve Activity and the experimental plans, ensuring a clear and repeatable framework. It also examines the creation and improvement of algorithms required for data analysis, offering insights into the technical details and procedural complexities of the research. In Chapter 4, the thesis provides Results and Discussion. This chapter is dedicated to assessing the data collected throughout the experimental processes. It examines burst analysis and AP extraction, looking for relationships and implications. This part not only offers the empirical findings but also conducts a comprehensive discussion. Finally, Chapter 5 provides Conclusions. It synthesizes the research findings, reflecting on the implications and contributions of the study to the field of neuroscience. It revisits the research questions and hypotheses, providing a cogent summary of the results and discussing the broader 7 impact of the study. Additionally, this chapter proposes directions for future research, considering the limitations encountered and the potential avenues for further exploration in the realm of AP detection and analysis. 8 Chapter 2: Literature Review 2.1 General Physiology of Nervous System 2.1.1 Functions of the Nervous System The nervous system, a complex and intricate network, plays a pivotal role in the coordination and execution of various bodily functions. It is divided into two parts: the Central Nervous System (CNS) and the Peripheral Nervous System (PNS). The nervous system's primary function is to receive, process, and respond to sensory information, a task that requires a complex interplay of various neural structures (Thau et al., 2019). It includes both the central and peripheral nervous systems, with the brain and spinal cord constituting the CNS. The CNS not only regulates essential physiological processes such as breathing, heart rate, hormone production, and body temperature, but it also controls thoughts, movements, and emotions. Furthermore, the cortex, a key component of the CNS, contains primary, secondary, and tertiary sensory and motor areas, integrating diverse signals for coherent functioning (Ludwig et al., 2017). This integration is crucial for sustaining homeostasis, which ensures that the body's internal environment stays stable and efficient in the face of external disturbances. As a result, the nervous system is essential to both our deliberate and involuntary behaviors, defining our interactions with the environment and maintaining our internal physiological equilibrium. The nervous system is important in controlling many physiological activities, including the complex process of blood pressure management. The sympathetic nervous system (SNS), which controls both immediate and long-term blood pressure alterations, is crucial to this regulation. This system works by sending APs from the brainstem via the nervous system, prompting the release of neurotransmitters like norepinephrine. These neurotransmitters modulate the functioning of the 9 heart, blood vessels, and other organs in order to maintain stable blood pressure. Notably, the SNS's reaction to blood pressure variations is mediated in part by the baroreflex mechanism, an essential feedback loop. Extensive studies, such as Cowley's work on long-term arterial pressure regulation (Cowley Jr, 1992), Osborn's hypothesis (Osborn, 2005), and Schlaich et al.'s exploration of sympathetic augmentation in hypertension (Schlaich et al., 2004), offer valuable insights into these complex processes. The nervous system's numerous activities, ranging from controlling blood pressure to supporting sophisticated cognitive processes, highlight its extraordinary complexity and efficiency. These functions are made possible via a complex network of components, each of which serves a distinct purpose in the overall system. The following section highlights important components of the nervous system. 2.1.2 Exploring the Components of the Nervous System: Structure, Function, and Connectivity The CNS, which includes the brain and spinal cord, is the principal coordinator of sensory input and controls body reactions. It evaluates incoming sensory information and gives orders based on previous experience (Ikegami & Ijima, 2021). The PNS is a complex network made up of sensory receptors, neurons, ganglia, and plexuses. It serves as a vital communication link between the CNS and the rest of the body, delivering sensory information to the CNS and transferring motor orders from the CNS to peripheral tissues (Damodaran, 2014). The PNS connects the CNS to limbs and organs, serves as a relay between the brain, spinal cord, and the body, and is vital in delivering sensory axon impulses (Hornemann, 2021). 10 Within the CNS, the brain stands out as the command center, in charge of interpreting sensory data, aiding cognitive activities, and making decisions. The hippocampus and prefrontal cortex, for example, are regions of the brain that specialize in memory processing and decision-making (Daly et al., 2012). The spinal cord, an intrinsic part of the CNS, serves as a major information route. It controls muscular responses and sends sensory information back to the brain, resulting in a continual feedback-response loop. The spinal cord, as a component of the CNS, is vital for the integration and coordination of complex sensory, motor, and autonomic processes (Hochman, 2007). As we move on to the PNS, it is divided into somatic and autonomic systems. The somatic system controls voluntary movements and sensory reception, interacting with the outside world through motor neurons. In contrast, the autonomic nervous system, which is divided into sympathetic and parasympathetic branches, governs internal organs autonomously. Central to these operations are APs, the electrical signals that facilitate neural communication. These APs emphasize the importance of electrical impulses in both deliberate and involuntary responses. The following sections will focus on the mechanism of APs, explaining how these signals are essential to the operation of both somatic and autonomic systems (Damodaran, 2014). 2.1.3 The Vital Role of APs in Neural Communication The AP, also known as the 'language of neurons,' is a fundamental biophysical phenomenon that underpins nervous system function. It is a quick, transitory, self-regenerating voltage change across the neuron's membrane. The controlled opening and closing of ion channels allow ions to flow in and out of the neuron, creating a wave of electrical activity. A significant stimulation causes depolarization, which results in a brief reversal of the membrane potential, followed by 11 repolarization, which returns it to its resting condition. This mechanism is fundamental for information transmission across large neural networks and is a cornerstone in our knowledge of neuronal excitability and the complex communications within the nervous system (McCormick, 2004). APs are crucial for neural communication within the nervous system. When a neuron receives adequate stimulation, often near the axon hillock, the process of AP production begins. This causes depolarization, which is characterized by an influx of positively charged ions and a brief reversal of the electrical charge of the membrane. The AP then spreads throughout the axon, keeping its intensity constant by a continuous cycle of depolarization and repolarization. The arrival of an AP at the synaptic connection causes the release of neurotransmitters, bridging the gap to the next cell and ensuring signal transmission. This sophisticated mechanism emphasizes the pivotal role of APs in connecting diverse areas of the nervous system and facilitating intricate brain operations (Wang et al., 2016). APs are essential for the operation of both the CNS and PNS. APs are required in the CNS for sensory information processing, sophisticated thinking processes, and reaction initiation. They are essential in the PNS for signal transmission to and from the CNS, allowing for both involuntary reflex actions and deliberate movements. This dynamic interaction exemplifies the practical importance of APs in everyday physiological operations. The next sections will elaborate on the significance of measuring APs, examining how these measurements are essential for understanding and evaluating the complicated workings of the neurological system, as well as the methods utilized to collect this key data. This study will not only highlight the significance of APs 12 but will also pave the way for comprehending recent advances and approaches in neurological research. 2.2 The Importance of Measuring AP 2.2.1 The Central Role of APs in Cardiovascular Regulation APs are important electrical signals that govern cardiac function and vascular tone in the cardiovascular system. These APs coordinate the rhythmic contractions of the heart muscle, ensuring that blood circulates efficiently throughout the body. They also alter blood vessel tone, which is essential for maintaining stable blood pressure. This is accomplished by a complex balance of electrical impulses that control cardiac cycles and vascular responses while responding to the body's changing needs. Understanding the mechanisms of APs in this setting is vital because it sheds light on their essential role in cardiovascular health. This understanding lays the groundwork for a more in-depth examination of APs specific roles in heart rate regulation and blood vessel tone, which is essential for a thorough understanding of cardiovascular function. (Wahler, 1995) APs are important in managing blood pressure by modulating blood vessel tone. APs influence the contraction and relaxation of smooth muscle cells in blood vessel walls in the vascular system (Brozovich et al., 2016). An increase in APs triggers muscle contraction, causing the vessels to constrict (vasoconstriction) and subsequently elevating blood pressure. Conversely, a reduction in APs facilitates muscular relaxation, prompting the arteries to widen (vasodilate) and thereby lowering blood pressure. The ability of APs to adjust blood vessel tone guarantees that the body may regulate blood pressure adaptively in response to various physiological demands. 13 The principles of APs in cardiovascular control provide insights that can be applied to other physiological systems, emphasizing their universal importance in bodily functioning. APs are essential in other control loops, such as neuronal communication and muscular contractions, much as they are in heart rhythm and blood vessel tone regulation (Scott & Santana, 2010). This broad applicability highlights the importance of APs beyond cardiovascular health, highlighting their role in the larger context of physiological regulation. Understanding these various roles opens the door to investigating APs in specific regulatory circuits like as blood pressure regulation, where their effect is especially direct and significant. 2.2.2 The Pivotal Role of APs in Regulating Blood Pressure As previously mentioned, APs are important in controlling numerous physiological control loops, one of which is blood pressure regulation. The autonomic nervous system, which includes sympathetic and parasympathetic branches, regulates blood pressure. The SNS encourages the heart to beat faster and with greater force by releasing neurotransmitters such as norepinephrine, while simultaneously causing blood vessels to contract. The parasympathetic nervous system, on the other hand, produces acetylcholine, primarily through the vagus nerve, to reduce the heart rate and dilate the blood vessels (Benarroch, 2014). This careful balance between the two branches, which is mediated by the frequency and amplitude of APs, keeps blood pressure within a healthy range. The baroreceptor reflex is a famous illustration of how APs play an important role in this regulatory loop. Baroreceptors are stretch-sensitive nerve endings found in the carotid sinus and aortic arch that detect changes in blood pressure. When blood pressure rises, the baroreceptors stretch, and the frequency of APs transmitted to the brain increases. In reaction, the brain increases 14 parasympathetic activity while decreasing sympathetic activity, resulting in a decrease in heart rate and blood vessel constriction, lowering blood pressure. When blood pressure falls, baroreceptors fire APs at a lesser frequency, causing an increase in heart rate and vessel constriction to raise blood pressure (Parati & Bilo, 2012). Thus, the AP functions as an important messenger in this regulatory loop, turning mechanical stimuli (blood vessel strain) into electrical signals that the central nervous system can interpret and act on. This emphasizes the crucial significance of APs not only in blood pressure regulation but also as a key component in various physiological regulatory loops. The role of APs in maintaining homeostasis is extensive and vital, ranging from controlling body temperature to coordinating muscular movement. 2.3 Current and potential Methods for Measuring AP In examining the nervous system, we've analyzed the essential roles of APs in both central and peripheral systems, highlighting their crucial function in neural transmission and regulation. This insight emphasizes the importance of accurately measuring APs to unravel the complexities of neurological dynamics. The journey of quantifying APs traces back to the foundational work of Luigi Galvani in 1771, who discovered that electrical stimulation of muscular tissue produces contraction and force. However, the electrical response of this phenomenon was not quantified until advancements in recording equipment, such as the cathode ray oscilloscope by Gasser and Erlanger in 1922 and the amplifier and oscillograph by Matthews in 1928, were made. The era of modern Electromyography (EMG) machines began with Jasper's construction at McGill University in 1942. Pioneering work by Adrian and Bronk in 1929 made it possible to record a single motor unit action potential 15 (MUAP) in humans by connecting concentric and bipolar needle electrodes to an amplifier and a loudspeaker. By the late 1930s, Denny-Brown and Pennybacker had recorded and described fasciculations, or APs of single contracting or spontaneously firing motor units, distinguishing them from the APs of single denervated muscle fibers, thus describing fibrillations. Following World War II, Larrabee, Hodes, and German began measuring the combined potential of individual muscle fibers from the muscle surface after stimulating the supplying nerve percutaneously, later known as the compound muscle action potential (CMAP). They quantified the differences in CMAP waveforms when stimulating above and below the site of peripheral nerve injury in war victims. In 1957, Lambert and Eaton described the electrophysiologic features of a newly recognized disease they called the myasthenic syndrome sometimes associated with lung carcinoma. The contributions of these researchers laid the groundwork for our current understanding and measurement of APs, shaping the field of neurophysiology and its applications in clinical settings (Kazamel et al., 2013). Building upon this history, our current understanding of AP measurement is shaped by both technological and methodological innovations. As we delve deeper into this topic, we encounter a spectrum of sophisticated methodologies, from Burst Analysis to CWT. These methods not only capture the complexity of APs but also underscore the challenges in recording these transient neural events with precision. In this section, we will dissect these techniques, shedding light on their unique contributions and limitations in the broader landscape of neuroscience research. 2.3.1 Integrated SNA: Burst Analysis The late twentieth century was an important turning point in the study of neural encoding, motivated by advances in neural recording and encoding methods. The researchers' focus shifted 16 from the rate of APs to their pattern, with a particular emphasis on burst firing. The invention of multi-electrode arrays transformed the field by allowing for the simultaneous recording of several neurons. This technological advancement provided important insights into the complicated dynamics of the brain, allowing for a better knowledge of neural transmission and network activity (Sher et al., 2007). The combination of strong computational methodology with cutting-edge recording technology stimulated novel approaches to brain data interpretation, such as SNA analysis, providing unparalleled accuracy and depth in understanding neural systems (Paninski & Cunningham, 2018). Understanding SNA patterns, particularly through burst analysis, became essential for explaining the brain processes and its underlying functions. Integrated SNA (iSNA) burst analysis, entails processing and integrating raw nerve recordings in order to identify and evaluate sympathetic activity bursts. This approach, which allows for the quantification of sympathetic fluxes to various arterial beds and organs, is invaluable in dissecting the neural control of cardiovascular processes. The burst analysis of iSNA is conducted in two distinct stages to interpret the underlying neural patterns, as illustrated in Figure 2.1. Initially, the procedure begins with the rectification of raw nerve recordings, transforming the raw data into a rectified signal expressed as 𝑉(𝑡). Subsequently, 𝑉(𝑡) is integrated over a predetermined time window Δ(𝑡), which allows for the quantification of the signal's magnitude within specific intervals, thereby providing a detailed temporal profile of neural activity. 𝐼(𝑡) = ∫ 𝑉(τ)𝑡+Δ𝑡𝑡 𝑑τ (2.1) This two-stage process is essential for a comprehensive analysis of the iSNA's behavior and its implications in physiological responses. 17 Figure 2.1: Stages of iSNA Burst Analysis - Transition from Raw Nerve Recordings to Rectified and Time-Integrated Signals Following that, the integrated signal 𝐼(𝑡) is examined for burst detection, often using a threshold level 𝐼threshold, with a burst found when: 𝐼(𝑡) ≥ 𝐼threshold (2.2) The methodology used to detect bursts, which can range from simple threshold-based methods to complicated machine learning algorithms, plays an important role in this process. Following the identification of bursts, their parameters such as amplitude, frequency, and length are examined, revealing insights into the brain control processes at work. While iSNA burst analysis has been a basis in understanding SNA patterns, it is not without limits, prompting the investigation of other approaches. One significant limitation is the inability of burst parameters to distinguish whether a large burst originates from a few large amplitude sympathetic spikes, potential artifacts, or many small amplitude spikes occurring rapidly. This ambiguity is compounded by the variable signal-to-noise ratio (SNR) in recordings, which affects 18 the integration of passband noise into each burst and challenges the consistency of burst amplitude and area comparisons across subjects (Brychta, et al., 2007b). Moreover, crucial AP information, such as the count and morphology of spikes contributing to a sympathetic burst, is lost during the integration process. Furthermore, choosing optimal parameters for signal processing and burst detection, such as the integration window Δ(𝑡) and the detection threshold 𝐼threshold, frequently necessitates thorough tuning for various experimental setups. This process is pivotal; however, it introduces subjective elements that can affect the reproducibility of the experiments. These limits underscore the need for more refined methods in neural signal analysis that can provide better clarity and depth. We'll look at advanced approaches like CWT, and DWT, as we go. These methods have the potential to overcome some of the constraints of iSNA burst analysis, providing broader insights into neural encoding, and potentially revealing new dimensions in our understanding of the autonomic nervous system. 2.3.2 Wavelet Transform 2.3.2.1 General Principles of Wavelet Transform The WT has emerged as an important tool in signal processing, particularly for its ability to analyze non-stationary signals such as APs. Unlike classic methods like the Fourier transform, which only provides frequency information, the WT provides a more dynamic perspective of a signal by capturing both time and frequency features. Because of this dual element, it is particularly useful in neuroscience, where understanding the subtle, time-varying nature of neuronal impulses is essential. 19 Historically, the development of WT can be traced back to the early 20th century, though its theoretical foundations and widespread application gained momentum in the 1980s (Polikar, 1999). The term "wavelet" itself implies a small wave, signifying the method's ability to break down a complex signal into these 'small waves' or wavelets. This decomposition allows for the analysis of various components of the signal at different scales, offering a granular perspective that is particularly beneficial for signals whose characteristics change over time, such as the electrical activity of neurons. The evolution of WT has been a collaborative effort spanning mathematics, engineering, and physics, reflecting its interdisciplinary impact and versatility in various scientific and practical applications. The WT's foundational equation, provides a framework for signal decomposition: 𝑊(𝑎, 𝑏) =1√𝑎∫𝑥(𝑡)ψ∗ (𝑡 − 𝑏𝑎)𝑑 𝑡 (2.3) In this equation, 𝑊(𝑎, 𝑏) represents the wavelet coefficients, essential for understanding the signal at various scales and positions. The function 𝑥(𝑡) is the original signal, while ψ is the wavelet function. The parameter 𝑎 is a scaling factor that modifies the wavelet for analyzing different frequency components of the signal, and 𝑏 is a shifting factor that moves the wavelet across the time axis to investigate other regions of the data. This procedure is like employing a zoom lens on a camera. The scaling factor 𝑎 acts similarly to zoom, allowing us to perceive a broad picture or zoom in on small details. The shifting factor 𝑏, equivalent to moving the camera lens around, allows us to investigate different parts of the signal in depth. The WT is distinguished by its dual scaling and shifting capabilities, which provide a diverse approach to signal processing. Continuing the investigation of the WT as a tool for detailed signal analysis, similar to a camera's zoom lens, we now shift our attention to the fundamental concepts that support this technique. The 20 upcoming section will utilize the uncertainty principle to contrast the WT with the Fourier Transform (FT) and Short-Time Fourier Transform (STFT). This principle highlights the intrinsic compromises in signal processing, especially regarding wavelet time-frequency localization. This comparison aims to reveal the special characteristics of WT, notably their comprehensive capacity to capture both time and frequency information, offering a more profound understanding of complex, dynamic signals than traditional methods commonly used in neuroscience. 2.3.2.2 Integrating the Uncertainty Principle with FT and STFT to Advance Wavelet Analysis The Uncertainty Principle is a key notion in signal analysis that points to the inherent constraints in detecting a signal's temporal and frequency information with high precision. This approach, which is based on quantum mechanics, has profound implications in signal processing. It argues that there is a fundamental trade-off between the precision of measuring a signal's time and frequency: increasing precision in one domain ultimately leads to a loss of precision in the other. This is stated mathematically as: σ𝑡 ⋅ σω ≥12 (2.4) In this expression, 𝜎𝑡, the standard deviation in time, and 𝜎𝜔, the standard deviation in frequency, represent the uncertainties in time and frequency measurements, respectively. In Figure 2.2a, we observe a representation with a broader time uncertainty and a narrower frequency uncertainty, characteristic of a signal with a clear frequency that persists over an extended period. Conversely, Figure 2.2b illustrates a scenario with a narrower time uncertainty and a broader frequency uncertainty, typical of a brief signal that contains a wider range of frequencies. According to the 21 concept, the product of these uncertainties is bounded from below, which means that as we increase accuracy in one domain, we lose it in the other. (a) (b) Figure 2.2: Time-Frequency Localization. (a) High time uncertainty and low frequency uncertainty. (b) Low time uncertainty and high frequency uncertainty (Ghaderpour et al., 2021) For example, with perfect time resolution, we may detect when an event occurs but have no knowledge of its frequency content. In contrast, with complete frequency resolution, we know all of the frequencies involved but cannot predict when they will occur. This is especially important for evaluating complicated, dynamic signals, such as those seen in neuroscience, where APs are characterized by rapid transients, or music, where notes change over time. Understanding the uncertainty principle takes us to the foundation of the Fourier Transform (FT) and Short-Time Fourier Transform (STFT) techniques. The following equations express the FT: x(t) = ∑ ckejk(2π/T)t∞k=−∞, ck =1T∫x(t)e−jk(2π/T)tdtT (2.5) that deconstruct a signal into a sequence of sinusoidal functions of varying frequencies, resulting in a full spectrum analysis. Here, 𝑐𝑘 represents the complex amplitude of the 𝑘-th frequency component, detailing how much of each frequency is present in the original signal 𝑥(𝑡). However, the FT's reliance on sinusoids, which are infinite in time, limits its capacity to analyze signals that 22 evolve over time, such as non-stationary signals. The sinusoids' perfect frequency domain support contrasts with their infinite time domain support, leading to a lack of temporal localization in the analysis. To overcome this issue, the STFT modifies the FT using localized windows, obtaining the equations: 𝑆(τ, 𝑓) = ∫𝑥(𝑡)𝑤∗(𝑡 − τ)𝑒−2𝑗π𝑓𝑡𝑑𝑡𝑥 (2.6) where 𝑤(𝑡) is the windowing function, 𝑓 and 𝜏 are frequency and translation (time) parameters respectively, * is the complex conjugate operator, and 𝑆(τ, 𝑓) is the STFT of 𝑥(𝑡) at frequency 𝑓 and translation 𝜏. This approach divides the signal into smaller pieces, which are then analyzed using the FT, which offers a localized time-frequency representation. The STFT can thus provide a representation of when specific frequencies occur, although at the cost of the time-frequency resolution trade-off given by the window size. Understanding the limits of the STFT, particularly regarding the Uncertainty Principle, the next step in our signal analysis journey led us to the WT. This approach stands out as an advanced option, providing a solution to the STFT's fixed resolution barrier. As shown in Figure 2.3, WT suggests applying different frequency bands for different frequency ranges. The frequency resolution and time resolution in this time-frequency representation are managed in such a way that it normally offers more frequency resolution for lower frequencies and more time resolution for higher frequencies. 23 Figure 2.3: Time-frequency resolution trade-off in WT: Stretched wavelets for detailed low-frequency analysis and compressed wavelets for precise high-frequency timing. (Ghaderpour et al., 2021) Lower-frequency wavelets are stretched to cover a longer time period, allowing the wavelet to collect more detail about a signal's low-frequency components. Because of the greater time coverage, the time resolution is poorer, but this is often acceptable because low-frequency components change more slowly over time. For higher frequencies, however, the wavelets are compressed, which means they cover a shorter time period. This compression improves temporal resolution, allowing for the identification of quick changes in the signal, but decreasing frequency resolution—a reasonable trade-off given the transient nature of high-frequency components. Based on the comprehensive descriptions and the similarity between the WT and a camera lens, one can conclude that the WT is a useful tool for denoising data. This is accomplished by focusing on the signals of interest while filtering out the unimportant higher frequencies, which are recognized as noise. As a result, this approach is beneficial in denoising neural signals and separating APs from background noise. In the following sections, we go deeper into two basic WT algorithms: the DWT and the CWT. Our discussion focuses on their applications in neural signal denoising and AP extraction, as well as a comparison of their different strengths and limitations. 24 2.3.2.3 Discrete Wavelet Transformations The DWT is a systematic method that decomposes a signal into its constituent frequency bands by utilizing both a low-pass filter (𝑔[𝑛]) and a high-pass filter (ℎ[𝑛]). The DWT processes the signal 𝑥[𝑛] through specific mathematical operations at each level of decomposition, enabling a multi-resolution analysis of the signal. This process is visually demonstrated in Figure 2.4, which depicts a three-level decomposition to illustrate the layered analysis approach. Figure 2.4: Three-level DWT decomposition process. (Chowdhury et al., 2019) Multi-resolution analysis method divides a signal into two sets of coefficients: approximation and detail. This division is done iteratively, with high-pass (ℎ[𝑛]) and low-pass filters (𝑔[𝑛]) used at each level of analysis. • Approximation Coefficients: These coefficients represent the signal's lower frequency components. At any given level, they capture the lower half of the signal's frequency range or derive from the previous level's coefficients. The approximation coefficients 𝐴 at level 𝑗 are given by: 𝐴𝑗[𝑛] =∑𝑔[𝑘 − 2𝑛]𝑘⋅ 𝐴𝑗−1[𝑘] (2.7) 25 • Detail Coefficients: In contrast, these coefficients indicate higher frequency components, which include the upper half of the frequency range of the signal at each level. The detail coefficients 𝐷 at level 𝑗 are calculated as: 𝐷𝑗[𝑛] =∑ℎ[𝑘 − 2𝑛]𝑘⋅ 𝐴𝑗−1[𝑘] (2.8) For a specified level the frequency bands for these coefficients are estimated as follows: High-pass frequency band = (𝐹𝑠2𝐿+1,𝐹𝑠2𝐿) (2.9) Low-pass frequency band = (0,𝐹𝑠2𝐿+1) (2.10) where 𝐹𝑠 is the sampling frequency and 𝐿 is the level of decomposition. These techniques for dividing signals into high-pass and low-pass frequency bands are based on fundamental wavelet theory and digital signal processing principles. They are specifically consistent with the octave band filtering concept, which is an essential concept in wavelet analysis (Mallat, 1999). This method ensures that each decomposition level in the WT precisely captures a specific frequency range of the signal, allowing for a thorough and complete examination of all its elements. An integral aspect of understanding the frequency band calculations is the Nyquist Theorem, a fundamental principle in digital signal processing. According to the Nyquist Theorem, a signal must be sampled at least twice the frequency of its highest frequency component in order to be accurately reconstructed. This theorem explains the division by two noticed in the DWT frequency band computations. As an example, let's consider a two-level decomposition of a signal having been recorded at a sampling frequency of 10 kHz. At the first level of decomposition, the signal undergoes filtering to produce two sets of coefficients: 26 • Detail Coefficients 𝐷1: These are the higher frequency features in the frequency band between 2.5 kHz and 5 kHz, which is half the distance from the midpoint to the Nyquist frequency. • Approximation Coefficients 𝐴1: These encompass the lower frequencies in the range of 0 to 2.5 kHz, up to the midpoint of the Nyquist frequency. For the second level of decomposition, the approximation coefficients 𝑎1 are further decomposed to yield: • Detail Coefficients 𝐷2: Capturing the frequency details between 1.25 kHz and 2.5 kHz. • Approximation Coefficients 𝐴2: Containing the frequencies from 0 to 1.25 kHz. In the context of the DWT, the selection of the mother wavelet is a pivotal decision that directly influences the outcome of the signal analysis. Each mother wavelet is associated with unique coefficients in its corresponding high-pass and low-pass filters. These coefficients dictate how the signal is partitioned into its detail and approximation components at every level of decomposition. Figure 2.5 displays a selection of the most commonly used mother wavelets. Figure 2.5: Selection of commonly used mother wavelets in WT analysis. 27 The diversity of mother wavelets available allows for a tailored approach to signal processing, enabling researchers and practitioners to select a wavelet that best matches the characteristics and requirements of their specific application. For AP extraction, one would attempt to use a mother wavelet that approximates the shape of an ideal AP as closely as possible. This ensures that the high pass filter acts similar to a matched filter. As a result, the magnitude of the high passed filter signal represents a measure of correlation between the raw signal and the ideal AP. More generally, a careful selection process ensures that the WT is optimized for capturing the essential features of the signal, facilitating accurate and meaningful analysis. Therefore, several implementation decisions appear to have a considerable impact on the efficacy of WT algorithms. These options include the mother wavelet, decomposition level, and threshold definition approach. Diedrich et al. (2003) have been recognized as pioneers in using wavelet analysis for AP extraction, particularly for their groundbreaking work on muscular sympathetic nerve activity (MSNA) signals. Their novel technique used the DWT with "Symlet 7" as the mother wavelet. This decision was made since the shape of "Symlet 7" resembles that of an AP spike, making it ideal for analyzing human data. In their method, DWT was used for the fast decomposition and reconstruction of the simulated, noiseless neurogram. Specifically, they decomposed the signal to the desired level, then thresholded each decomposition level, and identified the APs in the reconstructed signal from the thresholded coefficients. For the most effective analysis, they methodically investigated several decomposition levels to determine which provided the most accurate performance for their task. Building on this basis of careful evaluation, they focused their efforts on improving the denoising process for MSNA signals. Their approach to improve MSNA signal denoising involved modifying the threshold technique originally proposed by Donoho (Donoho & Johnstone, 1994, Donoho, 1995)as follows: 28 𝑇 = 𝜎√2 ln(𝑁) (2.11) where 𝜎 is the standard deviation of the Gaussian noise and 𝑁 is the number of samples in the signal. The 𝜎 for each level is estimated as the median absolute value of the wavelet coefficients divided by 0.6745, which is the 75th percentile of the standard normal distribution. Through a careful selection procedure similar to determining the appropriate decomposition level, they discovered that applying a correction factor of 𝐾 = 0.8 to the threshold considerably improved the efficacy of their technique. This improvement was followed by the adoption of a precise peak detection approach to ensure accurate AP identification. Figure 2.6 presents a schematic overview of the procedure. Figure 2.6: Schematic overview of the AP extraction procedure using DWT denoising techniques. There are two primary drawbacks associated with using this algorithm for extracting APs: • Firstly, their method for determining the level of decomposition is based on the maximum number of correctly extracted APs. This approach is heuristic and lacks a scientific basis. • Secondly, the approach involves using a correction factor for thresholding, which, despite being based on the formula suggested by Donoho et al., is adjusted through empirical observations. So, similar to decomposition level determination, empirically modifying the threshold with a correction factor introduces a level of subjectivity and potential for inconsistency. Reconstruct the signal DWT Raw Recording Sym 7 Detail Coefficients Thresholding 𝐾𝜎√2 ln(𝑁) AP’s Time Stamps Denoised Signal 29 • Thirdly, compared to the CWT, as used by other researchers in subsequent works, Diedrich’s algorithm exhibits less accuracy in detecting APs. The specifics of how CWT is applied and its advantages over DWT in the context of AP extraction will be discussed in detail in the following section. 2.3.2.4 Continuous Wavelet Transform The CWT, defined by Equation. (2.3), is an extension of the basic WT that offers a continuous, over-complete analysis of a signal. Unlike DWT discrete counterpart, which samples the scale and translation parameters at discrete intervals, CWT performs an exhaustive search over all possible scales and translations, providing a detailed map of a signal's time-frequency landscape. Because of its continuous nature, CWT can represent signals with finer detail, making it particularly effective in detecting small, temporary events in neurological signals, such as APs in neuroscience. This accuracy allows researchers to precisely investigate the temporal evolution of neuronal firing patterns and gain a deeper understanding of the brain's complicated communication mechanisms. The CWT technical details can be thought of as a multi-step procedure that transfers a signal into the wavelet domain. Here is a breakdown of the stages, with a technical but understandable explanation of each: i. Initial Comparison: The CWT begins by comparing a mother wavelet to the first segment of the signal. In this comparison, the wavelet is aligned with the beginning of the signal and their similarity is evaluated. 30 ii. Correlation Calculation: To measure the strength of correlation between the wavelet and this section of the signal, a numerical value, generally designated as 𝐶 in Figure 2.7a, is generated. A higher 𝐶 value suggests more similarity. 𝐶 can be read as the correlation coefficient between the signal and the wavelet if they are normalized to have unit energy. iii. Shifting the Wavelet: The wavelet is then slowly shifted to the right, and the process is repeated. A fresh correlation value is calculated at each shift, which measures the similarity at that time (Figure 2.7b). iv. Scaling the Wavelet: After comparing the wavelet throughout the full signal, it is scaled (stretched or compressed) to examine the signal at a new frequency or scale. The correlation calculation is repeated for this new scale over the entire signal (Figure 2.7c). v. Repetition Across Scales: This technique is iteratively repeated across a wide range of scales. The frequency of the wavelet is changed by scaling it: stretching it to investigate lower frequency components of the signal and compressing it to focus on higher frequencies. Similar to the DWT, the selection of the mother wavelet is crucial in the interpretation of CWT results for AP analysis. Some mother wavelets are especially effective at recognizing the quick spikes that indicate APs, while others excel at tracing slower, rhythmic oscillations. This necessitates the use of a mother wavelet that closely matches the form and frequency content of the raw signal's APs. 31 (a) (b) (c) Figure 2.7: Main steps of the CWT. At the same time, the wavelet's scale is important. It must be properly calibrated to correlate to the number of samples in the neurological signal that contain the frequency range of interest. The CWT is thus tailored to focus on the frequency components most relevant to APs. Salmanpour et al. (2010) proposed an innovative solution in which they designed a wavelet that matched the mean of APs that were extracted from a real raw MSNA signal using DWT in their prior research. By employing the CWT with this custom-designed wavelet, they were able to accurately indicate the locations of APs within the signal. This method improved the precision in identifying APs by leveraging the continuity and adaptability of CWT analysis. Figure 2.8 provides a schematic overview of Salmanpour et al.'s procedure, illustrating the steps involved in this refined approach to AP detection. 32 Figure 2.8: Overview of Salmanpour et al.'s procedure for AP detection using CWT with a custom-designed wavelet. The concept of designing a wavelet matched to a specific signal for detection purposes, an extension of the matched filter theory, has been explored and implemented in various studies (Szu et al., 1992, Laterza & Olmo, 1997, Olmo et al., 2000, Chapa & Rao, 2000,Guido et al., 2006, Shark & Yu, 2006). The detection procedure is made more efficient and reliable by customizing a wavelet to match the exact properties of the target signal. This strategy has been used in a variety of ways where particular signal recognition is central in helping to advance the field of signal analysis in complicated contexts such as neural signal processing. Although Salmanpour et al. increased the accuracy of extracting APs by using a custom mother wavelet instead of Symlet 7, a potential drawback of their method is that, like Diedrich’s algorithm, they used a subjective threshold in their work. Building upon the work of Salmanpour et al., the research team Shafer et al. (2022) sought to refine the Salmanpour team’s algorithm through an iterative approach. Shafer et al. adapted the methodology of Salmanpour et al., utilizing MATLAB's built-in function to apply the CWT with a Symlet 7 wavelet. Notably, the Symlet 7 wavelet in MATLAB's function spans 14 samples, which led Shafer et al. to initially extract only the large amplitude APs. Subsequently, they averaged these limited obtained APs to construct a mother wavelet template. This facilitated the repetition of the CWT across two or three iterations, aimed at capturing a more extensive and Threshold CWT Raw Recording Average of AP‘s from Diedrich’s Method Coefficients Thresholding 𝐾𝜎√2 ln(𝑁) AP’s Time Stamps 33 representative collection of APs for template generation. This iterative refinement process sought to overcome the initial template's limitations in representativeness, thereby enhancing the accuracy and reliability of AP detection. A block diagram of their algorithm is shown in Figure 2.9. Figure 2.9: Shafer et al's iterative refinement of the AP detection algorithm using CWT-based template creation. Although they made efforts to refine the algorithm by iterating the process of template creation using APs extracted from the CWT, a fundamental limitation persists. Since the initial template is derived from a limited number of APs, there is no guarantee that the iterative algorithm will converge towards the most accurate and representative template. This limitation arises from the initial selection bias; if the starting set of APs does not comprehensively capture the diversity and variability inherent in APs, subsequent iterations may not necessarily rectify this incompleteness and the algorithm extract only the APs that are similar to the incomplete template. Furthermore, like Salmanpour et al. and Diedrich’s algorithms, Shafer’s method uses the modified threshold for AP detection. Despite efforts to refine the template and improve the extraction process, the issue of relying on an empirically adjusted threshold remains. This approach to threshold setting, while adaptable, introduces a degree of subjectivity and potential variability in the detection sensitivity, making it difficult to ensure consistent performance across different datasets or signal conditions. CWT Raw Recording MATLAB’s Symlet 7 Coefficients Thresholdin𝐾𝜎√2 ln(𝑁) Threshold Averaging Refined MW AP’s Time Stamps 34 2.4 Challenges in AP Extraction from Rat Neural Recordings The investigation of APs in rat models has been instrumental in furthering our understanding of nervous system physiology. However, rat recordings pose additional challenges compared to human recordings due to the smaller nature of the subject. Dealing with increased noise in AP measurements is a major challenge. Noise can conceal essential data and cause misinterpretations, making accurate identification and analysis of APs difficult. Cruttenden et al. (2020) developed an Adaptive Virtual Referencing (AVR) algorithm to mitigate noise in high-noise environments, emphasizing the need for sophisticated noise compensation in AP data analysis (Cruttenden et al., 2020). Urekar and Sovilj (2017) highlighted the significance of accurate noise floor assessment, a principle crucial for precise AP measurements (Urekar & Sovilj, 2017). Additionally, Kim and Kim (2003) introduced a wavelet-based detector to improve AP detection in low signal-to-noise ratio settings, underlining the importance of innovative detection techniques (Kyung Hwan Kim & Sung June Kim, 2003). These findings support the importance of our refined algorithm and reliable determination of the noise floor (threshold) in addressing noise-related challenges in AP measurements. The shape and length of APs are important research topics because they have a substantial impact on neuronal communication and brain function. While AP length and frequency fluctuation is a fundamental aspect of neuronal activity, the shape of the AP is also important. For instance, modulation of AP shape, as demonstrated by external stimuli like ultrasound, impacts potassium currents, affecting neuronal signaling and offering potential treatment avenues for ion channel-related diseases (Lin et al., 2019). Moreover, the diversity in AP shapes and firing patterns, orchestrated by various voltage-dependent channels, significantly affects synaptic transmission 35 and neural computation (Panzera & Hoppa, 2019). As another example, the intricate relationship between AP shape and neural signaling was investigated by the dynamic encoding of stimulus history into the AP shape which emphasizing the importance of understanding AP dynamics in neural communication (de Polavieja et al., 2005). Therefore, improving methods to measure and understand the intricacies of AP shape and length is pivotal for advancing our knowledge of neural signaling and its implications in brain function. Moreover, understanding the shape and duration of APs is essential for our research in neural signal analysis and pattern extraction. This knowledge is key not only to selecting a suitable mother wavelet that closely aligns with the APs’ shape but also to scaling the mother wavelet and determining the appropriate frequency range for the signal of interest. By ensuring a meticulous match, we can extract patterns that are accurate and meaningful. Key studies in other areas of pattern detections emphasize this decision's relevance. For example, Sloukia, Bybi, and Drissi (2017) proved the influence of mother wavelet selection on forecasting bearings' remaining useful life (RUL), while Chaari et al (1996) emphasized how important it was for evaluating earth faults in power networks (Sloukia et al., 2017,Chaari et al., 1996). Furthermore, the complexity of selecting the optimal mother wavelet, especially when determining cracks in rotating shafts, where the nature and type of information signals are paramount was studied by Codruta, (2020). In the following chapter, we aim to present solutions and approaches specifically designed to overcome the challenges outlined previously. Our exploration will focus on innovative strategies and the refinement of existing methodologies to enhance both the accuracy and efficiency of neural signal analysis and pattern extraction. This will involve tackling the complexities of defining noise 36 levels, selecting the most appropriate mother wavelet by analyzing the shape and duration of APs, and addressing the limitations of the signal processing techniques discussed so far. 37 Chapter 3: Recording and Data Analysis of SNA Signals This chapter focuses on the experimental methods used for accurately recording SNA signals, as well as a thorough examination of both the preprocessing and postprocessing stages of data analysis. The application of advanced analytical techniques, notably the MODWT and CWT, will be central to our discussion. These methods are fundamental in our approach to effectively separate individual APs from the background noise. The goal of this chapter is to establish the framework for the development of an automated AP extraction algorithm. We hope to improve the accuracy and efficiency of SNA analysis by incorporating these complex techniques, thereby offering valuable insights into the field of neural signal processing. 3.1 Data Collection The careful preparation of the rat serves as the foundation of our experimental approach. It needs to be noted that the experiments were prepared and performed exclusively by members of Dr. West’s team at the faculty of Medicine at UBCO. The work in this thesis is based on data recordings provided by Dr. West’s team to the candidate. The sole reason for the data recording procedure in this thesis is to provide the necessary background for understanding the limitations of the data. In this study, we have recorded data from the left greater splanchnic nerve, which is an integral part of the SNS. As illustrated in Figure 3.1, the greater splanchnic nerve emerges from the thoracic sympathetic trunk, with fibers that typically originate between the fifth and the ninth or tenth thoracic ganglia. It descends across the diaphragm to provide sympathetic innervation to various 38 abdominal organs. Figure 3.1 clearly delineates the left greater splanchnic nerve’s origin, its descending path, and its terminal connections. Figure 3.1: Anatomical Diagram of the Sympathetic Nervous System Highlighting the Left Greater Splanchnic Nerve (Haroun, 2018) The preparation of the rodent starts with a surgical incision on the left femoral skin, which is required to isolate the femoral artery for catheterization. We may properly assess the Mean Arterial Pressure (MAP) and heart rate by placing a pressure catheter into this artery, providing important baseline cardiovascular data. Following that, a right femoral incision allows for catheterization of both the artery and vein. This procedure is necessary for the placement of two fluid lines, which are necessary for the administration of agents or blood collection as shown in Figure 3.2. Continuing with the preparation, the rat is intubated and ventilated to ensure consistent respiratory function throughout the experiment. The carotid artery is subsequently catheterized with a left ventricular (LV) catheter to obtain full cardiac data such as volume, pressure, phase, and magnitude. These characteristics are utilized to generate accurate pressure-volume (PV) loops of 39 the left ventricle. Our method, as described in Squair et al., (2018) and Poormasjedi-Meibod et al., (2019) emphasizes precision and completeness. The use of high-frequency ultrasound imaging equipment (Vevo 3100, VisualSonics, Canada) ensures that the catheter placement within the left ventricle is properly validated, improving the precision and reliability of our data collection. (a) (b) Figure 3.2: Comprehensive surgical setup. (a) Diagrammatic representation. (b) Actual laboratory procedure The animal is then carefully positioned on its back and secured in a stereotaxic frame to ensure stability throughout the intricate surgical process. The left greater splanchnic nerve is exposed and isolated via a left flank incision, as depicted in Figure 3.3. Anatomical landmarks such as the adrenal gland (AG), inferior vena cava (IVC), and aorta (Ao) are important for the precise location of the nerve. The celiac ganglion (CG) serves as a pivotal guide for the careful dissection of the nerve branch. The central objective of this surgical intervention is the acquisition of SNA data from the isolated nerve, as illustrated in the graph component of the figure. The graph presents the distinct electrical activity patterns of the left greater splanchnic nerve, providing valuable insights into SNS behavior during a variety of physiological states or experimental scenarios. 40 The decision to record sympathetic activity from rats' left greater splanchnic nerves was strategic and informed. The left greater splanchnic nerve, known for its accessibility and substantial innervation, is densely supplied with axons from the thoracic spinal intermediolateral nucleus. As indicated in investigations by Sapru et al., (1982) and Barioni et al., (2022), this anatomical detail makes it a suitable place for monitoring SNA. These nerves are important because they regulate blood supply to the splanchnic area, affecting key organs such as the liver, spleen, and intestines. This region containing about one-fourth of total blood volume for humans at rest, as evidenced by studies by Rowell et al., (1972) and Harper & Chandler, (2016). Thus, the choice of nerve is important in the study's aim of giving insights into cardiovascular control and neural dynamics. Figure 3.3: Surgical exposure of the rat's splanchnic sympathetic nerve (SSN) with real-time recording of SNA To record SNA, a suction electrode is used, which provides a soft yet tight hold on the nerve to enable accurate and constant signal recording. The active recording electrode (G1) is carefully 41 positioned along the splanchnic nerve and cover, as shown in Figure 3.4, while the reference electrode (G2) is put up to establish a differential recording. As a conductive link, a saline bridge is used to ensure the nerve's electrical activity is faithfully transmitted. The proper placement of these components plays an important role in obtaining unique AP waveforms, which are needed for a complete analysis of sympathetic nerve dynamics. After recording the SNA, the signals are amplified (gain 10 K, AM-1800; A-M Systems, Sequim, WA, USA) and filtered (100-1000 Hz) to achieve the best clarity and fidelity. The next stage is to gather and display this data using a PowerLab data acquisition device (sampling frequency: 10 KHz, ADInstruments, USA). Figure 3.4: Electrophysiological setup for nerve recording with active and reference electrodes 3.2 Experimental Plan In this section, we will look at the systematic method of SNA recording and how we might use variances in neural signals to improve our data processing. Our goal is not to research these differences for the sake of studying them, but rather to use them to gain deeper insights into our experimental data. To create a consistent and effective framework for our investigation, nerve recordings are divided into two distinct phases, which are outlined below. 42 • Phase I: Normal State Recording: Our main objective is to record SNA in the animal's normal physiological state. This phase involves carefully recording the nerve activity while the animal remains unaltered. This data contains the AP of interests and noise. It will later be contrasted with the modified condition generated by pharmaceutical intervention. • Phase II: Ganglionic Blockade and Threshold Analysis: The experiment proceeds to the second step once we have generated a detailed profile of SNA in its normal state. The injection of hexamethonium bromide causes a ganglionic blockade, thereby inhibiting the SNA. Hexamethonium bromide is a ganglionic blocker that works by antagonizing nicotinic acetylcholine receptors (nAChRs) at the autonomic ganglia. This action prevents acetylcholine, a neurotransmitter, from binding to its receptors on neurons within the ganglia. As a result, hexamethonium effectively inhibits the transmission of impulses through the sympathetic ganglia. The absence or reduction of APs during this phase is essential for our investigation since it will guide the threshold determination procedure both for burst analysis and extraction of APs. The blockade is expected to clearly define the SNA noise floor, allowing for exact threshold identification. We conducted these experiments on a total of nine animals and recorded data from them. In the following sections, we will elaborate on the analytical approach that uses both normal state and post-blockade recordings to extract APs from background noise. We will review how frequency analysis and WT are used. These advanced signal processing and noise reduction techniques, including the application of threshold-setting based on different SNA levels, will be 43 important in distinguishing real neural activity. This comprehensive approach looks to improve the accuracy of identified APs and provide a better understanding of the dynamics of sympathetic nerve transmission. 3.3 Algorithm Development 3.3.1 Burst Analysis In the previously provided burst analysis approach (Chapter 2.3.1), we used raw nerve recording rectification to produce the rectified signal 𝑉(𝑡). This signal is used to calculate the iSNA. To review and expand on our prior topic, consider Equation. (2.1), which defines iSNA: 𝐼(𝑡) = ∫ 𝑉(τ)𝑡+Δ𝑡𝑡 𝑑τ (2.1) An important component in this process is the selection of the time window Δ(𝑡) over which the integration occurs. We chose a time constant of Δ(𝑡) = 0.1 seconds for our investigation, aligning with values previously utilized in the literature (Brewer, 2013 and Salmanpour & Shoemaker, 2012). This time constant is important in determining the burst analysis's temporal resolution and sensitivity. It can also be thought as the temporal lens through which bursts of activity are measured. This integration window is chosen to align with the intrinsic duration of bursts, ensuring that the entirety of the nerve discharge is captured without interference from neighboring neural events or background noise. The selection of Δ(𝑡) thus directly effects the detection and characterization of bursts—too brief a window may shorten the burst, omitting crucial activity, whereas an overly long Δ(𝑡) may inadvertently combine separate neural firings into a single burst profile. Therefore, the strategic calibration of Δ(𝑡) is important for the accuracy of burst analysis and its optimization 44 is an important discussion point in the approach since it supports the accuracy and reliability of SNA evaluation. Once the signal is rectified and integrated, a threshold needs to be set for selecting regions that represent bursts. To guide us in setting this threshold, we use the Phase II recordings, where APs are suppressed by the administration of hexamethonium bromide. The threshold is determined by the following formula: Threshold = 𝜇Phase II + 2 × 𝜎Phase II (3.1) where μPhase II and σPhase II are the mean and standard deviation of the integrated signal 𝐼(𝑡) from Phase II, respectively. This threshold is then applied to the Phase I recordings to detect considerable SNA. In particular, the peak of any signal segment surpassing this threshold in Phase I data is designated as the peak time of a burst. This peak time is important for the subsequent phases of burst identification because it provides a time frame for defining the start and finish of a sympathetic nerve burst. To define bursts within the iSNA signal, the following criteria are applied post-thresholding: i. Identify the peak of each segment of the signal that exceeds the threshold level. ii. The start of a burst is marked by the local minimum preceding this peak. iii. The end of a burst is determined by the point where the signal descends and crosses below the threshold level again. The rationale for these three burst detection procedures stems from the desire to correctly identify the intervals where APs are more concentrated. The SNA signal integration serves as a translation from raw activity to a cumulative representation, with an increasing trend indicating an AP firing 45 phase. The first rise of the integrated signal indicates the start of AP activation. As a result, the beginning of each AP firing interval is recognized as the point at which the signal begins to climb. In contrast, the point at which the integrated signal declines and crosses the set threshold after reaching a peak denotes the end of a firing period. This methodology guarantees that the intervals of AP activity are accurately captured, representing real physiological events observed during the experiment. For a practical application of the criteria to define bursts within the iSNA signal, Figure 3.5 serves as an exemplar. In this figure, the iSNA is represented by the solid blue line, and the threshold is denoted by the horizontal dashed blue line. The criteria for identifying bursts are illustrated by segments where the blue iSNA line crosses above the threshold. The onset of each burst is marked by a green dashed vertical line indicating the local minimum before the peak, and the cessation is marked by a red dashed vertical line indicating where the signal returns below the threshold level. Figure 3.5: iSNA with burst detection based on threshold crossing 46 3.3.2 Analytical Techniques for Extraction of APs from Noisy SNA In this section, we are going to study how advanced techniques can be used to extract and analyze APs from raw SNA data. We are proposing a two-step AP extraction technique. In step 1 we use MODWT for extracting an initial set of APs. The second step allows for a refinement of the extracted APs. Here, we average the identified APs from step 1 to generate a template, referred as the mother wavelet. We integrate this unique mother wavelet into an iterative CWT approach similar to Shafer et al. (2022). However, in contrast to Shafer, we have a more robust approach for generating the initial mother wavelet. This method emphasizes the adaptability and efficacy of WT in the study of biological data, paving the way for more sophisticated studies in neural signal processing. 3.3.2.1 AP extraction using Maximal Overlap Discrete Wavelet Transform (MODWT) Similar to DWT, the MODWT is a multi-resolution analysis method that decomposes a signal into two sets of coefficients: approximation and detail. However, there is an important difference between DWT and MODWT, which we will discuss in this section. Unlike the DWT, which uses decimation at each level of decomposition, the MODWT keeps the original signal length by not down sampling the wavelet coefficients. The absence of decimation allows for a more detailed representation of the signal, which is especially useful for examining events that occur during both short and long time periods. The technical process of MODWT can be described by equations like DWT but without the down-sampling step. For each level 𝑗, the MODWT wavelet (𝑊𝑗) and scaling (𝑉𝑗) coefficients can be computed as: 47 Wj[n] =∑hkk⋅ Vj−1[n − k] (3.2) Vj[n] =∑gkk⋅ Vj−1[n − k] (3.3) Here, ℎ𝑘 and 𝑔𝑘 are the coefficients of the high-pass and low-pass filters, respectively, and the coefficients 𝑉𝑗−1[𝑛] refer to the scaling coefficients from the previous level or the original signal for the first level. The key difference from DWT is that the MODWT coefficients 𝑊𝑗[𝑛] and 𝑉𝑗[𝑛] are calculated over the entire length of the signal for each level 𝑗, without reducing the resolution by half. MODWT produces a more detailed analysis by using overlapping wavelets at each level of decomposition, keeping the temporal resolution lost in DWT caused by down-sampling. This makes MODWT especially useful for signals with non-stationary behavior at several scales, such as neural APs, where high resolution in both the time and frequency domains is essential for capturing the complete dynamics of the signal. In summary, MODWT differentiates itself through its capacity to breakdown signals with greater temporal accuracy. MODWT acts as a band-pass filter at each level of decomposition. Therefore, one of the main factors in utilizing MODWT is finding the appropriate level of decomposition. Inspired by Ren et al., 2006, who determined the desired level of decomposition based on the frequency characteristics of the target signal for denoising recorded myoelectric signals, we identify the optimal levels of decomposition by pinpointing the dominant frequency components of the APs. These components are crucial for gaining insights into the nature of neural signaling and processing.Ren et al. (2006) 48 In order to do this, we applied the Fast Fourier Transform (FFT) to the average of an array of manually extracted APs from raw neural data. We then select the decomposition levels of the MODWT whose mother wavelet best matches the frequency spectrum of the manually extracted AP’s. In our work we manually extracted an array consisting of 20 APs from raw neural data, focusing on segments of 45 samples each, equivalent to 4.5 milliseconds, which were chosen to represent the duration of a single AP. This duration was chosen to capture the whole waveform of the APs while also providing a consistent foundation for comparison throughout the dataset. After extracting the APs, we computed their average to obtain a representative waveform. This averaged waveform was then subjected to a magnitude spectrum analysis. As illustrated in Figure 3.6a and 3.6b, the first displays the averaged APs, while the latter presents the corresponding magnitude spectra. Analysis in the frequency domain of this averaged spike revealed a dominant frequency around 444.44 Hz, underscoring a significant and recurrent characteristic in the neural signals we studied. (a) (b) Figure 3.6: (a) Averaged AP waveform extracted from neural data, and (b) the corresponding magnitude spectrum, revealing the dominant frequency at 444.44 Hz. 49 Figure 3.6a displays the averaged waveform of the APs, highlighting both slower and faster fluctuating regions. Our primary interest lies in the faster fluctuation, particularly observed in the central portion, as it represents the most significant and distinctive fluctuation in comparison to the slower parts of the waveform. This middle section captures the core properties of the waveform. To gain a better understanding of how the dominant frequency shows itself in this waveform, we have undertaken a specific analysis illustrated in Figure 3.7. In this figure, a sine wave with the previously identified dominant frequency of 444.44 Hz is fitted to the middle section of the averaged waveform. This fitting visually demonstrates its influence in shaping the primary features of the neural signal. Figure 3.7: Overlay of a 444.44 Hz sine wave on the central portion of the averaged AP waveform, demonstrating the alignment with the dominant frequency. The correct level of decomposition requires that the dominant frequency of 444.4Hz is located in the region of the high pass filter. 50 For a signal sampled at 10 kHz, the approximate frequency bands for the approximation and detail coefficients across the first three levels of the MODWT decomposition are as follows: 1. Level 1 : • Low-pass (approximation) frequency band: 0 to 2,500 Hz • High-pass (detail) frequency band: 2,500 to 5,000 Hz 2. Level 2: • Low-pass (approximation) frequency band: 0 to 1,250 Hz • High-pass (detail) frequency band: 1,250 to 2,500 Hz 3. Level 3: • Low-pass (approximation) frequency band: 0 to 625 Hz • High-pass (detail) frequency band: 625 to 1,250 Hz 4. Level 4: • Low-pass (approximation) frequency band: 0 to 312.5 Hz • High-pass (detail) frequency band: 312.5 to 625 Hz By selecting the detail coefficients from the fourth level of decomposition, we are able to effectively filter out high and low frequency noises while retaining the frequencies pertinent to the APs we aim to analyze. This level of decomposition aligns with the dominant frequency of the APs indicated by our preliminary analysis, thereby ensuring the integrity of the extracted signals for subsequent analysis. After selecting the proper level of decomposition using MODWT, the next step is to select a suitable mother wavelet that closely replicates the signal of interest — in this example, the APs. Previous studies selected Symlet 7 for analyzing MSNA, due to its shape similarities with APs in human recordings. We chose Symlet 6 for its shape, which more closely resembles the APs 51 detected in our recordings. With their near-symmetric shape and tight support, Symlet wavelets provide an appropriate balance between time and frequency localization. This likeness is not accidental, but rather an intentional decision to improve the accuracy of our AP extraction. Figure 3.8a presents the profile of the Symlet 6 wavelet, illustrating its characteristic shape and behavior. Figure 3.8b displays a comparison between the average of manually extracted APs and the Symlet 6 wavelet, at fourth decomposition level. The overlay of these two curves demonstrates how the Symlet 6 wavelet closely mirrors the fundamental attributes of an AP, highlighting the wavelet’s effectiveness in capturing the essential features of neural signals. Therefore, by picking a mother wavelet that closely approximates the fundamental shape of our signal of interest, we optimize the WT's application for more effective AP detection and analysis. (a) (b) Figure 3.8: (a) Profile of the Symlet 6 wavelet, illustrating its wave-like structure (Shan et al., 2009) and (b) Comparative overlay of the averaged manually extracted APs with the Symlet 6 wavelet. In the sections that follow, we will analyze and explain each phase of APs extraction step by step. This step-by-step investigation will not only clarify the process but will also illuminate the 52 reasoning behind each decision, ensuring an in-depth understanding of how we extract and assess APs from noisy SNA recordings. Having selected a suitable mother wavelet and its decomposition level, we need to determine a suitable threshold. Similar to the burst analysis, we will be using Phase II recordings to select this threshold. Threshold = 𝜇Phase II + 4 × 𝜎Phase II (3.4) Here, 𝜇Phase II represents the mean, and 𝜎Phase II denotes the standard deviation of the detail coefficients at the fourth level of the MODWT decomposition from the Phase II recordings. By setting the threshold at 4 times the standard deviation from the mean, we ensure coverage of most of the data in Phase II, well over 99 percent, as shown in Figure 3.9. This specific threshold is derived from the rationale that in the absence of significant reduction of APs, the amplitude variations in the detail coefficients at the fourth level of the MODWT decomposition are primarily due to noise. Thus, by setting the threshold at this level, we effectively differentiate between the noise and any remaining semblance of neural activity in the recordings. This methodical approach ensures that our subsequent analysis is focused on the most relevant signal components, free from the distortive influence of noise. 53 Figure 3.9: The distribution of wavelet coefficients of noise with the selected threshold indicated by the red dashed lines. We now apply the approach used for Phase II recordings to our Phase I data. We utilize MODWT on Phase I data, just as we did in Phase II. The threshold determined in the preceding section for Phase II is now used to identify APs in the Phase I data. As previously stated, we determined that the duration of an individual AP includes 45 samples. As a result, in our study of the detail coefficients at the fourth level of MODWT decomposition, any local maximum that exceeds the set threshold is treated as the peak value of an AP. To record the whole AP waveform, we extract 45 samples centered on this local maximum, including 22 before and 22 after the peak. This methodology is visually represented in Figure 3.10 as an example, which illustrates a half-second segment of the thresholded approximation coefficients. The figure clearly demonstrates how each identified local maximum, surpassing the threshold, corresponds to the peak of an AP. More detailed results and additional examples will be provided in Chapter 4. 54 Figure 3.10: Half-second segment of thresholded approximation coefficients highlighting peaks corresponding to APs. To summarize the AP extraction process using the MODWT algorithm, there are four main steps: 1. Compute the detail coefficients at fourth level of MODWT for Phase II: Compute the detail coefficients from the fourth level of decomposition by applying MODWT, allowing for analysis of the post-ganglionic blockade data. 2. Threshold Determination: Utilize the formula provided in Equation (3.4) to establish a threshold that encompasses most of the data. 3. Phase I Analysis and Thresholding: Apply fourth-level decomposition to Phase I recordings and perform thresholding on the detail coefficients to isolate potential APs. 4. AP Identification and Extraction: Identify the local maxima in the thresholded detail coefficients from the fourth level of decomposition using MODWT, and appropriately collect the surrounding data. 55 This systematic approach allows us to accurately isolate and extract APs from the Phase I recordings. By applying a consistent threshold across both phases and considering the specific length of APs, we ensure a coherent and methodologically sound analysis, crucial for comparing and contrasting neural activities under different physiological states. Furthermore, if Phase II coefficients are of similar size to those of Phase I (Raw recording), it indicates an inability to distinguish noise from the main signal. This would necessitate enhancements in the measurement setup to better assess the signal-to-noise ratio. Figure 3.11 provides a schematic overview of our provided method, illustrating both steps in analysis of Phase I and Phase II recordings to extract APs utilizing MODWT. Figure 3.11: Overview of the presented procedure for AP detection utilizing MODWT. 3.3.2.2 Refining AP extraction using an iterative CWT approach. Following Shafer’s algorithm, we also utilize an iterative CWT approach in our second phase of the algorithm. The main difference between the two approaches is the mother wavelet used during the first initial CWT extraction. In this first initial iteration, Shafer’s algorithm uses MATLAB's built-in Symlet 7. This wavelet spans 14 samples and acts as a high-pass filter, allowing only the extraction of larger APs. In contrast, we generate our first template from an average of the extracted APs in first phase of the algorithm. Thus, our initial mother wavelet in Phase two better represents APs’ Time Stamps Phase I Recording MODWT Sym 6 Wavelet Coefficients Thresholdin𝜇Phase II + 4 × 𝜎Phase II Threshold MODWT Phase II Recording Wavelet Coefficients 56 the frequency content of actual APs, as shown in Figure 3.12, which presents two key visual representations. Figure 3.12a shows the template of the averaged APs, and Figure 3.12b displays its magnitude spectrum. Importantly, the magnitude spectrum in Figure 3.12b aligns with our expectations, showcasing its maximum peak at 444.44 Hz. This peak frequency confirms our earlier predictions regarding the dominant frequency components of the APs. (a) (b) Figure 3.12: (a) Averaged AP template, and (b) Corresponding magnitude spectrum with a peak at 444.44 Hz. To construct mother wavelet from averaged previously identified AP’s we must follow an essential requirement known as the wavelet admissibility condition (Misiti et al., 2013). This mathematical requirement is crucial for ensuring that our customized function qualifies as a wavelet and can provide an accurate evaluation when applied to CWT. The admissibility condition is described as follows: ψ real, ψ ∈ 𝐿1 ∩ 𝐿2, ∫|𝑡ψ(𝑡)|𝑅 𝑑𝑡 < +∞ and ∫ψ(𝑡)𝑅 𝑑𝑡 = 0 (3.5) To satisfy this admissibility condition, a function ψ(𝑡) (The constructed mother wavelet) must not only be real and belong to both 𝐿1 and 𝐿2 spaces, indicating that the wavelet has finite energy, but also must satisfy two additional integral conditions. Firstly, the integral of the product of 𝑡 and 57 ψ(𝑡), which is the weighted function by time, must be finite. Secondly, the function must have a zero average over the entire real line, as indicated by the integral of 𝜓(𝑡) over ℝ equating to zero. In practical applications and numerical calculations, we often employ a sampled version of an unknown function. Assume we want to estimate a function, 𝑓, with compact support within the interval [𝑎, 𝑏] and finite energy. When designing a wavelet for continuous analysis, we want to approximate this function as closely as possible. To achieve this, we apply the method of least squares, which optimizes the fit of our wavelet 𝜓(𝑡) to the sampled data points of 𝑓. The least squares technique minimizes the sum of the squares of the differences between the actual values, represented by 𝑓 samples, and the values predicted by our model, which is the wavelet ψ(𝑡). This optimization problem may be expressed mathematically as follows: minψ∑[ψ(𝑡𝑘) − 𝑦𝑘]2𝐾𝑘=1 (3.6) where 𝑡𝑘 are the sample points, and 𝑦𝑘 are the corresponding sampled values of the function 𝑓. However, for 𝜓(𝑡) to be a valid wavelet, it must also satisfy the admissibility condition. This condition requires that the wavelet has a zero mean over its entire domain, which can be formally stated as: ∫ψ(𝑡)𝑅 𝑑𝑡 = 0 (3.7) In essence, by meeting both the least squares criterion (Equation (3.6)) and the wavelet admissibility condition (3.7), we ensure that our constructed wavelet 𝜓(𝑡) is not only the best representation of the function 𝑓 within the constraints of our sampling, but also a mathematically sound tool for continuous wavelet analysis (Misiti et al., 2013). We have used the built-in 58 MATLAB function ‘pat2cwav’, specifying the least-squares fitting method as ‘orthconst’ and the boundary constraints as ‘continuous’ to create the constructed wavelet that meets criteria in Equations (3.6) and (3.7). Applying this approach to the averaged MODWT AP template, shown in Figure 3.12a, produces the results shown in Figure 3.13. As can be observed, despite some discrepancies between the constructed wavelet and the template, the wavelet captures the majority of the template's features and bears a strong resemblance to it. This demonstrates the efficacy of the wavelet construction method in approximating the essential characteristics of the signal of interest. Figure 3.13: Comparison of the constructed wavelet with the averaged AP template. Having successfully constructed a wavelet that approximates the averaged AP template, we are now able to employ this wavelet to extract APs from neural recordings. Our procedure is organized as follows: 59 1. Compute CWT Coefficients for Phase II: Initially, we apply CWT using our constructed wavelet to the Phase II recordings, allowing for a nuanced analysis of the data post-ganglionic blockade. 2. Threshold Determination Using Equation (3.4): We continue to use the threshold formula established in Equation (3.4), which has proven effective in distinguishing relevant signal components from noise. 3. Apply CWT to Phase I Recording and Perform Thresholding: The custom template wavelet is then applied to the Phase I recordings. The thresholding process is conducted on the resulting CWT coefficients to isolate potential APs. 4. Identify and Extract APs: We identify local maxima in thresholded CWT coefficients. Each local maximum that exceeds the threshold is considered indicate of an AP, and the surrounding data is collected appropriately. This tailored technique is expected to produce more distinct APs than Symlet 6. The use of a wavelet produced directly from the physiological data promises to improve the sensitivity and specificity of our analysis, perhaps exposing details in neuronal activity that preset wavelets may miss. Similarly to the approach utilized in generating the initial template, we will conduct an averaging process on all the APs extracted using the constructed wavelet to create an updated template. The comparison between the original and the newly constructed template is illustrated in Figure 3.14. 60 Figure 3.14: Comparative illustration of the original and updated AP templates. As can be discerned from Figure 3.14, there are minimal differences between the first and second templates. The second template is deemed more reliable since its construction incorporates a larger number of extracted APs, thereby providing a more robust representation. The third iteration of template refinement is displayed in Figure 3.15. The difference between the second and third templates is minimal, suggesting that the template is approaching convergence. Additionally, the number of APs detected using the second template is 3,586, whereas the third template identifies 3,617 APs. Given the negligible difference between the second and third iterations, it is concluded that further iterations are unnecessary. Thus, the second iteration's template can be adopted as the mother wavelet. 61 Figure 3.15: Comparison of the second and third template refinements indicating near convergence. The complete approach to extract the APs via the CWT and the refining algorithm is shown in Figure 3.16. Figure 3.16: Overview of applying CWT for AP detection utilizing the constructed mother wavelet and its refinement. In conclusion, this chapter has laid the foundational groundwork for burst analysis and our novel approach in extracting APs from neural recordings. The next chapter of this thesis shows more detailed results that provide empirical evidence supporting our approach. It will also highlight its performance in comparison to conventional methods. Through this comprehensive analysis, we CWT MW From Phase I AP Coefficients Thresholdin𝜇Phase II + 4 × 𝜎Phase II Threshold AP’s Time Stamp Averaging Refined MW CWT Phase II Recording Noise Coefficients Phase I Recording 62 aim to solidify the credibility of our algorithm and underscore its potential contribution to the field of neural signal processing. 63 Chapter 4: Results and Discussion The previous parts of this thesis set the framework for a thorough investigation of neural activity in rats, largely via the lens of burst analysis and the adaptation of AP detection algorithms. In addition to the burst analysis, the primary goal of our research was to dive into the detailed patterns of neuronal firing, especially by measuring the duration of APs and determining a reliable method for separating them from background noise. This chapter presents the findings from the examination of neural recordings collected from nine rats. Each dataset has been processed to extract APs, with a special emphasis on defining the length of APs and generating a clear noise floor for AP extraction. The findings not only demonstrate the usefulness of customized algorithms in extracting neural signals from noise, but also provide light on the temporal features of neural activity in these animals. Building on the methodological foundation described in Chapter 3.2, our experimental technique consisted of recording neural activity from anesthetized rats, which was divided into separate stages, namely Phase I and Phase II, to capture the whole range of neuronal dynamics. The significant attenuation of neural signals in Phase II compared to Phase I is used to find the thresholds that would allow to reliably distinguish bursts and APs from background noise. In the next section, we will carefully show and explain the detected bursts and extracted APs for each of the nine rats. 4.1 Neural Analysis Outcomes: Burst Detection In this part, we provide the results for each rat in an organized way, using the analytical procedures described in Sections 3.3.1 for burst analysis from noisy SNA. The analyses are based 64 on one-minute recordings for each rat. However, the robustness of our technique is not limited to this duration; it is adaptable and may be used for extended periods. The iSNA for each rat was calculated using Equation. (2.1), as explained in Section 3.3.1. This strategy serves as the industry standard baseline for measuring neural activity, providing a comparison for our more detailed AP identification in subsequent sections. The results of the burst analysis for Rats 1–9 may be seen in Figure 4.1, which gives a one-second snapshot of the iSNA and its corresponding raw SNA and demonstrates our analytical methodologies in capturing the subtle patterns of neurological activity by comparing them with their corresponding raw SNA data. These figures display two graphs, with the top graph representing iSNA and the bottom graph showing the raw SNA. In the top graph, the blue line illustrates the dynamic changes in iSNA, offering a visual representation of neural burst activity as it unfolds over time. Dashed green lines indicate the start of each neural burst, while dashed red lines denote the end points. The solid black lines connect these start and end points, delineating the duration of burst firings. A dashed horizontal blue line marks the threshold level for burst detection. The bottom graph provides a raw trace of SNA, which serves as the basis for the integrated analysis depicted above. Rat 1 Rat 2 65 Rat 3 Rat 4 Rat 5 Rat 6 Rat 7 Rat 8 66 Rat 9 Figure 4.1: One-Second Burst Analysis Snapshot for Rats 1-9 Consistent with the methods outlined in Section 3.3.1, the outset of each AP firing interval is distinguished by an initial increase in the integrated signal, indicating the start of neural activity. In contrast, the termination of a firing period is demarcated by red vertical lines, where the signal descends and intersects the established threshold, depicted by the dashed horizontal line. The threshold level is set based on criteria outlined by Equation. (3.1) and is instrumental in ensuring consistent burst detection. The figure's x-axis displays the time in sample points, reflecting the duration and frequency of the APs within the one-second snapshot, while the y-axis scale emphasizes the amplitude of the signal fluctuations. This visualization shows the synchronization between bursts and raw SNA signal. In other words, the burst intervals are associated with the concentration of APs in the raw data. 4.2 AP Extraction and Template Analysis In this section, we provide a thorough presentation of the analytical progression proposed AP extraction algorithm from the neural recordings of the first rat, following the protocol established in Section 3.3.2. This includes demonstrating the initial AP extraction via MODWT, the formation 67 of the first averaged template, the derivation of a customized mother wavelet, and the generation of a refined second template. For this initial subject, the sequence of figures and accompanying narrative will trace the journey from raw data to the polished second template. For the sake of brevity, the discussions on the remaining eight rats will only cover the end-stage comparison between the first and second templates. By taking this approach, we will highlight the templates' comparative outcomes, summarizing the essential findings while avoiding repetition in the previously described methodology explanation. 4.2.1.1 AP Detection Through MODWT Analysis As described in Section 3.3.2, we used MODWT to neural recordings collected during Phase II. Our emphasis is focused on the detail coefficients at the fourth level of the MODWT decomposition, which have been recognized as the most relevant for our AP’s. Utilizing the pronounced contrast in neural activity between Phase I and Phase II allows us to establish the thresholds for extracting APs from noisy SNA data—a process that applies Equation. (3.4) to the MODWT coefficients of Phase II. Following the proposed methodology in Figure 3.11, we use the same MODWT technique on the Phase I data, utilizing the threshold calculated from Phase II data by Equation. (3.4). In the fourth level of MODWT decomposition, we considered any local maximum of the detail MODWT coefficients that exceeded this threshold to be the peak of an AP. To capture the whole AP waveform, we extract a range of 45 samples centered on this peak—22 before it and 22 after it. 68 Figure 4.2 (a) demonstrates this method by presenting a half-second segment of the Phase I recordings' thresholded detail coefficients. Figure 4.2 (b) shows the extracted APs aligned and overlapped, offering a collective visual of the general shape of the extracted APs. (a) (b) Figure 4.2: (a) Half-second segment of thresholded MODWT approximation coefficients and (b) overlay of extracted APs from Phase I recordings. The x-axis in Figure 4.2(a) represents the number of samples, providing a temporal framework for the analysis. On the y-axis, we observe the amplitude of detail MODWT coefficients at the fourth level of MODWT decomposition. The continuous blue line illustrates the wavelet coefficients, capturing the essential features of the signal at this decomposition level. The APs are identified above the calculated threshold, depicted as a dashed black line, which serves to differentiate meaningful signal peaks from background noise. The red crosses pinpoint the temporal location of detected APs, highlighting significant neuronal activity within the raw data. By meticulously analyzing the peak times in these wavelet coefficients, we can extract the APs from the raw data. 69 4.2.1.2 Synthesis of the First Template and Mother Wavelet Construction To continue our study, we average all extracted APs, resulting in a composite template that depicts the standard AP configuration in our data. This averaging template aids in converting the many waveforms into a single, uniform form. Then, using this averaged template, a customized mother wavelet is constructed. The specific criteria and methodology employed in crafting this mother wavelet are detailed in Section 3.3.2. The new mother wavelet is designed to match the shape of our averaged AP template, improving the accuracy of future AP detections. Figure 4.3 depicts the results of this technique, displaying the averaged AP template alongside the custom-constructed mother wavelet. Figure 4.3: Comparison of the averaged AP template and the custom-constructed mother wavelet 70 The blue line represents the averaged AP template, showing the typical shape of APs as they are commonly observed in our neurophysiological measurements. The red line depicts the constructed wavelet, which has been tailored to match the main features of the AP template. The close alignment of the two curves suggests that the wavelet is well-suited for detecting the presence of APs within noisy data. 4.2.1.3 Enhanced Detection through CWT and AP Template Refinement Using the Constructed Mother Wavelet Let us now discuss the use of the newly generated mother wavelet in CWT analysis of neural recordings. As illustrated in Figure 3.16, we initially compute the CWT coefficients for the Phase II recordings, using the mother wavelet to transform the data into a coefficient vector that allows the application of Equation (3.4) to generate the threshold for detecting AP’s from CWT coefficients in Phase I recordings. After determining the threshold, we apply the CWT to the Phase I recordings, again utilizing the newly produced mother wavelet. By using the same threshold from the Phase II analysis to the Phase I CWT coefficients, we can isolate the APs more reliably than using traditional approaches relying solely on Phase I data. Figure 4.4 (a) provides a visualization of this process by displaying a half-second slice of the Phase I recordings' CWT coefficients. It shows how the mother wavelet interacts with neural signals to highlight possible APs that exceed the threshold level. Figure 4.4 (b) shows the collected APs aligned and centered using the CWT. This overlay depicts a collective profile of the APs, highlighting the consistency and variance of the waveforms derived from the recordings. 71 (a) (b) Figure 4.4: (a) Half-second segment of Phase I recordings' CWT coefficients using the custom mother wavelet and (b) aligned and overlapped extracted APs. Approximately 70 percent of the APs identified using the constructed mother wavelet corresponded with those detected in the initial MODWT analysis. This overlap indicates a consistency between the two methods, demonstrating the effectiveness of the mother wavelet in capturing the majority of AP events that were initially identified. By averaging the newly extracted APs using the custom-constructed mother wavelet, we have arrived at what we define as the second template. Figure 4.5 showcases a comparative visualization between the first and second templates. Although there are minimal differences discernible between the two templates, the second template is considered to be more reliable. The enhanced reliability of the second template is attributed to its derivation from a broader dataset of extracted APs. We have repeated this procedure one more time This comprehensive base of data offers a more robust representation, capturing the inherent variability of the APs with greater fidelity and thereby serving as a stronger foundation for subsequent analyses. 72 Figure 4.5: Comparative overlay of the first and second AP templates. The methods of analysis described in steps (a)–(c) were applied to neural recordings of the remaining eight rats in the cohort. This methodology ensured that each dataset received the same thorough examination, starting with MODWT-based AP extraction and moving to the creation of an averaged AP template, mother wavelet construction, and CWT computation for refined extraction. Figure 4.6 illustrates the result of these efforts, including a side-by-side comparison of the first and second templates for each rat. 73 Rat 2 Rat 3 Rat 4 Rat 5 Rat 6 Rat 7 74 Rat 8 Rat 9 Figure 4.6: Comparative analysis of first and second AP templates for Rats 2-9, highlighting improved extraction accuracy and AP morphology. Figure 4.6 shows that for certain subjects, such as Rats 2 and 5, the iterative process of template refinement and the subsequent generation of a second template likely improve the quality of our analysis since its shape more accurately reflects the anticipated shape of the APs. This enhancement aligns with our pre-existing knowledge and expectations of the AP morphology in rats. Furthermore, it is observed that the template for Rat 9 does not converge to an acceptable AP shape, even after refinement. Given that experimental errors are always a possibility, our algorithm could be utilized in future research to assess whether the data is reliable and suitable for analysis. 4.3 Temporal Correlation of APs with Burst Intervals in Neuronal Activity In this part, we address the fundamental question that supports our research: do all APs occur entirely inside burst periods, or do some occur outside of these bursts? This research is pivotal because detecting and accurately characterizing APs is essential for understanding the complicated dynamics of neuronal activity. This study, as described in previous chapters, inspired the use of the WT for the extraction of individual APs from background noise. 75 Figure 4.7 exemplifies this investigative process for Rat 1, showcasing a two-second snapshot of the data. The top panel shows the iSNA with the identified bursts, which are denoted by the start and end locations of increased activity. The lower panel of the image shows the raw signal, with red crosses representing APs recognized by the wavelet. Together, these panels allow the frequency of the detected APs with and outside of burst. This comparison allows us to examine the temporal relationship between burst-defined intervals and the incidence of APs, revealing the neural coding strategies used by the nervous system. It's important to note that this figure serves as an illustrative overview, giving a visual insight into how APs are distributed in relation to the detected bursts. Figure 4.7: iSNA with marked burst intervals (top) and the corresponding raw signal with APs detected by wavelet analysis (bottom) for Rat 1. Table 4.1 provides a quantitative assessment comparing our algorithm with Shafer et al.’s algorithm, detailing the percentages of APs extracted during bursts and the total number of extracted ones. 76 Table 0.1: Comparison of the percentage of APs occurring during burst intervals as detected by our algorithm versus that of Shafer et al. (2022), with the total number of detected APs for each rat indicated in parentheses. Percentage of APs During Bursts Rat ID Our algorithm Shafer et al.’s algorithm Rat 1 78% (3586) 74% (2849) Rat 2 69% (2783) 67% (2251) Rat 3 70% (2873) 75% (2635) Rat 4 73% (3140) 72% (2678) Rat 5 79% (3676) 73% (2752) Rat 6 75% (3319) 75% (2846) Rat 7 82% (3943) 84% (2954) Rat 8 78% (3594) 73% (2651) Rat 9 76% (3408) 79% (2876) The data shown in the table offers insight into the complicated interaction of neural activity in the investigated rats. The percentage of APs occurring inside bursts (results from our algorithm) ranges from 69% to 79%, with the average value being about 75.5%. This finding is in alignment with results obtained by Klassen & Shoemaker, (2021) for humans, where they reported that around 70% of APs are synchronized with bursts. This shows that APs generally synchronize with burst intervals. However, the occurrence of a significant number of APs outside of these intervals highlights the complex nature of neural signaling. These findings underscore the limits of relying only on burst analysis and emphasize the importance of combining sophisticated analytical tools like WT into our study to completely capture the breadth of neuronal communication. Comparing our results to Shafer’s we see that both show similar trends in the percentage of APs detected inside bursts. However, our algorithm consistently identifies more APs from within the noisy nerve recordings in rats. WT analysis gives us the refinement needed to discern specific neuronal events that burst analysis alone may miss. The incorporation of WT in our technique thus improves our inquiry by guaranteeing that we account for a wider spectrum of neural activity rather than just the regions that are easily seen through burst analysis. 77 Chapter 5: Conclusion In this thesis, we conducted a thorough investigation of AP detection and analysis within the intricate landscape of neural signaling, with a special emphasis on rat models., To our knowledge, there have been no previously published characterizations of a rat's AP, making our research the first of its kind. Previous studies have focused on human data and have some limitations. Here, significant progress has been made in adapting and benchmarking methodologies that improve the reliability and reduce the need for fine-tuning of our AP detection algorithm. We have characterized the complex dimensions of rat’s APs in terms of length and shape and our new algorithm defines the noise floor for AP extraction from actual measurements. Finally, we validated our algorithm’s, performance by correlating APs with burst intervals. These results open the door for future advances in neuroscience research and medicinal applications. As we draw this study to a close, it is important to reflect on the significant findings, accept the inherent problems and limits, and cast a visionary eye toward the future avenues that will emerge from this fundamental work. 5.1 Summary of Major Findings The noise floor setting employed in all of our AP extraction procedures differs from that used by previous researchers. Whereas they adapted the threshold defined by Donoho (1995), which introduces a level of subjectivity and potential for inconsistency. Analyzing rats’ SNA allows to develop a distinct approach by integrating a pharmacological intervention. Our recording procedure is methodically divided into two independent phases. In Phase I, termed 'Normal State Recording,' we record the subject's baseline physiological activity, representing the dataset that 78 contains the APs of interest. In Phase II, known as 'Ganglionic Blockade and Threshold Analysis,' we administer hexamethonium bromide to induce a ganglionic blockade. This blockage results in a significant reduction or total cessation of APs. By comparing the data from these two phases, we are able to finely calibrate the threshold for detecting APs from wavelet coefficients (both MODWT and CWT) and for identifying bursts in iSNA data. In the adaptation phase of AP detection, our method incorporated a novel combination of MODWT and CWT, which aims to improve the reliability of AP detection. Initially, we used MODWT as a multiresolution analysis approach to reduce the high and low-frequency components of the signal, thereby highlighting the APs against the background noise. This setup necessitated performing frequency and spectrum analysis to effectively select the appropriate decomposition level. Based on the frequency analysis of manually extracted APs, we determined that the detail coefficients at the fourth level of decomposition overlap with the dominant frequency of APs, and thus, we extracted them from fourth level. This initial denoising process is crucial for enhancing clarity, yet it carries the risk of altering the raw signal, which could result in the loss of some APs. To mitigate this issue and maintain the integrity of our detection method, we generated an averaged template from all APs retrieved using MODWT. We then constructed a wavelet tailored to meet the mathematical specifications of CWT. This custom-designed wavelet more precisely captures the characteristics of our targeted APs, and it can be used as a replacement for standard alternatives like Symlet 6. In the final step, the CWT is applied to the recorded data using the custom-designed wavelet, from which APs are extracted. By averaging these APs, we have refined the initially created mother wavelet. While Shafer et al. (2022) have implemented a similar refinement algorithm, in 79 their initial attempt they used MATLAB's built-in function to apply the CWT with a Symlet 7 wavelet. Notably, the Symlet 7 wavelet in MATLAB's toolbox extends over 14 samples, which resulted in Shafer et al. initially capturing only the larger APs. They then averaged these selectively obtained APs to construct their mother wavelet template. Building on the distinct noise floor setting and refined AP extraction techniques outlined, our research further delved into the relationship between burst intervals and APs. We noted a general synchronicity between APs and these intervals. Yet, as previously shown by Klassen & Shoemaker, (2021), the presence of a substantial number of APs outside these burst intervals serves as reminder of the limitations inherent in relying solely on burst analysis. While valuable, burst analysis alone may inadvertently overlook the neuronal communication between bursts. It is here that the sophistication of WT analysis becomes indispensable. By integrating WT into our analytical toolkit, we capture the finer nuances and specific neuronal events that might elude burst analysis. In Summary, the major innovations of this thesis are: • We characterized the length, shape and frequency content of rat’s AP’s • We used a noise floor recording to identify AP thresholds • We defined a new 2 step algorithm for extracting AP’s from nerve recordings. Given that a ground truth is not available for nerve recordings, we validated our findings by correlating extracted AP to bursts. This analysis indicates that the percentage of AP’s inside bursts aligns well with extended values from literature. The validation also shows that our new algorithm consistently extracts more APs from nerve recordings than other algorithms from the literature. 80 5.2 Challenges and Limitations AP extraction and analysis provides a series of challenges and constraints. A major problem is the intrinsic nature of wavelet coefficients as measures of similarity between the wavelet and our signal. This resemblance measure adds additional complexity: a large spike with modest similarity and a small spike with high similarity may yield comparable coefficients, potentially muddying the waters of our analysis. This phenomenon underscores the imperative need for continual refinement of our algorithm, enhancing its discernment capabilities to extract APs with heightened reliability and precision. Furthermore, the issues extend to data collecting, particularly during the recording process. The procedure of obtaining high-detail recordings from the nerves of tiny animals such as rats comes with difficulties. The delicate balance of protecting the subject's physiological integrity while assuring the quality of the recordings requires extraordinary precision. Any change might have a significant impact on the results, causing vibrations across the whole scope of our investigation. This mix of analytical and procedural complexities affects the landscape of our research, motivating us to go forward with creativity and rigor in our quest to solve the secrets of brain signaling. 5.3 Future Directions As we look ahead to future study, numerous attractive routes emerge, with the potential to further enhance and broaden our understanding of neural dynamics. An important step in improving the accuracy of AP identification is the smart clustering of retrieved APs based on size and shape. This method allows for the removal of abnormal extractions, making it easier to build a more refined 81 and representative template. Such a template offers a more exact representation of the AP landscape. Looking ahead, investigating the correlation between the size and frequency of APs and physiological events, such as changes in blood pressure, opens new avenues for understanding the intricate dynamics of neural activity. Furthermore, studying AP patterns in species beyond rats could uncover significant discoveries and provide valuable perspectives on the adaptability and robustness of neural systems. In data science, the methodologies refined through our AP analysis can inform advanced algorithms, enhancing the capacity to manage and interpret vast datasets with nuanced structures. 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Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NoDerivatives 4.0 International
Graduate
Seethaler, Rudolf
van Heusden, Klaske
Enhanced biological signal processing : integrating advanced frequency analysis and wavelet techniques for action potential detection in rats’ sympathetic nerve recordings
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