arXiv subjects

Gabriel A. Silva

Publications and source records attributed to Gabriel A. Silva.

At least 19 recordsLinked to original sources

arXiv2026

Neuronal Spike Trains as Functional-Analytic Distributions: Representation, Analysis, and Significance

The action potential constitutes the digital component of the signaling dynamics of neurons. But the biophysical nature of the full-time course of the action potential associated with changes in membrane potential is mathematically distinct from its representation as a discrete set of events that encode when action potentials are triggered in a collection of spike trains. In this paper, we develop from first principles a unified functional-analytic framework for neuronal spike trains, grounded in Schwartz distribution theory. We show how this representation provides an exact operational calculus for convolution, distributional differentiation, and distributional support, which enables closed-form analysis of spike train dynamics without discretization, rate approximation, or smoothing. We then analyze the framework in the context of a two-neuron reciprocal circuit with propagation latencies and refractoriness, deriving exact results for synaptic drive, spike timing sensitivity, and causal admissibility of inputs, quantities that are either ill-defined or require approximation in conventional treatments.

q-bio.NC↗

arXiv2025

Leveraging Quantum Superposition to Infer the Dynamic Behavior of a Spatial-Temporal Neural Network Signaling Model

The exploration of new problem classes for quantum computation is an active area of research. In this paper, we introduce and solve a novel problem class related to dynamics on large-scale networks relevant to neurobiology and machine learning. Specifically, we ask if a network can sustain inherent dynamic activity beyond some arbitrary observation time or if the activity ceases through quiescence or saturation via an epileptic-like state. We show that this class of problems can be formulated and structured to take advantage of quantum superposition and solved efficiently using a coupled workflow between the Grover and Deutsch-Jozsa quantum algorithms. To do so, we extend their functionality to address the unique requirements of how input (sub)sets into the algorithms must be mathematically structured while simultaneously constructing the inputs so that measurement outputs can be interpreted as meaningful properties of the network dynamics. This, in turn, allows us to answer the question we pose.

quant-ph↗

arXiv2025

A Categorical Framework for Quantifying Emergent Effects in Network Topology

Emergent effect is crucial to understanding the properties of complex systems that do not appear in their basic units, but there has been a lack of theories to measure and understand its mechanisms. In this paper, we consider emergence as a kind of structural nonlinearity, discuss a framework based on homological algebra that encodes emergence as the mathematical structure of cohomologies, and then apply it to network models to develop a computational measure of emergence. This framework ties the potential for emergent effects of a system to its network topology and local structures, paving the way to predict and understand the cause of emergent effects. We show in our numerical experiment that our measure of emergence correlates with the existing information-theoretic measure of emergence.

nlin.AO↗

arXiv2025

Advancing Neural Network Performance through Emergence-Promoting Initialization Scheme

Emergence in machine learning refers to the spontaneous appearance of complex behaviors or capabilities that arise from the scale and structure of training data and model architectures, despite not being explicitly programmed. We introduce a novel yet straightforward neural network initialization scheme that aims at achieving greater potential for emergence. Measuring emergence as a kind of structural nonlinearity, our method adjusts the layer-wise weight scaling factors to achieve higher emergence values. This enhancement is easy to implement, requiring no additional optimization steps for initialization compared to GradInit. We evaluate our approach across various architectures, including MLP and convolutional architectures for image recognition and transformers for machine translation. We demonstrate substantial improvements in both model accuracy and training speed, with and without batch normalization. The simplicity, theoretical innovation, and demonstrable empirical advantages of our method make it a potent enhancement to neural network initialization practices. These results suggest a promising direction for leveraging emergence to improve neural network training methodologies. Code is available at: https://github.com/johnnyjingzeli/EmergenceInit.

cs.LG↗

arXiv2024

Quantifying Emergence in Neural Networks: Insights from Pruning and Training Dynamics

Emergence, where complex behaviors develop from the interactions of simpler components within a network, plays a crucial role in enhancing neural network capabilities. We introduce a quantitative framework to measure emergence during the training process and examine its impact on network performance, particularly in relation to pruning and training dynamics. Our hypothesis posits that the degree of emergence, defined by the connectivity between active and inactive nodes, can predict the development of emergent behaviors in the network. Through experiments with feedforward and convolutional architectures on benchmark datasets, we demonstrate that higher emergence correlates with improved trainability and performance. We further explore the relationship between network complexity and the loss landscape, suggesting that higher emergence indicates a greater concentration of local minima and a more rugged loss landscape. Pruning, which reduces network complexity by removing redundant nodes and connections, is shown to enhance training efficiency and convergence speed, though it may lead to a reduction in final accuracy. These findings provide new insights into the interplay between emergence, complexity, and performance in neural networks, offering valuable implications for the design and optimization of more efficient architectures.

cs.LG↗

arXiv2023

Discovering a change point and piecewise linear structure in a time series of organoid networks via the iso-mirror

Recent advancements have been made in the development of cell-based in-vitro neuronal networks, or organoids. In order to better understand the network structure of these organoids, a super-selective algorithm has been proposed for inferring the effective connectivity networks from multi-electrode array data. In this paper, we apply a novel statistical method called spectral mirror estimation to the time series of inferred effective connectivity organoid networks. This method produces a one-dimensional iso-mirror representation of the dynamics of the time series of the networks which exhibits a piecewise linear structure. A classical change point algorithm is then applied to this representation, which successfully detects a change point coinciding with the neuroscientifically significant time inhibitory neurons start appearing and the percentage of astrocytes increases dramatically. This finding demonstrates the potential utility of applying the iso-mirror dynamic structure discovery method to inferred effective connectivity time series of organoid networks.

stat.AP↗

arXiv2021

A category theory approach using preradicals to model information flows in networks

Category theory has been recently used as a tool for constructing and modeling an information flow framework. Here, we show that the flow of information can be described using preradicals. We prove that preradicals generalize the notion of persistence in spaces where the underlying structure forms a directed acyclic graph. We show that a particular $α$ preradical describes the persistence of a commutative $G$-module associated with a directed acyclic graph. Furthermore, given how preradicals are defined, they are able to preserve the modeled system's underlying structure. This allows us to generalize the notions of standard persistence, zigzag persistence, and multidirectional persistence.

math.CT↗

arXiv2021

Information entropy re-defined in a category theory context using preradicals

Algebraically, entropy can be defined for abelian groups and their endomorphisms, and was latter extended to consider objects in a Flow category derived from abelian categories, such as $R\textit{-}Mod$ with $R$ a ring. Preradicals are endofunctors which can be realized as compatible choice assignments in the category where they are defined. Here we present a formal definition of entropy for preradicals on $R$-Mod and show that the concept of entropy for preradicals respects their order as a big lattice. Also, due to the connection between modules and complete bounded modular lattices, we provide a definition of entropy for lattice preradicals, and show that this notion is equivalent, from a functorial perspective, to the one defined for module preradicals.

math.CT↗

arXiv2021

Learning without gradient descent encoded by the dynamics of a neurobiological model

The success of state-of-the-art machine learning is essentially all based on different variations of gradient descent algorithms that minimize some version of a cost or loss function. A fundamental limitation, however, is the need to train these systems in either supervised or unsupervised ways by exposing them to typically large numbers of training examples. Here, we introduce a fundamentally novel conceptual approach to machine learning that takes advantage of a neurobiologically derived model of dynamic signaling, constrained by the geometric structure of a network. We show that MNIST images can be uniquely encoded and classified by the dynamics of geometric networks with nearly state-of-the-art accuracy in an unsupervised way, and without the need for any training.

cs.LG↗

arXiv2020

Generalizable Machine Learning in Neuroscience using Graph Neural Networks

Although a number of studies have explored deep learning in neuroscience, the application of these algorithms to neural systems on a microscopic scale, i.e. parameters relevant to lower scales of organization, remains relatively novel. Motivated by advances in whole-brain imaging, we examined the performance of deep learning models on microscopic neural dynamics and resulting emergent behaviors using calcium imaging data from the nematode C. elegans. We show that neural networks perform remarkably well on both neuron-level dynamics prediction, and behavioral state classification. In addition, we compared the performance of structure agnostic neural networks and graph neural networks to investigate if graph structure can be exploited as a favorable inductive bias. To perform this experiment, we designed a graph neural network which explicitly infers relations between neurons from neural activity and leverages the inferred graph structure during computations. In our experiments, we found that graph neural networks generally outperformed structure agnostic models and excel in generalization on unseen organisms, implying a potential path to generalizable machine learning in neuroscience.

cs.LG↗

arXiv2020

Stability and Complexity Analyses of Finite Difference Algorithms for the Time-Fractional Diffusion Equation

Fractional differential equations (FDEs) are an extension of the theory of fractional calculus. However, due to the difficulty in finding analytical solutions, there have not been extensive applications of FDEs until recent decades. With the advent of increasing computational capacity along with advances in numerical methods, there has been increased interest in using FDEs to represent complex physical processes, where dynamics may not be as accurately captured with classical differential equations. The time-fractional diffusion equation is an FDE that represents the underlying physical mechanism of anomalous diffusion. But finding tractable analytical solutions to FDEs is often much more involved than solving for the solutions of integer order differential equations, and in many cases it is not possible to frame solutions in a closed form expression that can be easily simulated or visually represented. Therefore the development of numerical methods is vital. In previous work we implemented the full 2D time-fractional diffusion equation as a Forward Time Central Space finite difference equation by using the Grünwald-Letnikov definition of the fractional derivative. In addition, we derived an adaptive time step version that improves on calculation speed, with some tradeoff in accuracy. Here, we explore and characterize stability of these algorithms, in order to define bounds on computational parameters that are crucial for performing accurate simulations. We also analyze the time complexity of the algorithms, and describe an alternate adaptive time step approach that utilizes a linked list implementation, which yields better algorithmic efficiency.

math.NA↗

arXiv2020

Construction of edge-ordered multidirected graphlets for comparing dynamics of spatial temporal neural networks

The integration and transmission of information in the brain are dependent on the interplay between structural and dynamical properties. Implicit in any pursuit aimed at understanding neural dynamics from appropriate sets of mathematically bounded conditions is the notion of an underlying fundamental structure-function constraint imposed by the geometry of the structural networks and the resultant latencies involved with transfer of information. We recently described the construction and theoretical analysis of a framework that models how local structure-function rules give rise to emergent global dynamics on a neural network. An important part of this research program is the requirement for a set of mathematical methods that allow us to catalog, theoretically analyze, and numerically study the rich dynamical patterns that result. One direction we are exploring is an extension of the theory of graphlets. In this paper we introduce an extension of graphlets and associated metric that maps the topological transition of a network from one moment in time to another at the same time that causal relationships are preserved.

q-bio.NC↗

arXiv2019

The Effect of Signaling Latencies and Node Refractory States on the Dynamics of Networks

We describe the construction and theoretical analysis of a framework derived from canonical neurophysiological principles that model the competing dynamics of incident signals into nodes along directed edges in a network. The framework describes the dynamics between the offset in the latencies of propagating signals, which reflect the geometry of the edges and conduction velocities, and the internal refractory dynamics and processing times of the downstream node receiving the signals. This framework naturally extends to the construction of a perceptron model that takes into account such dynamic geometric considerations. We first describe the model in detail, culminating with the model of a geometric dynamic perceptron. We then derive upper and lower bounds for a notion of optimal efficient signaling between vertex pairs based on the structure of the framework. Efficient signaling in the context of the framework we develop here means that there needs to be a temporal match between the arrival time of the signals relative to how quickly nodes can internally process signals. These bounds reflect numerical constraints on the compensation of the timing of signaling events of upstream nodes attempting to activate downstream nodes they connect into that preserve this notion of efficiency. When a mismatch between signal arrival times and the internal states of activated nodes occurs, it can cause a break down in the signaling dynamics of the network. In contrast to essentially all the current state of the art in machine learning, this work provides a theoretical foundation for machine learning and intelligence architectures based on the timing of node activations and their abilities to respond, rather than necessary changes in synaptic weights. At the same time, this work is guiding the discovery of experimentally testable new structure-function principles in the biological brain.

q-bio.NC↗

arXiv2017

The prevalence of small world networks explained by modeling the competing dynamics of local signaling events in geometric networks

Networks are ubiquitous throughout science and engineering. A number of methods, including some from our own group, have explored how one goes about computing or predicting the dynamics of networks given information about internal models of individual nodes and network connectivity, possibly with additional information provided by statistical or descriptive metrics that characterize the network. But what can be inferred about network dynamics when there is no knowledge or information about the internal model or dynamics of participating nodes? Here, we explore how connected subsets of nodes competitively interact in order to activate a common downstream node they connect into. We achieve this by assuming a simple set of rules borrowed from neurophysiology. The model we develop reflects a local process from which global network dynamics emerges. We call this model a competitive refractory dynanics model. It is derived from a consideration of spatial and temporal summation in biological neurons, whereby summating post synaptic potentials (PSPs) along the dendritic tree contribute towards the membrane potential at the initial segment reaching a threshold potential. We first show how the 'winning node' or set of 'winning' nodes that achieve activation of a downstream node is computable by the model. We then derive a formal definition of optimized network signaling within our framework. We define a ratio between the signaling latencies on the edges of the network and the internal time it takes individual nodes to process incoming signals. We show that an optimal ratio is one where the speed of information propagation between connected nodes does not exceed the internal dynamic time scale of the nodes. We then show how we can use these results to arrive at a unique interpretation for the prevalence of the small world network topology in natural and engineered systems.

q-bio.MN↗

arXiv2015

Algebraic identification of the effective connectivity of constrained geometric network models of neural signaling

Cellular neural circuit and networks consisting of interconnected neurons and glia are ulti- mately responsible for the information processing associated with information processing in the brain. While there are major efforts aimed at mapping the structural and (electro)physiological connectivity of brain networks, such as the White House BRAIN Initiative aimed at the devel- opment of neurotechnologies capable of high density neural recordings, theoretical and compu- tational methods for analyzing and making sense of all this data seem to be further behind. Here, we propose and provide a summary of an approach for calculating effective connectivity from experimental observations of neuronal network activity. The proposed method operates on network-level data, makes use of all relevant prior knowledge, such as dynamical models of individual cells in the network and the physical structural connectivity of the network, and is broadly applicable to large classes of biological and non-biological networks.

q-bio.NC↗

arXiv2015

Efficient computation of the Grunwald-Letnikov fractional diffusion derivative using adaptive time step memory

Computing numerical solutions to fractional differential equations can be computationally intensive due to the effect of non-local derivatives in which all previous time points contribute to the current iteration. In general, numerical approaches that depend on truncating part of the system history while efficient, can suffer from high degrees of error and inaccuracy. Here we present an adaptive time step memory method for smooth functions applied to the Grunwald-Letnikov fractional diffusion derivative. This method is computationally efficient and results in smaller errors during numerical simulations. Sampled points along the system history at progressively longer intervals are assumed to reflect the values of neighboring time points. By including progressively fewer points backward in time, a temporally weighted history is computed that includes contributions from the entire past of the system, maintaining accuracy, but with fewer points actually calculated, greatly improving computational efficiency.

math.NA↗

arXiv2010

A framework for simulating and estimating the state and functional topology of complex dynamic geometric networks

We present a framework for simulating signal propagation in geometric networks (i.e. networks that can be mapped to geometric graphs in some space) and for developing algorithms that estimate (i.e. map) the state and functional topology of complex dynamic geometric net- works. Within the framework we define the key features typically present in such networks and of particular relevance to biological cellular neural networks: Dynamics, signaling, observation, and control. The framework is particularly well-suited for estimating functional connectivity in cellular neural networks from experimentally observable data, and has been implemented using graphics processing unit (GPU) high performance computing. Computationally, the framework can simulate cellular network signaling close to or faster than real time. We further propose a standard test set of networks to measure performance and compare different mapping algorithms.

cond-mat.dis-nn↗

arXiv2010

Computational efficiency of fractional diffusion using adaptive time step memory

Numerical solutions to fractional differential equations can be extremely computationally intensive due to the effect of non-local derivatives in which all previous time points contribute to the current iteration. In finite difference methods this has been approximated using the 'short memory effect' where it is assumed that previous events prior to some certain time point are insignificant and thus not calculated. Here we present an adaptive time method for smooth functions that is computationally efficient and results in smaller errors during numerical simulations. Sampled points along the system's history at progressively longer intervals are assumed to reflect the values of neighboring time points. By including progressively fewer points as a function of time, a temporally 'weighted' history is computed that includes contributions from the entire past of the system, resulting in increased accuracy, but with fewer points actually calculated, which ensures computational efficiency.

math-ph↗