arXiv subjects

Anca Radulescu

Publications and source records attributed to Anca Radulescu.

30 records · Page 2Linked to original sources

arXiv2015

Ebola impact and quarantine in a network model

Much effort has been directed towards using mathematical models to understand and predict contagious disease, in particular Ebola outbreaks. Classical SIR (susceptible-infected-recovered) compartmental models capture well the dynamics of the outbreak in certain communities, and accurately describe the differences between them based on a variety of parameters. However, repeated resurgence of Ebola contagions suggests that there are components of the global disease dynamics that we don't yet fully understand and can't effectively control. In order to understand the dynamics of a more widespread contagion, we placed SIR models within the framework of dynamic networks, with the communities at risk of contracting the virus acting as nonlinear systems, coupled based on a connectivity graph. We study how the effects of the disease (measured as the outbreak impact and duration) change with respect to local parameters, but also with changes in both short-range and long-range connectivity patterns in the graph. We discuss the implications of optimizing both these measures in increasingly realistic models of coupled communities.

q-bio.PE↗

arXiv2015

An energy balance model of carbon's effect on climate change

Due to climate change, the interest of studying our climatic system using mathematical modeling has become tremendous in recent years. One well-known model is Budyko's system, which represents the coupled evolution of two variables, the ice-line and the average earth surface temperature. The system depends on natural parameters, such as the earth albedo, and the amount A of carbon in the atmosphere. We introduce a 3-dimensional extension of this model in which we regard A as the third coupled variable of the system. We analyze the phase space and dependence on parameters, looking for Hopf bifurcations and the birth of cycling behavior. We interpret the cycles as climatic oscillations triggered by the sensitivity in our regulation of carbon emissions at extreme temperatures.

physics.ao-ph↗

arXiv2015

A mathematical model of reward and executive circuitry in obsessive compulsive disorde

The neuronal circuit that controls obsessive and compulsive behaviors involves a complex network of brain regions (some with known involvement in reward processing). Among these are cortical regions, the striatum and the thalamus (which compose the CSTC pathway), limbic areas such as the amygdala and the hippocampus, and well as dopamine pathways. Abnormal dynamic behavior in this brain network is a hallmark feature of patients with increased anxiety and motor activity, like the ones affected by OCD. There is currently no clear understanding of precisely what mechanisms generates these behaviors. We attempt to investigate a collection of connectivity hypotheses of OCD by means of a computational model of the brain circuitry that governs reward and motion execution. Mathematically, we use methods from ordinary differential equations and continuous time dynamical systems. We use classical analytical methods as well as computational approaches to study phenomena in the phase plane (e.g., behavior of the system's solutions when given certain initial conditions) and in the parameter space (e.g., sensitive dependence of initial conditions). We find that different obsessive-compulsive subtypes may correspond to different abnormalities in the network connectivity profiles. We suggest that it is combinations of parameters (connectivity strengths between regions), rather the than the value of any one parameter taken independently, that provides the best basis for predicting behavior, and for understanding the heterogeneity of the illness.

q-bio.NC↗

arXiv2015

Neural network spectral robustness under perturbations of the underlying graph

Recent studies have been using graph theoretical approaches to model complex networks (such as social, infrastructural or biological networks), and how their hardwired circuitry relates to their dynamic evolution in time. Understanding how configuration reflects on the coupled behavior in a system of dynamic nodes can be of great importance, for example in the context of how the brain connectome is affecting brain function. However, the connectivity patterns that appear in brain networks, and their individual effects on network dynamics, are far from being fully understood. We study the connections between edge configuration and dynamics in a simple oriented network composed of two interconnected cliques (representative of brain feedback regulatory circuitry). In this paper, our main goal is to study the spectra of the graph adjacency and Laplacian matrices, with a focus on three aspects in particular: (1) the sensitivity/robustness the spectrum in response to varying the intra and inter-modular edge density, (2) the effects on the spectrum of perturbing the edge configuration, while keeping the densities fixed and (3) the effects of increasing the network size. We study some tractable aspects analytically, then simulate more general results numerically. This paper aims to clarify, from analytical and modeling perspectives, the underpinnings of our related work, which further addresses how graph properties affect the network's temporal dynamics and phase transitions. We propose that this type of results may be helpful when studying small networks such as macroscopic brain circuits. We suggest potential applications to understanding synaptic restructuring in learning networks, and the effects of network configuration to function of emotion-regulatory neural circuits.

q-bio.NC↗

arXiv2015

Symbolic template iterations of complex quadratic maps

The behavior of orbits for iterated logistic maps has been widely studied since the dawn of discrete dynamics as a research field, in particular in the context of the complex quadratic family. However, little is is known about orbit behavior if the map changes along with the iterations. We investigate in which ways the traditional theory of Fatou-Julia may still apply in this case, illustrating how the iteration pattern (symbolic template) can affect the topology of the Julia and Mandelbrot sets. We briefly discuss the potential of this extension towards a variety of applications in genetic and neural coding, since it investigates how an occasional or a reoccurring error in a replication or learning algorithm may affect the dynamic outcome.

math.DS↗

arXiv2015

Nonlinear network dynamics under perturbations of the underlying graph

Many natural systems are organized as networks, in which the nodes (be they cells, individuals or populations) interact in a time-dependent fashion. The dynamic behavior of these networks depends on how these nodes are connected, which can be understood in terms of an adjacency matrix, and connection strengths. The object of our study is to relate connectivity to temporal behavior in networks of coupled nonlinear oscillators. We investigate the relationship between classes of system architectures and classes of their possible dynamics, when the nodes are coupled according to a connectivity scheme that obeys certain constrains, but also incorporates random aspects. We illustrate how the phase space dynamics and bifurcations of the system change when perturbing the underlying adjacency graph. We differentiate between the effects on dynamics of the following operations that directly modulate network connectivity: (1) increasing/decreasing edge weights, (2) increasing/decreasing edge density, (3) altering edge configuration by adding, deleting or moving edges. We discuss the significance of our results in the context of real life networks. Some interpretations lead us to draw conclusions that may apply to brain networks, synaptic restructuring and neural dynamics.

q-bio.NC↗

arXiv2012

Input Statistics and Hebbian Crosstalk Effects

As an extension of prior work, we study inspecific Hebbian learning using the classical Oja model. We use a combination of analytical tools and numerical simulations to investigate how the effects of inspecificity (or synaptic "cross-talk") depend on the input statistics. We investigated a variety of patterns that appear in dimensions higher than 2 (and classified them based on covariance type and input bias). The effects of inspecificity on the learning outcome were found to depend very strongly on the nature of the input, and in some cases were very dramatic, making unlikely the existence of a generic neural algorithm to correct learning inaccuracy due to cross-talk. We discuss the possibility that sophisticated learning, such as presumably occurs in the neocortex, is enabled as much by special proofreading machinery for enhancing specificity, as by special algorithms.

q-bio.NC↗

arXiv2012

Hebbian Crosstalk and Input Segregation

Purpose: We previously proposed that Hebbian adjustments that are incompletely synapse specific ("crosstalk") might be analogous to genetic mutations. We analyze aspects of the effect of crosstalk in Hebbian learning using the classical Oja model. Methods: In previous work we showed that crosstalk leads to learning of the principal eigenvector of EC (the input covariance matrix pre-multiplied by an error matrix that describes the crosstalk pattern), and found that with positive input correlations increasing crosstalk smoothly degrades performance. However, the Oja model requires negative input correlations to account for biological ocular segregation. Although this assumption is biologically somewhat implausible, it captures features that are seen in more complex models. Here, we analyze how crosstalk would affect such segregation. Results: We show that for statistically unbiased inputs crosstalk induces a bifurcation from segregating to non-segregating outcomes at a critical value which depends on correlations. We also investigate the behavior in the vicinity of this critical state and for weakly biased inputs. Conclusions: Our results show that crosstalk can induce a bifurcation under special conditions even in the simplest Hebbian models and that even the low levels of crosstalk observed in the brain could prevent normal development. However, during learning pairwise input statistics are more complex and crosstalk-induced bifurcations may not occur in the Oja model. Such bifurcations would be analogous to "error catastrophes" in genetic models, and we argue that they are usually absent for simple linear Hebbian learning because such learning is only driven by pairwise correlations.

q-bio.NC↗

arXiv2010

Mechanisms explaining transitions between tonic and phasic firing in neuronal populations as predicted by a low dimensional firing rate model

Several firing patterns experimentally observed in neural populations have been successfully correlated to animal behavior. Population bursting, hereby regarded as a period of high firing rate followed by a period of quiescence, is typically observed in groups of neurons during behavior. Biophysical membrane-potential models of single cell bursting involve at least three equations. Extending such models to study the collective behavior of neural populations involves thousands of equations and can be very expensive computationally. For this reason, low dimensional population models that capture biophysical aspects of networks are needed. \noindent The present paper uses a firing-rate model to study mechanisms that trigger and stop transitions between tonic and phasic population firing. These mechanisms are captured through a two-dimensional system, which can potentially be extended to include interactions between different areas of the nervous system with a small number of equations. The typical behavior of midbrain dopaminergic neurons in the rodent is used as an example to illustrate and interpret our results. \noindent The model presented here can be used as a building block to study interactions between networks of neurons. This theoretical approach may help contextualize and understand the factors involved in regulating burst firing in populations and how it may modulate distinct aspects of behavior.

q-bio.CB↗

arXiv2008

A multi-etiology model of systemic degeneration in schizophrenia

We discuss the possibility of multiple underlying etiologies of the condition currently labeled as schizophrenia. We support this hypothesis with a theoretical model of the prefrontal-limbic system. We show how the dynamical behavior of this model depends on an entire set of physiological parameters, representing synaptic strengths, vulnerability to stress-induced cortisol, dopamine regulation and rates of autoantibody production. Malfunction of different such parameters produces similar outward dysregulation of the system, which may readily lead to diagnosis difficulties in a clinician's office. We further place this paradigm within the contexts of pathophysiology and of antipsychotic pharmacology. We finally propose brain profiling as the future quantitative diagnostic toolbox that agrees with a multiple etiologies hypothesis of schizophrenia.

q-bio.NC↗

arXiv2008

Hebbian Inspecificity in the Oja Model

Recent work on Long Term Potentiation in brain slices shows that Hebb's rule is not completely synapse-specific, probably due to intersynapse diffusion of calcium or other factors. We extend the classical Oja unsupervised model of learning by a single linear neuron to include Hebbian inspecificity, by introducing an error matrix E, which expresses possible crosstalk between updating at different connections. We show the modified algorithm converges to the leading eigenvector of the matrix EC, where C is the input covariance matrix. When there is no inspecificity, this gives the classical result of convergence to the first principal component of the input distribution (PC1). We then study the outcome of learning using different versions of E. In the most biologically plausible case, arising when there are no intrinsically privileged connections, E has diagonal elements Q and off- diagonal elements (1-Q)/(n-1), where Q, the quality, is expected to decrease with the number of inputs n. We analyze this error-onto-all case in detail, for both uncorrelated and correlated inputs. We study the dependence of the angle theta between PC1 and the leading eigenvector of EC on b, n and the amount of input activity or correlation. (We do this analytically and using Matlab calculations.) We find that theta increases (learning becomes gradually less useful) with increases in b, particularly for intermediate (i.e. biologically-realistic) correlation strength, although some useful learning always occurs up to the trivial limit Q = 1/n. We discuss the relation of our results to Hebbian unsupervised learning in the brain.

q-bio.NC↗

arXiv2005

The Connected Isentropes Conjecture in a Space of Quartic Polynomials

This note is a shortened version of my dissertation thesis, defended at Stony Brook University in December 2004. It illustrates how dynamic complexity of a system evolves under deformations. The objects I considered are quartic polynomial maps of the interval that are compositions of two logistic maps. In the parameter space $P^{Q}$ of such maps, I considered the algebraic curves corresponding to the parameters for which critical orbits are periodic, and I called such curves left and right bones. Using quasiconformal surgery methods and rigidity I showed that the bones are simple smooth arcs that join two boundary points. I also analyzed in detail, using kneading theory, how the combinatorics of the maps evolves along the bones. The behavior of the topological entropy function of the polynomials in my family is closely related to the structure of the bone-skeleton. The main conclusion of the paper is that the entropy level-sets in the parameter space that was studied are connected.

math.DS↗