arXiv subjects

Anca Radulescu

Publications and source records attributed to Anca Radulescu.

At least 19 recordsLinked to original sources

arXiv2026

A distributed-delay Wilson-Cowan model of sleep-related rhythms in the corticothalamic system

The corticothalamic circuit supports rhythms with timescales that differ by orders of magnitude: sleep spindles, the sigma-band events of non-rapid-eye-movement (NREM) sleep, and infra-slow fluctuations near 0.02Hz that organize when spindles occur. Because the anatomy is the same in both cases, architecture alone cannot determine which rhythm the circuit expresses. We ask whether the temporal structure of the circuit's own feedback can. In a four-population Wilson--Cowan model comprising cortical excitatory and inhibitory populations, thalamic relay cells, and the thalamic reticular nucleus (TRN), we first establish how connectivity controls access to oscillatory behavior, and then introduce temporal coupling as either a weak Gamma distributed delay or a discrete delay. We investigate three distinct connectivity levels: recurrent cortical excitation gates whether the circuit can oscillate at all, the reciprocal relay-TRN pair determines where the oscillation lies and how it is configured, sustained, and terminated, and reticular self-inhibition limits its extent. We then examine how these connectivity-dependent regimes are affected by delayed coupling. Although delay does not change the equilibria themselves, it can substantially alter their stability and the organization of the resulting oscillatory dynamics. Under weak Gamma integration, short delays support spindle-compatible oscillations in the sigma band, while longer delays give rise to a much slower regime near 0.02Hz. The discrete-delay formulation produces a qualitatively different and more complex bifurcation structure. Together, these results show that the dynamics of the corticothalamic circuit depend not only on its connectivity, but also on the temporal organization of interactions within the circuit.

q-bio.NC↗

arXiv2025

Fractal geometry predicts dynamic differences in structural and functional connectomes

Understanding the intricate architecture of brain networks and its connection to brain function is essential for deciphering the underlying principles of cognition and disease. While traditional graph-theoretical measures have been widely used to characterize these networks, they often fail to fully capture the emergent properties of large-scale neural dynamics. Here, we introduce an alternative approach to quantify brain networks that is rooted in complex dynamics, fractal geometry, and asymptotic analysis. We apply these concepts to brain connectomes and demonstrate how quadratic iterations and geometric properties of Mandelbrot-like sets can provide novel insights into structural and functional network dynamics. Our findings reveal fundamental distinctions between structural (positive) and functional (signed) connectomes, such as the shift of cusp orientation and the variability in equi-M set geometry. Notably, structural connectomes exhibit more robust, predictable features, while functional connectomes show increased variability for non-trivial tasks. We further demonstrate that traditional graph-theoretical measures, when applied separately to the positive and negative sub-networks of functional connectomes, fail to fully capture their dynamic complexity. Instead, size and shape-based invariants of the equi-M set effectively differentiate between rest and emotional task states, which highlights their potential as superior markers of emergent network dynamics. These results suggest that incorporating fractal-based methods into network neuroscience provides a powerful tool for understanding how information flows in natural systems beyond static connectivity measures, while maintaining their simplicity.

nlin.CD↗

arXiv2025

A model of predation and survival in a system of three interacting species

The study of interactions between multiple species in an ecosystem is an active and impactful direction of inquiry. This is true in particular for fragile systems in which even small perturbations of their functional parameters can produce dramatic effects like species endangerment or extinction, leading the system to enter an unsustainable regime and eventually collapse. In this context, it is important to understand which factors can lead to such effects and for which systems, so that one can act proactively and timely to prevent them. We built and studied a mathematical model that captures the natural interactions between three species, in which two species are predators of the third, but such that one of the predators also consumes the other (to which we refer as Owls, Snakes and Mice). The nonlinear components of the model were documented on existing literature and assembled as a system of Lotka-Volterra ordinary differential equations. Our analytical computations and numerical exploration explorations revealed sequences of transcritical and Hopf bifurcations that underlie counterintuitive transitions of the system into regions of vulnerability to external noise. We conclude that, in order to avoid extinction,one needs to rigorously prescribe a well-documented, prediction-based approach to population control.

q-bio.PE↗

arXiv2023

Modeling the effects of adherence to vaccination and health protocols in epidemic dynamics by means of an SIR model

Susceptible-Infected-Recovered (SIR) models have been used for decades to understand epidemic outbreak dynamics. We develop an SIR model specifically designed to study the effects of population behavior with respect to health and vaccination protocols in a generic epidemic. Through a collection of parameters, our model includes the traditional SIR components: population birth, death, infection, recovery and vaccination rates, as well as limited immunity. We first use this simple setup to compare the effects of two vaccination schemes, one in which people are vaccinated at a rate proportional with the population, and one in which vaccines are administered to a fraction of the susceptible people (both of which are know strategies in real life epidemics). We then expand on the model and the analysis by investigating how these two vaccination schemes hold under two scenarios of population behavior: one in which people abide by health protocols and work towards diminishing transmission when infection is high; one in which people relax health protocols when infection is high. We illustrate these two aspects (vaccination and adherence to health protocols) act together to control the epidemic outbreak. While it is ideal that the tow components act jointly, we also show that tight observance of health protocols may diminish the need for vaccination in the effort to clear or mitigate the outbreak. Conversely, an efficient vaccination strategy can compensate for some degree of laxity in people's behavior.

q-bio.PE↗

arXiv2023

Stability of coupled Wilson-Cowan systems with distributed delays

Building upon our previous work on the Wilson-Cowan equations with distributed delays, we study the dynamic behavior in a system of two coupled Wilson-Cowan pairs. We focus in particular on understanding the mechanisms that govern the transitions in and out of oscillatory regimes associated with pathological behavior. We investigate these mechanisms under multiple coupling scenarios, and we compare the effects of using discrete delays versus a weak Gamma delay distribution. We found that, in order to trigger and stop oscillations, each kernel emphasizes different critical combinations of coupling weights and time delay, with the weak Gamma kernel restricting oscillations to a tighter locus of coupling strengths, and to a limited range of time delays. We finally illustrate the general analytical results with simulations for two particular applications: generation of beta-rhytms in the basal ganglia, and alpha oscillations in the prefrontal-limbic system.

math.DS↗

arXiv2023

Complex dynamics in two-dimensional coupling of quadratic maps

This paper examines the structure and limitations of equi-M sets in two-dimensional Complex Quadratic Networks (CQNs). In particular, we aim to describe the relationship between the equi-M set and the parameter domains where the critical orbit converges to periodic attractors (pseudo-bulbs). The two-node case serves as a foundational testbed: its analytical tractability enables the identification of critical phenomena and their dependence on coupling, while offering insight into more general principles. The two-node case is also simple enough to allow for explicit coupling conditions that govern phase transitions between synchronized and desynchronized behavior. Using a combination of analytical and numerical methods, the study reveals that while the period-1 pseudo-bulb closely tracks the boundary of the equi-M set near its main cusp, this correspondence breaks down for higher periods and in regions supporting coexisting attractors. These discrepancies highlight key differences between single-map and coupled dynamics, where equi-M sets no longer provide a full encoding of system combinatorics. These findings clarify the topological and dynamical behavior of low-dimensional CQNs and point toward a sharp increase in complexity as the number of nodes grows, laying the groundwork for future studies of high-dimensional dynamics.

nlin.CD↗

arXiv2022

Computing brain networks with complex dynamics

One important question in neuroscience is how global behavior in a brain network emerges from the interplay between network connectivity and the neural dynamics of individual nodes. To better understand this theoretical relationship, we have been exploring a simplified modeling approach in which we equip each node with discrete quadratic dynamics in the complex plane, and we study the emerging behavior of the resulting complex quadratic network (CQN). The long-term behavior of CQNs can be represented by asymptotic fractal sets with specific topological signatures going far beyond those described in traditional single map iterations. In this study, we illustrate how topological measures of these asymptotic sets can be used efficiently as comprehensive descriptors and classifiers of dynamics in tractography-derived connectomes for human subjects. We investigate to what extent the complex geometry of these sets is tied to network architecture (on one hand) and to the network behavior (on the other). This helps us understand the mechanics of the relationship between the subject's brain function, physiology and behavior and their underlying connectivity architecture.

q-bio.NC↗

arXiv2022

Gap junctions and synchronization clusters in the Thalamic Reticular Nuclei

The Thalamic Reticular Nuclei (TRN) mediate processes like attentional modulation, sensory gating and sleep spindles. The GABAergic inter neurons in the TRN are know to exhibit widespread synchronized activity patterns. One known contribution to shaping synchronization and clustering patterns in the TRN is coming from the presence of gap junctions. These are organized in specific connectivity architectures, that have been identified empirically through dye and electrical coupling studies. Our study uses a computational model in conjunction to implement realistic connectivity schemes in a small network. We explored the potential effects of the size, strength and distribution of gap junctional clusters on the synchronization patterns in TRN, and how these effects are modulated by other factors, such as the level of background inhibition.

q-bio.NC↗

arXiv2022

Mandelbrot sets for fixed template iterations

We study the dynamics of template iterations, consisting of arbitrary compositions of functions chosen from a finite set of polynomials. In particular, we focus on templates using complex unicritical maps in the family $\{ z^d + c, c \in \mathbb{C}, d \ge 2 \}$. We examine the dependence on parameters of the connectedness locus for a fixed template and show that, for most templates, the connectedness locus moves upper semicontiuously. On the other hand, one does not in general have lower semicontinuous dependence, and we show this by means of a counterexample.

math.DS↗

arXiv2022

Synchronization and clustering in complex quadratic networks

In continuation of prior work, we investigate ties between a network's connectivity and ensemble dynamics. This relationship is notoriously difficult to approach mathematically in natural, complex networks. In our work, we aim to understand it in a canonical framework, using complex quadratic node dynamics, coupled in networks which we call complex quadratic networks (CQNs). After previously defining extensions of the Mandelbrot and Julia sets for networks, we currently focus on the behavior of the node-wise projections of these sets, and on defining and analyzing the phenomena of node clustering and synchronization. We investigate the mechanisms that lead to nodes exhibiting identical or different Mandelbrot set. We propose that clustering is strongly determined by the network connectivity patterns, with the geometry of these clusters further controlled by the connection weights. We then illustrate the concept of synchronization in an existing set of whole brain, tractography-based networks obtained from 197 human subjects using diffusion tensor imaging. Synchronization and clustering are well-studied in the context of networks of oscillators, such as neural networks. Understanding the similarities to how these concepts apply to CQNs contributes to our understanding of universal principles in dynamic networks, and may help extend theoretical results to natural, complex systems.

nlin.CD↗

arXiv2020

Effects of local mutations in quadratic iterations

We introduce mutations in replication systems in which the intact copying mechanism is performed by discrete iterations of a complex quadratic map in the family $f_c(z) = z^2+c$. More specifically, we consider a "correct" function $f_{c_1}$ acting on the complex plane (representing the RNA to be copied). A "mutation" $f_{c_0}$ is a different ("erroneous") map acting on a locus of given radius $r$ around a mutation focal point $\xi^*$. The effect of the mutation is interpolated radially to eventually recover the original map $f_{c_1}$ when reaching an outer radius $R$. We call the resulting map a "mutated" map. In the theoretical framework of mutated iterations, we study how a mutation (replication error) affects the temporal evolution of the system, in the context of cellular differentiation. We use the prisoner set of the system to quantify simultaneously the long-term behavior of the entire space under mutated maps. We analyze how the position, timing and size of the mutation can alter the system's long-term evolution (as encoded in the topology of its prisoner set). In the context of genetics, this framework may increase our understanding of the factors and mechanisms that shape the genetic expression, in a specialized cell, in the process of differentiation from a stem cell.

math.DS↗

arXiv2020

Course of the first month of the COVID 19 outbreak in the New York State counties

We illustrate and study the evolution of reported infections over the month from March 1st to April 1st in the New York State as a whole, as well as in each individual county. We search for exponential trends, and try to understand whether there is any correlation of the timing and dynamics of these trends with state mandated measures on social distancing and testing. We conclude that the reports on April 1st may be dramatically under-representing the actual number of state-wide infections, and we propose reassessment of the data over the coming weeks, to monitor for effects of the PAUSE directive, and for the increasing number of casualties as a validating measure.

q-bio.PE↗

arXiv2020

Management strategies in a SEIR model of COVID 19 community spread

The 2019 Novel Corona virus infection (COVID 19) is an ongoing public health emergency of international focus. Significant gaps persist in our knowledge of COVID 19 epidemiology, transmission dynamics, investigation tools and management, despite (or possibly because of) the fact that the outbreak is an unprecedented global threat. On the positive side, enough is currently known about the epidemic process to permit the construction of mathematical predictive models. In our work, we adapt a traditional SEIR epidemic model to the specific dynamic compartments and epidemic parameters of COVID 19, as it spreads in an age-heterogeneous community. We analyze management strategies of the epidemic course (as they were implemented through lockdown and reopening procedures in many of the US states and countries worldwide); however, to more clearly illustrate ideas, we focus on the example of a small scale college town community, with the timeline of control measures introduced in the state of New York. We generate predictions, and assess the efficiency of these control measures (closures, mobility restrictions, social distancing), in a sustainability context.

physics.soc-ph↗

arXiv2019

Stability and bifurcations in Wilson-Cowan systems with distributed delays, and an application to basal ganglia interactions

The traditional Wilson-Cowan model of excitatory and inhibitory mean field interactions in neuronal populations considers a weak Gamma distribution of time delays when processing inputs, and is obtained via a time-coarse graining technique that averages the population response. Previous analyses of the stability of the Wilson-Cowan model focused on more simplified cases, where the delays were either not present, constant or were of a specific type. Since these simplifications may significantly alter the behavior of the model, we focus on understanding the behavior of the system before time-course graining, and for a wider range of delay distributions. For these generalized delay equations, we perform stability and bifurcation analyses with respect to parameters that capture both the coupling profile, and the time delay. The investigation is done through the examination of the system's associated characteristic equation. Under mild assumptions, we give complete mathematical proofs of our theoretical results, for the model with general delay distributions and prove the transversality condition for the possible Hopf bifurcations, in a generalized context. The stability region in this parameter space is described theoretically for several types of delay kernels, and numerical simulations are presented to substantiate the theoretical results. We illustrate these theoretical principles in an application to a basal ganglia circuit, in which $\beta$-band oscillations have been associated with Parkinson's Disease.

math.DS↗

arXiv2019

A pharmacokinetic model of lead absorption and calcium competitive dynamics

Lead is a naturally-occurring element. It has been known to man for a long time, and it is one of the longest established poisons. The current consensus is that no level of lead exposure should be deemed "safe." New evidence regarding the blood levels at which morbidities occur has prompted the CDC to reduce the screening guideline of 10 $\mu$g/dl to 2 $\mu$g/dl. Measurable cognitive decline (reduced IQ, academic deficits) have been found to occur at levels below 10mg/dl. Knowledge of lead pharmacology allows us to better understand its absorption and metabolization, mechanisms that produce its medical consequences. Based upon an original and very simplified compartmental model of Rabinowitz (1973) with only three major compartments (blood, bone and soft tissue), extensive biophysical models sprouted over the following two decades. However, none of these models have been specifically designed to use new knowledge of lead molecular dynamics to understand its deleterious effects on the brain. We build and analyze a compartmental model of lead pharmacokinetics, focused specifically on addressing neurotoxicity. We use traditional phase space methods, parameter sensitivity analysis and bifurcation theory to study the transitions in the system's behavior in response to various physiological parameters. We conclude that modeling the complex interaction of lead and calcium along their dynamic trajectory may successfully explain counter-intuitive effects on systemic function and neural behavior which could not be addressed by existing linear models. Our results encourage further efforts towards using nonlinear phenomenology in conjunction with empirically driven system parameters, to obtain a biophysical model able to provide clinical assessments and predictions.

q-bio.QM↗

arXiv2017

Networks of coupled quadratic nodes

We study asymptotic dynamics in networks of coupled quadratic nodes. While single map complex quadratic iterations have been studied over the past century, considering ensembles of such functions, organized as coupled nodes in a network, generate new questions with potentially interesting applications to the life sciences. We investigate how traditional Fatou-Julia results may generalize in the case of networks. We discuss extensions of concepts like escape radius, Julia and Mandelbrot sets (as parameter loci in $\mathbb{C}^n$, where $n$ is the size of the network). We study topological properties of these asymptotic sets and of their two-dimensional slices in $\mathbb{C}$ (defined in previous work). We find that, while network Mandelbrot sets no longer have a hyperbolic bulb structure, some of their geometric landmarks are preserved (e.g., the cusp always survives), and other properties (such as connectedness) depend on the network structure. We investigate possible extensions of the relationship between the Mandelbrot set and the Julia set connectedness loci in the case of network dynamics. We discuss possible classifications of asymptotic behavior in networks based on their underlying graph structure, using the geometry of Julia or Mandelbrot sets as a classifier. Finally, we propose a method for book-keeping asymptotic dynamics simultaneously over many networks with a common graph-theoretical property. \emph{Core} Julia and Mandelbrot sets describe statistically average asymptotic behavior of orbits over an entire collection of configurations.

math.DS↗

arXiv2016

Template iterations of quadratic maps and hybrid Mandelbrot sets

As a particular problem within the field of non-autonomous discrete systems, we consider iterations of two quadratic maps $f_{c_0}=z^2+c_0$ and $f_{c_1}=z^2+c_1$, according to a prescribed binary sequence, which we call a \emph{template}. We study the asymptotic behavior of the critical orbits, and define the Mandelbrot set in this case as the locus for which these orbits are bounded. However, unlike in the case of single maps, this concept can be understood in several ways. For a fixed template, one may consider this locus as a subset of the parameter space in $(c_0,c_1) \in \mathbb{C}^2$; for fixed quadratic parameters, one may consider the set of templates which produce a bounded critical orbit. In this paper, we consider both situations, as well as \emph{hybrid} combinations of them, we study basic topological properties of these sets and interpret them in light of potential applications.

math.DS↗

arXiv2016

Real and complex behavior for networks of coupled logistic maps

Many natural systems are organized as networks, in which the nodes interact in a time-dependent fashion. The object of our study is to relate connectivity to the temporal behavior of a network in which the nodes are (real or complex) logistic maps, coupled according to a connectivity scheme that obeys certain constrains, but also incorporates random aspects. We investigate in particular the relationship between the system architecture and possible dynamics. In the current paper we focus on establishing the framework, terminology and pertinent questions for low-dimensional networks. A subsequent paper will further address the relationship between hardwiring and dynamics in high-dimensional networks. For networks of both complex and real node-maps, we define extensions of the Julia and Mandelbrot sets traditionally defined in the context of single map iterations. For three different model networks, we use a combination of analytical and numerical tools to illustrate how the system behavior (measured via topological properties of the Julia sets) changes when perturbing the underlying adjacency graph. We differentiate between the effects on dynamics of different perturbations that directly modulate network connectivity: increasing/decreasing edge weights, and altering edge configuration by adding, deleting or moving edges. We discuss the implications of considering a rigorous extension of Fatou-Julia theory known to apply for iterations of single maps, to iterations of ensembles of maps coupled as nodes in a network.

math.DS↗