Synchronization and Clustering in Complex Quadratic Networks.

Synchronization and clustering are well studied in the context of networks of oscillators, such as neuronal networks. However, this relationship is notoriously difficult to approach mathematically in natural, complex networks. Here, we aim to understand it in a canonical framework, using complex quadratic node dynamics, coupled in networks that we call complex quadratic networks (CQNs). We review previously defined extensions of the Mandelbrot and Julia sets for networks, focusing on the behavior of the node-wise projections of these sets and on describing the phenomena of node clustering and synchronization. One aspect of our work consists of exploring ties between a network's connectivity and its ensemble dynamics by identifying mechanisms that lead to clusters of nodes exhibiting identical or different Mandelbrot sets. Based on our preliminary analytical results (obtained primarily in two-dimensional networks), we propose that clustering is strongly determined by the network connectivity patterns, with the geometry of these clusters further controlled by the connection weights. Here, we first explore this relationship further, using examples of synthetic networks, increasing in size (from 3, to 5, to 20 nodes). We then illustrate the potential practical implications of synchronization in an existing set of whole brain, tractography-based networks obtained from 197 human subjects using diffusion tensor imaging. 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.

Estimating the glutamate transporter surface density in distinct sub-cellular compartments of mouse hippocampal astrocytes.

Glutamate transporters preserve the spatial specificity of synaptic transmission by limiting glutamate diffusion away from the synaptic cleft, and prevent excitotoxicity by keeping the extracellular concentration of glutamate at low nanomolar levels. Glutamate transporters are abundantly expressed in astrocytes, and previous estimates have been obtained about their surface expression in astrocytes of the rat hippocampus and cerebellum. Analogous estimates for the mouse hippocampus are currently not available. In this work, we derive the surface density of astrocytic glutamate transporters in mice of different ages via quantitative dot blot. We find that the surface density of glial glutamate transporters is similar in 7-8 week old mice and rats. In mice, the levels of glutamate transporters increase until about 6 months of age and then begin to decline slowly. Our data, obtained from a combination of experimental and modeling approaches, point to the existence of stark differences in the density of expression of glutamate transporters across different sub-cellular compartments, indicating that the extent to which astrocytes limit extrasynaptic glutamate diffusion depends not only on their level of synaptic coverage, but also on the identity of the astrocyte compartment in contact with the synapse. Together, these findings provide information on how heterogeneity in the spatial distribution of glutamate transporters in the plasma membrane of hippocampal astrocytes my alter glutamate receptor activation out of the synaptic cleft.

A mathematical model of reward and executive circuitry in obsessive compulsive disorder.

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, as 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 generate 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 a combination of parameters (connectivity strengths between regions), rather than the value of any one parameter taken independently, that provide the best basis for predicting behavior, and for understanding the heterogeneity of the illness.

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.

Insights into pathological mechanisms and interventions revealed by analyzing a mathematical model for cone metabolism.

This work analyzes a mathematical model for the metabolic dynamics of a cone photoreceptor, which is the first model to account for energy generation from fatty acids oxidation of shed photoreceptor outer segments (POS). Multiple parameter bifurcation analysis shows that joint variations in external glucose, the efficiency of glucose transporter 1 (GLUT1), lipid utilization for POS renewal, and oxidation of fatty acids affect the cone's metabolic vitality and its capability to adapt under glucose-deficient conditions. The analysis further reveals that when glucose is scarce, cone viability cannot be sustained by only fueling energy production in the mitochondria, but it also requires supporting anabolic processes to create lipids necessary for cell maintenance and repair. In silico experiments are used to investigate how the duration of glucose deprivation impacts the cell without and with a potential GLUT1 or oxidation of fatty acids intervention as well as a dual intervention. The results show that for prolonged duration of glucose deprivation, the cone metabolic system does not recover with higher oxidation of fatty acids and requires greater effectiveness of GLUT1 to recover. Finally, time-varying global sensitivity analysis (GSA) is applied to assess the sensitivity of the model outputs of interest to changes and uncertainty in the parameters at specific times. The results reveal a critical temporal window where there would be more flexibility for interventions to rescue a cone cell from the detrimental consequences of glucose shortage.

Dynamic coupling between the COVID-19 epidemic timeline and the behavioral response to PAUSE in New York State counties.

Behavioral epidemiology suggests that there is a tight dynamic coupling between the timeline of an epidemic outbreak, and the social response in the affected population (with a typical course involving physical distancing between individuals, avoidance of large gatherings, wearing masks, etc). We study the bidirectional coupling between the epidemic dynamics of COVID-19 and the population social response in the state of New York, between March 1, 2020 (which marks the first confirmed positive diagnosis in the state), until June 20, 2020. This window captures the first state-wide epidemic wave, which peaked to over 11,000 confirmed cases daily in April (making New York one of the US states most severely affected by this first wave), and subsided by the start of June to a count of consistently under 1,500 confirmed cases per day (suggesting temporary state-wide control of the epidemic). In response to the surge in cases, social distancing measures were gradually introduced over two weeks in March, culminating with the PAUSE directive on March 22nd, which mandated statewide shutdown of all nonessential activity. The mandates were then gradually relaxed in stages throughout summer, based on how epidemic benchmarks were met in various New York regions. In our study, we aim to examine on one hand, whether different counties exhibited different responses to the PAUSE centralized measures depending on their epidemic situation immediately preceding PAUSE. On the other hand, we explore whether these different county-wide responses may have contributed in turn to modulating the counties' epidemic timelines. We used the public domain to extract county-wise epidemic measures (such as cumulative and daily incidence of COVID-19), and social mobility measures for different modalities (driving, walking, public transit) and to different destinations. Our correlation analyses between the epidemic and the mobility time series found significant correlations between the size of the epidemic and the degree of mobility drop after PAUSE, as well as between the mobility comeback patterns and the epidemic recovery timeline. In line with existing literature on the role of the population behavioral response during an epidemic outbreak, our results support the potential importance of the PAUSE measures to the control of the first epidemic wave in New York State.

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 of March in New York State as a whole, as well as in each individual county in the state. We identify piecewise exponential trends, and search for correlations between the timing and dynamics of these trends and statewide mandated measures on testing and social distancing. We conclude that the reports on April 1 may be dramatically under-representing the actual number of statewide infections, an idea which is supported by more recent retroactive estimates based on serological studies. A follow-up study is underway, reassessing data until June 1, using additional measures for validation and monitoring for effects of the PAUSE directive, and of the reopening timeline.

Management strategies in a SEIR-type 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.