
Michael Reed
Most biological questions are network questions because biological systems are large and complicated. Stochastic processes arise naturally (1) as external forcing to which the system must adapt; (2) as an external probe of system dynamics; (3) as a representation of underlying biological diversity; (4) as a fundamental mechanism for a biological object to achieve a specific purpose. Examples will be given. Biological networks have myriad control mechanisms that ensure that some variables remain stable while other fluctuate wildly. How can one tell from the network (and the dynamics) which variables are the stable ones? Many examples of (4) will be given including gene expression and volume transmission in the brain. To find exciting and new mathematical questions, one should focus on how specific biological systems work.

James Keener
No abstract provided.

Thomas Kurtz
Beginning with the simple derivation of the (deterministic) law of mass action from Markov chain models of chemical reaction networks, we will illustrate the derivation of deterministic, piecewise deterministic, and stochastically perturbed deterministic models from increasingly complex stochastic models. The arguments exploit asymptotic properties of stochastic equations and limits of exchangeable systems. The last method will be applied to obtain results of Robert and Touboul for a neural network model.

Yao Li
Proposed in systems biology, systematic measures of complex biological systems, including degeneracy, complexity, and robustness, have been frequently used by biologists. Measuring the ability of structurally different components to perform the same function, degeneracy is known to have close ties with both structural complexity and robustness of complex systems. In this talk I will report our efforts of quantifying these systematic measures. In addition, we will also discuss our results about connections among degeneracy,complexity, and robustness.

Igor Belykh
In many biological networks, the individual nodes composing the network communicate via short onoff stochastic interactions. Pulsecoupled neuronal networks and ecological metapopulations with sporadic dispersal are important examples. In this talk, we present a general rigorous theory of stochastically switched dynamical networks and apply rigorous mathematical techniques to investigate the interplay between overall system dynamics and the stochastic switching process. If the switching time is fast, with respect to the characteristic time of the individual node dynamics, we expect the switching network to follow the averaged system where the dynamical law is given by the expectation of the stochastic variables. However, there are exceptions, especially in multistable networks where the trajectory may escape to another (wrong) attractor with small probability. Using the Lyapunov function method, we derive explicit bounds for these probabilities and relate them to the switching frequency and intrinsic parameters. Going beyond fast switching, we consider ecological networks and reveal an unexpected range of intermediate switching frequencies where synchronization becomes stable in a network which switches between two nonsynchronous dynamics.

Barbel Finkenstadt
Time series relating to gene expression are now routinely measured in various important biological experiments such as microarrays, nanostring, etc as well as experiments based on bioluminescent imaging. One of the most important aim is to gain an understanding of the transcriptional regulation of genes, i.e. what determines their activation. A natural model is to assume that gene activation is a constant rate birth process but that the rate may change to different levels at unknown time points leading to the piecewise linear â€œswitchâ€? model. Statistical inference for such a model poses interesting and challenging problems, in particular since experiments can only measure events downstream whereas the processes of interest remain unobserved. We will give an overview of our experience with fitting such switch models to real data where, depending on the type of experiment, our assumptions range from stochastic differential equations to the use of ordinary differential equations, along with realistic stochastic formulations of the measurement processes. We will also present further results on extending this nonlinear approach to the multivariate case of identifying networks of interacting genes. Here we find that the concept of â€œthresholdingâ€? constitutes a simple yet realistic and very effective modelling device to help identifying network connections. Joint work with Dafyd Jenkins, Kirsty Hey, George Minas and David Rand.

Kresimir Josic
Simultaneous recordings from large neural populations are becoming increasingly common. Correlations in neural activity measured in such recordings can reveal important aspects of neural network organization and function. However, estimating and interpreting large correlation matrices is challenging. Moreover, the network mechanisms that modulate these changes are also not fully understood. I will discuss how estimation of correlations can be improved by regularization, i.e. by imposing a structure on the estimate. I will illustrate this approach by analyzing the activity of 150â€“350 cells in mouse visual cortex. I will show that activity in this network is best explained by a combination of a sparse graph of pairwise partial correlations representing local interactions, and a lowrank component representing common fluctuations and external inputs.
Correlated activity can also be modulated by a number of factors, from changes in arousal and attentional state to learning and task engagement. I will review recent theoretical results that identify three separate biophysical mechanisms that modulate spike train correlations: changes in input correlations, internal fluctuations, and the transfer function of single neurons. Along with the statistical approaches discussed in the first part of the talk, such mechanistic constraints on the modulation of population activity will be important in analyses of high dimensional neural data.

Jay Newby
I will present theoretical support for a hypothesis about cellcell contact, which plays a critical role in immune function. A fundamental question for all cellcell interfaces is how receptors and ligands come into contact, despite being separated by large molecules, the extracellular fluid, and other structures in the glycocalyx. The cell membrane is a crowded domain filled with large glycoproteins that impair interactions between smaller pairs of molecules, such as the T cell receptor and its ligand, which is a key step in immunological information processing and decisionmaking. A first passage time problem allows us to gauge whether a reaction zone can be cleared of large molecules through passive diffusion on biologically relevant timescales. I combine numerical and asymptotic approaches to obtain a complete picture of the first passage time, which shows that passive diffusion alone would take far too long to account for experimentally observed cellcell contact formation times. The result suggests that cellcell contact formation may involve previously unknown active mechanical processes.

Jonathan Touboul
Multielectrode recordings have revealed complex and coordinated population activity in neuronal networks. Among the variety of reported patterns, a central regime of synchronous activity is thought to serve important function and the disruptions of which could be associated to serious condition. A more debated regime is concerned with the distribution of avalanches, defined as chunks of population activity separated by silences, in population (LFPs) or spike recordings; powerlaw distributions of avalanches are sometimes interpreted as revealing that the brain functions at a critical regime. I will present some thoughts and mathematical developments on simple models of largescale stochastic networks, in order to uncover the complex interplay between stochastic and nonlinear dynamics in the emergence of these regimes. I will show that network structure, as well as increased noise levels, interact with nonlinear neurons activity to induce synchronization in largescale systems, a phenomenon already reported in the biological literature on epilepsy. As for avalanche analysis, we will show that the data analysis methods used do not univocally reveal the presence of criticality: powerlaw scalings in LFPs avalanches may only be due to the noise in the LFP recordings, and powerlaws in spiking data to Boltzmannâ€™s molecular chaos (or propagation of chaos), a universal phenomenon in statistical systems.

Peter Kramer
We will describe how an idealized form of synchronized firing in a network of integrateandfire neurons can be represented in terms of the Watts model for a network cascade. Both meanfield and treelike (branching process) approximations are employed to characterize the probability that a particular statistical network model will sustain the idealized synchronous firing pattern. We describe some adaptations to these widely used analytical approximations that are indicated for this neuroscience application.

Peter Thomas
Schmandt and GalÃ¡n (Phys. Rev. Lett., 2012) introduced the Stochastic Shielding Approximation (SSA) as a fast, accurate simplification of randomly gated ion channel models. Viewing the channel as a discrete process on a directed graph, driven by an independent noise source for each edge, the SSA accurately represents the process using independent noise sources for only a small subset of the edges. Schmidt and Thomas (J. Math. Neurosci., 2014) showed the variance of the channel conductance decomposes into a sum of contributions from each directed edge, providing a metric for ranking the relative importance of each edge. Moreover they showed that SSA is equivalent to a dimensionreducing projection acting on the sample space, rather than on the state space. The SSA preserves the mean field behavior while selectively incorporating only the independent underlying noise sources that contribute the most significantly to observable system behavior. Thus the stochastic shielding heuristic provides an analytically tractable example of incorporating fluctuations "beyond the mean field" in a manner relevant to the network's physiological function.

Jae Kyoung Kim
The nonelementary reaction functions (e.g. MichaelisMenten or Hill functions) are used to reduce the determinsitic models of biochemical networks. Such deterministic reductions are frequently a basis for heuristic stochastic models in which nonelementary reaction functions are used as propensities of Gillespie algorithm. Despite the popularity of this heuristic stochastic simualtions, it remains unclear when such stochastic reductions are valid. In this talk, I will present conditions under which the stochastic models with the nonelementary propensity functions accurately approximate the full stochastic models. If the validity condition is satisfied, we can perform accurate and computationally inexpensive stochastic simulation without converting the nonelementary functions to the elementary functions (e.g. mass action kinetics).

Leonid Bunimovich
Hepatitis C virus (HCV) has the propensity to cause chronic infection. HCV affects an estimated 170 million people worldwide. Immune escape by continuous genetic diversification is commonly described using a metaphor of "arm race" between virus and host. We developed a mathematical model that explained all clinical observations which could not be explained by the "arm race theory". The model applied to network of crossimmunoreactivity suggests antigenic cooperation as a mechanism of mitigating the immune pressure on HCV variants. Crossimmunoreactivity was observed for dengue, influenza, etc. Therefore antigenic cooperation is a new target for therapeutic and vaccine development strategies. Joint work with P.Skums and Yu. Khudyakov (CDC)

Tiago Pereira
We study isolation as a means to control epidemic outbreaks in complex networks, focusing on the consequences of delays in isolating infected nodes. Our analysis uncovers a tipping point: if infected nodes are isolated before a critical day dc, the disease is effectively controlled, whereas for longer delays the number of infected nodes climbs steeply. We show that dc can be estimated explicitly in terms of network properties and disease parameters, connecting lowered values of dc explicitly to heterogeneity in degree distribution. Our results reveal also that initial delays in the implementation of isolation protocols can have catastrophic consequences in heterogeneous networks. As our study is carried out in a general framework, it has the potential to offer insight and suggest proactive strategies for containing outbreaks of a range of serious infectious diseases. This is a joint work with LaiSang Young.