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There is enormous interest -- both mathematically and in diverse applications -- in understanding the dynamics of coupled oscillator networks. The real-world motivation of such networks arises from studies of the brain, the heart, ecology, and more. It is common to describe the rich emergent behavior in these systems in terms of complex patterns of network activity that reflect both the connectivity and the nonlinear dynamics of the network components. Such behavior is often organized around phase-locked periodic states and their instabilities. However, the explicit calculation of periodic orbits in nonlinear systems (even in low dimensions) is notoriously hard, so network-level insights often require the numerical construction of some underlying periodic component. In this paper, we review powerful techniques for studying coupled oscillator networks. We discuss phase reductions, phase-amplitude reductions, and the master stability function for smooth dynamical systems. We then focus in particular on the augmentation of these methods to analyze piecewise-linear systems, for which one can readily construct periodic orbits. This yields useful insights into network behavior, but the cost is that one needs to study nonsmooth dynamical systems. The study of nonsmooth systems is well-developed when focusing on the interacting units (i.e., at the node level) of a system, and we give a detailed presentation of how to use \textit{saltation operators}, which can treat the propagation of perturbations through switching manifolds, to understand dynamics and bifurcations at the network level. We illustrate this merger of tools and techniques from network science and nonsmooth dynamical systems with applications to neural systems, cardiac systems, networks of electro-mechanical oscillators, and cooperation in cattle herds., Comment: 63 pages, 26 figures

Predictive variability due to data ambiguities has typically been addressed via construction of dedicated models with built-in probabilistic capabilities that are trained to predict uncertainty estimates as variables of interest. These approaches require distinct architectural components and training mechanisms, may include restrictive assumptions and exhibit overconfidence, i.e., high confidence in imprecise predictions. In this work, we propose a post-hoc sampling strategy for estimating predictive uncertainty accounting for data ambiguity. The method can generate different plausible outputs for a given input and does not assume parametric forms of predictive distributions. It is architecture agnostic and can be applied to any feed-forward deterministic network without changes to the architecture or training procedure. Experiments on regression tasks on imaging and non-imaging input data show the method's ability to generate diverse and multi-modal predictive distributions, and a desirable correlation of the estimated uncertainty with the prediction error.

We present a formal mathematical multi-agent modeling framework for autonomously combating a wildland fire with unmanned aerial vehicles. The problem is formulated as a collaboration between a drone and a helicopter equipped with a tanker. The modeling solutions are designed to capture the communication between agents and the information processes between the agents and their environment. The drone is used to make partial observations and use them to update probability distributions over the uncertain state of the world. We design a parameterized direct lookahead approximation policy to guide the drone through the region and promote active learning. The helicopter is used to extinguish the fire. We design a helicopter policy which anticipates the zones which will burn through the belief modeling and extinguish them. We simulate the modeling and policy solutions using a wildland fire simulation in an area of Northern California., Comment: 42 pages, 6 figures

The pandemic caused by the SARS-CoV-2 virus has exposed many flaws in the decision-making strategies used to distribute resources to combat global health crises. In this paper, we leverage reinforcement learning and optimization to improve upon the allocation strategies for various resources. In particular, we consider a problem where a central controller must decide where to send testing kits to learn about the uncertain states of the world (active learning); then, use the new information to construct beliefs about the states and decide where to allocate resources. We propose a general model coupled with a tunable lookahead policy for making vaccine allocation decisions without perfect knowledge about the state of the world. The lookahead policy is compared to a population-based myopic policy which is more likely to be similar to the present strategies in practice. Each vaccine allocation policy works in conjunction with a testing kit allocation policy to perform active learning. Our simulation results demonstrate that an optimization-based lookahead decision making strategy will outperform the presented myopic policy.

Cancer cells exhibit increased motility and proliferation, which are instrumental in the formation of tumours and metastases. These pathological changes can be traced back to malfunctions of cellular signalling pathways, and calcium signalling plays a prominent role in these. We formulate a new model for cancer cell movement which for the first time explicitly accounts for the dependence of cell proliferation and cell-cell adhesion on calcium. At the heart of our work is a non-linear, integro-differential (non-local) equation for cancer cell movement, accounting for cell diffusion, advection and proliferation. We also employ an established model of cellular calcium signalling with a rich dynamical repertoire that includes experimentally observed periodic wave trains and solitary pulses. The cancer cell density exhibits travelling fronts and complex spatial patterns arising from an adhesion-driven instability (ADI). We show how the different calcium signals and variations in the strengths of cell-cell attraction and repulsion shape the emergent cellular aggregation patterns, which are a key component of the metastatic process. Performing a linear stability analysis, we identify parameter regions corresponding to ADI. These regions are confirmed by numerical simulations, which also reveal different types of aggregation patterns and these patterns are significantly affected by \ca. Our study demonstrates that the maximal cell density decreases with calcium concentration, while the frequencies of the calcium oscillations and the cell density oscillations are approximately equal in many cases. Furthermore, as the calcium levels increase the speed of the travelling fronts increases, which is related to a higher cancer invasion potential. These novel insights provide a step forward in the design of new cancer treatments that may rely on controlling the dynamics of cellular calcium., Comment: 27 pages; 8 figures

Neural mass models are ubiquitous in large scale brain modelling. At the node level they are written in terms of a set of ODEs with a nonlinearity that is typically a sigmoidal shape. Using structural data from brain atlases they may be connected into a network to investigate the emergence of functional dynamic states, such as synchrony. With the simple restriction of the classic sigmoidal nonlinearity to a piecewise linear caricature we show that the famous Wilson-Cowan neural mass model can be analysed at both the node and network level. The construction of periodic orbits at the node level is achieved by patching together matrix exponential solutions, and stability is determined using Floquet theory. For networks with interactions described by circulant matrices, we show that the stability of the synchronous state can be determined in terms of a low-dimensional Floquet problem parameterised by the eigenvalues of the interaction matrix. This network Floquet problem is readily solved using linear algebra, to predict the onset of spatio-temporal network patterns arising from a synchronous instability. We consider the case of a discontinuous choice for the node nonlinearity, namely the replacement of the sigmoid by a Heaviside nonlinearity. This gives rise to a continuous-time switching network. At the node level this allows for the existence of unstable sliding periodic orbits, which we construct. The stability of a periodic orbit is now treated with a modification of Floquet theory to treat the evolution of small perturbations through switching manifolds via saltation matrices. At the network level the stability analysis of the synchronous state is considerably more challenging. Here we report on the use of ideas originally developed for the study of Glass networks to treat the stability of periodic network states in neural mass models with discontinuous interactions.

The theoretical description of non-renewal stochastic systems is a challenge. Analytical results are often not available or can only be obtained under strong conditions, limiting their applicability. Also, numerical results have mostly been obtained by ad-hoc Monte--Carlo simulations, which are usually computationally expensive when a high degree of accuracy is needed. To gain quantitative insight into these systems under general conditions, we here introduce a numerical iterated first-passage time approach based on solving the time-dependent Fokker-Planck equation (FPE) to describe the statistics of non-renewal stochastic systems. We illustrate the approach using spike-triggered neuronal adaptation in the leaky and perfect integrate-and-fire model, respectively. The transition to stationarity of first-passage time moments and their sequential correlations occur on a non-trivial timescale that depends on all system parameters. Surprisingly this is so for both single exponential and scale-free power-law adaptation. The method works beyond the small noise and timescale separation approximations. It shows excellent agreement with direct Monte Carlo simulations, which allows for the computation of transient and stationary distributions. We compare different methods to compute the evolution of the moments and serial correlation coefficients (SCC), and discuss the challenge of reliably computing the SCC which we find to be very sensitive to numerical inaccuracies for both the leaky and perfect integrate-and-fire models. In conclusion, our methods provide a general picture of non-renewal dynamics in a wide range of stochastic systems exhibiting short and long-range correlations., Comment: 14 pages, 12 figures, 1 appendix. Accepted for publication in Physical Review E

First-passage time problems are ubiquitous across many fields of study including transport processes in semiconductors and biological synapses, evolutionary game theory and percolation. Despite their prominence, first-passage time calculations have proven to be particularly challenging. Analytical results to date have often been obtained under strong conditions, leaving most of the exploration of first-passage time problems to direct numerical computations. Here we present an analytical approach that allows the derivation of first-passage time distributions for the wide class of non-differentiable Gaussian processes. We demonstrate that the concept of sign changes naturally generalises the common practice of counting crossings to determine first-passage events. Our method works across a wide range of time-dependent boundaries and noise strengths thus alleviating common hurdles in first-passage time calculations., Comment: 7 pages, 6 figures. Accepted for publication in Phys. Rev. E

We consider a leaky integrate-and-fire neuron with deterministic subthreshold dynamics and a firing threshold that evolves as an Ornstein-Uhlenbeck process. The formulation of this minimal model is motivated by the experimentally observed widespread variation of neural firing thresholds. We show numerically that the mean first passage time can depend non-monotonically on the noise amplitude. For sufficiently large values of the correlation time of the stochastic threshold the mean first passage time is maximal for non-vanishing noise. We provide an explanation for this effect by analytically transforming the original model into a first passage time problem for Brownian motion. This transformation also allows for a perturbative calculation of the first passage time histograms. In turn this provides quantitative insights into the mechanisms that lead to the non-monotonic behaviour of the mean first passage time. The perturbation expansion is in excellent agreement with direct numerical simulations. The approach developed here can be applied to any deterministic subthreshold dynamics and any Gauss-Markov processes for the firing threshold. This opens up the possibility to incorporate biophysically detailed components into the subthreshold dynamics, rendering our approach a powerful framework that sits between traditional integrate-and-fire models and complex mechanistic descriptions of neural dynamics., Comment: 8 pages, 7 figures. Accepted for publication in Physical Review E