Complex spatiotemporal dynamics of physicochemical processes are often modeled at a microscopic level (through e.g. atomistic, agent-based or lattice models) based on first principles. Some of these processes can also be successfully modeled at the macroscopic level using e.g. partial differential equations (PDEs) describing the evolution of the right few macroscopic observables (e.g. concentration and momentum fields). Deriving good macroscopic descriptions (the so-called "closure problem") is often a time-consuming process requiring deep understanding/intuition about the system of interest. Recent developments in data science provide alternative ways to effectively extract/learn accurate macroscopic descriptions approximating the underlying microscopic observations. In this paper, we introduce a datadriven framework for the identification of unavailable coarse-scale PDEs from microscopic observations via machine learning algorithms. Specifically, using Gaussian Processes, Artificial Neural Networks, and/or Diffusion Maps, the proposed framework uncovers the relation between the relevant macroscopic space fields and their time evolution (the right-hand-side of the explicitly unavailable macroscopic PDE). Interestingly, several choices equally representative of the data can be discovered. The framework will be illustrated through the data-driven discovery of macroscopic, concentration-level PDEs resulting from a fine-scale, Lattice Boltzmann level model of a reaction/transport process. Once the coarse evolution law is identified, it can be simulated to produce long-term macroscopic predictions. Different features (pros as well as cons) of alternative machine learning algorithms for performing this task (Gaussian Processes and Artificial Neural Networks), are presented and discussed.

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One of the most critical issues in epidemiology revolves around the bridging of the diverse space and time scales stretching from the microscopic scale, where detailed knowledge on the immune mechanisms, host-microbe and host-host interactions is often available, to the macroscopic population-scale where the epidemic emerges, the questions arise and the answers are required. In this paper we show how the so called Equation-Free approach, a novel computational framework for multi-scale analysis, can be exploited to efficiently analyze the macroscopic emergent behavior of complex epidemic models on certain type of networks by acting directly on the multi-scale simulation. The methodology can be used to bypass the need of derivation of closures for the emergent population-level equations providing a systematic computational strict approach for macroscopic-level analysis. We illustrate the methodology through a stochastic individual-based model with agents acting on two different networks: a random regular and an Erdős-Rényi network. We construct the macroscopic bifurcation diagrams and locate the critical points that mark the onset of emergent hysteresis behavior which are associated with disease outbreaks. Finally, we perform a rare-events analysis that may in principle be used to estimate the mean time of possible outbreaks of phenomenologically latent infectious diseases.

An important question in computational neuroscience is how to improve the efficacy of deep brain stimulation by extracting information from the underlying connectivity structure. Recent studies also highlight the relation of structural and functional connectivity in disorders such as Parkinson’s disease. Exploiting the structural properties of the network, we identify nodes of strong influence, which are potential targets for Deep Brain Stimulation (DBS). Simulating the volume of the tissue activated, we confirm that the proposed targets are reported as optimal targets (sweet spots) to be beneficial for the improvement of motor symptoms. Furthermore, based on a modularity algorithm, network communities are detected as set of nodes with high-interconnectivity. This allows to localise the neural activity, directly from the underlying structural topology. For this purpose, we build a large scale computational model that consists of the following elements of the basal ganglia network: subthalamic nucleus (STN), globus pallidus (external and internal parts) (GPe-GPi), extended with the striatum, thalamus and motor cortex (MC) areas, integrating connectivity from multimodal imaging data. We analyse the network dynamics under Healthy, Parkinsonian and DBS conditions with the aim to improve DBS treatment. The dynamics of the communities define a new functional partition (or segregation) of the brain, characterising Healthy, Parkinsonian and DBS treatment conditions.

A large-scale computational model of the basal ganglia network and thalamus is proposed to describe movement disorders and treatment effects of deep brain stimulation (DBS). The model of this complex network considers three areas of the basal ganglia region: the subthalamic nucleus (STN) as target area of DBS, the globus pallidus, both pars externa and pars interna (GPe-GPi), and the thalamus. Parkinsonian conditions are simulated by assuming reduced dopaminergic input and corresponding pronounced inhibitory or disinhibited projections to GPe and GPi. Macroscopic quantities are derived which correlate closely to thalamic responses and hence motor programme fidelity. It can be demonstrated that depending on different levels of striatal projections to the GPe and GPi, the dynamics of these macroscopic quantities (synchronisation index, mean synaptic activity and response efficacy) switch from normal to Parkinsonian conditions. Simulating DBS of the STN affects the dynamics of the entire network, increasing the thalamic activity to levels close to normal, while differing from both normal and Parkinsonian dynamics. Using the mentioned macroscopic quantities, the model proposes optimal DBS frequency ranges above 130 Hz.

Abstract. We IntroductionOver the past years there has been a rapid growth in biological knowledge even at the molecular/cell level concerning the physiology of neurons. Clinical studies and mathematical models have gone hand-in-hand enhancing our understanding and leading to breakthroughs in the field of neuroscience. One of the very first attempts in trying to systematically quantify the dynamics of nerve cells in mathematical terms can be traced back to 1907 when Lapicque introduced the modelling of a neuron's activity by an integrate and fire process using an electric circuit with a capacitor and resistor in parallel driven by a time-varying current (for a discussion see also Abbot, 1999). McCulloch and Pitts in 1943 introduced the concept of the artificial neuron while in 1952 Hodgkin and Huxley proposed the most celebrated model in the field describing the dynamics of action potentials. For their work in the area the two biophysicists won the Noble Prize in Physiology and Medicine in 1963. Other early benchmarks that shaped the field include the work of Rall [1959] who actually originated the development of compartmental models which incorporate spatial electro-physiological characteristics, the spatial distributed Fitzhugh-Nagumo model [Fithugh, 1961; Nagumo et al., 1962] describing the propagation of nerve pulses in the form of traveling waves, the work of Wiener and Rosenbluth [1946] and Greenberg and Hastings Cellular Automata [1978] modeling complex spatial patterns in excitable media with interconnected neurons, and models of spatially localized neural populations [Smith & Davidson, 1962;Griffith, 1963; Anninos et al., 1970;Wilson & Cowan, 1972]. A review on recent developments on the modelling of neuron dynamics can be found in [Inzikievitz, 2004;Herz et al., 2006].The studies have proceeded to the development of detailed state-of-the-art models aspiring to approximate the complex dynamics underlying the physics and mechanisms of a wide range of problems stretching from the description of the behaviour of certain neural tissues such as the cerebral neocortex [Marr, 1970], the cortical and thalamic [Wilson & Cowan, 1973], the CA3 hippocumpal [Traub, 1983] to visual hallucinations [Ermentrout & Cowan, 1979; Bressloff et al., 2001], phase transitions in human hand movements [Haken et al., 1985] and working memory mechanisms [Durstewitz et al., 2000; Brunel & Wang, 2001;Laing & Troy, 2003; Durstewitz and Seaman, 2006] and from neurological disorders such as schizophrenia [Cohen et al., 1996;Hoffman, 1997;Rolls et al., 2008] and epilepsy dynamics [Babloyantz & Destexhe, 1986; Theoden et al., 2004] to therapeutic surgical procedures such as deep-brain stimulation process [Rubin & Terman, 2004] to name just a few.These models are usually individual-based ones: they are composed by a large number of individual subunits interacting through a network, a caricature of the connections circuitry [Strogatz, 2001]. The dynamics of each single subunit are described in the spirit of representations such ...

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