Introduction to Focus Issue: Nonlinear science of living systems: From cellular mechanisms to functions

Noble and Noble (2018) apply nonlinear dynamics thinking and stochastic theory to address the question of how organisms make choices. They argue that the deterministic view of life cannot explain human decision making and that stochastic and chaotic processes are essential to the ability of making a choice. By using an analogy with the immune system, they demonstrate that stochasticity can be used to generate novel behavioral responses to face an unusual challenge.

Liljenström (2018) argues that one of the greatest challenges to science, in particular to neuroscience, is to understand how processes at different levels of organization are related to each other. In connection with this problem is the question of the functional significance of fluctuations, noise, and chaos. Computational models can be a useful tool in elucidating these types of issues, and the author briefly reviews some work of Hans Braun and colleagues and relates it to his own modeling efforts. The constructive role of noise and chaos in such systems is discussed and related to functions such as learning and associative memory, decision making, and transitions between different (un-)conscious states. Concluding, the author discusses downward causation and the problem of free will.

Methods of stochastic dynamics are used in modeling active particles and active media, ranging from interactive agents in chemical reactions to socio-dynamics. Noetel et al. (2018) describe a simple stochastic kinematic model of search and return, whereby an agent (e.g., an insect or a robot) searches for a target (e.g., a food source) in a noisy environment but is also required to navigate and return to a dock. The study underlines the importance of both the type and the intensity of random perturbations in achieving optimal search performance.

Disturbances at lower functional and structural levels of an organism often manifest in diseases at the behavioral level. Granitza et al. (2018) consider the case of sleep-disordered breathing called Chayne-Stokes respiration (CSR), where epochs of unusually deep and rapid breathing alternate with a complete lack of respiratory drive. They apply nonlinear dynamics methods to develop a novel index to capture and quantify oxygen in the blood and explore its potential to predict mortality in CSR patients.

Taking a systems-orientated approach, Tretter (2018) considers issues such as nonlinearity and complexity within the context of mental disorders. He considers how nonlinear science can contribute to an understanding of complex brain dynamics. He further discusses that for a deeper understanding of mental phenomena, a solely network-orientated neurobiology is not sufficient and argues that systemic approaches using nonlinear methods might help to bridge gaps between neurobiology and psychiatry.

Many systems across natural disciplines can be modeled as networks of networks. As even small networks are amenable to complex dynamics, a network of networks may show a richer spectrum of various dynamical regimes compared to its sub-network constituents. An important question is whether a network of networks can be reduced to a simple, but larger, network, without loss of its dynamical richness. Gorjao et al. (2018) address this issue by a numerical study of heterogeneous networks of FitzHugh-Nagumo elements. They show that reducing a network of networks to a single larger network might not be sufficient to reproduce the whole dynamical repertoire of the original system.

Communication and information transfer are essential at every level of living organisms, be it genetic feedback loops, intercellular communication, or social dynamics. Bettenworth et al. (2018) explore communication processes in bacterial colonies where bacteria exchange signaling molecules to convey information about the population size of a network of entities, and initiate population-wide changes in their behavior. They show that a combination of diffusive spreading of the signal molecules and an exponential growth of colonies releasing them leads to constant concentration spread in a front-like fashion.

Electroencephalography (EEG) is used to record electrical dynamics of large populations of cortical neurons that is measured through the skull. Quintero-Quiroz et al. (2018) apply the symbolic ordinal method of analysis to investigate the transition between resting states with eyes closed (EC) and eyes open (EO). They implement the analysis in two EEG datasets with different recording conditions using permutation entropy and an asymmetry coefficient to show that the method is capable of distinguishing between the two brain states from the raw data alone.

Boaretto et al. (2018) consider neurons coupled in a small-world topology displaying anomalous phase synchronization, which they show to be related to the behavior of the individual neurons. They identify strong correlation between the behavior of the neuronal inter-bursting intervals and the ability of the network to exhibit anomalous phase synchronization. They show that external perturbations that eliminate anomalous phase synchronization can also promote small changes in the individual dynamics of the neurons.

Tchaptchet (2018) introduces heterogeneity with randomized temperatures in Huber-Braun neurons as a scaling factor for the generation of a large variety of impulse patterns, all known to be experimentally observed in peripheral neurons as well as in the central nervous system. This heterogeneity is demonstrated to play a role in the output of the network, which ranges from silence to synchronous bifurcation transitions.

The framework of Boolean recurrent neural networks with emphasis on sequences of distinct attractors observed in limit cycles is applied by Cabessa and Villa (2018) to a simplified model of the basal ganglia-thalamocortical network circuit with each brain area represented by a neuronal node. The authors implement a new adaptive plasticity rule and show that an interactive feedback allows the network to switch between stable domains with highly discontinuous boundaries and very high levels of complexity.

Sun et al. (2018) consider two clustered neuronal networks with dense intra-synaptic links within each cluster and sparse inter-synaptic links between them. They show that intermediate intra- and inter-time delays are able to separately induce fast regular firing-spiking activity with a high firing rate as well as a high spiking regularity. Their results indicate that appropriately adjusted intra- and inter-time delays can facilitate fast regular firing in neuronal networks. They conjecture that this is most likely when the largest value of common divisors of the intra- and inter-time delays falls into a range where fast regular firings are induced by suitable intra- or inter-time delays alone.

Large-scale brain phenomena, such as MRI and EEG, are controlled at a lower level by the interaction between individual neurons. Orio et al. (2018) investigate synchronization dynamics in large neuronal networks where each neuron is capable of a variety of spiking patterns, including chaos and bursting, and is coupled to its neighbors via gap junctions. They demonstrate that a deterministic network of weakly coupled neurons shows multistable synchrony patterns. The range of coupling strengths at which multistability is observed increases if noise is introduced.

Unveiling the mechanisms underlying contrast enhancement, which might be used for the detection of a change in a sensory stimulus or environment, is of importance for the understanding of sensory encoding mechanisms in general. Han et al. (2018) study a new mechanism of firing rate contrast enhancement by means of synchronization by inhibition in excitatory/inhibitory neuronal networks. They demonstrate that synchronized inhibitory neurons provide global inhibition that can enhance the firing rate contrast of excitatory neurons in synchronized networks, suggesting a further functional role of inhibitory synchronization in neuronal systems.

Holmgren Hopkins et al. (2018) use a minimal model to simulate two thalamic neurons coupled via gap junction and driven by a synaptic input from a two-neuron model of sleep regulation by the hypothalamus. The model predicts that the transitions between sleep and wake happen via chaos, because a single thalamic neuron has to undergo a chaotic transition between regular tonic and bursting activities. The results of this study suggest that sleep- and wake-related dynamics in the thalamus may be generated at the level of gap junction-coupled clusters of thalamic neurons driven from the hypothalamus, which would then propagate throughout the thalamus and cortex via long-range axonal connections.

Mechanical and chemical traumatic injuries may alter normal firing patterns of a neuron by, for example, modifying the kinetics of the inward sodium ionic current. This modifies the excitability of a neuron and may lead to the spontaneous firing of the unstimulated neuron, which otherwise remains silent under normal conditions. The modeling study of Barlow et al. (2018) explores the possibilities to alleviate such pathological spontaneous firing by means of thermal stimulation. They show that cooling a neuron by just a few degrees shifts the bifurcation point of the onset of periodic firing, raising the firing threshold, and thus preventing spontaneous firing.

Different patterns of neuronal activity can result in strengthening or weakening connections between neurons in the brain. This phenomenon is known as synaptic plasticity and is thought to underlie the processes of memory and learning. In addition, time delays are often present in neuronal communication but are frequently neglected in the theoretical studies of synaptic plasticity. Madadi Asl et al. (2018) review the recent findings accounting for propagation delays in spike-timing-dependent plasticity (STDP) rules and show that these together allow to account for experimental findings that cannot be explained by STDP alone.

Synchronization of neurons in neuronal networks of varying size and of various coupling types is of major importance when trying to gain an understanding of different aspects of brain functions. Follmann et al. (2018) investigate a neuronal network with gap junction coupling and demonstrate in numerical simulations how neurons synchronize into different robust states evolving through interesting period-doubling cascades and transitions to chaos.

Consideration of interspike intervals with both linear and nonlinear measures is an essential part of the analysis of sensory systems and, indeed, this has also always been a major part of Hans Braun's work as an “nonlinear” physiologist. Kostal et al. (2018) take a fresh look at the statistics of interspike intervals by revisiting the old measure of the instantaneous firing rate. The authors demonstrate the compatibility of different notions of the firing rate using computer simulations of relevance for investigators dealing with neuronal spike trains in experimental and numerical studies.

Zhang et al. (2018) explore the roles of noise and the brain-derived neurotrophic factor (BDNF) on the generation of epileptic seizures in a model of coupled hippocampal neurons. They show that increase in both noise and activation of BDNF-modulated receptors induce high-frequency spiking and synchronization associated with seizures in temporal lobe epilepsy. The authors thus demonstrate how events at a single neuron level result in changes at the population and behavioral levels.

Grines et al. (2018) extend the concept of phase-response curves to the case of multistable oscillators with several stable limit cycles. Perturbations may cause transitions in the dynamics of the system from one oscillating mode to another, and phase transfer curves are used to describe phase shifts in such transitions. In this way, it is possible to construct one-dimensional maps that characterize periodically kicked multistable oscillators. These maps are shown to be good approximations of the full dynamics for large forcing periods.

The computational capabilities of the transcriptional regulatory networks of five evolutionary distant organisms are studied by Gabalda-Sagarra et al. (2018). The authors identify in all cases a cyclic recurrent structure essential for dynamical encoding and information integration. The recent history of the cell is projected nonlinearly into this recurrent reservoir of nodes, where it is encoded by its transient dynamics, while the rest of the network forms a readout layer devoted to decoding and interpreting the high-dimensional dynamical state of the recurrent core. These results suggest that recurrent nonlinear dynamics is a key element for the processing of complex time-dependent information by cells.

The experimental work of Patejdl and Noack (2018) uses confocal microscopy to study spatiotemporal oscillatory calcium dynamics in smooth muscle. They used auto- and cross-correlation analysis to evaluate inter-cellular coupling and its role in the formation of oscillations and calcium signal propagation. Interestingly, the authors observe anti-phase oscillations in specific tissue areas, distinct from the homogeneous muscle cell layers.

Intracellular spatiotemporal calcium dynamics in astrocytes is studied in computational modeling paper by Brazhe et al. (2018). Just as a neuron has a soma, axon, and dendritic arbor, astrocytes possess non-uniform morphological organization with a cell body with thick branches and thin branchlets and leaflets. These morphologically distinct compartments may have different excitability for calcium signaling. The authors develop a novel computational model based on the concept spatially-partitioned oscillators, which accounts for stable calcium oscillations.

Alternative communication pathways caused by bi-directional transport of substances between the cells and the intercellular space are used by Verisokin et al. (2018) as “volume transmission” known to appear in multidimensional quantitative models of cellular processes. The authors propose a simple model that allows the investigation of the features of volume transmission at various spatial scales, taking into account several inhomogeneities. The study has revealed a number of characteristic spatiotemporal types of behavior that include self-organizing bursting and phase-locked firing patterns, different scenarios of excitation spreading, noise-sustained target patterns, and long-living slowly moving wave segments.

Ju et al. (2018) use biologically plausible Hodgkin-Huxley type models of individual cells and reduced phenomenological models with slow and fast dynamics to study quasi-periodicity phenomena occurring at the transition between tonic spiking and bursting. They show that the transition is due to either a torus bifurcation or to the period-doubling bifurcation of a stable periodic orbit on the two-dimensional slow-motion manifold near a characteristic fold. The authors also examine various torus bifurcations including stable and saddle torus-canards, resonant tori, the co-existence of nested tori, and torus breakdown leading to the onset of complex and bistable dynamics.

Rosenblum and Pikovsky (2018) develop an approach for fast experimental inference of synchronization properties of an oscillator. The approach only requires several observations of a driven system, and by reconstructing the phase dynamics from data, the method successfully determines synchronization domains of noisy and chaotic oscillators. The technique is especially important for experiments with living systems, where an invasive action could be harmful and should be minimized. The method can be applied to data obtained at different functional levels, from genetic to behavioral, because oscillations are ubiquitous across all of them.

Master equations are used in many natural sciences, from physics and chemistry to cell and population biology, to describe stochastic dynamics of systems. Analytical solutions of master equations are very rare, and scientists rely on stochastic simulations and various approximations. Peralta and Toral (2018) review the existing expansion methods for the analytical solution of a master equation and then study a novel method which turns out to be more accurate than the traditional ones. They demonstrate the method on several models, including a model of gene transcription and on a model of epidemic spreading.

The papers in this Focus Issue provide a sampling of the vibrant, cutting-edge research that is being done on nonlinear science of living systems. They also point future trends, indicating that research over the coming years will likely focus on how systems across the different functional, structural, and temporal levels interact with one another. This approach will lead to a more complete understanding of living systems across the multiple levels of organization, from molecular to single and then multiple cells to the full organism, allowing the understanding of the multi-level process of regulation of living systems and disease development. New hypotheses will then be able to be tested in specifically tailored experiments, and this ability will open opportunities for the implementation of novel procedures to confront harmful pathologies.

We thank the referees for their excellent and timely work in reviewing the manuscripts for this Focus Issue. We thank Deborah Doherty, Matthew Kershis, and Kristen Overstreet in the Chaos Editorial Office for their continuous and reliable assistance during this editorial process. Finally, we thank Juergen Kurths, who graciously embraced the idea of this Focus Issue during the Marburg meeting, for his support and advice.