Nonlinear waves in biology contributions

Our objective is to develop a geometrical model of the primary visual cortex in accordance with the neural characteristics of the cortex and construct orientation maps by using the relevant model geometry. Our departure point is the visual cortex model of the orientation selective cortical neurons, which was presented in [1] by Citti and Sarti. We spatially extend this model to a five dimensional sub-Riemannian geometry and provide a novel geometric model of the primary visual cortex which models orientation-frequency selective, phase shifted cortical cell behavior and the associated neural connectivity. This model extracts orientation, frequency and phase information of any given two dimensional input image. We employ in particular an input image with uniformly distributed white noise as the mathematical interpretation of internal stimulation on the retinal plane. Then, we start from the very first step mechanisms of visual perception and by using our sub-Riemannian model in order to extract visual features from the noise image, we provide a neurally inspired geometric procedure for multi-feature orientation map construction.

Bibliography

[1] G. Citti and A. Sarti, “A cortical based model of perceptual completion in the roto-translation space,” Journal of Mathematical Imaging and Vision, vol. 24, no. 3, pp. 307–326, 2006.

Understanding how behavioral processes, inter-individual variability and interactions shape the spatial spread and dispersal of animal populations is a major challenge in ecology. Trichogramma parasitic waps are among the smallest insects in the world (less than 500 micrmeters long). They are grown and released by millions in the field to protect crops from insect pests, so that understanding their spatial propagation dynamics is critical to predict performance. I’ll present how a novel experimental system coupled with high-throughput tracking of individual movements by computer vision can give insight into the spatial spread of groups of parasitoid individuals over large temporal (one entire day) and spatial (six meters, ca. 12,000 body lengths) scales in the lab. In particular I’ll show how population spread is well described by heterogeneous diffusion, whereby individuals switch between two states dynamically (active versus sedentary) depending on their encounter with other individuals or with resource items. I’ll also show how these rather complex movement strategies ultimately generate a fairly simple Gaussian spatial distribution of host parasitism around the release point.

Since the pioneering work of the Hubel and Wiesel, the visual system is mostly conceived as a feed-forward hierarchical flow of sensory information. Accordingly, low-level visual information (such as position and orientation) is extracted locally within stationary receptive fields and is rapidly cascaded to downstream areas to encode more complex features. Such a framework implies that processing at each level of processing must be fast, efficient and mostly confined to network of neurons with overlapping receptive fields. In recent work, however, we have demonstrated that any local stationary stimulus is, in itself, generating waves propagating within each cortical steps of visual processing. Visual information thus does not stay confined to a particular retinotopic location but instead invades neighboring cortical territory, connecting neurons with neighboring receptive fields. What could be the computational advantage of cortical waves in the processing visual information? We have shown that, in response to a non-stationary sequence of visual stimuli, such as an object moving along a trajectory, these waves interact non-linearly with feedforward and feedback streams. They hereby shape the representation of moving stimuli within cortical retinotopic maps to encode accurately the object velocity.

Switch-like and oscillatory dynamical systems are widely observed in biology. We investigate the simplest biological switch that is composed of a single molecule that can be autocatalytically converted between two opposing activity forms. We propose that this single molecule system could work as a primitive biological sensor and show by steady state analysis of a mathematical model of the system that it could switch between possible states for changes in environmental signals. Particularly, we show that a single molecule phosphorylation-dephosphorylation switch could work as a nucleotide or energy sensor. We also notice that a given set of reductions in the reaction network can lead to the emergence of oscillatory behaviour. We propose that evolution could have converted this switch into a single molecule oscillator, which could have been used as a primitive timekeeper. I will discuss how the structure of the simplest known circadian clock regulatory system, found in cyanobacteria, resembles the proposed single molecule oscillator. Besides, I will speculate if such minimal systems could have existed in an RNA world. I will also present how the regulatory network of the cell cycle could have emerged from this system and what are the consequences of this possible evolution from a single antagonistic kinase-phosphatase network.

Most organisms have evolved a circadian timing system to adapt their physiology and behaviour to the daily environmental changes resulting from the rotation of the earth on its axis. This is achieved through a self-sustained oscillatory gene network present in virtually all cells and which temporally coordinates a plethora of molecular, cellular and physiological processes. Interestingly, daily synchronous rhythms of the cell division cycle are observed in many species including humans. This strongly suggests that the circadian clock and the cell cycle machineries are functionally connected. Consistently, several molecular mechanisms underlying this crosstalk have been uncovered during the last 10-15 years. However, despite this mechanistic knowledge, how the temporal organization of cell division at the single cell level produces coherent daily rhythm at the tissue level and how the clock and cell cycle dynamics are coordinated have remained elusive. Using multispectral fluorescent imaging of genetically modified single live cells, computational methods and mathematical modelling we have addressed this issue in mouse fibroblastic cells. This approach revealed that in unsynchronized cells, the cell cycle and circadian clock robustly phase-lock each other in a 1:1 fashion so that in an expanding cell population the two oscillators oscillate in a synchronized way with a common frequency. Further, pharmacological synchronization of cellular clocks reveals additional phase-locked clock states. The temporal coordination of cell division by phase-locking to the clock at a single cell level has significant implications because circadian disruption is increasingly being linked to the pathogenesis of many diseases including metabolic diseases and cancer.

Various propagating waves occur in the brain, at different spatial and temporal scales. We report here on a mixed theoretical and experimental study of propagating waves in visual cortex of the awake monkey. Optical imaging measurements of the primary visual cortex (V1) revealed that every visual stimulus is followed by a propagating waves at sub-millimeter scale and with a propagating velocity of about half a meter per second. When two propagating waves collide, their combined action is largely sublinear, which reveals suppressive effects. Mean-field models can reproduce this nonlinear interaction if inhibitory neurons have a higher gain than excitatory neurons, and if they interact via conductance-based mechanisms. Finally, an external decoder can correctly discern the two stimuli, but only if the propagating waves are suppressive. We conclude that the suppressive nonlinearity of propagating waves enable to disambiguate visual stimuli and thus participate to a finer visual discrimination. Supported by the CNRS, ANR and the EU (Human Brain Project).

The surface of northern and tropical peatland ecosystems frequently exhibits self-organized patterning of densely vegetated hummocks and more sparsely vegetated hollows. Theoretical studies so far suggest multiple alternative mechanisms that could be driving this pattern formation. The long time span associated with peatland surface pattern formation, however, limits possibilities for empirically testing cause-effect relationships through field manipulations. We present a reaction-advection-diffusion model that describes spatial interactions between vegetation, nutrients, hydrology, and peat. Modification of the model’s reaction terms and the hydraulic conductivity function enable the study of pattern formation as driven by three different mechanisms: peat accumulation, water ponding, and nutrient accumulation. By on-and-off switching of each mechanism, we created a full-factorial design to see how these mechanisms affected surface patterning (pattern of vegetation and peat height) and underlying patterns in nutrients and hydrology.

Results revealed that different combinations of structuring mechanisms lead to similar types of peatland surface patterning but contrasting underlying patterns in nutrients and hydrology. These contrasting underlying patterns suggested that the presence or absence of the structuring mechanisms can be identified by relatively simple short-term field measurements of nutrients and hydrology, meaning that longer-term field manipulations could be circumvented. Performing these empirical tests in similarly patterned peatland complexes along a Eurasian climatic gradient, we found that the underlying patterns in nutrients and hydrology reversed along the climatic gradient, corroborating the main prediction of the model framework.

This study follows the Appropriate Complexity Landscape Modelling approach, in that it explores multiple pattern-forming mechanisms in a model environment, and subsequently confront these predictions to empirical data. This approach may not only be useful for northern peatlands but for (sub)tropical peatlands as well. This notion is illustrated with current work in progress, in which we study multiple mechanisms that may drive peatland pattern formation in the Florida Everglades.

Whether a cell will grow and divide is a highly regulated decision that is controlled by a large and complex network of genes and proteins. Our understanding of how these network components collectively work together in space and time is still limited. In our lab, we combine theory of nonlinear dynamics and complex systems with biological experiments in order to gain new insights into cell cycle regulation. Here, I will discuss our work on cell division coordination in frog embryos. Upon fertilization, the early Xenopus leavis frog egg quickly divides about ten times to go from a single cell with a diameter of a millimeter to several thousands of cells of somatic cell size (tens of microns). Using frog cell-free extracts, one can reconstitute in vitro the biochemical reactions that regulate these clock-like cell divisions. On the one hand, such extract experiments allow us to identify how the presence of feedback loops in the molecular network ensures robust cell cycle oscillations. On the other hand, we find that cell division is spatially coordinated via biochemical waves, whose properties depend on the dimensions of the spatial environment. By carrying out experiments in Teflon tubes of varying diameter, we show that mitotic waves are driven by an internal pacemaker in thinner tubes, while they are boundary-driven in thicker tubes. We show how changing the spatial geometry of the system effectively tunes the relative strength of two pacemaker regions, thus reversing the direction of propagation of mitotic waves.

Competition for water or nutrients or interactions with herbivores drive spatial instabilities in landscapes of terrestrial plants, resulting in pattern formation phenomena that have been a subject of intense research in the last years. Observations from aerial images and side -scan sonar data have recently revealed analogous pattern forming phenomena in submerged vegetation in the Mediterranean Sea, mainly in meadows of seagrasses such as Posidonia oceanica and Cymodocea nodosa. Starting from growth rules of these clonal plants, we have derived a macroscopic model for the plant density able to provide an explanation to the observed submarine hexagonal patterns or isolated ‘fairy circles’, and landscapes of spots and stripes. The essential ingredient is a competitive interaction at a distance of 20-30m. Beyond a qualitative description of the observed patterns, and their prevalence under different meadow conditions, the model fits well measurements of the population density of Posidonia, which show great variability close to the coast, where patterns typically appear.

Circadian clocks are endogenous oscillators that drive ~24-hour rhythms in physiology, metabolism and behaviour of almost all life on earth. Circadian clocks are found at all levels - from cells, tissues and organs to the entire organism. In mammals, the master circadian clock resides in the hypothalamic suprachiasmatic nuclei (SCN) and coordinates daily rhythms of sleep and wakefulness, core body temperature and hormone secretion (such as

cortisol, melatonin and many others). It is synchronized to Earth’s rotation primarily by light- dark cycles – a process called `entrainment’, which is crucial for an organisms’ fitness. Little

is known about which oscillator qualities determine entrainment, i.e. entrainment range, phase and amplitude. Using mathematical modelling combined with experimental studies we found that coupling among single cell oscillators governs fundamental properties of circadian clock systems. In addition, we will present our recent development that allows the assessment of the phase of human circadian rhythms by a single time-point measurement using machine-learning algorithms at high dimensional time-series data from human blood cell transcriptomes. Since the internal circadian phase of humans is different for each individual and does not correspond to external clock time, such a precision medicine tool (BodyTime) enabling the personalization of healthcare according to the patient’s circadian clock is urgently needed.

Spreading depolarization (SD) refers to waves of abrupt, sustained mass depolarization in the gray matter of the central nervous system, observed in different pathological conditions. Cortical spreading depression (CSD) is a SD generated in well-nourished and oxygenated tissue, and characterized by transient intense neuronal firing leading to a long lasting depolarizing block of neuronal activity. CSD is a proposed pathological mechanism of migraine. Some molecular/cellular mechanisms of migraine with aura and of CSD have been identified studying a rare genetic form: familial hemiplegic migraine (FHM). FHM type 3 is caused by mutations of the SCN1A gene, leading to gain of function of NaV1.1 sodium channels, which are essential for GABAergic neurons’ excitability. I will present our recent results about mechanisms of induction of CSD caused by gain of function of Nav1.1. Acute activation of Nav1.1 with a selective toxin in brain slices, mimicking the effect of FHM3 mutations, induce SD selectively in the cerebral cortex. We tested the role of GABAergic neurons by activating them with optogenetic techniques. Hyperactivity of interneurons is sufficient to ignite CSD by spiking-induced extracellular K+ build-up in the cerebral cortex, but not in other brain structures. GABAergic and glutamatergic synaptic transmission was not required for CSD initiation, but glutamatergic transmission was implicated in CSD propagation in the cortex. These results reveal the key role of Nav1.1 and GABAergic neurons in a novel mechanism of CSD initiation, which can be relevant for FHM3 and possibly also for other types of migraine.

During early retina development, waves of activity propagate across the retina and play a key role in building the early visual system. In vertebrates species, upon maturation and before eye-opening, transient networks of cells generate these waves, characterized by $3$ consecutive stages. Here, we focus on the biophysical detailed modelling of the second stage (stage II), during which waves are controlled by directly interconnected specific cells, the cholinergic starburst amacrine cells (SACs) which are able to burst autonomously. We propose plausible underlying mechanisms for: i) waves generation at the single neuron level, ii) propagation at the network level in a landscape marked by previous waves prints and iii) waves termination. Based on a bifurcation analysis we show how biophysical parameters control retinal waves characteristics and we provide a theoretical condition for waves propagation and disappearance. Moreover, we show that the continuous decrease of the strength of the acetylcholine synaptic coupling, associated with the crossing of a synchronization transition, impacts dramatically the waves distribution. We report especially on the existence of power law distributions of the avalanche size not only at the synchronization threshold, but also for a whole range of coupling strength. This may play a key role in the ability of the retina to respond to visual stimuli by maximizing its dynamical range.

Chemical and electrical synapses shape the collective dynamics of neuronal networks. Numerous theoretical studies have investigated how, separately, each of these type of synapses contributes to the generation of neuronal oscillations, but their combined effect is less understood. This limitation is further magnified by the impossibility of traditional neuronal mean field models ---often referred to as firing rate models--- to account for electrical synapses. Here we perform a comparative analysis of the dynamics of heterogeneous populations of quadratic integrate-and-fire neurons with chemical, electrical, and both chemical and electrical coupling.

In the thermodynamic limit, we show that the population's mean-field dynamics is exactly described by a system of two ordinary differential equations for the center and the width of the distribution of membrane potentials -or, equivalently, for the population-mean membrane potential and firing rate. These firing rate equations describe, in a unified framework, the collective dynamics of the ensemble of spiking neurons, and reveal that both chemical and electrical coupling are mediated by the population firing rate. Furthermore, while chemical coupling shifts the center of the distribution of membrane potentials, electrical coupling tends to reduce the width of this distribution promoting the emergence of synchronization. The analysis of the firing rate equations allows us to obtain exact formulas for all Saddle-Node and Hopf boundaries, and to construct phase diagrams characterizing the dynamics of the original network of spiking neuron. In networks with instantaneous chemical synapses the phase diagram is characterized by a codimension-two Cusp point, and by the presence of persistent states for strong excitatory coupling. In contrast, the phase diagram for electrically coupled networks is determined by a Takens-Bogdanov codimension-two point, which entails the presence of oscillations and greatly reduces the possibility of persistent states. In this case oscillations arise either via a Saddle-Node-Invariant-Circle bifurcation, or through a supercritical Hopf bifurcation -as shown using weakly nonlinear stability analysis. Finally, we show that the Takens-Bogdanov bifurcation scenario is generically present in networks with both chemical and electrical coupling.

New recording technologies allow neuroscientists to record from cortex with high spatial and temporal resolution. For the first time, we can visualize the complex activity patterns in cortical populations during natural sensory behaviors. Because these imaging experiments occur in intact biological systems, however, certain restrictions are inevitable. In particular, the signal-to-noise ratio (SNR) remains low relative to other scientific imaging domains.

In our research, we have developed new signal processing techniques to analyze nonlinear waves in high-noise multisite data. With these tools, we have found unexpected structure in the dynamics of cortical populations during natural sensory behavior. First, we found that small visual stimuli evoke far-reaching propagating waves in the awake monkey. In recent work, we have found that spontaneous, internally-generated traveling waves modulate sensitivity to visual stimuli in the awake marmoset. These results indicate that traveling waves shape neural computations during normal vision and have more general implications for the way we think about noise in the brain.

The hearing organ of lizards -papilla- has been modelled as a chain of over-damped (inertia-less) bio-mechanical self-oscillators mutually coupled by a combination of viscous and elastic forces. In the extreme case when the elastic ones are negligible the combination of viscous coupling and overdamping leads to the study of unusual class of extended dynamical systems defined by a nonlocal spatial operator. In other words, the lack of inertia in the dynamics of the individual oscillators effectively mutates the original, locally defined coupling into one defined by a global, albeit exponentially weakening, prescription. In this talk we present a number of counterintuitive consequences of this phenomenon on the propagation of perturbations along the media, as well as on the expected synchronization behaviour of the chain. Other characteristics of papillae is tonotopy: the oscillators proper frequencies are arranged in an increasing order along the chain. The combination of different types of couplings and tonotopy, produces characteristic collective frequency spectra that one could associate with distinguishably stable spontaneous otoacoustic emissions observed in individual of certain lizards’ species like tokkai gecko for instance. We explore this phenomenon in simple settings.

Collective properties of oscillators are often analysed by running simulations for increasingly large ensembles of elements. Therefore, analytical approaches/results are definitely welcome as they play the role of references for validating the results on the various scenarios that are otherwise only numerically observed. Here we show that the well known model of mean-field coupled, Stuart-Landau oscillators can be semi- analytically studied at a macroscopic level in an intermediate regime, where the oscillators maintain some typical features of phase-oscillators (remaining aligned along a closed smooth curve), but amplitude oscillations manifest themselves as fluctuations of the curve itself. Our approach allows characterising the collective dynamics for different values of the coupling strengths and in particular to find evidence of self-consistent partial synchrony and an intriguing collective-chaos regime characterised by a small number of positive exponents and a seemingly high-dimensional dynamics.

I will first focus on scaling properties obtained from the analysis of Species Abundance Distributions (SADs) of planktonic organisms. Using the dataset gathered by the Tara Oceans expedition for marine microbial eukaryotes (protists) we explore how SADs of planktonic local communities vary across the global ocean. We find that the decay in abundance of more than the 99% of species is commonly governed by a power-law. Moreover, the power-law exponent varies by less than 10% across locations and does not show biogeographical signatures suggesting that large-scale ubiquitous ecological processes could govern the assembly of such communities.

I will then introduce a Network Theory framework developed for the characterization of fluid transport dynamics in the ocean. The discretization of the sea surface in small equal-sized cells brings to the construction of a new kind of network networks, called Lagrangian Flow Networks (LFNs), that describe water exchanges between different regions of the seascape. Using Network Theory concepts & tools we can study dispersion and mixing at both local and global scales evidencing relationships between network measures and dynamical properties of the flow. Among possible applications, such a framework provides a systematic characterization of the dispersal of planktonic life-stages of marine organisms which helps to understand the connectivity and structural complexity of marine populations.

I will finally discuss possible perspectives to investigate the effects of ocean transport and mixing on planktonic community assembly in the Mediterranean using the LFN methodology.

Spatial vegetation patterns with different morphologies (gaps, stripes/labyrinths, spots) have been observed in many drylands worldwide. These patterns are thought to be caused by a water flux from bare to vegetated areas.

Reaction(-advection)-diffusion models can help explain why these spatial patterns form. But how does the pattern morphology depend on the choice of model? And what does this imply for real ecosystems?

Intracranial electroencephalography is a standard tool in clinical evaluation of patients with focal epilepsy. Various early electrographic seizure patterns differing in frequency, amplitude, and waveform of the oscillations are observed in intracranial recordings. The pattern most common in the areas of seizure propagation is the so-called theta-alpha activity (TAA), whose defining features are oscillations in the theta-alpha range and gradually increasing amplitude. A deeper understanding of the mechanism underlying the generation of the TAA pattern is however lacking. We show by means of numerical simulation that the features of the TAA pattern observed on an implanted depth electrode in a specific epileptic patient can be plausibly explained by the seizure propagation across an individual folded cortical surface. In order to demonstrate this, we employ following pipeline: First, the structural model of the brain is reconstructed from the T1-weighted images, and the position of the electrode contact are determined using the CT scan with implanted electrodes. Next, the patch of cortical surface in the vicinity of the electrode of interest is extracted. On this surface, the simulation of the seizure spread is performed using The Virtual Brain framework. As a mathematical model the Epileptor model in its field formulation is employed. The simulated source activity is then projected to the sensors using the dipole model, and this simulated stereo-electroencephalograpic (SEEG) signal is compared with the recorded one. The results show that the simulation on the patient-specific cortical surface gives a better fit between the recorded and simulated signals than the simulation on generic surrogate surfaces. Furthermore, the results indicate that the spectral content and dynamical features might differ in the source space of the cortical gray matter activity and among the intracranial sensors, questioning the previous approaches to classification of seizure onset patterns done in the sensor space, both based on spectral content and on dynamical features. In conclusion, we demonstrate that the investigation of the seizure dynamics on the level of cortical surface can provide deeper insight into the large scale spatiotemporal organization of the seizure. At the same time it highlights the need for a robust techniques for inversion of the observed activity from sensor to source space that would take into account the complex geometry of the cortical sources and the position of the intracranial sensors.

We address the question of color-space interactions in the brain by proposing a neural field model of color perception with spatial context, for the visual area V1 of the cortex. Our framework reconciles two opposing perceptual phenomena, known as simultaneous contrast and chromatic assimilation. They have been previously shown to act synergistically, so that at some point in an image, the color seems perceptually more similar to that of the adjacent neighbors, while being more dissimilar from that of remote ones. Thus their combined effects are enhanced in the presence of a spatial pattern, and can be measured as larger shifts in color matching experiments. Our model supposes a hypercolumnar structure coding for colors in V1, and relies on the notion of color opponency introduced by Hering. The connectivity kernel of the neural field exploits the balance between attraction and repulsion in color and physical spaces, so as to reproduce the sign reversal in the influence of neighboring points. The color sensation at a point, defined from a steady state of the neural activities, is then extracted as a nonlinear percept conveyed by an assembly of neurons. It connects the cortical and perceptual levels, because we describe the search for a color match in asymmetric matching experiments as a mathematical projection of color sensations. We validate our color neural field alongside this color matching framework, by performing a multi-parameter regression to psychophysical data produced by Monnier & Shevell (2004, 2008), and ourselves. All the results show that we are able to explain the nonlinear behavior of shifts along one or two dimensions in color space, which cannot be done using a simple linear model.

The retina is able to perform complex tasks and general feature extraction, allowing the visual cortex to process visual stimuli with more efficiency. With regards to motion processing, an interesting and useful task performed by the retina is anticipation and trajectory extrapolation. Anticipation in the retina lies in the fact that the peak of retinal ganglion cells response is shifted, occurring before the object reaches the center of the receptive field, and can be explained by gain control mechanisms occurring at the level of bipolar and ganglion cells. Trajectory extrapolation on the other hand is related to a rise in the activity before the object enters the receptive field of the cell and is carried out through electrical synapses (gap junctions) connecting ganglion cells. This extrapolation has also been observed at the level of the primary visual cortex, where lateral propagation drives the activity ahead of the input, denoting predictive computations. Motion encoding in the retina also involves amacrine cells, which connect bipolar cells to either bipolar or ganglion cells, but their role has not been investigated yet in motion anticipation.

The first contribution of our work lies in the development of a generalized 2D model of the retina with three layers of ganglion cells : Fast OFF cells with gain control accounting for anticipation, direction selective cells connected via gap junctions, and Y-cells connected through amacrine cells, accounting for motion extrapolation.This model affords a mathematical analysis via dynamical systems theory and allows to outline the role of lateral connectivity (gap junctions and amacrine cells) in motion perception, anticipation and trajectory extrapolation. The second contribution is the use of the output of our retina model as an input to a mean field model of the primary visual cortex to reproduce motion anticipation as observed in VSDI recordings of V1. We present results of the integrated retino-cortical model for motion processing, and study how anticipation and extrapolation depend on stimuli parameters such as speed, shape and trajectory. Through the integrated retina-cortical model we emphasize the mechanisms defining motion anticipation, due to the cooperation of gain control and lateral connectivity at the level of the retina and lateral connectivity in the cortex. Moreover, we show how cortical nonlinearities due to a different gain between excitatory and inhibitory neurons shape the cortical response thus affecting object recognition.

Previous empirical work has hypothesised that tropical forest and savanna are two alternative stable states as a result of fire-vegetation feedbacks. The hysteresis associated with such dynamic implies that when an area of tropical forest is exposed to shocks such as deforestation or drought, it can remain locked into a savanna state unless it experiences large increases in rainfall. In my PhD, I have provided empirical and theoretical evidence that instead of two alternative stable states and hysteresis, there is only a predictable front, occurring at a single tipping point, the Maxwell point. This becomes clear after spatial heterogeneity and spatial interaction are taken into account.

In the presentation, I will start with some background on tropical tree cover bistability. Then, I use a simple reaction-diffusion equation with bistable reaction term to explain travelling wave fronts under homogeneous forcing and front pinning under heterogeneous forcing. After showing how the pinning location can be derived from data, I will briefly show the data analysis results. I will then finally introduce and analyse the reaction-diffusion model of Amazonian tree cover. It will become clear towards the end that spatial heterogeneity can lead to the false impression of bistability and hysteresis when in fact there is only a front.