Tobias Reichenbach

Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games

Biodiversity is essential to the viability of ecological systems. Species diversity in ecosystems is promoted by cyclic, non-hierarchical interactions among competing populations. Central features of such non-transitive relations are represented by the ‘rock–paper–scissors’ game, in which rock crushes scissors, scissors cut paper, and paper wraps rock. In combination with spatial dispersal of static populations, this type of competition results in the stable coexistence of all species and the long-term maintenance of biodiversity. However, population mobility is a central feature of real ecosystems: animals migrate, bacteria run and tumble. Here, we observe a critical influence of mobility on species diversity. When mobility exceeds a certain value, biodiversity is jeopardized and lost. In contrast, below this critical threshold all subpopulations coexist and an entanglement of travelling spiral waves forms in the course of time. We establish that this phenomenon is robust; it does not depend on the details of cyclic competition or spatial environment. These findings have important implications for maintenance and temporal development of ecological systems and are relevant for the formation and propagation of patterns in microbial populations or excitable media.

Noise and correlations in a spatial population model with cyclic competition.

Noise and spatial degrees of freedom characterize most ecosystems. Some aspects of their influence on the coevolution of populations with cyclic interspecies competition have been demonstrated in recent experiments [e.g., B. Kerr, Nature (London) 418, 171 (2002)10.1038/nature00823]. To reach a better theoretical understanding of these phenomena, we consider a paradigmatic spatial model where three species exhibit cyclic dominance. Using an individual-based description, as well as stochastic partial differential and deterministic reaction-diffusion equations, we account for stochastic fluctuations and spatial diffusion at different levels and show how fascinating patterns of entangled spirals emerge. We rationalize our analysis by computing the spatiotemporal correlation functions and provide analytical expressions for the front velocity and the wavelength of the propagating spiral waves.

The physics of hearing: fluid mechanics and the active process of the inner ear

Most sounds of interest consist of complex, time-dependent admixtures of tones of diverse frequencies and variable amplitudes. To detect and process these signals, the ear employs a highly nonlinear, adaptive, real-time spectral analyzer: the cochlea. Sound excites vibration of the eardrum and the three miniscule bones of the middle ear, the last of which acts as a piston to initiate oscillatory pressure changes within the liquid-filled chambers of the cochlea. The basilar membrane, an elastic band spiraling along the cochlea between two of these chambers, responds to these pressures by conducting a largely independent traveling wave for each frequency component of the input. Because the basilar membrane is graded in mass and stiffness along its length, however, each traveling wave grows in magnitude and decreases in wavelength until it peaks at a specific, frequency-dependent position: low frequencies propagate to the cochlear apex, whereas high frequencies culminate at the base. The oscillations of the basilar membrane deflect hair bundles, the mechanically sensitive organelles of the ears sensory receptors, the hair cells. As mechanically sensitive ion channels open and close, each hair cell responds with an electrical signal that is chemically transmitted to an afferent nerve fiber and thence into the brain. In addition to transducing mechanical inputs, hair cells amplify them by two means. Channel gating endows a hair bundle with negative stiffness, an instability that interacts with the motor protein myosin-1c to produce a mechanical amplifier and oscillator. Acting through the piezoelectric membrane protein prestin, electrical responses also cause outer hair cells to elongate and shorten, thus pumping energy into the basilar membranes movements. The two forms of motility constitute an active process that amplifies mechanical inputs, sharpens frequency discrimination, and confers a compressive nonlinearity on responsiveness. These features arise because the active process operates near a Hopf bifurcation, the generic properties of which explain several key features of hearing. Moreover, when the gain of the active process rises sufficiently in ultraquiet circumstances, the system traverses the bifurcation and even a normal ear actually emits sound. The remarkable properties of hearing thus stem from the propagation of traveling waves on a nonlinear and excitable medium.

Exclusion processes with internal states

We introduce driven exclusion processes with internal states that serve as generic transport models in various contexts, ranging from molecular or vehicular traffic on parallel lanes to spintronics. The ensuing nonequilibrium steady states are controllable by boundary as well as bulk rates. A striking polarization phenomenon accompanied by domain wall motion and delocalization is discovered within a mesoscopic scaling. We quantify this observation within an analytic description providing exact phase diagrams. Our results are confirmed by stochastic simulations.

The Spatial Pattern of Cochlear Amplification

Sensorineural hearing loss, which stems primarily from the failure of mechanosensory hair cells, changes the traveling waves that transmit acoustic signals along the cochlea. However, the connection between cochlear mechanics and the amplificatory function of hair cells remains unclear. Using an optical technique that permits the targeted inactivation of prestin, a protein of outer hair cells that generates forces on the basilar membrane, we demonstrate that these forces interact locally with cochlear traveling waves to achieve enormous mechanical amplification. By perturbing amplification in narrow segments of the basilar membrane, we further show that a cochlear traveling wave accumulates gain as it approaches its peak. Analysis of these results indicates that cochlear amplification produces negative damping that counters the viscous drag impeding traveling waves; targeted photoinactivation locally interrupts this compensation. These results reveal the locus of amplification in cochlear traveling waves and connect the characteristics of normal hearing to molecular forces.

A ratchet mechanism for amplification in low-frequency mammalian hearing

The sensitivity and frequency selectivity of hearing result from tuned amplification by an active process in the mechanoreceptive hair cells. In most vertebrates, the active process stems from the active motility of hair bundles. The mammalian cochlea exhibits an additional form of mechanical activity termed electromotility: its outer hair cells (OHCs) change length upon electrical stimulation. The relative contributions of these two mechanisms to the active process in the mammalian inner ear is the subject of intense current debate. Here, we show that active hair-bundle motility and electromotility can together implement an efficient mechanism for amplification that functions like a ratchet: Sound-evoked forces, acting on the basilar membrane, are transmitted to the hair bundles, whereas electromotility decouples active hair-bundle forces from the basilar membrane. This unidirectional coupling can extend the hearing range well below the resonant frequency of the basilar membrane. It thereby provides a concept for low-frequency hearing that accounts for a variety of unexplained experimental observations from the cochlear apex, including the shape and phase behavior of apical tuning curves, their lack of significant nonlinearities, and the shape changes of threshold tuning curves of auditory-nerve fibers along the cochlea. The ratchet mechanism constitutes a general design principle for implementing mechanical amplification in engineering applications.

Waves on Reissner's Membrane: A Mechanism for the Propagation of Otoacoustic Emissions from the Cochlea

Sound is detected and converted into electrical signals within the ear. The cochlea not only acts as a passive detector of sound, however, but can also produce tones itself. These otoacoustic emissions are a striking manifestation of the cochleas mechanical active process. A controversy remains of how these mechanical signals propagate back to the middle ear, from which they are emitted as sound. Here, we combine theoretical and experimental studies to show that mechanical signals can be transmitted by waves on Reissners membrane, an elastic structure within the cochlea. We develop a theory for wave propagation on Reissners membrane and its role in otoacoustic emissions. Employing a scanning laser interferometer, we measure traveling waves on Reissners membrane in the gerbil, guinea pig, and chinchilla. The results are in accord with the theory and thus support a role for Reissners membrane in otoacoustic emissions.

Traffic jams induced by rare switching events in two-lane transport

We investigate a model for driven exclusion processes where internal states are assigned to the particles. The latter account for diverse situations, ranging from spin states in spintronics to parallel lanes in intracellular or vehicular traffic. Introducing a coupling between the internal states by allowing particles to switch from one to another induces an intriguing polarization phenomenon. In a mesoscopic scaling, a rich stationary regime for the density profiles is discovered, with localized domain walls in the density profile of one of the internal states being feasible. We derive the shape of the density profiles as well as resulting phase diagrams analytically by a mean-field approximation and a continuum limit. Continuous as well as discontinuous lines of phase transition emerge, their intersections induce multi-critical behaviour.

Entropy Production of Cyclic Population Dynamics

Entropy serves as a central observable in equilibrium thermodynamics. However, many biological and ecological systems operate far from thermal equilibrium. Here we show that entropy production can characterize the behavior of such nonequilibrium systems. To this end we calculate the entropy production for a population model that displays nonequilibrium behavior resulting from cyclic competition. At a critical point the dynamics exhibits a transition from large, limit-cycle-like oscillations to small, erratic oscillations. We show that the entropy production peaks very close to the critical point and tends to zero upon deviating from it. We further provide analytical methods for computing the entropy production which agree excellently with numerical simulations.

Contribution of active hair-bundle motility to nonlinear amplification in the mammalian cochlea

The cochlea’s high sensitivity stems from the active process of outer hair cells, which possess two force-generating mechanisms: active hair-bundle motility elicited by Ca2+ influx and somatic motility mediated by the voltage-sensitive protein prestin. Although interference with prestin has demonstrated a role for somatic motility in the active process, it remains unclear whether hair-bundle motility contributes in vivo. We selectively perturbed the two mechanisms by infusing substances into the endolymph or perilymph of the chinchilla’s cochlea and then used scanning laser interferometry to measure vibrations of the basilar membrane. Blocking somatic motility, damaging the tip links of hair bundles, or depolarizing hair cells eliminated amplification. While reducing amplification to a lesser degree, pharmacological perturbation of active hair-bundle motility diminished or eliminated the nonlinear compression underlying the broad dynamic range associated with normal hearing. The results suggest that active hair-bundle motility plays a significant role in the amplification and compressive nonlinearity of the cochlea.