Mathematization of nature: how it is done.

Natural phenomena can be quantitatively described by means of mathematics, which is actually the only way of doing so. Physics is a convincing example of the mathematization of nature. This paper gives an answer to the question of how mathematization of nature is done and illustrates the answer. Here nature is to be taken in a wide sense, being a substantial object of study in, among others, large domains of biology, such as epidemiology and neurobiology, chemistry, and physics, the most outspoken example. It is argued that mathematization of natural phenomena needs appropriate core concepts that are intimately connected with the phenomena one wants to describe and explain mathematically. Second, there is a scale on and not beyond which a specific description holds. Different scales allow for different conceptual and mathematical descriptions. This is the scaling hypothesis, which has meanwhile been confirmed on many occasions. Furthermore, a mathematical description can, as in physics, but need not be universally valid, as in biology. Finally, the history of science shows that only an intensive gauging of theory, i.e., mathematical description, by experiment leads to progress. That is, appropriate core concepts and appropriate scales are a necessary condition for mathematizing nature, and so is its verification by experiment.

Active tympanic tuning facilitates sound localization in animals with internally coupled ears.

Earlier studies have reported that numerous vertebrate taxa have skeletal muscle(s) attaching directly, or indirectly, onto the tympanic membrane. The present study links these prior studies by quantitatively modeling the influence of skeletal muscle contraction on tympanic tension, tympanic dampening, and, ultimately, the fundamental frequency. In this way, the efficacy of these tympanic muscles to dynamically alter the sensory response of the vertebrate ear is quantified. Changing the tension modifies the eardrum's fundamental frequency, a key notion in understanding hearing through internally coupled ears (ICE) as used by the majority of terrestrial vertebrates. Tympanic tension can also be modulated by altering the pressure acting on the deep (medial) surface of the tympanum. Herein we use the monitor lizard Varanus as an example to demonstrate how active modulation of the pharyngeal volume permits tuning of an ICE auditory system. The present contribution offers a behaviorally and biologically realistic perspective on the ICE system, by demonstrating how an organism can dynamically alter its morphology to tune the auditory response. Through quantification of the relationships between tympanic surface tension, damping, membrane fundamental frequency, and auditory cavity volume, it can be shown that an ICE system affords a biologically relevant range of tuning.

Modeling underwater hearing and sound localization in the frog Xenopus laevis.

Animals that are small compared to sound wavelengths face the challenge of localizing a sound source since the main cues to sound direction-interaural time differences (ITD) and interaural level differences (ILD)-both depend on size. Remarkably, the majority of terrestrial vertebrates possess internally coupled ears (ICE) with an air-filled cavity connecting the two eardrums and producing an inherently directional middle-ear system. Underwater, longer wavelengths and faster sound-speed reduce both ITD and ILD cues. Nonetheless, many animals communicate through and localize underwater sound. Here, a typical representative equipped with ICE is studied: the fully aquatic clawed frog Xenopus laevis. It is shown that two factors improve underwater sound-localization quality. First, inflated lungs function as Helmholtz resonator and generate directional amplitude differences between eardrum vibrations in the high-frequency (1.7-2.2 kHz) and low-frequency (0.8-1.2 kHz) range of the male advertisement calls. Though the externally arriving ILDs practically vanish, the perceived internal level differences are appreciable, more than 10 dB. As opposed to, e.g., lizards with thin and flexible eardrums, plate-like eardrums are shown to be Xenopus' second key to successfully handling aquatic surroundings. Based on ICE, both plate-like eardrums and inflated lungs functioning as Helmholtz resonators explain the phonotaxis performance of Xenopus.

Animals and ICE: meaning, origin, and diversity.

ICE stands for internally coupled ears. More than half of the terrestrial vertebrates, such as frogs, lizards, and birds, as well as many insects, are equipped with ICE that utilize an air-filled cavity connecting the two eardrums. Its effect is pronounced and twofold. On the basis of a solid experimental and mathematical foundation, it is known that there is a low-frequency regime where the internal time difference (iTD) as perceived by the animal may well be 2-5 times higher than the external ITD, the interaural time difference, and that there is a frequency plateau over which the fraction iTD/ITD is constant. There is also a high-frequency regime where the internal level (amplitude) difference iLD as perceived by the animal is much higher than the interaural level difference ILD measured externally between the two ears. The fundamental tympanic frequency segregates the two regimes. The present special issue devoted to "internally coupled ears" provides an overview of many aspects of ICE, be they acoustic, anatomical, auditory, mathematical, or neurobiological. A focus is on the hotly debated topic of what aspects of ICE animals actually exploit neuronally to localize a sound source.

Internally coupled ears: mathematical structures and mechanisms underlying ICE.

In internally coupled ears (ICE), the displacement of one eardrum creates pressure waves that propagate through air-filled passages in the skull, causing a displacement of the opposing eardrum and vice versa. In this review, a thorough mathematical analysis of the membranes, passages, and propagating pressure waves reveals how internally coupled ears generate unique amplitude and temporal cues for sound localization. The magnitudes of both of these cues are directionally dependent. On the basis of the geometry of the interaural cavity and the elastic properties of the two eardrums confining it at both ends, the present paper reviews the mathematical theory underlying hearing through ICE and derives analytical expressions for eardrum vibrations as well as the pressures inside the internal passages, which ultimately lead to the emergence of highly directional hearing cues. The derived expressions enable one to explicitly see the influence of different parts of the system, e.g., the interaural cavity and the eardrum, on the internal coupling, and the frequency dependence of the coupling. The tympanic fundamental frequency segregates a low-frequency regime with constant time-difference magnification (time dilation factor) from a high-frequency domain with considerable amplitude magnification. By exploiting the physical properties of the coupling, we describe a concrete method to numerically estimate the eardrum's fundamental frequency and damping solely through measurements taken from a live animal.

Neuroscience from a mathematical perspective: key concepts, scales and scaling hypothesis, universality.

This article analyzes the question of whether neuroscience allows for mathematical descriptions and whether an interaction between experimental and theoretical neuroscience can be expected to benefit both of them. It is argued that a mathematization of natural phenomena never happens by itself. First, appropriate key concepts must be found that are intimately connected with the phenomena one wishes to describe and explain mathematically. Second, the scale on, and not beyond, which a specific description can hold must be specified. Different scales allow for different conceptual and mathematical descriptions. This is the scaling hypothesis. Third, can a mathematical description be universally valid and, if so, how? Here we put forth the argument that universals also exist in theoretical neuroscience, that evolution proves the rule, and that theoretical neuroscience is a domain with still lots of space for new developments initiated by an intensive interaction with experiment. Finally, major insight is provided by a careful analysis of the way in which particular brain structures respond to perceptual input and in so doing induce action in an animal's surroundings.

Supervised spike-timing-dependent plasticity: a spatiotemporal neuronal learning rule for function approximation and decisions.

How can an animal learn from experience? How can it train sensors, such as the auditory or tactile system, based on other sensory input such as the visual system? Supervised spike-timing-dependent plasticity (supervised STDP) is a possible answer. Supervised STDP trains one modality using input from another one as "supervisor." Quite complex time-dependent relationships between the senses can be learned. Here we prove that under very general conditions, supervised STDP converges to a stable configuration of synaptic weights leading to a reconstruction of primary sensory input.

Vector strength after Goldberg, Brown, and von Mises: biological and mathematical perspectives.

The vector strength, a number between 0 and 1, is a classical notion in biology. It was first used in neurobiology by Goldberg and Brown (J Neurophys 31:639-656, 1969) but dates back at least to von Mises (Phys Z 19:490-500, 1918). It is widely used as a means to measure the periodicity or lack of periodicity of a neuronal response to an outside periodic signal. Here, we provide a self-contained and simple treatment of a closely related notion, the synchrony vector, a complex number with the vector strength as its absolute value and with a definite phase that one can directly relate to a biophysical delay. The present analysis is essentially geometrical and based on convexity. As such it does two things. First, it maps a sequence of points, events such as spike times on the time axis, onto the unit circle in the complex plane so that for a perfectly periodic repetition, a single point on the unit circle appears. Second, events hardly ever occur periodically, so that we need a criterion of how to extract periodicity out of a set of real numbers. It is here where convex geometry comes in, and a geometrically intuitive picture results. We also quantify how the events cluster around a period as the vector strength goes to 1. A typical example from the auditory system is used to illustrate the general considerations. Furthermore, von Mises' seminal contribution to the notion of vector strength is explained in detail. Finally, we generalize the synchrony vector to a function of angular frequency, not fixed on the input frequency at hand and indicate its potential as a "resonating" vector strength.

Resonating vector strength: what happens when we vary the "probing" frequency while keeping the spike times fixed.

The synchrony vector, whose length stands for the vector strength (VS), is a means to quantify the amount of periodicity in a neuronal response to a given periodic signal, say, the stimulus. One usually chooses the input angular frequency and evaluates the synchrony vector as a weighted sum of exponentials taken at given experimental spike times of the neuronal response in combination with the driving input frequency. Given the experimental spike times, we replace the stimulus frequency by a variable probing frequency, study the synchrony vector in dependence upon this probing frequency, i.e., as a function of the frequency as a real variable, and exhibit both mathematically and experimentally a resonance behavior once the variable frequency is in the neighborhood of the stimulus frequency. Furthermore, a "resonating" VS is shown to be quite useful since one need not know the external frequency but can simply stick to the given spike times and analyze the ensuing resonance as the frequency varies, for example, to determine at the same time a "best" frequency and the corresponding VS. Finally, it is straightforward to determine the corresponding phase originating from, say, a delay as well.

Frequency selectivity emerging from spike-timing-dependent plasticity.

Periodic neuronal activity has been observed in various areas of the brain, from lower sensory to higher cortical levels. Specific frequency components contained in this periodic activity can be identified by a neuronal circuit that behaves as a bandpass filter with given preferred frequency, or best modulation frequency (BMF). For BMFs typically ranging from 10 to 200 Hz, a plausible and minimal configuration consists of a single neuron with adjusted excitatory and inhibitory synaptic connections. The emergence, however, of such a neuronal circuitry is still unclear. In this letter, we demonstrate how spike-timing-dependent plasticity (STDP) can give rise to frequency-dependent learning, thus leading to an input selectivity that enables frequency identification. We use an in-depth mathematical analysis of the learning dynamics in a population of plastic inhibitory connections. These provide inhomogeneous postsynaptic responses that depend on their dendritic location. We find that synaptic delays play a crucial role in organizing the weight specialization induced by STDP. Under suitable conditions on the synaptic delays and postsynaptic potentials (PSPs), the BMF of a neuron after learning can match the training frequency. In particular, proximal (distal) synapses with shorter (longer) dendritic delay and somatically measured PSP time constants respond better to higher (lower) frequencies. As a result, the neuron will respond maximally to any stimulating frequency (in a given range) with which it has been trained in an unsupervised manner. The model predicts that synapses responding to a given BMF form clusters on dendritic branches.

Temporal precision and reliability in the velocity regime of a hair-cell sensory system: the mechanosensory lateral line of goldfish, Carassius auratus.

Fish and aquatic frogs detect minute water motion by means of a specialized mechanosensory system, the lateral line. Ubiquitous in fish, the lateral-line system is characterized by hair-cell based sensory structures across the fish's surface called neuromasts. These neuromasts occur free-standing on the skin as superficial neuromasts (SN) or are recessed into canals as canal neuromasts. SNs respond to rapid changes of water velocity in a small layer of fluid around the fish, including the so-called boundary layer. Although omnipresent, the boundary layer's impact on the SN response is still a matter of debate. For the first time using an information-theoretic approach to this sensory system, we have investigated the SN afferents encoding capabilities. Combining covariance analysis, phase analysis, and modeling of recorded neuronal responses of primary lateral line afferents, we show that encoding by the SNs is adequately described as a linear, velocity-responsive mechanism. Afferent responses display a bimodal distribution of opposite Wiener kernels that likely reflected the two hair-cell populations within a given neuromast. Using frozen noise stimuli, we further demonstrate that SN afferents respond in an extremely precise manner and with high reproducibility across a broad frequency band (10-150 Hz), revealing that an optimal decoder would need to rely extensively on a temporal code. This was further substantiated by means of signal reconstruction of spike trains that were time shifted with respect to their original. On average, a time shift of 3.5 ms was enough to diminish the encoding capabilities of primary afferents by 70%. Our results further demonstrate that the SNs' encoding capability is linearly related to the stimulus outside the boundary layer, and that the boundary layer can, therefore, be neglected while interpreting lateral line response of SN afferents to hydrodynamic stimuli.

Tears of venom: hydrodynamics of reptilian envenomation.

In the majority of venomous snakes, and in many other reptiles, venom is conveyed from the animal's gland to the prey's tissue through an open groove on the surface of the teeth and not through a tubular fang. Here we focus on two key aspects of the grooved delivery system: the hydrodynamics of venom as it interacts with the groove geometry, and the efficiency of the tooth-groove-venom complex as the tooth penetrates the prey's tissue. We show that the surface tension of the venom is the driving force underlying the envenomation dynamics. In so doing, we explain not only the efficacy of the open groove, but also the prevalence of this mechanism among reptiles.

Testing resonating vector strength: auditory system, electric fish, and noise.

Quite often a response to some input with a specific frequency ν(○) can be described through a sequence of discrete events. Here, we study the synchrony vector, whose length stands for the vector strength, and in doing so focus on neuronal response in terms of spike times. The latter are supposed to be given by experiment. Instead of singling out the stimulus frequency ν(○) we study the synchrony vector as a function of the real frequency variable ν. Its length turns out to be a resonating vector strength in that it shows clear maxima in the neighborhood of ν(○) and multiples thereof, hence, allowing an easy way of determining response frequencies. We study this "resonating" vector strength for two concrete but rather different cases, viz., a specific midbrain neuron in the auditory system of cat and a primary detector neuron belonging to the electric sense of the wave-type electric fish Apteronotus leptorhynchus. We show that the resonating vector strength always performs a clear resonance correlated with the phase locking that it quantifies. We analyze the influence of noise and demonstrate how well the resonance associated with maximal vector strength indicates the dominant stimulus frequency. Furthermore, we exhibit how one can obtain a specific phase associated with, for instance, a delay in auditory analysis.

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks V: self-organization schemes and weight dependence.

Spike-timing-dependent plasticity (STDP) determines the evolution of the synaptic weights according to their pre- and post-synaptic activity, which in turn changes the neuronal activity on a (much) slower time scale. This paper examines the effect of STDP in a recurrently connected network stimulated by external pools of input spike trains, where both input and recurrent synapses are plastic. Our previously developed theoretical framework is extended to incorporate weight-dependent STDP and dendritic delays. The weight dynamics is determined by an interplay between the neuronal activation mechanisms, the input spike-time correlations, and the learning parameters. For the case of two external input pools, the resulting learning scheme can exhibit a symmetry breaking of the input connections such that two neuronal groups emerge, each specialized to one input pool only. In addition, we show how the recurrent connections within each neuronal group can be strengthened by STDP at the expense of those between the two groups. This neuronal self-organization can be seen as a basic dynamical ingredient for the emergence of neuronal maps induced by activity-dependent plasticity.

Representation of input structure in synaptic weights by spike-timing-dependent plasticity.

Spike-timing-dependent plasticity (STDP) has been shown to generate a synaptic weight structure that is determined by the timing of the pre- and postsynaptic spikes at the synapse. In this paper it is shown under what conditions a neuron stimulated by several pools of delta-correlated inputs encodes this input structure in its resulting weight structure. The analysis is carried out using Poisson neurons with weight-dependent STDP. The learning dynamics induced by STDP leads to both stabilization of the input weights and competition between the weights for a broad range of learning parameters. The results demonstrate how weight-dependent STDP can generate multimodal stable asymptotic distributions of the synaptic weights.

Analytical model of internally coupled ears.

Lizards and many birds possess a specialized hearing mechanism: internally coupled ears where the tympanic membranes connect through a large mouth cavity so that the vibrations of the tympanic membranes influence each other. This coupling enhances the phase differences and creates amplitude differences in the tympanic membrane vibrations. Both cues show strong directionality. The work presented herein sets out the derivation of a three dimensional analytical model of internally coupled ears that allows for calculation of a complete vibration profile of the membranes. The analytical model additionally provides the opportunity to incorporate the effect of the asymmetrically attached columella, which leads to the activation of higher membrane vibration modes. Incorporating this effect, the analytical model can explain measurements taken from the tympanic membrane of a living lizard, for example, data demonstrating an asymmetrical spatial pattern of membrane vibration. As the analytical calculations show, the internally coupled ears increase the directional response, appearing in large directional internal amplitude differences (iAD) and in large internal time differences (iTD). Numerical simulations of the eigenfunctions in an exemplary, realistically reconstructed mouth cavity further estimate the effects of its complex geometry.