On the laws of psychophysics

Experiments by S. S. Stevens (Stevens, 1957, and Stevens and Galanter, 1957) and his collaborators indicate that the so-called logarithmic Weber-Fechner Law is not realized in most human perceptions. Instead, a power law seems to emerge over a large number of sensory continua. This is important because for a long time the logarithmic law was looked upon as almost the only possible psychophysical law. The logarithmic law appeared desirable intuitively because it made the sensation depend on the relative values of the stimuli and not on their absolute values. This is, of course, useful for evolutionary reasons. Some other reasons are also discussed by Stevens (1961).     

A teleological explanation of Weber's law and the motor unit size law

It is shown that the Weber-Fechner law. which relates the response of a sensory biosystem to the intensity of the input stimulus, can be derived from a teleological principle of minimum transentropy (maximal noise reduction) provided the relative mean fluctuation (coefficient of variation) of the input intensity can be assumed to be (approximately) constant for all feasible mean input intensities. A law is then deduced from experimental results which quantifies the relationship existing between the relative amount of activated muscle mass and the “size” (which term is clearly defined) of a newly recruited motor unit. This law is found to be formally equivalent to the Weber-Fechner law when applied to motor unit recruitment. It is then shown that, in general, the ratio of the force increment upon recruitment, to the present force output does not obey Weber's law. Finally, it is proved that the “motor unit size law” as derived in this paper implies a fixed sequential order in the recruitment of motor units and that it may be viewed as the realization, by the mammalian neuromuscular system, of a general principle of maximum grading sensitivity.     

Cellular Sensory Mechanisms for Detecting Specific Fold-Changes in Extracellular Cues

Cellular sensory systems often respond not to the absolute levels of inputs but to the fold-changes in inputs. Such a property is called fold-change detection (FCD) and is important for accurately sensing dynamic changes in environmental signals in the presence of fluctuations in their absolute levels. Previous studies defined FCD as input-scale invariance and proposed several biochemical models that achieve such a condition. Here, we prove that the previous FCD models can be approximated by a log-differentiator. Although the log-differentiator satisfies the input-scale invariance requirement, its response amplitude and response duration strongly depend on the input timescale. This creates limitations in the specificity and repeatability of detecting fold-changes in inputs. Nevertheless, FCD with specificity and repeatability by cells has been reported in the context of Drosophila wing development. Motivated by this fact and by extending previous FCD models, we here propose two possible mechanisms to achieve FCD with specificity and repeatability. One is the integrate-and-fire type: a system integrates the rate of temporal change in input and makes a response when the integrated value reaches a constant threshold, and this is followed by the reset of the integrated value. The other is the dynamic threshold type: a system response occurs when the input level reaches a threshold, whose value is multiplied by a certain constant after each response. These two mechanisms can be implemented biochemically by appropriately combining feed-forward and feedback loops. The main difference between the two models is their memory of input history; we discuss possible ways to distinguish between the two models experimentally.     

Cellular Sensory Mechanisms for Detecting Specific Fold-Changes in Extracellular Cues

Cellular sensory systems often respond not to the absolute levels of inputs but to the fold-changes in inputs. Such a property is called fold-change detection (FCD) and is important for accurately sensing dynamic changes in environmental signals in the presence of fluctuations in their absolute levels. Previous studies defined FCD as input-scale invariance and proposed several biochemical models that achieve such a condition. Here, we prove that the previous FCD models can be approximated by a log-differentiator. Although the log-differentiator satisfies the input-scale invariance requirement, its response amplitude and response duration strongly depend on the input timescale. This creates limitations in the specificity and repeatability of detecting fold-changes in inputs. Nevertheless, FCD with specificity and repeatability by cells has been reported in the context of Drosophila wing development. Motivated by this fact and by extending previous FCD models, we here propose two possible mechanisms to achieve FCD with specificity and repeatability. One is the integrate-and-fire type: a system integrates the rate of temporal change in input and makes a response when the integrated value reaches a constant threshold, and this is followed by the reset of the integrated value. The other is the dynamic threshold type: a system response occurs when the input level reaches a threshold, whose value is multiplied by a certain constant after each response. These two mechanisms can be implemented biochemically by appropriately combining feed-forward and feedback loops. The main difference between the two models is their memory of input history; we discuss possible ways to distinguish between the two models experimentally.     

Neural circuit flexibility in a small sensorimotor system

Neuronal circuits underlying rhythmic behaviors (central pattern generators: CPGs) can generate rhythmic motor output without sensory input. However, sensory input is pivotal for generating behaviorally relevant CPG output. Here we discuss recent work in the decapod crustacean stomatogastric nervous system (STNS) identifying cellular and synaptic mechanisms whereby sensory inputs select particular motor outputs from CPG circuits. This includes several examples in which sensory neurons regulate the impact of descending projection neurons on CPG circuits. This level of analysis is possible in the STNS due to the relatively unique access to identified circuit, projection, and sensory neurons. These studies are also revealing additional degrees of freedom in sensorimotor integration that underlie the extensive flexibility intrinsic to rhythmic motor systems.     

Dynamical compensation in physiological circuits

Biological systems can maintain constant steady‐state output despite variation in biochemical parameters, a property known as exact adaptation. Exact adaptation is achieved using integral feedback, an engineering strategy that ensures that the output of a system robustly tracks its desired value. However, it is unclear how physiological circuits also keep their output dynamics precise—including the amplitude and response time to a changing input. Such robustness is crucial for endocrine and neuronal homeostatic circuits because they need to provide a precise dynamic response in the face of wide variation in the physiological parameters of their target tissues; how such circuits compensate their dynamics for unavoidable natural fluctuations in parameters is unknown. Here, we present a design principle that provides the desired robustness, which we call dynamical compensation (DC). We present a class of circuits that show DC by means of a nonlinear feedback loop in which the regulated variable controls the functional mass of the controlling endocrine or neuronal tissue. This mechanism applies to the control of blood glucose by insulin and explains several experimental observations on insulin resistance. We provide evidence that this mechanism may also explain compensation and organ size control in other physiological circuits.     

Coding of cognitive magnitude: compressed scaling of numerical information in the primate prefrontal cortex

Whether cognitive representations are better conceived as language-based, symbolic representations or perceptually related, analog representations is a subject of debate. If cognitive processes parallel perceptual processes, then fundamental psychophysical laws should hold for each. To test this, we analyzed both behavioral and neuronal representations of numerosity in the prefrontal cortex of rhesus monkeys. The data were best described by a nonlinearly compressed scaling of numerical information, as postulated by the Weber-Fechner law or Stevens' law for psychophysical/sensory magnitudes. This nonlinear compression was observed on the neural level during the acquisition phase of the task and maintained through the memory phase with no further compression. These results suggest that certain cognitive and perceptual/sensory representations share the same fundamental mechanisms and neural coding schemes.     

Comparing Apples and Oranges: Fold-Change Detection of Multiple Simultaneous Inputs

Sensory systems often detect multiple types of inputs. For example, a receptor in a cell-signaling system often binds multiple kinds of ligands, and sensory neurons can respond to different types of stimuli. How do sensory systems compare these different kinds of signals? Here, we consider this question in a class of sensory systems – including bacterial chemotaxis- which have a property known as fold-change detection: their output dynamics, including amplitude and response time, depends only on the relative changes in signal, rather than absolute changes, over a range of several decades of signal. We analyze how fold-change detection systems respond to multiple signals, using mathematical models. Suppose that a step of fold F₁ is made in input 1, together with a step of F₂ in input 2. What total response does the system provide? We show that when both input signals impact the same receptor with equal number of binding sites, the integrated response is multiplicative: the response dynamics depend only on the product of the two fold changes, F₁F₂. When the inputs bind the same receptor with different number of sites n₁ and n₂, the dynamics depend on a product of power laws, . Thus, two input signals which vary over time in an inverse way can lead to no response. When the two inputs affect two different receptors, other types of integration may be found and generally the system is not constrained to respond according to the product of the fold-change of each signal. These predictions can be readily tested experimentally, by providing cells with two simultaneously varying input signals. The present study suggests how cells can compare apples and oranges, namely by comparing each to its own background level, and then multiplying these two fold-changes.     

Predicting Chemical Environments of Bacteria from Receptor Signaling

Sensory systems have evolved to respond to input stimuli of certain statistical properties, and to reliably transmit this information through biochemical pathways. Hence, for an experimentally well-characterized sensory system, one ought to be able to extract valuable information about the statistics of the stimuli. Based on dose-response curves from in vivo fluorescence resonance energy transfer (FRET) experiments of the bacterial chemotaxis sensory system, we predict the chemical gradients chemotactic Escherichia coli cells typically encounter in their natural environment. To predict average gradients cells experience, we revaluate the phenomenological Weber''s law and its generalizations to the Weber-Fechner law and fold-change detection. To obtain full distributions of gradients we use information theory and simulations, considering limitations of information transmission from both cell-external and internal noise. We identify broad distributions of exponential gradients, which lead to log-normal stimuli and maximal drift velocity. Our results thus provide a first step towards deciphering the chemical nature of complex, experimentally inaccessible cellular microenvironments, such as the human intestine.     

The functional sense of central oscillations in walking

 Rhythmic motor output is generally assumed to be produced by central pattern generators or, more specific, central oscillators, the rhythmic output of which can be entrained and modulated by sensory input and descending control. In the case of locomotor systems, the output of the central system, i.e., the output obtained after deafferentation of sensory feedback, shows many of the temporal characteristics of real movements. Therefore the term fictive locomotion has been coined. This article concentrates on a specific locomotor behavior, namely walking; in particular walking in invertebrates. In contrast to the traditional view, an alternative hypothesis is formulated to interpret the functional sense of these central oscillations which have been found in many cases. It is argued that the basic function of the underlying circuit is to avoid cocontraction of antagonistic muscles. Such a system operates best with an inherent period just above the maximum period observed in real walking. The circuit discussed in this article (Fig. 2) shows several properties in common with results described as “fictive walking”. It furthermore could explain a number of properties observed in animals walking in different situations. According to this hypothesis, the oscillations found after deafferentation are side effects occurring in specific artificial situations. If, however, a parameter called central excitation is large enough, the system can act as a central oscillator that overrides the sensory input completely. Received: 18 May 2001 / Accepted in revised form: 20 November 2001     

Adaptive rescaling maximizes information transmission

Adaptation is a widespread phenomenon in nervous systems, providing flexibility to function under varying external conditions. Here, we relate an adaptive property of a sensory system directly to its function as a carrier of information about input signals. We show that the input/output relation of a sensory system in a dynamic environment changes with the statistical properties of the environment. Specifically, when the dynamic range of inputs changes, the input/output relation rescales so as to match the dynamic range of responses to that of the inputs. We give direct evidence that the scaling of the input/output relation is set to maximize information transmission for each distribution of signals. This adaptive behavior should be particularly useful in dealing with the intermittent statistics of natural signals.     

Neural circuits underlying circadian behavior in Drosophila melanogaster

Circadian clocks include control systems for organizing daily behavior. Such a system consists of a time-keeping mechanism (the clock or pacemaker), input pathways for entraining the clock, and output pathways for producing overt rhythms in behavior and physiology. In Drosophila melanogaster, as in mammals, neural circuits play vital roles in all three functional subdivisions of the circadian system. Regarding the pacemaker, multiple clock neurons, each with cell-autonomous pacemaker capability, are coupled to each other in a network. The outputs of different sets of clock neurons in this network combine to produce the normal bimodal pattern of locomotor activity observed in Drosophila. Regarding input, multiple sensory modalities (including light, temperature, and pheromones) use their own circuitry to entrain the clock. Regarding output, distinct circuits are likely involved for controlling the timing of eclosion and for generating the locomotor activity rhythms. This review summarizes work on all of these circadian circuits, and discusses the broader utility of studying the fly's circadian system.     

Order in spontaneous behavior

Brains are usually described as input/output systems: they transform sensory input into motor output. However, the motor output of brains (behavior) is notoriously variable, even under identical sensory conditions. The question of whether this behavioral variability merely reflects residual deviations due to extrinsic random noise in such otherwise deterministic systems or an intrinsic, adaptive indeterminacy trait is central for the basic understanding of brain function. Instead of random noise, we find a fractal order (resembling Lévy flights) in the temporal structure of spontaneous flight maneuvers in tethered Drosophila fruit flies. Lévy-like probabilistic behavior patterns are evolutionarily conserved, suggesting a general neural mechanism underlying spontaneous behavior. Drosophila can produce these patterns endogenously, without any external cues. The fly's behavior is controlled by brain circuits which operate as a nonlinear system with unstable dynamics far from equilibrium. These findings suggest that both general models of brain function and autonomous agents ought to include biologically relevant nonlinear, endogenous behavior-initiating mechanisms if they strive to realistically simulate biological brains or out-compete other agents.     

Functional organization of a neural network for aversive olfactory learning in Caenorhabditis elegans

Many animals use their olfactory systems to learn to avoid dangers, but how neural circuits encode naive and learned olfactory preferences, and switch between those preferences, is poorly understood. Here, we map an olfactory network, from sensory input to motor output, which regulates the learned olfactory aversion of Caenorhabditis elegans for the smell of pathogenic bacteria. Naive animals prefer smells of pathogens but animals trained with pathogens lose this attraction. We find that two different neural circuits subserve these preferences, with one required for the naive preference and the other specifically for the learned preference. Calcium imaging and behavioral analysis reveal that the naive preference reflects the direct transduction of the activity of olfactory sensory neurons into motor response, whereas the learned preference involves modulations to signal transduction to downstream neurons to alter motor response. Thus, two different neural circuits regulate a behavioral switch between naive and learned olfactory preferences.     

Thermodynamic principles of the behavior of biological systems

A theory of the behavior of biological systems is proposed which is an extension of the conception of biological evolution (Gladyshev, 1977, Gladyshev, 1978) based on classical (equilibrium) thermodynamics. A thermodynamic theory of homeostasis is presented, in accordance with which homeostatic mechanisms of regulation are connected with a compensative shift of a fundamental quasi-equilibrium. The principle of least compulsion is formulated on the basis of thermodynamic laws and describes behavior of biological systems. A fundamental thermodynamic equation of behavioral processes is introduced. The Weber-Fechner law is shown to be a corollary of the fundamental thermodynamic equation.     

Cyclic kinetics and mathematical expression of the primary immune response to soluble antigen

The injected dose of antigen determines not only the duration of its persistence in the injection site but also the intensity of plasma cell response in the regional lymph node. It was found that the logarithmic sum of antigen quantity in the injection site was related to the sum of cell response values, the correlation coefficient approaching 1. The antigen-lymphoid system interrelations appear to obey Weber-Fechner’s law for afferent systems of the organism. The sum of plasma cells appeared to be in direct connection with the logarithm of the dose injected, with antigen persistence in the injection site and also with the tangent of the acute angle adjoining the ordinate. The basic components of the primary immune response of the organism to soluble antigen,viz. logarithm of the dose injected, antigen persistence in the injection place, plasma cell quantity, tangent of the acute angle, transition modulus from antigen to plasma cells, are interconnected by rather simple equations, which represent the structural elements of the mathematical model described in the text.     

System analysis of the goldfish olfactory bulb: Spatio-temporal transfer properties of the mitral cell granule cell complex

In this paper the goldfish olfactory bulb is described from a systems theoretical point of view. A chain of nine interacting circuits, each one mitral cell and one granule cell, is modelled. Glomerular synapses are assumed to have variable strengths. The analysis of the model system leads to the following conclusions:

1. The temporal input pattern of a mitral cell—granule cell circuit is either maintained by the circuit or inverted (lateral inhibition effect). This property together with available receptor data allows the theoretical explanation of experimentally recorded mitral cell patterns.
2. The sensitivity of a mitral cell—granule cell circuit is a function of the input signal's frequency. This provides an explanation for mitral cell cluster activity patterns measured in experiments.
3. Given a spatial input pattern to adjacent mitral cell—granule cell circuits, the output pattern depends largely upon the ratio between the feedback parameter p and the similarity β of the inputs to adjacent circuits. For appropriate p and β a local order between the responses of single neighbouring circuits is established. This local order can lead to a globally ordered mapping of odours onto mitral cell activities, thus providing a coding concept for the bulb. Some consequences of this concept coincide well with the spatial activity patterns found in 2-DOG-studies.
4. Glomerular synapses endowed with plasticity could account for long term effects such as degeneration and sensitivity changes with respect to certain odours.

     

Modelling the Time Course of Self-thinning in Crowded Plant Populations

A logarithmic model for the self-thinning of plants is proposed.This model describes the time course of self-thinning very welland fits data from forest stands and yield tables, which followthe 3/2 power law. An approximated expression of this modelshows that plant density decreases with age along a Gompertzcurve. This appears to be a basic property of the time courseof self-thinning in plants. Pinus strobus L., Pinus densiflora Sieb, et Zucc., stand development, self-thinning, 3/2 power law, logarithmic model, mortality     

Modified Box-Cox transform for modulating the dynamic range of flow cytometry data

We describe an algorithm, Vout = Integer ([2(12)-1/2(12 lambda)-1] V lambda in-1) + 1; lambda greater than 0 based upon Box-Cox transformations as an alternative to nonlinear electronic amplifiers to expand or compress high- or low-amplitude flow cytometer-derived signals. If the indexing parameter lambda less than 1, input channels in the high-amplitude input range are compressed in the output range as occurs when an electronic logarithmic amplifier is used. However, if lambda greater than 1, input channels in the low-amplitude input range are compressed in the output range as occurs when an electronic power amplifier is used. Our modified Box-Cox transform can be implemented either during data collection or off-line for the transformation of previously collected raw data. The transform is the equivalent of an infinite class of nonlinear amplifiers. As the transform is implemented in software, it does not suffer from many of the disadvantages of nonlinear electronic amplifiers.