The response of a model neurone to a white noise input

We consider a simple model of a neurone in which the input voltage is integrated to form the somatic potential, and a pulse is emitted when this reaches a threshold; the somatic potential is then reset to its resting value. We subject this model to a white-noise input, and evaluate the cross correlation between input white noise and the output pulse train; this is proportional to the small-signal impulse response of the model. Some numerical estimations are presented.     

Threshold model for clustered firing of neurons

Two probabilistic threshold models for burst activity of cortical neurons are proposed. In model I every input impulse increases the summed effect of previous input impulses by one unit. The decay of the summed effect takes place in discrete steps of one unit. A response occurs on arrival of an input impulse, when a threshold value is attained.Although after a response the summed effect is not reset to zero, it cannot exceed the threshold either. The distribution of intervals can be resolved in two components, one for long and one for short intervals. In model II intervals of the short component are terminated by a multiple response instead of one response.     

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.     

Contextual determinants of temporal control: Behavioral contrast in a free-operant psychophysical procedure

The question of how temporal control of responding might be influenced by contingency changes in other contexts was investigated. Each of three pigeons first was exposed to a two-component multiple schedule in which a two-key free-operant psychophysical procedure operated in one component and a variable-interval schedule operated in the other component. The variable-interval schedule then was changed to extinction while the free-operant psychophysical procedure remained unchanged. Finally, the variable-interval schedule was reintroduced. Response rates on the left key and the estimated temporal threshold under the free-operant psychophysical procedure increased for each pigeon when the alternate component schedule was changed to extinction and then decreased again when the variable-interval schedule was reintroduced. The results suggest one way that temporal control is affected by its context, and may be interpreted through the direct effects of overall reinforcement rate on temporal control mechanisms or the disruptive effects of alternative sources of reinforcement on temporally controlled behavior.     

A first mathematical model of brood sorting by ants: Functional self-organization without swarm-intelligence

Brood sorting, observed in Leptothorax unifasciatus ant colonies, is a major example of social insects ability to solve problems at the collective level. Two processes characterize this phenomenon: a process of aggregation of all items in a single cluster, coupled with a process of segregation of items in concentric annuli, each containing items of different type and ordered such a way that the smallest are at the center, the largest at the periphery, and prepupæ dispersed in-between. In spite of its influence on algorithmic and robotic methods, no formal explanation of the brood-sorting phenomenon was ever given. We present a first mathematical model devoted to brood sorting. Our hypothesis about ants behavior are consciously minimal: we assume that random rules their acts, not only when they walk but also when they choose a brood item that they pick up, or beside which they deposit the one they carry. The first part of our work deals with the process of aggregation in a single cluster. The main subject of our study is the time evolution of a mathematical function linked to the notion of cluster. We prove that, whatever the number of ants acting, this function tends to decrease until it reaches a threshold that we compute: this threshold matches with the formation of the single cluster. The second part of our work deals with segregation in concentric annuli. Coupling the concept of virtual size of a brood item to the previous conclusions leads to a realistic explanation of the concentric structure observed in ant colonies. Finally, we prove the existence of a feed-back effect, so that our results suggest that brood sorting is a case of self-organization that does not involve swarm-intelligence.     

Effect of fenprostalene, a PGF(2)alpha analogue, on plasma levels of estradiol-17beta and progesterone in cyclic heifers

Following a 40-day acclimatization period, 12 cyclic beef heifers entered a 95- to 101-day test period. Prior to fenprostalene treatment, all animals were studied through two normal estrous cycles. Plasma samples were obtained daily from all animals during the course of the study and were assayed for estradiol-17beta and progesterone. Group 1 heifers (n=6) were then treated with fenprostalene at mid-cycle during two subsequent cycles. This treatment was accomplished by treating the animals 11 days after the first clinically observed signs of estrus following Study Day 21 and treating them again 11 days later. Each treatment consisted of a subcutaneous injection of 1.0 mg fenprostalene. The animals were studied through two or three estrous cycles following the second injection. The Group 2 animals (n=6) were maintained as untreated controls through a corresponding period. Fenprostalene induced estrus in five of six treated heifers within 5 d following the first injection and in five of six heifers within 3 d following the second injection. The mean time to estrus was 3.4 d (+/- 1.1 d SD) following the first injection and 2.2 d (+/-0.8 days SD) following the second injection. No significant differences were found in the plasma levels of estradiol-17beta and progesterone when comparing fenprostalene-induced cycles to those that occurred naturally. The fenprostalene injection reset the estrous cycle without changing the nature of the cycle. The time of clinically detected estrus usually coincided with a sharp peak in estradiol-17beta concentration.     

Extinction times for a birth-death process with two phases

Many populations have a negative impact on their habitat or upon other species in the environment if their numbers become too large. For this reason they are often subjected to some form of control. One common control regime is the reduction regime: when the population reaches a certain threshold it is controlled (for example culled) until it falls below a lower predefined level. The natural model for such a controlled population is a birth-death process with two phases, the phase determining which of two distinct sets of birth and death rates governs the process. We present formulae for the probability of extinction and the expected time to extinction, and discuss several applications.     

On a stochastic model for the firing sequence of a neuron

A stochastic model of a neuron with excitatories and inhibitories incident on it is studied. The excitatory and the inhibitory sequences are independent renewal processes. The effect of an excitatory is to increase the membrane potential by random amounts that are independently and identically distributed, while an inhibitory causes a reset of the potential to the rest level so that the accumulation must start anew. When the potential crosses a threshold level K, the neuron fires. Immediately after this, the membrane potential returns to the rest level. An expression for the probability density function of the interval between two successive firings is derived, and special cases worked out. Graphs of the mean and the mean − √variance versus the threshold level are presented and discussed.     

Lipase and alpha amylase of chick embryo pancreas and of chickens of various ages]

In chicken embryo pancreas on the 19th day the activity of alpha-amylase and especially of lipase is very low; it increases soon after until it reaches its highest level in 36 hours old chick. Then it falls considerably on the 2nd and 10th day of normal diet; finally it starts again: in fact in three months old chickens the activity of alpha-amylase is equal to thirty-six hours old chick's, while lipase's is lower than this last one.     

The nature of the Ag-staining of nucleolus organizer regions. Electron- and light-microscopic studies on human cells in interphase, mitosis, and meiosis.

Electron micrographs reveal that the Ag-stainable substance is located on the outside of NOR's or around them but not in the chromosomes themselves. In association figures, the Ag-positive material lies between the acrocentric chromosomes. Light-microscopic studies show that the Ag stainability of the nucleolus in interphase is correlated with the function of the NOR, as seen from inactive and activated lymphocytes. Much more Ag-positive material is seen in prophase than in meta- and anaphase. It starts to increase again in late telophase. In male meiosis the NOR's remain Ag-positive until pachytene. First and second metaphase figures are negative. Experiments using RNase, TCA, and trypsin indicate that the Ag-stainable substance is an acidic protein. The precipitation of Ag granules in interphase nuclei seen in the electron microscope is greatest over the fibrillar component of the nucleolus. The most likely interpretation is that the Ag-stainable material is a component of ribonucleic protein accumulating around active NOR's. In mitosis some of this material remains at the NOR's. In first meiosis it is completely removed before diakinesis.     

Decision Theory and the Concept of Threshold in Psychophysics

Psychophysics deals basically with how signals are perceived. It is assumed that a signal or stimulus is perceived after it reaches a certain threshold value, called the sensory threshold. Thus, there are two possible states: either a signal is perceived by the neuronal system, or it is not. Fechner deserves most of the credit for the concept of the sensory threshold and for detailing the various techniques for measuring it. A discussion of this cardinal concept may be found in his book Elemente der Psychophysik (1860), although the idea is actually of much earlier origins: Leibnitz and Herbart certainly spoke of it, and indeed it seems to have been considered even by ancient Greek philosophers. Methods for measuring the sensory threshold were known even before Fechner; his contribution was being the first to systematize and perfect them. It is for this reason that his book (Elemente) is considered basic.     

On some properties of stochastic information processes in neurons and neuron populations

The information in the nervous spike trains and its processing by neural units are discussed. In these problems, our attention is focused on the stochastic properties of neurons and neuron populations. There are three subjects in this paper, which are the spontaneous type neuron, the forced type neuron and the reciprocal inhibitory pairs.

1. The spontaneous type neuron produces spikes without excitatory inputs. The mathematical model has the following assumptions. The neuron potential (NP) has the fluctuation and obeys the Ornstein-Uhlenbeck process, because the N P is not so perfectly random as that of the Wiener process but has an attraction to the rest value. The threshold varies exponentially and the NP has the constant lower limit. When the NP reaches the threshold, the neuron fires and the NP is reset to a certain position. After a firing, an absolute refractory period exists. In discussing the stochastic properties of neurons, the transition probability density function and the first passage time density function are the important quantities, which are governed by the Kolmogorov's equations. Although they can be set up easily, we can rarely obtain the analytical solutions in time domain. Moreover, they cover only simple properties. Hence the numerical analysis is performed and a good deal of fair results are obtained and discussed.
2. The forced type neuron has input pulse trains which are assumed to be based on the Poisson process. Other assumptions and methods are almost the same as above except the diffusion approximation of the stochastic process. In this case, we encounter the inhomogeneous process due to the pulse-frequency-modulation, whose first passage time density reveals the multimodal distribution. The numerical analysis is also tried, and the output spike interval density is further discussed in the case of the periodic modulation.
3. Two types of reciprocal inhibitory pairs are discussed. The first type has two excitatory driving inputs which are mutually independent. The second type has one common excitatory input but it advances in two ways, one of which has a time lag. The neuron dynamics is the same as that of the forced type neuron and each neuron has an identical structure. The inputs are assumed to be based on the Poisson process and the inhibition occurs when the companion neuron fires. In this case, the equations of the probability density functions are not obtained. Hence the computer simulation is tried and it is observed that the stochastic rhythm emerges in spite of the temporally homogeneous inputs. Furthermore, the case of inhomogeneous inputs is discussed.

     

Effects of eel (Anguilla anguilla L.) removals from selected sites of a stream on its subsequent densities

We assessed the effect of eel (Anguilla anguilla) removal from three sites of a Cantabrian stream upon its subsequent densities. In the first sample (Sept. 1986) numbers and densities were estimated as 43, 45 and 84 ind and 3490, 3030 and 3750 ind ha ⁻¹. Removal of these eels reduced the subsequent numbers and densities which, except on two occasions, were never reached again during the two years (eleven estimates) of study. Highest densities were recorded in the uppermost site in May and July, 1987, coincident with a strong drought and the lowest densities occurred in 1988 during a normal wet year. We hypothesize first that, because of a selective underground homing behaviour of eels, electro-fishing is inefficient and results in underestimates of the population. Second, seasonal variations of water discharge and droughts may not influence the homing behaviour of'eels until a threshold of dryness is reached. If this occurs, eels abandon their refuges and move towards the stream bottom. It seems that in Arroyo Chabatchos this threshold was exceeded in the summer of 1987 when the highest densities were estimated. The re-colonization of these sites experimentally depleted of eels, is a slow procces that lasts for, at least, two years.     

Threshold stimulation and accommodation of the Hodgkin-Huxley axon.

The charge-duration and strength-duration relations for just threshold rectangular stimuli were numerically investigated for the Hodgkin-Huxley axons of different lengths and different membrane capacitances under normal conditions and blockage of the development of accommodative processes. Two linear portions could be distinguished on the charge-duration curve. One of them followed the Weiss law. The other one represented a portion of a straight line passing through the zero point of the coordinates. The slope of the second portion was determined by the charge for very short stimuli (Q0), the slope of the first portion, and the maximum time to excitation (tau max). The rheobase reflected the slope of the second portion. Upon varying the fibre length the slope of the first and the second linear portions and the rheobase changed. The membrane capacitance substantially affected both the value of Q0 (as in the case of myelinated fibres) and the rheobase. The accommodative processes affected the Q0, the slope of the first line, tau max, and, consequently, the rheobase. The effect of potassium activation was stronger than that of sodium inactivation. The slope of the first line, tau max, and the rheobase might be considered more comprehensive indicators of the accommodative processes than the usually used indicators.     

Effect of a sharp change of the incidence function on the dynamics of a simple disease

We investigate two cases of a sharp change of incidencec functions on the dynamics of a susceptible-infective-susceptible epidemic model. In the first case, low population levels have mass action incidence, while high population levels have proportional incidence, the switch occurring when the total population reaches a certain threshold. Using a modified Dulac theorem, we prove that this system has a single equilibrium which attracts all solutions for which the disease is present and the population remains bounded. In the second case, an increase of the number of infectives leads to a mass action term being added to a standard incidence term. We show that this allows a Hopf bifurcation to occur, with periodic orbits being generated when a locally asymptotically stable equilibrium loses stability.     

Interaction between injected Ca2+ and intracellular Ca2+ stores in Xenopus oocytes.

Upon two repetitive deep injections of Ca2+ into Xenopus oocyte (200-300 microns under the membrane), the amplitude of the transient Cl- current induced by the second injection is several-fold higher than that of the first one. This 'potentiation' persists even at 60-90 min intervals between injections. However, in oocytes permeabilized to Ca2+ by the ionophore A23187 in a Ca2(+)-free solution, the potentiation completely disappears after 30 min. It is proposed that the injected Ca2+ is largely taken up by the stores, whereas following the second injection, a higher proportion of Ca2+ reaches the membrane, since the stores are already loaded. In ionophore-treated oocytes, the stores lose the accumulated Ca2+ over several minutes and are then ready to take up Ca2+ again, hindering its arrival at the membrane.     

Analysis of a simple self-organizing process

This paper contains mathematical results relating to a recently discovered self-organizing process. In particular, an analysis of two partial processes is presented. In the first one, a set of numerical representations assumes the correct order, and in the second one the final map of the representations converges to its asymptotic form.     

Aortic diastolic caliber changes as a determinant for complete aortic baroreceptor resetting

It is well known that baroreceptors reset to operate at higher pressure in hypertension. The time course and mechanisms responsible for resetting are still unclear. There is a rapid or acute partial resetting that reaches its maximum within the first 5-15 min but changes little within the first hours. This resetting is, however, partial and becomes complete only if the pressure change is held permanently. Resetting is complete when the change in pressure threshold for baroreceptor activation matches the total pressure change. In the rat, complete resetting to hypo- or hypertension occurs in 48 h. The aortic caliber was studied in freely moving rats during the development of sustained hypertension produced by subdiaphragmatic aortic constriction. A striking coincidence was observed between the time taken for the diastolic caliber to reach maximal dilation and the time taken for complete resetting of the aortic baroreceptors. Moreover, during sudden pressure increases, the displacement of the diastolic caliber is much greater than the increase in pulsation, which indicates that in conscious rats the operational level of the resting diastolic caliber is an important factor for aortic baroreceptor distortion.     

Intensity and kinetics of the respiratory burst of human neutrophils in relation to receptor occupancy and rate of occupation by formylmethionylleucylphenylalanine

Studies on the relationship between the binding of fMet-Leu-Phe and the respiratory response in human neutrophils have been carried out under two different conditions of stimulus presentation, i.e., instantaneously and over a period of time. The main findings are as follows (1) Under the first condition the activation of the respiratory response reaches the maximum value very quickly, when the receptor occupancy is less than 20% that at equilibrium. After reaching this maximal value, the activated respiration progressively decreases, while the specific binding of the stimulant continues until equilibrium. (2) Under the second condition, i.e., when the stimulus to neutrophils is presented over a time of 1, 2 or 4 min, the respiratory response (and also the secretory one) is depressed or absent, and the initial rate of the binding (initial Vass) is lower, but the maximal values of the receptor occupancy at equilibrium and of the rate of receptor occupation (maximal Vass) are similar and only slightly lower than those reached under the condition of instantaneous presentation of the stimulus. (3) This form of desensitization is specific for fMet-Leu-Phe and does not consist of the inactivation of the target (NADPH oxidase), since neutrophils desensitized by the slow presentation of the peptide are able to respond to a second challenge with other stimulants. These results indicate that: (1) the efficacy of the stimulus-receptor complexes is short-lived; (2) the intensity of the respiratory response is dependent on the rate of reaching a threshold of binding; (3) when this initial rate is slow, owing to the slow presentation of the stimulus, a specific desensitization takes place, indicating the existence of a molecular mechanism, linked in some way to the initial rate of binding, that modulates the capacity of the stimulus-receptor complexes to transduce signals for cell responses. The physiological role of this type of desensitization is discussed.