A New Functional Equation Analogous to CAUCHT-PEXIDER Functional Equation and its Application

The CAUCHY-PEXIDER functional equation H (x±y)=F(x) G(y) is generalized to the form H ((x^c±y^c)^1/c) = F(x) G(y), c≠0, assuming the function H(x) possesses a measurable majorant on a set of positive measure. The result is used to obtain a characterization of WEIBULL distribution. This functional equation is generalized to functions of vector variables.     

A visualization of 3D proteome universe: Mapping of a proteome ensemble into 3D space based on the protein-structure composition

To visualize a bird’s-eye view of an ensemble of proteomes for various species, we recently developed a novel method of mapping a proteome ensemble into Three-Dimensional (3D) vector space. In this study, the “proteome” is defined as the entire set of all proteins encoded in a genome sequence, and these proteins were dealt with at the level of the SCOP Fold. First, we represented the proteome of a species s by a 1053-dimensional vector x(s), where its length ∣x(s)∣ represents the overall amount of all the SCOP Folds in the proteome, and its unit vector x(s)/∣x(s)∣ represents the relative composition of the SCOP Folds in the proteome and the size of the dimension, 1053, is the number of all possible Folds in the proteome ensemble given. Second, we mapped the vector x(s) to the 3D vector y(s), based on the two simple principles: (1) ∣y(s)∣ = ∣x(s)∣, and (2) the angle between y(s) and y(t) maximally correlates with the angle between x(s) and x(t). We applied to the mapping of a proteome ensemble for 456 species, which were retrieved from the Genomes TO Protein structures and functions (GTOP) database. As a result, we succeeded in the mapping in that the properties of the 1053-dimensional vectors were quantitatively conserved in the 3D vectors. Particularly, the angles between vectors before and after the mapping highly correlated with each other (correlation coefficients were 0.95–0.96). This new mapping method will allow researchers to intuitively interpret the visual information presented in the maps in a highly effective manner.     

Optimal Fixing of the Bandwidth-Parameter for the Empirical Regression1,2

The estimator ?₀(x) of the regression r(x) = E (Y | × = x) from measured points (x_i, y_i), i = 1(1) n, of a continuous two-dimensional random variable (X, Y) with unknown continuous density function f(x, y) and with moments up to the second order can be made with the help of a density estimation f?₀(x, y) (see e.g. SCHMERLING and PEIL, 1980). Here f?₀(x, y) still contains free parameters (so-called band-width-parameters), the values of which have to be optimally fixed in the concrete case. This fixing can be done by using a modification of the maximum-likelihood principle including jackknife techniques. The parameter values can be also found from the estimators for r(x). Here the cross-validation principle can be applied. Some numerical aspects of these possibilities for optimally fixing the bandwidth-parameter are discussed by means of examples. If ?₀(x) is used as a smoothing operator for time series the optimal choice of the parameter values is dependent on the purpose of application of the smoothed time series. The fixing will then be done by considering the so-called filter-characteristic of ?C₀(x).     

Visual cortical detector neurons in the Siberian chipmunkEutamias sibiricus

Three functional classes of neurons are described in the visual cortex of the Siberian chipmunk: neurons not selective for direction of movement and orientation, neurons selective for movement in a particular direction, and neurons selective for orientation. Unselective and directionally-selective neurons were activated maximally at speeds of movement of 100–500 deg/sec or more, most orientation-selective neurons at speeds of 10–50 deg/sec. For all three classes of neurons clear correlation was observed between selectivity for velocity of movement and character of responses to presentation of stimuli stationary in the receptive field. With reference to this sign the neurons were divided into two groups: phasic (fast) and tonic (slow). Phasic (fast) neurons predominate in the visual cortex ofEutamias sibiricus.A. N. Severtsov Institute of Evolutionary Morphology and Ecology of Animals, Academy of Sciences of the USSR, Moscow. Translated from Neirofiziologiya, Vol. 16, No. 6, pp. 807–814, November–December, 1984.     

Spatial organization of receptive fields and dynamic peculiarities of neurons of the extrastriate area 21a of the cat cerebral cortex

In neurons of the extrastriate area 21a of the cat cortex, we examined the mode of initiation and peculiarities of inhibitory components in responses of these units to visual stimulation. About 31% of the studied neurons generated complex responses to mobile visual stimuli; the parameters of inhibitory components in these responses (location and duration) were different and depended on the contrast, dimension, and shape of the visual stimuli presented. We compared in detail the stationary spatial organization of receptive fields (RFs) and parameters of neuronal responses to presentation of moving stimuli in order to estimate the correlation between static and dynamic characteristics of the activity generated by the studied neurons. Our experiments showed that in most cases the neurons possessing identical homogeneous static characteristics of the RFs with off, on-off, and on responses could demonstrate quite different patterns of responses to moving stimuli, which differed from each other both in localization of inhibitory zones and discharge centers within the RFs and in time parameters of the components of these responses. The obtained data allow us to hypothesize that the dynamic characteristics of visually sensitive neurons in the extrastriate associative cortical regions are formed due to complex processes of spatial interaction between their “classic” RFs and the surrounding visual space. Neirofiziologiya/Neurophysiology, Vol. 40, No. 2, pp. 119–129, March–April, 2008.     

Some Theorems Concerning Characterization of the WEIBULL Distribution

This paper presents a number of characterizations of the WEIBULL distribution. Some of these results are generalizations of the corresponding results for the exponential distribution. Some characterizing properties lead to a functional equation f(x) · f(y) = f((x^c+y^c)^i/c) which is analogous to the CAUCHY functional equation. While the first two characterizations assume somewhat less accessible information concerning the probability distribution, the third and fourth require more readily available information regarding the expected values.     

Reorganization of Receptive Fields of Cortical Field 21a Neurons and Formation of Responses to Moving Visual Stimuli

Using extracellular recording of spike activity from single neurons of field 21a of the cat neocortex, we examined in detail the spatial organization of receptive fields (RFs) of such cells after conditions of presentation of an immobile blinking light spot (a static RF) and moving visual stimuli (dynamic RFs). As was shown, the excitability of different RF subfields of a group of neurons possessing homogeneous on–off organization of the static RF changes significantly depended on the contrast, shape, dimension, orientation, and direction of movement of the applied mobile visual stimulus. This is manifested in changes in the number of discharge centers and shifts of their spatial localization. A hypothesis on the possible role of synchronous activation of the neurons neighboring the cell under study in the formation of an additional neuronal mechanism providing specialization of neuronal responses is proposed.     

On the ability of neurons in the barn owl's inferior colliculus to sense brief appearances of interaural time difference

Summary This paper investigates the ability of neurons in the barn owl's (Tyto alba) inferior colliculus to sense brief appearances of interaural time difference (ITD), the main cue for azimuthal sound localization in this species. In the experiments, ITD-tuning was measured during presentation of a mask-probe-mask sequence. The probe consisted of a noise having a constant ITD, while the mask consisted of binaurally uncorrelated noise. Collicular neurons discriminated between the probe and masking noise by showing rapid changes from untuned to tuned and back to untuned responses.The curve describing the relation between probe duration and the degree of ITD-tuning resembled a leaky-integration process with a time constant of about 2 ms. Many neurons were ITD-tuned when probe duration was below 1 ms. These extremely short effective probe durations are interpreted as evidence for neuronal convergence within the pathway computing ITD. The minimal probe duration necessary for ITD-tuning was independent of the bandwidth of the neurons' frequency tuning and also of the best frequency of a neuron. Many narrowly tuned neurons having different best frequencies converge to form a broad-band neuron. To yield the short effective probe durations the convergence must occur in strong temporal synchronism.Abbreviations ICc central nucleus of the inferior colliculus; - ICx external nucleus of the inferior colliculus; - ITD interaural time difference - LP Likelihood parameter     

Perfect harmony: the discrete dynamics of cooperation

We study the discrete model for cooperation as expressed through the dynamics of the family of noninvertible planar maps (x, y) (x exp(r(1 – x) + sy), y exp(r(1 – y) + sx)), with parameters r, s > 0. We prove that the map is proper in the open positive quadrant and describe its various stretching and folding actions. We determine conditions for a Hopf bifurcation — probably one of a cascade of double, quadruple, ... limit cycles, as a curve is followed in parameter space. For r > s an approximating version of the map is dissipative and permanent in the positive quadrant. We include the results of an extensive computer simulation, including a bifurcation diagram (y vs. r, with s fixed) through which is cut a number of x–y phase-plane plots; (an r–y curve penetrates each plot like a thread through cards). These indicate a complex dynamical evolution for cooperation, from stable cycle to strange attractor. A general conclusion is that the benefit of cooperation can be relatively high average values at the cost of oscillations of high amplitude.     

Use of Transformed Auxiliary Variable in Estimating the Finite Population Mean

For estimating the finite population mean Y- of the study character y, an estimator using a transformed auxiliary variable has been defined. The bias and mean-squared error (MSE) of the proposed estimator have been obtained. The regions of preference have been obtained under which it is better than usual unbiased estimator y-, the ratio estimator y-_R = y-X-/x-, Sisodia and Dwivedi (1981) estimator y-_s = y-(X- + C_x)/(x- + C_x) and Singh and Kakran (1993) estimator y-_k = y[X- + β₂(x)]/[x- + β₂(x)]. An empirical study has been carried out to demonstrate the superiority of the suggested estimator over the others.     

Receptive Fields of Visually Sensitive Neurons of the Extrastriate Associative Area 21b of the Cat Cerebral Cortex

Organization of the receptive fields (RFs) of neurons of the extrastriate associative region 21b of the cerebral cortex was studied in cats. Most neurons under study (63%) were “monocular,” while 37% of the cells were “binocular” units. Among 178 neurons examined in detail, heterogeneous RF functional organization was typical of about 76% of the units; point-to-point testing of the entire RF area by stationary stimuli resulted in the generation of various types of responses (on, off, or on-off). The rest of the neurons (24%) generated homogeneous responses. The dimension, form, and functional organization of RFs of the neurons under study depended to a certain extent on the parameters of visual stimuli used for the measurements. Examination of the influence of the visual space, which surrounded the RF, on responses of the neurons evoked by stimulation of the RF per se showed that darkening of the visual space adjacent to the RF inhibited neuronal responses to moving stimuli; in some cases the responses were totally suppressed. Analysis of spatial overlapping of the RF sequentially recorded in the course of each insertion of the electrode showed that the density of distribution of the overlapping RF areas of neighboring neurons with the RF of the examined neuron is irregular, and that the RF is of a mosaic nature. We hypothesize that the visual space surrounding the RF plays a significant role in the formation of responses of visually sensitive neurons to presentation of moving stimuli. Neirofiziologiya/Neurophysiology, Vol. 37, No. 3, pp. 223–234, May–June, 2005.     

A theory of cerebral learning regulated by the reward system

Hypothetical mechanisms of the neocorticohippocampal system are presented. Neurophysiological and neuroanatomical findings concerning the system are integrated to demonstrate how animals associate sensory stimuli with rewarding actions: (1) cortical plasticity regulated by cholinergic/noradrenergic inputs from the hypothalamic reward system reinforces association connections between the most activated columns in the cortex; (2) the repetitive reinforcement forms association pathways connecting sensory cortical columns activated by the stimuli with motor cortical columns producing the rewarding actions; (3) after the pathways are formed, the cortex is capable of temporarily memorizing the stimuli by producing long-term potentiation through the cortico-hippocampal circuits; and (4) the memory allows the cortex to extend correct association pathways even in an environment where sensory stimuli rapidly change. A mathematical model of parts of the nervous system is presented to quantitatively examine the mechanisms. Membrane characteristics of single neurons are given by the Hodgkin-Huxley electric circuit. According to anatomical data, neural circuits of the neocortico-hippocampal system are composed by connecting populations of the model neurons. Computer simulation using physiological data concerning ion channels demonstrates how the mechanisms work and how to test the hypotheses presented.A preliminary version of this paper was presented at the Fifteenth Annual Meeting of the Japan Neuroscience Society, Tokyo, December 1991     

Spectro-temporal receptive fields of auditory neurons in the grassfrog

The nature of the stimulus-response relation for single auditory neurons is reflected in the properties of the Pre-Event Stimulus Ensemble: the ensemble of stimuli, preceding the occurrence of an action potential (neural event). This paper describes methods to analyse the spectro-temporal properties of this ensemble. These methods are based on the analytic signal representation of acoustic signals and functionals derived from it: the instantaneous amplitude and instantaneous frequency and the dynamic power spectrum. The procedures have been applied to a number of extra-cellular single unit recordings from the grassfrog, Rana temporatia L., recorded during presentation of an ensemble of tonal stimuli. The outcome of this analysis describes the spectro-temporal receptive field of the neuron under the present stimulus conditions. The procedure, based on the dynamic power spectrum is applicable to an arbitrary stimulus ensemble, thus allowing a comparison of the spectrotemporal receptive fields for different types of stimuli.     

A Property of Second Order Differential Equations with Application to Formal Kinetics

It is shown that if x(t) is the solution of a second order differential equation, with real negative characteristic roots (not necessarily distinct), which exhibits an extremum at t = T, then T|x(T)|/|A| [UNK] 1/e where A is the area under the x(t) curve. This result is compared to a special case previously derived by M. Morales and applications of the theorem to formal kinetic problems are discussed.     

Action of the octavolateralis efferent system upon the lateral line of free-swimming toadfish,Opsanus tau

Summary The activation and action of the octavolateralis efferent system was studied by chronic recordings of discharge patterns from putative efferent and single primary afferent neurons in alert, free-swimming toadfish. Efferent axons isolated in the anterior lateral line nerve showed phasic discharges following touch stimuli applied to the head or trunk and demonstrated sustained discharges to visual stimuli. Resting discharge patterns of primary afferents were categorized into irregular, burster, regular, and silent classes. Afferent discharges were often modulated by low frequency (< 1 Hz) water movement around the head generated during respiratory movements. When fish with recording electrodes implanted in the lateral line nerve were visually stimulated, modulated peak discharges and average (DC) firing rates were inhibited in irregular-type units only. Inhibition of irregular-type afferent neurons also followed visual presentation of natural prey and persisted long after prey stimuli were removed from view. The inhibitory action upon lateralis afferents when activated by biologically significant visual stimuli leads to the hypothesis that the octavolateralis efferent system functions in the peripheral processing of information carried by the lateral line in natural settings.Abbreviations DC average - IO infraorbital - IPSPs inhibitory postynaptic potentials - MXC maxillary canal - OMC operculomandibular canal - SOC supraorbital canal     

Estimating nest locations of bumblebee <Emphasis Type="Italic">Bombus ardens</Emphasis> from flower quality and distribution

The central-place forager in a social-insect colony, e.g., the bumblebee, has been expected to maximize its net rate of energy gain to increase the success of its colony. In addition to foraging behavior, the nest location is an important factor for the success of the colony. The bumblebee’s nest location would be affected by the spatial distribution of flowers and their food quality. In this study, we constructed a model to estimate bumblebee nest sites, using the net energy intake rate at available food sites for workers foraging from the nest site. We hypothesized that the probability of colony establishment at a site in coordinates (x, y) was high as the sum of the net energy intake rate I(x, y) increased. To obtain I(x, y), nectar standing crop, sugar concentration, and foraging time were measured for ten plant species in the study site covering 6.25 km². As available flowers changed seasonally, I(x, y) was calculated for three periods: the end of April, the beginning of May, and the middle of May. To verify our hypothesis, we compared the estimations in our model with the actual nest sites of Bombus ardens found in the beginning of May and June by means of tracking bumblebees. From the results, we considered that the net energy intake rate at mid-May might represent the probability of colony establishment, because it could affect colony persistence and reproductive success.     

Neuronal selectivity,population sparseness,and ergodicity in the inferior temporal visual cortex

The sparseness of the encoding of stimuli by single neurons and by populations of neurons is fundamental to understanding the efficiency and capacity of representations in the brain, and was addressed as follows. The selectivity and sparseness of firing to visual stimuli of single neurons in the primate inferior temporal visual cortex were measured to a set of 20 visual stimuli including objects and faces in macaques performing a visual fixation task. Neurons were analysed with significantly different responses to the stimuli. The firing rate distribution of 36% of the neurons was exponential. Twenty-nine percent of the neurons had too few low rates to be fitted by an exponential distribution, and were fitted by a gamma distribution. Interestingly, the raw firing rate distribution taken across all neurons fitted an exponential distribution very closely. The sparseness a ^s or selectivity of the representation of the set of 20 stimuli provided by each of these neurons (which takes a maximal value of 1.0) had an average across all neurons of 0.77, indicating a rather distributed representation. The sparseness of the representation of a given stimulus by the whole population of neurons, the population sparseness a ^p, also had an average value of 0.77. The similarity of the average single neuron selectivity a ^s and population sparseness for any one stimulus taken at any one time a ^p shows that the representation is weakly ergodic. For this to occur, the different neurons must have uncorrelated tuning profiles to the set of stimuli.     

Sequential Classification Based on Regression

Let us consider m general populations π₁, …,π_m. Each object belonging to these populations is represented by (p ± 1) characteristics x₁, x₂,…,x_p,y. A certain object, which is an element of one of the m general populations π₁,…,π_m has to be classified into the correct population. It will be assumed that knowledge of the value of the characteristic y would permit its correct classification, but that the observation of this characteristic is expensive, difficult or dangerous, as e.g. in medical applications. y is correlated with a set of p characteristics x₁,x₂,…,x_p, which are observed sequentially. The classification procedure is based on the division of the space of the observed value of characteristics x₁,x₂,…,x_p into nonintersecting areas determined so as to minimize the value of BAYES' risk given by equation (3).     

Unit responses of the squirrel visual cortex to shaped visual stimuli

Visual cortical unit responses of the squirrelSciurus vulgaris to shaped visual stimuli (stationary and moving spots and bands) were studied. Neurons responding selectively to the direction of stimulus movement and orientation of lines and those not responding selectively to these features were distinguished. Many neurons, whether responding selectively or not to movement direction, were specifically sensitive to high speeds of movement, of the order of hundreds of degrees per second. This selectivity in neurons responding selectively to movement direction persisted at these high speeds, despite the short time taken by the stimulus to move across the receptive field. Neurons responding selectively to line orientation were sensitive to lower speeds of stimulus movement — from units to tens of degrees per second. Neuronal sensitivity to high speeds of stimulus movement is achieved through rapid summation of excitation from large areas of the receptive field crossed by the fast-moving stimulus. Selectivity of the response to movement direction is produced under these conditions with the aid of directed short-latency inhibition, inhibiting unit activity for stimulus movement in "zero" direction.