Contribution to the probabilistic theory of neural nets: IV. Various models for inhibition

Input-output formulas are derived for a neuron upon which converge single axones of two other neurons, which are subjected to a Poisson shower, where a number of different assumptions are made concerning the mechanism of inhibition. In one assumption so-called “bilateral pre-inhibition” is considered. That is to say, both neuronsN ₁ andN ₂ may exciteN ₃, but if the stimulus of one of them follows within a certain interval σ of the other, the second stimulus is not effective. This model is essentially no different from that involving two excitatory neurons acting upon a neuron having a refractory period. Another mechanism considered involves so-called “pre-and-post” inhibition, in which if two stimuli fromN ₁ andN ₂ fall within σ,both are ineffective. This case being mathematically much more involved than the preceding, an approximation method is used for deriving the input-output formula. Previous papers of this series are denoted by I, II, and III in this paper.     

A convergence theorem for linear threshold elements

“Linear threshold element”, is the generic term for a device, natural or artificial, of the input-output type which (1) has a finite numberk of input lines and one output line; (2) forms a linear combinationa ₁ x ₁+a ₂ x ₂+...+a _k x _k of the inputk-tuple (x ₁,x ₂,...,x _k) and compares it with a threshold quantity θ; and (3) yields one of two outputs depending on whether the linear combination does or does not exceed the threshold θ. Such elements find wide use as means for realizing linear discriminant functions and as analogs for certain features characterizing neurons. Adaptive classification of inputs into one of two categories, as designated by the two outputs, can sometimes be achieved by the suitable adjustment, based on experience, of the parametersa ₁,a ₂,...,a _k and θ. One such adjustment rule having the characteristics of a steepest descent algorithm is here investigated. Established is a theorem giving the necessary and sufficient conditions that the mean values of the adjusted parameters converge to a solution of the equations required for minimizing the mean square error relative to a quantity associated with classification performance when the element is exposed to a stationary sequence of independent inputs.     

Contribution to the probabilistic theory of neural nets: I. Randomization of refractory periods and of stimulus intervals

Aggregates of neurons are considered in which the frequency of occurrence of neurons with a specified value of the refractory period follows certain probability distributions. Input-output functions are derived for such aggregates. In particular, if input and output intensities are defined in terms of stimulus frequencies and firing frequencies per neuron respectively, it is shown that a rectangular distribution of refractory periods leads to a logarithmic input-output curve. If input and output are defined in terms of the total number of stimuli and firings in the aggregate, it is shown how the “mobilization” picture leads to the logarithmic input-output curve. By randomizing the intervals between stimuli received by a single neuron and by introducing an inhibitory neuron a very simple “filter net” can be constructed whose output will be sensitive to a particular range of the input, and this range can be made arbitrarily small.     

Bayesian image processing of data from constrained source distributions—I. non-valued,uncorrelated and correlated constraints

A series of Bayesian image processing algorithms which incorporate various classes ofa priori source information in treating data which obeys Poisson and Gaussian statistics is derived using maximum entropy considerations. The standard maximum likelihood equations are shown to be a special case of Bayesian image processing when thea priori information about a source distribution φ _j is solely that a non-vanishing probability for each element value φ _j exists only in some finite interval,a _j ≤φ _j ≤φ _j . Bayesian image processing equations for thea priori source information that all φ _j are finite -∞<φ _j <∞ and each φ _j distribution has a defined mean φ _j and a defined variance σ _j are derived. The Bayesian image processing equations are also derived when thea priori source information is that all φ _j ≥0 and that each φ _j distribution has a defined mean φ _j and a defined variance σ _j . The a priori source distribution constraint that a correlation exists among nearby elements is also considered. The results indicate improvement over standard methods.     

The carbon and nitrogen isotopic values of particulate organic material from the Florida Keys: a temporal and spatial study

The δ¹⁵N and δ¹³C values of particulate organic material (POM) were analyzed from 35 sites in the Florida Keys over the time interval 2000 to 2002. The sites within the study area were delineated into nine transects stretching from Key West to Key Largo. Each transect consisted of three to five sites extending from close to the Keys to the edge of the reef tract. The POM had mean δ¹⁵N and δ¹³C values of +3.6‰ (σ = ±3.2‰) and −19.9‰ (σ = ±0.6‰) respectively. Over the study period there were no statistically significant changes in δ¹⁵N, δ¹³C, or C:N. For the majority of the sampling dates, the δ¹³C values showed a distinct inshore (δ¹³C = −18.3‰, σ = ±1.0‰) to offshore gradient (δ¹³C = −21.4, σ = ±0.9‰). In contrast, the δ¹⁵N values showed no consistent patterns related to the distance from land. The more positive δ¹³C values of the nearshore samples suggest that the source of the carbon and the nitrogen in the POM in the nearshore was mainly derived from the degradation of seagrass detritus and not from the input of anthropogenically derived material from the Florida Keys. In contrast, the POM on the outer reef was dominated by marine plankton. As mineralization and nitrification of the organic nitrogen pool are major contributors to the dissolved inorganic nitrogen in the water column, it is unlikely that variations in the δ¹⁵N of the algae and other benthic organisms reported in the Florida Keys are related to the input of sewage.     

Evaluation of the maximum specific growth rate of a yeast indicating non-linear growth trends in batch culture

The maximum specific growth rate (μ_max) of an ethanolic D-xylose-fermenting yeast, Pichia stipitis, showing non-linear growth trends in batch culture, was calculated using the rate equation μ₂ = (1/Δt) ln(x ₂/x ₁). The absolute error Δμ, affecting μ₂, was derived using an equation given by Borzani (1994). Based on the assumption of linearity of growth curves between two closest time points, the relation between the two rate formulae, μ₁ = (1/xˉ)dx _t /dt and μ₂ = (1/Δt) ln(x ₂/x ₁) was established. In a particular condition, when μ₁ = μ₂, an equation has been developed, the roots of which are the specific growth rates at different time points. This revised version was published online in July 2006 with corrections to the Cover Date.     

Altitude trends in conifer leaf morphology and stable carbon isotope composition

The natural ratio of stable carbon isotopes (δ¹³C) was compared to leaf structural and chemical characteristics in evergreen conifers in the north-central Rockies, United States. We sought a general model that would explain variation in δ¹³C across altitudinal gradients. Because variation in δ¹³C is attributed to the shifts between supply and demand for carbon dioxide within the leaf, we measured structural and chemical variables related to supply and demand. We measured stomatal density, which is related to CO₂ supply to the chloroplasts, and leaf nitrogen content, which is related to CO₂ demand. Leaf mass per area was measured as an intermediate between supply and demand. Models were tested on four evergreen conifers: Pseudotsuga menziesii, Abies lasiocarpa, Picea engelmannii, and Pinus contorta, which were sampled across 1800 m of altitude. We found significant variation among species in the rate of δ¹³C increase with altitude, ranging from 0.91‰ km^–1 for A. lasiocarpa to 2.68‰ km^–1 for Pinus contorta. Leaf structure and chemistry also varied with altitude: stomatal density decreased, leaf mass per area increased, but leaf nitrogen content (per unit area) was constant. The regressions on altitude were particularly robust in Pinus contorta. Variables were derived to describe the balance between supply and demand; these variables were stomata per gram of nitrogen and stomata per gram of leaf mass. Both derived variables should be positively related to internal CO₂ supply and thus negatively related to δ¹³C. As expected, both derived variables were negatively correlated with δ¹³C. In fact, the regression on stomatal density per gram was the best fit in the study (r ²=0.72, P<0.0001); however, the relationships were species specific. The only general relationship observed was between δ¹³C and LMA: δ¹³C (‰)=–32.972+ 0.0173×LMA (r ²=0.45, P<0.0001). We conclude that species specificity of the isotopic shift indicates that evergreen conifers demonstrate varying degrees of functional plasticity across environmental gradients, while the observed convergence of δ¹³C with LMA suggests that internal resistance may be the key to understanding inter-specific isotopic variation across altitude. Received: 1 June 1999 / Accepted: 2 November 1999     

Contribution to the probabilistic theory of neural nets: II. Facilitation and threshold phenomena

The output curve of a single neuron with a threshold of response with respect to the frequency of the stimuli is derived. If the stimuli are regularly spaced in time, the output curve has discontinuities. If the threshold and/or refractory period are sufficiently large, the output curve approaches the “all-or-none” curve. In the case of completely randomized stimuli, the output curve is sigmoid. The equation of this curve is derived and some properties are studied. Threshold and “all-or-none” effects can be achieved by “pyramiding” neurons of this type to converge on neurons of higher order.     

Generalizations of the Roginsky-Zeldovich (or Elovich) equation for charge transport across biological surfaces

The Roginsky-Zeldovich (or Elovich) equation, which is −dx/dt=m exp (nx) (x=substrate concentration,t=time,m andn=constants), describes the kinetics of various biological electron and ion transport processes, and has been derived from the concept of charge transport across an activation energy barrier at an interface between dissimilar phases, driven by a difference in redox or ion potentials, with the simplifying assumptions that charge carrier concentration is constant, backward current across the interface is zero, and diffusion of substrate is fast. If charge carrier concentration is proportional to substrate concentration, then the kinetic equation is −dx/dt=mx exp (nx). If backward current is not zero, then −dx/dt=m ₁ exp (n ₁x) −m ₂ exp (n ₂ x), wherem ₁,m ₂,n ₁ andn ₂ are constants. Kinetic equations for interfacial charge transport in the presence of a significant substrate diffusion potential are also derived.     

The smallest value of the axon density for which ‘ignition’ can occur in a random net

As shown by A. Rapoport (1952), when a very brief stimulation or “instantaneous input” is applied to a random net, the subsequent events are determined by the parameters of the net as follows: If the axon densitya is sufficiently large and the fraction γ of the neurons initially stimulated exceeds a certain value γ₁ (theover-all threshold of the net for instantaneous stimulation), excitation will spread through the net until a steady state is reached in which a fraction γ₂ ⩾ γ₁ of the neurons is firing (“ignition phenomenon”). If γ < γ₁ the activity in the net dies out. However, if the axon density is too small, the activity will ultimately die out, no matter how large the fraction of initially stimulated neurons. Thus there exists a limiting valueA of the axon density below which the net cannot “ignite”. ThisA is a function ofh, theindividual threshold of the neurons constituting the net (we assume hereh≥2, since forh=1 the situation is essentially different). Geometrically γ₁ and γ₂ are determined as the two intersection points of a straight line with a sigmoid curve. Whena<A the two curves do not intersect and fora=A they are tangent. In this paper the “tangency case” is investigated and the general features of the functionA(h) are determined. It is shown thatA increases monotonically withh (as one would expect). For all values ofh>1 we haveA(h)>h, but the fractionA(h)/h and the derivativedA(h)/dh approach unity ash increases. An analytical expression of the functionA(h) valid for very large values ofh is derived.     

Does Bienertia cycloptera with the single-cell system of C(4) photosynthesis exhibit a seasonal pattern of delta (13)C values in nature similar to co-existing C (4) Chenopodiaceae having the dual-cell (Kranz) system?

Family Chenopodiaceae is an intriguing lineage, having the largest number of C₄ species among dicots, including a number of anatomical variants of Kranz anatomy and three single-cell C₄ functioning species. In some previous studies, during the culture of Bienertia cycloptera Bunge ex Boiss., carbon isotope values (δ¹³C values) of leaves deviated from C₄ to C₃−C₄ intermediate type, raising questions as to its mode of photosynthesis during growth in natural environments. This species usually co-occurs with several Kranz type C₄ annuals. The development of B. cycloptera morphologically and δ¹³C values derived from plant samples (cotyledons, leaves, bracts, shoots) were analyzed over a complete growing season in a salt flat in north central Iran, along with eight Kranz type C₄ species and one C₃ species. For a number of species, plants were greenhouse-grown from seeds collected from the site, in order to examine leaf anatomy and C₄ biochemical subtype. Among the nine C₄ species, the cotyledons of B. cycloptera, and of the Suaeda spp. have the same respective forms of C₄ anatomy occurring in leaves, while cotyledons of members of tribe Caroxyloneae lack Kranz anatomy, which is reflected in the δ¹³C values found in plants grown in the natural habitat. The nine C₄ species had average seasonal δ¹³C values of −13.9‰ (with a range between species from −11.3 to −15.9‰). The measurements of δ¹³C values over a complete growing season show that B. cycloptera performs C₄ photosynthesis during its life cycle in nature, similar to Kranz type species, with a seasonal average δ¹³C value of −15.2‰. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Toward an accurate statistics of gapped alignments

Sequence alignment has been an invaluable tool for finding homologous sequences. The significance of the homology found is often quantified statistically by p-values. Theory for computing p-values exists for gapless alignments [Karlin, S., Altschul, S.F., 1990. Methods for assessing the statistical significance of molecular sequence features by using general scoring schemes. Proc. Natl. Acad. Sci. USA 87, 2264–2268; Karlin, S., Dembo A., 1992. Limit distributions of maximal segmental score among Markov-dependent partial sums. Adv. Appl. Probab. 24, 13–140], but a full generalization to alignments with gaps is not yet complete. We present a unified statistical analysis of two common sequence comparison algorithms: maximum-score (Smith-Waterman) alignments and their generalized probabilistic counterparts, including maximum-likelihood alignments and hidden Markov models. The most important statistical characteristic of these algorithms is the distribution function of the maximum score S _max, resp. the maximum free energy F _max, for mutually uncorrelated random sequences. This distribution is known empirically to be of the Gumbel form with an exponential tail P(S _max > x) ∼ exp(−λx) for maximum-score alignment and P(F _max > x) ∼ exp(−λx) for some classes of probabilistic alignment. We derive an exact expression for λ for particular probabilistic alignments. This result is then used to obtain accurate λ values for generic probabilistic and maximum-score alignments. Although the result demonstrated uses a simple match-mismatch scoring system, it is expected to be a good starting point for more general scoring functions.     

Diurnal variations of needle water isotopic ratios in two pine species

Diurnal fluctuations of leaf water isotope ratios (δ¹⁸O and δD) were measured for Jeffrey (Pinus jeffreyi Balf.) and lodgepole (Pinus contorta Douglas ex Louden) pine. Two trees per species were sampled every few hours on 15–16 October 2005 and 19–20 June 2006. Diurnal gas exchange was measured during the summer sampling. In fall 2005, leaf water δ¹⁸O ranged from 0.7 to 9.0‰, and leaf water δD ranged from −70 to −50‰. In summer 2006, leaf water δ¹⁸O ranged from 7.7 to 20.7‰, and leaf water δD ranged from −61 to −24‰. Diurnal variation of leaf water isotope values typically reached a maximum in early afternoon, began decreasing around midnight, and reached a minimum in mid-morning. Both periods showed a high degree of enrichment relative to source water, with leaf water–source water enrichments ranging up to 37.8‰ for δ¹⁸O, and up to 95‰ for δD. Leaf water enrichment varied by season with summer enrichment being greater than fall enrichment. A steady-state model (i.e., modified Craig–Gordon modeling) for leaf water isotope compositions did not provide a good fit to measured values of leaf water. In summer, a non-steady state model provided a better fit to the measured data than the steady-state model. Our findings demonstrate substantial leaf water enrichment above source water and diurnal variations in the isotopic composition of leaf water, which has application to understanding short-term variability of atmospheric gases (water vapor, CO₂, O₂), climate studies based on the isotopic composition of tree rings, and ecosystem water fluxes.     

Effects of dark-light step transitions on circadian locomotor activity rhythms of the hagfish,Eptatretus burgeri

Captive hagfish,Eptatretus burgeri, were subjected to step transitions from continuous dark (DD) to continuous light (LL) and their locomotor activity patterns recorded. Free-running activity rhythms occurred in both the DD and LL regimes. The timing of the transition influenced the circadian period (τ), the relationship between the period response (δτ, or τ{inLL}–τ{inDD}, change in the circadian period) and the former being represented by a cosine curve δτ became most positive and most negative when hagfish underwent a DD-LL transition shortly before the beginning of the duration of activity (α) and shortly after the end of such, respectively. The phase response (δ-phase, change in the relative timing of the activity phase) was characterized by a delay in the activity phase after the DD-LL transition, although its magnitude bore no relation to the timing of the transition.     

Phase behaviour and crystallinity of plant cuticular waxes studied by Fourier transform infrared spectroscopy

The phase behaviour of cuticular waxes from leaves of Hedera helix L. and Juglans regia L. was studied by Fourier transform infrared spectroscopy. For this purpose reconstituted waxes, isolated cuticular membranes, dewaxed polymer matrix membranes and whole leaves were studied in the horizontal attenuated total reflection and transmission modes. Melting curves of cuticular waxes were derived from temperature-dependent changes in the absorption maximum of the symmetric stretching mode of CH₂ groups (ν_s, at approx. 2856–2848 cm⁻¹). With increasing temperature absorption band doublets due to CH₂ scissoring (δ_sciss) and rocking (δ_rock) movements (at approx. 1473–1471 and 730–720 cm⁻¹, respectively) indicative of an orthorhombic arrangement of alkyl chains merged into a single peak. The area ratio of the peaks at approx. 720 and 730 cm⁻¹ was used as a measure for aliphatic crystallinity of plant cuticular waxes at a given temperature. The investigations of reconstituted cuticular waxes and those still embedded in isolated cuticles or in situ on the leaf produced comparable results. The findings are discussed in terms of the properties of the cuticular transport barrier. Received: 21 March 1997 / Accepted: 25 April 1997     

Photosynthetic pathway of dominant plant species in the Qaidam Basin: evidence from foliar stable carbon isotope measurement

Foliar δ¹³C values of Calligogum kozlovi and Haloxylon ammodendron ranged from −13.13 to −15.11 ‰, while those of the rest 11 species were in the range of −22.22 to −27.73 ‰. This indicates that two of 13 dominant plant species in the Qaidam Basin possess a C₄ photosynthetic pathway. Significant differences were observed for the average foliar δ¹³C values between C₃ or C₄ plant communities, between grass and shrub communities, even between the same species derived from different sites. Precipitation accounted for the major part of the differences.     

Permeability and Reflection Coefficients of Urea and Small Amides in the Human Red Cell

Measurement of the transport parameters that govern the passage of urea and amides across the red cell membrane leads to important questions about transport of water. It had initially been thought that small protein channels, permeable to water and small solutes, traversed the membrane (see Solomon, 1987). Recently, however, very strong evidence has been presented that the 28 kDa protein, CHIP28, found in the red cell membrane, is the locus of the water channel (see Agre et al., 1993). CHIP28 transports water very rapidly but does not transport small nonelectrolytes such as urea. The irreversible thermodynamic parameter, σ _i , the reflection coefficient, is a measure of the relationship between the permeability of the solute and that of water. If a solute permeates by dissolution in the membrane, σ _i = 1.0; if it permeates by passage through an aqueous channel, σ _i < 1.0. For urea, Goldstein and Solomon (1960) found that σ_urea= 0.62 ± 0.03 which meant that urea crosses the red cell membrane in a water-filled channel. This result and many subsequent observations that showed that σ_urea < 1.0 are at variance with the observation that CHIP28 is impermeable to urea. In view of this problem, we have made a new series of measurements of σ _i for urea and other small solutes by a different method, which obviates many of the criticisms Macey and Karan (1993) have made of our earlier method. The new method (Chen et al., 1988), which relies upon fluorescence of the intracellular dye, fluorescein sulfonate, leads to the corrected value, σ_urea,corr= 0.64 ± 0.03 for ghosts, in good agreement with earlier data for red cells. Thus, the conclusion on irreversible thermodynamic and other grounds that urea and water share a common channel is in disagreement with the view that CHIP28 provides the sole channel for water entrance into the cell. Received: 6 February 1996/Revised: 20 May 1996     

Current progress of the biological single-ion microbeam at FUDAN

A biological microbeam for precisely positioned single-ion/single cell irradiation is built in the Institute of Modern Physics in Fudan University, Shanghai, China, based on the tandem accelerator (2 × 3MV) in the laboratory. In this paper, the developing progress of the FUDAN microbeam is reported, including the newly constructed beam line, the microbeam collimator, the ion detection system, and the cell-imaging and targeting systems. Statistical models are proposed for evaluating the spatial resolution and dosage precision of the microbeam. By taking the collimated ions as a Gaussian beam, the spatial resolution can be evaluated by the full width at half maximum of the 2-D Gaussian distribution, which is determined by fitting the proportions of peripheral pits outside specific radii in the pit clusters etched on ion track detectors to a 2-D Gaussian distribution. In the preset hitting of defined ion number, by taking the real delivered number of ions as an independent identically distributed random variable (iidrv), according to the Law of Large Numbers and Central Limit Theorem, the expected value μ and standard deviation σ of the real delivered ion number in a preset N-ion hitting can be determined by approaching the normal distribution of N (μ, σ ²/n) with the proportions of the mean counts of pits in multiple pit clusters on ion track detectors. By the values of μ, σ and additional assumptions, statistical dosage precision evaluations can be made on the preset hitting. From the linear fit curve of μ(N) and the power function fit curve of σ(N) on different preset ion numbers, characteristic factors k, b, A, p can be extracted for a precision evaluation independent of the specific preset ion number.     

Low resolution structure of subunit b (b (22-156)) of Escherichia coli F(1)F(O) ATP synthase in solution and the b-delta assembly

The first low resolution solution structure of the soluble domain of subunit b (b _22–156) of the Escherichia coli F₁F_O ATPsynthase was determined from small-angle X-ray scattering data. The dimeric protein has a boomerang-like shape with a total length of 16.2 ± 0.3 nm. Fluorescence correlation spectroscopy (FCS) shows that the protein binds effectively to the subunit δ, confirming their described neighborhood. Using the recombinant C-terminal domain (δ_91–177) of subunit δ and the C-terminal peptides of subunit b, b _120–140 and b _140–156, FCS titration experiments were performed to assign the segments involved in δ–b assembly. These data identify the very C-terminal tail b _140–156 to interact with δ_91–177. The novel 3D structure of this peptide has been determined by NMR spectroscopy. The molecule adopts a stable helix formation in solution with a flexible tail between amino acid 140 to 145.     

Mycelial cultivation of <Emphasis Type="Italic">Phellinus linteus</Emphasis> using cheese-processing waste and optimization of bioconversion conditions

A medicinal mushroom, Phellinus linteus, was successfully cultivated using a cheese-processing waste, whey, and the optimal bioconversion conditions for the maximum mycelial growth rate was also estimated through solid-state cultivation experiments. Response surface analysis with a face-centered design (center point replication = 5) was applied to statistically approximate the simultaneous effects of the three variables, i.e., substrate concentration (10–30 g lactose l⁻¹), temperature (20–30°C), and pH (4–6), on the mycelial growth rate of P. linteus. The following is a partial cubic model where η is the mycelial growth rate (K _r ) and x _k is the corresponding variable term (k = substrate concentration, temperature, and pH in order): η = −23.8 + 8.67 × 10⁻² x ₁ + 1.48x ₂ + 1.77x ₃ + 8.00 × 10⁻⁴ x ₁ x ₂ + 7.25 × 10⁻² x ₁ x ₃ + 5.13 × 10⁻² x ₂ x ₃ −1.28 × 10⁻² x ₁² –3.18 × 10⁻² x ₂². −2.64 × 10⁻¹ x ₃² −3.28 × 10⁻³ x ₁ x ₂ x ₃ + 4.68 × 10⁻⁴ x ₁² x ₂. The produced response surface model proved to be significant (r ² > 0.99, P-value <0.0001, coefficient of variation <5%) to describe the explored space. Temperature was found to be the most significant factor of dominant effects on the mycelial growth rate, and other variables such as temperature², pH, pH², and (substrate concentration² × temperature) also showed significant effects on the model output. The maximum mycelial growth rate was predicted to be 2.80 mm d⁻¹ at 29.7 g lactose l⁻¹, 26.2°C, and pH 5. Our results proved a good potential of whey to serve as an alternative growth medium for cultivating P. linteus mycelia. This may provide another potential for managing this nutrient-rich waste in a cost-effective way.