The Royal Society
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A model for vein formation in higher plants Experiments on vein regeneration (Jost 1942; Jacobs 1952) suggest that a signal of some kind, which can cause the differentiation of veins, flows from a source in the growing tissues of leaves towards the root. Sachs (1969, 1978) has given evidence that the capacity of a given pathway to transport this signal increases with the flux it carries. He suggests that this progress could cause the canalization of signal flow into a pattern of discrete strands, which subsequently differentiate into veins. I formulate here a mathematical model based on these assumptions, and show that it can generate well defined strands. Given plausible estimates for diffusion constants and polar transport rates, it appears that vein formation could occur by this mechanism over an appropriate distance within an acceptably short time. I show that this model can simulate Sachs’s experiments on vein formation. I also show that, with suitable assumptions about the distribution of source activity, the model can generate elaborate networks, with branches and loops of the kind seen in the leaves of higher plants.
The Royal Society - Tập 207 Số 1166 - Trang 79-109 - 1980
Predictive coding: a fresh view of inhibition in the retina Interneurons exhibiting centre-surround antagonism within their receptive fields are commonly found in peripheral visual pathways. We propose that this organization enables the visual system to encode spatial detail in a manner that minimizes the deleterious effects of intrinsic noise, by exploiting the spatial correlation that exists within natural scenes. The antagonistic surround takes a weighted mean of the signals in neighbouring receptors to generate a statistical prediction of the signal at the centre. The predicted value is subtracted from the actual centre signal, thus minimizing the range of outputs transmitted by the centre. In this way the entire dynamic range of the interneuron can be devoted to encoding a small range of intensities, thus rendering fine detail detectable against intrinsic noise injected at later stages in processing. This predictive encoding scheme also reduces spatial redundancy, thereby enabling the array of interneurons to transmit a larger number of distinguishable images, taking into account the expected structure of the visual world. The profile of the required inhibitory field is derived from statistical estimation theory. This profile depends strongly upon the signal: noise ratio and weakly upon the extent of lateral spatial correlation. The receptive fields that are quantitatively predicted by the theory resemble those of X-type retinal ganglion cells and show that the inhibitory surround should become weaker and more diffuse at low intensities. The latter property is unequivocally demonstrated in the first-order interneurons of the fly’s compound eye. The theory is extended to the time domain to account for the phasic responses of fly interneurons. These comparisons suggest that, in the early stages of processing, the visual system is concerned primarily with coding the visual image to protect against subsequent intrinsic noise, rather than with reconstructing the scene or extracting specific features from it. The treatment emphasizes that a neuron’s dynamic range should be matched to both its receptive field and the statistical properties of the visual pattern expected within this field. Finally, the analysis is synthetic because it is an extension of the background suppression hypothesis (Barlow & Levick 1976), satisfies the redundancy reduction hypothesis (Barlow 1961 a, b) and is equivalent to deblurring under certain conditions (Ratliff 1965).
The Royal Society - Tập 216 Số 1205 - Trang 427-459 - 1982
Atheroma and arterial wall shear - Observation, correlation and proposal of a shear dependent mass transfer mechanism for atherogenesis
On the basis of various observations, we argue that there is spatial variation of the time averaged wall shear rate in arteries, both overall and locally. From our own observations, and those of others, we show that the distribution of
early
atheroma in man is coincident with those regions in which arterial wall shear rate is expected to be relatively low, while the development of lesions is inhibited or retarded in those regions in which wall shear rate is expected to be relatively high. Such a correlation is inconsistent with a proposal, made by several workers, that there is a causative relation between arterial blood mechanics and the development of atheroma, i. e. that atheroma is associated with wall damage due to the motion of blood. Instead it immediately suggests that the process is associated with shear dependent mass transport phenomena. It has been demonstrated by others that mass transport, in the inner part of the arterial wall, is dominantly to and from blood flowing within the lumen. We review theory relevant to diffusional mass transport across such a sheared interface, and examine available experimental evidence, relating to normally occurring (quasi-steady state) and experimentally induced (transient-type) atheroma, as well as the distribution of cholesterol in arteries. These results are considered in the light of simple theoretical schemes which we develop for the movement of cholesterol, in particular, although the arguments may also be relevant to other diffusing species. Shear enhances mass transport by means of a steepening effect on the concentration gradient, thus diffusion of material from a wall is promoted when material which has already diffused is swept rapidly away, so that the concentration gradient leading to further diffusion remains steep. However, the influence of shear on the diffusion of a species, say, from just within the wall of an artery to fluid in the main stream, depends upon the relative resistances to its diffusion from within the wall to surface fluid (wall phase) and from surface fluid to fluid in the main stream (blood phase); diffusion is not appreciably shear dependent if the latter resistance is small compared with the former. Assuming simplified flow conditions and that as suggested by others cholesterol is transported in blood in association with plasma protein, we can estimate resistance for diffusion of this species in the blood phase, for different stations in the arterial system. However, we possess no definite comparable information for the wall phase; we conjecture that this resistance is relatively small, and assume shear dependence of diffusional transport of cholesterol between arterial walls and intraluminal blood. We find that a net flux of cholesterol from blood to wall, as has been suggested by others, cannot account, in terms of the proposed schemes, for the observed normally occurring (quasi-steady state) distribution of atheromatous lesions in man and in animals; mass transport is inhibited in low shear regions by the thick diffusional boundary layer. Instead it appears that cholesterol, which has been shown by others to be synthesized in arterial walls, accumulates in low shear regions because its local diffusional efflux from wall to blood is inhibited by the reduced concentration gradient. Given suitable values for relevant parameters, the theoretical schemes are also able to account for adequacy of supply of precursor to the wall for cholesterol synthesis, for the preferential occurrence that we now recognize of lesions in high shear regions in response to sudden natural or experimental elevation of blood cholesterol, and for the responses to administration of labelled cholesterol (transient type phenomena); it appears therefore possible, in terms of these schemes, to unify naturally occurring and experimentally induced atheroma. It is reported by others that platelets are associated only with advanced lesions; the correlation of naturally occurring atheroma with low shear regions, and transient type lesions with high shear regions, with the fluid mechanics being unaltered in the two situations, provides no support for the implication of platelets in the development of early atheroma. It appears that wall shear rate may be a major controlling factor in the development of atheroma, i.e. that high shear, such as is associated for example with increased cardiac output in exercise, will retard progression of the process. Its progression will also be retarded by any means which reduces the accumulation of atheromatous material, by influencing its rate of net production or diffusion.
The Royal Society - Tập 177 Số 1046 - Trang 109-133 - 1971
Non-stationarity of tree branching patterns and bifurcation ratios Branching patterns in organisms have been analysed by using branch ordering and bifurcation ratio techniques originally developed by geomorphologists to study stream drainage patterns. An underlying assumption of these methods is that branching patterns are stationary across boundaries of scale so that a subsample will accurately model a larger structure. This assumption of stationarity is not generally valid for tree branching patterns. Data from four species reveal two specific patterns of non-stationarity that are directly related to morphological patterns of shoot development. Average bifurcation ratios appear to be inappropriate descriptions of tree branching patterns, since they are based solely on relative branch position and ignore biologically important details of form and development. ‘Certain assumptions are liable, through long usage, to become part of the recognized currency of thought, although most of those who employ them have never made any effort to test their validity .’ Agnes Arber, 1933
The Royal Society - Tập 228 Số 1251 - Trang 187-194 - 1986
Understanding the intrinsic circuitry of the cat’s lateral geniculate nucleus: electrical properties of the spine-triad arrangement Electron-microscopic studies of relay cells in the lateral geniculate nucleus of the cat have shown that the retinal input to X-cells is associated with a special synaptic circuitry, termed the spine-triad. The retinal afferent makes an asymmetrical synapse with both a dendritic appendage of the X-cell and a geniculate interneuron. The interneuron contacts in turn the same dendritic appendage with a symmetrical synaptic profile. The retinal input to geniculate Y-cells is predominantly found on dendritic shafts without any triadic arrangement. We explore the integrative properties of X- and Y-cells resulting from this striking dichotomy in synaptic architecture. The basis of our analysis is the solution of the passive cable equation in a HRP-stained and reconstructed geniculate X-cell with known somatic input resistance. Recent evidence shows that geniculate interneurons stain for glutamic acid decarboxylase, the synthesizing enzyme for the inhibitory neuro-transmitterγ- aminobutyric acid (GABA). Under the assumption that the GABAergic inhibition has a reversal potential close to the resting potential of the cell, activation of the interneuron reduces very efficiently the excitatory postsynaptic potential induced by the retinal afferentwithout affecting the electrical activity in the rest of the cell. Therefore, the spine—triad circuit implements the analogue version of an AND—NOT gate. Functionally, this corresponds to a presynaptic, feed-forward type of inhibition of the optic tract terminal although inhibition actually occurs at a postsynaptic site. If the inhibition has a reversal potential well below the resting potential, the inhibitory postsynaptic potential would hyperpolarize large parts of the relay cell, abolishing the local character of the synaptic veto operation. Since Y-cells lack this structure, inhibition acts globally, reducing the general electrical activity of the cell. We propose that geniculate interneurons gate the flow of visual information into the X-system as a function of the behavioural state of the animal, enhancing the centre-surround antagonism and possibly mediating reciprocal lateral inhibition and eye-movement-related suppression.
The Royal Society - Tập 225 Số 1240 - Trang 365-390 - 1985
The projection from the lateral geniculate nucleus to the prestriate cortex of the macaque monkey By injecting the enzyme horseradish peroxidase into the prestriate cortex of the macaque monkey and examining the lateral geniculate nucleus (l. g. n.) for retrograde label, the presence of a direct projection from the l. g. n. to prestriate visual cortex (Brodmann’s areas 18 and 19) was confirmed. Labelled cells occurred in all layers of the l. g. n., distributed in a roughly columnar fashion. The large scatter in cell distribution indicated a lower retinotopic precision for this projection than for the one to area 17. Labelled cells are of a medium to large size and, in each section, a few were located near the laminar border or in interlaminar zones. The functional significance of this projection is discussed.
The Royal Society - Tập 213 Số 1190 - Trang 73-80 - 1981
Fine structural morphology of identified X- and Y-cells in the cat's lateral geniculate nucleus Four physiologically identified neurons in the A laminae of the cat’s dorsal lateral geniculate nucleus were filled with horseradish peroxidase and studied using the electron microscope. Two were X-cells and two were Y-cells. Each had electrophysiological properties appropriate for its X- or Y-cell class, and each also had an axon that projected into the optic radiation, indicative of a geniculocortical relay cell. Representative samples from about 10% of each neuron’s entire dendritic arbor (proximal and distal) were taken to obtain an estimate of the types and distributions of synapses contacting these arbors. One X-cell had a cytoplasmic laminar body, but there were no other significant cytological differences seen among the neurons. Common to each of the neurons were the following synaptic features: (i) retinal terminals (r. l. p.) were mostly or entirely restricted to proximal dendrites or dendritic appendages (< 100 μm from the soma). These terminals constituted about 15-25% of the synapses on the proximal dendrites, (ii) Terminals with flattened or pleomorphic synaptic vesicles (f. terminals) were predominant on the proximal dendrites (30-55% of the total synapses for that region) and were mainly located near the retinal terminals. A smaller percentage (10-20%) were also distributed onto the distal dendrites, (iii) Small terminals with round synaptic vesicles (r. s. d.), many presumably having a cortical origin, predominated (60-80%) on distal dendrites (> 100 μm), but also formed a large proportion (40-70%) of the synapses on the intermediate (50-150 μm) dendrites. Total synaptic contacts for one X-cell and one Y-cell were estimated at about 4000 and 5000, respectively. The major fine structural differences observed between X- and Y-cells were almost entirely related to the retinal afferents. First, the retinal synapses for X-cells were mostly made on to dendritic appendages (spines, etc.), whereas Y-cells had most of their retinal synapses onto the shafts of primary and proximal secondary dendrites (that is, near branch points). Second, the retinal terminals that contacted X-cell dendrites nearly always formed triadic arrangements that included nearby f. terminals, but those on Y-cells rarely did so. Finally, the main type of f. terminals associated with X-cells were morphologically different from most of those associated with the Y-cells, and this also related directly to the triadic arrangements; that is, f. terminals in the triadic arrangements were morphologically distinguishable from f. terminals that did not participate in triadic arrangements. Even though the present sample is quite small, these morphological differences between X- and Y-cells indicate that they might be the synaptic basis for some of the differential processing of information occurring for the two cell types in the lateral geniculate nucleus.
The Royal Society - Tập 221 Số 1225 - Trang 411-436 - 1984
A model of neuronal bursting using three coupled first order differential equations We describe a modification to our recent model of the action potential which introduces two additional equilibrium points. By using stability analysis we show that one of these equilibrium points is a saddle point from which there are two separatrices which divide the phase plane into two regions. In one region all phase paths approach a limit cycle and in the other all phase paths approach a stable equilibrium point. A consequence of this is that a short depolarizing current pulse will change an initially silent model neuron into one that fires repetitively. Addition of a third equation limits this firing to either an isolated burst or a depolarizing afterpotential. When steady depolarizing current was applied to this model it resulted in periodic bursting. The equations, which were initially developed to explain isolated triggered bursts, therefore provide one of the simplest models of the more general phenomenon of oscillatory burst discharge.
The Royal Society - Tập 221 Số 1222 - Trang 87-102 - 1984
The assembly of ionic currents in a thalamic neuron III. The seven-dimensional model
We replace our earlier three-dimensional
h
a
-model of a thalamic neuron (Rose & Hindmarsh,
Proc. R. Soc. Lond.
B 237, 289–312 (1989
b
)) by a seven-dimensional
h
a
-model. The stability and state diagrams for this seven-dimensional model are shown to be similar to those of the three-dimensional system with which it is compared. Two examples that illustrate how the seven-dimensional model can be related to experimental recordings are then discussed in detail. In each case we show the state and stability diagrams and the time courses of the different ionic currents during the burst response. We discuss how the various components of the state diagram could be determined experimentally. These experiments are illustrated by using our model to simulate the results of actual experiments. Finally the state and stability diagrams are transferred to a current–voltage diagram. This shows the advantage of the state diagram compared with the current–voltage diagram for representing the dynamics of the firing of a thalamic neuron.
The Royal Society - Tập 237 Số 1288 - Trang 313-334 - 1989
The assembly of ionic currents in a thalamic neuron. II. The stability and state diagrams
In the previous model of a thalamic neuron (R. M. Rose & J. L. Hindmarsh,
Proc. R. Soc. Lond
. B 237, 267-288 (1989)), which we referred to as the
z
-model, the burst response was terminated by the slow activation of a subthreshold outward current. In this paper we show that similar results can be obtained if the burst response is terminated by slow inactivation of the subthreshold inward current,
I
s
a
. We illustrate the use of this new model, which we refer to as the
h
a
-model, by using it to explain the response of a thalamic neuron to a double ramp current. The main aim of the paper is to show how the stability and state diagrams introduced previously can be used to explain various types of firing pattern of thalamic and other neurons. We show that increasing the threshold for the fast action potentials leads to low threshold spikes of increased amplitude. Also, addition of a second subthreshold inward current adds a new stability region, which enables us to explain the origin of plateau potentials. In addition, various types of subthreshold oscillation are produced by relocating a previously stable equilibrium point in an unstable region. Finally, we predict a sequence of responses to current steps from different levels of background current that extends the burst, rest, tonic sequence of thalamic neurons. The stability and state diagrams therefore provide us with a useful way of explaining further properties of thalamic neurons and appear to have further applications to other mammalian neurons.
The Royal Society - Tập 237 Số 1288 - Trang 289-312 - 1989
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