The impact of internodal segmentation in biophysical nerve fiber models

Implementation of double cable models to simulate the behavior of myelinated peripheral nerve fibers requires defining a segmentation of the internode between successive nodes of Ranvier. The number of internodal segments is a model parameter that is not well agreed on, with values in the literature ranging from 1 to more than 500. Moreover, a lot of studies also lack a sensitivity study or a rationale behind the implementation used. In a model of a myelinated nerve fiber developed in our group, the segmentation scheme (i.e., the number of segments and their individual morphology) strongly influenced model outcomes such as action potential shape and velocity, stimulation threshold and absolute refractory period. In the present study these influences were investigated systematically in homogeneous neurons with different diameters. Uniformly segmented internodes were found to require several hundreds of segments (and associated computational power) to reach model outcomes differing by less than 1 % from the asymptotic value. In fact, in the majority of segmentation schemes the main determinant is not the number of segments, but the length λ of the internodal segments directly adjacent to the nodes of Ranvier. If λ is larger than approximately 10 μm, model outcomes for the tested fibers are almost independent of the total number of segments. Furthermore, λ can be optimized to enable models using just three segments per internode, to reach physiologically relevant model outcomes with limited computational resources. However, to study anatomical or physiological details of the internode itself, an appropriately detailed segmentation scheme is crucial.     

Techniques for obtaining analytical solutions to the multicylinder somatic shunt cable model for passive neurones.

The somatic shunt cable model for neurones is extended to the case in which several equivalent cylinders, not necessarily of the same electrotonic length, emanate from the cell soma. The cable equation is assumed to hold in each cylinder and is solved with sealed end conditions and a lumped soma boundary condition at a common origin. A Green's function (G) is defined, corresponding to the voltage response to an instantaneous current pulse at an arbitrary point along one of the cylinders. An eigenfunction expansion for G is obtained where the coefficients are determined using the calculus of residues and compared with an alternative method of derivation using a modified orthogonality condition. This expansion converges quickly for large time, but, for small time, a more convenient alternative expansion is obtained by Laplace transforms. The voltage response to arbitrary currents injected at arbitrary sites in the dendritic tree (including the soma) may then be expressed as a convolution integral involving G. Illustrative examples are presented for a point charge input.     

Computation of Impulse Initiation and Saltatory Conduction in a Myelinated Nerve Fiber

A mathematical model of the electrical properties of a myelinated nerve fiber is given, consisting of the Hodgkin-Huxley ordinary differential equations to represent the membrane at the nodes of Ranvier, and a partial differential cable equation to represent the internodes. Digital computer solutions of these equations show an impulse arising at a stimulating electrode and being propagated away, approaching a constant velocity. Action potential curves plotted against distance show discontinuities in slope, proportional to the nodal action currents, at the nodes. Action potential curves plotted against time, at the nodes and in the internodes, show a marked difference in steepness of the rising phase, but little difference in peak height. These results and computed action current curves agree fairly accurately with published experimental data from frog and toad fibers.     

Derivation of cable parameters for a reduced model that retains asymmetric voltage attenuation of reconstructed spinal motor neuron dendrites

Spinal motor neurons have voltage gated ion channels localized in their dendrites that generate plateau potentials. The physical separation of ion channels for spiking from plateau generating channels can result in nonlinear bistable firing patterns. The physical separation and geometry of the dendrites results in asymmetric coupling between dendrites and soma that has not been addressed in reduced models of nonlinear phenomena in motor neurons. We measured voltage attenuation properties of six anatomically reconstructed and type-identified cat spinal motor neurons to characterize asymmetric coupling between the dendrites and soma. We showed that the voltage attenuation at any distance from the soma was direction-dependent and could be described as a function of the input resistance at the soma. An analytical solution for the lumped cable parameters in a two-compartment model was derived based on this finding. This is the first two-compartment modeling approach that directly derived lumped cable parameters from the geometrical and passive electrical properties of anatomically reconstructed neurons.     

The “Lillie Transition”: models of the onset of saltatory conduction in myelinating axons

Almost 90 years ago, Lillie reported that rapid saltatory conduction arose in an iron wire model of nerve impulse propagation when he covered the wire with insulating sections of glass tubing equivalent to myelinated internodes. This led to his suggestion of a similar mechanism explaining rapid conduction in myelinated nerve. In both their evolution and their development, myelinating axons must make a similar transition between continuous and saltatory conduction. Achieving a smooth transition is a potential challenge that we examined in computer models simulating a segmented insulating sheath surrounding an axon having Hodgkin-Huxley squid parameters. With a wide gap under the sheath, conduction was continuous. As the gap was reduced, conduction initially slowed, owing to the increased extra-axonal resistance, then increased (the “rise”) up to several times that of the unmyelinated fiber, as saltatory conduction set in. The conduction velocity slowdown was little affected by the number of myelin layers or modest changes in the size of the “node,” but strongly affected by the size of the “internode” and axon diameter. The steepness of the rise of rapid conduction was greatly affected by the number of myelin layers and axon diameter, variably affected by internode length and little affected by node length. The transition to saltatory conduction occurred at surprisingly wide gaps and the improvement in conduction speed persisted to surprisingly small gaps. The study demonstrates that the specialized paranodal seals between myelin and axon, and indeed even the clustering of sodium channels at the nodes, are not necessary for saltatory conduction.     

On shunting inhibition

The interaction between excitation and inhibition is analyzed for nerve cylinders when reversal potentials for synaptic action are included. Both impulsive and sustained conductance changes are employed to model synaptic action.Exact results, in terms of Green's functions are obtained for the solutions of the cable equation with reversal potentials when there are impulsive conductance changes. The amplification factor for an inhibitory input due to a prior excitatory input is found exactly. In the case of an infinite cylinder, the dependence of this factor on the spatial separation of the excitatory and inhibitory synapses is one plus a Gaussian density function. Similar results aply when excitation follows inhibition. There is shunting inhibition even for impulsive conductance changes in the cable, but it is very different from that for sustained conductance changes. The interaction of excitation and inhibition is also studied in the full cable equation with reversal potentials and sustained conductance changes. An exact result is obtained for the potential in response to simultaneous excitation and inhibition at the same space point in an infinite cable. The effects of timing and spatial separation of inputs is analyzed in a finite nerve cylinder by numerically integrating the cable equation by the Crank-Nicolson method. Shunting inhibition is found to be most effective, for the chosen parameter values, when inhibition quickly foolows excitation. The EPSP amplitude at the soma is found to be roughly proportional to the distance from the soma to the site of inhibition when the excitation is at the center of the nerve cylinder.Dedicated to Jane Pauley     

Segmented and "equivalent" representation of the cable equation

The linear cable theory has been applied to a modular structure consisting of n repeating units each composed of two subunits with different values of resistance and capacitance. For n going to infinity, i.e., for infinite cables, we have derived analytically the Laplace transform of the solution by making use of a difference method and we have inverted it by means of a numerical procedure. The results have been compared with those obtained by the direct application of the cable equation to a simplified nonmodular model with "equivalent" electrical parameters. The implication of our work in the analysis of the time and space course of the potential of real fibers has been discussed. In particular, we have shown that the simplified ("equivalent") model is a very good representation of the segmented model for the nodal regions of myelinated fibers in a steady situation and in every condition for muscle fibers. An approximate solution for the steady potential of myelinated fibers has been derived for both nodal and internodal regions. The applications of our work to other cases dealing with repeating structures, such as earthworm giant fibers, have been discussed and our results have been compared with other attempts to solve similar problems.     

浙江天童常绿阔叶林藤本植物的适应生态学研究 II.攀援能力和单株攀援效率

选用节间长、节数和支持木大小作为攀援能力的指标,单株攀援的支持木种类、数量及水平扩展面积作为单株攀援效率的指标,对浙江天童常绿阔叶林中5类12种大中型木质藤本的攀援能力及生态适应性和其中2种藤本单株攀援效率进行了分析,结果发现:1)不同攀援类型的节间长和节数存在显著差异,对支持木大小要求不同,反映出攀援能力在类型间的差异;没有一类藤本兼具所有优势,攀援能力是节间长、节数和支持木大小不同组合的结果;卷须类的广东蛇葡萄(Ampelopsiscantoniensis)和暗色菝葜(Smilaxlanceifoliavar.opaca)节间最长、节数居中、要求较小支持木;枝卷类的香港黄檀(Dalbergiamillettii)节数和支持木大小与卷须类接近,但节间较短;不定根类的中华常春藤(Hederanepalensisvar.sinensis)、扶芳藤(Euonymusfortunei)、薜荔(Ficuspumila)和络石(Trachlospermumjasmi noides)以及吸盘类的异叶爬山虎(Parthnocissusheterophylla)节间最短、但节数最多,要求较大支持木;茎缠绕类的白花野木瓜(Stauntonialeucantha)、羊角藤(Morindaumbellata)、忍冬(Lonicerajaponica)和薯莨(Dioscoreacirrhosa)节间长居中,节数最少,要求中等但幅度较宽的支持木。2 )同类型不同藤本的节间长、节数和支持木大小的差异反映出由非亲缘关系植     

The role of the leaves in the regulation of internode elongation in Phaseolus vulgaris

Time-course patterns of leaf and internode elongation were studied in bean plants. Each leaf started its main elongation period when the leaf below reached half of its final length. The onset of leaf unfolding was nearly synchronous with the initiation of the elongation of the subjacent internode. Excision of young leaves increased the rate of stem elongation as a result of an earlier unfolding of the next upper leaves and the concomitant advancement in the elongation of their subjacent internodes. IAA or NAA (1% in lanolin) suppressed the enhancement effects of leaf excision on leaf and internode elongation. The excision of a young leaf increased the final length of internodes located below it, and at the same time decreased the final length of the internodes located above the excised leaf. The reduction was greater the younger the internode. Differences in internode elongation after leaf excision were related to changes during internode ontogenesis in their relative response to the availability of assimilates on the one hand, and on the other hand to hormonal factors transported acropetally from the young leaves to the growing internodes.     

The Effect of Single and Repeated Gibberellic Acid Treatment on Internode Number and Length in Dwarf Peas

In long-term experiments comparing the results of single and repeated treatment of dwarf peas with gibberellic acid, both once-treated and weekly-treated plants formed more internodes than control plants. The initial high rate of internode formation in the once-treated peas dropped rapidly reaching that of control plants three weeks after treatment. Weekly-treated plants maintained a higher rate of internode formation. The ratio of internode lengths of once-treated plants to control plants was high (over 3) for the internodes which were elongating during or shortly after the single treatment but dropped rapidly and approached unity in later-developed internodes. The ratio of internode lengths of weekly-treated plants to control plants continued to be high throughout the experiment.     

Compensation of the low-pass filter properties of the current measuring internode in potential-clamped myelinated nerve fibres

The bandwidth of membrane current measurements in potential-clamped Ranvier nodes is limited by the low-pass filter properties of the internodes. If about 35% of the current measuring internode is grounded via an additional electrode, the bandwidth of this internode increases by a factor of about 40. Consequently, in potential clamp experiments the measured time course of the early ionic currents changes markedly, while the duration of the capacity current is shortened.     

Solutions for transients in arbitrarily branching cables: III. Voltage clamp problems.

Branched cable voltage recording and voltage clamp analytical solutions derived in two previous papers are used to explore practical issues concerning voltage clamp. Single exponentials can be fitted reasonably well to the decay phase of clamped synaptic currents, although they contain many underlying components. The effective time constant depends on the fit interval. The smoothing effects on synaptic clamp currents of dendritic cables and series resistance are explored with a single cylinder + soma model, for inputs with different time courses. "Soma" and "cable" charging currents cannot be separated easily when the soma is much smaller than the dendrites. Subtractive soma capacitance compensation and series resistance compensation are discussed. In a hippocampal CA1 pyramidal neurone model, voltage control at most dendritic sites is extremely poor. Parameter dependencies are illustrated. The effects of series resistance compound those of dendritic cables and depend on the "effective capacitance" of the cell. Plausible combinations of parameters can cause order-of-magnitude distortions to clamp current waveform measures of simulated Schaeffer collateral inputs. These voltage clamp problems are unlikely to be solved by the use of switch clamp methods.     

Origin of Nodes and lnternodes in Plant Shoots. I. Transverse Zonation of Apical Parts of the Shoot

Several plant species possessing at least two leaves at eachnode, or well anatomically distinguished nodes, were examinedfor potential differences in node and internode construction,and unusual histological features were sought as markers ofthe nodes and internodes. In this capacity idioblastic cellsfilled with phenolic compounds can serve, in either node (Polygonum)or internode (Sambucus). When constructed in a special way intercellularspaces can also be markers, e.g. in Rumex, as can glands innodes and internodes (Angelica, Ugustrum, Astrantia). Such histologicaldifferences exist exclusively in internodes or nodes examined,permitting inferences to be made about the origins of theseplant parts. Idioblasts, node, internode, intercellular spaces, phenolics     

The role of stolon internodes for ramet survival after clone fragmentation in Potentilla anserina

Stoloniferous herbs are common in disturbed habitats, where interconnected ramets frequently become separated by water, grazing animals or human activity. Stolon internodes serve as spacers and transport channels for resources, but they may also fulfil a storage function. We simulated physical disturbance and breakage of stolon connections in the stoloniferous herb Potentilla anserina, to test the hypothesis that juvenile ramets can survive clone fragmentation better if they remain attached to an internode after being severed from the rest of the clone. Juvenile, unrooted ramets were cut off from stolons with and without the preceding (basal) internode and placed on wet soil in a greenhouse. All ramets with an attached internode survived, while survival rates dropped to 37% for ramets from which internodes had been removed. The dry weight per unit volume of internodes decreased by about 20% during the experiment. We conclude that stolons are likely to fulfil a storage function that may be of ecological importance in disturbed habitats.     

Soluble and insoluble invertase activity in elongating Tulipa gesneriana flower stalks

Previously 'frozen' Tulipa gesneriana L. bulbs cv. Apeldoorn, were planted and grown at higher temperatures to study the role of invertase (EC 3.2.1.26) in the cold-induced elongation of the flower stalk internodes. After planting, flower stalks were left intact, or, the leaves and flower bud were both removed to inhibit internode elongation. In intact flower stalks, elongation of the internodes was accompanied by an accumulation of glucose and an initial decrease in the sucrose content g,⁻¹ dry weight. Insoluble invertase activity g,⁻¹ dry weight hardly changed, but soluble invertase activity showed a peak pattern, that was related, at least for the greater part, to the changes in the sugar contents. Peak activities of soluble invertase were found during (lower- and uppermost internodes) or around the onset of the rapid phase of internode elongation (middle internodes). Internode elongation and glucose accumulation immediately ceased when the leaves and flower bud were removed. Insoluble invertase activity g,⁻¹ dry weight remained at its initial level (lowermost internode) or increased more towards the upper internodes. Soluble invertase activity did not further increase (uppermost internode) or decreased abruptly to a low level. It is concluded that soluble invertase may be one of the factors contributing to glucose accumulation and internode elongation in the tulip flower stalk.     

Transient response in a somatic shunt cable model for synaptic input activated at the terminal

Rall's neuron model is extended by including a non-uniform time constant together with synaptic input modeled as a square step of conductance. An analytic solution (in series form) for the electrotonic potential is obtained. The major conclusion reached is that a lower somatic time constant attenuates the amplitude of the potential at the soma, brought about by the activation of a synapse located at the distal end of the dendritic cable in an initially polarized neuron.     

Analytical solution of the cable equation with synaptic reversal potentials for passive neurons with tip-to-tip dendrodendritic coupling

A passive cable model is presented for a pair of electrotonically coupled neurons in order to investigate the effects of tip-to-tip dendrodendritic gap junctions on the interaction between excitation and either pre or postsynaptic inhibition. The model represents each dendritic tree by a tapered equivalent cylinder attached to an isopotential soma. Analytical solution of the cable equation with synaptic reversal potentials is considered for each neuron to yield a system of Volterra integral equations for the voltage. The solution to the system of linear integral equations (expressed as a Neumann series) is used to determine the current spread within the two coupled neurons, and to re-examine the sensitivity of the soma potentials (in particular) to the coupling resistance for various loci of synaptic inputs. The model is actually posed generally, so that active as well as passive properties could be considered. In the active case, a system of non-linear integral equations is derived for the voltage.     

MITOTIC ACTIVITY IN THE CAMBIAL ZONE OF PINUS STROBUS

Mitotic activity in the cambial zone of 20-year-old Pinus strobus L. trees was determined quantitatively, using mitotic counts from serial tangential sections of sample pieces. Counts from nine cores 1.6 mm in diameter from each sample piece were averaged and expressed as the number of mitoses per core with the sampling error. Estimation of the number of cambial zone cells per core permitted calculation of mitotic indices. In the fourth internode from the ground the first mitoses were observed on April 15 and the last on September 15. The number of mitoses per sample piece was comparable throughout each internode at any one sampling, but the number was higher in internodes within the crown than in those below it. Mitotic index was not appreciably higher within the crown, but it decreased from 3.7 in spring to 2.0 in fall. An internode in the crown sampled May 11–13 showed afternoon peaks in mitotic activity, but an internode below the crown showed no peaks, nor did other internodes sampled later in the season.     

Elongation of Stem Internodes in the Japanese Morning Glory (Pharbitis nil) in Relation to Auxin Destruction

The relationship between auxin destruction and stem internode elongation was investigated in the vines of the Japanese morning glory (Pharbitis nil Choisy). In young plants an age-dependent gradient was demonstrated in which the decreasing rate of elongation of older internodes correlated with an increasing ability of such tissue to destroy indoleacetic acid. Fragments of tissue from old internodes when incubated with indoleacetic acid (IAA), destroyed the hormone immediately and rapidly; in contrast, young, rapidly elongating internode tissue destroyed IAA only after a lag of several hours. In older plants the gradient was more erratic towards the middle of the plant but old and young tissue behaved as in young plants, i.e., old internodes destroyed IAA rapidly whereas young internodes did not. It appears reasonable to conclude that cessation of elongation in maturing internodes is brought about by developing an internal environment in which auxin is rapidly destroyed.