Quantitative relations between transient BOLD responses, cortical energetics, and impulse firing in different cortical regions.

The blood oxygen level-dependent (BOLD) functional magnetic resonance imaging signal arises as a consequence of changes in blood flow (cerebral blood flow) and oxygen usage (cerebral metabolic rate of oxygen) that in turn are modulated by changes in neuronal activity. Much attention has been given to both theoretical and experimental aspects of the energetics but not to the neuronal activity. Here we use our previous theory relating the steady-state BOLD signal to neuronal activity and amalgamate it with the standard dynamic causal model (DCM, Friston) theory to produce a quantitative model relating the time-dependent BOLD signal to the underlying neuronal activity. Unlike existing treatments, this new theory incorporates a nonzero baseline activity in a completely consistent way and is thus able to account for both positive and negative BOLD signals. It can reproduce a wide variety of experimental BOLD signals reported in the literature solely by adjusting the neuronal input activity. In this way it provides support for the claim that the main features of the signals, including poststimulus undershoot and overshoot, are principally a result of changes in neuronal activity.NEW & NOTEWORTHY A previous model relating the steady-state blood oxygen level-dependent (BOLD) signal to neuronal activity, both above and below baseline, is extended to account for transient BOLD signals. This allows for a detailed investigation of the role neuronal activity can play in such signals and also encompasses poststimulus undershoot and overshoot. A wide variety of experimental BOLD signals are reproduced solely by adjusting the neuronal input activity, including recent results regarding the BOLD signal in patients with schizophrenia.

Quantitative relations between BOLD responses, cortical energetics, and impulse firing.

The blood oxygen level-dependent (BOLD) functional magnetic resonance imaging signal arises as a consequence of changes in blood flow and oxygen usage that in turn are modulated by changes in neural activity. Much attention has been given to both theoretical and experimental aspects of the energetics but not to the neural activity. Here we identify the best energetic theory for the steady-state BOLD signal on the basis of correct predictions of experimental observations. This theory is then used, together with the recently determined relationship between energetics and neural activity, to predict how the BOLD signal changes with activity. Unlike existing treatments, this new theory incorporates a nonzero baseline activity in a completely consistent way and is thus able to account for both sustained positive and negative BOLD signals. We also show that the increase in BOLD signal for a given increase in activity is significantly smaller the larger the baseline activity, as is experimentally observed. Furthermore, the decline of the positive BOLD signal arising from deeper cortical laminae in response to an increase in neural firing is shown to arise as a consequence of the larger baseline activity in deeper laminae. Finally, we provide quantitative relations integrating BOLD responses, energetics, and impulse firing, which among other predictions give the same results as existing theories when the baseline activity is zero. NEW & NOTEWORTHY We use a recently established relation between energetics and neural activity to give a quantitative account of BOLD dependence on neural activity. The incorporation of a nonzero baseline neural activity accounts for positive and negative BOLD signals, shows that changes in neural activity give BOLD changes that are smaller the larger the baseline, and provides a basis for the observed inverse relation between BOLD responses and the depth of cortical laminae giving rise to them.

Origins of the BOLD changes due to synaptic activity at astrocytes abutting arteriolar smooth muscle.

We have recently provided a detailed model that links glutamatergic synaptic activity to volume and blood flow changes in nearby arterioles [Bennett, M.R., Farnell, L., Gibson, W.G., 2008. Origin of blood volume change due to glutamatergic synaptic activity at astrocytes abutting on arteriolar smooth muscle cells. J. Theor. Biol. 250, 172-185]. This neurovascular coupling model is used in the present work to predict changes in deoxyhemoglobin (Hbr) in capillaries, arterioles, venules and veins due to glutamatergic synaptic activity and hence the changes in the blood oxygen level dependent (BOLD) signals recorded by functional magnetic resonance imaging. The model provides a quantitative account of Hbr changes observed in each of the vascular compartments following stimulation of somatosensory cortex and visual cortex and of the BOLD signal following stimulation of motor and visual cortex.

Origins of blood volume change due to glutamatergic synaptic activity at astrocytes abutting on arteriolar smooth muscle cells.

The cellular mechanisms that couple activity of glutamatergic synapses with changes in blood flow, measured by a variety of techniques including the BOLD signal, have not previously been modelled. Here we provide such a model, that successfully accounts for the main observed changes in blood flow in both visual cortex and somatosensory cortex following their stimulation by high-contrast drifting grating or by single whisker stimulation, respectively. Coupling from glutamatergic synapses to smooth muscle cells of arterioles is effected by astrocytes releasing epoxyeicosatrienoic acids (EETs) onto them, following glutamate stimulation of the astrocyte. Coupling of EETs to the smooth muscle of arterioles is by means of potassium channels in their membranes, leading to hyperpolarization, relaxation and hence an increase in blood flow. This model predicts a linear increase in blood flow with increasing numbers of activated astrocytes, but a non-linear increase with increasing glutamate release.

Mechanisms of calcium sequestration during facilitation at active zones of an amphibian neuromuscular junction.

The calcium transients (Delta[Ca(2+)](i)) at active zones of amphibian (Bufo marinus) motor-nerve terminals that accompany impulses, visualized using a low-affinity calcium indicator injected into the terminal, are described and the pathways of subsequent sequestration of the residual calcium determined, allowing development of a quantitative model of the sequestering processes. Blocking the endoplasmic reticulum calcium pump with thapsigargin did not affect Delta[Ca(2+)](i) for a single impulse but increased its amplitude during short trains. Blocking the uptake of calcium by mitochondria with CCCP had little effect on Delta[Ca(2+)](i) of a single impulse but greatly increased its amplitude during short trains. This present compartmental model is compatible with our previous Monte Carlo diffusion model of Ca(2+) sequestration during facilitation [Bennett, M.R., Farnell, L., Gibson, W.G., 2004. The facilitated probability of quantal secretion within an array of calcium channels of an active zone at the amphibian neuromuscular junction. Biophys. J. 86(5), 2674-2690], with the single plasmalemma pump in that model now replaced by separate pumps for the plasmalemma and endoplasmic reticulum, as well as the introduction of a mitochondrial uniporter.

A quantitative model of purinergic junctional transmission of calcium waves in astrocyte networks.

A principal means of transmitting intracellular calcium (Ca2+) waves at junctions between astrocytes involves the release of the chemical transmitter adenosine triphosphate (ATP). A model of this process is presented in which activation of purinergic P2Y receptors by ATP triggers the release of ATP, in an autocrine manner, as well as concomitantly increasing intracellular Ca2+. The dependence of the temporal characteristics of the Ca2+ wave are shown to critically depend on the dissociation constant (K(R)) for ATP binding to the P2Y receptor type. Incorporating this model astrocyte into networks of these cells successfully accounts for many of the properties of propagating Ca2+ waves, such as the dependence of velocity on the type of P2Y receptor and the time-lag of the Ca2+ wave behind the ATP wave. In addition, the conditions under which Ca2+ waves may jump from one set of astrocytes across an astrocyte-free lane to another set of astrocytes are quantitatively accounted for by the model. The properties of purinergic transmission at astrocyte junctions may determine many of the characteristics of Ca2+ propagation in networks of these cells.

A quantitative description of the contraction of blood vessels following the release of noradrenaline from sympathetic varicosities.

A model is presented that highlights the principal factors determining the form and extent of contraction in arteries upon stimulation of their sympathetic nerve supply. This model incorporates a previous quantitative model of the process of noradrenaline (NAd) diffusion into the vascular media and reuptake into sympathetic varicosities during nerve stimulation (J. Theor. Biol. 226 (2004) 359). It is also dependent on a model of how the subsequent activation of metabotropic receptors initiates a G-protein cascade, resulting in the production of inositol trisphosphate (IP3) and an increase in intracellular calcium concentration, [Ca2+]i, in the smooth muscle cells (J. Theor. Biol. 223 (2003) 93). In the present work we couple this rise in [Ca2+]i to the increase in phosphorylated myosin bound to actin in the cells and hence determine the force development in arteries due to nerve stimulation. The model accounts for force development as a function of [Ca2+]i and for the rate of change of force as a function of the rate of change of [Ca2+]i in single smooth muscle cells. It also accounts for the characteristic time course of the force developed by the media of the rat-tail artery upon nerve stimulation. This consists of a rapid rise to a transient peak followed by a sustained plateau of contraction during the stimulation period, after which the contraction slowly decays back to baseline at a rate dependent on the strength of the stimulation. The model indicates that the transient peak is primarily due to the partial block of the IP3 receptor by the rise in [Ca2+]i and that the main determinant of the equilibrium condition indicated by the plateau phase is the rate of pumping of calcium into the sarcoplasmic reticulum. The relatively slow decline of contraction at the end of nerve stimulation is primarily a consequence of the slow rates of removal of NAd from the media by diffusion and reuptake into the sympathetic varicosities. The model thus provides a quantitative account of vascular smooth muscle contraction upon sympathetic nerve stimulation.

Calcium mobilization and spontaneous transient outward current characteristics upon agonist activation of P2Y2 receptors in smooth muscle cells.

A quantitative model is provided that links the process of metabotropic receptor activation and sequestration to the generation of inositol 1,4,5-trisphosphate, the subsequent release of calcium from the central sarcoplasmic reticulum, and the consequent release of calcium from subsarcolemma sarcoplasmic reticulum that acts on large-conductance potassium channels to generate spontaneous transient outward currents (STOCs). This model is applied to the case of STOC generation in vascular A7r5 smooth muscle cells that have been transfected with a chimera of the P2Y(2) metabotropic receptor and green fluorescent protein (P2Y(2)-GFP) and exposed to the P2Y(2) receptor agonist uridine 5'-triphosphate. The extent of P2Y(2)-GFP sequestration from the membrane on exposure to uridine 5'-triphosphate, the ensuing changes in cytosolic calcium concentration, as well as the interval between STOCs that are subsequently generated, are used to determine parameter values in the model. With these values, the model gives a good quantitative prediction of the dynamic changes in STOC amplitude observed upon activation of metabotropic P2Y(2) receptors in the vascular smooth muscle cell line.

The facilitated probability of quantal secretion within an array of calcium channels of an active zone at the amphibian neuromuscular junction.

A Monte Carlo analysis has been made of the phenomenon of facilitation, whereby a conditioning impulse leaves nerve terminals in a state of heightened release of quanta by a subsequent test impulse, this state persisting for periods of hundreds of milliseconds. It is shown that a quantitative account of facilitation at the amphibian neuromuscular junction can be given if the exocytosis is triggered by the combined action of a low-affinity calcium-binding molecule at the site of exocytosis and a high-affinity calcium-binding molecule some distance away. The kinetic properties and spatial distribution of these molecules at the amphibian neuromuscular junction are arrived at by considering the appropriate values that the relevant parameters must take to successfully account for the experimentally observed amplitude and time course of decline of F1 and F2 facilitation after a conditioning impulse, as well as the growth of facilitation during short trains of impulses. This model of facilitation correctly predicts the effects on facilitation of exogenous buffers such as BAPTA during short trains of impulses. In addition, it accounts for the relative invariance of the kinetics of quantal release due to test-conditioning sequences of impulses as well as due to change in the extent of calcium influx during an impulse.

A quantitative description of the diffusion of noradrenaline in the media of blood vessels following its release from sympathetic varicosities.

A quantitative model is provided which describes how noradrenaline (NAd), released from varicosities at the adventitial surface of an artery, either diffuses into the media of the vessel to reach the intimal surface, diffuses into the volume of solution surrounding the artery, or is removed by the uptake 1 process in the varicosities. These predictions are then compared with experimental evaluations of the extent of changes in NAd to be found at the adventitial and intimal surfaces of the rat-tail artery, during and after trains of impulses, as determined using amperometry. In the model of the blood vessel there is a sequential decrease in the diffusion constant of NAd from the surrounding solution, to the adventitia, to the media, to the endothelium, to rise again in the lumen of the vessel; there is also an uptake 1 NAd pump in the varicosities described by Michaelis-Menten kinetics. This model is shown to provide a quantitative account of the spatial and temporal changes in NAd observed following trains of impulses at different frequencies of stimulation (5-40 Hz) for different periods of times (10-40 s). Changes in the spatio-temporal distribution of NAd observed following block of the uptake 1 NAd pump were also successfully predicted by the model. It is concluded that, within the context of the model, there is no need to evoke special mechanisms of buffering at the sympathetic varicosities, nor distinctions on the basis that only secreting varicosities utilize the uptake 1 mechanism, in order to describe the dynamics of NAd distribution in arteries during nerve activity.

Metabotropic receptor activation, desensitization and sequestration-I: modelling calcium and inositol 1,4,5-trisphosphate dynamics following receptor activation.

A mathematical account is given of the processes governing the time courses of calcium ions (Ca2+), inositol 1,4,5-trisphosphate (IP(3)) and phosphatidylinositol 4,5-bisphosphate (PIP(2)) in single cells following the application of external agonist to metabotropic receptors. A model is constructed that incorporates the regulation of metabotropic receptor activity, the G-protein cascade and the Ca2+ dynamics in the cytosol. It is subsequently used to reproduce observations on the extent of desensitization and sequestration of the P(2)Y(2) receptor following its activation by uridine triphosphate (UTP). The theory predicts the dependence on agonist concentration of the change in the number of receptors in the membrane as well as the time course of disappearance of receptors from the plasmalemma, upon exposure to agonist. In addition, the extent of activation and desensitization of the receptor, using the calcium transients in cells initiated by exposure to agonist, is also predicted. Model predictions show the significance of membrane PIP(2) depletion and resupply on the time course of IP(3) and Ca2+ levels. Results of the modelling also reveal the importance of receptor recycling and PIP(2) resupply for maintaining Ca2+ and IP(3) levels during sustained application of agonist.

Metabotropic receptor activation, desensitization and sequestration-II: modelling the dynamics of the pleckstrin homology domain.

Recent observations have been made regarding the generation of inositol 1,4,5-trisphosphate (IP(3)), using chimeras of green fluorescent protein and the pleckstrin homology domain of phospholipase C-delta. In this paper a model is presented giving the quantitative relations between the green fluorescent protein-pleckstrin homology domain (GFP-PHD) construct and membrane phosphatidylinositol 4,5-bisphosphate (PIP(2)) levels as well as the concentration of IP(3), the product of hydrolysis of PIP(2). The model can correctly reproduce the dependence of cytosolic GFP-PHD fluorescence on IP(3) concentration. This model extends a previous one (Metabotropic receptor activation, desensitization and sequestration-I: modelling calcium and inositol 1,4,5-trisphosphate dynamics following receptor activation, in this issue) dealing with the processes governing the production of IP(3) and the subsequent calcium (Ca2+) changes in cells following activation of metabotropic receptors. This model is applied to the case of purinergic P(2)Y(2) receptor activation in Madin-Darby Canine Kidney (MDCK) cells with adenosine triphosphate (ATP) (Science 284 (1999) 1527). It is shown that it can correctly reproduce the dependence of GFP-PHD fluorescence on the concentration of P(2)Y(2) receptor ligand, as well as the temporal changes of GFP-PHD fluorescence following application of ligand.

Potential fields in vascular smooth muscle generated by transmitter release from sympathetic varicosities.

A quantitative model is provided of how current flow occurs in the media of blood vessels upon the release of transmitter from autonomic varicosities onto ionotropic receptors located on smooth muscle cells at the adventitial surface of the vessel. In particular, the extent to which potential generated in cells at the adventitial surface (AS) conducts through to cells at the intimal surface (IS) is investigated. Experimental tests of the model have been made for the case of the rat tail artery. The model of the media is an extension of the discrete bidomain syncytium to the case where the smooth muscle syncytium is bounded on two sides by a volume conductor, as is the case with the media of blood vessels. The amplitudes and temporal characteristics of excitatory junction potentials (EJPs), recorded throughout this syncytium following the release of ATP from varicosities located on one side of the syncytium, are predicted by the theory. Current injection into a single cell at the AS will not give rise to a detectable membrane potential at the IS; however, simultaneous injection of current into all the cells at the AS can give rise to a membrane potential at the IS that has an amplitude of about 50% of that at the AS, in agreement with experimental findings. In addition, the effects of perturbing the electrical couplings between cells in the syncytium on the EJPs recorded at different sites in the syncytium are also predicted. This work shows that the discrete bidomain model of the syncytium gives a quantitative description of the current and potential fields that occur throughout the smooth muscle of the media of blood vessels following the release of transmitter from varicosities at the adventitial surface of the vessels. The theory can be applied to the media of blood vessels of any size to determine the relative effectiveness of sympathetic nerves in controlling the excitability of smooth muscle cells through the media.

Neuronal cell death, nerve growth factor and neurotrophic models: 50 years on.

Viktor Hamburger has just died at the age of 100. It is 50 years since he and Rita Levi-Montalcini laid the foundations for the study of naturally occurring cell death and of neurotrophic factors in the nervous system. In a period of less than 10 years, from 1949 to 1958, Hamburger and Levi-Montalcini made the following seminal discoveries: that neuron cell death occurs in dorsal root ganglia, sympathetic ganglia and the cervical column of motoneurons; that the predictions arising from this observation, namely that survival is dependent on the supply of a trophic factor, could be substantiated by studying the effects of a sarcoma on the proliferation of ganglionic processes both in vivo and in vitro; and that the proliferation of these processes could be used as an assay system to isolate the factor. This work provides a short review mostly of the early history of this subject in the context of the Hamburger/Levi-Montalcini paradigm. This acts as an introduction to a consideration of models that have been proposed to account for how the different sources of growth factors provide for the survival of neurons during development. It is suggested that what has been called the 'social-control' model provides the most parsimonious quantitative description of the contribution of trophic factors to neuronal survival, a concept for which we are in debt to Viktor Hamburger and Rita Levi-Montalcini.

Quantal current fields around individual boutons in sympathetic ganglia.

The release of a quantum of neurotransmitter from an active zone of a bouton is accompanied by the flow of extracellular current that creates a potential field about the site of transmitter action beneath the bouton. It is shown theoretically that the density of the field at the peak of the quantal current gives rise to an extracellular potential that declines to values of less than 5 microV at 1.3 microm distance in the circumferential direction around the neuron and equally rapidly in the radial direction away from the neuron. A loose-patch electrode placed over a bouton distorts the quantal field about the bouton and calculations show that under current-clamp conditions, potentials of over 40 microV can be recorded with an electrode of tip diameter 2 microm, provided the separation between the tip and the neuron's surface is about 0.1 microm. Quantal release recorded from visualized boutons on rat monopolar pelvic ganglion cells with loose-patch electrodes is in agreement with the properties of the quantal potential field given in the theoretical analysis.

Quantal and non-quantal current and potential fields around individual sympathetic varicosities on release of ATP.

The electrical phenomena that occur at sympathetic varicosities due to the release of ATP include spontaneous and evoked excitatory junction potentials (SEJPs and EJPs; recorded with an intracellular electrode) as well as fast and slow excitatory junctional currents (EJCs; recorded with a loose-patch electrode placed over varicosities). The electrical analysis of these transients is hampered by lack of a detailed theory describing how current and potential fields are generated upon the release of a quantum of ATP. Here, we supply such a theory and develop a computational model for the electrical properties of a smooth muscle syncytium placed within a volume conductor, using a distributed representation for the individual muscle cells. The amplitudes and temporal characteristics of both SEJPs and fast EJCs are predicted by the theory, but those of the slow EJCs are not. It is shown that these slow components cannot arise as a consequence of propagation of fast quantal components from their site of origin in the muscle syncytium to the point of recording. The possibility that slow components arise by a mechanism of transmitter secretion that is different from quantal release is examined. Experiments that involve inserting peptide fragments of soluble N-ethylmaleimide-sensitive fusion attachment protein (alpha-SNAP) into varicosities, a procedure that is known to block quantal release, left the slow component of release unaffected. This work provides an internally consistent description of quantal potential and current fields about the varicosities of sympathetic nerve terminals and provides evidence for a non-quantal form of transmitter release.

The probability of quantal secretion near a single calcium channel of an active zone.

A Monte Carlo analysis has been made of calcium dynamics and quantal secretion at microdomains in which the calcium reaches very high concentrations over distances of <50 nm from a channel and for which calcium dynamics are dominated by diffusion. The kinetics of calcium ions in microdomains due to either the spontaneous or evoked opening of a calcium channel, both of which are stochastic events, are described in the presence of endogenous fixed and mobile buffers. Fluctuations in the number of calcium ions within 50 nm of a channel are considerable, with the standard deviation about half the mean. Within 10 nm of a channel these numbers of ions can give rise to calcium concentrations of the order of 100 microM. The temporal changes in free calcium and calcium bound to different affinity indicators in the volume of an entire varicosity or bouton following the opening of a single channel are also determined. A Monte Carlo analysis is also presented of how the dynamics of calcium ions at active zones, after the arrival of an action potential and the stochastic opening of a calcium channel, determine the probability of exocytosis from docked vesicles near the channel. The synaptic vesicles in active zones are found docked in a complex with their calcium-sensor associated proteins and a voltage-sensitive calcium channel, forming a secretory unit. The probability of quantal secretion from an isolated secretory unit has been determined for different distances of an open calcium channel from the calcium sensor within an individual unit: a threefold decrease in the probability of secretion of a quantum occurs with a doubling of the distance from 25 to 50 nm. The Monte Carlo analysis also shows that the probability of secretion of a quantum is most sensitive to the size of the single-channel current compared with its sensitivity to either the binding rates of the sites on the calcium-sensor protein or to the number of these sites that must bind a calcium ion to trigger exocytosis of a vesicle.

The probability of quantal secretion within an array of calcium channels of an active zone.

A Monte Carlo analysis has been made of calcium dynamics in submembranous domains of active zones in which the calcium contributed by the opening of many channels is pooled. The kinetics of calcium ions in these domains has been determined using simulations for channels arranged in different geometries, according to the active zone under consideration: rectangular grids for varicosities and boutons and lines for motor-nerve terminals. The effects of endogenous fixed and mobile buffers on the two-dimensional distribution of free calcium ions at these active zones are then given, together with the extent to which these are perturbed and can be detected with different affinity calcium indicators when the calcium channels open stochastically under an action potential. A Monte Carlo analysis of how the dynamics of calcium ions in the submembranous domains determines the probability of exocytosis from docked vesicles is also presented. The spatial distribution of exocytosis from rectangular arrays of secretory units is such that exocytosis is largely excluded from the edges of the array, due to the effects of endogenous buffers. There is a steeper than linear increase in quantal release with an increase in the number of secretory units in the array, indicating that there is not just a local interaction between secretory units. Conditioning action potentials promote an increase in quantal release by a subsequent action potential primarily by depleting the fixed and mobile buffers in the center of the array. In the case of two parallel lines of secretory units exocytosis is random, and diffusion, together with the endogenous calcium buffers, ensures that the secretory units only interact over relatively short distances. As a consequence of this and in contrast to the case of the rectangular array, there is a linear relationship between the extent of quantal secretion from these zones and their length, for lengths greater than a critical value. This Monte Carlo analysis successfully predicts the relationship between the size and geometry of active zones and the probability of quantal secretion at these, the existence of quantal versus multiquantal release at different active zones, and the origins of the F1 phase of facilitation in synapses possessing different active zone geometries.

Quantal potential fields around individual active zones of amphibian motor-nerve terminals.

The release of a quantum from a nerve terminal is accompanied by the flow of extracellular current, which creates a field around the site of transmitter action. We provide a solution for the extent of this field for the case of a quantum released from a site on an amphibian motor-nerve terminal branch onto the receptor patch of a muscle fiber and compare this with measurements of the field using three extracellular electrodes. Numerical solution of the equations for the quantal potential field in cylindrical coordinates show that the density of the field at the peak of the quantal current gives rise to a peak extracellular potential, which declines approximately as the inverse of the distance from the source at distances greater than about 4 microm from the source along the length of the fiber. The peak extracellular potential declines to 20% of its initial value in a distance of about 6 microm, both along the length of the fiber and in the circumferential direction around the fiber. Simultaneous recordings of quantal potential fields, made with three electrodes placed in a line at right angles to an FM1-43 visualized branch, gave determinations of the field strengths in accord with the numerical solutions. In addition, the three electrodes were placed so as to straddle the visualized release sites of a branch. The positions of these sites were correctly predicted on the basis of the theory and independently ascertained by FM1-43 staining of the sites. It is concluded that quantal potential fields at the neuromuscular junction that can be measured with available recording techniques are restricted to regions within about 10 microm of the release site.

Quantal secretion and nerve-terminal cable properties at neuromuscular junctions in an amphibian (Bufo marinus).

The effect of a conditioning depolarizing current pulse (80-200 micros) on quantal secretion evoked by a similar test pulse at another site was examined in visualized motor-nerve terminal branches of amphibian endplates (Bufo marinus). Tetrodotoxin (200 nM) and cadmium (50 microM) were used to block voltage-dependent sodium and calcium conductances. Quantal release at the test electrode was depressed at different distances (28-135 microm) from the conditioning electrode when the conditioning and test pulses were delivered simultaneously. This depression decreased when the interval between conditioning and test current pulses was increased, until, at an interval of approximately 0.25 ms, it was negligible. At no time during several thousand test-conditioning pairs, for electrodes at different distances apart (28-135 microm) on the same or contiguous terminal branches, did the electrotonic effects of quantal release at one electrode produce quantal release at the other. Analytic and numerical solutions were obtained for the distribution of transmembrane potential at different sites along terminal branches of different lengths for current injection at a point on a terminal branch wrapped in Schwann cell, in the absence of active membrane conductances. Solutions were also obtained for the combined effects of two sites of current injection separated by different time delays. This cable model shows that depolarizing current injections of a few hundred microseconds duration produce hyperpolarizations at approximately 30 microm beyond the site of current injection, with these becoming larger and occurring at shorter distances the shorter the terminal branch. Thus the effect of a conditioning depolarizing pulse at one site on a subsequent test pulse at another more than approximately 30 microm away is to substantially decrease the absolute depolarization produced by the latter, provided the interval between the pulses is less than a few hundred microseconds. It is concluded that the passive cable properties of motor nerve terminal branches are sufficient to explain the effects on quantal secretion by a test electrode depolarization of current injections from a spatially removed conditioning electrode.