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.

Conformation of histamine derivatives. 5. Molecular orbital calculation of the H1-receptor "essential" conformation of histamine.

Conformational energies of histamine and 4-methylhistamine monocations are calculated using the EHT molecular orbital procedure; the results are expressed as potential energy surfaces in which bond rotations (theta1 for ring-Cbeta, theta2 for Cbeta-Calpha) are measured along the axes, and energy variation is indicated by contours. Using the classical Boltzmann partition function and Simpson's rule for normalization, corresponding probability surfaces are generated which take account of the potential surface entropy. Comparing the two surfaces provides regions which are within a given probability contour of histamine but outside this contour for 4-methylhistamine. Thus, at the 99% probability level, three conformational regions defined by the bond rotation angles are indicated as possible "H1-essential" conformations of histamine: viz. trans (theta1=290-330 degrees, theta2=150-210 degrees) and gauche (theta1=260-280 degrees, theta2=30-90 degrees and theta1=290-320 degrees, theta2=270-320 degrees). This procedure provides a quantitative basis for comparison with other histamine derivatives and may have a general value for studying relationships between conformation and biological activity of closely related small molecules.

On the origin of skewed distributions of spontaneous synaptic potentials in autonomic ganglia.

The histograms of spontaneous synaptic potentials at synapses in autonomic ganglia are described by distributions consisting of mixtures of Gaussians, rather than by single Gaussian distributions. The possible origin of these mixed distributions is investigated, using Monte-Carlo simulations of the action of spontaneously released units of transmitter. A single unit of acetylcholine of fixed size, released from an active zone with receptor patches both beneath and adjacent to the zone, does not give rise to the observed histograms. But if the unit is of variable size, consisting of integer multiples of smaller units, and release is from an active zone onto either the receptor patch beneath, or in addition onto adjacent patches, then the histogram is well described by a mixture of Gaussians. However, this explanation is unlikely to be correct as present evidence suggests that in most cases the released unit of transmitter saturates the postsynaptic receptor patch beneath the active zone. The final case considered is where a unit of transmitter is spontaneously released from an active zone, simultaneously with a unit in an adjacent zone less than one micron away. The histogram of potentials then conforms to those observed even when there are differences in the sizes of the receptor patches. It is suggested that this kind of release could provide an explanation for distributions of spontaneous potentials that are mixtures of Gaussians.

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 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.

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.

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.

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.

MULTICOMP: a program for preparing sequence data for phylogenetic analysis.

MULTICOMP is a program that assists in the phylogenetic analysis of DNA sequences. It streamlines sequence handling and analysis. Input is from either individual sequence files or a file of aligned sequences. It produces data on variation at DNA and amino acid sequence level and can also convert sequences to data formats suitable for PHYLIP, PAUP and MacClade phylogenetic inference programs. Further, two tree-building programs, NEIGHBOR and DNAPARS, of PHYLIP can be directly run from within it. MULTICOMP performs Sawyer's algorithm for detection of gene conversion. The program facilitates analysis using only part of the data of a data set or using two or more combined data sets.

Quantal transmission at purinergic synapses: stochastic interaction between ATP and its receptors.

Monte Carlo methods are used to analyze the stochastic interaction between ATP, released in a packet at a bouton, and the underlying patch of purinoceptors. The time-course of the average quantal current recorded with an intracellular electrode in the peripheral and central nervous systems is reconstructed, given the geometry of the synapse and the known kinetics of ATP action. This leads to certain restrictions on the possible numbers of ATP molecules in a quantum and the density of purinoceptors at the synapses. The addition of an ectoenzyme into the synaptic cleft with the known kinetics of ATPase gives rise to a quantal current of appropriate time-course if the number of ATP molecules in a quantum is increased over that in the absence of the ATPase. The stochastic variability in the quantal current is less than 0.1 for a given size quantum.

Quantal transmitter release at somatic motor-nerve terminals: stochastic analysis of the subunit hypothesis.

Here we analyze the problem of determining whether experimentally measured spontaneous miniature end-plate currents (MEPCs) indicate that quanta are composed of subunits. The properties of MEPCs at end plates with or without secondary clefts at the neuromuscular junction are investigated, using both stochastic and deterministic models of the action of a quantum of transmitter. It is shown that as the amount of transmitter in a quantum is increased above about 4000 acetylcholine (ACh) molecules there is a linear increase in the size of the MEPC. It is possible to then use amplitude-frequency histograms of such MEPCs to detect a subunit structure, as there is little potentiation effect above 4000 ACh molecules. Autocorrelation and power spectral analyses of such histograms establish that their subunit structure can be detected if the coefficient of variation of the subunit size is less than about 0.12 or, if electrical noise is added, about 0.1. Positive gradients relate the rise time and half-decay times of MEPCs to their amplitude, even in the absence of potentiating effects; these gradients are shallower at motor nerve terminals that possess secondary clefts. The effect of asynchronous release of subunits is also investigated. The criteria determined by this analysis for identifying a subunit composition in the quantum are applied to an amplitude-frequency histogram of MEPCs recorded from a small group of active zones at a visualized amphibian motor-nerve terminal. This did not provide evidence for a subunit structure.

Quantal transmitter release onto different patterns of receptor distributions at somatic neuromuscular junctions.

The effect of different distributions of acetylcholine receptors at the active zone of somatic motor-nerve terminals on the amplitude-frequency histogram of spontaneous excitatory currents due to the secretion of a quantum transmitter has been analysed by numerically solving the diffusion and binding equations. The time course and amplitude of the spontaneous currents of different arrangements of receptors has been determined. Organization of receptors into stripes of different patterns beneath the active zone, which is known to occur, does not affect the amplitude or temporal characteristics of the spontaneous currents. Increases in the amount of transmitter in a quantum released onto the stripes does not introduce any subunit structure into the spontaneous currents. It is concluded that if subunits are detectable in measured amplitude-frequency histograms of spontaneous potentials they arise because of presynaptic rather than postsynaptic factors.

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.

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.

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.

Quantal transmission at purinergic junctions: stochastic interaction between ATP and its receptors.

The time course of most quantal currents recorded with a small diameter electrode placed over visualized varicosities of sympathetic nerve terminals that secrete ATP was determined: these had a time to reach 90% of peak of 1.3-1.8 ms and a time constant of decay of 12-18 ms; they were unaffected by blocking ectoenzymes or the uptake of adenosine. Monte Carlo methods were used to analyze the stochastic interaction between ATP, released in a packet from a varicosity, and the underlying patch of purinoceptors, to reconstitute the time course of the quantal current. This leads to certain restrictions on the possible number of ATP molecules in a quantum (about 1000) and the density of purinoceptors at the junctions (about 1000 microns-1), given the known geometry of the junction and the kinetics of ATP action. The observed quantal current has a relatively small variability (coefficient of variation < 0.1), and this stochastic property is reproduced for a given quantum of ATP. Potentiation effects (of about 12%) occur if two quanta are released from the same varicosity because the receptor patch is not saturated even by the release of two quanta. The simulations show that quantal currents have a characteristically distinct shape for varicosities with different junctional cleft widths (50-200 nm). Finally, incorporation of an ectoenzyme with the known kinetics of ATPase into the junctional cleft allows for a quantal current of the observed time course, provided the number of ATP molecules in a quantum is increased over the number in the absence of the ATPase.