Low-rate firing limit for neurons with axon, soma and dendrites driven by spatially distributed stochastic synapses.

Analytical forms for neuronal firing rates are important theoretical tools for the analysis of network states. Since the 1960s, the majority of approaches have treated neurons as being electrically compact and therefore isopotential. These approaches have yielded considerable insight into how single-cell properties affect network activity; however, many neuronal classes, such as cortical pyramidal cells, are electrically extended objects. Calculation of the complex flow of electrical activity driven by stochastic spatio-temporal synaptic input streams in these structures has presented a significant analytical challenge. Here we demonstrate that an extension of the level-crossing method of Rice, previously used for compact cells, provides a general framework for approximating the firing rate of neurons with spatial structure. Even for simple models, the analytical approximations derived demonstrate a surprising richness including: independence of the firing rate to the electrotonic length for certain models, but with a form distinct to the point-like leaky integrate-and-fire model; a non-monotonic dependence of the firing rate on the number of dendrites receiving synaptic drive; a significant effect of the axonal and somatic load on the firing rate; and the role that the trigger position on the axon for spike initiation has on firing properties. The approach necessitates only calculating the mean and variances of the non-thresholded voltage and its rate of change in neuronal structures subject to spatio-temporal synaptic fluctuations. The combination of simplicity and generality promises a framework that can be built upon to incorporate increasing levels of biophysical detail and extend beyond the low-rate firing limit treated in this paper.

Transmission of temporally correlated spike trains through synapses with short-term depression.

Short-term synaptic depression, caused by depletion of releasable neurotransmitter, modulates the strength of neuronal connections in a history-dependent manner. Quantifying the statistics of synaptic transmission requires stochastic models that link probabilistic neurotransmitter release with presynaptic spike-train statistics. Common approaches are to model the presynaptic spike train as either regular or a memory-less Poisson process: few analytical results are available that describe depressing synapses when the afferent spike train has more complex, temporally correlated statistics such as bursts. Here we present a series of analytical results-from vesicle release-site occupancy statistics, via neurotransmitter release, to the post-synaptic voltage mean and variance-for depressing synapses driven by correlated presynaptic spike trains. The class of presynaptic drive considered is that fully characterised by the inter-spike-interval distribution and encompasses a broad range of models used for neuronal circuit and network analyses, such as integrate-and-fire models with a complete post-spike reset and receiving sufficiently short-time correlated drive. We further demonstrate that the derived post-synaptic voltage mean and variance allow for a simple and accurate approximation of the firing rate of the post-synaptic neuron, using the exponential integrate-and-fire model as an example. These results extend the level of biological detail included in models of synaptic transmission and will allow for the incorporation of more complex and physiologically relevant firing patterns into future studies of neuronal networks.

Modeling microelectrode biosensors: free-flow calibration can substantially underestimate tissue concentrations.

Microelectrode amperometric biosensors are widely used to measure concentrations of analytes in solution and tissue including acetylcholine, adenosine, glucose, and glutamate. A great deal of experimental and modeling effort has been directed at quantifying the response of the biosensors themselves; however, the influence that the macroscopic tissue environment has on biosensor response has not been subjected to the same level of scrutiny. Here we identify an important issue in the way microelectrode biosensors are calibrated that is likely to have led to underestimations of analyte tissue concentrations. Concentration in tissue is typically determined by comparing the biosensor signal to that measured in free-flow calibration conditions. In a free-flow environment the concentration of the analyte at the outer surface of the biosensor can be considered constant. However, in tissue the analyte reaches the biosensor surface by diffusion through the extracellular space. Because the enzymes in the biosensor break down the analyte, a density gradient is set up resulting in a significantly lower concentration of analyte near the biosensor surface. This effect is compounded by the diminished volume fraction (porosity) and reduction in the diffusion coefficient due to obstructions (tortuosity) in tissue. We demonstrate this effect through modeling and experimentally verify our predictions in diffusive environments.NEW & NOTEWORTHY Microelectrode biosensors are typically calibrated in a free-flow environment where the concentrations at the biosensor surface are constant. However, when in tissue, the analyte reaches the biosensor via diffusion and so analyte breakdown by the biosensor results in a concentration gradient and consequently a lower concentration around the biosensor. This effect means that naive free-flow calibration will underestimate tissue concentration. We develop mathematical models to better quantify the discrepancy between the calibration and tissue environment and experimentally verify our key predictions.

Intracellular soluble α-synuclein oligomers reduce pyramidal cell excitability.

KEY POINTS: The presynaptic protein α-synuclein forms aggregates during Parkinson's disease. Accumulating evidence suggests that the small soluble oligomers of α-synuclein are more toxic than the larger aggregates appearing later in the disease. The link between oligomer toxicity and structure still remains unclear. In the present study, we have produced two structurally-defined oligomers that have a similar morphology but differ in secondary structure. These oligomers were introduced into neocortical pyramidal cells during whole-cell recording and, using a combination of experimentation and modelling, electrophysiological parameters were extracted. Both oligomeric species had similar effects on neuronal properties reducing input resistance, time constant and increasing capacitance. The net effect was a marked reduction in neuronal excitability that could impact on network activity. ABSTRACT: The presynaptic protein α-synuclein (αSyn) aggregates during Parkinson's disease (PD) to form large proteinaceous amyloid plaques, the spread of which throughout the brain clinically defines the severity of the disease. During early stages of aggregation, αSyn forms soluble annular oligomers that show greater toxicity than much larger fibrils. These oligomers produce toxicity via a number of possible mechanisms, including the production of pore-forming complexes that permeabilize membranes. In the present study, two well-defined species of soluble αSyn oligomers were produced by different protocols: by polymerization of monomer and by sonication of fibrils. The two oligomeric species produced were morphologically similar, with both having an annular structure and consisting of approximately the same number of monomer subunits, although they differed in their secondary structure. Oligomeric and monomeric αSyn were injected directly into the soma of pyramidal neurons in mouse neocortical brain slices during whole-cell patch clamp recording. Using a combined experimental and modelling approach, neuronal parameters were extracted to measure, for the first time in the neocortex, specific changes in neuronal electrophysiology. Both species of oligomer had similar effects: (i) a significant reduction in input resistance and the membrane time constant and (ii) an increase in the current required to trigger an action potential with a resultant reduction in the firing rate. Differences in oligomer secondary structure appeared to produce only subtle differences in the activity of the oligomers. Monomeric αSyn had no effect on neuronal parameters, even at high concentrations. The oligomer-induced fall in neuronal excitability has the potential to impact both network activity and cognitive processing.

Experimentally Verified Parameter Sets for Modelling Heterogeneous Neocortical Pyramidal-Cell Populations.

Models of neocortical networks are increasingly including the diversity of excitatory and inhibitory neuronal classes. Significant variability in cellular properties are also seen within a nominal neuronal class and this heterogeneity can be expected to influence the population response and information processing in networks. Recent studies have examined the population and network effects of variability in a particular neuronal parameter with some plausibly chosen distribution. However, the empirical variability and covariance seen across multiple parameters are rarely included, partly due to the lack of data on parameter correlations in forms convenient for model construction. To addess this we quantify the heterogeneity within and between the neocortical pyramidal-cell classes in layers 2/3, 4, and the slender-tufted and thick-tufted pyramidal cells of layer 5 using a combination of intracellular recordings, single-neuron modelling and statistical analyses. From the response to both square-pulse and naturalistic fluctuating stimuli, we examined the class-dependent variance and covariance of electrophysiological parameters and identify the role of the h current in generating parameter correlations. A byproduct of the dynamic I-V method we employed is the straightforward extraction of reduced neuron models from experiment. Empirically these models took the refractory exponential integrate-and-fire form and provide an accurate fit to the perisomatic voltage responses of the diverse pyramidal-cell populations when the class-dependent statistics of the model parameters were respected. By quantifying the parameter statistics we obtained an algorithm which generates populations of model neurons, for each of the four pyramidal-cell classes, that adhere to experimentally observed marginal distributions and parameter correlations. As well as providing this tool, which we hope will be of use for exploring the effects of heterogeneity in neocortical networks, we also provide the code for the dynamic I-V method and make the full electrophysiological data set available.

Localized adenosine signaling provides fine-tuned negative feedback over a wide dynamic range of neocortical network activities.

Although the patterns of activity produced by neocortical networks are now better understood, how these states are activated, sustained, and terminated still remains unclear. Negative feedback by the endogenous neuromodulator adenosine may potentially play an important role, as it can be released by activity and there is dense A1 receptor expression in the neocortex. Using electrophysiology, biosensors, and modeling, we have investigated the properties of adenosine signaling during physiological and pathological network activity in rat neocortical slices. Both low- and high-rate network activities were reduced by A1 receptor activation and enhanced by block of A1 receptors, consistent with activity-dependent adenosine release. Since the A1 receptors were neither saturated nor completely unoccupied during either low- or high-rate activity, adenosine signaling provides a negative-feedback mechanism with a wide dynamic range. Modeling and biosensor experiments show that during high-rate activity increases in extracellular adenosine concentration are highly localized and are uncorrelated over short distances that are certainly<500 µm. Modeling also predicts that the slow rise of the purine waveform cannot be from diffusion from distal release sites but more likely results from uptake and metabolism. The inability to directly measure adenosine release during low-rate activity, although it is present, is probably a consequence of small localized increases in adenosine concentration that are rapidly diminished by diffusion and active removal mechanisms. Saturation of such removal mechanisms when higher concentrations of adenosine are released results in the accumulation of inosine, explaining the strong purine signal during high-rate activity.

Matched pre- and post-synaptic changes underlie synaptic plasticity over long time scales.

Modifications of synaptic efficacies are considered essential for learning and memory. However, it is not known how the underlying functional components of synaptic transmission change over long time scales. To address this question, we studied cortical synapses from young Wistar rats before and after 12 h intervals of spontaneous or glutamate-induced spiking activity. We found that, under these conditions, synaptic efficacies can increase or decrease by up to 10-fold. Statistical analyses reveal that these changes reflect modifications in the number of presynaptic release sites, together with postsynaptic changes that maintain the quantal size per release site. The quantitative relation between the presynaptic and postsynaptic transmission components was not affected when synaptic plasticity was enhanced or reduced using a broad range of pharmacological agents. These findings suggest that ongoing synaptic plasticity results in matched presynaptic and postsynaptic modifications, in which elementary modules that span the synaptic cleft are added or removed as a function of experience.

Fine spatiotemporal activity in contracting myometrium revealed by motion-corrected calcium imaging.

Successful childbirth depends on the occurrence of precisely coordinated uterine contractions during labour. Calcium indicator fluorescence imaging is one of the main techniques for investigating the mechanisms governing this physiological process and its pathologies. The effective spatiotemporal resolution of calcium signals is, however, limited by the motion of contracting tissue: structures of interest in the order of microns can move over a hundred times their width during a contraction. The simultaneous changes in local intensity and tissue configuration make motion tracking a non-trivial problem in image analysis and confound many of the standard techniques. This paper presents a method that tracks local motion throughout the tissue and allows for the almost complete removal of motion artefacts. This provides a stabilized calcium signal down to a pixel resolution, which, for the data examined, is in the order of a few microns. As a byproduct of image stabilization, a complete kinematic description of the contraction-relaxation cycle is also obtained. This contains novel information about the mechanical response of the tissue, such as the identification of a characteristic length scale, in the order of 40-50 µm, below which tissue motion is homogeneous. Applied to our data, we illustrate that the method allows for analyses of calcium dynamics in contracting myometrium in unprecedented spatiotemporal detail. Additionally, we use the kinematics of tissue motion to compare calcium signals at the subcellular level and local contractile motion. The computer code used is provided in a freely modifiable form and has potential applicability to in vivo calcium imaging of neural tissue, as well as other smooth muscle tissue.

Linear and nonlinear integrate-and-fire neurons driven by synaptic shot noise with reversal potentials.

The steady-state firing rate and firing-rate response of the leaky and exponential integrate-and-fire models receiving synaptic shot noise with excitatory and inhibitory reversal potentials is examined. For the particular case where the underlying synaptic conductances are exponentially distributed, it is shown that the master equation for a population of such model neurons can be reduced from an integrodifferential form to a more tractable set of three differential equations. The system is nevertheless more challenging analytically than for current-based synapses: where possible, analytical results are provided with an efficient numerical scheme and code provided for other quantities. The increased tractability of the framework developed supports an ongoing critical comparison between models in which synapses are treated with and without reversal potentials, such as recently in the context of networks with balanced excitatory and inhibitory conductances.

Adenosine A1 receptor activation mediates the developmental shift at layer 5 pyramidal cell synapses and is a determinant of mature synaptic strength.

During the first postnatal month glutamatergic synapses between layer 5 pyramidal cells in the rodent neocortex switch from an immature state exhibiting a high probability of neurotransmitter release, large unitary amplitude and synaptic depression to a mature state with decreased probability of release, smaller unitary amplitude and synaptic facilitation. Using paired recordings, we demonstrate that the developmental shift in release probability at synapses between rat somatosensory layer 5 thick-tufted pyramidal cells is mediated by a higher and more heterogeneous activation of presynaptic adenosine A1 receptors. Immature synapses under control conditions exhibited distributions of coefficient of variation, failure rate and release probability that were almost coincident with the A1 receptor blocked condition; however, mature synapses under control conditions exhibited much broader distributions that spanned those of both the A1 receptor agonized and antagonized conditions. Immature and mature synapses expressed A1 receptors with no observable difference in functional efficacy and therefore the heterogeneous A1 receptor activation seen in the mature neocortex appears due to increased adenosine concentrations that vary between synapses. Given the central role demonstrated for A1 receptor activation in determining synaptic amplitude and the statistics of transmission between mature layer 5 pyramidal cells, the emplacement of adenosine sources and sinks near the synaptic terminal could constitute a novel form of long-term synaptic plasticity.

The dynamics of single spike-evoked adenosine release in the cerebellum.

The purine adenosine is a potent neuromodulator in the brain, with roles in a number of diverse physiological and pathological processes. Modulators such as adenosine are difficult to study as once released they have a diffuse action (which can affect many neurones) and, unlike classical neurotransmitters, have no inotropic receptors. Thus rapid postsynaptic currents (PSCs) mediated by adenosine (equivalent to mPSCs) are not available for study. As a result the mechanisms and properties of adenosine release still remain relatively unclear. We have studied adenosine release evoked by stimulating the parallel fibres in the cerebellum. Using adenosine biosensors combined with deconvolution analysis and mathematical modelling, we have characterised the release dynamics and diffusion of adenosine in unprecedented detail. By partially blocking K+ channels, we were able to release adenosine in response to a single stimulus rather than a train of stimuli. This allowed reliable sub-second release of reproducible quantities of adenosine with stereotypic concentration waveforms that agreed well with predictions of a mathematical model of purine diffusion. We found no evidence for ATP release and thus suggest that adenosine is directly released in response to parallel fibre firing and does not arise from extracellular ATP metabolism. Adenosine release events showed novel short-term dynamics, including facilitated release with paired stimuli at millisecond stimulation intervals but depletion-recovery dynamics with paired stimuli delivered over minute time scales. These results demonstrate rich dynamics for adenosine release that are placed, for the first time, on a quantitative footing and show strong similarity with vesicular exocytosis.

Firing-rate response of a neuron receiving excitatory and inhibitory synaptic shot noise.

The synaptic coupling between neurons in neocortical networks is sufficiently strong so that relatively few synchronous synaptic pulses are required to bring a neuron from rest to the spiking threshold. However, such finite-amplitude effects of fluctuating synaptic drive are missed in the standard diffusion approximation. Here exact solutions for the firing-rate response to modulated presynaptic rates are derived for a neuron receiving additive excitatory and inhibitory synaptic shot noise with exponential amplitude distributions. The shot-noise description of the neuronal response to synaptic dynamics is shown to be richer and qualitatively distinct from that predicted by the diffusion approximation. It is also demonstrated how the framework developed here can be generalized to multiplicative shot noise so as to better capture effects of the inhibitory reversal potential.

Dynamics of populations and networks of neurons with voltage-activated and calcium-activated currents.

The profile of transmembrane-channel expression in neurons is class dependent and a crucial determinant of neuronal dynamics. Here, a generalization of the experimentally verified exponential integrate-and-fire model is introduced that includes biophysical, nonlinear gated conductance-based currents, and a spike shape. A Fokker-Planck-based method is developed that allows for the rapid numerical calculation of steady-state and linear-response properties for recurrent networks of neurons with gating-variable dynamics slower than that of the voltage. This limit includes many cases of biological interest, particularly under in vivo conditions of high synaptic conductance. The utility of the method is illustrated by applying it to two biophysically detailed models adapted from the literature: an entorhinal layer-II cortical neuron and a neuron featuring both calcium-activated and voltage-activated spike-frequency-adaptation currents. The framework generalizes to networks comprised of different neuronal classes and so will allow for the modeling of emergent states in neural tissue at significantly increased levels of biological detail.

Introduction of Tau Oligomers into Cortical Neurons Alters Action Potential Dynamics and Disrupts Synaptic Transmission and Plasticity.

Tau is a highly soluble microtubule-associated protein that acts within neurons to modify microtubule stability. However, abnormally phosphorylated tau dissociates from microtubules to form oligomers and fibrils which associate in the somatodendritic compartment. Although tau can form neurofibrillary tangles (NFTs), it is the soluble oligomers that appear to be the toxic species. There is, however, relatively little quantitative information on the concentration-dependent and time-dependent actions of soluble tau oligomers (oTau) on the electrophysiological and synaptic properties of neurons. Here, whole-cell patch clamp recording was used to introduce known concentrations of oligomeric full-length tau-441 into mouse hippocampal CA1 pyramidal and neocortical Layer V thick-tufted pyramidal cells. oTau increased input resistance, reduced action potential amplitude and slowed action potential rise and decay kinetics. oTau injected into presynaptic neurons induced the run-down of unitary EPSPs which was associated with increased short-term depression. In contrast, introduction of oTau into postsynaptic neurons had no effect on basal synaptic transmission, but markedly impaired the induction of long-term potentiation (LTP). Consistent with its effects on synaptic transmission and plasticity, oTau puncta could be observed in the soma, axon and in the distal dendrites of injected neurons.

Spike-train spectra and network response functions for non-linear integrate-and-fire neurons.

Reduced models have long been used as a tool for the analysis of the complex activity taking place in neurons and their coupled networks. Recent advances in experimental and theoretical techniques have further demonstrated the usefulness of this approach. Despite the often gross simplification of the underlying biophysical properties, reduced models can still present significant difficulties in their analysis, with the majority of exact and perturbative results available only for the leaky integrate-and-fire model. Here an elementary numerical scheme is demonstrated which can be used to calculate a number of biologically important properties of the general class of non-linear integrate-and-fire models. Exact results for the first-passage-time density and spike-train spectrum are derived, as well as the linear response properties and emergent states of recurrent networks. Given that the exponential integrate-fire model has recently been shown to agree closely with the experimentally measured response of pyramidal cells, the methodology presented here promises to provide a convenient tool to facilitate the analysis of cortical-network dynamics.

Extracting non-linear integrate-and-fire models from experimental data using dynamic I-V curves.

The dynamic I-V curve method was recently introduced for the efficient experimental generation of reduced neuron models. The method extracts the response properties of a neuron while it is subject to a naturalistic stimulus that mimics in vivo-like fluctuating synaptic drive. The resulting history-dependent, transmembrane current is then projected onto a one-dimensional current-voltage relation that provides the basis for a tractable non-linear integrate-and-fire model. An attractive feature of the method is that it can be used in spike-triggered mode to quantify the distinct patterns of post-spike refractoriness seen in different classes of cortical neuron. The method is first illustrated using a conductance-based model and is then applied experimentally to generate reduced models of cortical layer-5 pyramidal cells and interneurons, in injected-current and injected- conductance protocols. The resulting low-dimensional neuron models-of the refractory exponential integrate-and-fire type-provide highly accurate predictions for spike-times. The method therefore provides a useful tool for the construction of tractable models and rapid experimental classification of cortical neurons.

Spike-triggered averages for passive and resonant neurons receiving filtered excitatory and inhibitory synaptic drive.

A path-integral approach is developed for the analysis of spike-triggered average quantities in neurons with voltage-gated subthreshold currents. Using a linearization procedure to reduce the models to the generalized integrate-and-fire form, analytical expressions are obtained in an experimentally relevant limit of fluctuation-driven firing. The influences of voltage-gated channels as well as excitatory and inhibitory synaptic filtering are shown to affect significantly the neuronal dynamics prior to the spike. Analytical forms are given for all relevant physiological quantities, such as the mean voltage triggered to the spike, mean current flowing through voltage-gated channels, and the mean excitatory and inhibitory conductance waveforms prior to a spike. The mathematical results are shown to be in good agreement with numerical simulations of the underlying nonlinear conductance-based models. The method promises to provide a useful analytical tool for the prediction and interpretation of the temporal structure of spike-triggered averages measured experimentally.

Acute ethanol exposure has bidirectional actions on the endogenous neuromodulator adenosine in rat hippocampus.

BACKGROUND AND PURPOSE: Ethanol is a widely used recreational drug with complex effects on physiological and pathological brain function. In epileptic patients, the use of ethanol can modify seizure initiation and subsequent seizure activity with reports of ethanol being both pro- and anticonvulsant. One proposed target of ethanol's actions is the neuromodulator adenosine, which is released during epileptic seizures to feedback and inhibit the occurrence of subsequent seizures. Here, we investigated the actions of acute ethanol exposure on adenosine signalling in rat hippocampus. EXPERIMENTAL APPROACH: We have combined electrophysiology with direct measurements of extracellular adenosine using microelectrode biosensors in rat hippocampal slices. KEY RESULTS: We found that ethanol has bidirectional actions on adenosine signalling: depressant concentrations of ethanol (50 mM) increased the basal extracellular concentration of adenosine under baseline conditions, leading to the inhibition of synaptic transmission, but it inhibited adenosine release during evoked seizure activity in brain slices. The reduction in activity-dependent adenosine release was in part produced by effects on NMDA receptors, although other mechanisms also appeared to be involved. Low concentrations of ethanol (10-15 mM) enhanced pathological network activity by selectively blocking activity-dependent adenosine release. CONCLUSIONS AND IMPLICATIONS: The complex dose-dependent actions of ethanol on adenosine signalling could in part explain the mixture of pro-convulsant and anticonvulsant actions of ethanol that have previously been reported.

Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive.

Integrate-and-fire models are mainstays of the study of single-neuron response properties and emergent states of recurrent networks of spiking neurons. They also provide an analytical base for perturbative approaches that treat important biological details, such as synaptic filtering, synaptic conductance increase, and voltage-activated currents. Steady-state firing rates of both linear and nonlinear integrate-and-fire models, receiving fluctuating synaptic drive, can be calculated from the time-independent Fokker-Planck equation. The dynamic firing-rate response is less easy to extract, even at the first-order level of a weak modulation of the model parameters, but is an important determinant of neuronal response and network stability. For the linear integrate-and-fire model the response to modulations of current-based synaptic drive can be written in terms of hypergeometric functions. For the nonlinear exponential and quadratic models no such analytical forms for the response are available. Here it is demonstrated that a rather simple numerical method can be used to obtain the steady-state and dynamic response for both linear and nonlinear models to parameter modulation in the presence of current-based or conductance-based synaptic fluctuations. To complement the full numerical solution, generalized analytical forms for the high-frequency response are provided. A special case is also identified--time-constant modulation--for which the response to an arbitrarily strong modulation can be calculated exactly.

Reducing Extracellular Ca2+ Induces Adenosine Release via Equilibrative Nucleoside Transporters to Provide Negative Feedback Control of Activity in the Hippocampus.

Neural circuit activity increases the release of the purine neuromodulator adenosine into the extracellular space leading to A1 receptor activation and negative feedback via membrane hyperpolarization and inhibition of transmitter release. Adenosine can be released by a number of different mechanisms that include Ca2+ dependent processes such as the exocytosis of ATP. During sustained pathological network activity, ischemia and hypoxia the extracellular concentration of calcium ions (Ca2+) markedly falls, inhibiting exocytosis and potentially reducing adenosine release. However it has been observed that reducing extracellular Ca2+ can induce paradoxical neural activity and can also increase adenosine release. Here we have investigated adenosine signaling and release mechanisms that occur when extracellular Ca2+ is removed. Using electrophysiology and microelectrode biosensor measurements we have found that adenosine is directly released into the extracellular space by the removal of extracellular Ca2+ and controls the induced neural activity via A1 receptor-mediated membrane potential hyperpolarization. Following Ca2+ removal, adenosine is released via equilibrative nucleoside transporters (ENTs), which when blocked leads to hyper-excitation. We propose that sustained action potential firing following Ca2+ removal leads to hydrolysis of ATP and a build-up of intracellular adenosine which then effluxes into the extracellular space via ENTs.