Modeling the response of homogeneous and heterogeneous cerebral capillary networks to local changes in vessel diameters.

While microvascular cerebral capillary networks are known to be highly heterogeneous, previous computational models have predicted that heterogeneous cerebral capillary flow patterns result in lower brain tissue partial oxygen pressures. Moreover, as blood flow increases, the flux among capillaries homogenizes. This homogenization of flow is expected to improve the efficiency of oxygenation extraction from the blood. In this work, we use mathematical modeling to explore a possible functional role for the high degree of heterogeneity observed in cerebral capillary networks. Our results suggest that heterogeneity allows for a greater response of tissue oxygen levels to local changes in vessel diameters due to neuronal activation. This result is confirmed for a full 3-dimensional model of capillary networks that includes oxygen diffusion within the tissue region and a reduced model that accounts for changes in capillary blood flow.

Modeling the effect of cerebral capillary blood flow on neuronal firing.

Adequate cerebral blood flow has long been recognized as essential for the maintenance of the neuronal function while interruption of cerebral blood flow for durations as short as minutes can result in permanent brain damage. A primary goal of this work is to determine how a neuron's ability to respond to synaptic input depends on parameters that control cerebral blood flow. A complex mathematical model is constructed that integrates detailed biophysical models of neuronal action potentials, mitochondrial ATP production and cerebral capillary blood flow. The model also provides insights of the role of astrocytes in maintaining neuronal responses, as well as the impact of elevated cytosolic calcium, due to increased synaptic activity, on mitochondrial ATP production. Both dynamical systems analysis and numerical simulations are used to determine how the maximum frequency at which the neurons can respond to synaptic input depends on capillary blow flow, as well as the ability of astrocytes to buffer extracellular potassium and cytosolic calcium handling. Results are presented for both the cases of homogenous and heterogeneous capillary networks. These results demonstrate, through this interconnected model, that heterogeneity of the capillary flow results in a decrease in the ability of neurons to respond to synaptic stimulation and that intact glial function provides a further protective role for the neurons.

On the electrical passivity of astrocyte potassium conductance.

Predominant expression of leak-type K+ channels provides astrocytes a high membrane permeability to K+ ions and a hyperpolarized membrane potential that are crucial for astrocyte function in brain homeostasis. In functionally mature astrocytes, the expression of leak K+ channels creates a unique membrane K+ conductance that lacks voltage-dependent rectification. Accordingly, the conductance is named ohmic or passive K+ conductance. Several inwardly rectifying and two-pore domain K+ channels have been investigated for their contributions to passive conductance. Meanwhile, gap junctional coupling has been postulated to underlie the passive behavior of membrane conductance. It is now clear that the intrinsic properties of K+ channels and gap junctional coupling can each act alone or together to bring about a passive behavior of astrocyte conductance. Additionally, while the passive conductance can generally be viewed as a K+ conductance, the actual representation of this conductance is a combined expression of multiple known and unknown K+ channels, which has been further modified by the intricate morphology of individual astrocytes and syncytial gap junctional coupling. The expression of the inwardly rectifying K+ channels explains the inward-going component of passive conductance disobeying Goldman-Hodgkin-Katz constant field outward rectification. However, the K+ channels encoding the outward-going passive currents remain to be determined in the future. Here, we review our current understanding of ion channels and biophysical mechanisms engaged in the passive astrocyte K+ conductance, propose new studies to resolve this long-standing puzzle in astrocyte physiology, and discuss the functional implication(s) of passive behavior of K+ conductance on astrocyte physiology.

Mathematical modeling of cerebral capillary blood flow heterogeneity and its effect on brain tissue oxygen levels.

Maintaining cerebral blood flow is critical for adequate neuronal function. Previous computational models of brain capillary networks have predicted that heterogeneous cerebral capillary flow patterns result in lower brain tissue partial oxygen pressures PO2). However, these previous models have often considered simple capillary networks in terms of their geometric properties. In this current work, we developed and analyzed computational models of brain capillary networks to determine how perturbations of network properties impact tissue oxygen levels. The models include variabilities in both their geometric (segment lengths and diameters) and three-dimensional, topological structure. Two classes of capillary network models are considered. The first consists of equations for the oxygen partial pressure, PO2, in both a capillary network and the surrounding tissue. In order to gain insight into the behavior of this detailed model, we also consider a reduced model for changes in PO2 in just the capillary network. The main result is that for a general class of networks, random perturbations of either segment diameters or conductances will always, on average, decrease the average tissue oxygen levels. This result is supported through both simulations of the models and mathematical analysis. Our results promise to expand our understanding of cerebral capillary blood flow and its impact on the brain function in health and disease.

Resurgent Na+ Current Offers Noise Modulation in Bursting Neurons.

Neurons utilize bursts of action potentials as an efficient and reliable way to encode information. It is likely that the intrinsic membrane properties of neurons involved in burst generation may also participate in preserving its temporal features. Here we examined the contribution of the persistent and resurgent components of voltage-gated Na+ currents in modulating the burst discharge in sensory neurons. Using mathematical modeling, theory and dynamic-clamp electrophysiology, we show that, distinct from the persistent Na+ component which is important for membrane resonance and burst generation, the resurgent Na+ can help stabilize burst timing features including the duration and intervals. Moreover, such a physiological role for the resurgent Na+ offered noise tolerance and preserved the regularity of burst patterns. Model analysis further predicted a negative feedback loop between the persistent and resurgent gating variables which mediate such gain in burst stability. These results highlight a novel role for the voltage-gated resurgent Na+ component in moderating the entropy of burst-encoded neural information.

Gain control with A-type potassium current: IA as a switch between divisive and subtractive inhibition.

Neurons process and convey information by transforming barrages of synaptic inputs into spiking activity. Synaptic inhibition typically suppresses the output firing activity of a neuron, and is commonly classified as having a subtractive or divisive effect on a neuron's output firing activity. Subtractive inhibition can narrow the range of inputs that evoke spiking activity by eliminating responses to non-preferred inputs. Divisive inhibition is a form of gain control: it modifies firing rates while preserving the range of inputs that evoke firing activity. Since these two "modes" of inhibition have distinct impacts on neural coding, it is important to understand the biophysical mechanisms that distinguish these response profiles. In this study, we use simulations and mathematical analysis of a neuron model to find the specific conditions (parameter sets) for which inhibitory inputs have subtractive or divisive effects. Significantly, we identify a novel role for the A-type Potassium current (IA). In our model, this fast-activating, slowly-inactivating outward current acts as a switch between subtractive and divisive inhibition. In particular, if IA is strong (large maximal conductance) and fast (activates on a time-scale similar to spike initiation), then inhibition has a subtractive effect on neural firing. In contrast, if IA is weak or insufficiently fast-activating, then inhibition has a divisive effect on neural firing. We explain these findings using dynamical systems methods (plane analysis and fast-slow dissection) to define how a spike threshold condition depends on synaptic inputs and IA. Our findings suggest that neurons can "self-regulate" the gain control effects of inhibition via combinations of synaptic plasticity and/or modulation of the conductance and kinetics of A-type Potassium channels. This novel role for IA would add flexibility to neurons and networks, and may relate to recent observations of divisive inhibitory effects on neurons in the nucleus of the solitary tract.

Syncytial isopotentiality: A system-wide electrical feature of astrocytic networks in the brain.

Syncytial isopotentiality, resulting from a strong electrical coupling, emerges as a physiological mechanism that coordinates individual astrocytes to function as a highly efficient system in brain homeostasis. However, whether syncytial isopotentiality occurs selectively to certain brain regions or is universal to astrocytic networks remains unknown. Here, we have explored the correlation of syncytial isopotentiality with different astrocyte subtypes in various brain regions. Using a nonphysiological K+ -free/Na+ electrode solution to depolarize a recorded astrocyte in situ, the existence of syncytial isopotentiality can be revealed: the recorded astrocyte's membrane potential remains at a quasi-physiological level due to strong electrical coupling with neighboring astrocytes. Syncytial isopotentiality appears in Layer I of the motor, sensory, and visual cortical regions, where astrocytes are organized with comparable cell densities, interastrocytic distances, and the quantity of directly coupled neighbors. Second, though astrocytes vary in their cytoarchitecture in association with neuronal circuits from Layers I-VI, the established syncytial isopotentiality remains comparable among different layers in the visual cortex. Third, neurons and astrocytes are uniquely organized as barrels in Layer IV somatosensory cortex; interestingly, astrocytes both inside and outside of the barrels do electrically communicate with each other and also share syncytial isopotentiality. Fourth, syncytial isopotentiality appears in radial-shaped Bergmann glia and velate astrocytes in the cerebellar cortex. Fifth, although fibrous astrocytes in white matter exhibit a distinct morphology, their network syncytial isopotentiality is comparable with protoplasmic astrocytes. Altogether, syncytial isopotentiality appears as a system-wide electrical feature of astrocytic networks in the brain.

A computational analysis of signal fidelity in the rostral nucleus of the solitary tract.

Neurons in the rostral nucleus of the solitary tract (rNST) convey taste information to both local circuits and pathways destined for forebrain structures. This nucleus is more than a simple relay, however, because rNST neurons differ in response rates and tuning curves relative to primary afferent fibers. To systematically study the impact of convergence and inhibition on firing frequency and breadth of tuning (BOT) in rNST, we constructed a mathematical model of its two major cell types: projection neurons and inhibitory neurons. First, we fit a conductance-based neuronal model to data derived from whole cell patch-clamp recordings of inhibitory and noninhibitory neurons in a mouse expressing Venus under the control of the VGAT promoter. We then used in vivo chorda tympani (CT) taste responses as afferent input to modeled neurons and assessed how the degree and type of convergence influenced model cell output frequency and BOT for comparison with in vivo gustatory responses from the rNST. Finally, we assessed how presynaptic and postsynaptic inhibition impacted model cell output. The results of our simulations demonstrated 1) increasing numbers of convergent afferents (2-10) result in a proportional increase in best-stimulus firing frequency but only a modest increase in BOT, 2) convergence of afferent input selected from the same best-stimulus class of CT afferents produced a better fit to real data from the rNST compared with convergence of randomly selected afferent input, and 3) inhibition narrowed the BOT to more realistically model the in vivo rNST data. NEW & NOTEWORTHY Using a combination of in vivo and in vitro neurophysiology together with conductance-based modeling, we show how patterns of convergence and inhibition interact in the rostral (gustatory) solitary nucleus to maintain signal fidelity. Although increasing convergence led to a systematic increase in firing frequency, tuning specificity was maintained with a pattern of afferent inputs sharing the best-stimulus compared with random inputs. Tonic inhibition further enhanced response fidelity.

A mathematical model of recurrent spreading depolarizations.

A detailed biophysical model for a neuron/astrocyte network is developed in order to explore mechanisms responsible for the initiation and propagation of recurrent cortical spreading depolarizations. The model incorporates biophysical processes not considered in the earlier models. This includes a model for the Na+-glutamate transporter, which allows for a detailed description of reverse glutamate uptake. In particular, we consider the specific roles of elevated extracellular glutamate and K+ in the initiation, propagation and recurrence of spreading depolarizations.

Sickle cells produce functional immune modulators and cytotoxics.

Sickle erythrocytes' (SSRBCs) unique physical adaptation to hypoxic conditions renders them able to home to hypoxic tumor niches in vivo, shut down tumor blood flow and induce tumoricidal responses. SSRBCs are also useful vehicles for transport of encapsulated drugs and oncolytic virus into hypoxic tumors with enhanced anti-tumor effects. In search of additional modes for arming sickle cells with cytotoxics, we turned to a lentiviral ß-globin vector with optimized Locus Control Region/ß-globin coding region/promoter/enhancers. We partially replaced the ß-globin coding region of this vector with genes encoding T cell cytolytics, perforin and granzyme or immune modulating superantigens SEG and SEI. These modified vectors efficiently transduced Sca+ ckit- Lin- hematopoietic stem cells (HSCs) from humanized sickle cell knockin mice. Irradiated mice reconstituted with these HSCs displayed robust expression of transgenic RNAs and proteins in host sickle cells that was sustained for more than 10 months. SSRBCs from reconstituted mice harboring SEG/SEI transgenes induced robust proliferation and a prototypical superantigen-induced cytokine reaction when exposed to human CD4+/CD8+ cells. The ß-globin lentiviral vector therefore produces a high level of functional, erythroid-specific immune modulators and cytotoxics that circulate without toxicity. Coupled with their unique ability to target and occlude hypoxic tumor vessels these armed SSRBCs constitute a potentially useful tool for treatment of solid tumors.

Neuroprotective Role of Gap Junctions in a Neuron Astrocyte Network Model.

A detailed biophysical model for a neuron/astrocyte network is developed to explore mechanisms responsible for the initiation and propagation of cortical spreading depolarizations and the role of astrocytes in maintaining ion homeostasis, thereby preventing these pathological waves. Simulations of the model illustrate how properties of spreading depolarizations, such as wave speed and duration of depolarization, depend on several factors, including the neuron and astrocyte Na(+)-K(+) ATPase pump strengths. In particular, we consider the neuroprotective role of astrocyte gap junction coupling. The model demonstrates that a syncytium of electrically coupled astrocytes can maintain a physiological membrane potential in the presence of an elevated extracellular K(+) concentration and efficiently distribute the excess K(+) across the syncytium. This provides an effective neuroprotective mechanism for delaying or preventing the initiation of spreading depolarizations.

Gap junction coupling confers isopotentiality on astrocyte syncytium.

Astrocytes are extensively coupled through gap junctions into a syncytium. However, the basic role of this major brain network remains largely unknown. Using electrophysiological and computational modeling methods, we demonstrate that the membrane potential (VM) of an individual astrocyte in a hippocampal syncytium, but not in a single, freshly isolated cell preparation, can be well-maintained at quasi-physiological levels when recorded with reduced or K(+) free pipette solutions that alter the K(+) equilibrium potential to non-physiological voltages. We show that an astrocyte's associated syncytium provides powerful electrical coupling, together with ionic coupling at a lesser extent, that equalizes the astrocyte's VM to levels comparable to its neighbors. Functionally, this minimizes VM depolarization attributable to elevated levels of local extracellular K(+) and thereby maintains a sustained driving force for highly efficient K(+) uptake. Thus, gap junction coupling functions to achieve isopotentiality in astrocytic networks, whereby a constant extracellular environment can be powerfully maintained for crucial functions of neural circuits.

Modeling the neuroprotective role of enhanced astrocyte mitochondrial metabolism during stroke.

A mathematical model that integrates the dynamics of cell membrane potential, ion homeostasis, cell volume, mitochondrial ATP production, mitochondrial and endoplasmic reticulum Ca(2+) handling, IP3 production, and GTP-binding protein-coupled receptor signaling was developed. Simulations with this model support recent experimental data showing a protective effect of stimulating an astrocytic GTP-binding protein-coupled receptor (P2Y1Rs) following cerebral ischemic stroke. The model was analyzed to better understand the mathematical behavior of the equations and to provide insights into the underlying biological data. This approach yielded explicit formulas determining how changes in IP3-mediated Ca(2+) release, under varying conditions of oxygen and the energy substrate pyruvate, affected mitochondrial ATP production, and was utilized to predict rate-limiting variables in P2Y1R-enhanced astrocyte protection after cerebral ischemic stroke.

Alternative splicing regulates kv3.1 polarized targeting to adjust maximal spiking frequency.

Synaptic inputs received at dendrites are converted into digital outputs encoded by action potentials generated at the axon initial segment in most neurons. Here, we report that alternative splicing regulates polarized targeting of Kv3.1 voltage-gated potassium (Kv) channels to adjust the input-output relationship. The spiking frequency of cultured hippocampal neurons correlated with the level of endogenous Kv3 channels. Expression of axonal Kv3.1b, the longer form of Kv3.1 splice variants, effectively converted slow-spiking young neurons to fast-spiking ones; this was not the case for Kv1.2 or Kv4.2 channel constructs. Despite having identical biophysical properties as Kv3.1b, dendritic Kv3.1a was significantly less effective at increasing the maximal firing frequency. This suggests a possible role of channel targeting in regulating spiking frequency. Mutagenesis studies suggest the electrostatic repulsion between the Kv3.1b N/C termini, created by its C-terminal splice domain, unmasks the Kv3.1b axonal targeting motif. Kv3.1b axonal targeting increased the maximal spiking frequency in response to prolonged depolarization. This finding was further supported by the results of local application of channel blockers and computer simulations. Taken together, our studies have demonstrated that alternative splicing controls neuronal firing rates by regulating the polarized targeting of Kv3.1 channels.

Frequency separation by an excitatory-inhibitory network.

We consider a situation in which individual features of the input are represented in the neural system by different frequencies of periodic firings. Thus, if two of the features are presented concurrently, the input to the system will consist of a superposition of two periodic trains. In this paper we present an algorithm that is capable of extracting the individual features from the composite signal by separating the signal into periodic spike trains with different frequencies. We show that the algorithm can be implemented in a biophysically based excitatory-inhibitory network model. The frequency separation process works over a range of frequencies determined by time constants of the model's intrinsic variables. It does not rely on a "resonance" phenomenon and is not tuned to a discrete set of frequencies. The frequency separation is still reliable when the timing of incoming spikes is noisy.

Ultrastructural view of astrocyte arborization, astrocyte-astrocyte and astrocyte-synapse contacts, intracellular vesicle-like structures, and mitochondrial network.

The complexity of astrocyte morphology and syncytial coupling through gap junctions are crucial for astrocyte function in the brain. However, the ultrastructural details of astrocyte arborization and interactions between neighboring astrocytes remain unknown. While a prevailing view is that synapses selectively contact peripheral astrocyte processes, the precise spatial-location selectivity of synapses abutting astrocytes is unresolved. Additionally, knowing the location and quantity of vesicles and mitochondria are prerequisites to answer two emerging questions - whether astrocytes have a signaling role within the brain and whether astrocytes are highly metabolically active. Here, we provided structural context for these questions by tracing and 3D reconstructing three neighboring astrocytes using serial block-face scanning electron microscopy. Our reconstructions reveal a spongiform astrocytic morphology resulting from the abundance of reflexive and leaflet processes. At the interfaces, varying sizes of astrocyte-astrocyte contacts were identified. Inside an astrocyte domain, synapses contact the entire astrocyte, and synapse-astrocyte contacts increase from soma to terminal leaflets. In contrast to densely packed vesicles at synaptic boutons, vesicle-like structures were scant within astrocytes. Lastly, astrocytes contain dense mitochondrial networks with a mitochondrial volume ratio similar to that of neurites. Together, these ultrastructural details should expand our understanding of functional astrocyte-astrocyte and astrocyte-neuron interactions.

The low conductance mitochondrial permeability transition pore confers excitability and CICR wave propagation in a computational model.

Mitochondria have long been known to sequester cytosolic Ca(2+) and even to shape intracellular patterns of endoplasmic reticulum-based Ca(2+) signaling. Evidence suggests that the mitochondrial network is an excitable medium which can demonstrate independent Ca(2+) induced Ca(2+) release via the mitochondrial permeability transition. The role of this excitability remains unclear, but mitochondrial Ca(2+) handling appears to be a crucial element in diverse diseases as diabetes, neurodegeneration and cardiac dysfunction that also have bioenergetic components. In this paper, we extend the modular Magnus-Keizer computational model for respiration-driven Ca(2+) handling to include a permeability transition based on a channel-like pore mechanism. We demonstrate both excitability and Ca(2+) wave propagation accompanied by depolarizations qualitatively similar to those reported in cell and isolated mitochondria preparations. These waves depend on the energy state of the mitochondria, as well as other elements of mitochondrial physiology. Our results support the concept that mitochondria can transmit state dependent signals about their function across the mitochondrial network. Our model provides the tools for predictions about the internal physiology that leads to this qualitatively different Ca(2+) excitability seen in mitochondria.

A quantitative real time PCR method to analyze T cell receptor Vbeta subgroup expansion by staphylococcal superantigens.

BACKGROUND: Staphylococcal enterotoxins (SEs), SE-like (SEl) toxins, and toxic shock syndrome toxin-1 (TSST-1), produced by Staphylococcus aureus, belong to the subgroup of microbial superantigens (SAgs). SAgs induce clonal proliferation of T cells bearing specific variable regions of the T cell receptor beta chain (Vbeta). Quantitative real time PCR (qRT-PCR) has become widely accepted for rapid and reproducible mRNA quantification. Although the quantification of Vbeta subgroups using qRT-PCR has been reported, qRT-PCR using both primers annealing to selected Vbeta nucleotide sequences and SYBR Green I reporter has not been applied to assess Vbeta-dependent expansion of T cells by SAgs. METHODS: Human peripheral blood mononuclear cells were stimulated with various SAgs or a monoclonal antibody specific to human CD3. Highly specific expansion of Vbeta subgroups was assessed by qRT-PCR using SYBR Green I reporter and primers corresponding to selected Vbeta nucleotide sequences. RESULTS: qRT-PCR specificities were confirmed by sequencing amplified PCR products and melting curve analysis. To assess qRT-PCR efficiencies, standard curves were generated for each primer set. The average slope and R2 of standard curves were -3.3764 +/- 0.0245 and 0.99856 +/- 0.000478, respectively, demonstrating that the qRT-PCR established in this study is highly efficient. With some exceptions, SAg Vbeta specificities observed in this study were similar to those reported in previous studies. CONCLUSIONS: The qRT-PCR method established in this study produced an accurate and reproducible assessment of Vbeta-dependent expansion of human T cells by staphylococcal SAgs. This method could be a useful tool in the characterization T cell proliferation by newly discovered SAg and in the investigation of biological effects of SAgs linked to pathogenesis.

A model of direction selectivity in the starburst amacrine cell network.

Displaced starburst amacrine cells (SACs) are retinal interneurons that exhibit GABA( A ) receptor-mediated and Cl (-) cotransporter-mediated, directionally selective (DS) light responses in the rabbit retina. They depolarize to stimuli that move centrifugally through the receptive field surround and hyperpolarize to stimuli that move centripetally through the surround (Gavrikov et al, PNAS 100(26):16047-16052, 2003, PNAS 103(49):18793-18798, 2006). They also play a key role in the activity of DS ganglion cells (DS GC; Amthor et al, Vis Neurosci 19:495-509 2002; Euler et al, Nature 418:845-852, 2002; Fried et al, Nature 420:411- 414, 2002; Gavrikov et al, PNAS 100(26):16047-16052, 2003, PNAS 103(49):18793-18798, 2006; Lee and Zhou, Neuron 51:787-799 2006; Yoshida et al, Neuron 30:771-780, 2001). In this paper we present a model of strong DS behavior of SACs which relies on the GABA-mediated communication within a tightly interconnected network of these cells and on the glutamate signal that the SACs receive from bipolar cells (a presynaptic cell that receives input from cones). We describe how a moving light stimulus can produce a large, sustained depolarization of the SAC dendritic tips that point in the direction that the stimulus moves (i.e., centrifugal motion), but produce a minimal depolarization of the dendritic tips that point in the opposite direction (i.e., centripetal motion). This DS behavior, which is quantified based on the relative size and duration of the depolarizations evoked by stimulus motion at dendritic tips pointing in opposite directions, is robust to changes of many different parameter values and consistent with experimental data. In addition, the DS behavior is strengthened under the assumptions that the Cl(-) cotransporters Na( + )-K( + )-Cl( -) and K( + )-Cl( -) are located in different regions of the SAC dendritic tree (Gavrikov et al, PNAS 103(49):18793-18798, 2006) and that GABA evokes a long-lasting response (Gavrikov et al, PNAS 100(26):16047-16052, 2003, PNAS 103(49):18793-18798, 2006; Lee and Zhou, Neuron 51:787-799, 2006). A possible mechanism is discussed based on the generation of waves of local glutamate and GABA secretion, and their postsynaptic interplay as the waves travel between cell compartments.

A mathematical model of the sleep/wake cycle.

We present a biologically-based mathematical model that accounts for several features of the human sleep/wake cycle. These features include the timing of sleep and wakefulness under normal and sleep-deprived conditions, ultradian rhythms, more frequent switching between sleep and wakefulness due to the loss of orexin and the circadian dependence of several sleep measures. The model demonstrates how these features depend on interactions between a circadian pacemaker and a sleep homeostat and provides a biological basis for the two-process model for sleep regulation. The model is based on previous "flip-flop" conceptual models for sleep/wake and REM/NREM and we explore whether the neuronal components in these flip-flop models, with the inclusion of a sleep-homeostatic process and the circadian pacemaker, are sufficient to account for the features of the sleep/wake cycle listed above. The model is minimal in the sense that, besides the sleep homeostat and constant cortical drives, the model includes only those nuclei described in the flip-flop models. Each of the cell groups is modeled by at most two differential equations for the evolution of the total population activity, and the synaptic connections are consistent with those described in the flip-flop models. A detailed analysis of the model leads to an understanding of the mathematical mechanisms, as well as insights into the biological mechanisms, underlying sleep/wake dynamics.