D M Rojas Castro

A Revel

M Ménard

A Robust Neural Robot Navigation Using a Combination of Deliberative and Reactive Control Architectures

This paper proposes a hybrid neural-based control architecture for robot indoor navigation. This architecture preserves all the advantages of reactive architectures such as rapid responses to unforeseen problems in dynamic environments while combining them with the global knowledge of the world used in deliberative architectures. In order to take the right decision during navigation, the reactive module allows the robot to corroborate the dynamic visual perception with the a priori knowledge of the world gathered from a previously examined floor plan. Experiments with the robot functioning based on the proposed architecture in a simple navigation scenario prove the feasibility of the approach.

Introduction

Navigation strategies allowing mobile robots to autonomously travel from a starting point to a goal are extremely diverse [START_REF] Siegwart | Introduction to autonomous mobile robots[END_REF]. Control architectures are the essential component of such navigation. They define the capacities of the robot to plan a path trajectory, to undertake autonomous decision-making and to execute the appropriate reaction according to the perceived environment information. Three types of control architecture approaches of mobile robots have been proposed in the literature: deliberative [START_REF] Latombe | Robot Motion Planning[END_REF], reactive [START_REF] Arkin | Integrating Behavioral, Perceptual and World Knowledge in Reactive Navigation[END_REF] and hybrid [START_REF] Qureshi | Gillett -The cognitive controller: a hybrid, deliberative/reactive control architecture for autonomous robots[END_REF] [START_REF] Low | A hybrid mobile robot architecture with integrated planning and control[END_REF].

This paper proposes a hybrid control architecture where the robot emulates the cognition process of a human brain when navigating an unknown building by "reading" a map or a floor plan using its camera and "remembering" a sequence of signs to follow in order to reach its goal destination while overcoming the challenges presented in dynamic environments. Our approach addresses the control in a manner that is significantly different from existing hybrid architectures mainly in two ways. Firstly, the a priori global knowledge of the environment is gathered by the robot from a floor plan of the building and it is used to only corroborate the dynamic visual information instead of directly controlling the actions of the robot. Secondly, instead of using a complete motion path, this architecture makes use of navigation signs (and their expected sequence in the route), the associated directional implications of which may be learnt as a consequence of a stimulus-response model during navigation.

Hybrid Neural-Based Control Architecture

The overall architecture proposed in this paper (see Fig. 1) is based on the PerAc ("Perception-Action") architecture proposed by Gaussier and Zrehen [START_REF] Gaussier | Perac: A neural architecture to control artificial animals[END_REF] as an organized neural structure allowing learning of sensory-motor associations.

It follows the same neural mechanism of evolution by robot-environment interaction. However, unlike the PerAc architecture, it uses an a priori knowledge of the environment in order to corroborate the dynamic visual information perceived during navigation. Therefore, it is composed of two modules (Fig. 1): A deliberative module, corresponding to the processing chain in charge of computing a path plan by extracting a sequence of navigation signs expected to be found in the environment; and a reactive module, which integrates the said sequence information and constantly uses it in order to control online navigation. This latter reactive module is composed of two data streams corresponding to perception and action flows, similar to the PerAc architecture. The first level (dotted purple line region in Fig. 1) uses a reflex mechanism that controls directly the robot's action based on the information extracted from the perceived input. The second level (dashed orange line region in Fig. 1) uses a cognitive mechanism performing recognition of the aforementioned perceptive flow and allows learning of the associations between the recognition of a particular sign and the realization of a particular action.

Deliberative Module: Route planning and sign sequence

The global knowledge of the world is represented by a paper-based document (e.g. floor plan) that is placed in front of the robot's camera just once, before the navigation activity starts (Fig. 2a). It contains the important information to define a potential navigation trajectory. In this work, this information is represented by some navigation signs used as reference points that are expected to be seen by the robot in the real world navigation (Fig. 2b). By means of computer vision methods, the robot is able to "read" the image of the floor plan acquired from its camera, generate an optimal plan to reach the goal, extract and memorize a sequence of signs arranged from the closest point to the furthest with respect to the starting point. This process is achieved through four main stages (Fig. 1) the details of which are omitted here for the sake of brevity, in favor of detailing the more relevant reactive module. Once the sequence of signs has been extracted, the robot is ready to start navigating the building as described in the following paragraphs.

Reactive Module: Real world navigation

This module is composed of two levels similar to PerAc. However, it has a nested PerAc architecture within its own second level, and hence, is composed of three layers (Fig. 1). At the start of the exploration the robot may or may not know the meaning of each sign in terms of the instruction it represents with respect to way-finding. The architecture is designed such that if the sign is unknown or not detected at all, a "reflex" exploratory behavior gradually leads it to the correct direction and then the association between the sign and the movement performed is learnt (see 2.2.1 (b)). The learning is conditioned by a reinforcement signal which reflects the success or failure of the robot and whose information is transmitted by a modulation connection. Hence, if the same sign appears again and it has already been associated to a particular movement, the robot knows which direction to take and it executes the related movement (see 2.2.1 (a)).

Additionally, the architecture also performs a target approaching behavior when the robot is too far from a sign in order to be able to read it (see 2.2.1 (c)). Since the whole system works in parallel, a competitive mechanism decides on the best behavior (from among alternatives) for controlling the robot according to the stimulus received. Hence, the neural interconnection is done by either excitatory or inhibitory connections allowing or preventing the activation of group of neurons respectively.

Layers description a) Signs Recognition and Verification (SRV):

Once the robot begins exploration, this level enables the robot to perform a movement based on the combination of the acquired static data (sign sequence) and the dynamic visual perception of the robot's camera. This layer is composed of six neural groups as shown in Fig. 3. Each group has a number of neurons greater than the number of signs that can be recognized by the robot, each neuron representing a unique sign. The output direction group is composed of two neurons for left and right movements respectively. As the robot interacts with its environment, dynamic visual information is constantly fed to the Sign detection group. However, this group is only activated if one or more signs appear in the robot's view activating their corresponding neurons. The Sequenced sign group stores the sequenced signs from the floor plan. The expected sign from the sequence activates its corresponding neuron and once the robot has approached it, the sequence is rescanned to obtain the next expected sign. The corresponding neuron of the Sign merged detector group is then activated only when both inputting neurons (from the two neural groups mentioned above) are active. Then, the Short term memory group stores the activation value of the detected current sign in order to associate it, at a later stage, to the movement that would lead to the detection of the next sign. The WTA group enables the neuron with the highest activation value to stay active whereas all the others are set to zero by competitive "winner-takes-all" mechanism. The resulting activated neuron represents the current sign, to be associated to a particular action. The interconnectivity is made in such a way that it allows learning of the said association conditioned by the activation of a reinforcement signal set in the reflex level (see (b) below). Finally, the Output Direction group receives input from both layers (the current one and the Direction Determination Reflex Behavior layer (DDRB)).

b) Direction Determination Reflex Behavior (DDRB)

This layer is in charge of making the robot explore the environment by rotating in its place (to its left by design) using small reflex movements, to look for the next expected sign in the pre-captured sequence and then learn the association between the sign and the movement performed. This occurs in one of the following cases:

1. The received visual input from the camera does not correspond to the expected sign. In this case, the robot continues to search for it using the aforementioned rotational reflex movements. If the sign is found, the SRV layer (explained above in (a)) and TARB layer (explained below in (c)) are activated.

2. The expected sign is detected but has not yet been associated with a specific movement. In this case, the robot searches for the next sign by rotational movements. Once this next sign is found, the angle of rotation undergone is allocated to the current sign as its associated movement in that direction (left by default). If this angle is greater than 180º, the movement associated is a turn in the opposite direction (right).

Thereafter, the reinforcement signal is activated to learn the association in the SRV layer. The DDRB layer is mainly composed of seven neural groups (Fig. 4): The Sign Detector receives the input coming from the SRV group. Then, the reflex movement is triggered or inhibited by the Trigger Reflex group. The Reflex Output Direction group sends the information to the motor output to perform small leftwards rotational reflex movements. The angle of each rotational movement is stored in the Memory Angle group and then added to itself as many times as is required to find the next symbol. Once the symbol is found, the resulting angle is transmitted forward to the Direction group to be compared to a threshold and then reset to 0 for the next calculation. The resulting comparison allows the activation of the neuron corresponding to the movement, either a left or right by excluding each other. Thereafter, the Direction Result group activates the neuron corresponding to the resulting movement as well as the reinforcement signal R (set to 1) in the WTA group of SRV. Finally, once R is set to 1, the association between the current sign and the movement is learnt in the Output Direction group.

c) Target Approaching Reflex Behavior (TARB)

This layer is triggered when the robot is far from the sign. It directs an approach towards the sign by keeping it in the center of the robot's vision. If, for instance, the sign is situated at the left side in the robot's visual space, the movement needs to be performed towards the left. It is important for the robot to approach the target sign to avoid premature turns with respect to the intended point of turn for that sign. This layer is composed of the following three neural groups as shown in Fig. 5. Each group comprises of three neurons corresponding to a single movement each: walking to the left, walking to the right or walking straight ahead.

In the Reflex sign position group, each neuron has a position (x, y) in the robot's visual space so as to be compared to the position (within the same space) of the detected sign. They all behave as neural fields that can be calculated by a Gaussian function. The closer the neuron is in relation to the sign position, the higher the resulting value. All three values are sent to the WTA group that enables the neuron with the highest activation value to stay active whereas all the other neurons are set to zero. The Reflex output position group sends the resulting movement of the activated neuron to the motor output.

Convergence of layers

The three layers described above converge towards the motor output group which comprises of six neurons corresponding respectively to six possible movements: turning left, turning right, walking left, walking right, walking straight ahead and turning left as a reflex movement (Fig. 6). The activation of one excludes the others' depending on inhibitory and excitatory signal connections. When the robot is close to a sign (proximity sensor on), the activation of the reflex output position is inhibited and conversely if the robot is far from a sign, the direction movements are inhibited. The reset neuron inhibits the activation of the motor output group when it is activated by DDRB since the movement has already been performed by the reflex movement.

Verification Results

Fig. 7 shows a summary of results obtained (by following the path from Fig. 2b) in the form of the activation of the output neural group corresponding to the six possible movements that can be performed by the robot (so far), over time. The movements were a result of either recognition, proximity or absence of any signs from the extracted sign sequence. The activation of the reinforcement signal allowing the association learning is also shown. In each of the (a, t) plots shown, a is the binary activation of each neural group or the reinforcement signal and t the time seconds in terms of a PerAc cycle. Overall, it was observed that while navigating the environment, the robot was successfully able to perform the intended actions. For instance, when the robot was close to an expected but unknown sign, 'sign A', at t8, it performed reflex movements from t9 to t13 to search for the next sign, figured out the associated direction and learnt it at t14. When it got close to the same sign at t32, it was able to recall the learning and perform the associated movement.

Conclusions and Perspectives

A combination of reactive and deliberative architectures into a neural system for robot control navigation has been presented in this paper. As a result, a robot is capable of performing autonomous navigation, online learning of sensory-motor associations, parallel processing, decision-making and rapidly responding in environments prone to change. Implementation of the architecture in a simple indoor navigation scenario shows feasibility of this approach.

A potential drawback of the described implementation using the robot's camera is that the expected information might be partly occluded or not be visible at all. Consequently, the robot may get lost in the environment or be led to the wrong destination. A solution can be to implement principally the same architecture with other types of sensors to detect other relevant visual or non-visual cues in the environment. This would be possible by simply adding new layers without modifying the already in-built components or layers. Such a robust and complete architecture could even allow the robot to not only achieve its goal destination but also recreate an updated map according to the information assimilated from the environment.

Figure 2 .

 2 Figure 2. (2a) Nao robot reading the map; (2b) Generated route by following the navigation signs Figure 1. Schematic representation of the hybrid architecture

Figure 3 .

 3 Figure 3. Sign Recognition & Verification (SRV)

Figure 4 .

 4 Figure 4. Direction Determination Reflex Behavior (DDRB)

Figure 6 .Figure 5 .

 65 Figure 6. Convergence of layers towards Motor Output

* Research work supported by European regional development Funds(Contract35053) and the Poitou Charente Region


Harold Couchoux

Bruno Allard

Claude Legrand

Vincent Jacquemond

Christine Berthier
email: christine.berthier@univ-lyon1.fr

Loss of caveolin-3 induced by the dystrophy-associated P104L mutation impairs L-type calcium channel function in mouse skeletal muscle cells

Keywords: L-type calcium channel, caveolin, skeletal muscle

Caveolins are membrane scaffolding proteins which associate with-and regulate a variety of signalling proteins, including ion channels. A deficiency in caveolin-3 (Cav-3), the major striated muscle isoform, is responsible for skeletal muscle disorders, such as limb girdle muscular dystrophy 1C (LGMD 1C). The molecular mechanisms leading to the muscle wasting that characterizes this pathology are poorly understood. Here we show that a loss of Cav-3 induced by the expression of the LGMD 1C-associated mutant P104L (Cav-3 P104L ) provokes a reduction by half of the maximal conductance of the voltage-dependent L-type Ca 2+ channel in mouse primary cultured myotubes and foetal skeletal muscle fibers.

Confocal immunomiscrocopy indicated a colocalization of Cav-3 and Ca v 1.1, the poreforming subunit of the L-type Ca 2+ channel, at the surface membrane and in the developing T-tubule network in control myotubes and foetal fibers. In myotubes expressing Cav-3 P104L , the loss of Cav-3 was accompanied by a 66% reduction in Ca v 1.1 mean labelling intensity.

Our results suggest that Cav-3 is involved in proper L-type Ca 2+ channel membrane function and localization in skeletal muscle cells and that an alteration of L-type Ca 2+ channels could be involved in the physiopathological mechanisms of caveolinopathies.

Introduction

Caveolins are integral membrane proteins associated with cholesterol-rich omegashaped plasma membrane pits called caveolae [START_REF] Parton | Lipid rafts and caveolae as portals for endocytosis: new insights and common mechanisms[END_REF][START_REF] Cohen | Role of caveolae and caveolins in health and disease[END_REF]). The mammalian caveolin protein family comprises caveolin-1 and -2, expressed in many non-muscle cells and the muscle-specific caveolin-3 (Cav-3) [START_REF] Way | M-caveolin, a muscle-specific caveolin-related protein[END_REF][START_REF] Tang | Molecular cloning of caveolin-3, a novel member of the caveolin gene family expressed predominantly in muscle[END_REF]. In the recent years, point mutations and deletions in the human CAV3 gene were shown to be responsible for several muscle disorders [START_REF] Cohen | Role of caveolae and caveolins in health and disease[END_REF][START_REF] Fulizio | Molecular and muscle pathology in a series of caveolinopathy patients[END_REF]. One of the best characterized mutation consists in the P104L substitution, associated with limb girdle muscular dystrophy 1C (LGMD 1C) [START_REF] Minetti | Mutations in the caveolin-3 gene cause autosomal dominant limb-girdle muscular dystrophy[END_REF][START_REF] Betz | Mutations in CAV3 cause mechanical hyperirritability of skeletal muscle in rippling muscle disease[END_REF]. Cav-3 P104L exerts a dominant negative effect presumably through the formation of aggregates with the wild type Cav-3 which accumulate in the Golgi complex and are degraded through a proteasome-dependent mechanism (Galbiati et al. 1999a;Galbiati et al. 1999b;[START_REF] Sunada | Transgenic mice expressing mutant caveolin-3 show severe myopathy associated with increased nNOS activity[END_REF][START_REF] Carozzi | Inhibition of lipid raft-dependent signalling by a dystrophy-associated mutant of caveolin-3[END_REF].

How a defective Cav-3 expression leads to muscle disorders remains very mysterious, largely due to the poor understanding of the role played by Cav-3 in normal skeletal muscle function and to controversial related series of results described in the literature. Results obtained using antisense inhibition of Cav-3 expression in cultured C2C12 skeletal muscle cells suggest a role for Cav-3 in promoting myoblast fusion (Galbiati et al. 1999a) whereas opposite conclusions were drawn from results obtained on muscle cell lines derived from Cav-3 null mice and Cav-3 over-expressing mice [START_REF] Volonte | Modulation of myoblast fusion by caveolin-3 in dystrophic skeletal muscle cells: implications for Duchenne muscular dystrophy and limb-girdle muscular dystrophy-1C[END_REF]. Also, recent results obtained on zebrafish embryos after Cav-3 knock-down suggested that Cav-3 is required for correct muscle intracellular organization and myogenesis [START_REF] Nixon | Zebrafish as a model for caveolin-associated muscle disease; caveolin-3 is required for myofibril organization and muscle cell patterning[END_REF]. Along this line, immunolabeling experiments revealed a close association of Cav-3 with the T-tubule network in developing and mature skeletal muscle fibers [START_REF] Parton | Caveolin-3 associates with developing Ttubules during muscle differentiation[END_REF][START_REF] Galbiati | Caveolin-3 null mice show a loss of caveolae, changes in the microdomain distribution of the dystrophin-glycoprotein complex, and t-tubule abnormalities[END_REF][START_REF] Ralston | Caveolin-3 is associated with the T-tubules of mature skeletal muscle fibers[END_REF].

Additionally, T-tubule structure was shown to be altered in Cav-3 deficient muscles suggesting that the protein may play a role in proper T-tubule network organization [START_REF] Galbiati | Caveolin-3 null mice show a loss of caveolae, changes in the microdomain distribution of the dystrophin-glycoprotein complex, and t-tubule abnormalities[END_REF][START_REF] Minetti | Impairment of caveolae formation and Tsystem disorganization in human muscular dystrophy with caveolin-3 deficiency[END_REF].

Although multiple molecular investigations have shown that Cav-3 may associate with a number of structural or signalling proteins including dystroglycan [START_REF] Song | Expression of caveolin-3 in skeletal, cardiac, and smooth muscle cells. Caveolin-3 is a component of the sarcolemma and co-fractionates with dystrophin and dystrophin-associated glycoproteins[END_REF][START_REF] Sotgia | Caveolin-3 directly interacts with the Cterminal tail of beta -dystroglycan. Identification of a central WW-like domain within caveolin family members[END_REF], M-phosphofructokinase [START_REF] Scherer | Association of phosphofructokinase-M with caveolin-3 in differentiated skeletal myotubes. Dynamic regulation by extracellular glucose and intracellular metabolites[END_REF], dysferlin [START_REF] Matsuda | The sarcolemmal proteins dysferlin and caveolin-3 interact in skeletal muscle[END_REF] or nitric oxide synthase [START_REF] Feron | Endothelial nitric oxide synthase targeting to caveolae. Specific interactions with caveolin isoforms in cardiac myocytes and endothelial cells[END_REF][START_REF] Venema | Interaction of neuronal nitric-oxide synthase with caveolin-3 in skeletal muscle. Identification of a novel caveolin scaffolding/inhibitory domain[END_REF], there is still no explanation as to how Cav-3 deficiency leads to the progressive and irreversible skeletal muscle wasting that characterizes muscular caveolinopathies. Alteration of intracellular Ca 2+ handling is the source of various cell disorders and is highly suspected to be involved in a number of the most devastating degenerative muscle pathologies [START_REF] Mallouk | Elevated subsarcolemmal Ca 2+ in mdx mouse skeletal muscle fibers detected with Ca 2+ -activated K + channels[END_REF][START_REF] Rizzuto | When calcium goes wrong: genetic alterations of a ubiquitous signalling route[END_REF]. Yet, up to now, the consequences of a down-regulation of Cav-3 on skeletal muscle Ca 2+ homeostasis have never been investigated. Interestingly we previously showed that depletion of membrane cholesterol in foetal skeletal fibers induced a disruption of caveolae and T-tubules and a marked reduction in the L-type Ca 2+ channel function of the dihydropyridine receptor (DHPR), a critical element in the regulation of intracellular Ca 2+ in skeletal muscle (Pouvreau et al. 2004a). This altered L-type voltagedependent Ca 2+ channel function caused by cholesterol depletion could have resulted from the associated loss of Cav-3.

Here, we investigated the properties of voltage-dependent Ca 2+ channels in mouse primary cultured myotubes and in foetal muscle fibers expressing the LGMD 1C-associated Cav-3 P104L mutant.

We demonstrate that the loss of Cav-3 induced by expression of the Cav-3 P104L mutant induces (i) a drastic reduction in the maximum conductance of the voltage-activated L-type Ca 2+ channel in both models, and (ii) a decrease in the labelling intensity of the pore-forming alpha1S subunit of the DHPR protein (Ca v 1.1).

Part of this work has been published in abstract form [START_REF] Couchoux | Downregulation of caveolin-3 affects L-type Ca2+ channel functional expression in mouse skeletal muscle[END_REF].

Methods

All experiments were carried out at room temperature unless otherwise stated.

Isolation and culture of skeletal muscle cells

A suspension of muscle cells was first obtained from ribcages of 18-day-old mouse foetuses (Swiss OF1, Charles River, France) using enzymatic treatment, as previously described (Pouvreau et al. 2004a). Briefly, pregnant mice were killed by cervical dislocation and the foetuses were extracted after laparotomy and killed by decapitation, in accordance with local ethical guidelines laid down by the French directives for care of laboratory animals (decree 87-848). The two halves of the ribcage of each foetus were dissected in normal Tyrode solution containing (mM): 140 NaCl, 5 KCl, 2.5 CaCl 2 , 1 MgCl 2 , 10 Hepes-NaOH, pH 7.4. The tissues were incubated at 37 °C for 15 min in normal Tyrode solution containing 3 mg ml -1 collagenase (type I, Sigma) and 1 mg ml -1 trypsin (type III, Sigma). Fibers were isolated by trituration with a sterile pipette and then allowed to sediment for 10 min according to a procedure described by Rahkila and collaborators [START_REF] Rahkila | Transport pathway, maturation, and targetting of the vesicular stomatitis virus glycoprotein in skeletal muscle fibers[END_REF], plated on poly-D-lysine-coated dishes and cultured in survival medium (DMEM with 2% decomplemented horse serum).

For primary cultured skeletal myotubes, the supernatant from the sedimentation step was twice dispersed in dishes coated with calf skin collagens (Calbiochem) and allowed to sediment for 15 min to remove fibroblasts. Myoblasts from the supernatant were grown in proliferation medium consisting of DMEM with 10% foetal bovine serum, 10% horse serum and 2% chick embryo extract until transfection. All cells were cultivated at 37°C in 5% CO2.

Transfection

Plasmids encoding YFP-tagged wild type Cav-3 (Cav-3 wt -YFP) and HA-tagged Cav-3 P104L were generously provided by R. Parton (University of Queensland, Australia).

YFP-tagged Cav-3 P104L (Cav-3 P104L -YFP) was generated by subcloning into pEYFP (Clontech) using Kpn I/BamHI restriction sites.

Foetal fibers were transfected after 2 days in survival conditions using CombiMag (OzBiosciences) combined with FuGene 6 (Roche Diagnostics). Primary myoblasts were transfected at ~70% confluency using the same method and were allowed subsequently to differentiate by switching to DMEM with 10% foetal bovine serum. Immunostaining and electrophysiology experiments were performed 5 days post-transfection.

Immunofluorescence staining

Cells were fixed with 1% paraformaldehyde-0.01% glutaraldehyde in PBS (10 minutes) and permeabilized with 1% Triton X100-50 mM glycine in PBS (5 minutes). Nonspecific sites were blocked with 3% normal goat serum (NGS)-0.1% bovine serum albumin in PBS (30 minutes) and endogenous biotins were saturated using the Avidin/Biotin Blocking kit (Vector Laboratories). Cells were then incubated for 2 hours with anti-Cav-3 rabbit polyclonal antibody (PA1-066, Affinity BioReagents) or anti-Ca v 1.1 mouse monoclonal antibody (mAb427, Chemicon International) diluted respectively 1:1,500 and 1:500 in PBS with 1% NGS. Cells were subsequently incubated for 30 minutes with biotinconjugated goat anti-rabbit or goat anti-mouse IgG (Jackson Immunoresearch Laboratories) diluted 1:1,000 in PBS with 1% NGS and treated for 30 minutes with Cy3-or Alexa 488conjugated streptavidin (Jackson Immunoresearch Laboratories or Molecular Probes, respectively) diluted 1:1,000 in PBS. For sequential labelling, cells were first labelled with the anti-Cav-3 primary antibody then labelled with the anti-Ca v 1.1 primary antibody after performing a mouse non specific sites blocking step (MOM Blocking kit, Vector Laboratories). Preparations were all mounted in anti-fading medium (Vectashield) and examined using a LSM 510 confocal fluorescence microscope (Zeiss). Excitation was performed with argon and helium/neon lasers at 488 or 533 nm and optical sections were collected using a 63x oil immersion objective and appropriate band pass filters.

Image Analysis

Ca v 1.1 labelling was quantified by numerical analysis of Z-series of confocal images taken from Cav-3 P104L -YFP transfected primary cultures after immunolabeling of Ca v 1.1. The confocal Z-step size was set to 0.5 µm. Settings including photomultiplier gain, offset, aperture, and laser power were maintained rigorously identical to allow comparison of the labelling intensity between different samples.

Field-specific analysis was performed by measuring Ca v 1.1 immunoreactivity of Cav-3 P104L -YFP transfected myotubes and of non-transfected control myotubes present on a same RGB image. RGB images corresponding to each optical section of a given Z-stack were analyzed using the Image J software (National Institute of Health, Bethesda, MA, USA). Analysis was performed after extracting layers of each RGB image: the red layer corresponded to the Cav1.1 labelling (Cy3 image) and the green layer corresponded to the Cav-3 P104L -YFP signal. In order to quantify the labelling relative to the cell surface area, the cell outer limits had to be estimated from the fluorescence signals. In control myotubes this was achieved on the Cy3 image whereas in Cav-3 P104L -YFP transfected myotubes the Cav-3 P104L -YFP signal was used because of the very weak, unreliable Ca v 1.1 labelling in some of these myotubes. Non-labelled regions that clearly corresponded to nuclei were excluded from the area measurement.

Within the thus selected regions, the Ca v 1.1 immunoreactivity of Cav-3 P104L -YFP expressing myotubes and control myotubes was quantified on the Cy3 image by measuring the sum of the values of the pixels (integrated density) after optimally setting the threshold to eliminate background signal.

Mean Ca v 1.1 labelling intensity was determined for each control and transfected myotube present on a given image by normalizing the integrated density to the cell area.

These measurements were performed on all images of the confocal stack and seven different stacks of images were analyzed.

Electrophysiology

Membrane currents were recorded in the whole cell configuration on cultured myotubes and foetal fibers using a patch clamp amplifier (RK 400, Bio-Logic). Pipettes were filled with (in mM): 110 Cs-aspartate, 20 TEACl, 2 MgCl 2 , 5 MgATP, 10 EGTA, 10 HEPES, pH 7.2. The bath solution contained (in mM): 140 TEAMeSO 3 , 10 CaCl 2 , 2 MgCl 2 , 0.001 TTX (Sigma), 10 HEPES, pH 7.2. Voltages were not corrected for liquid junction potentials calculated to be less than 5 mV with the solutions used. Data acquisition and generation of command voltage pulses were done using the pClamp9 software driving the Digidata 1322A A/D, D/A converter (Axon Instruments Inc.). Cell capacitance, used to calculate the density of currents (A/F), was determined by integration of a control current trace obtained with a 10-mV depolarizing pulse from -80 mV. Leak currents were subtracted from all recordings using a 10-mV depolarizing pulse from the holding potential supposing a linear evolution of leak current with depolarization. The amplitude of the currents was measured at the peak of the currents and individual curves of the voltage dependence of I Ca, L density were fitted with the following equation (1):

I(V) = G max (V- V rev )/{1+exp[(V 0.5 -V)/k]}, where I(V)
is the density of the current measured, V, the test pulse, G max , the maximum conductance per capacitance, V rev , the apparent reversal potential, V 0.5 , the half-activation voltage, and k, a steepness factor. Individual curves of the voltage dependence of the normalized conductance were obtained by dividing I(V) by

G max (E -E rev ).

Statistics

Non-linear least-squares fits were performed using a Marquadt-Levenberg algorithm routine included in Microcal Origin. Data values are presented as means ± S.E.M. In the figures, values of standard errors are in some cases smaller than data symbols and are therefore not apparent. Data were statistically analyzed using Student t-test. Values were considered significant when P < 0.05.

Results

Cav-3 P104L expression leads to the loss of endogenous Cav-3 in mouse primary cultured myotubes

We first examined Cav-3 imunolocalization in both control and Cav-3 P104Lexpressing primary myotubes. The immunolabelling patterns shown on Fig. 1 are representative of results obtained in several independent experiments (n=5 for control and n=6 for Cav-3 P104L -transfected myotubes).

In control myotubes (Fig. 1A), Cav-3 labelling (red) was intense at the surface plasma membrane and was also characteristically present more discretely throughout the cell where it exhibited a punctate or reticular pattern consistent with the developing Ttubule membrane system [START_REF] Flucher | Biogenesis of transverse tubules in skeletal muscle in vitro[END_REF][START_REF] Parton | Caveolin-3 associates with developing Ttubules during muscle differentiation[END_REF].

As shown in Fig. 1B, Cav-3 P104L -expressing myotubes displayed a different pattern of immunostaining (bottom panel). Yellow fluorescent protein (YFP)-tagged Cav-3 P104L (top panel) localized in perinuclear vesicles likely belonging to the Golgi apparatus. The corresponding Cav-3 labelling indicated that endogenous Cav-3 no longer reached the sarcolemma, being most often barely detectable (Fig. 1B). Our data thus confirm that expression of the mutant Cav-3 P104L in primary myotubes induces a drastic loss of endogenous membrane-associated Cav-3, as previously shown in adult skeletal muscle of transgenic mouse and LGMD-1C patients [START_REF] Minetti | Mutations in the caveolin-3 gene cause autosomal dominant limb-girdle muscular dystrophy[END_REF][START_REF] Sunada | Transgenic mice expressing mutant caveolin-3 show severe myopathy associated with increased nNOS activity[END_REF]. Golgiassociated mutant or endogenous caveolin was not detected by the PA1-066 antibody, probably due to epitope-masking of the corresponding N-terminal epitope when in the Golgi compartment as previously suggested [START_REF] Luetterforst | Molecular characterization of caveolin association with the Golgi complex: identification of a cis-Golgi targeting domain in the caveolin molecule[END_REF].

The amplitude of the L-type Ca 2+ current (I Ca, L ) is reduced in Cav-3 P104L -expressing mouse primary cultured myotubes

The properties of voltage-dependent Ca 2+ channels were studied in control and Cav-3 P104L -expressing myotubes. Two types of voltage-dependent Ca 2+ currents typical of cultured skeletal myotubes were recorded. I Ca, T , displaying fast activating and inactivating kinetics, developed for depolarizations within the -40 to 0 mV range and I Ca, L , displaying slow activating and inactivating kinetics, was recorded for depolarizations to values more positive than -20 mV (Fig. 2A). We found that the amplitude of I Ca, L was strongly reduced in Cav-3 P104L -expressing myotubes as compared to control myotubes while I Ca, T remained unchanged (Figs 2A,B). For each cell, the current-voltage relationship of I Ca, L was fitted using equation 1 (see methods). Mean values for G max , V 0.5 , V rev and k were 197 ± 12 S/F, 9 ± 0.8 mV, 65 ± 1.6 mV and 5 ± 0.4 mV for control myotubes (n = 40) and 88 ± 13 S/F, 11 ± 1.9 mV, 55 ± 2 mV and 5 ± 0.5 mV for Cav-3 P104L -expressing myotubes (n = 21), respectively (Fig. 2C). In Cav-3 P104L -expressing myotubes, there was a mean reduction of 55% of G max and a 10 mV shift of V rev towards negative potentials (P < 0.0001 and 0.001, respectively). The slight shift of E rev was likely due to unmasking of a contaminating outward current component as a consequence of the decreased amplitude of the inward Ca 2+ current. In myotubes transfected with YFP alone, G max (191 ± 10, n = 6) was not significantly changed. Finally, we found that the mean capacitance was not significantly different in control myotubes (218 ± 20 pF, n = 40) and in Cav-3 P104L -expressing myotubes (229 ± 27 pF, n = 34).

Expression of Cav-3 P104L in myotubes alters the distribution and mean intensity of Ltype voltage-dependent Ca 2+ channel immunolabelling

A possible reason for the drastic reduction in the amplitude of I Ca, L induced by Cav-3 P104L expression is that loss of Cav-3 leads to a down-expression or defective targeting of the DHPR.

To explore this possibility, control and Cav-3 P104L -expressing myotubes were immunostained with anti-Cav-3 and anti-Ca v 1.1 antibodies. Representative images are shown on Figs 3 and4. In control myotubes (Fig. 3) both proteins essentially colocalized, this being particularly apparent at the level of the surface plasma membrane. In the internal cellular compartment, such a colocalization is also visible although it appears more restricted to specific areas (arrowheads). The Ca v 1.1 labelling corresponded to an extended array of regularly distributed dots at the level of which Cav-3 labelling was, although not systematically, also concentrated. As for Fig. 1, these labelled internal structures likely belong to the nascent T-tubule system as previously documented [START_REF] Flucher | Biogenesis of transverse tubules in skeletal muscle in vitro[END_REF][START_REF] Flucher | Triad formation: organization and function of the sarcoplasmic reticulum calcium release channel and triadin in normal and dysgenic muscle in vitro[END_REF][START_REF] Parton | Caveolin-3 associates with developing Ttubules during muscle differentiation[END_REF].

In Cav-3 P104L -expressing myotubes, Ca v 1.1 labelling was clearly reduced (Fig. 4A).

In addition, part of the residual labelling appeared to redistribute around the nuclei, likely within the Golgi compartment, as indicated by the colocalization with Cav-3 P104L -YFP (merged image).

Ca v 1.1 labelling intensity (mean pixel value per pixel of cell area) was measured in individual control and in Cav-3 P104L -expressing myotubes using numerical analysis of confocal images. As shown in Fig. 4B, the mean Ca v 1.1 labelling intensity in Cav3 P104Lexpressing myotubes (23 ± 2.4 mean pixel value/pixel, n = 56) was significantly reduced by 66% as compared to control myotubes (68.2 ± 3.3, n = 134) (P < 0.0001). Comparing the labelling densities on pairs of control and Cav-3 P104L -expressing myotubes present in a same field image, so as to limit any bias introduced by variability within the different sampled images, yielded essentially identical results (not shown).

Cav-3 P104L expression in foetal skeletal muscle fibers also leads to the loss of endogenous Cav-3 and to a reduction of I Ca, L amplitude

Since Cav-3 is thought to be involved in muscle developmental processes, one of the possibilities is that the reduction of the voltage-dependent Ca 2+ channel function in Cav-3 P104L -expressing cultured myotubes could result from abnormal myotube differentiation. To test for this, we explored the channel function in foetal skeletal muscle fibers maintained in survival conditions and transfected with YFP-tagged Cav-3 P104L . This cell system, although not fully mature, possesses a well differentiated skeletal muscle phenotype and is suitable for transfection.

As shown on Fig. 5A (top panel), Cav-3 immunolabeling was present at the level of both the surface and T-tubule membrane in the control fibers (cell on the right) while the labelling intensity was strongly decreased in the Cav-3 P104L -expressing foetal fibers (cell on the left) indicating that, as in cultured myotubes, expression of the mutant Cav-3 P104L induced an important loss of endogenous Cav-3. It is also worth noting that in foetal fibers Cav-3 P104L -YFP also accumulates perinuclearly. We then measured Ca 2+ currents in Cav-3 P104L -expressing foetal fibers. Under survival conditions, I Ca, T vanished after a few days in culture while robust I Ca, L could still be recorded (Fig. 5B, left and middle traces). This current was typical of foetal skeletal muscle fibers (Pouvreau et al. 2004a) with slow kinetics of activation, a threshold at -10 mV, a peak at +20 mV and a reversal potential close to +80 mV (Fig. 5C). Its amplitude was strongly decreased in Cav-3 P104L -expressing fibers as compared to control fibers. Mean values for G max , V 0.5 , V rev and k were 203 ± 11 S/F, 10 ± 0.9 mV, 77 ± 1.1 mV and 4 ± 0.2 mV in control fibers (n = 31) and 85 ± 18 S/F, 16 ± 3.5 mV, 63 ± 5 mV and 4 ± 0.7 mV in Cav-3 P104L -expressing fibers (n = 8), respectively. G max was significantly reduced by 58% (P < 0.0001) and V rev and V 0.5 were significantly shifted by -14 mV (P < 0.0001) and +6 mV (P < 0.02) respectively, in Cav-3 P104L -expressing fibers as compared to control ones. The voltage shift of V 0.5 is clearly shown in the right panel of figure 5C, where the mean normalized conductance versus voltage relationship in control and Cav-3 P104L expressing foetal fibers is presented. Fitting the individual normalized conductance versus voltage relationship with a Boltzman equation gave mean values for V 0.5 and k of 12 ± 4.7 mV and 3.4 ± 2 mV in control fibers and of 17.7 ± 2.4 mV and 3.2 ± 0.6 mV in Cav-3 P104L expressing foetal fibers, respectively.

The mean capacitance of Cav-3 P104L -expressing fibers (182 ± 13, n = 8) did not significantly differ from the one of control fibers (175 ± 21, n = 31).

Over-expression of wild type Cav-3 in foetal skeletal muscle fibers does not affect I Ca, L

We also tested the effect of over-expression of wild type Cav-3 on the functional expression of the Ca 2+ channels. For this, expression of an YFP-tagged wild type Cav-3 was achieved after transfection of foetal muscle fibers in survival conditions. Fig. 5A (bottom panel) shows that under these conditions YFP-tagged wild type Cav-3 appears concentrated at the surface plasma membrane as well as in discrete intracellular clusters presumably corresponding to T-tubules. The measured I Ca, L yielded similar properties in Cav-3 over-expressing fibers and in control ones (Figs 5B,C). Mean values for the parameters fitted from the individual I/V curves in Cav-3 over-expressing fibers were 188 ± 16 S/F, 8 ± 1.6 mV, 72 ± 1.4 mV and 4 ± 0.3 mV for G max , V 0.5 , V rev and k respectively (n = 13 16). The only difference as compared to the control fibers was a slight 5 mV shift of V rev towards negative potentials (P = 0.01).

Discussion

The present study demonstrates that expression of the LGMD-1C associated Cav-3 P104L mutant in skeletal muscle cells induces a very significant alteration of both the L-type voltage-dependent Ca 2+ current and the cellular distribution of the corresponding channels in mouse skeletal muscle cells. Altogether, our results strongly suggest that a marked alteration of the function of the L-type voltage-dependent Ca 2+ channel results from Cav-3 deficiency in skeletal muscle. Such a deficiency, revealed by an almost complete loss of immunolabeling, was induced here by the expression of Cav-3 P104L mutant in transfected primary cultured myotubes and foetal fibers, thereby reproducing the pathological loss of caveolin observed in skeletal muscle of LGMD-1C patients and transgenic mice bearing the P104L mutation [START_REF] Minetti | Mutations in the caveolin-3 gene cause autosomal dominant limb-girdle muscular dystrophy[END_REF][START_REF] Sunada | Transgenic mice expressing mutant caveolin-3 show severe myopathy associated with increased nNOS activity[END_REF]. Upon expression of Cav-3 P104L in cultured myotubes and foetal fibers, mutant as well as of endogenous Cav-3 did not reach the sarcolemma and we observed a strong intracellular accumulation of Cav-3 P104L , presumably at the level of the Golgi apparatus, consistent with what was previously described in fibroblasts, C2C12, and LGMD-1C biopsies [START_REF] Minetti | Mutations in the caveolin-3 gene cause autosomal dominant limb-girdle muscular dystrophy[END_REF]Galbiati et al.1999b;[START_REF] Carozzi | Inhibition of lipid raft-dependent signalling by a dystrophy-associated mutant of caveolin-3[END_REF].

The reduction of I Ca, L amplitude in Cav3 P104L -expressing myotubes (55%) was accompanied by a similar reduction in the Ca v 1.1 labelling intensity (66%). It is thus likely that the reduced I Ca, L mainly results from a reduced number of channels rather than from an alteration of the functional properties of the channel. It was recently suggested that Cav-3 is implicated in trafficking since its down-regulation was shown to lead to Golgi accumulation of dysferlin [START_REF] Hernandez-Deviez | Aberrant dysferlin trafficking in cells lacking caveolin or expressing dystrophy mutants of caveolin-3[END_REF]. Trafficking and/or membraneanchoring of channels such as store-operated Ca 2+ channels were moreover found to be dependent on caveolin-1 [START_REF] Brazer | Caveolin-1 contributes to assembly of store-operated Ca 2+ influx channels by regulating plasma membrane localization of TRPC1[END_REF]. In this respect, the partial Golgi redistribution of the Ca v 1.1 labelling and its marked reduction upon Cav-3 P104L expression suggests that Cav-3 may also act as a post Golgi trafficking partner for the DHPR. Cav-3 could exert such a role from the earliest stages of muscle differentiation since both proteins display similar temporal appearance patterns, as shown on cultured muscle cells. Several studies have indeed demonstrated that the mRNA and corresponding L-type voltage-dependent Ca 2+ channel protein appear at a detectable level as myoblasts fuse into myotubes [START_REF] Romey | Ontogenesis and localization of Ca2+ channels in mammalian skeletal muscle in culture and role in excitation-contraction coupling[END_REF][START_REF] Varadi | Developmental regulation of expression of the alpha 1 and alpha 2 subunits mRNAs of the voltage-dependent calcium channel in a differentiating myogenic cell line[END_REF][START_REF] Cognard | Appearance and evolution of calcium currents and contraction during the early postfusional stages of rat skeletal muscle cells developing in primary culture[END_REF][START_REF] Kyselovic | Temporal differences in the induction of dihydropyridine receptor subunits and ryanodine receptors during skeletal muscle development[END_REF]; likewise, Cav-3 expression was reported to be strongly induced upon differentiation of cultured mouse skeletal muscle cells [START_REF] Way | M-caveolin, a muscle-specific caveolin-related protein[END_REF][START_REF] Song | Expression of caveolin-3 in skeletal, cardiac, and smooth muscle cells. Caveolin-3 is a component of the sarcolemma and co-fractionates with dystrophin and dystrophin-associated glycoproteins[END_REF][START_REF] Tang | Molecular cloning of caveolin-3, a novel member of the caveolin gene family expressed predominantly in muscle[END_REF][START_REF] Parton | Caveolin-3 associates with developing Ttubules during muscle differentiation[END_REF].

Cav-3 is also believed to participate to skeletal muscle T-tubules biogenesis. This hypothesis first originated from ultrastuctural and immunofluorescence studies [START_REF] Ishikawa | Formation of elaborate networks of T-system tubules in cultured skeletal muscle with special reference to the T-system formation[END_REF][START_REF] Franzini-Armstrong | Simultaneous maturation of transverse tubules and sarcoplasmic reticulum during muscle differentiation in the mouse[END_REF][START_REF] Parton | Caveolin-3 associates with developing Ttubules during muscle differentiation[END_REF][START_REF] Ralston | Caveolin-3 is associated with the T-tubules of mature skeletal muscle fibers[END_REF]. Furthermore, skeletal muscles from Cav-3 knock-out mice and Cav-3 P104L expressing LGMD-1C patients were both reported to harbour T-tubule structure abnormalities, although not identical ones [START_REF] Galbiati | Caveolin-3 null mice show a loss of caveolae, changes in the microdomain distribution of the dystrophin-glycoprotein complex, and t-tubule abnormalities[END_REF][START_REF] Minetti | Impairment of caveolae formation and Tsystem disorganization in human muscular dystrophy with caveolin-3 deficiency[END_REF]. Although Cav-3 does not appear indispensable for the formation of the T-tubule network, these data suggest that Cav-3 may play a role in its proper organization. One possible hypothesis could thus be that the observed impairment of the DHPR Ca 2+ channel function induced by Cav3 P104L -expression could result from an altered T-tubule biogenesis. In order to test this hypothesis, the mean electrical capacitance, which is a reliable index of the total membrane surface, was determined and found not significantly altered in Cav-3 P104L -expressing myotubes. This indicates that Cav-3 P104L expression did not alter the amount of the electrically active T-tubular membrane but rather specifically targeted the DHPR. This was further confirmed by results obtained in foetal muscle fibers in which a comparable decrease in maximal conductance of the L-type Ca 2+ channel was observed upon Cav-3 P104L expression. Under these conditions, and in contrast to cultured myotubes, Cav-3 P104L expression was achieved at a maturation stage at which well formed T-tubules are already present [START_REF] Takekura | Sequential docking, molecular differentiation, and positioning of T-Tubule/SR junctions in developing mouse skeletal muscle[END_REF]) so that an effect related to the disruption of T-tubules organization could hardly be raised. Besides, membrane capacitance was as well found unchanged in Cav-3 P104L -expressing foetal fibers. Furthermore, the lack of effect of Cav-3 P104L expression on I Ca, T in primary cultured myotubes provides a critical internal control in this set of experiments and further argues in favour of a selective effect of the loss of Cav-3 on DHPR.

Our immunolabeling experiments show that Cav-3 and DHPR proteins colocalize at both the surface and T-tubular membranes and are in agreement with previous studies [START_REF] Jorgensen | Subcellular distribution of the 1,4-dihydropyridine receptor in rabbit skeletal muscle in situ: an immunofluorescence and immunocolloidal gold-labelling study[END_REF][START_REF] Parton | Caveolin-3 associates with developing Ttubules during muscle differentiation[END_REF][START_REF] Ralston | Caveolin-3 is associated with the T-tubules of mature skeletal muscle fibers[END_REF]. However, it is not entirely clear whether Cav-3 directly physically interacts with DHPR and/or indirectly serves to organize ion channel regulating molecules or membrane environment for proper ion channel targeting. A number of other ion channels have been shown to be specifically targeted to caveolin-rich microdomains in different tissues [START_REF] Martens | Targeting of ion channels to membrane microdomains: localization of K V channels to lipid rafts[END_REF]). In cardiac muscle, Cav-3 has been shown to coimmunoprecipitate with the cardiac isoform of the pore-forming subunit of the L-type Ca 2+ channel and to play a role in the beta2-adrenergic response [START_REF] Balijepalli | Localization of cardiac Ltype Ca 2+ channels to a caveolar macromolecular signalling complex is required for beta(2)-adrenergic regulation[END_REF]. In endothelial cells, the Ca 2+ -activated K + channel was found to directly interact with caveolin-1 [START_REF] Wang | Caveolae targeting and regulation of large conductance Ca 2+ -activated K + channels in vascular endothelial cells[END_REF]) and caveolin-1 was also found to bind to the store-operated Ca 2+ TRPC1 channels [START_REF] Brazer | Caveolin-1 contributes to assembly of store-operated Ca 2+ influx channels by regulating plasma membrane localization of TRPC1[END_REF]. In our study, no reliable quantitative comparison of the Ca v 1.1 labelling between control and Cav-3 P104Lexpressing foetal fibers could be achieved, due to the much lower transfection efficiency in this system. We thus cannot rule out the possibility that, in this cell system, the observed reduction in L-type Ca 2+ channel conductance would result from mechanism(s) other than down-expression or mis-membrane-targeting or anchoring of the channel protein. Along this line, Western blot analysis showed that the amount of Ca v 1.1 was unchanged in fully matured Cav-3-null mouse muscles [START_REF] Galbiati | Caveolin-3 null mice show a loss of caveolae, changes in the microdomain distribution of the dystrophin-glycoprotein complex, and t-tubule abnormalities[END_REF]. These data hence suggest that, unlike in myotubes, the decrease of Ca 2+ channel conductance in the more differentiated foetal fibers does not result from a reduced density of the protein but is rather primarily due to an altered function of the L-type Ca 2+ channel associated with the loss of Cav-3. In agreement with this hypothesis, we found that the voltage dependence of the Ca 2+ current was significantly shifted towards positive membrane potentials in Cav-3 P104L expressing foetal fibers but not in myotubes. This result thus strengthens the possibility that the reduced amplitude of I Ca,L in Cav-3 P104L expressing cells mainly results from a disruption of the Ca 2+ channel function in foetal fibers whereas it mainly results from a reduced number of channels in myotubes. Finally, our observation that Cav-3 over-expression in foetal fibers did not have any drastic effect on I Ca, L may indicate that under physiological conditions the available amount of endogenous Cav-3 already allows optimal operating of the L-type Ca 2+ channel.

Our results provide important clues to explain the physio-pathological consequences of Cav-3 deficiency in skeletal muscle. A reduction of function of the DHPR may be associated with muscle weakness since it is directly involved in excitation-contraction coupling [START_REF] Rios | Voltage sensor of excitation-contraction coupling in skeletal muscle[END_REF]. This may account for the muscle weakness which characterizes LGMD-1C [START_REF] Woodman | Caveolinopathies: mutations in caveolin-3 cause four distinct autosomal dominant muscle diseases[END_REF]). The disease is also associated with progressive muscle wasting [START_REF] Woodman | Caveolinopathies: mutations in caveolin-3 cause four distinct autosomal dominant muscle diseases[END_REF], and, as suspected for a number of muscular dystrophies [START_REF] Rizzuto | When calcium goes wrong: genetic alterations of a ubiquitous signalling route[END_REF], chronic intracellular Ca 2+ overload may act in this case as the primary process of necrosis. Along this line, transgenic mice expressing Cav-3 P104L were shown to over-produce NO [START_REF] Sunada | Transgenic mice expressing mutant caveolin-3 show severe myopathy associated with increased nNOS activity[END_REF] which in turn causes a redox-sensitive continuous leak of Ca 2+ from the sarcoplasmic reticulum (Pouvreau et al. 2004b). In this way, the reduction in I Ca,L possibly together with a reduction in the voltage-controlled calcium release in Cav-3 deficient muscle may be seen as a compensatory or protective mechanism to preserve normal Ca 2+ levels.

Figure 1 .

 1 Figure 1. Cav-3 P104L -YFP expression leads to the loss of Cav-3 in primary cultured myotubes. A, in control myotubes, Cav-3 immunolabeling strongly outlines the surface plasma membrane. It localizes as well intracellularly in small dot-shaped clusters (arrowheads) as best seen on the partial 2X enlargement (inset). B, Cav-3 P104L -YFP accumulates preferably around the nuclei in vesicular structures (top image). Cav-3 immunolabeling is dramatically decreased, if not totally absent, in the Cav-3 P104L -YFP-expressing myotube (arrow) whereas membrane staining is seen at an extremity of a nearby non-transfected myotube (bottom image). Scale bars = 10 µm.

Figure 2 .

 2 Figure 2. I Ca, L is reduced in Cav-3 P104L -expressing primary cultured myotubes. A, representative currents elicited by applying 500 ms-lasting voltage pulses from a holding potential of -80 mV to the indicated values in the two kinds of cells. B, mean currentvoltage relationships established for the fast transient current (I Ca, T ) and for the slow activating current (I Ca, L ) in 40 control and 21 Cav-3 P104L -expressing myotubes. C, mean current-voltage relationships of isolated I Ca, L on the same control and mutant myotubes.The superimposed curves were calculated using the mean parameters obtained from fitting the voltage dependence in each fibre (see text for details).

Figure 3 .

 3 Figure 3. Double immunolabeling reveals a strong colocalization of Cav-3 and Ca v 1.1 in primary cultured myotubes. Cav-3 (red) and Ca v 1.1 (green) strongly colocalizes as clearly shown on the merged images. The colocalization is particularly pronounced at the surface plasma membrane, where both proteins accumulate (arrows). Additionally, the Ca v 1.1 labelling appears characteristically concentrated within internal punctate structures distributed rather uniformly throughout the cell and likely belonging to the developing T-tubule network. Cav-3 labelling appears generally weaker in this internal

Acknowledgments

We are very grateful to Dr R.G. Parton (University of Queensland, Australia) for the gift of the wild type and P104L caveolin cDNAs and to Dr L.E. Espinosa (University of Marseille, France) for his expert advice and help regarding image analysis. This work was supported by grants from CNRS, Agence Nationale de la Recherche, Université Lyon 1 and Association Française contre les Myopathies.

compartment except for specific areas, where it co-localizes with Ca v 1.1 (arrowheads).

Scale bar = 10 µm. A, in this representative Cav-3 P104L -YFP expressing myotube (green), Ca v 1.1 labelling (red) appears much dimmer as compared to the non-transfect neighbouring myotubes and is in addition partly redistributed to the perinuclear region where it colocalizes with Cav-3 P104L , as shown on the merged images (arrow). Scale bar = 10 µm. B, numerical image analysis of Ca v 1.1 labelling intensity. Field-specific analysis was performed on confocal Z-stacks by measuring Ca v 1.1 immunoreactivity of control myotubes (Ct) and Cav-3 P104L -expressing myotubes (P104L) present on a same image. Mean labelling intensity appeared to be reduced by a mean 66% percent in Cav-3 P104L -expressing myotubes (P <0.0001).


Tao Sun
email: tao@taosphysics.net

Green Function Theory of Strongly Correlated Electron Systems

A novel effective Hamiltonian in the subspace of singly occupied states is obtained by applying the Gutzwiller projection approach to a generalized Hubbard model with the interactions between two nearestneighbor sites. This model provides a more complete description of the physics of strongly correlated electron systems. The system is not necessarily in a ferromagnetic state as temperature T → 0 at any doping level δ ≥ 0. The system, however, must be in an antiferromagnetic state at the origin of the doping-temperature (δ-T ) plane (T → 0, δ = 0). Moreover, the model exhibits superconductivity in a doped region at sufficiently low temperatures. We summarize the studies and provide a phase diagram of the antiferromagnetism and the superconductivity of the model in the δ-T plane here. Details will be presented in subsequent papers.

Introduction

Since the discovery of high-temperature superconductivity in 1986 [1], a tremendous number of studies have been devoted to the understanding of its underlying mechanism. It is generally believed that some essential physics of this strongly correlated electron system can be described by the Hubbard model [2,3] and its strong-coupling limit, the t-J model [START_REF]A personal perspective from the pioneering times[END_REF][START_REF] Zhang | [END_REF]. The Hubbard model is the simplest approximation of the general Hamiltonian of the interacting electron systems, in which all the Coulomb interaction terms except the onsite term (U -term) are neglected. It is possible that the neglected interaction terms may play important roles in the understanding of the physics of the strongly correlated electron systems. That is, some important physics might have been missed in the Hubbard model. As a result, the t-J model, which is a projection of the Hubbard model to the subspace of singly occupied states, may not be sufficient to describe the essential physics of the strong coupling systems. In this study, we will apply the Gutzwiller projection scheme to a more general model, in which, in addition to the Hubbard on-site term, all of the two-site interaction terms between nearest-neighbor sites are retained in the approximation of the Coulomb potential energy. Our study shows that this novel model provides a more complete understanding of the properties of high-temperature superconductors.

The Model

The general Hamiltonian describing the dynamics of electrons in Wannier representation can be expressed as [2,6,[START_REF] Mahan | Many-Particle Physics[END_REF]]

H = ij,σ T ij c † iσ c jσ + 1 2 ijkl,σσ ′ ij| 1 r |kl c † iσ c † jσ ′ c lσ ′ c kσ , (1)
where c † iσ and c iσ are the creation and annihilation operators for an electron with spin σ in a Wannier orbital localized at site i, T ij is the Fourier transform of the band energy ǫ k

T ij = 1 N k ǫ k e ik•(Ri-Rj ) , (2)
and the Wannier representation matrix element is given by

ij| 1 r |kl = e 2 dxdx ′ φ * (x -R i )φ(x -R k )φ * (x ′ -R j )φ(x ′ -R l ) |x -x ′ | , (3)
where φ(x -R i ) and φ * (x -R i ) are the Wannier functions localized around lattice site i. In the most general cases where i = j = k = l, the Wannier matrix elements ij| 1 r |kl are four-center integrals and the corresponding terms in the series of the Coulomb interaction in Eq (1) are the so-called four-site terms. Since the Wannier function φ(x -R i ) goes to zero rapidly when x is away from R i , the matrix element (3) is not negligible only when the sites i, j, k, l are close enough so that the overlaps between the Wannier functions are sufficiently large. Therefore, the Coulomb interaction energy can be conveniently approximated by a number of its leading terms of the series. The biggest interaction term is the so-called on-site term (i = j = k = l)

1 2 U iσ n iσ n iσ , (4)
where n iσ = c † iσ c iσ and U = ii| 1 r |ii . If only this on-site term is taken into account, the Hubbard model is obtained. Apparently, the next leading terms are the ones where the set {i, j, k, l} actually consists of only one pair of nearestneighbor sites, which may be referred to as the two-site interaction terms [8]. In this work, we take one step further beyond the Hubbard model. We retain all of the terms up to the two-site interaction terms in the approximation of the Coulomb interaction. Such a Hamiltonian can be written as

H ≃ T 0 iσ n iσ -t ij,σ γ ij c † iσ c jσ + U i n i↑ n i↓ + 1 2 V ij,σσ ′ γ ij n iσ n jσ ′ + X ij,σ γ ij c † iσ c jσ n iσ + n j σ + 1 2 Y ij,σ γ ij c † iσ c jσ c † iσ c j σ + σ ′ c † iσ c † jσ ′ c iσ ′ c jσ , (5)
where γ ij = 1 for nearest-neighbor sites i, j, and 0 otherwise, which restricts the summation over the nearest-neighbor pairs, and

V = ij| 1 r |ij , X = ii| 1 r |ij = ii| 1 r |ji = ij| 1 r |ii = ji| 1 r |ii , Y = ii| 1 r |jj = ij| 1 r |ji . (6)
In fact, the X and V terms have been discussed in literature previously [6,[START_REF] Sun | [END_REF], and the matrix element Y has been mentioned in [2]. Please note that in Eq (5) only the constant and nearest-neighbor hopping terms of the kinetic energy are considered for consistency.

Using the Gutzwiller projection operator technique, the Hamiltonian (5) can be projected to a subspace of the Hilbert space where only empty and singly occupied sites are allowed [3,[START_REF] Hubac | Brillouin-Wigner Methods for Many-Body Systems[END_REF][START_REF] Vollhardt | Proceedings of the International School of Physics "Enrico Fermi" Course CXXI[END_REF]. To the leading order of the perturbation expansion, the effective Hamiltonian in the subspace of singly occupied states can be written as

H s = T 0 iσ ñiσ + ij,σ γ ij -t c † iσ cjσ - 1 2 J c † iσ c † j σ c iσ c jσ + 1 2 p ñiσ ñjσ + 1 2 q ñiσ ñjσ , (7)
where c †

iσ = c † iσ (1 -n iσ ), ciσ = c iσ (1 -n iσ ), ñiσ = σ c † iσ ciσ
, and the coefficients are given by the following equations

p = V -Y, q = V -2J 0 , J 0 = (t -X) 2 /U ′ , U ′ = U + z ′ (2V -Y ), J = p -q, (8)
where 0 ≤ z ′ ≤ z. When V =X=Y =0, Eq [START_REF] Mahan | Many-Particle Physics[END_REF] reduces to the t-J model [START_REF]A personal perspective from the pioneering times[END_REF]. Rescaling H s with t, this model has only three dimensionless independent material parameters T0 = T 0 /t, p = p/t, and q = q/t. One may expect that the system will exhibit ferromagnetism when q is sufficiently larger than p. On the other hand, the system will exhibit antiferromagnetism when p is sufficiently larger than q. Our studies show that this is true. More interestingly, we will show that this model will exhibit superconductivity for some parameter range of p > q.

Green Function Method

We use the Zubarev Green function technique [2,[START_REF] Zubarev | [END_REF] to study the physics of the model established above. Define the grand canonical Hamiltonian operator [START_REF] Fetter | Quantum Theory of Many-Particle Systems[END_REF]

K = H s -µN, ( 9
)
where N is the total number operator, H s is the Hamiltonian given in Eq [START_REF] Mahan | Many-Particle Physics[END_REF], and µ is the chemical potential of electrons. The corresponding grand partition function may be written as

Z G = Tre -βK = e -βΩ , β = 1 kT , ( 10
)
where k is the Boltzmann's constant and Ω is the thermodynamic potential of the system. Therefore, the ensemble average of any operator O can be calculated by

O = Tr Oe β(Ω-K) . (11)
Note here that O and K are operators, while Ω is a c-number. The ensemble average O is assumed to be the measurement value of the observable O in experiments. The retarded Green function of two Fermion operators A(t) and B(t ′ ) in the Heisenberg representation can be defined as

≪ A(t)|B(t ′ ) ≫= -iθ(t -t ′ ) {A(t), B(t ′ )} , (12)
where {A, B} = AB + BA, the Fermion anticommutation relation, θ(t) is the usual step function, and ... means the ensemble average with the grand partition function Z G , defined in Eq [START_REF] Vollhardt | Proceedings of the International School of Physics "Enrico Fermi" Course CXXI[END_REF]. Since ≪ A(t)|B(t ′ ) ≫ is a function of t -t ′ , it is convenient to define the Fourier transform

≪ A|B ≫ ω = ∞ -∞ d(t -t ′ ) ≪ A(t)|B(t ′ ) ≫ e iω(t-t ′ ) . ( 13
)
It can be shown that the retarded Green function in frequency space satisfies the following equation of motion

ω ≪ A|B ≫ ω = {A, B} + ≪ [A, K]|B ≫ ω , (14)
where again {A, B} is the anticommutator of operators A and B, and [A, K] is the commutator of the operators A and K. The ensemble average B(t)A(t ′ ) can be calculated by

B(t)A(t ′ ) = i ∞ -∞ dω 2π e -iω(t-t ′ ) e βω + 1 ≪ A|B ≫ ω+i0 + -≪ A|B ≫ ω-i0 + . ( 15
)
Eqs ( 14)-( 15) are applicable to all subsequent studies of the ferromagnetism, antiferromagnetism, and superconductivity of the model given in Eq [START_REF] Mahan | Many-Particle Physics[END_REF].

Due to the complexity of the model, we only consider the simplest situation in this study: a three-dimensional square lattice system. The two-dimensional case and other complicated factors are left as future work.

Ferromagnetism

We first study the ferromagnetism of the model using the standard Zubarev Green function technique. For a uniform system, we assume that the ensemble average of the electron number operator n lσ is independent of lattice site l, but dependent on spin σ. Denoting n σ = n lσ , the average number of electrons per site and magnetization can be defined as

n = n ↑ + n ↓ , (16)
m = n ↑ -n ↓ , (17)
from which, n σ can be expressed as

n σ = 1 2 (n + σm), (18)
where σ = 1(-1) in the expression for spin subscript ↑ (↓). Here n is a known quantity when the doping level is known, from which the chemical potential µ is determined. For electron-doped materials, n = 1 + δ, while for hole-doped cases, n = 1 -δ, where δ is the doping concentration. We will focus on the hole-doping only in this study.

The analysis above shows that the central task in the study of the magnetic properties is to calculate the ensemble average of the number operator of electrons, n lσ . According to Eq [START_REF]line should be determined by either 1) all of the electrons in the system cannot 'see' the energy gap (n * = n = 1 -δ) or 2) the energy gap disappears (∆ = 0, and therefore no electron can[END_REF], n lσ can be evaluated from the Green function ≪ c lσ |c † l ′ σ ≫. The retarded Green function in frequency space satisfies the following equation of motion

ω ≪ c lσ |c † l ′ σ ≫ ω = δ ll ′ + ≪ [c lσ , K]|c † l ′ σ ≫ ω , (19)
where [c lσ , K] is the commutator of the operators c lσ and K, and K is given in Eq [START_REF] Sun | [END_REF]. The Green function ≪ c lσ |c † l ′ σ ≫ ω is a function of ω and R l ′ -R l for systems with translational invariance.

Carrying out the commutation relation [c lσ , K], to the lowest order of the decoupling approximation of the sequence of Green function equations, we obtain

≪ c lσ |c † l ′ σ ≫ ω = 1 N k e ik•(R l -R l ′ ) 1 ω + µ -za σ -b σ γ(k) , ( 20
)
where z is the coordination number and the σ dependent coefficients a σ and b σ are given by the following equations

a σ = a 1σ + σ m a 2σ , b σ = b 1σ + σ m b 2σ , (21)
a 1σ = T 0 δ + t(1 + δ) + (p + q)δ 1 4 (1 -δ 2 ) + Λ 2 σ + 1 4 m 2 , a 2σ = -T 0 -t + 1 4 (3p -q) -(p + q) δ 2 4 + Λ 2 σ - 1 4 m 2 , b 1σ = t - 1 4 (1 + δ) 2 + Λ 2 σ + 2Λ σΛ σ - 1 4 m 2 -(p -q)Λ σ + - 1 2 (p -q) -2(p + q) 1 4 δ 2 -Λ 2 σ + 1 4 m 2 Λ σ , b 2σ = - 1 2 t(1 + δ) -(p + q)δΛ σ , (22)
and the dispersion relation γ(k) is defined as

γ(k) = a e k•a , ( 23
)
where a is the lattice space vector and the summation is over all the nearestneighbor sites. In Eq (22), Λ σ represents the value of the nonzero formal ensemble average c † l ′ σ c lσ for the nearest-neighbor sites l and l ′ . Λ σ will be calculated self-consistently [START_REF]The ensemble average can only be performed formally at this stage, be[END_REF].

Using Eqs [START_REF]line should be determined by either 1) all of the electrons in the system cannot 'see' the energy gap (n * = n = 1 -δ) or 2) the energy gap disappears (∆ = 0, and therefore no electron can[END_REF] and (20), the ensemble average c † l ′ σ c lσ can be calculated as follows

c † l ′ σ c lσ = 1 N k e ik•(R l -R l ′ ) 1 e β(Eσ(k)-µ) + 1 , (24)
where the elementary excitations energy spectrum is given by

E σ (k) = za σ + b σ γ(k). (25)
Eq (24) is the basic formula for the ensemble average c † l ′ σ c lσ , from which we are able to obtain the self-consistent equations of n σ and Λ σ

n σ = z 1 -1 dxρ(x) 1 e βzbσ (x-x0σ) + 1 , (26)
Λ σ = z 2 z 1 1 -1 dxρ(x) x e βzbσ (x-x0σ) + 1 , (27)
where x 0σ = (μ -a σ )/b σ with μ = µ/z, and the density of state can be expressed as

ρ(x) = 1 3π 1 -x 2 , (28)
for three dimensional systems [2]. Here the summation over k has been converted to an integral over x → γ(k)/z and z 1 = z(1 -δ), the average number of the nonempty nearest-neighbor sites of the site l at doping level δ.

Eqs ( 26) and ( 27) are the self-consistent equations describing the ferromagnetism of the model described by Eq [START_REF] Mahan | Many-Particle Physics[END_REF]. Five unknowns, n ↑ , n ↓ , Λ ↑ , Λ ↓ , and µ need to be determined from the system of equations. When Eqs ( 26) and ( 27) are written down explicitly for each spin, one has four independent equations. The fifth equation is given in Eq (16), which indicates that there is only one independent unknown in the set of n ↑ and n ↓ . Given δ (or n = 1 -δ) and T (or β = 1/kT ), solving Eqs ( 16), (26), and (27) for n σ , Λ σ , and µ, we can obtain the information about the magnetization m as a function of δ and T .

Of course, m depends on the material parameters T0 = T 0 /t, p = p/t, and q = q/t as well.

The system can be solved numerically. We briefly outline the main features of the ferromagnetism as follows:

1. An increase in q or decrease in p is favorable for the magnetization m.

2. The magnetization m decreases with temperature.

3. The doping dependence of m is complicated. The favorability of doping for m depends on the values of p and q.

The detailed results will be presented in a subsequent paper. Our main purpose of discussing the ferromagnetism here is to examine the singularity of the self-consistent equations and make a comparison with that of the antiferromagnetism and superconductivity.

The most interesting feature of the system of equations ( 26) and ( 27) is that, when m = 0, there is no singularity as T → 0 (or β → ∞). This feature persists even for the situation of δ = 0 and T → 0. When β → ∞, the integrals in Eq (26) and ( 27) can be carried out analytically with the following results: For b σ > 0, integrating gives

n σ = 1 2 + 1 π x 0σ 1 -x 2 0σ + arcsin(x 0σ ) , Λ σ = - 4 πz 1 (1 -x 2 0σ ) 3/2 , (29)
while for b σ < 0, we have

n σ = 1 2 - 1 π x 0σ 1 -x 2 0σ + arcsin(x 0σ ) , Λ σ = 4 πz 1 (1 -x 2 0σ ) 3/2 , (30)
Indeed, Eqs (29) and (30) are not singular for any value of m even for the case of δ = 0. This means that the model of Eq [START_REF] Mahan | Many-Particle Physics[END_REF] does not necessarily have a ferromagnetic phase with nonzero m at the condition of T → 0. That is, m = 0 is a possible solution of the model at zero temperature. As we will see below, this is in remarkable contrast to the cases of the antiferromagnetism and superconductivity of this model.

Antiferromagnetism

We now proceed to study the antiferromagnetism of the model. As we will see below, this model exhibits much more interesting antiferromagnetic features than those of the ferromagnetism. The most striking one is that the system is necessarily in an antifierrormagnetic state as T → 0 at δ = 0.

The formalism is similar to the discussion of the ferromagnetism. It is convenient, however, to formally divide the original lattice into two sublattices so that the nearest-neighbor sites belong to different sublattices. We use i 1 , j 1 , ..., to label the sites in the first sublattice, and i 2 , j 2 , ..., to label the sites in the second sublattice. The central task is to calculate the average number of electrons per site for each spin direction in both sublattices. Denoting the mean number of electrons with spin σ in the sublattices as n 1σ = n i1σ and n 2σ = n i2σ respectively, we can define the mean number of electrons per site and magnetization for the first sublattice

n 1 = n 1↑ + n 1↓ , (31)
m 1 = n 1↑ -n 1↓ . (32)
Similarly, for the second sublattice, we have

n 2 = n 2↑ + n 2↓ , (33)
m 2 = n 2↑ -n 2↓ . (34)
Eqs ( 31)-(34) immediately lead to the following expressions for each sublattice

n 1σ = 1 2 (n 1 + σm 1 ), (35)
n 2σ = 1 2 (n 2 + σm 2 ). ( 36
)
We make two important assumptions: 1) n 1 = n 2 = n, and 2) m 1 = -m 2 = m. Therefore, Eqs (35) and (36) become

n 1σ = 1 2 (n + σm), (37)
n 2σ = 1 2 (n -σm), (38)
where n = 1 -δ and m will be calculated self-consistently. The above analysis shows that we only need to find the average electron number in one sublattice. For example, if n 1σ is known, then m = n 1↑ -n 1↓ is known, and therefor n 2σ is known.

Similar to the ferromagnism case, the ensemble averages n 1σ = n i1σ can be calculated from the Green function

≪ c l1σ |c † l ′ 1 σ ≫.
Nevertheless, further analysis shows that the two Green functions

≪ c l1σ |c † l ′ 1 σ ≫ and ≪ c l2σ |c † l ′ 1 σ
≫ are coupled to each other even at the lowest order of the decoupling approximation of the Green function equations. Therefore we need to discuss the two Green functions simultaneously. In frequency space, the two Green function satisfy the following equations of motion

ω ≪ c l1σ |c † l ′ 1 σ ≫ ω = δ ll ′ + ≪ [c l1σ , K]|c † l ′ 1 σ ≫ ω , (39)
ω ≪ c l2σ |c † l ′ 1 σ ≫ ω =≪ [c l2σ , K]|c † l ′ 1 σ ≫ ω , (40)
where the anticommutation relations {c l1σ , c

† l ′ 1 } = δ ll ′ and {c l2σ , c † l ′ 1 } = 0 have
been used. Using Eq [START_REF] Sun | [END_REF], the commutation relations [c l1σ , K] and [c l2σ , K] can be obtained exactly. Nonetheless, an approximation must be used to break off the chain of Green function equations. To the lowest order of the decoupling approximation, the Green functions can expressed as

≪ c l1σ |c † l ′ 1 σ ≫ ω = 2 N N/2 k e ik•(R l 1 -R l ′ 1
) g kσ (ω), (41)

≪ c l2σ |c † l ′ 1 σ ≫ ω = 2 N N/2 k e ik•(R l 2 -R l ′ 1 ) f kσ (ω), (42)
where

g kσ (ω) = A kσ ω + µ -E 1 (k) + B kσ ω + µ -E 2 (k) , (43)
f kσ (ω) = C k ω + µ -E 1 (k) + D k ω + µ -E 2 (k) , (44)
with the elementary excitation spectrum

E 1 (k) = za - zma ′ 2 + b ′ γ(k) 2 , (45)
E 2 (k) = za + zma ′ 2 + b ′ γ(k) 2 , (46)
and the partial-fraction decomposition coefficients

A kσ = 1 2 1 - σzma ′ zma ′ 2 + b ′ γ(k) 2 , (47)
B kσ = 1 2 1 + σzma ′ zma ′ 2 + b ′ γ(k) 2 , (48)
C k = - b ′ γ(k) 2 zma ′ 2 + b ′ γ(k) 2 = D k , (49)
where γ(k) is given in Eq (23), b ′ = b(1 -δ), and the coefficients a, a ′ , and b are given the following equations

a = T 0 z δ + t(1 + δ)Λ + (p + q)δ 1 4 (1 -δ 2 ) + Λ 2 + 1 4 m 2 , ( 50
)
a ′ = - T 0 z + tΛ + 1 4 (3q -p) + (p + q) - 1 4 δ 2 + Λ 2 + 1 4 m 2 , ( 51
) b = t - 1 4 (1 + δ) 2 + 3Λ 2 + 1 4 m 2 + -J - 1 2 (p -q) + 2(p + q)(- 1 4 δ 2 + Λ 2 + 1 4 m 2 ) Λ. ( 52
)
Here Λ is the value of the nonzero formal ensemble average c † l1σ c l2σ , where l 1 and l 2 are the nearest-neighbor sites. We have assumed that Λ is independent of spin σ. Similar to the ferromagnetism case, Λ will be calculated selfconsistently [START_REF]The ensemble average can only be performed formally at this stage, be[END_REF]. Please note that a, a ′ , and b are all independent of σ explicitly. As can be seen from Eqs (45) and (46), two branches of the elementary excitation spectrum exist in the system, both of which are spin σ independent. The coefficients A kσ and B kσ depend on spin σ explicitly, while C k and D k are spin independent. Therefore f kσ (ω) does not depend on σ explicitly, which results from the assumption that Λ is independent of σ.

Using Eqs (15), ( 41) and (42), we can obtain the following ensemble averages

c † l ′ 1 σ c l1σ = 2 N N/2 k e ik•(l1-l ′ 1 ) A kσ e β(E1(k)-µ) + 1 + B kσ e β(E2(k)-µ) + 1 , (53) c
† l ′ 1 σ c l2σ = 2 N N/2 k e ik•(l2-l ′ 1 ) C k e β(E1(k)-µ) + 1 + D k e β(E2(k)-µ) + 1 , (54)
where we have used

l 1 -l ′ 1 to represent R l1 -R l ′ 1 for conciseness. Setting l 1 = l ′
1 in Eq (53), we have

n 1σ = 2 N N/2 k A kσ e β(E1(k)-µ) + 1 + B kσ e β(E2(k)-µ) + 1 . (55)
Noticing n = n 1↑ + n 1↓ and m = n 1↑ -n 1↓ , and using Eqs (47)-(48), Eq (55) immediately leads to the following self-consistent equations for µ and m

n = 2 N N/2 k 1 e β(E1(k)-µ) + 1 + 1 e β(E2(k)-µ) + 1 , (56) 1
= 2 N N/2 k -za ′ zma ′ 2 + bγ(k) 2 1 e β(E1(k)-µ) + 1 - 1 e β(E2(k)-µ) + 1 . (57)
The self-consistent equation for Λ can be obtained from Eq (54) by setting l 2 and l ′ 1 as the nearest neighbors

z 1 Λ = 2 N N/2 k -b ′ γ(k) 2 2 zma ′ 2 + b ′ γ(k) 2 1 e β(E1(k)-µ) + 1 - 1 e β(E2(k)-µ) + 1 , (58)
where Eq (49) has been used. Here z 1 = z(1 -δ), the average number of the nonempty nearest neighbors of the site l 2 . Eqs (45) and (46) show that E 1 (k) ≤ E 2 (k). Thus from Eqs (57) and (58), we see that for meaningful solutions, 1) a ′ must be negative, and 2) Λ and b must have the opposite sign.

Finally, introducing a new variable x → γ(k)/z, and noticing that the integrand depends on x through x 2 , for the case of b > 0, the self-consistent equations can be written in the following integration form n = 2z

1 0 dxρ(x) 1 e βǫ1(x) + 1 + 1 e βǫ1(x) + 1 , (59)
Λ = - z 2 z 1 1 0 dxρ(x) x 2 √ m2 + x 2 1 e βǫ1(x) + 1 - 1 e βǫ1(x) + 1 , (60) b
′ = -2za ′ 1 0 dxρ(x) 1 √ m2 + x 2 1 e βǫ1(x) + 1 - 1 e βǫ1(x) + 1 , (61)
where the density of states ρ(x) is given in Eq (28) and the two branches of energy excitation spectrum take the form

ǫ 1 (x) = zb ′ -x 0 -m2 + x 2 , ( 62
)
ǫ 2 (x) = zb ′ -x 0 + m2 + x 2 , (63)
where x 0 = (µ/z -a)/b ′ and m2 = (ma ′ /b ′ ) 2 . Eqs (59)-(61) are the basic self-consistent equations for antiferromagnetism of the model in Eq [START_REF] Mahan | Many-Particle Physics[END_REF]. At a given condition of the doping level δ and temperature T , three unknowns,

x 0 , Λ, and m can be determined from the system of equations. Of course, x 0 , Λ, and m also depend on the dimensionless material parameters T0 = T 0 /t, p = p/t, q = q/t. In the following, we discuss the case of T 0 = 0.

The system is in ground state at absolute zero temperature. This is the δ-axis in the δ-T plane. When T → 0, the integral in Eq (59) can be carried out analytically with the result

δ = 2 π ξ 1 -ξ 2 + arcsin(ξ) , (64)
where ξ = x 2 0 -m2 . In the case of b > 0, x 0 < 0 is applicable to the case of n = 1 -δ (hole doping), while x 0 > 0 is applicable to the case of n = 1 + δ (electron doping). When T → 0, for the hole doping case (x 0 < 0), Eqs ( 60) and ( 61) can be written as

Λ = - z 2 z 1 1 ξ dxρ(x) x 2 √ m2 + x 2 , ( 65
) b ′ = -2za ′ 1 ξ dxρ(x) 1 √ m2 + x 2 , ( 66
)
where ξ is given by Eq (64). There is no singularity as long as either ξ or m is nonzero. The quantity ξ depends on δ only and Eq (64) is not coupled with Eqs (65) and (66). Thus ξ can be solved separately from Eq (64) for a given δ. When δ is small, to the leading order, Eq (64) can be solved with the result

ξ ≃ π 4 δ, (67)
which leads to an approximate relation between x 0 and m on the δ-axis (T → 0).

x 2 0 ≃ π 4 δ 2 + m2 . (68)
A more complete solution of the system of equations ( 64)-( 66) can be obtained numerically for given δ, p, and q. Now we discuss two special points on the δ-axis (T → 0).

An interesting point on the δ-axis is where m vanishes. At this point, ξ = |x 0 | and the integrals in the self-consistent equations ( 65)-(66) can be carried out analytically. Use δ 0 to denote the doping concentration at this point, which can be determined by the following equations

δ 0 = - 2 π x 0 1 -x 2 0 + arcsin(x 0 ) , ( 69
) Λ = - 4 πz 1 1 -x 2 0 3/2 , ( 70
) b ′ = 4 π a ′ 1 -x 2 0 + ln |x 0 | -ln 1 + 1 -x 2 0 . ( 71
)
To the leading order of δ 0 , x 0 can be solved from Eq (69) as

x 0 ≃ - π 4 δ 0 . ( 72
)
The term ln |x 0 | in Eq (71) shows that a meaningful δ 0 must be finite. Solving x 0 and Λ from Eqs (69) and (70), and then substituting their values into Eq (71), we obtain a single variable equation for δ 0 . Given p and q, this equation can be solved to obtain δ 0 (p, q). In practice, the value of δ 0 is known from experiments, so that δ 0 (p, q) specifies a relation between the dimensionless parameters p and q. For example, in the case of b > 0, for δ 0 = 0.05, q = 0.63 corresponds to r = p/q ≃ 2.5, while q = 1 corresponds to r = p/q ≃ 3.73.

The most interesting case is the ground state at half-filling. This is the origin of the δ-T plane. Eq (64) indicates that ξ = 0 and therefore x2 0 = m2 at this point. Noticing that at δ = 0, z 1 = z, and b ′ = b, the self-consistent equations become

Λ = -z 1 0 dxρ(x) x 2 √ m2 + x 2 , ( 73
) b = -2za ′ 1 0 dxρ(x) 1 √ m2 + x 2 . (74)
Apparently, the integral in Eq (74) is divergent when m = 0, which indicates that m = 0 is not a possible solution of the system as T → 0 at δ = 0. That is, the system must be in an antiferromagnetic state with a nonzero m at the origin of the δ-T plane. This is one of the most striking features of the model described by Eq [START_REF] Mahan | Many-Particle Physics[END_REF]. Since in this case T → 0 and δ = 0, the magnetization m depends on the two parameters q and p (or q and r = p/q = p/q ) only. Our results show that for a given value of q, a nonzero m starts with a threshold value r ≥ r 0 . Then m increases with r, and finally saturates to a maximum value of m max ≃ 0.6698 when r is sufficiently large. Generally, the greater the q, the greater the threshold value r 0 . The threshold value of r for a given q is related to the condition of a ′ < 0 and b ′ > 0.

In the simplest case where m = 0, for the case of b > 0, the self-consistent equations become

n = 2z 1 0 dxρ(x) 1 e βǫ1(x) + 1 + 1 e βǫ1(x) + 1 , (75)
Λ = - z 2 z 1 1 0 dxρ(x)x 1 e βǫ1(x) + 1 - 1 e βǫ1(x) + 1 , (76) b
′ = -2za ′ 1 0 dxρ(x) 1 x 1 e βǫ1(x) + 1 - 1 e βǫ1(x) + 1 , (77)
Figure 1: Antiferromagnetic phase boundary lines for typical values of q = q/t and r. The region below each line is the antiferromagnetic state.

where

ǫ 1 (x) = zb ′ -x 0 -x , (78)
ǫ 2 (x) = zb ′ -x 0 + x . (79)
Now there are only two unknowns x 0 and Λ, which can be determined from Eqs (75) and (76) when δ and T are given. The third equation (77) will give the Néel temperature at a doping concentration δ, T N (δ), in the δ-T plane, which is the boundary of the area where m = 0. Fig. 1 shows the antiferromagnetic phase diagrams for some typical values of q and r. As discussed above, Eq (77) is singular at the point of T → 0 and δ = 0. Therefore, the boundary of m = 0 must bypass the origin of the δ-T plane.

Superconductivity

In order to study the superconductivity of the model described by Eq [START_REF] Mahan | Many-Particle Physics[END_REF], one needs to calculate the ensemble average c l↓ c l ′ ↑ or its complex conjugate c † l ′ ↑ c † l↓ . These quantities provide the information about the state of the Cooper pairs, from which the properties of the superconducting and pseudogap state can be obtained. The ensemble average c † l ′ ↑ c † l↓ can be calculated from the Green function

≪ c † l↓ |c † l ′ ↑ ≫ . Further analysis shows that the Green function ≪ c † l↓ |c † l ′ ↑ ≫ is coupled with ≪ c l↑ |c † l ′ ↑
≫ even at the lowest order of decoupling approximation. Therefore, we need to deal with both ≪ c † l↓ |c † l ′ ↑ ≫ and ≪ c l↑ |c † l ′ ↑ ≫ simultaneously in the study of the superconductivity. In ωspace, the equations of motion for both Green functions

≪ c l↑ |c † l ′ ↑ ≫ ω and ≪ c † l↓ |c † l ′ ↑ ≫ ω are ω ≪ c l↑ |c † l ′ ↑ ≫ ω = δ ll ′ + ≪ [c l↑ , K]|c † l ′ ↑ ≫ ω , (80)
ω ≪ c † l↓ |c † l ′ ↑ ≫ ω =≪ [c † l↓ , K]|c † l ′ ↑ ≫ ω , (81)
where the Fermion anticommutation relations

{c l↑ , c † l ′ ↑ } = δ ll ′ and {c † l↓ , c † l ′ ↑ } = 0 have been taken into account. For translational invariant systems, ≪ c l↑ |c † l ′ ↑ ≫ ω and ≪ c † l↓ |c † l ′ ↑ ≫ ω are functions of frequency ω and R l -R l ′ .
To the lowest order of decoupling approximation, we obtain the Green functions as follows

≪ c l↑ |c † l ′ ↑ ≫ ω = 1 N k e ik•(R l -R l ′ ) A k ω -E k + B k ω + E k , (82)
≪ c † l↓ |c † l ′ ↑ ≫ ω = 1 N k e ik•(R l -R l ′ ) C k ω -E k + D k ω + E k , (83)
where the partial-fraction decomposition coefficients are given by

A k = 1 2 1 + -µ + za + b ′ γ k E k , (84)
B k = 1 2 1 - -µ + za + b ′ γ k E k , (85)
C k = 1 2 (zc + d ′ γ k )∆ * E k = -D k , (86)
and the elementary excitation energy spectrum takes the form

E k = (µ -za -b ′ γ k ) 2 + (zc + d ′ γ k ) 2 |∆| 2 . ( 87
)
Here the dispersion relation γ k is given in Eq (23) and b ′ = b(1 -δ)/(1 + r), d ′ = d(1 -δ)r/(1 + r), where r = p/q, and the coefficients a, b, c, and d are expressions of Λ and ∆

a = t(1 + δ)Λ + (p + q)δ 1 4 (1 -δ 2 ) + Λ 2 -|∆| 2 , ( 88
) b = t - 1 4 (1 + δ) 2 + 3Λ 2 + |∆| 2 + - 3 2 (p -q) + 2(p + q)(- 1 4 δ 2 + Λ 2 + |∆| 2 ) Λ, (89)
c = t(1 + δ) + 2(p + q)δΛ, (90)
d = t2Λ - 3 2 (p -q) + 2(p + q) 1 4 δ 2 + Λ 2 + |∆| 2 . ( 91
)
It is easy to check that the dimensionless coefficients ā = a/t, b = b/t, c = c/t, and d = d/t satisfy the following relations

ā = cΛ + (p + q)δ 1 4 (1 -δ 2 ) -Λ 2 -|∆| 2 , ( 92
) b = dΛ - 1 2 cδ - 1 4 (1 + δ) 2 + Λ 2 + |∆| 2 , (93)
where p = p/t and q = q/t. Here Λ and ∆ are the values of the nonzero formal ensemble averages c † lσ c l ′ σ and c l↓ c l ′ ↑ for the nearest-neighbor sites l and l ′ , which will be calculated self-consistently [START_REF]The ensemble average can only be performed formally at this stage, be[END_REF]. We have assumed that Λ and ∆ do not depend on spin σ explicitly in the discussion of superconductivity.

Using Eq [START_REF]line should be determined by either 1) all of the electrons in the system cannot 'see' the energy gap (n * = n = 1 -δ) or 2) the energy gap disappears (∆ = 0, and therefore no electron can[END_REF], the ensemble averages c † l ′ ↑ c l↑ and c † l ′ ↑ c † l↓ can be calculated by the following equations

c † l ′ ↑ c l↑ = 1 2 δ ll ′ - 1 2N k e ik•(l-l') -µ + za + b ′ γ k E k tanh 1 2 βE k , (94)
c † l ′ ↑ c † l↓ = - ∆ * 2N k e ik•(l-l') zc + d ′ γ k E k tanh 1 2 βE k . (95)
These are the general formulas of the ensemble averages for the study of superconductivity. Eqs (94) and (95) confirm that c † l ′ ↑ c l↑ , and c † l ′ ↑ c † l↓ are functions of R l -R l ′ , which is a result of the translational invariance of the system.

Three quantities µ, Λ, and ∆ need to be determined from this formalism. The self-consistent equations for µ, Λ, and ∆ can be obtained from Eqs (94) and (95) with the following form

δ = 1 N k -µ + za + b ′ γ k E k tanh 1 2 βE k . ( 96
) 2z 1 Λ = - 1 N k (-µ + za + b ′ γ k )γ k E k tanh 1 2 βE k , (97)
2z 2 = - 1 N k (zc + d ′ γ k )γ k E k tanh 1 2 βE k , (98)
where z 1 = z(1 -δ)/(1 + r) and z 2 = z(1 -δ)r/(1 + r), the approximate average numbers of the nearest neighbors of the site l with the same and opposite spin, respectively.

Furthermore, introducing a new variable x → γ k /z, the self-consistent equations can be written in the integration form

δ = zb ′ 1 -1 dxρ(x) x -x 0 E(x) tanh 1 2 zβE(x) , (99)
2z 1 Λ = -z 2 b ′ 1 -1 dxρ(x) (x -x 0 )x E(x) tanh 1 2 zβE(x) , (100)
2z 2 = -z 2 d ′ 1 -1 dxρ(x) (x -x ′ 0 )x E(x) tanh 1 2 zβE(x) , (101)
where x 0 = (µ/z -a)/b ′ , x ′ 0 = -c/d ′ , the density of states ρ(x) is given in Eq (28), and the elementary excitation energy can be written as

E(x) = b ′ 2 (x -x 0 ) 2 + d ′ 2 (x -x ′ 0 ) 2 ∆ 2 . ( 102
)
In the dimensionless system where the coefficients a, b, c, and d are rescaled with t, there are only two independent parameters p and q. Except for the expression of density of state (28), all equations are applicable to any dimensional systems.

Eqs (99)-( 101) are the basic self-consistent equations for the superconductivity state of the model. There are three unknowns, x 0 , Λ, and ∆, to be determined from the system of equations. The most interesting quantity is the energy gap parameter ∆. It is the order parameter of the superconducting state. The quantity x 0 , the effective chemical potential of the system, is relevant to the features of the Fermi surface. In the case where d = 0 and x 0 = x ′ 0 , a nonzero ∆ causes an energy gap at the Fermi surface, as shown in Eq (102). This is the most remarkable feature of the superconducting state. The system of equations ( 99)-(101) can be solved numerically. The phase diagram of the superconductivity for some typical values of q and r is shown in Fig. 2. Other detailed results, such as the δ and T dependence of the energy gap parameter ∆, will be presented in the forthcoming papers.

Pseudogap State

We believe that the pseudogap state observed in experiments can be described by the special case of the system of equations (99)-(101) where x 0 = x ′ 0 . It can be shown that b = Λd in this case, and therefore the self-consistent equations take the following form

δ = zr 1 Λ κ 1 -1 dxρ(x) tanh 1 2 zβκd(x -x 0 ) , (103) 2
= -z 1 κ 1 -1 dxρ(x)x tanh 1 2 zβκd(x -x 0 ) , (104)
Λ 2 + ∆ 2 = 1 4 (1 + δ) 2 + (p + q)δ 2 Λ, ( 105
)
where κ is given by

κ = (r 1 Λ) 2 + (r 2 ∆) 2 . ( 106
)
Note that there is no singularity in Eqs (103)-( 104) at any doping level as T → 0. At a given temperature T and doping level δ, the quantities x 0 , Λ, and ∆ are determined from Eqs (103)-(105).

In the pseudogap state, a new degree of freedom, n * , which is the number of electrons that cannot 'see' the energy gap, emerges. This is the density of the charge carriers contributing to the electrical conductivity in the pseudogap region. Explicitly, the condition of x 0 = x ′ 0 can expressed as

) 109
When n is the total number of the electrons, n = 1 -δ, Eq (107) describes the line of T * (δ) [START_REF]line should be determined by either 1) all of the electrons in the system cannot 'see' the energy gap (n * = n = 1 -δ) or 2) the energy gap disappears (∆ = 0, and therefore no electron can[END_REF]. If we require the condition (107) to be satisfied for T < T * , then the only possibility is that n < 1 -δ. We assume that the new degree of freedom n * is determined by Eq (107) when T < T * . That is, the condition x 0 = x ′ 0 determines the number of carries n * in the pseudogap state. Replacing n by n * in Eq (107) leads to a quadratic equation of n * , which can be solved with the result n * = 1 -δ * , δ * = -1 p + q r 2 x 0 1 + 2 p + q Λ + √ D , D = 1 + 2 p + q Λ 2 -2 p + q r 2 x 0 1 + 2 -3 4 p -q + Λ + p + q Λ 2 + ∆ 2 r 2 x 0 .

(110)

The symbol δ * should not be confused with the doping parameter. As Fig. 3 shows, at a doping level δ, n * increases linearly with temperature T for the range of higher T . This remarkable result indicates that the Hall coefficient R H = 1/en * has a temperature dependence of 1/T .

As expected, the condition of n * = 0 should give the metal-insulator (MI) transition line. A semi-phenomenological analysis can show that the resistivity in the pseudogap region ρ * ∼ -1/x 0 at the Fermi surface. Detailed discussion of the resistivity and the phase diagram of the pseudogap state (the MI transition line and T * line) will be presented in forthcoming papers.

Conclusion

We have established a novel model Hamiltonian for the strongly correlated electron systems, which contains the Coulomb potential energy terms up to the order of two-site interactions. The magnetic properties (ferromagnetism and antiferromagnetism) of this model have been studied. Our results indicate that the system is not necessarily in the ferromagnetic state as T → 0 at any doping level. Nevertheless, the system must be in an antiferromagnetic state with a nonzero magnetization at the state point of T → 0 and δ = 0. Moreover, the system exhibits a superconducting state in a doped region at sufficiently low temperatures. The phase diagrams for antiferromagnetism and superconductivity of the model are presented. This model also predicts the temperature dependence of 1/T of the Hall coefficient in the pseudogap state.

Figure 2 :

 2 Figure 2: Phase diagram of superconductivity state for typical values of q = q/t and r. The region below each phase boundary line (within each dome) is the superconductivity state.

c + x 0 r 2 d

 2 -n + 2(p + q)(1 -n)Λ, (108) d = 2Λ -p + 2q + 2(p + q) -1 4 n(2 -n) + Λ 2 + ∆ 2 . (

Figure 3 :

 3 Figure 3: Temperature dependence of the charge carriers density n * in pseudogap state for typical doping levels (q = 0.63, r = 2.5).